[Frost] Dilution method

[Michael Dugger]

Volker,

I could be wrong, but I think that once the errors have been properly 
propagated, there is not going to be an advantage in using the dilution 
method over the subtraction method in terms of final error bars.

In the subtraction method it is easy to propagate the errors from the 
start and so the results we show include these errors. For the dilution 
method, I think that people tend not to propagate the errors yet.

The question of assuming a smooth distribution only helps if the smooth 
distribution can be known better than things that we measure, otherwise we 
are just trading systematic errors for statistical uncertainties.

If it is possible to gain an advantage using an assumed smooth 
distribution, than we could use that smooth distribution for the 
subtraction method. There is no reason why the smooth distribution would 
only work for the dilution method.

I am very interested in any work done in determining an assumed smooth 
distribution. Please let us know when you have had a chance to study the 
issue.

Take care,
Michael

On Mon, 28 Feb 2011, Volker Crede wrote:

> Hi Michael,
>
> thanks for the nice summary, I fully agree with your email below. Just a brief comment:
we noticed that the Carbon statistics for the two-pion channel is only about 10-15% of the
Butanol statistics. For this reason, subtracting the Carbon distributions introduces
huge statistical errors without even considering the error of the scaling factor.
Since the helicity difference is likely small (at least for the most part) for ppi+pi- ,
we conclude that the subtraction method is not applicable for us; the final errors of
the observables after the subtraction (dominated by the statistical fluctuations of Carbon)
wash out any polarization effect we see for the Butanol alone. I guess that all of us have
more or less the same problem.
>
> The determination of the dilution factor also depends on the Carbon statistics, but our
hope is that we can extract sort of a function which smoothly describes the dilution factor.
  This will probably include some assumptions on the dependence of the dilution factor on
certain kinematic variables. At the moment, we see that we are limited by the Carbon
statistics and to a somewhat lesser extent by the Butanol statistics.
>
> Best wishes
>
>       Volker
>
>
> On Feb 28, 2011, at 2:51 PM, Michael Dugger wrote:
>
>>
>> Hi,
>>
>> There are two common methods of dealing with the bound nucleons:
>> 1) Subtraction method
>> 2) Dilution method.
>>
>> Either method should produce final results that are consistent with one
>> another. I want to make sure that we are all in agreement as to the use of
>> the dilution method. (the subtraction method is very simple and probably
>> does not need much in the way of explanation.)
>>
>>> From my discussion with Ken, I understand that (please correct me if I am
>> wrong) the dilution method can be summarized through the equation:
>>
>> O_free = [O_butanol*N_butanol - N_bound*O_carbon]/N_free,
>>
>> where
>> O_free = Observable on the free proton
>> O_butanol = Observable on the butanol
>> O_carbon = Observable on the carbon
>> N_free = Number of events on free proton
>> N_butanol = Number of events of interest on butanol
>> N_bound = Number of events of interest on bound nucleons within butanol
>>
>> If we introduce the dilution factor D = N_free/N_butanol, then we can
>> rewrite O_free as
>>
>> O_free = O_butanol/D - O_carbon*(1-D)/D
>>
>> There are two observables and a dilution factor that make up the
>> observable for free protons.
>>
>> For the special case where O_carbon = 0, we get the nice result that
>> O_free = O_butanol/D.
>>
>> The scale factor can be connected to the dilution factor through the
>> equation
>> D = 1 - sf*N_carbon/N_butanol,
>> as Sung does.
>>
>> It is important to note that for the Sigma, Cz and Cx observables O_carbon
>> is not expected to be = 0.
>> This means that for these observables you must use the full equation:
>> O_free = O_butanol/D - O_carbon*(1-D)/D
>>
>> Liam: It looks like it is possible for an incorrect dilution factor to
>> cause a sign change in observables. Perhaps you can vary the dilution
>> factors and see if you can get Cz and Cz observables to better agree with
>> the g1c data.
>>
>> Take care,
>> Michael
>>
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>
>
>