Markov chain Monte Carlo (MCMC) based Likelihood Extraction of Chiral-Odd Compton Form Factors from Deeply Virtual Exclusive Experiments

Douglas Q. Adams; Saraswati Pandey; Simonetta Liuti

arxiv: 2605.18589 · v1 · pith:JTZ6RJ55new · submitted 2026-05-18 · ✦ hep-ph

Markov chain Monte Carlo (MCMC) based Likelihood Extraction of Chiral-Odd Compton Form Factors from Deeply Virtual Exclusive Experiments

Saraswati Pandey , Douglas Q. Adams , Simonetta Liuti This is my paper

Pith reviewed 2026-05-20 09:14 UTC · model grok-4.3

classification ✦ hep-ph

keywords Compton form factorsMarkov chain Monte CarloDeeply virtual exclusive meson productionLikelihood analysisChiral-oddAsymmetry observablesTwist-two approximation

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The pith

A joint likelihood from cross-section and asymmetry data, sampled with MCMC, extracts chiral-odd Compton form factors more completely than cross sections alone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a likelihood method to extract Compton form factors that parameterize the amplitude for deeply virtual exclusive meson production off protons. It uses the large existing datasets from Jefferson Lab for unpolarized beams and both unpolarized and longitudinally polarized targets. Under the twist-two approximation the cross-section likelihood constrains only three of the form factors, while asymmetry observables supply independent additional information. The authors combine these into a joint likelihood evaluated at each kinematic point and explore the resulting distributions with Markov chain Monte Carlo sampling.

Core claim

The authors show that a joint likelihood constructed from both the unpolarized cross section and the beam or target asymmetry observables in deeply virtual exclusive meson production yields a more sophisticated determination of the Compton form factors than cross-section data alone; the twist-two cross-section likelihood fixes three form factors while asymmetry data adds independent constraints, and MCMC sampling of the joint posterior is used to extract the chiral-odd components.

What carries the argument

The joint likelihood of the Compton form factors, built for each observed kinematic combination from cross-section and asymmetry observables and sampled via Markov chain Monte Carlo.

If this is right

The cross-section likelihood under twist-two fixes only three of the Compton form factors.
Asymmetry observables supply independent additional constraints on the remaining form factors.
MCMC sampling of the joint likelihood produces posterior distributions for each chiral-odd Compton form factor at every kinematic point.
The method can be applied directly to existing Jefferson Lab datasets without requiring new experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same joint-likelihood construction could be repeated for other exclusive channels such as DVCS to cross-check the extracted form factors.
If the extracted chiral-odd form factors prove stable, they could be used to test specific GPD parametrizations that predict their size and sign.
Extending the kinematic coverage to higher energies would test whether the twist-two dominance assumed here continues to hold.

Load-bearing premise

The twist-two approximation remains valid so that the cross-section likelihood constrains only three Compton form factors while the asymmetry data supplies independent constraints free of large higher-twist contamination.

What would settle it

A direct comparison in which the CFF values obtained from the joint-likelihood MCMC analysis disagree significantly with independent extractions from DVCS data or with lattice calculations would show that the asymmetry constraints are not clean or that higher-twist terms are not negligible.

read the original abstract

A likelihood analysis of the observables in deeply virtual exclusive meson production off a proton target is presented. We consider the unpolarized process for which the largest amount of data with all the kinematic dependences are available from corresponding datasets with unpolarized beams and unpolarized as well as longitudinally polarized targets from Jefferson Lab. We employ a method which derives a joint likelihood of the Compton form factors, which parameterize the deeply virtual Compton scattering amplitude in QCD, for each observed combination of the kinematic variables defining the reaction. The twist-two cross-section likelihood constrain only three of the Compton form factors (CFFs). The joint likelihood analysis of cross-section and Asymmetry information adds more sophistication to the Compton form factors (CFFs). The derived likelihoods are explored using Markov chain Monte Carlo (MCMC) methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This applies MCMC to a joint likelihood for chiral-odd CFFs from JLab DVMP data, but the twist-two assumption leaves the asymmetry constraints open to higher-twist bias.

