Information Geometry of DGLAP Evolution: From Geodesic Tests to Emergent Small-Circle Geometry

Introduction

Parton distribution functions (PDFs) encode the longitudinal momentum structure of hadrons, and their resolution-scale dependence is governed by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) equations [1, 2, 3, 4]. This evolution is conventionally described through the changing PDFs and their Mellin moments [5, 6, 7]. For a conserved partonic sector, however, a positive distribution can also be normalized to a probability density, allowing its scale evolution to be viewed as a trajectory on a statistical manifold. This raises a complementary question: what geometric path does QCD evolution select in the space of normalized distributions? The Fisher–Rao metric provides a natural measure of the statistical distinguishability between neighboring states [8, 9, 10, 11].

Information-theoretic approaches to partonic structure have explored connections among small-$x$ dynamics, entanglement entropy, and final-state hadron production, including comparisons with DIS phenomenology [12, 13, 14, 15, 16, 17, 18, 19, 20]. These studies motivate broader information-theoretic questions in QCD, but their entropy observables are distinct from the statistical geometry of the evolving PDF considered here.

Geometric descriptions of renormalization-group flows and information-theoretic coarse graining provide another related context [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Such constructions often involve spaces of couplings, theories, or coarse-grained states. Here the manifold instead consists of normalized parton distributions. The DGLAP kernel determines their evolution, while the Fisher–Rao metric characterizes the geometry of the resulting path. The dynamical law and the statistical metric thus serve distinct, complementary roles.

The leading-order (LO) nonsinglet valence sector offers a direct setting for this analysis. Its evolution does not involve singlet–gluon mixing, and valence-number conservation permits the positive valence distributions considered here to be normalized as probability densities $p(x,Q^2)$. The square-root embedding $\Phi= 2\sqrt{p}$ maps these states isometrically into the positive sector of a Hilbert sphere of radius two [10, 33, 34]. DGLAP evolution therefore defines a curve whose length, curvature, and effective dimensionality can be studied in a common geometric framework.

Although DGLAP evolution is governed by a first-order equation, the geodesic character of its trajectory is determined by the acceleration induced by that evolution. We obtain the first and second derivatives of the Fisher state directly from the LO evolution operator. Removing the sphere-normal component and the component parallel to the instantaneous velocity isolates the transverse geodesic acceleration, whose norm relative to the squared Fisher speed defines the parametrization-invariant geodesic curvature. This construction tests the geodesicity of the QCD-selected path without introducing a new evolution equation.

We apply this framework to a GRV98-based totalvalence distribution evolved over $2 \le Q^2 \le2000\ \mathrm{GeV}^2$. The resulting Fisher trajectory is nongeodesic, yet its geodesic curvature varies only weakly across the analyzed interval. Arc-length-weighted principal-component analysis identifies a dominant two-dimensional affine sector, whose displaced plane defines a small-circle reference on the Fisher sphere. The curvature obtained directly from DGLAP-generated acceleration agrees closely with the geometric prediction of this construction.

An $x$-resolved analysis connects the trajectory geometry to the relative reshaping of the normalized valence distribution: the regions contributing most strongly to statistical displacement need not be those dominating its transverse bending. The Mellin representation provides a complementary perspective on the collective evolution underlying the effective low-dimensional trajectory. Together, these results identify a nongeodesic but near-small-circle geometry for the reference LO nonsinglet evolution over a finite scale interval.

Information-Geometric Formulation of LO DGLAP Evolution

We formulate LO nonsinglet DGLAP evolution as a trajectory on the Fisher–Rao manifold of normalized valence distributions. The statistical geometry is fixed by probability normalization, whereas the physical path is selected by the DGLAP kernel. We first construct the Fisher embedding and then derive the velocity, acceleration, and geodesic curvature generated by the QCD evolution. Details concerning reparameterization invariance and numerical implementation are given in the appendices.

Normalized valence distribution and Fisher sphere

We consider a nonsinglet valence distribution

with conserved valence number

For nonsinglet DGLAP evolution, $N_v$ is independent of $Q^2$ because the first Mellin moment of the nonsinglet splitting kernel vanishes [1, 2, 3, 4, 35]. We therefore define the normalized density

For the positive valence distributions considered here, $p(x,Q^2)$ may be treated as a probability density on $0<x<1$. Because $N_v$ is scale independent, the normalization removes only a conserved overall factor and leaves the shape evolution unchanged. The scale dependence therefore defines a trajectory

on the infinite-dimensional manifold of normalized positive densities. An infinitesimal tangent deformation satisfies

and the Fisher–Rao line element is

This metric provides the intrinsic local measure of statistical distinguishability on the probability manifold [8, 9, 10, 33, 34]. The spherical form of the metric follows directly from the square-root embedding

Normalization implies

while

gives

Thus $\Phi=2\sqrt{p}$ is an isometric embedding of the Fisher–Rao manifold into the positive sector of a Hilbert sphere [10, 33, 34, 36] with radius

For two normalized states $p_1(x)$ and $p_2(x)$, their embedded inner product is determined by the Bhattacharyya overlap [37],

Their angular separation on the Fisher sphere is therefore $\Theta_{12}=\arccos\mathcal{B}(p_1,p_2)$, and the corresponding Fisher–Rao geodesic distance is

This distance is fixed entirely by the endpoint distributions. DGLAP evolution, in contrast, selects a specific physical path between those endpoints. The distinction between the metric shortest path and the QCD-selected trajectory will be central to the analysis below.

