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
- 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
INTRODUCTION
*Free Electron-Positron Pair Production
Z1  Z 2  Z1  Z 2  e e
Ion 1
e
e
Ion 2
time
Pair Production
Emits photon
Emits photon
3
Z
e+
e-
Z
* Bound-Free Electron-Positron Pair
Production
Z1  Z 2  ( Z1  e )1s1 / 2 ,...  Z 2  e
Fig-1: Pair production with capture in relativistic heavy ion colliders [1].
5
Z
e+
e-
Z
Z-1
e-
* Pair Production by Nuclear Dissociation
- Free Pair Production;
 
Z1  Z2  Z  Z  e e
*
1
*
2
Fig-2: Free pair production accompanied by GDR in a relativistic heavy
ion collisions.
7
n
n
n
n
n
n
e+
e-
n
n
n
n
n
n
8
- Bound-Free Pair Production;
Z1  Z 2  ( Z1*  e )1s1 / 2 ,...  Z 2*  e
Fig-3: BFPP accompanied by GDR in a relativistic heavy ion collisions.
9
n
n
n
n
n
n
e+
e-
n
n
n
n
e-
n
n
10
- FORMULATION
* Formulation for Free Electron-Positron Pair Production
Semi-Classical Action Integral
S   d x (t ) : L0 ( x )  LI ( x )  LEM ( x ) : (t )
4
Free Fermion Lagrangian Density
L0 ( x )  ( x )(   i   m )( x )
Interaction Lagrangian Density

LI ( x )  e( x )  ( x ) A ( x )
EM Lagrangian Density
LEM   Fμν(x)F (x)
1
4
μν
11
LQED  L0  LI  LEM
 Ψ(  i   m)Ψ  eΨγμ ΨAμ  14 Fμν (x)F μν (x)

Dirac wave-function of electrons&positrons
A
Electromagnetic vector potential
F    A ( x )   A ( x )
Electromagnetic field tensor
12
Equations of Motion



( i   A ( x ))  1  ( x )  0
H ( x) K (t , t )  i t K (t , t )
'
'
H ( x)  H 0 ( x)  V ( x)
H 0 ( x)  i .   0 m
V ( x)   . A( x)  A0 ( x)
13
Perturbative Expansion
t
K  t ,    K 0  t ,    (i)  d K 0  t , V   K 0  ,  

t



(i) 2  d  d K 0  t , V   K 0  ,  V    K 0  ,    ...
K 0 ( t ,t )  e
'
 iH0 ( t t' )
14
4-Vector Potentials of Colliding Ions



A  A (1)  A ( 2 )

 b
 ( q0  qz )
0
2
2
A ( 1 )   8 Z 2
exp[iq . ]
2
2
2
qz   ( qx  q y )
2
A (1 )   A (1 )
z
0
A (1 )  A ( 2 )  0
x
x
A (1 )  A ( 2 )  0
y
y
15
The time-evolved vacuum state in the interaction
picture;
S 0  lim K 0 ( 0,t )K ( t ,t ) 0
t 
Total cross section for electron-positron pair production;
   d b 
2
k 0 q 0
()
k
S
()
q
2
16
Second order terms for direct and crossed diagrams;




 k(  ) S  q(  )  ( i )2  d  d '  k(  ) K 0 ( 0, )K 0 (  , )
 [ V1(  )K 0 (  , ' )V 2(  ' )  V2 (  )K 0 (  , ' )V1(  ' )]
 K 0 (  ' , )K 0 ( ,0 )  q(  )
S  S12  S21
17
* Cross Section Calculations for Bound-Free
Electron-Positron Pair Production
Captured electron (Darwin) wave function [2,3];
ψ
()

 
i
 1 

.  uΨ non r ( r )
 2m



1  Z

Ψ non  r ( r ) 
  aH



3/ 2
e  Zr / aH
i   1  Z 

()
 
ψ ( r )  1 
 . u
 2m
   aH 
3/ 2
e  Zr / aH
18
Positron (Sommerfeld-Maue) wave-function [1,4];
ψ
N  e
N
2
ψ
'
N
()
q
 N  [e
 a / 2
i q.r
u q  ψ ]
(1  ia )
2a
 2a
e  1
()
'
Ze
a 
v
2
Distortion (correction) term due to the large
charge of the ion.
Normalization constant.
19
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].
20
S- transition matrix element for direct
diagrams;
term of Feynman
ψ (  ) S12 ψq(  ) 
iN  1  Z 
 
2    aH 
3/ 2
2
d p
 (2 )2 e
i ( p 
q
). b
2
F (  p  : 1 ) F ( p   q  : 2 ) q ( p  :   )
21
Scalar parts of the fields as associated with ions 1 and 2;
F (  p  : 1 ) 
4 Z e
2 
 Z2
12
 2  2 2  p  
 aH  

