NNLO electron structure functions (PDFs) from SCET

Abstract

We calculate the electron structure functions, aka parton distribution functions (PDFs), to NNLO in QED. The calculation is based on the definition of the PDFs in terms of operator matrix elements in soft collinear effective theory (SCET) and directly performed in momentum space. The electron PDFs describe the universal effects of collinear initial state radiation (ISR) off the electron (or positron) in DIS or $e^+ e^-$ collision events. The parton collinear to the electron that enters the hard scattering process governed by the energy scale $Q$ can be a photon or a (anti)fermion with mass $m_f \sim m_e \ll Q$. Our SCET momentum-space calculation confirms earlier results obtained from a QED computation in Mellin space and extends them by taking more than one massive fermion flavor into account. The well-known factorization of sufficiently inclusive cross sections into PDFs with massive fermions and hard partonic cross sections with massless fermions at leading order in $m_e/Q$ is discussed from the SCET perspective. Analogies to the nonperturbative PDFs and collinear factorization in QCD are pointed out. Based on this factorization DGLAP-type resummation of large logarithms of the ratio $m_e/Q$ is straightforward.

Figures (4)

• Figure 1: Illustration of the hard matching calculation to determine the Wilson coefficient ($C$) of the hard-scattering operator in SCET. The LHS represents a squared full-QED matrix element where the (hard) integration over the full phase space of an arbitrary number of unresolved partons (symbolized by the blue ellipse) is carried out. The closed lines crossing the final state cut (indicated by the little tilted double lines) correspond to fragmenting partons with fixed momenta ($\sim Q$ in directions $n_i$). All external and cut lines are on-shell and do not emit (soft or collinear) radiation. LO contributions and virtual (loop) corrections to the scattering amplitude and its complex conjugate are depicted as gray circles. The RHS represents the corresponding forward-scattering matrix element with one insertion of the hard-scattering operator (black circle) at leading power in SCET. The labels ($n$, $\bar{n}$, $n_i$) denote the relevant collinear directions. For leading-power matching the LHS may be evaluated with $m_f=0$ (i.e. in the hard region) and thus amounts to our definition of the (bare) massless partonic cross section for a given parton channel, i.e. a given set of partons involved in the hard scattering process.
• Figure 2: Two-loop diagrams with a photon self-energy bubble. The black lines represent (w.l.o.g.) massive electrons, the blue lines represent massive fermions of arbitrary flavor. The $\otimes$ vertices symbolize the two field operators inside the PDF operator including the collinear Wilson lines according to eq. \ref{['eq:chiBperp']}, which can directly couple to photons as in diagrams a,c,f,g. Note that we draw the $\otimes$ vertices (unlike e.g. ref. Stewart:2010qs) with a gap between them, although they represent a local composite operator in SCET. This is to emphasize that the PDF can be interpreted in such a way that the two $\otimes$ vertices lie on the different sides of the final state cut. In this sense, diagram a yields a virtual correction, diagrams b,c represent wave function renormalization corrections, and diagrams e-g provide real corrections to $f_{e/e}^{(2)}$. Diagram d contributes to $f_{\gamma/e}^{(2)}$. Left-right mirror graphs (with adapted fermion flow) are not shown, but understood to contribute equally.
• Figure 3: Diagrams without photon self energy, but at least one real emission, contributing to $f_{e/e}^{(2)}$ (diagrams a-e), $f_{\bar{e}/e}^{(2)}$ (diagram f), and $f_{f/e}^{(2)}$ (diagram g). In contrast to figure \ref{['fig:DiagsBub']}, we only show diagrams that are nonzero in $n\cdot A_n=0$ (light-cone) gauge, where the Wilson lines in the PDF operators equal unity. In covariant gauges there are many more. Lines and symbols have the same meaning as in figure \ref{['fig:DiagsBub']} and we again do not show left-right mirror graphs, which yield equal contributions.
• Figure 4: One-loop diagrams contributing to $f_{e/e}^{(1)}$ and $f_{\gamma/e}^{(1)}$. Graphs with photons attached to Wilson lines are not shown.