Factorization of Twist-Four Gluon Operator Contributions

TL;DR

The paper extends the Ellis–Furmanski–Petronzio framework to gluonic twist-four contributions in deep inelastic scattering by analyzing diagrams with up to four t-channel gluons. It develops a gauge-invariant factorization where twist-four gluon-operator matrix elements couple to gluonic coefficient functions, using Ward identities and axial-gauge techniques to rewrite A-operators in terms of F^{μν}. The main results provide explicit gauge-invariant expressions for the two-, three-, and four-gluon contributions (H1, H2, H3), along with symmetry relations and a detailed treatment of the color structure, laying the groundwork for computing the relevant quark-loop amplitudes R and connecting to the OPE operator basis. The work clarifies the role of gluonic operators at twist four, discusses limitations, and outlines concrete steps needed to complete a phenomenological twist-four analysis, including operator mixing and anomalous dimensions in the gluon sector.

Abstract

We consider diagrams with up to four t-channel gluons in order to specify gluonic twist-four contributions to deep inelastic structure functions. This enables us to extend the method developed by R.K.Ellis, W.Furmanski, and R.Petronzio (EFP) to the gluonic case. The method is based on low-order Feynman diagrams in combination with a dimensional analysis. It results in explicitly gauge invariant expressions for the factorization of twist-four gluon-operator matrix elements and the corresponding coefficient functions.

Figures (4)

• Figure 1: For a systematic analysis of the twist-four contributions of gluonic operators it is sufficient to consider the contributions of the three diagrams shown here. The quark-loops symbolize the coupling of the gluons in all possible ways and the propagators of the gluon-lines belong to the lower blobs. We take all gluon momenta flowing upwards, i.e. into the quark-loops.
• Figure 2: Sample diagrams which contain three- and four-gluon vertices.
• Figure 3: Two sample diagrams which enter into $R_{\mu\sigma\nu}^{abc}$ at leading order in $\alpha_\mathrm{s}$. With an adequate choice of the loop integration variable the traces over Dirac-matrices give the same (up to the sign) momentum dependence for both diagrams. The first diagram contributes with a color factor $\mathop{\mathrm{tr}}\nolimits t^at^bt^c$ while the second one contributes with a color factor $\mathop{\mathrm{tr}}\nolimits t^ct^bt^a$.
• Figure 4: Two sample diagrams which enter into $R_{\mu\sigma\lambda\nu}^{abcd}$. They are identical if one sets $k_1=k_2'-k_1'$, $k_2=k_2'$, $k_3=k_3'$ and exchanges $\mu\leftrightarrow\sigma$ and $a\leftrightarrow b$.