Transcript No Slide Title

Incoherent pair background processes with full
polarizations at the ILC
Anthony Hartin
JAI, Oxford University Physics, Denys Wilkinson Building, Keble Road Oxford, UK
Introduction
Final pair polarization
The International Linear Collider will collide polarized
particle bunches to produce physics processes of interest
and background events. The polarization states of both
the bunch particles and the collective field of the bunch
are important parameters affecting cross-sections and
hence final particle states. There are three incoherent
background pair processes considered in background pair
generators such as the CAIN program. Initial photons
involved in these process are either real or virtual.
The polarizations of final states are specified by the e
polarization vector (ζ1,ζ2,ζ3). The components of this
vector can be written in terms of a sum over products of
initial photon polarization states and a function Fii'jj' of 4vector scalar products specified in a paper by V.N. Baier
and A.G. Grozin (hep-ph/0209361)
1
 i   F jji 0' j 'j ' where F   F jj00' j 'j '
F ijj'
jj '
The present version of CAIN deals with partial
polarizations for the real photons only. Most of the pairs
however are produced by processes involving virtual
photons. The flux of virtual photons is related to the bunch
fields via the usual Weizsacker-Williams method. The
polarization state of virtual photons is obtained with an
expression for the bunch electric field at the point of
production. An expression for the pair production crosssection containing all polarizations is also required. These
modifications lead to new characteristics for the produced
pairs which are essential for other background studies.
The number of pair particles for all seven ILC parameter sets with 500GeV
centre of mass energy. Inclusion of all polarizations result in 10-20% less
pairs than previously simulated by CAIN
The components of final e polarization vector are
strongly dependent on the extent of circular polarization
of initial photons. Consequently the pairs are produced
with polarization components almost zero
1st order
nonlinear Pair
Production
Bunch Fields
2nd order
nonlinear Pair
Production
Stokes Parameters
The photon polarization vector is defined with respect to
a basis vector (êx,êy,kz) and can be expressed in terms
of a density matrix ρ, Pauli matrices σ and stokes
parameters (ξ1, ξ2, ξ3)
1
~ ~
 ij  ei e
  (1   . )
2
For virtual photons, the stokes parameters can be
written in terms of the transverse bunch electric field
(Ex,Ey) at the point (x,y) of pair production. Since the
electric field for relativistic charge bunches is constant
and crossed, there is almost no circular polarization
component for the initial photon polarizations
  Eˆ Eˆ *  Eˆ Eˆ *
  ( Eˆ Eˆ *  Eˆ Eˆ * )
*
j
1
x
y
y
x
2
y
x
x
y
3  Eˆ x Eˆ x*  Eˆ y Eˆ *y
Incoherent pair particle energy, comparing partial polarizations originally in
CAIN and full polarizations. Full polarizations manifest as a reduction in low
energy pairs.
Breit-Wheeler cross-section
In order to take advantage of the full polarisation of initial
photons, the full Breit-Wheeler cross-section is required.
At present in CAIN the cross-section σcirc is written down
only for the product of circular polarisations ξ2ξ’2 of initial
photons k and k'. The full cross-section σfull is a sum over
all polarisation states and functions of final electron
energy ε and momentum p. With some algebraic
manipulation the two cross-sections can be written in
similar form.
2

2

1 
p
1 
circ
1

  21  h2 
sinh
p

3
h

1

 2
2 
2 
2 

 

h3 
h3 
2
p

full
1
'
  21  h2  2 (h1  h3 )  4  sinh p   3h2  1   3 3  2 

 

 

where h1  11'
h2   2 2'
h3  1   3   3'   3 3'
Virtual Photon Polarization
The stokes parameters for the virtual photons follow
from an explicit expression for the bunch electric field.
The field of a single relativistic charge can be expanded
in plane waves. For a Gaussian bunch of size (σx,σy) of
N charges, the collective field is an integration over
transverse wave vector (qx,qy) of a product of Gaussian
form factor and the single charge field
E x, y  
qx, y
F (q~ ) exp( ixqx  iyq y )dq x dq y
q x2  q y2
where F (q~ )  N exp  12 (q x x ) 2  12 (q y y ) 2


A numerical investigation of the two cross-sections in the
above equation reveal the usual peak at low energies.
However accounting for full polarizations reveals a
substantially reduced cross-section for electron energies
approximately less than 50 MeV (see below). It was
expected that such a reduction in the Breit-Wheeler
cross-section would result in less background pairs. It
was also considered important to determine any effect on
collision luminosity. So the modified CAIN program was
run for all seven 500 GeV centre of mass collider
parameter sets. There was a 10-20% overall reduction in
pair numbers (see above) with no discernible effect on
collision luminosity.
The integrations are performed by expanding 1/(qx2+qy2)
in a Taylor series. The integration variables can then be
separated and the Fourier transforms are easily
performed. Since the ILC uses flat beams (σx>>σy) only
the first term in the Taylor series is required.
E x  exp   2x 2 
x 

2

x 

 x3 

2
 y2 
exp   2   y
 2 y 
x2  1

E y  2 exp   2 2   x Erf
x 


 

2
Erf
 p ( x)  E p ( x)u p

Ep  1
e

2 ( kp )
where


( kx )
e
e

 exp   i ( px)  i 
k A
(
pA
)
2
(
kp
)
0

e
e
2

Ae 2 

The full cross-section for this process requires substantial
work to calculate but it retains the same form with respect
to polarization as its external field free equivalent. Much
of the foregoing analysis of polarization effects can be
directly applied to this stimulated Breit-Wheeler process,
and will be the subject of ongoing work
Conclusion
 The full polarizations of the incoherent
background pair processes at the interaction point
of the ILC have been investigated analytically and
numerically.
Virtual photon polarization is related to the bunch
electric field at the point of production and, like the
real beamstrahlung photons, have almost no
circular polarization component
The full Breit-Wheeler cross-section with all
polarization states can be written in similar form to
the cross-section with circular polarizations only
For all seven parameter sets contemplated for the
ILC, with 500 GeV centre of mass collision energy,
there is a 10-20% reduction in low energy pair
numbers with no discernible change in collision
luminosity
Extensions of this study
processes are envisaged





y
2 y
The magnitude of the y component of the bunch field is
much greater than that of the x component, consistent
with the beam being squeezed in y. Both components
are real, meaning that there is no circular polarisation of
virtual photons.
The pair production processes occur in the presence of
intense bunch fields and the possibility must be allowed
for coherent production of pairs to all orders. The Dirac
equation solutions in the presence of the bunch fields
Ae(k) are required and turn out to be a simple product of
Volkov function Ep and the usual bispinor up
Produced pairs have almost no polarization
 
y
2 y
Stimulated Breit-Wheeler process
to
the
coherent
Further information
[email protected]
Comparison of the existing CAIN Breit-Wheeler cross-section and the BreitWheeler cross-section for full polarizations of initial states
http://www-pnp.physics.ox.ac.uk/~hartin/theory
+44(0)1865273381