LO DGLAP flow on the statistical manifold

We now introduce the QCD dynamics that selects the physical trajectory on the Fisher–Rao manifold. Using

the LO nonsinglet DGLAP equation for the normalized valence density is

where

Because the first Mellin moment of $P_{qq}^{(0)}$ vanishes, this flow preserves the normalization of $p$ and therefore remains tangent to the statistical manifold.

For the geometric analysis it is convenient to absorb the running coupling into the evolution parameter,

so that

The LO evolution then takes the fixed-operator form

At LO, $\mathcal{L}$ is independent of $\tau$. The running coupling therefore determines the mapping between the physical scale $Q^2$ and the evolution parameter $\tau$, whereas the path in the space of normalized distributions is generated by the fixed operator $\mathcal{L}$.

Under the Fisher embedding $\Phi=2\sqrt{p}$, the corresponding velocity is

Normalization conservation gives

confirming that the DGLAP-induced velocity is tangent to the Fisher sphere. Its norm defines the Fisher–Rao speed in the $\tau$ parametrization,

The physical path length accumulated between two scales is then

Although the instantaneous speed depends on the parameter used to traverse the trajectory,

the accumulated Fisher–Rao length is invariant,

The explicit reparameterization properties of the curvature are derived in Appendix A.

Acceleration and direct geodesic test

Geodesicity is determined by the acceleration of a curve rather than by its velocity alone. Differentiating the fixed-operator evolution in Eq. (19), and using the $\tau$ independence of $\mathcal{L}$ at LO, gives

The second derivative therefore involves only repeated action of the same LO evolution operator and does not introduce a higher-order splitting kernel.

For the Fisher embedding $\Phi=2\sqrt{p}$, the corresponding acceleration is

The second term originates from the nonlinear square-root embedding and is therefore geometric in origin even though the evolution equation for $p$ is linear.

The spherical constraint

implies

Hence the radial component of the acceleration is fixed entirely by the sphere geometry,

Removing this radial contribution leaves the acceleration tangent to the Fisher sphere,

The parameter $\tau$ is not generally an affine parameter along the curve, so $A_{\mathrm{tan}}^{(\tau)}$ may still contain a component parallel to the velocity. Such a component changes only the rate at which the trajectory is traversed and does not bend the path. Removing it defines the transverse geodesic residual,

By construction,

so that $\mathcal{R}_{\mathrm{geo}}^{(\tau)}$ measures only the transverse departure of the physical trajectory from Fisher–Rao geodesic motion. The intrinsic geodesic curvature is therefore

A Fisher–Rao geodesic satisfies

Although the velocity and acceleration separately depend on the parameter used to traverse the curve, the combination in Eq. (34) is intrinsic. Its invariance under a monotonic reparameterization is shown explicitly in Appendix A.

Great-circle and small-circle geometry

The local geodesic test can be connected to a simple global geometry. Suppose that the Fisher trajectory is confined, exactly or approximately, to a two-dimensional affine plane

where $e_{1}$ and $e_{2}$ are orthonormal directions within the plane,

The vector $C$ therefore points from the center of the Fisher sphere to the closest point of the affine plane, and its norm

defines the plane offset. Because the Fisher sphere has radius $R_{\mathrm{FR}}$, the intersection of $\mathcal{P}$ with the sphere satisfies

The intersection is therefore a circle with radius

If $h=0$, the affine plane passes through the sphere center and the intersection is a great circle, corresponding to a Fisher–Rao geodesic. For $h\ne0$, the intersection is a small circle.

For an exact small circle, the intrinsic geodesic curvature is constant and is fixed entirely by the sphere radius and plane offset,

Thus the great-circle limit $h\to0$ gives $k_{\mathrm{sc}}\to0$, as required for geodesic motion. If the trajectory spans an angular interval $\Delta\phi$ within the affine plane, the corresponding circular arc length is

The quantities $r$, $k_{\mathrm{sc}}$, and $L_{\mathrm{sc}}$ are therefore predictions of the affine-plane geometry rather than independent fit parameters. In the numerical analysis below, they will be compared directly with the radius, geodesic curvature, and accumulated Fisher–Rao length of the DGLAP trajectory.

