Reduced Pseudo-Ioffe-Time Distributions (RpITDs)

• Reduced pseudo-Ioffe-time distributions (RpITDs) are coordinate-space ratios built from spacelike hadron matrix elements that cancel Wilson-line UV divergences.
• They preserve the key Ioffe-time dependence needed for short-distance factorization while suppressing nonleading z²-dependence in lattice extractions.
• Their application across nucleon, pion, and gluon analyses enhances PDF reconstruction accuracy and guides perturbative and discretization corrections.

Searching arXiv for recent and foundational papers on reduced pseudo-Ioffe-time distributions. Reduced pseudo-Ioffe-time distributions (RpITDs) are coordinate-space ratios built from spacelike bilocal hadron matrix elements, usually written as

$$\mathfrak{M}(\nu,z^2)=\frac{\mathcal{M}(\nu,z^2)}{\mathcal{M}(0,z^2)},$$

with Ioffe time $\nu=p\cdot z$. In the pseudo-distribution program, $\mathcal{M}(\nu,z^2)$ is the pseudo-Ioffe-time distribution (pseudo-ITD), its Fourier transform in $\nu$ defines the pseudo-PDF, and the reduced ratio is the renormalization-improved observable matched to light-cone distributions. The central rationale is that the ratio cancels multiplicative Wilson-line renormalization factors, suppresses part of the nonleading $z^2$-dependence, and preserves the short-distance logarithms needed for perturbative matching to $\overline{\rm MS}$ PDFs (Radyushkin, 2017). For nonsinglet quarks the ratio obeys $\mathfrak{M}(0,z^2)=1$, whereas in the gluon case the analogous reduced ITD is normalized to the gluon momentum fraction rather than to a conserved charge (Balitsky et al., 2021).

1. Definition, notation, and historical placement

The formal starting point is the gauge-invariant bilocal operator with a straight Wilson line,

$$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$

or, in equivalent conventions,

$$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$

For lattice applications one usually chooses a purely spatial separation and boost along the same direction, so that $\nu=p_3 z_3$ in practice. The pseudo-ITD $\nu=p\cdot z$0 is then the spacelike continuation of the light-cone ITD, while the pseudo-PDF $\nu=p\cdot z$1 is defined by Fourier transform in $\nu=p\cdot z$2 and retains canonical support $\nu=p\cdot z$3 for all $\nu=p\cdot z$4 (1705.01488).

The reduced ratio appeared in the early pseudo-distribution literature under the names “reduced Ioffe function” and “reduced ITD,” before the label “reduced pseudo-Ioffe-time distribution” became standard. Those early papers already identified the essential construction

$\nu=p\cdot z$5

as the natural coordinate-space observable for lattice extractions, both because it divides out the rest-frame spatial distribution and because it cancels the multiplicative gauge-link renormalization (Radyushkin, 2017).

Object	Symbol	Role
Light-cone ITD	$\nu=p\cdot z$6 or $\nu=p\cdot z$7	Fourier transform of the light-cone PDF
Pseudo-ITD	$\nu=p\cdot z$8	Spacelike coordinate-space correlator
RpITD	$\nu=p\cdot z$9	Reduced ratio at fixed $\mathcal{M}(\nu,z^2)$0
Pseudo-PDF	$\mathcal{M}(\nu,z^2)$1	Fourier transform of $\mathcal{M}(\nu,z^2)$2 in $\mathcal{M}(\nu,z^2)$3
PDF	$\mathcal{M}(\nu,z^2)$4, $\mathcal{M}(\nu,z^2)$5, $\mathcal{M}(\nu,z^2)$6	Light-cone parton distribution

For quark PDFs the reduced ratio is typically built from the same matrix element at nonzero and zero Ioffe time. In some lattice implementations it is realized as a double ratio that also normalizes the local current. For gluons the reduced object is

$\mathcal{M}(\nu,z^2)$7

where $\mathcal{M}(\nu,z^2)$8 is the invariant amplitude tied to the gluon PDF in the light-cone limit (Balitsky et al., 2019).

2. Renormalization logic and the rationale for reduction

The unreduced bilocal operator contains UV singularities generated by the straight Wilson line. In cutoff language these include a linear divergence and logarithmic endpoint divergences; in dimensional regularization they appear as UV poles and $\mathcal{M}(\nu,z^2)$9-terms of ultraviolet origin. The crucial observation is that these factors are multiplicative and depend on the line length and shape, not on the Ioffe time. In the ratio $\nu$0, the Wilson-line self-energy and other multiplicative UV factors cancel (Radyushkin, 2017).

This cancellation is only part of the story. The reduced ratio does not remove the physical short-distance logarithms associated with perturbative evolution. The standard distinction is between UV artifacts of the Wilson line, which the ratio removes, and genuine short-distance logarithms, which remain because they encode DGLAP evolution and scheme conversion. That distinction is central to the theoretical status of RpITDs: they are cleaner observables, but not “matching-free” observables (Zhang et al., 2018).

