Heavy Quark Threshold Research Articles

An improved description of charm fragmentation data

We consider the fragmentation of heavy quarks into heavy-flavoured hadrons, specifically the production of charmed mesons in e+e-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$e^+e^-$$\\end{document} collisions, at different centre-of-mass energies. We focus our attention on the ratio of moments of the D∗+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{*+}$$\\end{document} energy spectrum measured by ALEPH and CLEO. This ratio is believed to provide us with a direct test of perturbative QCD evolution because hadronisation effects should cancel between the numerator and denominator. However, state-of-the-art calculations based on standard (final-state) collinear factorisation fail to describe the experimental data. We show that this discrepancy is considerably reduced if heavy-quark threshold effects are accounted for not only in DGLAP evolution, as it is usually done, but also in the resummed coefficient functions.

Low-energy matrix elements of heavy-quark currents

In QCD at energies well below a heavy-quark threshold, the heavy-quark vector current can be represented via local operators made of the lighter quarks and of the gluon fields. We extract the leading perturbative matching coefficients for the two most important sets of operators from known results. As an application, we analytically determine the O(αs3)mc2/mb2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{O}(\\alpha _s^3)m_c^2/m_b^2$$\\end{document} effect of the bottom quark current on the R(s) ratio below the bottom but above the charm threshold. For the low-energy representation of the charm quark current, the two most important operators are given by the total divergence of dimension-six gluonic operators. We argue that the charm magnetic moment of the nucleon is effectively measuring the forward matrix elements of these gluonic operators and predict the corresponding bottom magnetic moment. Similarly, the contribution of the charm current to R(s≈1GeV2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(s\\approx 1\\,\ extrm{GeV}^2)$$\\end{document}, which is associated with quark-disconnected diagrams, is dominantly determined by the decay constants of the ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} and ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document} mesons with respect to the two gluonic operators.

Effects of the quark flavour thresholds in the hadronic vacuum polarization contributions to the muon anomalous magnetic moment

The equivalent representations for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment in the presence of the quark flavour thresholds are studied. Specifically, the explicit relations between the contributions given by the integration over a finite kinematic interval to expressed in terms of the hadronic vacuum polarization function, Adler function, and the R-ratio of electron–positron annihilation into hadrons are derived. It is shown that the quark flavour thresholds of the hadronic vacuum polarization function generate additional contributions to expressed in terms of the Adler function and the R-ratio and the explicit expressions for such contributions are obtained. The commonly employed dispersion relations, which bind together hadronic vacuum polarization function, Adler function, and R-ratio, are extended to account for the effects due to the quark flavour thresholds. The examined additional contributions due to the heavy quark thresholds to expressed in terms of the R-ratio appear to be quite sizable, that can be of a particular relevance for the data-driven method of assessment of the hadronic part of the muon anomalous magnetic moment.

Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops

We obtain generalised power series expansions for a family of planar two-loop master integrals relevant for the QCD corrections to Higgs + jet production, with phys- ical heavy-quark mass. This is achieved by defining differential equations along contours connecting two fixed points, and by solving them in terms of one-dimensional generalised power series. The procedure is efficient, and can be repeated in order to reach any point of the kinematic regions. The analytic continuation of the series is straightforward, and we present new results below and above the physical thresholds. The method we use allows to compute the integrals in all kinematic regions with high precision. For example, per- forming a series expansion on a typical contour above the heavy-quark threshold takes on average O(1 second) per integral with worst relative error of O(10−32), on a single CPU core. After the series is found, the numerical evaluation of the integrals in any point of the contour is virtually instant. Our approach is general, and can be applied to Feynman integrals provided that a set of differential equations is available.

