Valence Pion GPDs: Insights from Theory & Lattice

• Valence pion GPDs are mathematical functions that encode the pion's quark structure by correlating longitudinal momentum fraction with momentum transfer and transverse spatial distribution.
• They are derived using various methods such as Dyson–Schwinger equations, light-front overlap models, and lattice-QCD extractions, ensuring consistency with sum rules, support, and polynomiality conditions.
• These approaches connect the pion’s parton distribution, electromagnetic form factor, and three-dimensional tomography, revealing nontrivial x–t correlations and tighter transverse localization at high x.

Searching arXiv for recent and foundational papers on valence pion GPDs to ground the article in the literature. Valence pion generalized parton distributions describe the leading-twist quark content of the pion in off-forward kinematics, correlating longitudinal momentum fraction with momentum transfer and, at zero skewness, with transverse spatial structure. For a spin-0 target there is a single chiral-even quark GPD, usually denoted $H_\pi^q(x,\xi,t)$, with the forward limit reducing to the pion parton distribution function and the lowest Mellin moment reproducing the electromagnetic form factor. In the valence sector, the literature combines symmetry-preserving continuum constructions based on Dyson–Schwinger and Bethe–Salpeter equations, covariant chiral quark models, light-front overlap representations, and direct lattice-QCD extractions at zero skewness, yielding a technically coherent but method-dependent picture of pion structure (Mezrag, 2015, Lin, 2023).

1. Operator definition and kinematics

For a spinless target, the quark GPD is defined through the light-front bilocal vector correlator. In one standard convention,

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$

with average momentum $P=(P'+P)/2$, momentum transfer $\Delta=P'-P$, skewness

$\xi=-\frac{\Delta^+}{2P^+},$

and invariant momentum transfer $t=\Delta^2$ (Mezrag, 2015). The support is $x\in[-1,1]$. The regions $|x|>\xi$ and $|x|<\xi$ are the DGLAP and ERBL domains, respectively, and time-reversal invariance enforces evenness in $\xi$, $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$0 (Son et al., 2024).

At zero skewness, $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$1 admits the standard impact-parameter interpretation. The corresponding transverse density is

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$2

so that $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$3 is the kinematic limit in which longitudinal momentum and transverse position can be discussed simultaneously (Lin, 2023).

2. Exact constraints and limiting relations

Three structural constraints organize the theory of valence pion GPDs. The first is the forward limit,

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$4

which identifies the positive-$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$5 region with quarks and the negative-$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$6 region with antiquarks (Mezrag, 2015). The second is the electromagnetic sum rule,

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$7

which is $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$8-independent by polynomiality (Mezrag, 2015). The third is polynomiality of Mellin moments,

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$9

which, for the spin-0 pion, yields even polynomials in $P=(P'+P)/2$0 of degree at most $P=(P'+P)/2$1 (Mezrag, 2015).

The soft-pion limit is a further nontrivial constraint. In the form used in Dyson–Schwinger studies,

$P=(P'+P)/2$2

so maximal skewness connects the pion GPD to the pion distribution amplitude (Mezrag, 2015). A central point in the continuum literature is that this relation is not recovered by a simplified algebraic triangle model unless the axial-vector Ward–Takahashi identity is enforced; rainbow–ladder-consistent numerical DSE/BSE solutions restore the theorem (Mezrag, 2015). In the nonlocal chiral quark model, the strict chiral-limit statement at $P=(P'+P)/2$3 is that the isoscalar pion GPD vanishes pointwise, which yields a vanishing second Mellin moment in the soft-pion-theorem context, consistent with Polyakov–Weiss results (Son et al., 2024).

These constraints also delimit recurrent misconceptions. A model may reproduce a plausible forward PDF or form factor while still violating the soft-pion theorem or breaking valence symmetry. Conversely, continuity at the crossover $P=(P'+P)/2$4, correct support, and $P=(P'+P)/2$5-polynomiality are not optional technical details but defining consistency conditions.

3. Dyson–Schwinger and Bethe–Salpeter constructions

A major continuum route begins by computing Mellin moments from the triangle diagram in an impulse approximation and then reconstructing the full GPD through double distributions. In the algebraic DSE/BSE-inspired model, the moments are generated from

$P=(P'+P)/2$6

with a dressed-quark propagator

$P=(P'+P)/2$7

and a pion Bethe–Salpeter vertex

$P=(P'+P)/2$8

with $P=(P'+P)/2$9 chosen to recover the asymptotic pion distribution amplitude (Mezrag, 2015).

