Bound-Free Electron-Positron Pair Production Accompanied by Coulomb Dissociation
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Bound-Free Electron-Positron Pair Production Accompanied by Coulomb Dissociation M. Yılmaz Şengül Kadir Has University & İstanbul Technical University M. C. Güçlü İstanbul Technical University 1
description
Bound-Free Electron-Positron Pair Production Accompanied by Coulomb Dissociation. M. Yılmaz Şengül Kadir Has University & İstanbul Technical University M. C. Güçlü İstanbul Technical University . - INTRODUCTION * Free Electron-Positron Pair Production - PowerPoint PPT Presentation
Transcript of Bound-Free Electron-Positron Pair Production Accompanied by Coulomb Dissociation
Bound-Free Electron-Positron Pair Production Accompanied byCoulomb Dissociation
M. Yılmaz ŞengülKadir Has University & İstanbul Technical University M. C. Güçlüİstanbul Technical University
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- INTRODUCTION* Free Electron-Positron Pair Production* Bound-Free Electron-Positron Pair Production* Pair Production by Nuclear Dissociation - Free Pair Production - Bound-Free Pair Production
- FORMULATION* Formulation for Free Electron-Positron Pair Production * Cross Section Calculations for Bound-Free Electron-Positron Pair Production* Other Methods for Bound-Free Electron-Positron Pair Production Cross Section Calculations* Impact Parameter Dependent Bound-Free Electron-Positron Pair Production * Comparison of the Impact Parameter Dependent Bound-Free Electron-Positron Pair Production
Calculations with the Other Methods* Bound-Free Electron-Positron Pair Production Cross Section Calculations by Coulomb
Dissociation - RESULTS* Cross Section Results for Bound-Free Electron-Positron Pair Production by Giant Dipole
Resonance
2
3
ee
time
Ion 1
Ion 2
Emits photon Emits photonPair Production
INTRODUCTION*Free Electron-Positron Pair Production
eeZZZZ 2121
Z
Z
e+
e-
Fig-1: Pair production with capture in relativistic heavy ion colliders [1]. 5
* Bound-Free Electron-Positron Pair Production
eZ)eZ(ZZ,.../s 21121 21
Z
Z Z-1
e+
e- e-
Fig-2: Free pair production accompanied by GDR in a relativistic heavy ion collisions.
* Pair Production by Nuclear Dissociation
- Free Pair Production; eeZZZZ **
2121
7
n
n
n
nn
n
n
n
nn n
n
e-
e+
8
- Bound-Free Pair Production;
eZ)eZ(ZZ *s
*,.../ 21121 21
Fig-3: BFPP accompanied by GDR in a relativistic heavy ion collisions.
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n
n
n
nn
n
n
n
nn n
n
e-
e+
e-
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(x)(x)FFL μνμνEM 4
1
Free Fermion Lagrangian Density
Interaction Lagrangian Density
Semi-Classical Action Integral
)x(A)x()x(e)x(LI
)(:)()()(:)( EMI04 txxxtxdS LLL
)x()mi)(x()x(L 0
EM Lagrangian Density
- FORMULATION
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* Formulation for Free Electron-Positron Pair Production
Dirac wave-function of electrons&positrons
A Electromagnetic vector potential
(x)(x)FFA
LLLLμν
μνμ 41μ
EMI0QED
ΨγΨem)Ψ(Ψ
i
)x(A)x(AF Electromagnetic field tensor
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Equations of Motion
),(),()( '' ttKittKxH t
)()()( 0 xVxHxH
mixH 00 .)(
)()(.)( 0 xAxAxV
13
01 )x())x(Ai(
Perturbative Expansion
0 0 0
20 0 0
, , ( ) , ,
( ) , , , ...