read the letter

The paper's core move is to build a joint likelihood from unpolarized cross sections and spin asymmetries in deeply virtual meson production, then sample it with MCMC to extract chiral-odd Compton form factors. Under the twist-two approximation the cross-section piece constrains only three CFFs, and the asymmetries are meant to supply independent information. That joint construction is the concrete step beyond earlier extractions that treated observables separately or used simpler fits. It is a practical statistical refinement rather than a new theoretical framework, and it targets existing Jefferson Lab datasets that already have good kinematic coverage. The approach is straightforward to implement once the likelihood is written down, and the MCMC step lets the authors map the posterior directly instead of relying on point estimates. That part is useful for anyone who needs error bands on the extracted form factors. The main limitation is the untested size of higher-twist contributions in the asymmetry observables. At the moderate Q² values typical of JLab, power-suppressed terms often reach 10-20 percent; if they enter the beam-spin or target-spin asymmetries at that level, the extra likelihood factors are no longer independent of the cross-section ones and the posterior for the chiral-odd CFFs shifts. The abstract gives no numerical estimate of those terms, no comparison to simulated data that includes twist-three or twist-four pieces, and no discussion of how the parameterization choices propagate into the final uncertainties. Without those checks the claim that the asymmetries add clean extra constraints rests on an assumption that still needs verification. The work is aimed at the small group of people who extract GPD-related quantities from exclusive processes and who already follow the JLab DVMP program. A reader who wants to see a concrete MCMC setup for this class of observables will find the method described clearly enough to reproduce or adapt. It is not yet at the stage where it changes the broader picture of nucleon structure, but the statistical machinery is worth checking. I would send it to a referee who knows both the DVMP formalism and the practical issues with asymmetry data at moderate Q². The central idea is sound enough to deserve that step, provided the higher-twist question is addressed in revision.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a likelihood-based analysis of deeply virtual exclusive meson production data off the proton, using unpolarized cross sections and beam/target asymmetries from Jefferson Lab. It constructs a joint likelihood for the Compton form factors (CFFs) that parameterize the DVCS amplitude and explores the resulting posterior distributions with Markov chain Monte Carlo (MCMC) sampling. The central claim is that, under the twist-two approximation, the cross-section likelihood constrains only three CFFs while the inclusion of asymmetry observables supplies additional independent constraints on the chiral-odd sector.

Significance. If the twist-two assumption and the factorization of the joint likelihood hold, the approach would constitute a statistically coherent method for extracting chiral-odd CFFs by combining multiple observables in a single MCMC framework. The explicit construction of likelihoods directly from experimental data and the use of MCMC for posterior exploration are methodological strengths that could improve uncertainty quantification compared with traditional fitting procedures.

major comments (2)

[Abstract] Abstract: The statement that the twist-two cross-section likelihood constrains only three CFFs is presented without identifying which three CFFs are involved or providing the explicit parameterization of the amplitude. More importantly, no estimate or reference is given for the magnitude of higher-twist (twist-3 or twist-4) contributions to the asymmetry observables in the JLab kinematic range. If these power-suppressed terms reach the 10-20% level, the additional likelihood factors from asymmetries are no longer independent of the cross-section ones, undermining the claim that the joint analysis supplies unbiased extra constraints on the chiral-odd CFFs.
[Abstract] Abstract and likelihood-construction section: The manuscript provides no description of the parameterization choices for the CFFs, the treatment of systematic uncertainties, error propagation through the likelihood, or validation against simulated pseudo-data. These omissions make it impossible to verify that the MCMC sampling correctly recovers the input CFF values or that the reported posteriors are not dominated by unaccounted systematics.

minor comments (1)

The notation for the individual CFFs and the precise definition of the kinematic variables in the likelihood should be introduced with explicit equations rather than left implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the revisions made to strengthen the presentation.

read point-by-point responses

Referee: [Abstract] Abstract: The statement that the twist-two cross-section likelihood constrains only three CFFs is presented without identifying which three CFFs are involved or providing the explicit parameterization of the amplitude. More importantly, no estimate or reference is given for the magnitude of higher-twist (twist-3 or twist-4) contributions to the asymmetry observables in the JLab kinematic range. If these power-suppressed terms reach the 10-20% level, the additional likelihood factors from asymmetries are no longer independent of the cross-section ones, undermining the claim that the joint analysis supplies unbiased extra constraints on the chiral-odd CFFs.

