[Sane-analysis] New fit to world A1 and d2 update

[O. A. Rondon]

To address the questions at the last analysis meeting about the Q^2
dependence of the global fits to SLAC and SANE A1 and A2 data, I have
updated my reports as follows:

- I have plotted the data and fits to A1(W,Q^2) with Q^2 on the x axis.
The fits match the data for each subset of beam energy and angle very
well, see top plots on pp. 4 and 7 of the report here
https://userweb.jlab.org/~rondon/analysis/asym/world/fits_143_155_13.pdf

- to disentangle the contributions of the W and Q^2 dependence, I have
also plotted the data for the range 2 <= W < 3 GeV, and the fits both at
each point's kinematics and at fixed <W> = 2.46 GeV, which is the mean
of the data in the range, versus Q^2, bottom plots on p. 4 and 7.

The resulting linear dependence on Q^2 (purple line) for constant W has
excellent agreement with the data's trend in Q^2. Moreover, in this W
range dominated by SANE, SANE anchors the low Q^2 end of the fits, and
largely determines the linear slope parameter b1, which is flatter for
SLAC data alone (so having the right normalization of our data is
important).

The combined fit dependence on W and Q^2 is displayed by the green curve
segments, which are dominated by the W dependence. The bottom plot on p.
7 shows the W values of the 5.5 deg. SLAC data, illustrating this dominance.

- to try decreasing the minimum W for which the fit to A1 can be used, I
tried a new form with 1/W^c. The new fit has c < 1, which extends the
range for which A1<=1 down to W = 1.095 GeV at Q^2 =0. Considering the
errors of the fit parameters, a0, a1, and b1, the fit is consistent with
unitarity down to the pion threshold, for Q^2 up to 4 GeV^2. The Chi
squared also improves a bit, see details on p. 6.

The off-diagonal elements of the covariance matrix, also on p. 6, are
quite small, indicating little correlation among parameters. The largest
is the correlation between the constant term a0 and the power c1
(cvm{4,1]), but it is still reasonably small.

- I have updated the calculation of the Nachtmann moment for d2 using
the new fit, see bottom plot on p. 4. of the report on d2. There is a
small change to more positive d2 values at all Q^2, which comes in part
from the improved A1 fit being also used to get g2.
https://userweb.jlab.org/~rondon/analysis/asym/world/d2.pdf

In summary, I think the fits are a reasonable, empirical model
representation of the data with smooth functions which, therefore, can
and should be used as inputs to calculate the internal radiative
corrections.

The fits correctly average over the point to point fluctuations of the
data, and depend on solid invariant quantities W and Q^2, which are
valid at any kinematics and in any coordinate system (unlike nu, or even
x, which is really meaningful only in the infinite momentum frame). It
should be kept in mind that any model, physics or phenomenology driven,
that depends on two variables, will also have discontinuous jumps when
calculated at the kinematics of each data point. And any additional
normalization of the data can be easily absorbed in updated fit parameters.

I encourage the analysis to use this method, although I'm sure other,
probably better, fit forms can also be found.

Cheers,

Oscar

O. A. Rondon wrote:
> I found bugs in the calculation of the Q^(3/2)trend curves (actually
> plotted was Q^3, which, in any case, doesn't make much sense anyway).
> 
> An updated file, with more sensible 1/sqrt(Q^2) trend curves and an
> extra plot including the inelastic piece of RSS d2, is now posted.
>  https://userweb.jlab.org/~rondon/analysis/asym/world/d2.pdf
> 
> Cheers,
> 
> Oscar
> 
> O. A. Rondon wrote:
>> Here is d2 calculated using both C-N and Natchmann moments. The top plot
>> on p. 3 shows d2 calculated as before using AAC 2003 g1 and g1 from my
>> A1, A2(W, Q^2) fits to SANE and SLAC data, but computing the M_2^3
>> Nachtmann moment, instead of the C-N one (The C-N results are on p. 1,
>> with updated labels).
>>
>> The difference between d2 for the two input g1's is smaller for
>> Nachtmann than for C-N (about 1/2). At 5 GeV^2, the agreement with the
>> published C-N SLAC average also improves.
>> https://userweb.jlab.org/~rondon/analysis/asym/world/d2.pdf
>>
>> The expression for the M_2^n=3 moment is taken from eq. 3 of the RSS PRL
>> on twist-3,
>> http://prl.aps.org/abstract/PRL/v105/i10/e101601
>>
>> Since the OPE moments are based on measured SSF's, target mass effects
>> (as opposed to Target Mass corrections) were applied to the g1 SSF from
>> the (asymptotic) AAC PDF's, to make it approximate the experimental
>> values, in order to combine it with the g2 SSF from the A1, A2 fits to
>> data. (The NMC F1 used to get the SSF's from the SA's is a
>> parameterization of experimental data, so it does not need target mass
>> effects; this should be kept in mind if F1 from PDF's is used at some
>> point). The bottom plot on p. 3 shows d2 for AAC g1 with and without
>> target mass effects. It's a  small but not totally negligible difference
>> (~20% at 5 GeV^2).
>>
>> Cheers,
>>
>> Oscar
>>
>>
>>
>>
>>
>>
> 
> 
> 
>