NUMERICAL INVESTIGATION OF BOUND RELATIVISTIC TWO-PARTICLE SYSTEMS WITH ONE-BOSON EXCHANGE POTENTIAL Yu.A. Grishechkin and V.N. Kapshai Francisk Skorina Gomel State University
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NUMERICAL INVESTIGATION OF BOUND RELATIVISTIC TWO-PARTICLE SYSTEMS WITH ONE-BOSON EXCHANGE POTENTIAL
Yu.A. Grishechkin and V.N. Kapshai
Francisk Skorina Gomel State University
• Two-particle equations of quasipotential type
• Integral equations for bound s-states in the relativistic configurational representation (RCR)
• One-boson exchange potentials and their superposition
• Numerical method of solving
• Spectrum of the orthopositronium
• Decay width of the parapositronium
Plan of the talk
Two-particle equations of quasipotential type
( ) ( ) 3 ( )3
( , ) ( , ) (2 , , ) ( , )(2 )
j j jE p G E p d kV E p k E k
(1)2 2
( , )p
mG E p
E E
− Logunov-Tavkhelidze
equation
− Kadyshevsky
equation
− Logunov-Tavkhelidze
modified equation
− Kadyshevsky
modified equation
(2) ( , )
2 p p
mG E p
E E E
(3)2 2
( , ) p
p
EG E p
E E
(4) 1
( , )2 p
G E pE E
2E – two-particle system energy2 2
pE m p
Integral equations for bound states in the momentum representation
Relativistic configurational representation
RCR is as expansion over functions
1
( , ) , imr
pE pnp r r rn
m
Transformations of the two-particle wave function of relative motion
33
1( , ) ( , ) ( , )
(2 ) p
mE r p r E p d p
E
– from the momentum representation to the RCR
* 3( , ) ( , ) ( , )E p p r E r d r
– from the RCR to the momentum representation
For s-states this transformation analogous the Fourier transformation
0 0
( , ) sin ( , ), ( , ) sin ( , ),2
E r d mr E E dr mr E r
where χ – is the rapidity, connected with momentum by relation sinhp m
Integral two-particle equations for bound s-states in the RCR
( ) ( ) ( )
0( , ) ( , , ) ( ) ( , )j j jw r dr g w r r V r w r
j = 1
( ) ( ) ( )( , , ) ( , ) ( , )j j jg w r r g w r r g w r r Green functions in the RCR:
(1) 1 sinh[( 2 ) ],
sin 2 sinh[ 2]
w mrg w r
m w mr
j = 2 1
(2) (4 cos ) 1 sinh[( ) ],
cosh[ 2] sin 2 sinh[ ]
m w w mrg w r
mr m w mr
j = 3 (3) 1 cosh[( 2 ) ],
2 sin cosh[ 2]
w mrg w r
m w mr
(4) 1 sinh[( ) ],
2 sin sinh[ ]
w mrg w r
m w mr
j = 4
− Logunov-Tavkhelidze
equation
− Kadyshevsky
equation
− Logunov-Tavkhelidze
modified equation
− Kadyshevsky
modified equation
2 2 cosE m wr – is radius-vector modulus in the RCR
2( )
0
( , ) 1jdr w r
– normalization condition of wave function
Two-particle equations for bound s-states in the momentum representation
22 2
0
2(2 , ) (2 , , ) (2 , ) (2 , )p
k
m dkE E p V E p k E k E E p
E
Logunov-Tavkhelidze equation:
Kadyshevsky equation:
2
0
22 (2 , ) (2 , , ) (2 , ) 2 (2 , )p
p k
m dkE E p V E p k E k E E p
E E
Normalization conditions of wave functions:
32
0 0 0
2(2 , ) (2 , ) (2 , ) (2 , , ) 1
(2 )p p k
m m dp dkdp E p E p E k V E p k
E E E E E
32
0 0 0
2(2 , ) (2 , ) (2 , ) (2 , , ) 1
(2 )p k
m m dp dkdp E p E p E k V E p k
E E E E E
One-boson exchange potentials in the momentum representation
1
(2 , , )2p k
V E- E E - E
p kp k p k
One of the first one-boson exchange potentials for two scalar particles obtained in framework quasipotential approach is
This potential was obtained on the basis of diagram technique of the quantum field theory Hamiltonian formulation [1]. Mass of exchange boson is equal to zero.
