Pressure Inside Hadrons: Criticism, Conjectures, and All That

This paper is a conceptual and technical review of Maxim Polyakov’s influential proposal to interpret the hadronic energy–momentum tensor (EMT) “D-term” form factor, usually denoted as the function of momentum transfer, as encoding mechanical properties of hadrons—specifically spatial distributions of isotropic pressure and shear forces. The central research question is not a new computation of a particular observable, but rather: are the mechanical interpretations of the EMT form factor physically sound, and can the criticisms raised in the literature be reconciled with the interpretation? This matters because the D-term is increasingly used as a bridge between QCD matrix elements and intuitive “inside the hadron” pictures, and because it is a target for lattice QCD and phenomenology (and, indirectly, for experimental extractions of EMT-related quantities).

The authors begin by recalling the formal framework. For a nucleon, the symmetric EMT matrix element is parametrized by gravitational form factors, including the D-term. In the Breit frame, Polyakov defined a static EMT distribution via a three-dimensional Fourier transform of the EMT matrix element, yielding spatial densities energy and stress . For spin-0 and spin- systems, the spatial stress tensor can be decomposed into two radial functions: an isotropic part, interpreted as pressure, and a traceless anisotropic part, interpreted as pressure anisotropy and linked to shear forces. EMT conservation implies a differential constraint between these functions. The D-term is then related to moments of the pressure and shear distributions; in particular, the paper emphasizes that the D-term can be written as an integral weighted by the shear function and equivalently by a second moment of the pressure, provided the integrals converge.

The review then organizes and addresses several classes of criticisms.

First, there is the concern that Breit-frame three-dimensional densities suffer from relativistic recoil corrections and may not admit a clean mechanical interpretation. The authors argue that this is not unique to the pressure interpretation: similar issues arise in the interpretation of electromagnetic form factors as spatial charge distributions. They present several ways the Breit-frame picture can be justified or mitigated, including a phase-space/quasi-probabilistic interpretation, a heavy-target limit, and a more formal justification in the large-c limit. They also mention alternative constructions (infinite-momentum frame or light-front) and the possibility of defining 2D densities there, with 3D densities recoverable after appropriate averaging.

Second, the paper addresses the apparent conflict with thermodynamic intuition: in standard thermodynamics, pressure is typically positive, whereas the pressure distribution inferred from the D-term can be negative. The authors show that this is expected for any bound, closed system due to a relativistic analogue of the virial theorem, encoded in the von Laue condition. EMT conservation implies that the spatial integral of the trace of the spatial stress must vanish, which translates into the statement that the radial moment $${\int }_0^{\infty }dr{r}^{2}p(r)=0.$$ This condition forces the pressure function to change sign (i.e., to have at least one node). The authors connect this to historical ideas of Poincar stresses: without non-electromagnetic stresses, a charged object would not be stable. Thus, negative pressure is not a pathology but a sign that attractive and confining forces must balance repulsive contributions.

Third, the authors discuss mechanical stability more strongly than the von Laue condition. They review a local stability criterion used in the literature: the radial pressure combination $$p_r(r)=p(r)+\frac{2}{3}s(r)\ge 0,$$ which is interpreted as the requirement that the normal stress along the radial direction cannot become mechanically unstable. Under additional assumptions (notably that the relevant densities decay sufficiently fast so that the D-term integral exists), this criterion implies that the D-term must be negative. The paper highlights that this provides a more compelling explanation for the negative sign conjecture than the von Laue condition alone, because the von Laue condition is necessary but not sufficient for stability.

Fourth, the authors address the “negative D-term sign conjecture” itself. They recall that Polyakov and Weiss proved a soft-pion theorem in the chiral limit relating the quark contribution to the pion D-term to the pion gravitational form factor at zero momentum transfer, yielding a total pion D-term of $$D_{\text{pion}}=-1.$$ They note that hadronic models (e.g., bag model, chiral quark-soliton model) and other approaches (lattice QCD, dispersion relations) generally find negative D-terms for hadrons, and that Maxim also predicted negative nuclear D-terms with scaling behavior such as $$D\propto -A^{7/3}$$ within liquid-drop-like reasoning.