Numerical Evolution and Geometric Observables

We now construct a reference LO nonsinglet trajectory in $x$ space and evaluate the Fisher–Rao observables introduced in Sec. II. The underlying DGLAP evolution is validated independently before the geometric analysis is performed. Numerical discretization, conservation corrections, and detailed validation tests are collected in Appendix B.

Reference LO evolution

For the reference trajectory, we use the total valence combination

from the LO GRV98 parametrization [38]. The evolution is initialized at the original GRV98 input scale,

and first evolved to

This pre-evolution constructs the initial statistical state used in the information-geometric analysis. No geometric interpretation is assigned to the evolution below $Q_{0}^{2}$; the lower-scale evolution is used only to construct the GRV98-based reference state at $Q_0^2 = 2\ \mathrm{GeV}^2$.

At $Q_0^2$, the normalized density is defined by

In the continuum valence normalization the denominator equals three. In the numerical calculation, however, the normalization is evaluated with the same quadrature used for the evolution, as described in Appendix B.

The Fisher–Rao trajectory is studied over

Thus $Q_{\mathrm{GRV}}^2$ is the dynamical input scale of the GRV98 parametrization, whereas $Q_0^2$ defines the starting point of the geometric trajectory.

The reference evolution uses the LO nonsinglet splitting function

where $C_F=4/3$. This is equivalent to the conventional LO Altarelli–Parisi form. The plus distribution is defined through

Consequently,

which expresses valence-number conservation.

The running coupling is treated consistently at one-loop order,

The flavor thresholds, the corresponding values of $\Lambda_{\mathrm{QCD}}$, and the numerical discretization are specified in Appendix B.

The continuum evolution and its first two derivatives follow Eqs. (19) and (26). In the numerical calculation, the velocity and acceleration are constructed from the corresponding discrete DGLAP convolution, with the valence-normalization constraint imposed on the grid as described in Appendix B 2. The geometric derivatives are generated from this discrete evolution operator rather than by finite differences of the sampled trajectory.

Validation of the reference trajectory

Before constructing the geometric observables, we verify the underlying $x$-space evolution against the analytic LO nonsinglet solution in Mellin space. Defining

the LO evolution becomes diagonal,

with solution

The conserved first moment, $\gamma_1^{(0)}=0$, provides an additional normalization check. For the first five Mellin moments, the moments reconstructed from the numerical $x$-space evolution reproduce Eq. (54) over the full analysis interval. The maximum relative discrepancy among the nontrivial moments is

After imposing the normalization constraint on the numerical grid, the maximum deviation from unit discrete normalization is

The evolved density remains positive throughout the trajectory, so that the Fisher embedding $\Phi=2\sqrt{p}$ is well defined over the complete numerical domain. The local geodesic curvature is reconstructed from the operator-generated first and second derivatives of the Fisher state. As a complementary geometric check, we compare it with a finite-chord curvature estimator constructed from neighboring points of the trajectory. The two reconstructions agree at the few-parts-per-million level, with a maximum relative mismatch of

The Mellin-space comparison, discrete normalization check, and finite-chord reconstruction provide complementary validations of the reference evolution and its geometric implementation. The numerical prescriptions and diagnostics are collected in Appendix B.

Geometric observables

We characterize the Fisher–Rao trajectory at three complementary levels. First, principal-component analysis (PCA) [39] determines whether the evolution is effectively confined to a low-dimensional affine sector. Second, the DGLAP-generated acceleration gives the local geodesic curvature. Third, the sphere–plane geometry provides global predictions for the circle radius and accumulated path length. All inner products are evaluated in the Fisher representation,

For the sampled Fisher states $\Phi_i = \Phi(\cdot,Q_i^2)$, we assign normalized arc-length weights $\omega_i$ using the trapezoidal discretization of the accumulated Fisher–Rao length,

The weighted mean state is

and the centered trajectory is

The weighted covariance operator,

defines orthonormal principal directions $e_a$ with eigenvalues $\lambda_a$. The fraction of the total trajectory variance captured by the first two directions is

When $f_2$ is close to unity, the trajectory is effectively confined to the affine plane

Let $\Phi_i^{\mathrm{proj}}$ denote the orthogonal projection of $\Phi_i$ onto this plane. We define the pointwise out-of-plane displacement

and the arc-length-weighted RMS size of the full trajectory,

The normalized RMS departure from the dominant affine sector is

For the two-component PCA used here,

Writing the projected trajectory as

where $C$ is the point of $\mathcal{P}_{\mathrm{PCA}}$ closest to the Fisher-sphere center, the plane offset is

The sphere–plane geometry of Sec. II D then determines the reference circle radius and curvature through Eqs. (40) and (41).

