Hi Michael,
Thanks for elucidating the discussion at the BLAST meeting; that
was the point I was trying to make. Of course in practice, your
formula is implemented by the simpler one:
< theta_s > \equiv \sum_i (w_i * theta_s(z_i)) / \sum_i (w_i)
where 'i' runs over the event sample, and the weight of each event
'w_i' can be taken as unity.
It would also be helpful to provide 'th_nominal' for each of the
fieldmaps as a reference. So if I understand, the main point is that
we need to compare different extractions / measurements in terms of
'th_nominal' to avoid dependence of the event distribution.
One final thing, I don't think the event distribution does not
make much difference, since I got pretty much the same result for
each Q^2 bin, left and right sector, but if you look at the z-
distributions for each, they are wildly different. There are even
gaping holes in some.
--Chris
TA-53/MPF-1/D111 P-23 MS H803
LANL, Los Alamos, NM 87545
505-665-9804(o) 665-4121(f) 662-0639(h)
On May 5, 2006, at 22:52:46, Michael Kohl wrote:
> Hi,
>
> -On the discussion of "average" vs "nominal" spin angle:
> In Doug's routine, the spin profile parametrized as a 9th order
> polynomial is evaluated relative to a "nominal" spin angle (which
> happens to be theta(z=0)).
> Doug's profile:
> th(z) = th_nominal + \sum_i a_i z^i, i=1..9
> = th_nominal + dth(z)
> The yield-weighted average spin angle (over the allowed length of the
> target) of Doug's profile is thus
> <th>_yield = 1/40 * \int_{-20}^{20}[dz \rho(z) th(z)]
> = th_nominal + <dth>_yield,
> where \rho(z) is the target density distribution along z (normalized
> to 1), and <dth>_yield is the yield-weighted average of the
> polynomial terms.
> While theta_nominal is a quantity of the detector that does not
> depend on the reaction channel, the quantities \rho(z), <th>_yield
> and <dth>_yield are dependent on the particular considered reaction,
> or even on the sector.
>
> The average angle <th>_yield can be determined from both ed elastic
> and ep elastic asymmetry analysis; as such, this number is not yet
> useful for any other reaction unless it is converted into the
> "nominal" angle. In order to do this for the given reaction,
> <dth>_yield = 1/40*\sum_i{\int_{-20}^{20}[dz \rho(z) a_i z^i]},
> i=1..9
> needs to be determined which is a simple number.
> This said, comparing the spin angles from ed elastic and ep elastic
> must only be done for the resulting theta_nominal, but not for the
> yield-averaged numbers!
>
> Chris has evaluated <th>_yield and <dth>_yield for ep elastic:
> ep elastic(47): <th>_yield = 45.8 +- ??? degrees
> <dth>_yield = -0.8
> -> theta_nominal = 46.6
> ed elastic(32): <th>_yield = 31.4
> <dth>_yield = ???
>
> There is deuterium data for both 47 (2005) and 32 (2004) degree
> settings. Only the latter is available so far with the latest
> recrunch, the former is being crunched right now.
>
> For each of the 32 and 47 degree settings, the same theta_nominal
> ought to be used by every analysis. The resulting <th>_yield for each
> reaction channel is dependent on the specific yield distribution
> \rho(z) which may be different in each channel. It may even depend on
> the sector. In order to calculate the average angle in any reaction
> channel from a given "universal" map, it needs to be averaged
> over the specific yield (=evaluating <dth>_yield). Eugene has done
> this for 32-deg=2004 d(e,e'n): <dth>_yield = <th>_yield -
> theta_nominal
> en quasielastic(32): <dth>_yield (left sector) = -2.27 deg
> <dth>_yield (right sector) = -2.42 deg
>
> The resulting average <th>_yield is a number which can be quoted in a
> paper for the considered reaction, but it is of no further use if the
> extraction of observables relies on the same common spin angle map.
>
> Nevertheless it is a good idea to compare the various <dth>_yield of
> the various reaction channels (e,e'), d(e,e'p), (e,e'pi+), ...
> Comparing ep elastic with en quasielastic, there seems to be some
> significant difference, -0.8 deg vs -2.4 deg.
> One should do the same exercise for MC-generated (cross-section
> weighted) target distributions rho(z) for each channel, in order to
> exclude that there are any surprises. Keep in mind that the
> Q2bin-by-Q2bin target distributions in ep elastic from reconstructed
> data looked rather funny, and they are prone to systematics in the
> reconstruction. In MC, the tossed and accepted z distribution can be
> used directly, without the reconstruction uncertainty.
>
>
> Regards,
>
> Michael
>
>
>
>
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