Parton Distribution Functions from Large Momentum Expansion of Current-Current Correlators
Pith reviewed 2026-05-08 17:27 UTC · model grok-4.3
The pith
Parton distribution functions can be computed from the large-momentum expansion of current-current correlators.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The universality of the large momentum expansion allows parton distribution functions to be obtained from current-current correlators with appropriate large-momentum Fourier components. This yields an expansion formula up to next-to-leading order that avoids linear power divergences and features straightforward renormalization, though it demands computation of four-point functions in lattice QCD.
What carries the argument
The large-momentum expansion of current-current correlators, which expands the Euclidean four-point function in inverse powers of the large momentum to recover the PDF moments and distributions.
If this is right
- PDFs become accessible without the linear power divergences that affect quasi-PDF extractions.
- Renormalization of the correlator remains simpler because only local operators appear.
- The same universality permits cross-checks by applying the expansion to multiple types of Euclidean correlators.
- Lattice computations must handle four-point functions but can target next-to-leading order accuracy with controlled errors.
Where Pith is reading between the lines
- The method may allow direct use of existing lattice setups designed for short-distance expansions of moments.
- If four-point function precision improves, the approach could become competitive for extracting x-dependent PDFs rather than only moments.
- Testing the expansion on known distributions in quenched QCD would provide an immediate numerical check before full dynamical simulations.
Load-bearing premise
The large momentum expansion applies to current-current correlators in the same universal way as to other operators without introducing new complications, and lattice four-point functions can be computed with enough precision to make the next-to-leading order terms practical.
What would settle it
A lattice calculation in which the next-to-leading-order large-momentum expansion of a current-current correlator fails to reproduce known parton distributions obtained from other methods would show that the approach does not work as claimed.
Figures
read the original abstract
The universality of the large momentum expansion allows computing parton distribution functions (PDFs) starting from any Euclidean correlator with appropriate large momentum Fourier Components. Here we consider current-current correlators which have been used in short-distance expansion to obtain moments of PDFs. The advantage of such correlators is that they have simple renormalization properties and do not have linear power divergences as in quasi-PDF. However, in lattice calculations, four-point functions are needed. Here we present an expansion formula with current-current correlators up to the next-to-leading order, and preliminary numerical calculations with four-point functions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes computing parton distribution functions (PDFs) via the large-momentum expansion applied to Euclidean current-current correlators. It emphasizes advantages including simple renormalization and the absence of linear power divergences that affect quasi-PDF approaches, derives an explicit next-to-leading-order (NLO) matching formula, and reports preliminary lattice results based on four-point functions.
Significance. If the numerical feasibility is established, the approach would provide a useful alternative route to PDFs that builds on existing short-distance techniques while avoiding certain lattice artifacts. The explicit NLO formula constitutes a concrete, reusable contribution that enables systematic improvement beyond leading order.
major comments (1)
- Numerical results section: The preliminary four-point function calculations are presented without quantitative error budgets, statistical uncertainties, or tests demonstrating that NLO corrections can be extracted above noise and systematics. This directly affects the claimed practical advantage, as the skeptic note correctly identifies that insufficient precision would render the NLO terms unusable.
minor comments (1)
- Abstract: The statement that the expansion 'allows computing PDFs starting from any Euclidean correlator' should be qualified by the specific kinematic requirements on the Fourier components, to avoid overstatement.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major point raised below.
read point-by-point responses
-
Referee: Numerical results section: The preliminary four-point function calculations are presented without quantitative error budgets, statistical uncertainties, or tests demonstrating that NLO corrections can be extracted above noise and systematics. This directly affects the claimed practical advantage, as the skeptic note correctly identifies that insufficient precision would render the NLO terms unusable.
Authors: We agree that the numerical results are exploratory and currently lack a complete quantitative error budget. In the revised manuscript we have added the statistical uncertainties obtained from our lattice ensembles and included a brief discussion of the dominant systematic uncertainties (finite-volume effects, discretization artifacts, and renormalization). We have also clarified in the text that these calculations serve only to illustrate the practical feasibility of the method on current lattices and do not yet claim a precision sufficient to extract NLO corrections above noise. The central result of the paper—the explicit NLO matching formula between the current-current correlator and the PDFs—remains independent of the numerical precision achieved in this initial study. A full demonstration of NLO extraction will require substantially larger statistics and volumes, which we intend to pursue in follow-up work. revision: partial
Circularity Check
No circularity detected in derivation of NLO expansion for current-current correlators
full rationale
The paper states the universality of the large-momentum expansion as a premise drawn from prior literature and then derives and presents an explicit NLO matching formula for PDFs extracted from current-current correlators, along with preliminary lattice four-point function results. No equations in the provided text reduce by construction to fitted parameters renamed as predictions, self-definitions, or load-bearing self-citations whose content collapses into the present work. The central contribution—the NLO formula and its application—remains independent of any internal fitting loop or tautological renaming, making the derivation self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Universality of the large momentum expansion for any Euclidean correlator with appropriate large momentum Fourier components
Reference graph
Works this paper leans on
-
[1]
The sink is real- ized using the sequential source technique again employ- ing momentum smearing
to obtain a point-to-all propagator. The sink is real- ized using the sequential source technique again employ- ing momentum smearing. The propagator between the two currents is evaluated using the stochastic source tech- nique, which is accompanied by improvements exploiting the hopping parameter expansion for Wilson fermions. This allows us to efficient...
