Hi Chi,
thank you for the detailed arguments. I believe now that the variance in
the map does not matter for the determination of the mean.
May I conclude or summarize that for the average spin angles
<thd_T20> -> from T20/T21 analysis
and
<thd_June2005> -> from edel-yield weighted average of June 2005 map
the best resulting spin angle map thd_corrected(z) would be
thd_corrected(z) = thd_June2005(z) + <thd_T20> - <thd_June2005> ?
We could take any other map thd_orig(z) as well (Jan2005,Jun2005,Mar2006):
thd_corrected(z) = thd_orig(z) + <thd_T20> - <thd_orig>,
where <thd_orig> is the yield-weighted average of any given survey map.
For each original map, the resulting corrected maps thd_corrected(z) would
finally agree with each other within a small band. The systematic error on
the spin angle would be determined by the variance of the yield-weighted
average of the three resulting corrected maps (insignificant) and the
error on <thd_T20> (dominant). The latter error could be reduced somewhat
if also ep elastic is used (for 47 degree data). If we believe the
physics, the uncertainty will be ~0.5 degrees.
Chi, if you agree to the above, could you generate a corrected spin angle
map? Could you do this with each Jan2005, June2005 and March2006 map and
see what the variance between the three corrected maps is (plot the
difference between each two). The final map could be the average of the
three corrected maps.
Can you also distribute it in the end such that it can be used equally in
each Montecarlo?
Best regards,
Michael
On Fri, 31 Mar 2006, Chi Zhang wrote:
>
> Hi Michael, please see comments below
>
> Hi Bill, atatched are two plots showing the field profile obtained from the
> mapping, T20, and Zilu's TOSCA simulation. strikingly, ZIlu's results agree
> with T20 fairly reasonably. within +-10cm where edel had good statistics, T20
> map and TOSCA are within 1-2 sigma. both these two float above the field
> maps.
>
> granted, TOSCA is not real measurements.
>
> On Thu, 30 Mar 2006, Michael Kohl wrote:
>
>>
>> +Spin angle map with 3d hall probe has been repeated in March 2006.
>> Comparison of June2005 map (Blast field on) with March2006 map
>> (Blast field off): Profile is reproduced but shifted by ~1 degree in
>> average, hence increasing the discrepancy with CZ to 2.5 degrees.
>> Constancy of the difference along z could point to an alignment
>> issue with the hall probe (would be good to check the constancy of
>> the shift between two measured maps!!).
>
> mis-alignment is likely to be random though. we misaligned +1 today, -1 next
> time. So it is surprising that ever since the July 2004 field map, the
> surveyed profile shifted toward lower angle everytime we measure:
> 07/2004 > 01/2005 > 06/2005 > 03/2006
> which requires big coincidences in misalignments.
>
> But hey, I ain't no experts on this matter. :)
>
>> +My suggestion: The fact that the spin angle z-profiles are shifted by
>> more or less constant amounts relative to each other raises doubts in
>> the absolute alignment while relative measurements on the same jig are
>> consistent. Hence, Chi's determination of the average spin angle from
>> the tensor asymmetry analysis should be based on a comparison with
>> the Montecarlo that uses a measured profile. (So far the used
>> Montecarlo has assumed a flat distribution). Chi's subsequent
>> analysis result for the average spin angle can then be compared with
>> the ed-elastic yield weighted average of the initially used
>> profile and will then reveal by how much the measured profile needs
>> to be shifted in order to become realistic (=in agreement with the
>> physics). The method needs to be cross-checked with the ep-elastic
>> analysis (so far both ed and ep analyses have given consistent
>> results). Result should be a corrected spin angle map that can be
>> equally used by every analysis. Please let me know what you think.
>>
>
> I thought I explained this issue Wednesday to you and Doug, so surprised that
> this came up again yesterday and here.
>
> As I explained, it was a concern of mine too that the MC used a flat average
> value. However this concern had long been addressed by cross checking with
> yet a 3rd method to determine Pzz and theta_spin.
>
> Namely, in the analysis, I used the "shifted" profile to obtain the kinematic
> coefficients such as the 1.5*cos^2(theta*) - 0.5 in front of T20. I verified
> that when I shift the Jan 2005 profile up to 47.7, T20 and T21 values below
> 2fm-1 are pinned right onto model predictions. The model dependency of spin
> angle obtained is small as the models tend to differ in the same way in T20
> and T21, i.e. if model A predicts a bigger T20 than model B, it does the same
> to T21 too.
>
> Effectively this is another fitting procedure where I fitted at low Q2,
> T20(pzz,theta_spin) and T21(pzz,theta_spin)
> to models by varying pzz, theta_spin. The results reproduced very well the
> pzz and theta_spin extracted from comparison to MC.
>
> Further, as Genya pointed out, the spin angle extraction can be viewed as
> finding the root of.
> A_para / <A_perp(theta_spin)>
> ------------------------------- = 1
> A_perp / <A_para(theta_spin)>
> where A_* are the observed asymmetries, <A> are the MC values.
>
> using a flat profile effectively replaces <A(theta)> with A(<theta>), the
> discrepancy between these two is:
> 0.5* A" * (d_theta)^2.
> The mean square error of theta is < 2 deg due to the profile shape. do
> d_theta^2 ~ 0.001 rad^2. so using a flat profile Monte Carlo introduces an
> error which effectively change the above root finding to
> A_para / <A_perp(theta_spin)>
> -------------------------------
> A_perp / <A_para(theta_spin)>
>
> A_para / <A_perp(theta_spin)> /A_para" A_perp"\
> = 1 - 0.5 ----------------------------- |------ - ------ | d_theta^2
> A_perp / <A_para(theta_spin)> \A_para A_perp /
>
> /A_para" A_perp"\
> = 1 - 0.5 |------ - ------ | d_theta^2
> \A_para A_perp /
>
> A" arround 32 and 47 degrees are both around 1/rad^2, so all in all the
> correction term on the right hand side to 1 is on the order of 10^-3.
>
> So now if
> f(theta) = 1
> f(theta + d) = 1 - D,
> it is easy to see that: d = D/f'(theta)
>
> here D~10^-3,
>
> A_para / <A_perp(theta_spin)>
> f(theta) = ------------------------------- ~10 /rad around both 32 and 47
> A_perp / <A_para(theta_spin)>
>
> This can be easily seen from my plots of the two Pzz's vs. theta_d. at 31.7
> for example, this ratio is 1, at 28 degree, it is already 1.4. so the
> derivative is about (1.4-1)/0.07 ~ 6.
>
> so d ~ 10^-4 rad which is small by all standards pertained to the problem at
> hand.
>
> To conclude, using a flat MC is no problem unless we care about 0.01 degree
> level errors. This is verified but both a analytical derivation and the
> T20/T21 analysis based on a shifted profile.
>
> Chi
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