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Different decompositions of the nucleon mass, in terms of the masses and energies of the underlying constituents, have been proposed in the literature. We explore the corresponding sum rules in quantum electrodynamics for an electron at one-loop order in perturbation theory. To this aim we compute the form factors of the energy-momentum tensor, by paying particular attention to the renormalization of ultraviolet divergences, operator mixing and scheme dependence. We clarify the expressions of all the proposed sum rules in the electron rest frame in terms of renormalized operators. Furthermore, we consider the same sum rules in a moving frame, where they become energy decompositions. Finally, we discuss some implications of our study on the mass sum rules for the nucleon.

We discuss the twist-three, unpolarized, chiral-odd, transverse momentum dependent parton distribution (TMD) e q (x, k ⊥ ) within a light-front model. We review a model-independent decomposition of this TMD, which follows from the QCD equations of motion and is given in terms of a leading-twist mass term, a pure interaction-dependent contribution, and singular terms. The leading-twist and pure twist-three terms are represented in terms of overlap of light-front wave functions (LFWFs), taking into account the Fock states with three valence quark (3q) and three-quark plus one gluon (3q + g). The 3q and 3q + g LFWFs with total orbital angular momentum zero are modeled using a parametrization derived from the conformal expansion of the proton distribution amplitudes, with parameters fitted to reproduce available phenomenological information on the unpolarized leading-twist quark and gluon collinear parton distributions. Numerical predictions for both the quark TMD e q (x, k ⊥ ) and the collinear parton distribution e q (x) are presented, discussing the role of the quark-gluon correlations in the proton. * Electronic address: barbara.pasquini@pv.infn.it † Electronic address: simone.rodini01@ateneopv.it (LFWFs), that provide a convenient framework for modelling parton distribution functions [38][39][40]. We focus on the twist-three, unpolarized, chiral-odd quark TMD e q (x, k ⊥ ) [6], which is constructed as overlap integrals between LFWFs with the minimum (valence) and next-to-minimum (one extra gluon) parton content. The LFWFs are modeled in the same spirit of Ref. [41], where the calculation was restricted to the leading-twist PDFs and to the twist-three polarized structure function g q 2 (x). In particular, we consider only the Fock states with zero partons' orbital angular momentum, which are expected to be the dominant contribution for unpolarized distribution functions. These LFWF components for the 3q and 3q + g Fock states are related, in the light-front limit (zero transverse separation), to the nucleon twist-three and twist-four distribution amplitudes (DAs), respectively. We then use the lattice and QCD sum rule results for the proton DAs as guideline to parametrize the dependence of the LFWFs on the longitudinal momenta of the partons. For the dependence on the partons' transverse momentum k ⊥ we adopt a Gaussian form, modified according to the Brodsky-Huang-Lepage prescription [42] to take into account a non-vanishing mass of the partons. The mass of the partons along with the other parameters modelling the proton DAs are then fitted to reproduce the results for the quark and gluon unpolarized twist-two PDFs from available phenomenological parametrizations. Having specified the LFWFs, we calculate the e q (x, k ⊥ ) TMD and e q (x) PDF, discussing the role of the twist-two and the genuine twist-three contributions, and compare our predictions with available phenomenological information.The work is organized as follows: in Sec. II we introduce the definitions of the unpolarized twist-two TMDs of quark and g...

We review and examine in detail recent developments regarding the question of the nucleon mass decomposition. We discuss in particular the virial theorem in quantum field theory and its implications for the nucleon mass decomposition and mechanical equilibrium. We reconsider the renormalization of the QCD energy-momentum tensor in minimal-subtraction-type schemes and the physical interpretation of its components, as well as the role played by the trace anomaly and Poincaré symmetry. We also study the concept of “quantum anomalous energy” proposed in some works as a new contribution to the nucleon mass. Examining the various arguments, we conclude that the quantum anomalous energy is not a genuine contribution to the mass sum rule, as a consequence of translation symmetry.

We present an in-depth analysis of transverse momentum dependent (TMD) distributions of twist-three. In particular, we focus on evolution equations, symmetry relations, parameterization, interpretation, small-b asymptotic behaviour and the structure of singularities. The starting point of discussion are the correlators with the definite TMD-twist. By considering suitable combinations of these correlators, we introduce physical TMD distribution of twist-three that can be used for practical applications. We also establish relations with generic TMD distribution of twist-three, and demonstrate that their evolution equations are autonomous in the large-Nc limit.

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Different decompositions of the nucleon mass, in terms of the masses and energies of the underlying constituents, have been proposed in the literature. We explore the corresponding sum rules in quantum electrodynamics for an electron at one-loop order in perturbation theory. To this end we compute the form factors of the energy-momentum tensor, by paying particular attention to the renormalization of ultraviolet divergences, operator mixing and scheme dependence. We clarify the expressions of all the proposed sum rules in the electron rest frame in terms of renormalized operators. Furthermore, we consider the same sum rules in a moving frame, where they become energy decompositions. Finally, we discuss some implications of our study on the mass sum rules for the nucleon.