GPD Uniqueness from Partial DGLAP Knowledge, and its Coexistence with Shadow GPDs #
Two recent results about GPD reconstruction point in opposite directions:
- Non-uniqueness. There is an infinite family of shadow GPDs producing identical
Compton form factors at a fixed order and scale
(
Deconvolution.hasShadow_of_isLeadingOrderDvcs, arXiv:2303.12006). - Uniqueness up to a D-term. Knowing a GPD at vanishing and low skewness inside the
DGLAP region determines it on its whole support up to a D-term
(
exists_dTerm_of_agreeOnLowSkewnessDglap, arXiv:2401.12013).
Both are right, and this module is about why. The two statements take different inputs. The first is about what Compton form factor data at one order and scale determines; the second is about what GPD values in a kinematic region determine. The purpose of the module is to make the difference a theorem rather than a remark.
The reconciliation, in three proved steps #
EpsilonEridani.Particles.Parton.GPD.gpdOfDTerm_eq_zero_of_inDglapRegion— a pure D-term vanishes identically on the DGLAP region. So the ambiguity left by the uniqueness theorem is, by construction, invisible to the data that theorem uses.not_isShadow_of_eq_gpdOfDTerm— a shadow GPD is never a pure D-term, provided the Compton map is faithful on D-terms (SeparatesDTerms). So the ambiguity left by the Compton map is not of the kind the uniqueness theorem tolerates.shadow_visible_in_lowSkewness_dglap— consequently, under the uniqueness statement, every shadow GPD is nonzero somewhere in the DGLAP region at accessible skewness. A shadow hides from Compton form factors at one order and scale; it does not hide from a direct determination of the GPD at low skewness.
Steps 1-3 are proved. The two headline statements they reconcile are not: each is a
tagged sorry naming what it needs.
Why SeparatesDTerms is the right hypothesis and not a placeholder #
The D-term is the subtraction constant of the fixed-t dispersion relation for the DVCS
amplitude (Diehl and Ivanov, arXiv:0712.3533; Polyakov and Schweitzer,
arXiv:1801.05858). Faithfulness of the Compton map on D-terms is therefore a property the
physical map is believed to have, not an arbitrary convenience: it says that the real
part of the form factor fixes the subtraction constant. It is stated as a concrete
implication between quantified equations, not as an opaque Prop field.
Deviation from the source statement #
arXiv:2401.12013 reports reconstructions with a maximal skewness as low as 20% of the
longitudinal momentum fraction, i.e. a data region of the shape |ξ| ≤ c |x|. The
hypothesis used here, |ξ| ≤ ξ₀ with ξ₀ > 0 fixed, is a neighbourhood of vanishing
skewness rather than a cone. The two coincide near |x| = 1 and the cone is the weaker
assumption at small |x|; the choice made here is the one under which the analytic
argument (Radon inversion from a neighbourhood of the ξ = 0 slice) is stated most
directly. See NOTES.md of task/frontier-open-targets.
References #
- Y. Guo, X. Ji, M. G. Santiago, J. Yang and H.-C. Zhang (listing; primary source not consulted), Unraveling Generalized Parton Distributions Through Lorentz Symmetry and Partial DGLAP Knowledge, arXiv:2401.12013.
- E. Moffat et al., Shedding light on shadow generalized parton distributions, Phys. Rev. D 108 (2023) 036027 (arXiv:2303.12006).
- M. Diehl and D. Yu. Ivanov, Dispersion representations for hard exclusive reactions (arXiv:0712.3533).
The first two entries were checked against public listings. The next two were written from memory when this file was drafted and have since been verified (13 September 2026); the residual gap is stated explicitly below:
- N. Chouika, C. Mezrag, H. Moutarde and J. Rodríguez-Quintero, Covariant Extension of
the GPD overlap representation at low Fock states, Eur. Phys. J. C 77 (2017) 906
(arXiv:1711.05108). Verified: §III presents "an original discussion based on Radon
transform properties of covariant extensions of GPDs from the DGLAP to the ERBL
region", and the paper states the physics problem is solved by inverting the Radon
transform. It also discusses the D-term ambiguity as the obstruction to the
DGLAP → ERBL converse, which is the ambiguity isolated in
GPD/Ambiguity.lean. - J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J. 55 (1987) no. 4, DOI 10.1215/S0012-7094-87-05547-5. Verified against the Crossref record for that DOI: title, both author names, journal, volume 55, issue 4 and year 1987 all confirmed. The page range often quoted for this article (943–948) is not confirmed — the Crossref record carries no page field, and no retrieved source stated it — so it is omitted here rather than asserted.
Residual gap. Both references are confirmed to exist and to concern what they are cited for. What was not checked is whether arXiv:1711.05108 itself invokes Boman–Quinto for its uniqueness step, as opposed to some other incomplete-data uniqueness result; the pairing of the two in the sketch below is this development's own reading. No equation number is attributed to either.
The data of a low-skewness DGLAP determination #
Two GPD models are indistinguishable by DGLAP-region data at skewness below ξ₀.
This is the input of the uniqueness statement: the values of H in the region
|ξ| < |x| ≤ 1 for |ξ| ≤ ξ₀. Note that it is a hypothesis about the distribution,
not about an observable — which is exactly what separates this result from the
shadow-GPD obstruction, whose input is Compton form factor data.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Uniqueness up to a D-term #
The ambiguity the uniqueness statement leaves is exactly the ambiguity its data cannot see: adding a D-term to a model changes nothing in the DGLAP region.
Proved, and the reason exists_dTerm_of_agreeOnLowSkewnessDglap cannot be strengthened
by any argument using only AgreeOnLowSkewnessDglap.
Faithfulness of the Compton map on D-terms #
The Compton map at this order and scale separates D-terms: a D-term whose Compton form factors vanish identically is itself zero.
Physically this is the statement that the D-term is fixed by the real part of the Compton
form factor, being the subtraction constant of the fixed-t dispersion relation
(arXiv:0712.3533, arXiv:1801.05858). It is carried as a hypothesis because it is a
property of the physical coefficient function, which is not written down here.
- dTerm_faithful (dt : Particles.Parton.GPD.DTerm Flavor) (i : Flavor) : (∀ (xi t : ℝ), cffOfGpd K (Particles.Parton.GPD.gpdOfDTerm dt) i xi t = 0) → ∀ (u t : ℝ), dt.D i u t = 0
A D-term in the kernel of the Compton map is the zero D-term.
Instances For
The reconciliation #
The two ambiguities are disjoint. A shadow GPD is never a pure D-term.
This is the first half of the reconciliation: the object the shadow-GPD result produces is not the object the uniqueness result tolerates. Its proof is the whole argument — a pure D-term in the kernel of a map that separates D-terms is the zero D-term, hence the zero GPD, contradicting nontriviality.
The reconciliation. Under the uniqueness statement and faithfulness of the
Compton map on D-terms, every shadow GPD is nonzero somewhere in the DGLAP region at
skewness below ξ₀.
In words: a shadow hides from Compton form factors at one order and scale, but it does not hide from a direct determination of the GPD at low skewness. The two frontier results therefore constrain complementary data and do not collide.
The uniqueness statement is taken as an explicit hypothesis huniq rather than pulled in
from exists_dTerm_of_agreeOnLowSkewnessDglap, so that this theorem is free of sorry
and the logical dependency is visible in its signature.