Parton Degrees of Freedom from the Path-Integral Formalism

TL;DR

This paper reframes deep inelastic scattering within the Euclidean path-integral formalism, revealing a threefold gauge-invariant decomposition of parton degrees of freedom into valence, connected sea (CS), and disconnected sea (DS). It develops a path-integral-based operator product expansion that cleanly separates CI (valence+CS) and DI (DS) contributions, introducing distinct operator moments and renormalization/evolution patterns for CS and DS. Crucially, CS behaves like valence under evolution and is identified as the primary source of Gottfried sum-rule violation, while DS remains flavor-singlet and gluon-mixed; together, CS and DS yield different small-$x$ behaviors. The work implies that in the nucleon, $ar{u}$ and $ar{d}$ receive both CS and DS components whereas $ar{s}$ is DS-only, signaling that global parton distribution fits should explicitly account for CS/DS separation to improve high-energy predictions, including those relevant for the LHC.

Abstract

We formulate the hadronic tensor $W_{μν}$ of deep inelastic scattering in the path-integral formalism. It is shown that there are 3 gauge invariant and topologically distinct contributions. Besides the valence contribution, there are two sources for the sea -- one in the connected insertion and the other in the disconnected insertion. The operator product expansion is carried out in this formalism. The operator rescaling and mixing reveal that the connected sea partons evolve the same way as the valence, i.e. their evolution is decoupled from the disconnected sea and the gluon distribution functions. We explore the phenomenological consequences of this classification in terms of the small x behavior, Gottfried sum rule violation, and the flavor dependence. In particular, we point out that in the nucleon $\bar{u}$ and $\bar{d}$ partons have both the connected and disconnected sea contributions; whereas, $\bar{s}$ parton has only the disconnected sea contribution. This difference between $\bar{u} + \bar{d}$ and $\bar{s}$, as far as we know, has not been taken into account in the fitting of parton distribution functions to experiments.

Figures (3)

• Figure 1: Quark skeleton diagrams in the Euclidean path integral formalism for evaluating $W_{\mu\nu}$ from the four-point function defined in Eq. (\ref{['wmunu_tilde']}). These include the lowest twist contributions to $W_{\mu\nu}$. (a) and (b) are connected insertions and (c) is a disconnected insertion.
• Figure 2: Quark skeleton diagrams similar to those in Fig. 1, except that the two current insertions are on different quark lines. They give higher twist contributions to $W_{\mu\nu}$.
• Figure 3: Quark skeleton diagrams in the Euclidean path integral formalism considered in the evaluation of matrix elements for the sum of local operators from the operator product expansion of $J_{\mu}(x) J_{\nu}(0)$. (a), (b) and (c) corresponds to the operator product expansion from Fig. 1(a), 1(b) and 1(c) respectively.