For s-states this potential has the form
| | 21(2 , , ) ln
2 2
p k
p k
E E p k EV E p k
E E p k E
[1] Kadyshevsky V.G. Quasipotential type equation for the relativistic scattering amplitude /
Nucl. Phys. – 1968.– V.B6, №1. – P.125-148.
In article [2] one-boson exchange potentials were obtained on the basis of retarded and causal Green functions calculation.
| | 2 | |1(2 , , ) (2 , , ) ln (2 , , ) ln
2 2
1 1(2 , , )
| |,
p pk kret
p pk k
p pk k
E E p k E E E p kV E p k A E p k B E p k
E E p k E E E p k
C E p kE E p k E E p k
where
2
2(2 , , ) (2 , , ) 2 (2 , , ),,p k
p k
E E EA E p k B E p k A E p k
E E E
2( )( )(2 , , ) .p kE E E E
C E p kE
1(2 , , ) ( )( ) (2 , , ) ( )( ) ( 2 , , )
4c p ret p retk k
p k
V E p k E E E E V E p k E E E E V E p kE E
[2] Капшай В.Н., Саврин В.И., Скачков Н.Б. О зависимости квазипотенциала от полной энергии двухчастичной системы / – ТМФ, 1986. – Т.69, №3. – С. 400-410.
Potentials of two fermions interaction in the cases of different total spin and total angular momentum values were obtained in article [3] as coefficient of general three-dimensional potential for two fermions [4] expansion into the spherical spinors.
22
| | 21(2 , , ) 2 ln
2 2
p kp k
p k
E E p k EV E p k E E m
m E E p k E
Total spin of system is equal to zero:
[3] Двоеглазов В.В, Скачков Н.Б., Тюхтяев Ю.Н., Худяков С.В. Релятивистские парциальные интегральные уравнения для волновой функции системы двух фермионов / – ЯФ, 1991. – Т.54, №3. – С. 658-668.
[4] Архипов А.А. Приближение одноглюонного обмена для квазипотенциала взаимодействия двух кварков в квантовой хромодинамике / – ТМФ, 1990. – Т.83, №3. – С. 358-373.
Total spin of system is equal to one:
2
1 22 2
| | 21(2 , , ) (2 , , ) ln (2 , , ) ln
6 2
p pk k
p pk k
E E p k E E E m pkV E p k F E p k F E p k
m E E p k E E E m pk
2 2 2 2 2 2
3
2 2 2 2(2 , , ) arctg arctg
| |,p pk kE E m p k E E m p k
F E p kp k p k
where
2 2 2 2 2 22
1 2 2 2 2
2 ( )( ( 2 ) )(2 , , ) 4 2
( 2 ) 2( ),p pk k
p kp pk k
p k E E m p k E E EF E p k E E m
p k E E E E E m
22 2 2
2 2 2 2 2(2 , , ) 4
2 ( 2 )
p k
p pk k
p k E E mF E p k
E E m p k E E E
3 2 2 2 2
2 2(2 , , ) (2 , , )
2 2
p k
p k
E E EF E p k F E p k
E E m p k
Potentials in the RCR
cosh( )
sinh
mrV r
r mr
One-boson exchange potential
2 2cos 1 2m Parameter α is associated with mass of exchange boson µ
Mass of exchange boson is equal to zero coth
( )mr
V rr
Superposition of two one-boson exchange potentials with masses are equal to zero and 2m
coth 1 tanh( )
sinh
mr mrV r
r r mr r
Coulomb potential in the RCR1
( )V rr
Numerical method of equations solving in the momentum representation
For solving of equations the rectangles quadrature method was used after replaced the variable
(1 ) , 0k m Cx x C
Nonlinear energy eigenvalue problem
for algebraic equation systems. 2
1(2 )K E E 2 (2 ) 2K E E
For solving this problem the iteration method was used [5]. Initial value energy 2E(0)
[5] T.M. Solov’eva Numerical Calculation of the Energy Spectrum of a Two-Fermion System / Comp. Phys. Comm., 136 (2001), p. 208-2011.
