Hi,
another long minutes because people seem to like it ...
Let me know your comments.
Minutes:
Attending NM, VZ, AS, BT, DH, MK
-Deuteron recoil polarimetry: BT reports about a new paper from
Juelich which demonstrates that scattering of unpolarized deuterons
from an unpolarized target may tensor-polarize the transmitted
deuterons as the d-p cross section for m=0 is different from m=+-1.
This could have an impact on the deuteron recoil polarimetry results
of e-d scattering.
-Radiative Montecarlo
Nik and Vitaliy are working on implementing MASCARAD into
BLASTMC. Goal is to have radiative processes in the generator.
One task is to prove that the elastic and quasielastic
asymmetries are affected only little (at the 1% level) by
radiation. Equivalently expressed, 99% of the radiative effect would
not depend on the helicity, and would hence cancel in the formation of
the asymmetry.
Second, it is desirable to have a kinematical match between Montecarlo
and measured yields as a function of energy or invariant mass. The
elastic (quasielastic) peak is broadened, possibly shifted, and skewed
as it gets a tail on the high side.
Another task is to provide an estimate of the elastic (quasielastic)
radiation tail (which carries the asymmetry of elastic scattering)
into the Delta region (which has a different asymmetry). Only a
quantitative Montecarlo that accounts for the elastic radiative tail
may provide access to asymmetries of the NDelta transition.
With MASCARAD all radiation processes that occur in elastic
scattering are accurately calculated without approximations. While
the soft photon part of the code is reasonably fast, in the
evaluation of the hard photon part complicated integrals occur, which
are expensive in terms of CPU. These integrals may be tabulated.
For our purpose, sticking with soft photons (i.e. Bethe-Heitler and
vertex correction), even in peaking approximation is very likely good
enough for our generator. Of course the hard part needs to be
evaluated for the proof that the asymmetries are not affested much by
radiation.
There was also some discussion on radiation by the pions in the NDelta
channel. While the radiation by pions is an order of magnitude more
likely than by protons in the elastic scattering process, it may still
be a small number compared to the Bethe-Heitler process. The pion
radiation is again likely to be mostly independent of helicity and
would therefore cancel in the NDelta asymmetries.
The bottom line is, the MC generator should be made to work with soft
photons for elastic scattering; everything beyond this is a correction
to a correction.
-Combination of asymmetries
We discussed the issue of how to get a correct asymmetry (and error or
figure of merit) if the polarization (or polarization product)
changes significantly between two datasets:
Let the measured asymmetry A1exp for dataset 1 be
A1exp = (N1+ - N1-)/(N1+ + N1-) = P1*A1, likewise for A2exp and A2
for dataset 2, where P1(P2) are the average polarizations of dataset
1(2).
The physics observable is the asymmetry A which is probed by both
datasets 1 and 2 statistically independently: A = A1 = A2 for N1,N2
to infinity.
The usual approach to extract A from the total dataset is to evaluate
the total asymmetry and normalize to the average polarization:
<A> = (1/<P>)*(N+ - N-)/(N+ + N-)
= (1/<P>)*(1/(N1+N2))*(N1+ - N1- + N2+ - N2-)
= (1/<P>)*(1/(N1+N2))*(N1*A1exp + N2*A2exp)
= (1/<P>)*(1/(N1+N2))*(N1*P1*A1 + N2*P2*A2),
where <P> = (N1*P1 + N2*P2) / (N1 + N2) and hence
<A> = (N1*P1*A1 + N2*P2*A2) / (N1*P1 + N2*P2) (1)
= (N1*A1exp + N2*A2exp) / (N1*P1 + N2*P2)
i.e. the average asymmetry would be obtained by weighting the
individual asymmetries by the individual yield times polarization.
The statistical error amounts to
1/dA^2 = (N1+N2)*<P>^2 = FOM
= (N1 + N2)*(N1*P1 + N2*P2)^2 / (N1 + N2)^2
= (N1*P1 + N2*P2)^2 / (N1 + N2) (2)
This implies e.g. that if dataset 2 is unpolarized (P2=0), the figure
of merit would become worse than the FOM1 of dataset 1 alone.
The more natural approach would be to consider A1 and A2 as two
independent measurements of the same quantity A, and thus write
the average asymmetry as an error-weighted mean:
<A> = (A1/dA1^2 + A2/dA2^2) / (1/dA1^2 + 1/dA2^2),
where 1/dA1^2 = N1*P1^2 = FOM1, 1/dA2^2 = N2*P2^2 = FOM2 and thus
1/dA^2 = N1*P1^2 + N2*P2^2 = FOM (figure of merit). (3)
Therefore,
<A> = (N1*P1^2*A1 + N2*P2^2*A2) / (N1*P1^2 + N2*P2^2) (4)
= (N1*P1*A1exp + N2*P2*A2exp) / (N1*P1^2 + N2*P2^2)
This way, the overall FOM can only grow as soon as a dataset with any
nonzero polarization is added, and it cannot become smaller, even if an
unpolarized dataset is added.
Comparing Eqs. (1) and (4), we see that once the polarizations enter
linearly and once quadratically. In fact, as long as the variation of
P is small between two datasets (or within Gaussian statistics), both
equations (1) and (4) give the same result, and so do Eqs. (2) and (3)
for the resulting error of FOM.
However, as soon as the polarizations P1 and P2 differ significantly,
the equations for both the asymmetry and for the error give different
results, in fact the asymmetry error from the first approach is
always larger than the one from the second (correct) approach.
Equivalently, as soon as <P^2> != <P>^2, it starts to matter.
Note that there is no need at all for P to be Gaussian-distributed!
For 2004 we are lucky that hPz and Pzz are mostly constant and high.
However, this year's data has a lot more variation (hPz varies between
0.3 and 0.5, alomost a factor 2!)
It is suggested to bin the data into datasets of piecewise constant
polarization and pursue the second approach to obtain a resulting asymmetry.
Regards,
Michael
--+-------------------------------------+--------------------------+ | Office: | Home: | |-------------------------------------|--------------------------| | Dr. Michael Kohl | Michael Kohl | | Laboratory for Nuclear Science | 5 Ibbetson Street | | MIT-Bates Linear Accelerator Center | Somerville, MA 02143 | | Middleton, MA 01949 | U.S.A. | | U.S.A. | | | - - - - - - - - - - - - | - - - - - - - - -| | Email: kohlm@mit.edu | K.Michael.Kohl@gmx.de | | Work: +1-617-253-9207 | Home: +1-617-629-3147 | | Fax: +1-617-253-9599 | Mobile: +1-978-580-4190 | | http://blast.lns.mit.edu | | +-------------------------------------+--------------------------+
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