
Formalism of extraction of longitudinal spin transfer DLL
' in semiinclusive  production in 3D formalism and flickering coordinate
systems.
S.Belostotski and D.Veretennikov
Abstract.

Final version of formalism of spin transfer vector DLL ' extraction is presented in details. Using moment method, most

general case of DLL ' extraction without requirement of exact helicity balance is considered. It is shown that the spin
transfer vector components evaluated independently in two typically used in  analysis coordinate systems may approximately be connected with a unitary transformation from one coordinate system to another, correspondently, the

length of DLL ' vector is only approximately preserved.
I.

DLL
' extraction using the moment method

Extraction of the spin-transfer DLL
' is based on moment method [1]. The polarized and unpolarized
moments are calculated using only experimental data sample. As long as the beam polarization changes
periodically the sign, no Monte Carlo simulation of the spectrometer acceptance is needed for the proce
dure of DLL
' extraction. All the considerations below are in the Λ rest frame.

Polarization ( DLL
' ) of the produced Λ hyperon can be measured using weak   p decay by asymmetry of proton emission in respect to the Λ polarization vector in the Λ rest frame. In 4  acceptance
case, the decay probability is given by a factor
d
d p
1   P k p .
(1)
   
,   and   is Λ production cross section with spin projection up or down in re
spect to (unknown) direction of the polarization vector,        is unpolarized cross section for Λ
Here P 
production, k p is the unit vector along the proton momentum in the  rest frame,  is decay analyzing
power ,   0.642  0.013 for  decay ( 0.642  0.013 for ) decay. The  polarization can be written
as
P  DLL ' D  y  PB ,
where PB is the beam polarization, D  y  
(2)
y 2  y
is depolarization factor, y is fractional energy
y  2 1  y 
2
of the scattered positron and DLL ' is vector of spin transfer with components DLx , DLy , DLz , L is direction
of virtual photon polarization vector assumed to be along its momentum. Scalar product P k p can be
written as D  y   PB

i x, y, z
DLi cos  pi , where  pi (i  x, y, z ) is the angle between the proton momentum and
a coordinate system axis x, y or z.
1
Introducing notations di   DLi and c pi  D( y ) cos  pi one can rewrite Eq.1 as
d
d p
1  PB d  c p  1  PB 

k  x, y , z
(3)
dk c pk
Introduce spectrometer acceptance function
a  a(c p , var) ,
(4)
where var is set of variables including Λ production kinematics (kin), production V prod and decay V dec
vertex positions, i.e., var  kin, V prod , V dec . Acceptance function 0  a  1 gives probability to detect the
decay proton and pion for a given Λ production kinematics and event topology. Note that here we ignore
smearing effects.
One can use Eq.3 as probability to detect the proton emitted within d p (in coincidence with the piT
on) for a data taking period 0  t  T with integrated luminosity LT   L (t )dt .
0
d  (c p , t )
d  p dt
 F  L(t )  [1  PB (t ) d  c p ]  a(c p )  F  L(t )  [1  PB (t ) 

k  x, y,z
d k c pk ]  a(c p )
(5)
where F is normalization constant and a (c p ) is in the “var bin” (see comment on that at the end of paragraph). Both for moment method and for maximum likelihood method, it is very important that F would
by correctly define. Normalizing integral is evaluated as
T
I   dt  d  p F  L(t )  [1  PB (t )d  c p ]  a (c p ) 
0
T
F  [ LT  a(c p )d  p  d   PB (t )L(t )  dt   c p a (c p )d  p 
(6)
0
F  LT   a (c p )d  p [1  PB  d  c p ]
0
Here PB 
1
LT
T
 P (t )L(t )dt
B
is luminosity weighted beam polarization and c p
0
0

 c a(c )d 
 a(c )d 
p
p
p
p
is
p
unpolarized first moment of c p weighted with acceptance function only (i.e., not affected by the beam
polarization). Normalized to unity probability (Eq.3) is rewritten as
d  (c p , t )
d  p dt
Here
d 0 ( c p , t )
d  p dt

L(t )  a (c p )
LT  a (c p ) d  p

d 0 (c p , t )
d  p dt

1  PB (t )d  c p
1  PB  d  c p
(7)
0
is normalized to unity unpolarized probability to detect Λ event.
2
g
Using Eq.7 one may calculate an average (moment) of a correlation PBs  c pif c pn
, i  x, y, z :
d  (c p , t )
T
g
g
PBs  c pif c pn
  PBs (t )  c pif c pn
d  p dt
0
PBs
g
c pif c pn
g
 PBs 1  d  c p  c pif c pn
0
1  PB  d  c p
s
B
P
s 1
B
 P
f g
pi pn 0
c c

dtd  p 
0

(8)
0

k  x, y ,z
g
d k c pk  c pif c pn
1  PB  d  c p
0
0
For practical calculations, there are two important cases:
s=1, f=1, g=0
PB
c pi
PB  c pi 
0
 PB2 