F ( p   q  : 2 ) 
4 Ze 2  2
 2   2  2 ( p  q ) 2 
 2





22
The virtual photons frequency of ion 1 & ion 2;
1 
2 
E
()
 Eq
()
  qz
2
Eq
()
E
()
2
  qz
  pz
  ( pz  qz )
23
Transition amplitudes;
 

1
. p
 q ( p  :   )  
1 

()
()
2m 
 
E  Eq
s  p  (s)
q
z
 Ep  (
) 


2
2


 u (1   Z ) u( sp)  u( s ) (1   Z ) u( q ) 
p
24
After making all the simplifications, the final form of
the BFPP cross section can be expressed as;
   d b ψ
2
()
Sψ
q 0

N
4
2
2
3
2

d qd p 
1 Z 3
( ) 

5
 aH  q
(2 )
()
q
2

2
()
(q : p  )  
()
(q : q   p  )
Some proper products of the transition amplitudes and
scalar parts of the fields as associated with ions 1 and 2.


()
( q : p  )  F (  p  : 1 ) F ( p   q  : 2 ) q ( p  :   )
()
(q : q   p  )  F ( p   q  :  2 ) F ( p  : 1 ) q (q   p  :   )
25
* 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]
 BFPP  A ln   B
 BFPP  11.2 ln   24 (barn) Au  Au
 BFPP  14.3ln   31 (barn) Pb  Pb
26
* Impact Parameter Dependent Bound-Free
Electron-Positron Pair Production [6,7]
d BFPP 
  dq q b J 0 ( qb ) (q)
db
0

( q ) 
8 2
2 1  Z

N
  aH




3
2
dqz d 2 Kd 2 Q
d q 


( 2 )7
q 0

 1

 1

F
(
Q

q
);

F

K
;



(
Q

q
);


1
2
q
 2

 

 2



1
  F  ( Q  q );  F  K ;  K ; 

1
2
q



2



 


 1

 1

F
(
Q

q
);

F

K
;



(
Q

q
);


1
2
q
 2
 
 

 2


1
  F  ( Q  q );  F  K ;  K ; 

1
2
q



2


 

27

d BFPP
ab
a q
 (0) dq q b J 0 ( qb ) e  PBFPP( b )   BFPP 2
2 3/ 2
db
(
a

b
)
0
(q)  (0) eaq   BFPP eaq
Fig-5: The function ( q ) / ( 0 ) 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 1.35C .
28
Comparison of the Impact Parameter
Dependent Bound-Free Electron-Positron Pair
Production Calculations with the Other Methods
*
178.246
P(b) 
(1.72134 1010  b 4 )1 / 2
RHIC-Au+Au
P(b) 
228.807
(1.72134 1010  b 4 )1 / 2
RHIC-Pb+Pb
PBFPP ( b )   BFPP
a
2  ( a 2  b 2 )3 / 2
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].
29
* Bound-Free Electron-Positron Pair
Production Cross Section Calculations by
Coulomb Dissociation
PBFPP (b)   BFPP
PC (1n ) (b)  P
1
C (1n )
PC1 (b)  S / b 2
a
2  ( a 2  b2 )3 / 2
(b) e
 PC1 ( b )
the probability of bound-free electron-positron
pair production probability;
the probability of GDR excitation in one ion;
the probability of a simultaneous nuclear excitation as
a function of impact parameter;
2 2 Z 3 N
S
 5.45  105 Z 3 NA2 / 3 fm2
AmN GDR
30

2
db
b
P
(
b
)
P
BFPP
C (1n ) ( b)

GDR
 BFPP
 2
bmin
the total cross section for BFPP with mutual nuclear excitation;
PC ( Xn ) (b)  1  e
(  PC1 ( Xn ) ( b ))
the probability for at least one Coulomb excitation;

GDR
 BFPP
 2  db b PBFPP (b)PC2( Xn ) (b)
bmin
the total cross section for BFPP with mutual nuclear excitation at least one
Coulomb excitation;
31
- RESULTS
* Cross Section Results for Bound-Free ElectronPositron 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.
32
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].
33
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].
34
Fig-8: The differential cross section as function of energy 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 longitudinal momentum of the produced positrons in the graph
(a) for RHIC and (b) for LHC [7].
35
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].
36
Finally;
1)
2)
3)
4)
5)
By using semi-classical two photon method, we have obtained bound-free
electron-positron pair production cross section.
By using this method, we have calculated impact parameter dependent boundfree electron-positron pair production probability.
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.
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.
We are planning to implement this method to calculate the cross section of
different particles such as mesons, heavy leptons and anti-hydrogen.
37
- 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!
38