The reconstructed GPD is then written in double-distribution form,

$\Delta=P'-P$0

This implementation guarantees support, polynomiality, continuity at $\Delta=P'-P$1, and the form-factor sum rule by construction, and no separate D-term is introduced in that particular model (Mezrag, 2015). The resulting $\Delta=P'-P$2 and forward valence PDF were reported to agree very well with available data.

A related DSE treatment emphasized that the impulse approximation is incomplete in the valence sector. In that analysis, the triangle approximation alone breaks the $\Delta=P'-P$3 symmetry of the forward valence PDF, and additional terms corresponding to derivative insertions at the pion vertex are required to restore the symmetry analytically (Mezrag, 2015). An earlier rainbow–ladder study sharpened this point further by arguing that the impulse approximation is generally invalid owing to omission of contributions from the gluons which bind dressed-quarks into the pion, and by proposing an improved $\Delta=P'-P$4 construction that yields a positive-definite impact-parameter distribution and a qualitatively sound picture of the pion’s dressed-quark structure at a hadronic scale (Mezrag et al., 2014).

Within that $\Delta=P'-P$5 construction, the valence GPD is written as

$\Delta=P'-P$6

which makes explicit the nontrivial $\Delta=P'-P$7–$\Delta=P'-P$8 correlation that is absent in a purely factorized $\Delta=P'-P$9 ansatz (Chang et al., 2015). The same line of work also relates the $\xi=-\frac{\Delta^+}{2P^+},$0 limit to the pion distribution amplitude through the soft-pion theorem.

4. Chiral and light-front quark-model realizations

The Nambu–Jona-Lasinio model provides a covariant Bethe–Salpeter realization of pion GPDs with an explicit DD$\xi=-\frac{\Delta^+}{2P^+},$1D-term structure. In the chiral limit and at $\xi=-\frac{\Delta^+}{2P^+},$2, the model yields the simple piecewise expression

$\xi=-\frac{\Delta^+}{2P^+},$3

and the isoscalar GPD obeys $\xi=-\frac{\Delta^+}{2P^+},$4 (Courtoy, 2010). The model also makes polynomiality explicit through tabulated generalized form factors and supplies a low matching scale, $\xi=-\frac{\Delta^+}{2P^+},$5 and $\xi=-\frac{\Delta^+}{2P^+},$6, from which LO and NLO evolution can be performed (Courtoy, 2010).

A complementary nonlocal chiral quark model analyzes valence pion GPDs across DGLAP and ERBL regions with two nonlocal topologies: a type-(a) contribution entering both regions and a type-(b) contribution entering only ERBL. The resulting valence GPD is continuous at $\xi=-\frac{\Delta^+}{2P^+},$7, although its first derivative exhibits kinks, and the model reports a total valence momentum fraction of $\xi=-\frac{\Delta^+}{2P^+},$8 at the model scale, with the deviation from unity attributed to nonlocal-current gauge-invariance issues (Son et al., 2024). The same framework performs one-loop evolution with \texttt{APFEL++} to $\xi=-\frac{\Delta^+}{2P^+},$9 and $t=\Delta^2$0, finding that produced sea quarks and gluons are strongly suppressed as $t=\Delta^2$1 becomes nonzero and are largely confined within the ERBL region.

Light-front constituent approaches usually restrict the analysis to $t=\Delta^2$2. In a light-cone quark model with Brodsky–Huang–Lepage wave functions and Melosh–Wigner rotation, the pion GPD is represented by a diagonal overlap,

$t=\Delta^2$3

with starting scale $t=\Delta^2$4 (Kaur et al., 2021). In that model, $t=\Delta^2$5 peaks at $t=\Delta^2$6 at $t=\Delta^2$7, while increasing $t=\Delta^2$8 decreases the magnitude and shifts the peak toward higher $t=\Delta^2$9.

Basis light-front quantization with holographic confinement, longitudinal confinement, and color-singlet NJL interactions gives another valence-only realization. For the pion, the overlap formulas at $x\in[-1,1]$0 generate both $x\in[-1,1]$1 and the chiral-odd $x\in[-1,1]$2, and Mellin moments define generalized form factors

$x\in[-1,1]$3

At $x\in[-1,1]$4, the model reports $x\in[-1,1]$5, $x\in[-1,1]$6, and an average transverse shift $x\in[-1,1]$7, in reasonable agreement with lattice-QCD benchmarks quoted in the same work (Adhikari et al., 2021).