k  x, y ,z
1  PB  d  c p
d k c pk c pi
0
(9)
0
and s=0, f=g=1
c pi c pn
c pi c pn 
0
 PB 

k  x, y , z
d k c pk c pi c pn
1  PB  d  c p
0
(10)
0
At first look, as our experimental data sample is “always polarized,” the unpolarized moments ... 0
cannot be extracted directly from the experiment. On the other hand , as far as the polarized moments can
be expressed through the unpolarized ones (Eq.8, 9, 10) one may try to find equations not comprising unpolarized moments at all. This can be done using Eq.7 which we rewrite as
1
1  PB  d  c p
d0 (c p , t )
0
d  p dt

d  (c p , t )
1
1  PB (t )d  c p d  p dt
(11)
After integration of Eq.11 over t and  p one obtains
1
1  PB  d  c p
Evaluate unpolarized moments ci
0
1
1  PB d  c p

0
and ck ci
0
using Eq.11
1
c pi
0
1

1  PB d  c p
(12)
c pi

1  PB d  c p
3
(13)
c pi c pk
0
1

1  PB d  c p
1

c pi c pk
(14)
1  PB d  c p
Last two comments to this section. We have considered the case of var=const, i.e., it is assumed that Λ
production kinematics is restricted by a narrow bin and vertex configuration is fixed. Note first that vertex
topology affects only acceptance function, and thus the formalism does not change if instead of
a1 (c p , kin,V prod ,V dec ) will be used a(c p , kin) which is a1 (c p , kin,V prod ,V dec ) integrated other various vertex configurations.
d
and spin transfer d
dkin
within the bin must be taken into account. This however again does not affect Eq.8 except that
d
n
c npi a(c p )
d  p dkin
c
a
(
c
)
d


pi
p
p

dkin
and in similar way spin transfer must be averaged over the kin
d

a
(
c
)
d

 p p
 a(c p ) dkind  p dkin
ematical bin.
Second, if kinematical bin is wide a variation of the production cross section
Helicity balanced data sample. The data sample with Pb  0 is helicity balanced. It is equivalent to
unpolarized beam in which the number of beam particles with helicity +1 is equal to that with −1. In this
case, as follows from Eq. 10 c pk c pi  c pk c pi and Eq.9 is simplified to
0

k  x, y , z
PB c pi
c pk c pi d k 
(15)
PB2
which can be rewritten in matrix form
 c 2px

 c py c px

 c pz c px

c px c py
c 2py
c pz c py
c px c pz  d
 x 
1
 
c py c pz   d y   2
 
PB
c 2pz   d z 
 PB c px

 PB c py

 PB c pz







(16)
If Λ spin is known to be directed along z-axis (one dimensional case) Eq.15 is reduced to
N
DLz 
dz


PB c pz
 PB2 c 2pz

[ P D( y) cos ]
l 1
B
z l
(17)
N
 P
2
B
[ D ( y) cos
2
l 1
2
 z ]l
If the data sample is not helicity balanced one can evaluate formalism of the spin transfer extraction
similar to helicity balance case (Eqs. 16, 17) by replacing unknown unpolarized moments with the help of
Eqs.13, 14. One then obtains
4

k  x, y,z
c pk c pi
1  PB

j  x, y,z
d j c pj
dk 
PB c pi  PB
PB2  PB
c pi
2
i  x, y , z
(18)
It is easy to see that in the case of helicity balance Eq.18 reduce to Eq.15. It is also evident that “im2
balance” must not be very large, otherwise PB2  PB (analyzing power) is getting small and statistical
error bar of extracted spin transfer unreasonably large.
II.
 polarization (spin transfer) vector in two different coordinate
systems.
 polarization or spin transfer is extracted from a data sample using values measured for each event in
the  rest frame (LRF). For 3D analysis a coordinate system in the LRF must be defined. The latter can
be done using two physical vectors reconstructed for each Λ event: the momentum of virtual photon and
the Λ momentum in lab. frame. In the LRF, unit vector along virtual photon momentum e and unit vector along direction of Λ momentum (in lab. frame) e form a plane. Two Cartesian coordinate systems
XYZ and X’Y’Z’ can be built, respectively, with Y=Y’ axis perpendicular to the e , e plane and with
either Z axis of XYZ along e or Z’ axis of X’Y’Z’ along e . X and X’ axes are taken perpendicular to
the ZY and Z’Y’ planes, respectively. The X’Y’Z’ system is rotated around Y=Y’ axis in respect to the
XYZ by an angle  zz ' where cos zz '  e e . An important feature of a  data sample is that the angle

spin transfer
 zz ' is not stable but changing on event basis. Typical zz ' distribution for the recent DLL
analysis [2] is shown in Fig.1
Fig. 1 Probability distribution for angle between z and z' axes.
General considerations. It is commonly believed that the length of a physical vector P stays constant
in various coordinate systems. This is however not true if the vector under discussion is defined as an average over experimental data set and transformation matrix is not stable but changing on event basis
(flickering coordinate systems).
5
Assume that a physical vector P is defined as an average
P
1
N
N
k ,


