Effective field theory approach to heavy quark fragmentation

TL;DR

The paper develops a comprehensive EFT framework combining SCET and bHQET to extract the $b$-quark fragmentation function from $e^+e^-$ data at the $Z$ pole. It achieves NNLO fixed-order accuracy with NNLL resummation of DGLAP logs and approximate NNNLL resummation of endpoint logs, enabling a controlled separation of perturbative and nonperturbative effects via a shape-function formalism. The analysis demonstrates improved theoretical precision over previous work, reveals convergence of the expansion, and highlights potential HQET power corrections in the tail region. The results are consistent with prior extractions while offering a transparent path to include higher-order corrections and to apply the HQFF in collider phenomenology, with the nonperturbative aspects presented in a renormalon-subtracted, short-distance-mass framework.

Abstract

Using an approach based on Soft Collinear Effective Theory (SCET) and Heavy Quark Effective Theory (HQET) we determine the $b$-quark fragmentation function from electron-positron annihilation data at the $Z$-boson peak at next-to-next-to leading order, with next-to-next-to leading log resummation of DGLAP logarithms, and next-to-next-to-next-to leading log resummation of endpoint logarithms. This analysis improves, by one order, the previous extraction of the $b$-quark fragmentation function. We find that while the addition of the next order in the calculation does not much shift the extracted form of the fragmentation function, it does reduce theoretical errors indicating that the expansion is converging. Using an approach based on effective field theory allows us to systematically control theoretical errors. While the fits of theory to data are generally good, the fits seem to be hinting that higher order correction from HQET may be needed to explain the $b$-quark fragmentation function at smaller values of momentum fraction.

Figures (12)

• Figure 1: Virtual and real $\mathcal{O}(\alpha_s)$ corrections to the bHQET shape function. Double-dashed lines denote the boosted heavy quark. Springs denote soft-collinear gluons.
• Figure 2: Profile functions. In the left panel (a), the blue and orange lines denote $\mu_H$ and $\mu_J$. In the right panel (b) the green and red lines denote $\mu_M$ and $\mu_S$. We plotted the profile function for the default values of the parameters $e_H$, $e_M$, $\mu_0^j$, $\mu_0^s$, $x_1$ and $x_2$. The points $x_1$ and $x_2$, which mark the transition between different regions, are denoted by vertical dashed lines.
• Figure 3: The differential cross section as a function of $x$: the blue curve is the N${}^2$LO endpoint cross section with N${}^3$LL resummation, the orange curve is the endpoint result at fixed order ( i.e. no resummation), the green curve is the N${}^2$LO QCD cross section, and the red curve is the combined cross section, Eq. \ref{['eq:ext.1']}.
• Figure 4: $\chi^2/dof$ and best fit value of $\lambda$ for the N$^2$LO + N$^3$LL fits to the ALEPH, DELPHI, and SLD data. The black dot denotes the fit with default values of theory parameters.
• Figure 5: Distributions of $\lambda$ and $\sigma_\lambda$ in the N$^2$LO + N$^3$LL fits. Color coding is the same as in Figure \ref{['chi2vslambda']}.