Hidden self-energy contributions of collinear functions in SCET

TL;DR

This paper investigates how quark self-energy contributions on external legs manifest in SCET under different operator bases. By contrasting a direct-QCD-derived basis with a Wilson-line-based modified basis, it shows that self-energy topologies are explicit in the former but can be hidden in Wilson-line diagrams in the latter, starting at subleading power in $\lambda$. The authors perform a detailed one-loop analysis: full QCD yields $i\mathcal{A}_{QCD}$ for external-leg self-energy, which is reproduced by direct-QCD SCET building blocks, while the modified basis exhibits a mismatch that is resolved only after including Wilson-line contributions that carry hidden self-energy pieces. The results highlight the need for careful treatment of external-leg self-energies in SCET and suggest that similar hidden structures may arise in other EFTs when Wilson lines are used to redefine operator bases.

Abstract

Motivated by the requirement of the LSZ reduction formula to remove self-energy contributions on external legs, we examine quark self-energy contributions in soft-collinear effective (SCET) theory. We examine an operator basis that follows directly from full quantum chromodynamics (QCD) (upon application of the SCET equations of motion to express small Dirac components in terms of large Dirac components). We find that, for this basis, the self-energy contributions can be identified from their diagrammatic topologies, as in full QCD. However, for an alternative operator basis that is obtained from the direct-QCD basis by an application of Wilson-line identities, interactions are shifted from a covariant derivative to a Wilson line. Consequently, some self-energy contributions are hidden in diagrams involving Wilson lines, making their identification subtle.

Figures (4)

• Figure 1: A diagram for a self-energy correction on an outgoing external quark-antiquark (${\cal Q} \bar{\cal Q}$) meson. The solid lines represent a quark or an antiquark, the wavy line represents a gluon, the rectangular region that is labeled "${\cal Q}\bar{\cal Q}$" represents the external ${\cal Q}\bar{\cal Q}$ state, and the circle labeled "hard" represents the hard subdiagram.
• Figure 2: Self-energy Feynman diagrams that could contribute to collinear functions. Note that, in this figure, and throughout this paper, we display Wilson lines explicitly, instead of incorporating them into definitions of operator vertices.
• Figure 3: The Wilson-line diagram that contributes to the quark self-energy part of $F(u_1,u_2)$.
• Figure 4: The Wilson-line diagram that contributes to the quark self-energy part of $G(u_1,u_2)$. The part of the diagram in the cyan dashed box originates from $\bar{\chi}_n(s_1\bar{n})$, while the part of the diagram in the red dashed box originates from the longitudinally polarized gluon field that is contained in $\slashed{\mathcal{G}}_{n\perp}^\dagger(s_1\bar{n})$. The crossed circle represents the contribution from the ordinary derivative in the factor $-i\overleftarrow{D}_{n\perp\nu}$ in ${\cal G}_{n\perp\nu}^\dagger$. This contribution yields, in momentum space, a factor of the negative of longitudinal-gluon momentum that flows into the Wilson-line, which is $-(\slashed{k}_\perp^\nu - \slashed{p}_\perp^\nu)$, in the present case.