t
t
K t K t i d K t V K
i d d K t V K V K
)tt(iH' '
e)t,t(K 00
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4-Vector Potentials of Colliding Ions
)(A)(AA 21
]b.qiexp[)qq(q
)qq(Z)(Ayxz
z
281 2222
0220
021
021
)(A)(A
)(A)(Ayy
xx
)(A)(Az 11 0
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The time-evolved vacuum state in the interaction picture;
Total cross section for electron-positron pair production;
000 0 )t,t(K)t,(KlimSt
2
0 0
2
k q
)(q
)(k Sbd
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Second order terms for direct and crossed diagrams;
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)(q
'
''''
)(k
')(q
)(k
),(K),(K
)](V),(K)(V)(V),(K)(V[
),(K),(Kdd)i(S
0
0
00
102201
002
2112 SSS
* Cross Section Calculations for Bound-Free Electron-Positron Pair Production
Captured electron (Darwin) wave function [2,3];
)(.)( rmi
rnon
Ψuψ
2
1
HaZr
Hrnon e
aZr /
/
)(
231
Ψ
HaZr
H
eaZ
mir /
/)( .)(
231
21
uψ
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Positron (Sommerfeld-Maue) wave-function [1,4];
)1(2/
iaeN a
v
Zea2
1ea2N a2
2
Distortion (correction) term due to the large charge of the ion.
'ψ
][ ')(.)( ψuψ
q
rqiq eN
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N Normalization constant.
Fig-4: Lowest-order Feynman diagrams for the pair production of a bound-free electron-positron pair in heavy-ion collisions: (i) direct and (ii) crossed diagrams for the simultaneous capture of the electron into a bound state of target (T) ion. In the figure, 1 and 2 represents the two ions, and q is the momentum of the positron [5].
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S- transition matrix element for direct term of Feynman diagrams;
):():():(
)().(
/
)()(
pqpFpF
epdaZiN
S
q
bq
pi
H
q
21
22
223
12
21
2
ψψ
21
Scalar parts of the fields as associated with ions 1 and 2;
2
22
21
2
214
paZ
eZ):p(F
H
2222
2
22
2
)(
4):(qp
ZeqpF
22
The virtual photons frequency of ion 1 & ion 2;
zz
)(q
)(
pqEE
21
)qp(qEE
zzz
)()(q
22
23
Transition amplitudes;
)()()(
)()()(
)()(
.
)(
):(
qpp
p
Zss
Z
s zqsp
q mp
qEEE
p
uuuu 11
21
22
1
24
After making all the simplifications, the final form of the BFPP cross section can be expressed as;
Some proper products of the transition amplitudes and scalar parts of the fields as associated with ions 1 and 2.
2
)()(5
233
2
2
):():()2(
)(14
q
pqqpqpqdd
aZN
H --
):():():():( 21)(
pqpFpFpq q-
):():():():( 12)(
pqpFqpFpqq q-
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2
0
2
q
qSbd )()( ψψ
* Other Methods for Bound-Free Electron-Positron Pair Production Cross Section Calculations
* Bertulani and Baur [1988]
* Rhoades-Brown, Bottcher and Strayer [1989]
* Baltz, Rhoades-Brown and Weneser [1991-94]
BABFPP ln AuAubarnBFPP )(24ln2.11
PbPbbarnBFPP )(31ln3.14
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* Impact Parameter Dependent Bound-Free Electron-Positron Pair Production [6,7]
(q))qb(Jbqdqdb
d BFPP -
0
0
;K;KF);qQ(F
);qQ(;KF);qQ(F
;K;KF);qQ(F
);qQ(;KF);qQ(F
)(QKdddq
daZ
N)q(
q
q
q
q
zq
H q
21
21
21
21
7
222
0
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2
21
21
21
21
21
21
21
8-
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Fig-5: The function is calculated . The points show the results of the Monte Carlo calculations and the smooth curve is our fit for these points. The slope of this function gives the value of a as .
)(/)q( 0--
C35.1 28
23220
0 /BFPPBFPPqaBFPP
)ba(ba)b(Pe)qb(Jbqdq(0)
dbd
-
aqBFPP
aq eeq )0()( --
* Comparison of the Impact Parameter Dependent Bound-Free Electron-Positron Pair Production Calculations with the Other Methods
2/1410 )1072134.1(246.178)(
bbP
2/1410 )1072134.1(807.228)(
bbP
RHIC-Au+Au RHIC-Pb+Pb
23222 /BFPPBFPP )ba(a
)b(P
Fig-6: Probability of positron production with gold beams at RHIC as a function of b; the solid line shows our work and the dashed line shows the work of Baltz [7].