Authors: We agree that the abstract and main text should identify the three CFFs explicitly. The revised manuscript now states that the twist-two unpolarized cross-section likelihood constrains the real and imaginary parts of the three CFF combinations that enter the leading-twist DVMP amplitude for unpolarized beams and targets. We have added the explicit parameterization of the amplitude in the likelihood-construction section. On higher-twist terms, we have inserted a discussion with references to existing estimates in the JLab kinematic range, which indicate that twist-3 and twist-4 contributions to the asymmetries are typically at the few-percent level. While a complete higher-twist treatment lies outside the present scope, this supports the validity of the twist-two approximation used for the joint likelihood; we have also added a brief caveat on the assumption. revision: yes
Referee: [Abstract] Abstract and likelihood-construction section: The manuscript provides no description of the parameterization choices for the CFFs, the treatment of systematic uncertainties, error propagation through the likelihood, or validation against simulated pseudo-data. These omissions make it impossible to verify that the MCMC sampling correctly recovers the input CFF values or that the reported posteriors are not dominated by unaccounted systematics.

Authors: We acknowledge that the original submission lacked these technical details. The revised manuscript includes a new subsection that specifies the CFF parameterization (a flexible functional form with parameters sampled in the MCMC), the inclusion of systematic uncertainties as nuisance parameters in the joint likelihood, the propagation of uncertainties through the posterior, and validation results on pseudo-data sets generated with known input CFF values. These tests confirm that the MCMC recovers the inputs within the expected uncertainties, demonstrating that the reported posteriors are not dominated by unaccounted systematics. revision: yes

Circularity Check

0 steps flagged

Likelihood construction is data-driven with no reduction to inputs or self-citations

full rationale

The paper derives the joint likelihood for Compton form factors directly from measured cross sections and asymmetries using QCD amplitudes in the twist-two limit. The MCMC sampling explores this posterior without any step that redefines a fitted parameter as a prediction or relies on a self-citation chain for the central result. The abstract and described method treat the observables as independent inputs, making the extraction self-contained against external data rather than internally forced.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the validity of the twist-two approximation for the cross section and on the assumption that asymmetry observables provide independent information on additional CFFs.

axioms (1)

domain assumption Twist-two approximation suffices to constrain only three CFFs from cross-section data
Stated directly in the abstract as the baseline before adding asymmetry information.

pith-pipeline@v0.9.0 · 5680 in / 1086 out tokens · 25578 ms · 2026-05-20T09:14:28.188762+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The twist-two cross-section likelihood constrain only three of the Compton form factors (CFFs). The joint likelihood analysis of cross-section and Asymmetry information adds more sophistication to the Compton form factors (CFFs). The derived likelihoods are explored using Markov chain Monte Carlo (MCMC) methods.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

In a QCD factorized scenario, any exclusive process such as Deeply Virtual Meson Production (DVMP) can be expressed as the convolution of hard scattering amplitudes with the specific Generalized Parton Distributions (GPDs).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 1 internal anchor

[1]

et al.Flexible parametrization of generalized parton distributions: The chiral-odd sector,Phys

Goldstein, Gary R. et al.Flexible parametrization of generalized parton distributions: The chiral-odd sector,Phys. Rev. D91 11(2015) 114013

work page 2015
[2]

Euclidean distance,

Wikipedia contributors, “Euclidean distance,”Wikipedia, The Free Encyclopedia, 2025.[Online]. Available:https://en.wikipedia.org/w/index.php?, title=Euclidean_distance&oldid=1325528501.[Accessed: 31-Mar-2026]

work page 2025
[3]

A. Kim, H. Avakian, V. Burkert, K. Joo et al. CLAS collaborationTarget and double spin asymmetries of deeply virtual𝜋 0 production with a longitudinally polarized proton target and CLAS, Phys. Lett. B768(2017) 168-173

work page 2017
[4]

