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Register a new userxFitter 2.0.0: Heavy quark matching scales: Unifying the FFNS and VFNS
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xFitter is an open-source package that provides a framework for the determination of the parton distribution functions (PDFs) of the proton for many different kinds of analyses in Quantum Chromodynamics (QCD). It incorporates experimental data from a wide range of experiments including fixed-target, Tevatron, HERA, and LHC. xFitter version 2.0.0 has recently been released, and offers an expanded set of tools and options. The new xFitter 2.0.0 program links to the APFEL code which has implemented generalized matching conditions that enable the switch from $N_F$ to $N_F+1$ active flavors at an arbitrary matching scale $mu_m$. This enables us to generalize the transition between a FFNS and a VFNS and essentially vary continuously between the two schemes; in this sense the matching scale $mu_m$ allows us to unify the FFNS and VFNS in a common framework. This paper provides a brief overview of xFitter with emphasis of these new features.
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xFitter is an open-source package that provides a framework for the determination of the parton distribution functions (PDFs) of the proton for many different kinds of analyses in Quantum Chromodynamics (QCD). xFitter version 2.0.0 has recently been released, and offers an expanded set of tools and options. It incorporates experimental data from a wide range of experiments including fixed-target, Tevatron, HERA, and LHC. xFitter can analyze this data up to next-to-next-to-leading-order (NNLO) in perturbation theory with a variety of theoretical calculations including numerous methodological options for carrying out PDF fits and plotting tools which help visualize the results. While primarily based on the collinear factorization foundation, xFitter also provides facilities for fits of dipole models and transverse-momentum dependent (TMD) PDFs. The package can be used to study the impact of new precise measurements from hadron colliders, and also assess the impact of future colliders. This paper provides a brief overview of xFitter with emphasis of the new version 2.0.0 features.
We investigate the impact of displaced heavy quark matching scales in a global fit. The heavy quark matching scale $mu_{m}$ determines at which energy scale $mu$ 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 $alpha_S(mu)$. 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 $mu_{m}$ 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 $mu_m=mu_{c}$ dependence of the global fit $chi^2$ 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 $mu_{m}=mu_b$ dependence of the $chi^2$ 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.
The QCD/HQET matching coefficient for the heavy-quark field is calculated up to four loops. It must be finite; this requirement produces analytical results for some terms in the four-loop on-shell heavy-quark field renormalization constant which were previously only known numerically. The effect of a non-zero lighter-flavor mass is calculated up to three loops. A class of on-shell integrals with two masses is analyzed in detail. By specifying our result to QED, we obtain the relation between the electron field and the Bloch--Nordsieck field with four-loop accuracy.
We show that one can re-arrange the Heavy Quark Expansion for inclusive weak decays of charmed hadrons in such a way that the resulting expansion is an expansion in $Lambda_{rm QCD} / m_c$ and $alpha_s (m_c)$ with order-one coefficients. Unlike in the case of the bottom quark, the leading term includes not only the contribution of the free-quark decay, but also a tower of terms related to matrix elements of four quark operators.
We show how to obtain a ``heavy meson effective lagrangian for the case where the number of heavy particles is not conserved. We apply the method in a simple example at tree level and perform then the reduction for the case of vector mesons in Chiral Perturbation Theory in a manifestly chiral invariant fashion. Some examples showing that ``heavy meson effective theory also works at the one-loop level are included.
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