2( ) ( 1)1(2 )s sK E E ( ) ( 1)
2 (2 ) 2s sK E E 0, 1, 2,...s
To find the eigenvalues of the matrices one can use standard methods.
Process has to be continued until the condition holds, ε – accuracy.
( 1) ( )| |s sE E
Then, the Richardson extrapolation process is applied to energy values 2E, and
normalized wave functions, obtained on two grids N and 2N.
)2( )(2,1
sEK
Numerical method of equations solving in the RCR
To find solutions of integral equations in the RCR we used the composite Gaussquadrature method.
Nonlinear energy eigenvalue problem
for algebraic equation systems. sin( )K w w
For solving this problem the iteration method was used too.
Initial value w(0) (energy 2E(0)=2m cos(w(0)) )
0, 1, 2,...s
Process has to be continued until the condition holds, ε – accuracy.
( 1) ( )| |s sw w
( 1)( ) sin( )ssK w w
Results of equations solving in the momentum representation
Wave functions for Logunov-Tavkhelidze equation for potential (2 , , )cV E p k
Coupling constant is equal to the fine structure constant
37.2973525698 10
0.005, 1000C N
Results for spectrum
Binding energy 5(2 2) 10E m
Potential n Logunov-Tavkhelidze equation Kadyshevsky equation
800N 1600N 3200N 800N 1600N 3200N
Scalar particles
1 2 3 4
-1,266335 -0,300382 -0,124172 -0,064006
-1,266331 -0,300378 -0,124168 -0,064003
-1,266331 -0,300377 -0,124168 -0,064003
-1,266304 -0,300378 -0,124171 -0,064006
-1,266300 -0,300374 -0,124167 -0,064003
-1,266300 -0,300374 -0,124167 -0,064002
Scalar particles Vret
1 2 3 4
-1,266458 -0,300401 -0,124178 -0,064009
-1,266454 -0,300397 -0,124174 -0,064005
-1,266453 -0,300397 -0,124173 -0,064005
-1,266427 -0,300398 -0,124177 -0,064009
-1,266423 -0,300394 -0,124173 -0,064005
-1,266422 -0,300393 -0,124173 -0,064005
Scalar particles Vc
1 2 3 4
-1,266441 -0,300434 -0,124214 -0,064045
-1,266396 -0,300391 -0,124175 -0,064008
-1,266392 -0,300387 -0,124171 -0,064004
-1,266410 -0,300430 -0,124214 -0,064045
-1,266365 -0,300388 -0,124174 -0,064007
-1,266361 -0,300384 -0,124170 -0,064004
Spinor particles S=0
1 2 3 4
-1,266519 -0,300406 -0,124178 -0,064008
-1,266515 -0,300402 -0,124174 -0,064005
-1,266515 -0,300402 -0,124174 -0,064004
-1,266488 -0,300403 -0,124177 -0,064008
-1,266484 -0,300399 -0,124173 -0,064004
-1,266484 -0,300398 -0,124173 -0,064004
400N 800N 1600N 400N 800N 1600N
Spinor particles S=1
1 2 3 4
-1,266482 -0,300439 -0,124215 -0,064045
-1,266437 -0,300397 -0,124176 -0,064008
-1,266432 -0,300393 -0,124172 -0,064004
-1,266451 -0,300436 -0,124215 -0,064045
-1,266406 -0,300394 -0,124175 -0,064008
-1,266402 -0,300389 -0,124171 -0,064004
Convergence of the energy spectrum for scalar particles
Transition frequency, MHz
Potential Logunov-Tavkhelidze equation Kadyshevsky equation 1600N 3200N , % 1600N 3200N , %
Scalar particles 1193521990 1193521987 72, 5 10 1193488134 1193488121 61,1 10 Scalar particles
Vret 1193649720 1193649714
75, 0 10 1193615764 1193615758 75, 0 10
Spinor particles S=0
1193719570 1193719559 79, 2 10 1193685433 1193685431
71, 7 10
800N 1600N , % 800N 1600N , % Scalar particles
Vc 1193585705 1193585495
51,8 10 1193551800 1193551589 51,8 10
Spinor particles S=1
1193629089 1193628884 51, 7 10 1193595151 1193594932
51,8 10
Experimental date for frequency of transition from the ground state
to the first excited for orthopositronium: 1233607216.4(3.2)МHz.