For each projected state we define the in-plane radius

and quantify the radial departure from the sphere–plane prediction by the arc-length-weighted residual

Thus $f_2$ and $\epsilon_{\mathrm{plane}}$ characterize the effective dimensional reduction, whereas $\epsilon_{\mathrm{rad}}$ tests circular closure within the dominant affine plane.

The local bending of the physical trajectory is characterized by the geodesic curvature in Eq. (34). Its comparison with $k_{\mathrm{sc}}$ tests whether the local nongeodesic bending is consistent with the geometry inferred independently from the dominant affine plane.

For comparison with a constant-curvature reference, we define the Fisher–Rao arc-length average

The corresponding arc-length-weighted standard deviation is

The global path geometry is characterized by the accumulated Fisher–Rao length

and by the small-circle arc length defined in Eq. (42), where $\Delta\varphi$ is the angular displacement within the dominant affine plane. For the corresponding small-circle reconstruction, we define

This provides the small-circle reference used in the comparison with the physical DGLAP trajectory. We also define the geodesic length ratio

where $d_{\mathrm{FR}}$ is the Fisher–Rao geodesic distance between the endpoints. A minimizing Fisher–Rao geodesic has $\eta_G = 1$, while $\eta_G < 1$ measures the excess length of the QCD-selected path relative to the metric shortest path.

To resolve the Fisher motion in momentum fraction, we define

Its contribution per logarithmic interval in $x$ is

For an interval $[x_a,x_b]$, the fractional contribution to the squared Fisher speed is

The transverse bending is resolved by the acceleration $\mathcal{A}_{\perp}^{(\tau)} \equiv\mathcal{R}_{\mathrm{geo}}^{(\tau)}$ (transverse to the Fisher trajectory, not to the hadron momentum), with

The corresponding regional fraction is

Finally, the zero of the local scale deformation,

defines the pivot $x_p$, while the zeros of the transverse geodesic acceleration,

define the characteristic momentum fractions $x_{\kappa,1}$ and $x_{\kappa,2}$. These quantities distinguish the momentum-fraction structure associated with motion along the statistical trajectory from that associated with its transverse bending.

Geometric Structure of the DGLAP Trajectory

From PDF redistribution to Fisher motion

We begin with the underlying redistribution of the normalized valence density before examining the geometry of the resulting trajectory. Figure 1 shows $x\,p(x,Q^2)$ at representative scales over

As the resolution scale increases, the intermediate- and large-$x$ valence structure is progressively suppressed and the distribution is shifted toward lower momentum fractions. Valence-number conservation ensures that each scale defines a normalized statistical state along the Fisher trajectory. The geometric question is therefore not only how the distribution moves in $x$, but how its changing shape generates motion in the space of statistical states.

The local contribution to this motion, per logarithmic interval in momentum fraction, is given by the Fisher-speed density in Eq. (79). Figure 2(a) shows that the regions contributing most strongly to the Fisher motion need not coincide with those carrying the largest probability weight. The Fisher metric is sensitive to the relative deformation of the normalized distribution, rather than to its local magnitude alone.

A pronounced zero occurs at the scale-dependent pivot $x_p$, defined in Eq. (83). At this momentum fraction the distribution is instantaneously unchanged under the scale evolution, while probability is redistributed on either side of it. The pivot moves smoothly from

to

The transverse bending of the Fisher trajectory contains a different momentum-fraction structure, shown in Figure 2(b) and defined by Eq. (81). Its two characteristic zeros evolve from approximately

at the initial scale to

at the upper endpoint. As summarized in Figure 2(c), $x_p$, $x_{\kappa,1}$, and $x_{\kappa,2}$ all migrate toward lower $x$ while remaining clearly separated. The momentum-fraction structure governing motion along the trajectory is therefore distinct from that governing its transverse bending.

The same separation appears after integrating over finite momentum-fraction regions. At $Q^2 = 2\ \mathrm{GeV}^2$, the interval $0.3 < x < 0.7$ contains about 14.7% of the normalized probability but contributes approximately 51.0% of the squared Fisher speed and 21.0% of the transverse-acceleration budget. At $Q^2 = 2000\ \mathrm{GeV}^2$, the same interval contains only about 7.0% of the probability while still contributing about 40.2% of the Fisher-speed budget and 33.1% of the transverse bending. Conversely, the region $0.01 < x < 0.1$ carries about 41.5% of the probability at the upper endpoint but contributes only about 7.5% of the squared Fisher speed.

The Fisher trajectory is therefore governed by the relative scale-dependent reshaping of the normalized PDF rather than by the location of its largest probability weight.

The distinct momentum-fraction contributions to statistical displacement and transverse bending provide a local picture of the evolution. We next examine how the overall rate of Fisher motion changes with scale, before turning to the intrinsic curvature and global geometry of the trajectory.

Rate of traversal along the Fisher trajectory

Unlike the intrinsic geometry of a trajectory, its instantaneous Fisher speed depends on the parameter used to describe the evolution.