Comments on Global Parton Analyses

We discuss the causes which can limit the accuracy of the predictions based on the conventional PDFs when including in global parton analyses the data at moderate scales $\mu$. The first is the existence of power corrections ${\cal O} (Q_0^2/\mu^2)$ due to the double counting of contributions arising from the region below the input scale $Q_0$. The second concerns the possible inclusion of the BFKL re-summation of the $(\alpha_s\ln (1/x))^n$ terms. The third is the treatment of the heavy-quark thresholds. We show how to include the heavy-quark masses ($m_h$ with $h=c,b,t$) in DGLAP evolution which provides the correct {\it smooth} behaviour through the threshold regions and how to subtract the low parton virtuality $|k^2|<Q^2_0$ contributions from the DIS and Drell-Yan NLO coefficient functions in order to avoid the double counting.

Heavy quark-philic scalar dark matter with a vector-like fermion portal

The absence of confirmed signal in dark matter (DM) direct detection (DD) may suggest weak interaction strengths between DM and the abundant constituents inside nucleon, i.e. gluons and valence light quarks. In this work we consider a real scalar dark matter S interacting only with SU(2)L singlet Up-type quarks Ui = uR,cR,tR via a vector-like fermion ψ which has the same quantum number as Ui. The DM-nucleon scattering can proceed through both h-mediated Higgs portal (HP) and ψ-mediated vector-like portal (VLP), in which HP can receive sizable radiative corrections through the new fermions. We first study the separate constraints on the new Yukawa couplings yi and find that the constraints of XENON1T results are strong on y1 from VLP scattering and on y3 from its radiative contributions to HP scattering. Since both DM-light quark interactions and HP have been well studied in the existing literature, we move forward to focus on DM-heavy quark interactions. Since there is no valence c,t quark inside nucleons at μhad ∼ 1 GeV, y2,y3 interactions are manifested in DM-gluon scattering at loop level. We find that renormalization group equation (RGE) and heavy quark threshold effects are important if one calculates the DM-nucleon scattering rate σSIp at μhad ∼ 1 GeV while constructing the effective theory at μEFT ∼ mZ. For the benchmarks y3 = 0.5, y2 = 0.5, 1, 3, combined results from ΩDM h2 ≃ 0.12, XENON1T, Fermi-LAT, 13 TeV LHC data have almost excluded mS < mt/2 when only DM-{c,t} interactions are considered. FCNC of top quark can be generated at both tree level t → ψ(*)S → cSS and loop level t → c+γ/g/Z, of which the branching fractions are typically below 10−9 after passing the other constraints, which are still safe from the current top quark width measurements.

Standard Model anatomy of WIMP dark matter direct detection. II. QCD analysis and hadronic matrix elements

Models of Weakly Interacting Massive Particles (WIMPs) specified at the electroweak scale are systematically matched to effective theories at hadronic scales where WIMP-nucleus scattering observables are evaluated. Anomalous dimensions and heavy quark threshold matching conditions are computed for the complete basis of lowest-dimension effective operators involving quarks and gluons. The resulting QCD renormalization group evolution equations are solved. The status of relevant hadronic matrix elements is reviewed and phenomenological illustrations are given, including details for the computation of the universal limit of nucleon scattering with heavy $SU(2)_W\times U(1)_Y$ charged WIMPs. Several cases of previously underestimated hadronic uncertainties are isolated. The results connect arbitrary models specified at the electroweak scale to a basis of $n_f=3$ flavor QCD operators. The complete basis of operators and Lorentz invariance constraints through order $v^2/c^2$ in the nonrelativistic nucleon effective theory are derived.

Treatment of heavy quarks in QCD

We show that to correctly describe the effects of the heavy-quark mass, m_h, in DGLAP evolution, it is necessary to work in the so-called `physical' scheme. In this way, we automatically obtain a smooth transition through the heavy-quark thresholds. Moreover, we show that to obtain NLO accuracy, it is sufficient to account for the heavy-quark mass, m_h, just in the LO (one-loop) splitting function. The use of the MS-bar factorisation scheme is not appropriate, since at NLO we deal with a mixture of quarks and gluon (that is, the mass of the heavy parton is not well-defined). The formulae for the explicit m_h dependence of the splitting functions, and for alpha_s, are presented.