A second motivation is structural rather than purely renormalization-theoretic. In the early coordinate-space analyses, $\nu$1 was interpreted as the part encoding the hadron’s rest-frame spatial distribution or primordial momentum-distribution effects. Dividing by it removes much of the soft $\nu$2-dependence without altering the longitudinal $\nu$3-dependence that determines the PDF shape. This is why RpITDs were proposed as more economical observables than quasi-PDFs, whose Fourier transform entangles $\nu$4- and $\nu$5-dependence (1705.01488).

The normalization differs between channels. In the quark nonsinglet case, the denominator at $\nu$6 is tied to a conserved charge, so $\nu$7. In the gluon case,

$\nu$8

so the reduced gluon ITD fixes the shape of $\nu$9 only up to the independently determined gluon momentum fraction (Balitsky et al., 2021).

3. Short-distance factorization, evolution, and matching

At short spacelike separations, RpITDs admit a coordinate-space factorization onto light-cone ITDs. For quarks the standard relation used in lattice phenomenology is

$z^2$0

with one-loop kernel

$z^2$1

$z^2$2

Equivalent inversion formulas are then used to obtain the $z^2$3 light-cone ITD from the lattice RpITD (Joó et al., 2019).

The reduced ratio also obeys a short-distance evolution equation,

$z^2$4

which is the coordinate-space counterpart of nonsinglet DGLAP evolution. A central practical point is that the pseudo-distribution formalism requires sufficiently small $z^2$5, not an $z^2$6 limit taken first. Large hadron momentum is useful because it extends the accessible $z^2$7-range, but the fundamental control parameter is short distance, $z^2$8 (Joó et al., 2019).

A persistent misconception in the early literature was that approximate $z^2$9-scaling of the reduced ratio might justify identifying the RpITD directly with the light-cone ITD. The factorization analysis of the reduced Ioffe-time distribution corrected that interpretation: the ratio removes renormalization factors, but it is still an intermediate observable that must be perturbatively matched before one extracts the PDF (Zhang et al., 2018).

The same logic extends beyond unpolarized quark PDFs. In the gluon case, one-loop coordinate-space matching relates the reduced gluon ITD to the light-cone gluon ITD and includes singlet quark mixing through the gluon-from-quark kernel $\overline{\rm MS}$0. The preferred lattice projection is

$\overline{\rm MS}$1

because it isolates the invariant amplitude $\overline{\rm MS}$2 entering the gluon PDF with manageable renormalization properties at one loop (Balitsky et al., 2021).

4. Lattice construction and reconstruction of PDFs

In practice, lattice calculations determine nonlocal matrix elements from hadron two-point and three-point functions, extract the ground-state amplitude, and only then form the reduced ratio. Different studies have used the summation method, the Feynman–Hellmann method, Gaussian and momentum smearing, and, more recently, distillation. The common aim is the same: reduce excited-state contamination while obtaining broad coverage in $\overline{\rm MS}$3 at controllable $\overline{\rm MS}$4 (Joó et al., 2020).

One of the main empirical validations of the formalism is approximate $\overline{\rm MS}$5-scaling. In the nucleon study “Parton Distribution Functions from Ioffe time pseudo-distributions” (Joó et al., 2019), data that vary when plotted against $\overline{\rm MS}$6 or $\overline{\rm MS}$7 collapse onto a nearly universal curve when replotted as a function of $\overline{\rm MS}$8, with weak residual $\overline{\rm MS}$9-dependence. The same broad pattern appears in pion calculations and in later physical-point nucleon studies (Joó et al., 2019).

The inverse problem remains ill-posed because lattice data cover only a finite and discrete set of Ioffe times. Two standard strategies emerged. The first uses the small-$\mathfrak{M}(0,z^2)=1$0 expansion of the RpITD to extract low moments through the OPE. The second fits a parametric PDF ansatz directly to ITD or RpITD data instead of attempting a direct inverse Fourier transform. Physical-point nucleon work used both strategies and reported that the one-loop matching correction is of order $\mathfrak{M}(0,z^2)=1$1, while the reconstructed valence PDF remained larger than phenomenology for $\mathfrak{M}(0,z^2)=1$2 (Joó et al., 2020).

The most systematic fitting study in this direction embedded reduced pseudo-ITDs into the NNPDF machinery. There, fits based on statistical uncertainties only gave $\mathfrak{M}(0,z^2)=1$3 for the fine ensemble, while inclusion of continuum-limit, finite-volume, pion-mass, and additional Bayesian systematics reduced the fit quality measure to $\mathfrak{M}(0,z^2)=1$4. That study also made explicit that the real part of the RpITD constrains one nonsinglet combination and the imaginary part constrains another, through cosine- and sine-type kernels (Debbio et al., 2020).

High-precision distillation-based work sharpened a different point. It showed that the reduced ratio can still contain short-distance discretization effects that mimic a violation of the expected DGLAP-type evolution. In that study, the apparent discrepancy was largely resolved only after introducing an explicit discretization term proportional to $\mathfrak{M}(0,z^2)=1$5, which restored the expected short-distance behavior of the matched ITD (Egerer et al., 2021).