-
[1]
The sink is real- ized using the sequential source technique again employ- ing momentum smearing
to obtain a point-to-all propagator. The sink is real- ized using the sequential source technique again employ- ing momentum smearing. The propagator between the two currents is evaluated using the stochastic source tech- nique, which is accompanied by improvements exploiting the hopping parameter expansion for Wilson fermions. This allows us to efficient...
- [2]
- [3]
- [4]
- [5]
-
[6]
X. Ji, Sci. China Phys. Mech. Astron.57, 1407 (2014)
work page 2014
- [7]
-
[8]
K. Cichy and M. Constantinou, Adv. High Energy Phys. 2019, 3036904 (2019)
work page 2019
- [9]
- [10]
- [11]
- [12]
- [13]
- [14]
- [15]
-
[16]
T. Ishikawa, Y.-Q. Ma, J.-W. Qiu, and S. Yoshida, Phys. Rev. D96, 094019 (2017)
work page 2017
- [17]
-
[18]
M. Constantinou and H. Panagopoulos, Phys. Rev. D96, 054506 (2017)
work page 2017
-
[19]
I. W. Stewart and Y. Zhao, Phys. Rev. D97, 054512 (2018)
work page 2018
-
[20]
C. Alexandrou, K. Cichy, M. Constantinou, K. Had- jiyiannakou, K. Jansen, H. Panagopoulos, and F. Stef- fens, Nucl. Phys. B923, 394 (2017)
work page 2017
-
[21]
T. Izubuchi, X. Ji, L. Jin, I. W. Stewart, and Y. Zhao, Phys. Rev. D98, 056004 (2018)
work page 2018
- [22]
-
[23]
L.-B. Chen, W. Wang, and R. Zhu, Phys. Rev. Lett. 126, 072002 (2021)
work page 2021
-
[24]
X. Ji, Y. Liu, A. Sch¨ afer, W. Wang, Y.-B. Yang, J.-H. Zhang, and Y. Zhao, Nucl. Phys. B964, 115311 (2021)
work page 2021
-
[25]
X. Gao, K. Lee, S. Mukherjee, C. Shugert, and Y. Zhao, Phys. Rev. D103, 094504 (2021)
work page 2021
- [26]
-
[27]
Y. Su, J. Holligan, X. Ji, F. Yao, J.-H. Zhang, and R. Zhang, Nucl. Phys. B991, 116201 (2023)
work page 2023
- [28]
-
[29]
X. Ji, Y. Liu, and Y. Su, JHEP08, 037 (2023)
work page 2023
- [30]
-
[31]
X. Ji, Y. Liu, Y. Su, and R. Zhang, JHEP03, 045 (2025)
work page 2025
- [32]
- [33]
-
[34]
J.-W. Chen, X. Ji, and J.-H. Zhang, Nucl. Phys. B915, 1 (2017)
work page 2017
- [35]
- [36]
-
[37]
A. J. Chambers, R. Horsley, Y. Nakamura, H. Perlt, P. E. L. Rakow, G. Schierholz, A. Schiller, K. Somfleth, R. D. Young, and J. M. Zanotti, Phys. Rev. Lett.118, 242001 (2017)
work page 2017
- [38]
- [39]
-
[40]
G. S. Bali, V. M. Braun, B. Gl¨ aßle, M. G¨ ockeler, M. Gru- ber, F. Hutzler, P. Korcyl, A. Sch¨ afer, P. Wein, and J.-H. Zhang, Phys. Rev. D98, 094507 (2018)
work page 2018
- [41]
-
[42]
R. S. Sufian, C. Egerer, J. Karpie, R. G. Edwards, B. Jo´ o, Y.-Q. Ma, K. Orginos, J.-W. Qiu, and D. G. Richards, Phys. Rev. D102, 054508 (2020)
work page 2020
-
[43]
G. S. Bali, M. Diehl, B. Gl¨ aßle, A. Sch¨ afer, and C. Zim- mermann, JHEP09, 106 (2021)
work page 2021
- [44]
- [45]
- [46]
- [47]
- [48]
-
[49]
G. S. Bali, B. Lang, B. U. Musch, and A. Sch¨ afer, Phys. Rev. D93, 094515 (2016)
work page 2016
- [50]
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.