(19)
1
where k is physical vector found for an event  from a data set   1, 2,...N in Cartesian coordinate system x, y, z .
For each  event
k  ak .
(20)
Here k is physical vector found for an event  from a data set   1, 2,...N in coordinate system
x ', y ', z ' and a is unitary transformation matrix connecting coordinate systems x ', y ', z ' and x, y, z . If
mutual positions of the coordinate system is stable over the data set, i.e. a does not depend on  we have
promptly:
P 
1
N
N
1
N
k   ak  aP ,

N

1
(21)
1
i.e. average over data set vector P may be transformed from one coordinate system to another like individual vector k defined on event basis. Correspondently, the vector length P must stay unchanged:
PP  aPaP  Pa † aP  Pa 1aP  PP
(22)
For flickering coordinate systems their mutual position is changing from event to event, i.e. a  a . In
this case Eq.22 is in general no longer valid. We may however conditionally write
P 
where a 
1
N
1 N
1 N
1 N
1 N

k

a
k

a
k  a P
  N
  N
 
N  1
N  1
 1
 1
(23)
N
a


is averaged over data set transformation matrix. Eq.23 is only approximately cor-
1
rect provided probability functions for a and k do not strongly correlating, and, correspondently the
vector length is not exactly preserved, as in general a 1  a 
1 N 1 1 N
a  1 (compare to
 a  N 
N  1
 1
Eq.22).
III.
Practical example.
Here we consider some numerical results taken from recent analysis of spin transfer to the Λ hyperon

in DIS [2]. According to above considerations for the spin transfer DLL
' vector found independently in
two coordinate systems one can write (for simplicity we omit indices LL  and Λ)
6
D  a D
(24)
Eq.24 is approximately valid provided angle  zz ' and components of unit vector k p of emitted in Λ
decay proton do not practically correlate. The latter is reasonably fulfilled for the measured data set:
cos  px  zz '  16.82  0.11
cos  py  zz '  0.51  0.13
cos  pz  zz '  10.65  0.12
cos  px
cos  py
cos  pz
 zz '  15.71  0.10
 zz '  0.48  0.12
 zz '  13.07  0.11
Here  pi , i  x, y, z is polar angle of the emitted proton. One can see that difference between ... and
... ... is not large which indicates that correlation between  zz ' and projections of the unit vector in the
direction of emitted proton is not too strong.
Three component of D vector Dx , Dy , Dz are measured independently, the length of D vector is given
as usual by D  Dx  Dy  Dz with statistical error bar
2
D 
D 
D 
 D   D   x   Dx2   y   Dy2   z   Dz2 .
D
 D
 D
2
2
2
(25)
Having in mind that Dy  0 [2], the reduced transformation matrix for Dx and Dz components contains two parameters:
 cos  zz '
  sin  zz '
 
sin  zz '   0.776 0.539 
,

cos  zz '   0.539 0.776 
which are taken from distributions shown below in Figs.2,3
Figs 2, 3.
7
(26)
 0.776 0.539 

 one may
 0.539 0.776 
compare results obtained in two coordinate systems directly and by means of transformation from one coordinate system to another. The results are presented in Table 1. Note that for flickering coordinate sys†
1
tem a  a . This can be shown by direct calculation:
Finally, using direct and (conjugate) transpose transformation matrix a
a
1
†
 0.869 0.603 

.
 0.603 0.869 
(27)
Nevertheless in a coarse approximation the transformation is “roughly” unitary, and the length of spin
transfer vector may be assumed approximately as preserved.
Table 1
Evaluated in XYZ
system
Transformed from
X’Y’Z’ system
DX
-0.016±0.042
0.024±0.037
DZ
0.186±0.040
D
0.187±0.040
IV.
Evaluated in
X’Y’Z’ system
Transformed from
XYZ system
DX '
0.133±0.039
0.088±0.038
0.186±0.036
DZ '
0.147±0.038
0.153±0.039
0.188±0.036
D
0.197±0.039
0.177±0.040
Conclusion.
It has been explained why the spin transfer vectors evaluated in two coordinate system independently
using above formalism are in general not identical, i.e., not connected with a unitary transformation, and
correspondently, the length of vector is not preserved. It has been shown however that in the case of re
cently done analysis of the spin transfer DLL
' to the Λ [2], because of reaction kinematics (weak correlation between  zz ' and projections of the unit vector in the direction of emitted proton) the two vectors are
approximately identical and the vector length is approximately preserved.
V.
Publications.
[1] S.Belostotski in Proc.of the IX-Workshop on High Energy Spin Physics (Spin-01), Dubna
2001, preprint JINR E1,2-2002-103,192 (2002)
[2] S.Belostotski, Yu.Naryshkin, D.Veretennikov, Longitudinal Spin Transfer in Semi-Inclusive
Lambda Production, release report April 2010.
8