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

[Oscar Rondon]

I have posted Seonho's TN about d2, g2WW and the low x (DIS)
contribution to the moments
https://userweb.jlab.org/~rondon/analysis/d2-TN-S-Choi.pdf

It's from an old fax I got from Seonho, with annotations (his and mine,
mine are the lighter print ones). The second page is distorted because
the fax machine was jamming. The original might be available somewhere
in Seonho's archives or some old Temple web page.

In any case, it has the derivation of the identity between the 3*g2WW
and -2g1 x^2 moments, and a useful discussion of how using g2WW affects
the low x contribution to the d2 integrals.

Cheers,

Oscar




Oscar Rondon wrote:
> 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
>>
>>
>>
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