Impact of the heavy quark matching scales in PDF fits

TL;DR

This study investigates how displaced heavy-quark matching scales μ_m in the VFNS affect global PDF fits to the combined HERA data, framing μ_m as a theoretical scheme parameter that randomizes the location of threshold discontinuities. Using xFitter with APFEL, it compares NLO and NNLO results, showing that charm-related μ_c induces large χ^2 variations at NLO that are greatly reduced at NNLO, while bottom-related μ_b exhibits milder sensitivity at both orders. The work demonstrates that NNLO reduces scheme dependence, enables shifting thresholds away from data-rich regions, and introduces N_F-dependent PDFs as a flexible tool for multi-scheme analyses. These findings provide practical guidance for robust PDF extractions and offer a quantitative framework to study heavy-flavor threshold implementations in global fits.

Abstract

We investigate the impact of displaced heavy quark matching scales in a global fit. The heavy quark matching scale $μ_{m}$ determines at which energy scale $μ$ the QCD theory transitions from $N_{F}$ to $N_{F}+1$ in the Variable Flavor Number Scheme (VFNS) for the evolution of the Parton Distribution Functions (PDFs) and strong coupling $α_S(μ)$. We study the variation of the matching scales, and their impact on a global PDF fit of the combined HERA data. As the choice of the matching scale $μ_{m}$ effectively is a choice of scheme, this represents a theoretical uncertainty; ideally, we would like to see minimal dependence on this parameter. For the transition across the charm quark (from $N_{F}=3$ to $4$), we find a large $μ_m=μ_{c}$ dependence of the global fit $χ^2$ at NLO, but this is significantly reduced at NNLO. For the transition across the bottom quark (from $N_{F}=4$ to $5$), we have a reduced $μ_{m}=μ_b$ dependence of the $χ^2$ at both NLO and NNLO as compared to the charm. This feature is now implemented in xFitter 2.0.0, an open source QCD fit framework.

Figures (12)

• Figure 1: An illustration of the separate $N_{F}$ renormalization sub-schemes which define a VFNS. Historically, the matching scales $\mu_{m}$ were chosen to be exactly the mass values $m_{c,b,t}$ as in Figure-a. Figure-b is a generalized case where the matching scales $\mu_{m}$ are chosen to be different from the mass values.
• Figure 2: The comparison of the DGLAP evolved PDF $f_b(x,\mu)$ and the perturbatively calculated $\widetilde{f}_b(x,\mu)$ as a function of $\mu$ for selected $x$ values. For $\mu\to m_b$ we find the functions match precisely: $\widetilde{f}_b(x,\mu) \to f_b(x,\mu)$. We have used NNPDF30_lo_as_118_nf_6 as the base PDF set.
• Figure 3: We display the b-quark PDF $x\, f_b^{(5)}(x,\mu)$ for different choices of the matching scales $\mu_{m}=\{m_b/2,m_b,2m_b \}$ (indicated by the vertical lines) computed at NLO (Fig.-a) and NNLO (Fig.-b).
• Figure 4: We display$F_2^b(x,Q)$ for different choices of the matching scales $\mu_{m}=\{m_b/2,m_b,2m_b \}$ (indicated by the vertical lines) computed at NLO (Fig.-a) and NNLO (Fig.-b). Here, we have chosen $\mu=Q$. For details on the FONNL calculation see Ref. Forte:2010ta.
• Figure 5: $\chi^{2}$ vs. the charm matching scale $\mu_{c}$ at a) NLO and b) NNLO for all data sets. The bin boundaries for the HERA data set "HERA1+2 NCep 920" are indicated by the vertical lines.