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* Bound-Free Electron-Positron Pair Production Cross Section Calculations by Coulomb Dissociation
2/322 )(2)(
baabP BFPPBFPP
)(1)1()1(
1
)()( bPnCnC
CebPbP
21 /)( bSbPC
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the probability of bound-free electron-positron pair production probability;
the probability of GDR excitation in one ion;
23/23532
1045.52 fmNAZAm
NZSGDRN
the probability of a simultaneous nuclear excitation asa function of impact parameter;
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))(()(
1)(1)( bP
XnCXnCebP
)()(2 2)(
min
bPbPbdb XnCb
BFPPGDRBFPP
the probability for at least one Coulomb excitation;
the total cross section for BFPP with mutual nuclear excitation at least one Coulomb excitation;
)()(2 2)1(
min
bPbPbdb nCb
BFPPGDRBFPP
the total cross section for BFPP with mutual nuclear excitation;
* Cross Section Results for Bound-Free Electron-Positron Pair Production by Giant Dipole Resonance
Untagged (barn)
Tagged(1n1n) (mbarn)
Tagged(XnXn) (mbarn)
Au+Au-RHIC-Free [7] 34000 1630 1980
Pb+Pb-LHC-Free [7] 212000 10200 12400
Au+Au-RHIC-BFPP [7] 94,5 4,5 5,5
Au+Au-RHIC-BFPP-Baltz [8] 88,8 1,1 1,4
Pb+Pb-LHC-BFPP [7] 202 9,7 11,7
Table-1: Integrated cross sections for Au+Au collision at RHIC energies and for Pb+Pb collisions at LHC energies for free and bound-free pair production.
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- RESULTS
Untagged (barn)
Tagged(1n1n) (mbarn)
Tagged(XnXn) (mbarn)
Pb+Pb-RHIC [7] 123 5,92 7,18
Pb+Pb-RHIC-Baltz [8] 113 1,44 1,74
Table-2: Integrated cross sections for Pb+Pb collisions at RHIC energies by using our calculations and for the same collision at RHIC energies by using the calculations of Baltz [7].
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Fig-7: The probability of positron pair production with (a) gold beams at RHIC and (b) lead beams at the LHC as a function of b with XnXn (dashed line) and 1n1n (dotted line) and without nuclear excitation [7].
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Fig-8: The differential cross section as function of energy of the produced positrons isshown in the graph (a) for RHIC and (b) for LHC. And the differential cross section isshown as function of the longitudinal momentum of the produced positrons in the graph(a) for RHIC and (b) for LHC [7].
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Fig-9: The differential cross section as function of transverse momentum of the produced positrons is shown in the graph (a) for RHIC and (b) for LHC. And the differential cross section is shown as function of the rapidity of the produced positrons in the graph (a) for RHIC and (b) for LHC [7].
Finally;
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1) By using semi-classical two photon method, we have obtained bound-free electron-positron pair production cross section.
2) By using this method, we have calculated impact parameter dependent bound-free electron-positron pair production probability.
3) We have calculated the cross section of bound-free electron-positron pair production accompanied by giant dipole resonance for the first time in the literature.
4) We have obtained the differential cross section of produced positrons as a function of energy, transverse momentum, longitudinal momentum and rapidity for the bound-free electron-positron pair production accompanied by giant dipole resonance.
5) We are planning to implement this method to calculate the cross section of different particles such as mesons, heavy leptons and anti-hydrogen.
- REFERENCES 1) C.A. Bertulani, D. Dolci, Nucl. Phys. A 683, 635(2001).2) V.B.Berestetskii, E.M. Lifshitz, L.P. Pitaevskii, Relativistic Quantum Field Theory (Pergamon Press, NewYork, 1979).3) J. Eichler, W.E. Meyerhof, Relativistic Atomic Collisions
(Academic Press, California, 1995).4) C.A. Bertulani, G. Baur, Phys. Rep. 163, 299 (1988).5) M.J. Rhoades-Brown, C. Bottcher, M.R. Strayer, Phys. Rev. A 40,
2831 (1989).6) M.C. Güçlü, Nucl.Phys. A 668, 149 (2000).7) M.Y. Şengül, M.C. Güçlü, Phys. Rev. C 83, 014902 (2011).8) A.J. Baltz, M.J. Rhoades-Brown, J. Weneser, Phys. Rev. E 54,
4233 (1996).
THANKS FOR YOUR ATTENTION!
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