[Sane-analysis] Scaling fit to neutral pi photoproduction

[O. A. Rondon]

Since the Alanakian et al. inclusive pi+ photoproduction on C data
displays good scaling in PT for pion angles from 20 to 60 deg., and the
pi- 40 deg data agrees with the pi+ within about 25%, I have digitized
all the data for both pi+ and pi- to combine them to get a scaling fit
to the neutral pion cross section

To be able to do an apples to apples comparison to pi-
electroproduction, I have also digitized the pi- electroproduction on C
at 5 GeV and 13 deg data, of unknown origin, and the epc output included
in Fig. 1 of O'Connell's Wiser report.

The results are posted on my wiki section on inclusive pion production.
https://hallcweb.jlab.org/experiments/sane/wiki/index.php/Inclusive_pion_and_nucleon_electroproduction

The following items are relevant
- plot of all pi+ data, plus 40 deg. pi- data, showing that all data for
20 to 60 deg. and the 90 deg. data at low PT scale
https://userweb.jlab.org/~rondon/incl-pi_epc-pt_scaling.pdf

- plots of pi- electroproduction at 5 GeV (O'Connell) and
photoproduction at 4.5 GeV on C (Yerevan), along with scaling fit and
epc output.

I normalized the fit to photoproduction data by the ratio of the
averages of electro to photoproduction data. I'm sure the normalization
would agree much better if we knew the actual C target thickness used
for electroproduction, i.e. probably ~3%, instead of the 5.5% ratio of
the averages.
https://userweb.jlab.org/~rondon/incl-pim_epc-C_electrp.pdf

- scaling fit to combined pi+ and pi- to get pi0 cross section. The fit
form and parameters are given below the figure showing the pi- comparison.

The digitization error of the fit is 5% or less. It only affects the
cross sections. The kinematics are taken directly from the paper.

I do recommend we forget about Wiser, epc, etc. and just use this fit in
GEANT. For nitrogen and He we just multiply it by the ratio of atomic
masses. For H, we can safely assume it's C/12, since our minimum pion
momentum 500 MeV/c is well above the pion production threshold.

To use it, the only thing we need to do is to convert the pion
kinematics (p, theta, phi) to the corresponding PT and calculate the
cross section.

To get yields, we use the GEANT bremsstrahlung flux for the real photon
part times the fit's cross section.

For electroproduction we just replace Wiser with this cross section.

Since the original Yerevan data are in a copyrighted paper, I can't post
it. Unfortunately, it's not available as a digital file on INSPIRE, but
the hard copy is available at the JLAB library and at other physics
dept. libraries.

Cheers,

Oscar