Mechanical properties of proton in the momentum space

Abstract: We study the parametrization of the energy-momentum tensor for the case of a proton in momentum space in terms of gravitational transverse momentum-dependent distributions (TMDs). These gravitational TMDs are investigated with the inclusion of higher-twist contributions to predict the mechanical properties, specifically the transverse pressure and shear force distributions, along with the polarization-dependent $Π^q_S$ and $Π^q_A$ terms. The corresponding distributions are computed individually for both $u$ and $d$ quark flavors. The calculations have been performed in the light-cone framework using the spectator diquark model. A strong binding contribution to the transverse pressure is observed in the low-momentum space for both quark flavors of the proton.

• The paper demonstrates that gravitational TMDs enable determination of the proton's mechanical properties, differentiating between u and d quark contributions.
• It employs a spectator diquark model with light-cone wave functions to compute both leading- and higher-twist TMDs, detailing transverse pressure and shear force distributions.
• The analysis highlights distinct flavor and polarization-dependent structures, advancing understanding of nonperturbative QCD effects in hadron mechanics.

Mechanical Properties of the Proton in Momentum Space: TMD EMT Analysis

Overview

This work addresses the theoretical characterization of the proton's mechanical properties in momentum space through the formalism of gravitational transverse momentum-dependent distributions (TMDs). The proton is analyzed at the quark-parton level using a light-cone spectator diquark model, with explicit attention given to higher-twist effects. The study provides a quantitative mapping between the energy-momentum tensor (EMT) decomposed in terms of TMDs and intrinsic mechanical variables, such as transverse pressure, shear force, and additional polarization-dependent components for both $u$ and $d$ quark flavors. Notably, it extends prior TMD applications by systematically considering higher-twist T-odd contributions and establishing concrete numerical predictions for momentum-space distributions.

Theoretical Framework

The EMT operator for quarks in the proton is parametrized in the context of QCD via gravitational TMDs—quantities linked to generalized, gauge-invariant canonical (gic) versions of the quark EMT. The gic EMT enables a transparent connection to canonical momentum and, critically, a decomposition into 22 independent TMDs: 10 unpolarized, 16 transverse, and 6 longitudinal polarization-dependent distributions. This construction incorporates not only the leading twist (twist-2) but also subleading (twist-3, twist-4) contributions, addressing the longstanding deficiency regarding higher-twist effects in mechanical observables.

The mapping between these TMDs and mechanical variables, such as the transverse pressure $\sigma^q$, shear force, and the polarization-dependent quantities $\Pi^q_S$ and $\Pi^q_A$, is established by explicit tensor decomposition of the EMT in momentum space. These relations, in particular the dependence on intrinsic transverse momentum $\mathbf{k}_\perp$ and longitudinal momentum fraction $x$, allow the study of detailed spatial and dynamical correlations.

The computation utilizes a light-cone spectator model for the proton, incorporating both scalar and axial-vector diquark configurations. Model parameters are extracted from a fit to the unpolarized TMD $f_1^q$, ensuring normalization and phenomenological consistency. T-odd TMDs, sensitive to final-state interactions (FSI), are generated via an explicit one-gluon exchange kernel.

Results: Flavor-Decomposed Mechanical Distributions

Transverse Pressure Behavior

The transverse pressure distribution $\sigma^q$ is evaluated as a function of $\mathbf{k}_\perp$ for fixed $d$0 and as a function of $d$1 for fixed $d$2 values. For both $d$3 and $d$4 quarks, the pressure exhibits a pronounced peak at low $d$5, situated in the negative region, indicative of a confining (attractive) force.

Figure 1: The transverse pressure distribution $d$6 of (a) $d$7 and (b) $d$8 quark flavors of proton as a function of transverse momentum (GeV) at fixed values of $d$9.

The magnitude of the confining pressure is significantly greater for $\sigma^q$0 quarks relative to $\sigma^q$1 quarks and decreases with increasing $\sigma^q$2, with the distribution's peak shifting toward smaller $\sigma^q$3. This implies a stronger binding in the low-momentum region, particularly for $\sigma^q$4 quarks—a nontrivial flavor asymmetry not captured in leading-twist-only analyses.

