Resonance states of relativistic two-particle systems

Kapshai V.N., Grishechkin Yu.A.

F. Scorina Gomel State University

Plan of the talk• resonance states in nonrelativistic QM• equations of quasipotential type• integral equations in the RCR and scattering amplitudes• integral equations for resonance states in the RCR and in the MR• solving method• complex scaling in the RCR and in the MR• results of solving• conclusion

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Resonance states of two-particle systems in quantum theoryNonrelativistic theory

Scatteringstates

Bound states

Resonant states

-complex value0 0

0 Re q

Im q

Bound states

Resonances

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2

2q

- complex value

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Relativistic theory

Bound states

Scatteringstates

Resonant states

2 0E 0 2 2E m 2E

Relativistic two-particle equations of quasipotential type

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Quasipotential equations

In the momentum representation (MR)

In the relativistic configurational

representation (RCR)

integral form difference form integral form

Integral equations in the MR

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Three dimensional equations for scattering states

j=1 – the Logunov-Tavkhelidze equationj=2 – the Kadyshevsky equation j=3 – the modified Logunov-Tavkhelidze equationj=4 – the modified Kadyshevsky equation

(1) (2)2 2 2

(3) (4)2 2

1 1( , ) ( , )0 2 2 0

1 1 1( , ) ( , )0 2 2 0

q qp q p p q p

q qp q p q p

m mG E p G E pE E i E E E i E

G E p G E pE E i E E i E

Green functions (GF)

2 2pE p m

2Eq – two-particle system energym – mass of each particle

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( ) ( ) ( ) 3( , ) (2 ) ( ) ( , ) ( , , ) ( , )(2 )

pq qj j j

E d kq p p q G E p V E p k q km

( ) (0) 2 2

1( , ) ( , )0jG q p G q p

p q i

– non-relativistic limit

Integral equations in the RCR

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1

( , ) pimrE p n

p rm

r r n

r – coordinate in the RCRTransformation of wave function

3( ) ( )3

1( , ) , ( , )(2 )j j

p

mq r p r q p d pE

3*( ) ( )( , ) , ( , )j jq p p r q r d r

Three-dimensional equations for scattering states in the RCR 3

( ) ( ) ( )( , ) , ( , , ) ( ) ( , )qj j jq r q r G E r r V r q r d r

V(r) – spherically symmetric potential in the RCR

RCR is as expansion over functions

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Partial equations in the RCR for the s-states

( ) ( ) ( )0

( , ) sin( ) ( , , ) ( ) ( , )q q q qj j jr mr G r r V r r dr

Equations for the scattering s-states

χq>0 – rapidityGreen functions in the RCR ( ) ( ) ( )( , , ) ( , ) ( , )j q j q j qG r r G r - r G r + r

coshsinh

q q

q

E mq m

(1)sinh[( 2 ) ]

( , )sinh 2 sinh( 2)

qq

q

i mriG rm mr

1

(2)

(4 cosh ) sinh[( ) ]( , )

cosh( 2) sinh 2 sinh( )q q

qq

m i mriG rmr m mr

(3)cosh[( 2 ) ]

( , )2 sinh cosh( 2)

qq

q

i mriG rm mr

(4)sinh[( ) ]

( , )2 sinh sinh( )

qq

q

i mriG rm mr

— the Logunov-Tavkhelidze

— the Kadyshevsky

— the modified Logunov-Tavkhelidze

— the modified Kadyshevsky

( ) (0)

sin exp( ), ;1( , , ) ( , , )sin exp( ), .qj

qr iqr r rG r r G q r r

q qr iqr r r

– non-relativistic limit (0) (0) (0)

0

( , ) sin( ) ( , , ) ( ) ( , )q r qr G q r r V r q r dr

Green functions in the RCR

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(1)sinh[( 2 ) ]

( , )sinh 2 sinh( 2)

qq

q

i mriG rm mr

1

(2)(4 cosh ) sinh[( ) ]

( , )cosh( 2) sinh 2 sinh( )

q qq

q

m i mriG rmr m mr

(3)cosh[( 2 ) ]

( , )2 sinh cosh( 2)

qq

q

i mriG rm mr

(4)sinh[( ) ]

( , )2 sinh sinh( )

qq

q

i mriG rm mr

Green functions ( ) ( ) ( )( , , ) ( , ) ( , )j q j q j qG r r G r - r G r + r

( ) ( )

2( , , ) sin expj q q qjrq

G r r mr i mrK

(1) (2) sh 2q q qK K m (3) (4) 2 shq q qK K m

( ) ( )( , , ) ( , , )j jq qG r r G r r 1)

2) (1,3) (1,3)( , , ) ( , , )q qG r r G i r r

(2,4) (2,4)( , , ) (2 , , )q qG r r G i r r

Symmetry properties Asymptotic behavior

— the Logunov-Tavkhelidze

— the Kadyshevsky

— the modified Logunov-Tavkhelidze

— the modified Kadyshevsky

Integral equations in the RCR and scattering

amplitude

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( ) ( ) ( )0

( , ) sin( ) ( , , ) ( ) ( , )q q q qj j jr mr G r r V r r dr

( ) ( )( , ) sin( ) ( )exp( )q q q qj jrr mr q f i mr

( ) ( )( )0

2( ) sin( ) ( ) ( , )q q qj jjq

f dr mr V r rqK

At r → ∞ using GF asymptotic one obtain

Scattering amplitude

Scattering cross section

2( )0( ) ( ) 4 ( )j qj q f

2( ) (0) (0)0

1( ) ( ) sin( ) ( ) ( , )qjf f q dr qr V r q rq

Non-relativistic limit

Partial equations for s-states in the MR

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Equations for scattering s-states ( ) ( ) ( )