5. Lattice-QCD determinations at zero skewness

Direct $x\in[-1,1]$8-dependent lattice determinations of the valence pion GPD use Large-Momentum Effective Theory. An exploratory first study computed the valence quasi-GPD at $x\in[-1,1]$9 on a $|x|>\xi$0-flavor HISQ ensemble with $|x|>\xi$1, $|x|>\xi$2, $|x|>\xi$3, and boosts up to $|x|>\xi$4 (Chen et al., 2019). It reproduced the $|x|>\xi$5 PDF limit and found that $|x|>\xi$6 agrees with the pion form factor from the $|x|>\xi$7 matrix element within $|x|>\xi$8, but the errors were too large to discriminate among different $|x|>\xi$9–$|x|<\xi$0 parametrizations.

The first physical-pion-mass $|x|<\xi$1-dependent calculation at $|x|<\xi$2 used clover valence fermions on $|x|<\xi$3 HISQ ensembles with $|x|<\xi$4, $|x|<\xi$5, and boost momentum $|x|<\xi$6, together with hybrid renormalization and NNLO LaMET matching to $|x|<\xi$7 at $|x|<\xi$8 (Lin, 2023). The calculation covered $|x|<\xi$9, reproduced $\xi$0, predicted higher moments $\xi$1 and $\xi$2, and provided the first lattice-QCD pion tomography $\xi$3. A central result was that the transverse profile narrows as $\xi$4 increases, indicating stronger transverse localization of large-$\xi$5 valence partons.

A subsequent lattice study with $\xi$6 and valence pion mass $\xi$7 analyzed both symmetric and asymmetric momentum-transfer frames, showed the equivalence of both at the level of Lorentz-invariant amplitudes, and matched quasi-GPDs with NNLO perturbative corrections plus leading renormalon and renormalization-group resummations (Ding et al., 2024). In that work, the matched $\xi$8 was stated to be reliable for $\xi$9 at $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$00, and its $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$01-dependence at fixed $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$02 was well described by a monopole form,

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$03

The effective transverse radius extracted from this monopole decreases with $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$04, again supporting the picture that high-$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$05 quarks are more localized in the transverse plane (Ding et al., 2024).

6. Form factors, evolution, and phenomenological status

Electromagnetic and gravitational form factors are the most direct Mellin-moment observables associated with valence pion GPDs. In the nonlocal chiral quark model, the first Mellin moment is numerically $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$06-independent and yields the pion electromagnetic form factor with charge normalization $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$07, while the second Mellin moment gives $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$08 and $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$09; the model reports

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$10

a charge radius $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$11, and gravitational radii $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$12 and $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$13 (Son et al., 2024). In that same framework, the negative D-term is interpreted as consistent with chiral expectations, while the omission of meson loops is noted as a likely source of underestimated D-term slopes.

A more explicitly phenomenological program constructs pion GPDs from forward PDFs and a controlled off-forward extension that satisfies support, positivity, forward limits, polynomiality, and continuity by construction. In the DGLAP region, the quark GPD takes the master-equation form

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$14

with $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$15 and $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$16, and the soft-pion theorem is used to fix the odd-in-$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$17 D-term (Chavez et al., 2021). That analysis also extends to gluon GPDs and Compton form factors, finding that next-to-leading order corrections are significant even in the so-called valence region.

At zero skewness, a recent global QCD analysis of pion electromagnetic form-factor data adopts the factorized ansatz

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$18

with

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$19

Using xFitter, JAM21, and MAP23 pion PDFs at $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$20, the study finds that only two parameters are sufficient to describe the data and adopts the MAP23-based fit with

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$21

and total $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$22 (Collaboration et al., 20 Aug 2025). In impact-parameter space this ansatz yields

$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$23

making the transverse-width interpretation completely explicit.

Across approaches, several limitations recur. Valence-only constructions neglect explicit sea-quark and gluon degrees of freedom at the input scale; simplified algebraic DSE models may fail to enforce the axial-vector Ward–Takahashi identity; $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$24 light-front models do not by themselves test full polynomiality; and some continuum studies do not perform detailed positivity-bound checks or impact-parameter analyses (Mezrag, 2015). These are not contradictions so much as delimitations of scope. Taken together, the literature supports a stable qualitative picture: the valence pion GPD has the correct forward and form-factor limits, exhibits nontrivial $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$25–$$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$26 correlations, narrows in transverse space as $$H(x,\xi,t) = \frac{1}{2} \int \frac{\textrm{d}z^-}{2\pi} \, e^{i x P^+ z^-} \left\langle P+\frac{\Delta}{2} \left| \bar{q}\left(-\frac{z}{2}\right)\gamma^+q\left(\frac{z}{2}\right) \right |P-\frac{\Delta}{2}\right\rangle_{z^+=0,\,z_\perp=0},$$27 increases, and provides the common framework linking pion PDFs, form factors, distribution amplitudes, and three-dimensional tomography.