General-mass treatment for deep inelastic scattering at two-loop accuracy

We present a next-to-next-to-leading order (NNLO) realization of a general quark mass scheme (S-ACOT-$\ensuremath{\chi}$) in deep inelastic scattering and explore the impact of NNLO terms on heavy-quark structure functions ${F}_{2,L}^{c}(x,Q)$. An amended QCD factorization theorem for DIS is discussed that validates the S-ACOT-$\ensuremath{\chi}$ scheme to all orders in the QCD coupling strength. As a new feature, kinematical constraints on collinear production of heavy quarks that are crucial near the heavy-quark threshold are included in the amended factorization theorem. An algorithmic procedure is outlined for implementing this scheme at NNLO by using mass-dependent and massless results from literature. At two loops in QCD cut diagrams, the S-ACOT-$\ensuremath{\chi}$ scheme reduces scale dependence of heavy-quark DIS cross sections as compared to the fixed-flavor number scheme.

FAPT: A Mathematica package for calculations in QCD Fractional Analytic Perturbation Theory

We provide here all the procedures in Mathematica which are needed for the computation of the analytic images of the strong coupling constant powers in Minkowski (Āν(s;nf) and Aνglob(s)) and Euclidean (Āν(Q2;nf) and Aνglob(Q2)) domains at arbitrary energy scales (s and Q2, correspondingly) for both schemes — with fixed number of active flavours nf=3,4,5,6 and the global one with taking into account all heavy-quark thresholds. These singularity-free couplings are inevitable elements of Analytic Perturbation Theory (APT) in QCD, proposed in [10,69,70], and its generalization — Fractional APT, suggested in [42,46,43], needed to apply the APT imperative for renormalization-group improved hadronic observables. Program summaryProgram title: FAPTCatalogue identifier: AENJ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AENJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 1985No. of bytes in distributed program, including test data, etc.: 1895776Distribution format: tar.gzProgramming language: Mathematica.Computer: Any work-station or PC where Mathematica is running.Operating system: Windows XP, Mathematica (versions 5 and 7).Classification: 11.5.Nature of problem:The values of analytic images Āν(Q2) and Āν(s) of the QCD running coupling powers αsν(Q2) in Euclidean and Minkowski regions, correspondingly, are determined through the spectral representation in the QCD Analytic Perturbation Theory (APT). In the program FAPT we collect all relevant formulas and various procedures which allow for a convenient evaluation of Āν(Q2) and Āν(s) using numerical integrations of the relevant spectral densities.Solution method:FAPT uses Mathematica functions to calculate different spectral densities and then performs numerical integration of these spectral integrals to obtain analytic images of different objects.Restrictions:It could be that for an unphysical choice of the input parameters the results are without any meaning.Running time:For all operations the run time does not exceed a few seconds. Usually numerical integration is not fast, so that we advise the use of arrays of precalculated data and then to apply the routine Interpolate(as shown in supplied example of the program usage, namely in the notebook FAPT_Interp.nb).

DIS heavy-flavor contributions at two loops in a general mass scheme

We discuss an NNLO realization of the general mass scheme S-ACOT-χ for treatment of heavy-flavor production in neutral current deep-inelastic scattering. The connection of this scheme to the QCD factorization theorem with massive quarks is elucidated. As a new feature, kinematical constraints on collinear production of heavy quarks that are crucial near the heavy-quark threshold are included in the amended QCD factorization theorem. Practical implementation of the NNLO calculation is illustrated on the example of semi-inclusive structure functions F2c(x, Q) and FLc(x, Q).