In the $\tau$ parametrization, the squared Fisher speed decreases smoothly from approximately 2.70 at $Q^2 = 2\ \mathrm{GeV}^2$ to 2.04 at $Q^2 = 2000\ \mathrm{GeV}^2$, corresponding to a reduction of about 24%.

To characterize this decrease, we define

For the reference trajectory, $D_{\tau}$ remains positive throughout the evolution interval and decreases from approximately 3.60 to 2.13.

As shown in Figure 3, the decreasing Fisher speed indicates that a further increment of $\tau$ produces a progressively smaller statistical displacement along the reference trajectory. This behavior reflects the changing response of the normalized valence distribution to the evolution operator. It does not, by itself, determine how the trajectory bends within the Fisher manifold. That distinction is addressed next through the geodesic curvature.

Nongeodesicity and regular geometric bending

Having characterized the rate of statistical motion, we turn to the intrinsic bending of the Fisher trajectory. The central question is whether the DGLAP-selected path follows a Fisher–Rao geodesic. The numerical evolution shows a nonzero geodesic curvature throughout the analyzed interval. More importantly, this curvature varies only weakly with scale, despite the changing rate of traversal established in the preceding subsection.

Figure Figure 4(a) shows the curvature reconstructed directly from the DGLAP-generated velocity and acceleration. Its Fisher–Rao arc-length average is

with a relative variation of order $10^{-3}$. The departure from geodesic motion is therefore highly regular rather than an arbitrary bending of the trajectory.

The same trajectory remains close to the metric shortest path in an integrated sense. The geodesic length ratio defined in Eq. (77), shown in Figure 4(b), reaches

at the upper endpoint. The physical DGLAP path therefore exceeds the endpoint geodesic distance by only about 0.44% in relative length. Its local curvature is nevertheless clearly nonzero, showing that a small excess in total path length can coexist with a well-resolved geometric bending.

Panel (c) shows the finite-chord curvature comparison described in Sec. III B. Its few-parts-per-million agreement with the differential reconstruction provides a complementary geometric cross-check. The nonzero and nearly scale-independent curvature is therefore a stable property of the reference trajectory. We next show that its magnitude and global path geometry are naturally accounted for by the same affine small-circle construction.

Emergent small-circle closure

The effective dimensionality provides the link between the observed curvature and a simple global geometry. The first two principal directions capture a fraction

of the arc-length-weighted trajectory variance. Equivalently, Eq. (68) gives

The evolution is therefore confined to a narrow neighborhood of a two-dimensional affine plane.

The position of the plane relative to the Fisher-sphere center fixes the corresponding sphere–plane intersection. For the reference trajectory, the plane offset is $h \simeq1.7469$, which gives

and

The predicted curvature differs from the arc-averaged DGLAP value in Eq. (91) by only about $1.6 \times10^{-3}$ in relative terms. Thus the local nongeodesic bending and the global position of the dominant affine plane point to the same small-circle geometry.

Figure Figure 5(a) tests this construction directly in the dominant plane. The projected DGLAP states trace a smooth arc with best-fit radius $r_{\mathrm{fit}}\simeq0.973795$, in agreement with $r_{\mathrm{plane}}\simeq0.973796$ at the $10^{-6}$ relative level.

The two residual measures exhibit a clear hierarchy. Compared with the plane residual in Eq. (94), the in-plane radial residual is

The dominant departure from the idealized construction is associated with the weak nonplanarity of the full trajectory. In their respective normalized measures, the residual in-plane noncircularity is more than two orders of magnitude smaller than the out-of-plane deviation.

The same closure is visible in the accumulated path length. As shown in Figure 5(c), the DGLAP trajectory closely follows the exact-circle relation in Eq. (42) throughout the evolution interval. At the upper endpoint, the angular span is $\Delta\varphi\simeq21.27^\circ$, while

Their relative difference is

The radius, local curvature, radial closure, and accumulated path length therefore provide mutually consistent tests of the same geometry. Over the finite scale interval considered here, the LO DGLAP trajectory follows a short arc close to the small circle selected by its dominant affine plane.

Global Fisher-sphere geometry

Figure 6 summarizes the resulting geometry in coordinates adapted to the trajectory. The directions $u_1$ and $u_2$ span the dominant affine plane, while $u_\perp$ is chosen along the direction from the Fisher-sphere center $O$ to the closest point $C$ of that plane. The ideal affine plane is therefore located at $u_\perp=h$.

The geometric parameters obtained from the small-circle reconstruction determine the sphere–plane configuration shown in Figure 6(a). Since the dominant plane does not pass through the sphere center, its intersection with the Fisher sphere is a small circle rather than a great circle.