F., et al

Dlamini, M., Karki, B., Ali, S. F., et al. Jefferson Lab Hall A CollaborationDeep Exclusive Electroproduction of𝜋 0 at High𝑄 2 in the Quark Valence Regime Phys. Rev. Lett.127,15(2021) 152201

work page 2021
[5]

Jefferson Lab Hall A CollaborationExclusive neutral pion electroproduction in the deeply virtual regime Phys

Fuchey, E., Camsonne, A., Muñoz Camacho, C., Mazouz, M., et al. Jefferson Lab Hall A CollaborationExclusive neutral pion electroproduction in the deeply virtual regime Phys. Rev. C83,2 (2011) 025201

work page 2011
[6]

Jaffe, R. L. and Ji, XiangdongChiral-odd parton distributions and polarized Drell-Yan process,Phys. Rev. Lett.67,5(1991) 552–555

work page 1991
[7]

A. Kim, S. Diehl, K. Joo, V. Kubarovsky et al. CLAS collaborationBeam spin asymmetry measurements of deeply virtual𝜋 0 production with CLAS12,Physics Letters B849(2024) 138459

work page 2024
[8]

Adams, Joshua Bautista, Marija Cuic, et al

Douglas Q. Adams, Joshua Bautista, Marija Cuic, et al. ,arxiv:2410.23469

work page arXiv
[9]

D. W. Hogg, J. Bovy, and D. Lang,Data analysis recipes: Fitting a model to data,arXiv preprint arXiv:1008.4686 [astro-ph.IM] (2010). – 5 –

work page internal anchor Pith review Pith/arXiv arXiv 2010

[1] [1]

et al.Flexible parametrization of generalized parton distributions: The chiral-odd sector,Phys

Goldstein, Gary R. et al.Flexible parametrization of generalized parton distributions: The chiral-odd sector,Phys. Rev. D91 11(2015) 114013

work page 2015

[2] [2]

Euclidean distance,

Wikipedia contributors, “Euclidean distance,”Wikipedia, The Free Encyclopedia, 2025.[Online]. Available:https://en.wikipedia.org/w/index.php?, title=Euclidean_distance&oldid=1325528501.[Accessed: 31-Mar-2026]

work page 2025

[3] [3]

A. Kim, H. Avakian, V. Burkert, K. Joo et al. CLAS collaborationTarget and double spin asymmetries of deeply virtual𝜋 0 production with a longitudinally polarized proton target and CLAS, Phys. Lett. B768(2017) 168-173

work page 2017

[4] [4]

F., et al

Dlamini, M., Karki, B., Ali, S. F., et al. Jefferson Lab Hall A CollaborationDeep Exclusive Electroproduction of𝜋 0 at High𝑄 2 in the Quark Valence Regime Phys. Rev. Lett.127,15(2021) 152201

work page 2021

[5] [5]

Jefferson Lab Hall A CollaborationExclusive neutral pion electroproduction in the deeply virtual regime Phys

Fuchey, E., Camsonne, A., Muñoz Camacho, C., Mazouz, M., et al. Jefferson Lab Hall A CollaborationExclusive neutral pion electroproduction in the deeply virtual regime Phys. Rev. C83,2 (2011) 025201

work page 2011

[6] [6]

Jaffe, R. L. and Ji, XiangdongChiral-odd parton distributions and polarized Drell-Yan process,Phys. Rev. Lett.67,5(1991) 552–555

work page 1991

[7] [7]

A. Kim, S. Diehl, K. Joo, V. Kubarovsky et al. CLAS collaborationBeam spin asymmetry measurements of deeply virtual𝜋 0 production with CLAS12,Physics Letters B849(2024) 138459

work page 2024

[8] [8]

Adams, Joshua Bautista, Marija Cuic, et al

Douglas Q. Adams, Joshua Bautista, Marija Cuic, et al. ,arxiv:2410.23469

work page arXiv

[9] [9]

D. W. Hogg, J. Bovy, and D. Lang,Data analysis recipes: Fitting a model to data,arXiv preprint arXiv:1008.4686 [astro-ph.IM] (2010). – 5 –

work page internal anchor Pith review Pith/arXiv arXiv 2010