Resummation in QCD fractional analytic perturbation theory

We describe the generalization of Analytic Perturbation Theory (APT) for QCD observables, initiated by Radyushkin, Krasnikov, Pivovarov, Shirkov and Solovtsov, to fractional powers of coupling -- Fractional APT (FAPT). The basic aspects of FAPT is shortly summarized. We describe how to treat heavy-quark thresholds in FAPT and then show how to resum perturbative series in both the one-loop APT and FAPT. As an application we consider FAPT description of the Higgs boson decay $H^0\to b\bar{b}$. The main conclusion is: To achieve an accuracy of the order of 1% it is enough to take into account up to the third correction.

Threshold resummation for top-pair hadroproduction to next-to-next-to-leading log

We derive the threshold-resummed total cross section for heavy quark production in hadronic collisions accurate to next-to-next-to-leading logarithm, employing recent advances on soft anomalous dimension matrices for massive pair production in the relevant kinematic limit. We also derive the relation between heavy quark threshold resummations for fixed pair kinematics and the inclusive cross section. As a check of our results, we have verified that they reproduce all poles of the color-averaged $q\overline{q}\ensuremath{\rightarrow}t\overline{t}$ amplitudes at two loops, noting that the latter are insensitive to the color-antisymmetric terms of the soft anomalous dimension.

Global fractional analytic perturbation theory in QCD with selected applications

We give the generalization of Fractional Analytic Perturbation Theory (FAPT) for QCD observables, recently developed both for the Euclidean and Minkowski regions of squared momentum transfer q^2, which takes into account heavy-quark thresholds. The original analytic approach to QCD, initiated by Jones, Solovtsov and Shirkov, is shortly summarized. We also shortly consider the basic aspects of FAPT and then concentrate on the accounting for the heavy-quark thresholds problem and the construction of global version of FAPT. We discuss what one should use as an analytic coupling in the timelike region q^2=s>0 for the e^{+}e^{-}-annihilation and the pion form factor, and consider applications to phenomenologically relevant processes (the factorizable part of the pion form factor and the Higgs boson decay into a b\bar{b} pair), as well as to the summation of perturbative series.

A Higher Order Perturbative Parton Evolution Toolkit (HOPPET)

This document describes a Fortran 95 package for carrying out DGLAP evolution and other common manipulations of parton distribution functions (PDFs). The PDFs are represented on a grid in x-space so as to avoid limitations on the functional form of input distributions. Good speed and accuracy are obtained through the representation of splitting functions in terms of their convolution with a set of piecewise polynomial basis functions, and Runge–Kutta techniques are used for the evolution in Q. Unpolarised evolution is provided to NNLO, including heavy-quark thresholds in the MS ¯ scheme, and longitudinally polarised evolution to NLO. The code is structured so as to provide simple access to the objects representing splitting functions and PDFs, making it possible for a user to extend the facilities already provided. A streamlined interface is also available, facilitating use of the evolution part of the code from F77 and C/C++. Program summary Program title: HOPPET Catalogue identifier: AEBZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEBZ_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU Public License No. of lines in distributed program, including test data, etc.: 61 001 No. of bytes in distributed program, including test data, etc.: 270 312 Distribution format: tar.gz Programming language: Fortran 95 Computer: All Operating system: All RAM: ≲10 MB Classification: 11.5 Nature of problem: Solution of the DGLAP evolution equations up to NNLO (NLO) for unpolarised (longitudinally polarised) PDFs, and provision of tools to facilitate manipulation (convolutions, etc.) of PDFs with user-defined coefficient and splitting functions. Solution method: Representation of PDFs on a grid in x, adaptive integration of splitting functions to reduce them to a discretised form, obtaining fast convolutions that are equivalent to integration with an interpolated form of the PDFs; Runge–Kutta solution of the Q evolution, and its caching so as to speed up repeated evolution with different initial conditions. Restrictions: PDFs should be smooth on the scale of the discretisation in x. Running time: A few seconds for initialisation, then ∼10 ms for creating a tabulation with a relative accuracy of 10 −4 from a new initial condition (on a 3.4 GHz Pentium IV processor). Further details in Section 9.2.

