NNLO evolution of deep-inelastic structure functions: the singlet case

TL;DR

This work extends perturbative QCD analysis to NNLO for flavour-singlet parton densities and structure functions in deep-inelastic scattering by leveraging partial three-loop splitting-function information to construct accurate $x$-dependent parametrizations and to quantify residual uncertainties. It provides compact two-loop singlet coefficient-function expressions and develops approximate three-loop splitting-function representations with quantified errors, enabling NNLO evolution of $oldsymbol{ m extstyle extstyle extSigma}$ and $g$ alongside $F_{2,S}$. Numerical results show NNLO corrections stabilize scale dependence and reduce theoretical uncertainty across a broad $x$ range, with particularly large coefficient-function-driven effects at high $x$ and gluon-dominated effects at small $x$, thereby improving determinations of quark and gluon densities. The study also discusses scheme choices, the limitations at very small $x$, and supplies practical Fortran subroutines to implement the parametrizations for phenomenological analyses.

Abstract

We study the next-to-next-to-leading order (NNLO) evolution of flavour singlet parton densities and structure functions in massless perturbative QCD. Present information on the corresponding three-loop splitting functions is used to derive parametrizations of these quantities, including Bjorken-x dependent estimates of their residual uncertainties. Compact expressions are also provided for the exactly known, but in part rather lengthy two-loop singlet coefficient functions. The size of the NNLO corrections and their effect on the stability under variations of the renormalization and mass-factorizations scales are investigated. Except for rather low values of the scales, the residual uncertainty of the three-loop splitting functions does not lead to relevant effects for x > 10^-3. Inclusion of the NNLO contributions considerably reduces the theoretical uncertainty of determinations of the quark and gluon densities from deep-inelastic structure functions.

Figures (14)

• Figure 1: Approximations of the $N_f^1$ part $P_{qg,1}^{(2)}$ of the three-loop splitting function $P_{qg}^{(2)}(x)$ in Eq. (2.8), as obtained from the four moments (4.1) by means of Eqs. (4.9) and (4.12). The full and dashed curves represent the functions selected for further consideration. The upper group of curves in the right plot is for $\lambda=0$ in Eq.(4.12), the lower group for $\lambda = 4$.
• Figure 2: Left: as Fig. 1, but for the $N_f^2$ contribution $P_{qg,2}^{(2)}(x)$ using Eqs. (4.9) and (4.13). Right: convolution of the approximations A--D of Fig. 1, combined for $N_f = 4$ with the result selected for $P_{qg,2}^{(2)}$ in the left part, with a typical gluon density of the proton.
• Figure 3: The large-$x$ behaviour of our selected approximations of the three-loop splitting functions $P_{ij}^{(2)}(x)$ for $N_f=4$. $P_{qq}^{(2)}$ is obtained by adding the pure singlet (PS) term (also shown separately) to the non-singlet contribution of ref. NV1 according to Eq. (2.5).
• Figure 4: As Fig. 3, but for the small-$x$ behaviour of $P_{ij}^{(2)}(x)$. The difference between $P_{qq}^{(2)}$ and $P_{\rm PS}^{(2)}$ is negligible for $x < 0.1$, hence the latter quantity is not separately shown here.
• Figure 5: The perturbative expansion of the scale derivative, $\dot{\Sigma} \equiv d \ln \Sigma / d\ln \mu_f^2$, of the singlet quark density at $\mu_f^2 = \mu_{f,0}^2 \simeq 30 \hbox{GeV}^2$. The initial conditions are specified in Eqs. (5.1) and (5.2).