Panel (b) resolves the remaining departure from the ideal affine geometry. The dominant evolution occurs along $u_1$ and $u_2$, while the displacement in the orthogonal direction remains much smaller. With the residuals normalized to the Fisher-sphere radius as defined in Appendix B.§ 0.9.4, the arc-length-weighted RMS and maximum reconstruction errors are $\epsilon_{\mathrm{3D}}^{\mathrm{RMS}\simeq6.4\times10^{-5}}$ and $\epsilon_{\mathrm{3D}}^{\max}\simeq1.7\times10^{-4}$, respectively.

The three-dimensional reconstruction provides a geometric visualization of the near-small-circle structure identified from the local curvature and affine-plane analysis. We next examine how this organization is related to the underlying DGLAP evolution dynamics.

Dynamical Interpretation

The preceding analysis reveals two related features of the LO nonsinglet DGLAP trajectory: its effective confinement to a low-dimensional affine sector and its nonzero, nearly constant geodesic curvature. We examine these observations from two complementary perspectives. The Mellin-space evolution characterizes the spectral organization of the changing parton distribution, while the

Fisher-embedded acceleration connects the local DGLAP dynamics to the geometry of the trajectory.

Spectral perspective on effective dimensional reduction

At LO, the nonsinglet Mellin moments evolve independently according to Eq. (54), with the anomalous dimensions $\gamma_N^{(0)}$ determined by the splitting kernel [40, 41]. The conserved mode $\gamma_1^{(0)}=0$ fixes the valence normalization, whereas the higher moments describe the scale-dependent reshaping of the distribution.

The effective two-dimensional geometry found in § 0.4 does not imply a two-mode truncation of this Mellin hierarchy. Each nontrivial moment retains its own evolution, and the full distribution contains contributions from infinitely many moments. The observed low dimensionality instead characterizes the collective trajectory sampled over the finite evolution interval.

This distinction can be made explicit through the fixed-operator form of LO evolution,

The successive terms belong to the Krylov sequence generated by $\mathcal{L}$ and the initial distribution $p_0$. Although this sequence is not generally finite dimensional, its contributions to the evolving distribution may be strongly correlated over a restricted interval in $\tau$.

The Fisher representation introduces the nonlinear transformation $\Phi=2\sqrt{p}$. Consequently, the principal directions extracted from the Fisher trajectory are not individual Mellin eigenmodes. They describe collective variations of the normalized distribution after the square-root embedding.

The strong two-component dominance observed numerically is therefore a finite-window property of the complete Fisher trajectory rather than a reduction of the underlying DGLAP equation to two independent degrees of freedom. The Mellin and Krylov representations provide a dynamical perspective on this effective compression, while the nearly circular form of the trajectory requires additional information about its acceleration.

Geometric organization of the DGLAP acceleration

The geodesic curvature measures how the direction of statistical motion changes along the Fisher trajectory. For an arc-length parameter $s$, the acceleration on the Fisher sphere satisfies

where $N_{\mathrm{FR}}$ is the unit direction of the geodesic-curvature component. The first term is fixed by the spherical normalization constraint, while the second describes the intrinsic bending of the path away from geodesic motion.

For an exact small circle with center $C$ and radius $r$, the acceleration instead takes the form

with the constant geodesic curvature $k_{\mathrm{sc}}=\lvert h\rvert/(R_{\mathrm{FR}}r)$ from Eq. (41). An exact small circle thus combines a fixed affine plane with a constant intrinsic curvature.

The numerical DGLAP trajectory displays a close realization of this geometry. Its dominant affine plane is displaced from the Fisher-sphere center, while the geodesic curvature reconstructed directly from the evolution-generated acceleration varies only weakly with scale. The agreement between $k_{\mathrm{FR}}$ and the corresponding small-circle value $k_{\mathrm{sc}}$ connects the local bending of the physical trajectory to its global affine geometry.

This correspondence is particularly informative because spherical normalization alone does not require either near-planarity or nearly constant geodesic curvature. These properties characterize the particular path selected by the LO evolution and its initial distribution.

The connection to DGLAP dynamics follows from Eqs. (19) and (26). The velocity and acceleration of the Fisher state are therefore determined by successive actions of the same nonsinglet evolution operator, together with the nonlinear square-root embedding. In this representation, the geodesic curvature characterizes the changing direction of PDF reshaping, rather than the magnitude of the redistribution alone.

Physical scope and further directions

The $x$-resolved results provide a complementary partonic picture. Different momentum-fraction regions contribute to the Fisher speed and to the transverse geodesic acceleration, showing that the regions driving statistical displacement need not be those primarily responsible for its bending.

The distinction between the shape of the path and its rate of traversal is also important. The running coupling determines the mapping from $Q^{2}$ to $\tau$ and contributes to the physical-scale dependence of the Fisher speed. Even in the $\tau$ parametrization, however, the speed changes as the normalized distribution evolves. By contrast, the geodesic curvature is invariant under monotonic reparameterization. The decreasing Fisher speed and the nearly constant curvature therefore describe different aspects of the same DGLAP trajectory.