Results for spectrum
Method was tested for solving modified Logunov-Tavkhelidse equation in the case of potential
tanh( )
mrV r
r
Exact expression for quantization condition of energy
2 2 22 4nE m n
Eigenvalues 2 2E m of bound states for potential
Number of state n 1 2 3 4
Exact value 51.331288 10 50.332821 10 50.147920 10 50.083205 10
Approach value at:
50N
100N
200N
400N
800N
51.333222 10 51.331751 10 51.331375 10 51.331300 10 51.331288 10
50.333782 10 50.333048 10 50.332860 10 50.332823 10 50.332817 10
50.148488 10 50.148001 10 50.147876 10 50.147850 10 50.147846 10
50.083111 10 50.082741 10 50.082644 10 50.082625 10 50.082622 10
1( ) tanh 2V r r mr
Results of equations solving in the RCR
Experimental date for frequency of transition from the ground state
to the first excited for orthopositronium: 1233607216.4(3.2)МHz.
Transition frequency 1 tanh 2r mr 1 cothr mr 1r
1j 400N 800N
1233673256MHz 1233673569MHz
1233695644MHz 1233695415MHz
1233688627MHz 1233688064MHz
2j 400N 800N
1233651095MHz 1233651664MHz
1233673463MHz 123367351MHz
1233666466MHz 1233666159MHz
3j 400N 800N
1233708292MHz 1233701226MHz
1233730674MHz 1233723067MHz
1233723659MHz 1233715718MHz
4j 400N 800N
1233682565MHz 1233679188MHz
1233704950MHz 1233701031MHz
1233697934MHz 1233693681MHz
Frequency of transition obtained by solving the Schrödinger equation with the Coulomb potential : 1233690736MHz.
Frequency of transition obtained from the
quantization condition 2En=(4m2-λ2/n2)1/2: 1233695868MHz.
The solution of the Kadyshevsky equation with the potential V(r)=tanh(πmr/2)/r gives the best agreement with experimental data.
Decay width for two-particle systems
The decay width for systems of two scalar particles into two photons [6]:
2 22 2
( ) ( )20
0
8 4ln (2 , ) ( , )p
j jp r
E pm dpE p w r
E m rm
Decay width parapositronium , µs-1
Potential Logunov-Tavkhelidze equation Kadyshevsky equation 1000N 2000N , % 1000N 2000N , %
Scalar particles 7117,719 7117,730 0,0001 7103,877 7103,883 0,0001 Scalar particles
Vret
7148,564 7148,592 0,0004 7131,901 7131,915 0,0002
Scalar particles Vc
7130,939 7130,953 0,0002 7116,073 7116,081 0,0001
Spinor particles S=0
7196,282 7196,448 0,0023 7172,333 7172,449 0,0016
[6] Г.А. Козлов О распаде связанного состояния µ+µ--пары в e+e - далитц-пару и γ - квант / ТМФ, 60 (1984), №1, с. 24-36.
The substitution of non-relativistic wave function for Coulomb potential intothis formula gives value:
18032.5 s
The substitution of exact wave function for potential gives value:
tanh( )
mrV r
r
Decay width parapositronium Γ, μs-1
1 tanh 2r mr 1 cothr mr 1r
1j 800N 1600N
7938.39 7938.47
7966.34 7966.68
7956.08 7956.45
2j 800N 1600N
7918.82 7918.91
7942.44 7942.71
7933.91 7934.23
3j 800N
1600N 7995.24 7995.03
8059.11 8060.68
8031.52 8032.98
4j 800N
1600N 7966.67 7966.60
8012.48 8013.42
7993.61 7994.52
The experimental value for parapositronium: 17990.9(17) s
17995.18 s
Conclusions
• Our methods allows to solve effectively two-particle integral equations in the momentum representation and in the RCR for slowly decreasing potentials like Coulomb potential
• The wave functions in the RCR allow to calculate decay width simply
• The results of numerical solution for energy spectrum and decay width in the case of one-boson exchange potential and superposition of such potentials give good agreement with the experimental values for positronium. Herewith results for phenomenological potentials coincide with experimental values better then results for potentials which derived strongly.
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