Predictions for high energy neutrino cross-sections from the ZEUS global PDF fits

We have updated predictions for high energy neutrino and antineutrino charged current cross-sections within the conventional DGLAP formalism of NLO QCD using a modern PDF fit to HERA data, which also accounts in a systematic way for PDF uncertainties deriving from both model uncertainties and from the experimental uncertainties of the input data sets. Furthermore the PDFs are determined using an improved treatment of heavy quark thresholds. A measurement of the neutrino cross-section much below these predictions would signal the need for extension of the conventional formalism as in BFKL resummation, or even gluon recombination effects as in the colour glass condensate model.

QCD decoupling at four loops

We present the matching condition for the strong coupling constant α s ( n f ) at a heavy quark threshold to four loops in the modified minimal subtraction scheme. Our results lead to further decrease of the theoretical uncertainty of the evolution of the strong coupling constant through heavy quark thresholds. Using a low energy theorem we furthermore derive the effective coupling of the Higgs boson to gluons (induced by a virtual heavy quark) in four- and (partially) through five-loop approximation.

New 5-flavor LO analysis and parametrization of parton distributions in the real photon

New, radiatively generated, leading order (LO) quark $(u,d,s,c,b)$ and gluon densities in a real, unpolarized photon are presented. We perform a global 3-parameter fit, based on LO Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations, to all available data for the structure function ${F}_{2}^{\ensuremath{\gamma}}{(x,Q}^{2}).$ We adopt a new theoretical approach called ACOT(\ensuremath{\chi}), originally introduced for the proton, to deal with the heavy-quark thresholds. This defines our basic model [Cornet-Jankowski-Krawczyk-Lorca (CJKL) model], which gives a very good description of the experimental data on ${F}_{2}^{\ensuremath{\gamma}}{(x,Q}^{2}),$ for both ${Q}^{2}$ and x dependences. For comparison we perform a standard fit using the fixed flavor-number scheme $({\mathrm{FFNS}}_{\mathrm{CJKL}}$ model), updated with respect to the previous fits of this type. We show the superiority of the CJKL fit over the ${\mathrm{FFNS}}_{\mathrm{CJKL}}$ one and other LO fits to the ${F}_{2}^{\ensuremath{\gamma}}{(x,Q}^{2})$ data. The CJKL model gives also the best description of the CERN LEP data on the ${Q}^{2}$ dependence of the ${F}_{2}^{\ensuremath{\gamma}},$ averaged over various x regions, and the ${F}_{2,c}^{\ensuremath{\gamma}},$ which were not used directly in the fit. Finally, a simple analytic parametrization of the resulting parton densities obtained with the CJKL model is given.

Charm in the nucleon

A next-to-leading order analysis of inelastic electroproduction of charm is performed using an interpolating scheme which maps smoothly onto massless QCD evolution at large \(Q^2\) and photon–gluon fusion at small \(Q^2\). In contrast with earlier analyses, this scheme allows the inclusion of quark and target mass effects and heavy quark thresholds, as well as possible non-perturbative, or intrinsic, charm contributions. We find no conclusive evidence in favor of an intrinsic charm component in the nucleon, although several data points which disagree with perturbative QCD expectations will need to be checked by future experiments.

Charm in the nucleon

A next-to-leading order analysis of inelastic electroproduction of charm is performed using an interpolating scheme which maps smoothly onto massless QCD evolution at large $Q^2$ and photon–gluon fusion at small $Q^2$ . In contrast with earlier analyses, this scheme allows the inclusion of quark and target mass effects and heavy quark thresholds, as well as possible non-perturbative, or intrinsic, charm contributions. We find no conclusive evidence in favor of an intrinsic charm component in the nucleon, although several data points which disagree with perturbative QCD expectations will need to be checked by future experiments.