5. Extensions beyond forward quark PDFs

RpITDs are not restricted to forward nonsinglet nucleon PDFs. In pion structure studies they became the central lattice observable for extracting the valence PDF. The pion calculation based on two $\mathfrak{M}(0,z^2)=1$6-flavor ensembles at $\mathfrak{M}(0,z^2)=1$7 fm found that the reduced ratio exhibited little visible $\mathfrak{M}(0,z^2)=1$8-dependence, and it used one-loop matched RpITDs to extract both the valence PDF and the lowest four moments through “OPE without OPE” (Joó et al., 2019).

The same object was later incorporated directly into global QCD fits. In the combined experimental-and-lattice pion analysis, the real part of the reduced pseudo-ITD entered alongside Drell–Yan and leading-neutron data, with explicit models for higher-twist, discretization, and finite-volume corrections: $\mathfrak{M}(0,z^2)=1$9 That analysis concluded that RpITDs significantly decrease the uncertainties on the pion PDFs and support a large-$$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$0 behavior $$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$1 with $$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$2 (Barry et al., 2022).

In the gluon sector, the reduced ITD is even more central because the underlying operator structure is more complicated. The forward gluon analyses identified the relevant invariant amplitude $$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$3, showed that UV terms cancel in the reduced ITD, and derived the one-loop matching relation between the reduced gluon ITD and the light-cone gluon ITD, including singlet mixing (Balitsky et al., 2019). More recently, a machine-learning study took 35 lattice gluon RpITD values as input features and used a variational autoencoder inverse mapper to decode the proton gluon PDF, finding consistency with phenomenological global fits within uncertainties, particularly in the intermediate-to-high-$$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$4 region where the lattice data are most constraining (Kriesten et al., 23 Jul 2025).

Off-forward generalizations also exist. A zero-skewness GPD study implemented a ratio-scheme-renormalized coordinate-space observable,

$$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$5

which plays the same structural role as a forward RpITD while accommodating $$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$6-dependence and the off-forward decomposition of $$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$7 and $$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$8 (Bhattacharya et al., 2024). In the Nambu–Jona-Lasinio model, reduced generalized ITDs were written explicitly as

$$M^\alpha(p,z)=\langle p|\bar\psi(z)\gamma^\alpha U(z;0)\psi(0)|p\rangle,$$9

providing an analytic off-forward analogue of the RpITD and illustrating how reduction removes much of the purely separation-dependent structure (Shastry et al., 2022).

6. Misconceptions, limitations, and open issues

The first recurring misconception is that approximate collapse of data at fixed $$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$0 implies that matching can be omitted. The reduced-ratio factorization analysis showed that this is not correct: the RpITD is not itself the PDF or the light-cone ITD, even when its residual $$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$1-dependence looks mild (Zhang et al., 2018).

The second limitation is practical rather than formal. The reliable short-distance window is not universal. Studies typically identify it empirically by approximate $$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$2-independence after matching or by fixed-$$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$3 consistency across different $$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$4 combinations, rather than by a single numerical cut valid for all ensembles and channels. Discretization, finite-volume effects, and heavy pion masses can all be mild at low $$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$5 and still become serious at larger $$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$6 (Joó et al., 2020).

Perturbative truncation is another unresolved issue. Several studies use only NLO matching and state explicitly that NNLO or nonperturbative matching is required to assess truncation effects quantitatively (Joó et al., 2020). For gluons one must additionally track singlet quark mixing and independently determine $$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$7 (Balitsky et al., 2021).

A further caveat concerns the large-$$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$8 tail. A high-energy analysis of pseudo-ITDs in the BFKL approximation did not define the reduced ratio explicitly, but it showed that the unreduced pseudo-ITD acquires nontrivial large-Ioffe-time behavior controlled by all-twist physics rather than by leading-twist DGLAP alone. This suggests that an RpITD built from the same numerator inherits physically important large-$$M^\alpha(z,p)=2p^\alpha \mathcal{M}(\nu,z^2)+2z^\alpha \mathcal{N}(\nu,z^2), \qquad \nu=p\cdot z.$$9 structure, so aggressive extrapolation of finite-$\nu=p_3 z_3$0 lattice data with DGLAP-only assumptions can miss the correct small-$\nu=p_3 z_3$1 behavior (Chirilli, 2023).

Taken together, these developments place RpITDs in a specific methodological role. They are not alternate PDFs, and they are not merely a cosmetic reparameterization of matrix elements. They are UV-cleaned, coordinate-space observables that preserve the partonic Ioffe-time dependence needed for short-distance factorization. Their success across nucleon, pion, gluon, and off-forward applications explains why they have become the central observables of the pseudo-distribution approach, while their remaining sensitivity to discretization, higher twists, finite $\nu=p_3 z_3$2-coverage, and perturbative truncation continues to define the frontier of the method (Joó et al., 2019).