A key part of the review is the reconciliation of these expectations with cases where the D-term is found positive. The most prominent example discussed is the hydrogen atom, where the D-term is reported to be positive. The authors argue that this does not invalidate the hadronic pressure interpretation because the sign is sensitive to long-range forces and to whether the mechanical-continuum picture is appropriate. They show that electromagnetic long-range tails can make the D-term ill-defined (divergent) in the strict sense: in a classical proton model including QED, the asymptotic behavior of the shear and pressure functions behaves like a power law, leading to divergent integrals for the D-term. They emphasize that the D-term form factor remains well-defined at nonzero momentum transfer, but its sign and magnitude can differ from the short-range-force case. In hydrogen, the system is overall neutral and the long-range QED singularities from proton and electron contributions cancel, yielding a finite and positive D-term. The authors further stress that comparing atoms and hadrons may amount to comparing “apples and oranges”: hadrons are extremely dense and governed by strong QCD forces, whereas atoms are dilute and effectively described by a single electron wavefunction in a long-range electromagnetic potential.

The paper also provides model-based illustrations. In the bag model, the authors explain how the absence of Poincar stresses leads to an unstable system with positive pressure and an exploding nucleon (the mass minimized by taking the bag radius to infinity). Adding a bag constant introduces negative pressure contributions that restore stability and satisfy the von Laue condition. They also discuss a subtle consistency check involving the EMT form factor bar C : quark contributions alone do not sum to zero, but including the bag contribution restores EMT conservation, and the 3D EMT interpretation provides a way to compute the missing bag contribution.

In terms of methodology, the paper is a narrative review rather than an empirical study: it synthesizes theoretical derivations (EMT decomposition, von Laue condition, local stability arguments), and it uses illustrative model calculations (bag model, chiral quark-soliton model, classical proton model with scalar/vector meson fields plus electromagnetism) to demonstrate how sign patterns and convergence issues arise. The “data” are therefore theoretical constraints and previously published numerical/model results rather than new datasets.

Because no new statistical analysis is performed, the paper’s quantitative content is mostly in the form of explicit formulae and representative numerical values from cited models. For example, in the classical proton model, the von Laue integral contributions from scalar, vector, and electromagnetic fields are given as $${\int }_0^{\infty }dr{r}^{2}p(r{)}_S=-10.916\text{MeV},{\int }_0^{\infty }dr{r}^{2}p(r{)}_V=+10.891\text{MeV},{\int }_0^{\infty }dr{r}^{2}p(r{)}_{em}=+0.025\text{MeV},$$ showing near-cancellation consistent with EMT conservation. The authors also quote a conservative upper bound on the nucleon D-term from long-distance-insensitive chiral arguments, $$D\le -(0.20\pm 0.02),$$ which supports negativity.

Limitations are acknowledged implicitly through the reliance on assumptions: the sign arguments for negative D-terms require that pressure and shear distributions decay sufficiently fast (so that the D-term integral exists) and that the local stability criterion is applicable. The authors also note that the stability criterion is not equivalent to the von Laue condition, and that quantum-field-theoretic justification of some continuum-mechanics assumptions for hadrons (and especially the translation of classical reasoning to QFT) remains an open area.

Practically, the paper’s implications are that the mechanical interpretation of the hadronic EMT stress tensor via the D-term remains viable and is consistent with fundamental constraints (EMT conservation) and with stability-inspired sign expectations, provided one accounts for relativistic-frame issues and, crucially, for the role of long-range forces and convergence. Researchers using D-term-based pressure pictures should be careful when comparing systems with different force ranges (e.g., hadrons vs atoms) and when interpreting the D-term in theories with massless fields (photons) where the strict spatial moments may diverge. The work should matter to QCD phenomenologists, lattice QCD practitioners extracting or constraining D-terms, and to anyone using EMT form factors to infer “internal mechanics” of hadrons, because it clarifies which criticisms are genuine and which are resolved by understanding the domain of validity of the continuum/stability interpretation.

Overall, the paper’s core contribution is a reconciliation: it argues that none of the raised criticisms truly invalidates Polyakov’s mechanical interpretation for hadrons, while also delineating when and why sign expectations can change—especially in the presence of long-range electromagnetic forces and when the D-term moment becomes sensitive to asymptotic tails.