Taken together, the local curvature and global small-circle reconstruction indicate that the scale-dependent reshaping of the reference valence distribution follows a highly organized geometric path. The approximately constant bending is an observed feature of this evolution, consistent with the low-dimensional structure of its Fisher representation.

The Fisher-sphere construction follows generally from the normalization of a positive probability density. The near-small-circle structure, however, is a property of the particular evolution trajectory rather than a consequence of normalization alone. The present results concern the GRV98-based total-valence distribution evolved with the LO nonsinglet kernel over $2 \leq Q^{2} \leq2000\ \mathrm{GeV}^{2}$.

Within the fixed-operator LO formulation, the running coupling controls the relation between the physical scale and the evolution parameter $\tau$. For a fixed initial distribution, this reparametrization leaves the geometric path unchanged. By contrast, changing the initial valence distribution changes the trajectory itself. Separate evolution of $u_{v}$ and $d_{v}$, or alternative valence inputs, can therefore probe the dependence of the observed geometry on the initial partonic structure.

Further extensions involve singlet–gluon evolution, where the coupled matrix kernel introduces additional dynamical components [42], and higher-order splitting functions [43, 44]. Modern global PDF determinations provide additional reference distributions for such comparisons [45, 46, 47].

These developments offer a systematic way to examine how the information geometry changes with the initial state and the evolution dynamics. More broadly, the present construction demonstrates that Fisher–Rao geometry can resolve both the local bending and the global organization of a QCD evolution trajectory, complementing its conventional description through PDFs and Mellin moments.

Summary and Conclusions

We have investigated the information geometry of LO nonsinglet DGLAP evolution by treating normalized valence distributions as points on a Fisher–Rao statistical manifold. Through the isometric embedding $\Phi= 2\sqrt{p}$, the scale evolution defines a trajectory on a Hilbert sphere of radius two. The DGLAP-generated velocity and acceleration provide a direct measure of its geodesic curvature, allowing the geometry of the QCD-selected path to be examined using the standard LO nonsinglet DGLAP kernel.

For the GRV98-based total-valence trajectory over $2 \le Q^2 \le2000\ \mathrm{GeV}^2$, we find that the evolution is not a Fisher–Rao geodesic, but exhibits a nonzero, weakly scale-dependent geodesic curvature. More than 99.999% of the trajectory variance is captured by two affine directions. Their associated plane is displaced from the Fisher-sphere center and defines a small-circle reference geometry. The curvature predicted by this construction closely agrees with that obtained directly from the DGLAP-generated acceleration, while the projected trajectory and accumulated path length are consistent with the corresponding circular geometry. These results characterize the reference evolution as a highly regular, near-small-circle trajectory over the finite scale interval.

The $x$-resolved analysis further shows that Fisher motion reflects the relative reshaping of the normalized PDF, rather than its probability weight alone. Different momentum-fraction regions contribute to the statistical displacement and transverse bending of the trajectory, providing a partonic interpretation of its local geometry. The Mellin representation offers a complementary dynamical perspective: the effective low-dimensional Fisher trajectory is associated with the collective evolution of the full spectral hierarchy, without requiring a truncation to a small number of moments.

The present study thus establishes a quantitative framework for examining the geometry of QCD scale evolution and identifies an organized near-small-circle structure in a representative LO nonsinglet trajectory. Its dependence on the initial parton distribution and evolution dynamics can be explored through alternative valence inputs, singlet–gluon mixing, and higher perturbative orders. More generally, the Fisher–Rao description complements the conventional analysis of PDFs and Mellin moments by characterizing the geometric path selected by QCD evolution.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12547118 and 12605167).

Reparameterization Invariance

We briefly establish the parameterization invariance of the geodesic curvature used in Sec. II.

Consider two monotonic parameters $\lambda$ and $\widetilde{\lambda}$ related by

The derivatives of the Fisher state transform as

Substituting these relations into the transverse geodesic residual defined in Eq. (32), the term proportional to $\mathrm{d}a/\mathrm{d}\lambda$ cancels under the tangential projection. Consequently,

Since $v_{\lambda}^{2} = a^{2}v_{\widetilde{\lambda}}^{2}$, the geodesic curvature and arc-length element satisfy

Thus the transformation from $t = \ln(Q^{2}/Q_{0}^{2})$ to $\tau$ changes the rate of traversal but leaves the geometric path, its accumulated length, and its geodesic curvature unchanged.

Numerical Implementation and Validation

This appendix collects the numerical prescriptions used to construct the reference LO nonsinglet trajectory and its Fisher–Rao observables.

