The energy-momentum tensor of spin-1 hadrons: formalism

Abstract

We provide the complete decomposition of the local gauge-invariant energy-momentum tensor for spin-1 hadrons, including non-conserved terms for the individual parton flavors and antisymmetric contributions originating from intrinsic spin. We state sum rules for the gravitational form factors appearing in this decomposition and provide relations for the mass decomposition, work balance, total and orbital angular momentum, mass radius, and inertia tensor. Generalizing earlier work, we derive relations between the total and orbital angular momentum and the Mellin moments of twist-2 and 3 generalized parton distributions, accessible in hard exclusive processes with spin-1 targets. Throughout the work, we comment on the unique features in these relations originating from the spin-1 nature of the hadron, being absent in the lower spin cases. * wim.cosyn@ugent.be † sabrina.cotogno@polytechnique.edu ‡ afreese@anl.gov § cedric.lorce@polytechnique.edu 1 We call the form factors "gravitational" since the EMT is usually understood as the source of gravitational interactions. However, we do not measure in practice these form factors through gravity and we do not know the exact form that a theory of quantum gravity will take. Thus, it is unclear whether gravitation sees the symmetric, Belinfante-improved EMT as in general relativity or an asymmetric EMT as in Einstein-Cartan theory [3]. Despite this, we refer to all form factors appearing in either the symmetric or asymmetric EMT as "gravitational" form factors.

Poincaré constraints on the gravitational form factors for massive states with arbitrary spin

“…These terms are crucial in the study of the mechanical properties of hadrons, and receive different contributions from quarks and gluons. In particular, a general expression for Ji's relation which is valid for quarks and gluons separately would require the inclusion of such additional terms, as observed in [10,11]. One approach to derive these terms is to write a parametrisation of the EMT for arbitrary spin states as an expansion in terms of spin multipoles 9 .…”

“…(19) it follows immediately that B(0) = 0, and hence with this interpretation the AGM must vanish for massive particles of any spin. However, as previously outlined, this constraint arises purely from the Poincaré invariance of the theory, and does not in fact rely on any knowledge of the external gravitational interactions 10 . Einstein's equivalence principle is therefore not necessary to derive the constraint B(0) = 0.…”

Covariant multipole expansion of local currents for massive states of any spin

“…M ρ∆, ··· = M ρσ, ··· ∆ σ . Note that since covariant multipoles are symmetric under the exchange of bi-indices, the independent contractions can always be put in the canonical form (43). For convenience we shall use the notation…”

Report on scipost_201910_00011v3

“…Furthermore, we know that the energy-momentum tensor (EMT) T µν of the system relates to the gravitational form factors (GFFs) as [24,25]…”

“…where A 0,1 (t), D 0,1 (t), J(t), and E(t) are the six energy-momentum conserved GFFs of the spin-1 system, and · · · denotes the other contributions from the energy-momentum non-conserved form factors. GFFs can be extracted from the moments of GPDs [24,25]. Therefore, we can get the energy-momentum tensor as well as the mechanical properties of the system, like the mass distributions, shear forces, pressures, and the D−term.…”