[Hadstruct] Some Updates on the GPD Effort

[Colin Egerer]

Hi All,

My apologies, first, for having missed today's group meeting. I had some unexpected car trouble that needed tending to. I figured an 8am appointment would be ample time, but that was clearly not the case with limited staff.

Anyways, I wanted to give an update today on my efforts to resolve, or at least better understand, the curious behavior in the generalized pseudo-ITDs that I had last shown some ~6 weeks back. As a reminder, the \gamma_4 insertion between nucleon states of differing momenta induces the F_1 and F_2 form factors, which are a limiting case of the generalized pseudo-ITDs (what I denote as 'M' and 'L', respectively containing info on the H & E GPDs) induced when a displacement is included between the insertion's quark fields.

At the time, there were a couple issues in the M & L I had shown. The first were cases in which the form factor limit (i.e. z=0) was not being reproduced consistently. That is, in some pf/pi frames the limit was realized, yet in other rotationally equivalent frames the limit was not realized. This then relates to the bigger issue that was realized - rotational symmetry was not seen in M & L for pf/pi frames wherein the values of skewness and momentum transfer were identical. We traced this back to the data from the edbs and saw the same lack of manifest rotational symmetry, which caused me to cancel any weekly discussions. For reference and a reminder, I've included a tarball (deemed rubbish.tar.gz) of some of these purported M & L's for a few frames.

Long in short, I believe the data generated (and certainly the genprops!) are correct. The problem(s) was in my analysis, which did not correctly construct the continuum nucleon helicity spinors. This then translated into incorrect subduced spinors, and an incorrect kinematic matrix that I use to solve for M & L via a singular value decomposition. I now have the spinors correct, and can correctly and consistently realize charges and electromagnetic form factors for a variety of pf/pi frames using the off-forward genprops (hence my confidence above). In fact, the apparent lack of rotational symmetry in the data pulled from the edbs is precisely the point - it is a reflection of how the relativistic spinors, and by extension their subduced analogues, change as one introduces different amounts of momenta to each component of the 3-momentum. For instance, for some initial momentum pi with pf = (0,1,1) vs. pf = (1,0,1), although rotationally equivalent, will shuffle around factors of 'i' giving the appearance of dramatically different correlators. I've included what I believe are now correct determinations of M & L in the same frames contained in the rubbish tarball. You can see F_1 and F_2 (i.e z=0) are borne out for a given value of momentum transfer. Further, as the momentum transfer is increased you can see the amplitudes flatten out, which reflects a characteristic falloff of F_1 and F_2 with increasing momentum transfer.

You'll note what seems to be inconsistency in the values of momentum transfer. This is because I read in fitted values of the nucleon energy when computing the momentum transfer for these plots. I suppose a more consistent approach would be to use the ground state mass and the relativistic dispersion relation to express a given value of the momentum transfer. Rest assured, for pf/pi frames where you would expect the momentum transfers to be equal - they are. As last word, each plot contains three types of points: circle, triangle, square. These indicate the time series fit to in the summed ratio data that I use to pull out the matrix elements: circle (T = 4-14), triangle (T= 6-14), square (T = 8-14).

It's unfortunate this will not be ready for the lattice conference, but at least it seems we/I haven't wasted more than two years of effort with these data.
I am away next week, but happy to discuss some this week yet if there is interest.

Best,
Colin



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