[Sane-analysis] g2WW follow up to Re: Reminder of Sane meeting at 3:00pm Wed Jan 11th in L210A

[Oscar Rondon]

Hoyoung's comparisons of the Bjorken x dependence of the d_2 C-N
integrands show that

2g_1(x) + 3g_2(x) .ne. 3g_2(x) - 3g_2^WW(x)

https://hallcweb.jlab.org/experiments/sane/hykang/HMS_asym/20170111/SANE1_d2_integrand.png
https://hallcweb.jlab.org/experiments/sane/hykang/HMS_asym/20170111/RSS_d2_integrand.png

This is as expected. Only the integrals are equal, not the integrands.

The OPE for the moments of g_1 and g_2, as derived in many papers, e.g.
WW original paper's eq.(2), states that

Int dx x^(J-1) [(J-1)/J g_1(x) + g_2(x)) = d_J

where d_J are twist-3 matrix elements. No assumptions are made on the
twist contents of the left hand side. For J=3, the OPE looks like the
familiar

Int dx x^2 [2 g_1(x) + 3 g_2(x)] = 3 d_2

WW then made the approximation that twist-3 d_J<<twist-2 a_J to get
their expression for g_2^WW(x), which for its third moment J=3 gives

Int dx x^2 3g_2^WW(x) = -Int dx x^2 2g_1(x)

which, when substituted in the OPE gives
Int dx x^2 [-3 g_2^WW(x) + 3 g_2(x)]

but only the combined integrals 3(g_2 - g_2WW) and 2g_1 + 3g2_ are
equal, and only for J=3, but not the integrands.

As we know, g_1(x) has HT contributions suppressed by (M/Q)^(J=2, 4, ..)
which include matrix elements like the TMC a_2, twist-3 d_2, twist-4 f_2
and higher orders of (M/Q)^J. So it seems to me that there even is a
further problem  with getting d_2 from moments of 3(g_2(x) - g_2^WW(x))
if one takes g_2^WW as coming from pure twist_2 g_1, because the
original form of the OPE involves the full g_1, with all its twists.

Otherwise, if the g_1 in the OPE were only twist-2 it would not be
possible to make a twist expansion of Gamma_1, the first moment of g_1,
which indeed has the HT terms. The exact same full twist g_1 used to get
Gamma_1 must be the one used to get d_2. This implies that to get g_2^WW
one needs to also use the full g_1, not just twist-2 g_1, as Accardi et
al. do. They explicitly show that g_2WW is different if calculated with
only twist-2 g_1 or full g_1.

To be consistent with the OPE, it should be g_2WW(full g_1), i.e. the
equality Int dx x^2 3g_2^WW(x) = -Int dx x^2 2g_1(x) can't be used to
substitute for the OPE's g_1 term unless g_1 is full g_1.

On the other hand, the method of getting d_2 from g_T = g_1 + g_2 =
F_1*A_2 doesn't have that problem as long as the full g_1 is subtracted
from the main term 3F_1*A_2.

It should also be mentioned that the Nachtmann integrals reduce to the
C-N ones for (M/Q)^2 << 0 only if both Nachtmann and C-N moments involve
full twist g_1 and g_2. There is no Nachtmann form involving g_2WW.

I have posted Linda Stuart's exact kinematics derivation of g_2WW,
including target mass. At first sight, her result doesn't seem to agree
with Bluemlein and Tkabladze's result that the WW form works for TMC's
too, see Stuart's final equation. So I would suggest we only use
g_2WW(x) to look for the x-dependence of twist-3 g_2, not for the
moments, and as I said above, calculate g_2WW using the full g_1.
https://userweb.jlab.org/~rondon/analysis/g2ww-lstuart.pdf

Cheers,

Oscar



Oscar Rondon wrote:
> We had a productive meeting that cleared an important point about g2WW.
> I checked the Accardi et. al. paper (JHEP 0911 (2009) 093) which
> clarifies, as Whit and Zein-Eddine also pointed out, that the input to
> g2WW needs to be leading twist g1_LT, not experimental g1, as discussed
> in their fig. 1 bottom panel, where they show the difference in both
> approaches. I was thinking that experimental g1 was the input.
> 
> I also checked that what I had requested Linda Stuart to calculate for
> E143 was g2WW for finite (2Mx)^2/Q^2 > 0 (= 0 is the WW approximation),
> which she did. I will scan and post her report on the SANE wiki. With
> Linda's formula, the experimental g1 should work as input to g2WW.
> 
> This means that for the resonances, the d2 integrand of C-N integrals
> cannot be calculated using g2bar, because there is no g1_LT for the
> resonances to get g2WW. Maybe some g1_LT like LSS 2006 could be evolved
> down to the HMS data's mean Q^2, as I believe Zein-Eddine mentioned, but
> that would imply application of not just global, but local duality,
> which doesn't work for g1. Even global duality works only above ~ 1.8
> GeV^2. So HMS C-N integrals should be calculated with 2g1+3g2 only.
> 
> Next meeting will be next Wednesday 1/18 at 3:00 PM.
> 
> Cheers,
> 
> Oscar
> 
> 
> 
> 
> Mark Jones wrote:
>> Reminder of Sane meeting at 3:00pm Wed Jan 11th in L210A 
>>
>>
>> https://bluejeans.com/240199984 
>>
>>
>> Unfortunately I won't be at JLab again.
>> I will try and call in.
>> Oscar can chair the meeting with code 6720. 
>>
>>
>>
>> Cheers, Mark
>> _______________________________________________
>> Sane-analysis mailing list
>> Sane-analysis at jlab.org
>> https://mailman.jlab.org/mailman/listinfo/sane-analysis
> 
> 
> 
>