When assessed as a function of $\sigma^q$5, at fixed low values of $\sigma^q$6, the transverse pressure remains negative and vanishes as $\sigma^q$7. For $\sigma^q$8 quarks, the distribution saturates and vanishes faster than for $\sigma^q$9 quarks, reinforcing the dominance of $\Pi^q_S$0-quark binding effects across a broad $\Pi^q_S$1 range.

Figure 2: The transverse pressure distribution $\Pi^q_S$2 of (a) $\Pi^q_S$3 and (b) $\Pi^q_S$4 quark flavors of proton as a function of $\Pi^q_S$5 at fixed values of transverse momentum (GeV).

Shear Force and Higher-Twist Polarization Structures

The study examines the polarization-dependent EMT components, specifically $\Pi^q_S$6, which isolate T-odd, twist-3 contributions. The distribution of $\Pi^q_S$7 for $\Pi^q_S$8 quarks is positive at small $\Pi^q_S$9, crosses zero, and becomes negative at larger $\Pi^q_A$0; for $\Pi^q_A$1 quarks, the sign is inverted, with a negative peak at low $\Pi^q_A$2 followed by a positive maximum.

Figure 3: The distribution of $\Pi^q_A$3 of (a) $\Pi^q_A$4 and (b) $\Pi^q_A$5 quark flavors of proton as a function of transverse momentum (GeV) at fixed values of $\Pi^q_A$6.

The nodal structure observed for $\Pi^q_A$7 quark $\Pi^q_A$8 highlights nontrivial interference patterns between low- and high-momentum domains, sensitive to the underlying FSI dynamics. The $\Pi^q_A$9-dependence shows that both flavors' $\mathbf{k}_\perp$0 vanish in the $\mathbf{k}_\perp$1 limit, but the sign and location of zero crossings are strongly flavor and $\mathbf{k}_\perp$2 dependent.

Figure 4: The distribution of $\mathbf{k}_\perp$3 of (a) $\mathbf{k}_\perp$4 and (b) $\mathbf{k}_\perp$5 quark flavors of proton as a function of $\mathbf{k}_\perp$6 at fixed values of transverse momentum (GeV).

These higher-twist, polarization-dependent distributions reflect the intricate T-odd structure governed by gluon exchange mechanisms and are directly related to the mechanical response of the proton under external perturbations.

Implications and Future Perspectives

The formalism advanced in this work enables a systematic analysis of the proton's internal mechanical landscape in the three-dimensional momentum domain. The explicit links drawn between gravitational TMDs and mechanical observables provide a robust framework to interpret the dynamical content of the EMT, going beyond the constraints of leading-twist density interpretations.

The strong flavor-separation observed in the low-$\mathbf{k}_\perp$7 pressure distributions implies that future phenomenology—particularly lattice QCD and experimental extractions of TMD-sensitive observables—must account for substantial $\mathbf{k}_\perp$8 asymmetries inherent in the mechanical sector. The quantification of higher-twist (especially T-odd) contributions suggests that deeply virtual, semi-inclusive, or longitudinally polarized processes could provide indirect constraints on the mechanical properties via their sensitivity to gravitational TMDs.

Despite the non-direct measurability of momentum-space EMT components, the developed methodology paves the way for a deeper understanding of the correlations between QCD binding, hadron structure, and the space-momentum duality of mechanical variables. Extensions to include strangeness, sea quarks, and gluonic contributions are natural next steps. Furthermore, integration with lattice QCD calculations of off-forward matrix elements of the EMT may ultimately synergize model-driven insight with nonperturbative ab initio results.

Conclusion

This study presents a comprehensive mapping of the proton's mechanical properties in the momentum domain through the computation of gravitational TMDs, emphasizing the decisive roles of higher-twist and T-odd effects. Key findings underline the dominance of $\mathbf{k}_\perp$9 quark contributions to transverse confinement, significant flavor dependence, and intricate polarization-dependent structures. The formalism and results establish a foundation for future exploration of the dynamical QCD origins of hadronic mechanical stability.