0

1 2( , ) ( ) ( , ) ( , , ) ( , )2q q qq qj j jm

m G d V E

sinh sinhp m k m

Green functions

(1) 2 2 2

1( , )(cosh cosh 0)q

q

Gm i

(3) 2 2 2

cosh( , )(cosh cosh 0)q

q

Gm i

(2) 2

1( , )2 cosh cosh 0 coshq

q

Gm i

(4) 2

1( , )2 cosh cosh 0q

q

Gm i

— the Logunov-Tavkhelidze

— the Kadyshevsky

— the modified Logunov-Tavkhelidze

— the modified Kadyshevsky

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Equations for bound and resonance s-states in the RCREquations for bound states

Equations for resonance states

( ) ( ) ( )0

( , ) ( , , ) ( ) ( , )q q qj j jiw r G iw r r V r iw r dr

q qiw Rapidity – is imaginary value

V(r)

o r

Bound states

Resonance states

Rapidity – is complex value

q q qiw

(0) 1 2 (0) 1 2 (0) 1 20

( , ) ( , , ) ( ) ( , )q iq r dr G q iq r r V r q iq r

( ) ( ) ( )0

( , ) ( , , ) ( ) ( , )j q q j q q j q qiw r dr G iw r r V r iw r

qm q

- non-relativistic limit

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( ) ( ) ( )0

2( , ) ( , ) ( , ) ( , )j q q j q q j q qmiw G iw d V iw

( ) ( ) ( )0

2( , ) ( , ) ( , ) ( , )j j jq q qGmiw iw d V iw

(0) 1 2 (0) 1 2 (0) 1 20

2( , ) ( , ) ( , ) ( , )q iq p G q iq p dkV p k q iq k

Partial equations for s-states in the MREquations for bound states

Equations for resonance states

- non-relativistic limit

Solving method

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( ) ( ) ( )0

( , ) ( , , ) ( ) ( , )j q q j q q j q qiw r dr G iw r r V r iw r

( ) ( ) ( )1

( , ) ( , , ) ( ) ( , )N

q q qj k j k k j kk

r W G r r V r r

Quadrature formula 1

( ) ( )b N

k kka

dxf x W f x

1

0N

nk kk

M

( , , ) ( )n k n k k q n k kM W G r r V r

( ) ( )( ) det ( ) 0j q j qd M

( ) ( ) 0j jM

Model potential

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22( ) e rV r V r В РКП

В ИП

2 22 2 2 2

2 3 32 22 2 2 2

3 3,

m mV V

m m

Nonrelativistic case

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Solving of the Schrödinger equation in the coordinate representation at m=1, V2 =15, α = 1

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Relativistic case

Cross section for j=1 at m=1, V2 =15, α = 1

Zeros of determinant

Complex scaling method

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0 Re r

Im r

r → r ei θ

θ

Scaling in the RCR

( ) ( ) ( ) ( )( ) ( ) ( )

0

( , ) ( , , ) ( ) ( , )j q q j q q j q qiw r dr G iw r r V r iw r

exp( ) exp( )z r i z r i

( ) ( ) ( )( ) ( ) ( ) ( )( , ) ( , ) ( ) exp( ) ( ) ( , , ) ( , , )j q j q j q j qr z V r i V z G r r G z z

( ) ( )( ) det ( ) 0j q j qd M

( ) ( ) 0j jM

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Scaling in the MR (non-relativistic case)

p→pe-iθ

θ

Im p

Re p0

exp( ) exp( )p i k i π π

( ) ( ) ( ) ( )(0) 1 2 (0) 1 2 (0) 1 2

0

2( , ) ( , ) ( , ) ( , )q iq p G q iq p dkV p k q iq k

( ) ( ) ( )(0) (0) (0) (0)( , ) ( , ) ( , ) exp( ) ( , ) ( , ) ( , )q p q V p k i V G q p G q π π π π

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χq →e-iθ

θ

Im χq

Re χq0

exp( ) exp( )i i

( ) ( ) ( ) ( )( ) ( ) ( )

0

2( , ) ( , ) ( , ) ( , )j q q j q q j q qmiw G iw d V iw

( ) ( ) ( )( ) ( ) ( ) ( )( , ) ( , ) ( , ) exp( ) ( , ) ( , ) ( , )j q j q j q j qV i V G G

Scaling in the MR (relativistic case)

Results

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Zeros of determinant after complex scaling on 30º

Solving in the RCR

Solving in the MR

Resonance rapidity in the case of the Kadyshevsky equation

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θ=0 Решение в РКП

Решение в ИП

θ=30º

Resonances not determine

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Resonance rapidy in the case of modified Logunov-Tavkhelidze equation

Solving in the RCR

Solving in the MR θ=0

θ=30º

Resonances not determine

Conclusion

Complex scaling method can be used for resonance states finding on the basis relativistic two-particle equations solving both in the RCR and in the MR.

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Thank you for your attention!

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