Input distributions and evolution settings

At the original GRV98 LO input scale $Q_{\mathrm{GRV}}^{2} = 0.26\ \mathrm{GeV}^{2}$, we use [38]

The input is evolved to $Q_{0}^{2} = 2\ \mathrm{GeV}^{2}$ before the geometric analysis. The one-loop running coupling follows Eq. (51), with

The active flavor number changes from three to four at $Q^2 = 1.96\ \mathrm{GeV}^2$ and from four to five at $Q^2 = 20.25\ \mathrm{GeV}^2$. The full geometric-analysis interval corresponds to

The reference $x$ grid contains $N_x = 800$ points. Logarithmic spacing is used for $10^{-6} \le x < 10^{-2}$, linear spacing for $10^{-2} \le x < 0.9$, and logarithmic refinement in $1-x$ near the upper endpoint $x_{\max} = 1-10^{-8}$. These regions contain 40%, 35%, and 25% of the grid points, respectively.

Integration on the $x$ grid uses trapezoidal quadrature weights. The convolution uses $N_z = 128$ Gauss–Legendre nodes and interpolation at off-grid arguments $x/z$. The plus-prescription subtraction is implemented directly in the convolution, without an explicit endpoint regulator. Evolution is integrated with a fourth-order Runge–Kutta method using $\Delta\tau_{\max} = 2.5 \times10^{-4}$. The resulting trajectory is sampled at 241 uniformly spaced values of $\tau$.

Discrete normalization and derivatives

Let $\widetilde{\mathcal{L}}_d$ denote the discrete LO convolution operator, and let $w_j$ be the quadrature weights on the represented $x$ interval. Define

For a normalized discrete density, $\sum_j w_jp_j = 1$, the evolution velocity is

This prescription enforces $\sum_j w_jV_j = 0$ on the numerical grid.

Differentiating the same normalized flow gives the discrete second derivative

The Fisher velocity and acceleration are constructed from $V[p]$ and $B[p]$, rather than by finite differences of sampled states.

The correction in Eq. (113) defines a normalization-preserving discrete flow. Its continuum interpretation requires distinguishing the normalization on the represented interval from that of the full distribution on $0 < x < 1$.

Discrete Fisher geometry

The quadrature-weighted Fisher state is represented by

so that ordinary Euclidean inner products of the vectors $Z_i$ reproduce the discretized Fisher–Rao inner product.

The PCA weights are assigned from the accumulated Fisher arc length. Writing $\Delta s_i = s_{i+1}-s_i$ and $L = \sum_i \Delta s_i$, the trapezoidal weights are

They satisfy $\sum_i\omega_i = 1$ and define the weighted covariance operator introduced in Eq. (62).

The characteristic momentum fractions $x_p$ and $x_{\kappa,i}$ are determined from sign changes of the corresponding evolution derivatives, with their zeros interpolated in $\ln x$.

Three-dimensional reconstruction

For Figure 6, we retain the two dominant PCA directions $e_1,e_2$ and the normalized plane-offset direction $\widehat{C}=C/\lVert C\rVert$.

The reconstructed state is

The reconstruction residuals, normalized to the Fisher-sphere radius, are

The corresponding numerical values are reported in Sec. IV E.

Numerical cross-checks

The numerical evolution is benchmarked against the analytic LO nonsinglet solution in Mellin space. For the splitting-kernel convention used here, the anomalous dimensions are

The first five Mellin moments are used in the comparison, including the conserved first moment $\gamma_1^{(0)}=0$.

The discrete normalization is monitored both before and after applying the conservative correction in Eq. (113). The corrected flow preserves the discrete normalization to machine precision, while the uncorrected first-moment drift provides a diagnostic of quadrature and interpolation errors.

Quantity	Reference value
Numerical settings
$N_x$	800
$N_z$	128
$x_{\min}$	$10^{-6}$
$1-x_{\max}$	$10^{-8}$
$\Delta\tau_{\max}$	$2.5 \times10^{-4}$
Number of sampled states	241
$\Delta\tau$ ($Q^2$: 2 $\to$ 2000 GeV$^2$)	0.2365413
Validation diagnostics
Maximum normalization error	$2.22 \times10^{-16}$
Minimum sampled density $p(x,Q^2)$	$3.04 \times10^{-35}$
Maximum uncorrected first-moment drift	$4.88 \times10^{-4}$
Maximum Mellin relative error	$3.46 \times10^{-4}$
Maximum finite-chord curvature mismatch	$3.28 \times10^{-6}$

As a geometric cross-check, the differential geodesic curvature is compared with a finite-chord estimator constructed from neighboring Fisher states. Five sampled states on either side of each central point are used to define the local arc and its endpoint chord. This comparison tests the consistency between the operator-generated derivatives and the geometry of the sampled trajectory.

The numerical settings and principal validation diagnostics are summarized in Table I.

These checks establish the internal consistency of the reference numerical implementation. The reported diagnostics characterize the specified discretization and do not constitute a systematic continuum-extrapolation study.