Hadroproduction of open heavy flavour for PDF analyses

TL;DR

This paper introduces a new theoretical scheme, SACOT-mT, for calculating open heavy-flavour hadroproduction cross sections, improving the reliability of perturbative QCD predictions across all kinematic ranges in high-energy collisions.

Contribution

The paper presents SACOT-mT, a novel scheme within the general-mass variable-flavour-number formalism, enhancing the accuracy of heavy-flavour production calculations in QCD.

Findings

Improved description of heavy-flavour cross sections across all pT ranges.

Enhanced robustness of LHC heavy-flavour measurements for PDF analyses.

Introduction of a scheme analogous to SACOT-χ for deep-inelastic scattering.

Abstract

Due to the large masses of the charm and bottom quarks, their production cross sections are calculable within the perturbative QCD. This makes the heavy-quark mesons important observables in high-energy collisions of protons and nuclei. However, the available calculations for heavy-flavored-meson hadroproduction have been somewhat problematic in reliably describing the cross sections across the full kinematic range from zero to very high $p_{\rm T}$. This has put some question marks on the robustness of LHC heavy-flavored-meson measurements in studying the partonic structure of the colliding hadrons and nuclei. Here, we introduce SACOT-$m_{\rm T}$ - a novel scheme for open heavy-flavour hadroproduction within the general-mass variable-flavour-number formalism that solves this problem. The introduced scheme is an analogue of the SACOT-$\chi$ scheme often used for deeply-inelastic scattering in global analyses of PDFs.

Figures

Equations

$$\frac{d\sigma (p+p\to Q+X)}{d{p}_{\mathrm{T}}dy}=$$

$${\tau }_1\equiv p_1\cdot p_3/p_1\cdot p_2=m_{\mathrm{T}}{e}^{-y}/(\sqrt{s}{x}_2),{\tau }_2\equiv p_2\cdot p_3/p_1\cdot p_2=m_{\mathrm{T}}{e}^y/(\sqrt{s}{x}_1),m_{\mathrm{T}}^2=p_{\mathrm{T}}^2+m^2,$$

$$\frac{d\sigma (p+p\to {\mathrm{D}}^0+X)}{d{P}_{\mathrm{T}}dY}=$$

$$p_{\mathrm{T}}^2$$

$$y$$

$$\int \frac{dz}{z}d{x}_{1}d{x}_2{f}_Q^p(x_1,{\mu }_{\mathrm{fact}}^2)\frac{d{\hat{\sigma }}^{Qg\to Q+X}({\tau }_1,{\tau }_2)}{d{p}_{\mathrm{T}}dy}{f}_g^p(x_2,{\mu }_{\mathrm{fact}}^2)D_{Q\to {\mathrm{D}}^0}(z).$$

$$f_Q^p(x,{\mu }_{\mathrm{fact}}^2)=\left(\frac{{\alpha }_s}{2\pi }\right)\log \left(\frac{{\mu }_{\mathrm{fact}}^2}{m^2}\right){\int }_x^1\frac{\mathrm{d}\ell }{\ell }{P}_{qg}\left(\frac{x}{\ell }\right)f_g^p(\ell ,{\mu }_{\mathrm{fact}}^2)+\mathcal{O}({\alpha }_s^2),$$

$$\frac{d{\hat{\sigma }}^{Qg\to Q+X}({\tau }_1,{\tau }_2)}{d{p}_{\mathrm{T}}dy}\stackrel{p_{\mathrm{T}}\to \infty }{\to }\frac{d{\hat{\sigma }}^{qg\to q+X}({\tau }_1^0,{\tau }_2^0)}{d{p}_{\mathrm{T}}dy},{\tau }_{1,2}^0={\tau }_{1,2}\stackrel{m\to 0}{\to }{p}_{\mathrm{T}}{e}^{\mp y}/(\sqrt{s}{x}_{2,1}).$$

$$\int \frac{dz}{z}d{x}_{1}d{x}_2{f}_g^p(x_1,{\mu }_{\mathrm{fact}}^2)\frac{d{\hat{\sigma }}^{gg\to g+X}({\tau }_1,{\tau }_2)}{d{p}_{\mathrm{T}}dy}{f}_g^p(x_2,{\mu }_{\mathrm{fact}}^2)D_{g\to {\mathrm{D}}^0}(z,{\mu }_{\mathrm{frag}}^2).$$

$$D_{g\to {\mathrm{D}}^0}(x,{\mu }_{\mathrm{frag}}^2)=\left(\frac{{\alpha }_s}{2\pi }\right)\log \left(\frac{{\mu }_{\mathrm{frag}}^2}{m^2}\right){\int }_x^1\frac{\mathrm{d}\ell }{\ell }{P}_{qg}\left(\frac{x}{\ell }\right)D_{Q\to {\mathrm{D}}^0}(\ell )+\mathcal{O}({\alpha }_s^2).$$

$$\frac{d{\sigma }^{pp\to {\mathrm{D}}^0+X}}{d{P}_{\mathrm{T}}dY}=\sum _{ijk}\int \frac{dz}{z}d{x}_{1}d{x}_2$$

$$f_j^p(x_2,{\mu }_{\mathrm{fact}}^2)D_{k\to {\mathrm{D}}^0}(z,{\mu }_{\mathrm{frag}}^2).$$

Full text

1 Introduction

The hadroproduction of open heavy flavour – D mesons in particular – has recently been advocated as a promising constraint for proton parton distribution functions (PDFs) [1, 2]. The theoretical description is typically based on the fixed flavour-number scheme (FFNS) [3], fonll code [4], or FFNS matched to parton showers [5]. However, the use of e.g. FFNS calculation in conjunction with PDFs defined in variable flavour-number schemes (the commonly used general-purpose PDFs) may be too restrictive, and in this sense calculations within the framework of general-mass variable-flavour-number scheme (GM-VFNS) would be more natural. Here, we will discuss our novel implementation of the GM-VFNS, the so-called SACOT-$m_{\rm T}$ scheme [6].

2 D-meson production in fixed flavour-number scheme

Within FFNS – assuming no intrinsic heavy-quark content in the proton – the massive quarks $Q$ are always produced in pairs by the partonic processes, $g+g\rightarrow Q\overline{Q}+X,\,q+\overline{q}\rightarrow Q\overline{Q}+X,\,q+g\rightarrow Q\overline{Q}+X\,$. The cross section for inclusive $Q$ production can be written as an integral of PDFs $f_{i}^{p}(x_{1},\mu^{2}_{\rm fact})$ and partonic cross sections $d\hat{\sigma}^{ij}$,

[TABLE]

where $y$ and $p_{\rm T}$ denote the rapidity and transverse momentum of the produced heavy quark. The factorization and renormalization scales are marked by $\mu^{2}_{\rm fact}$ and $\mu^{2}_{\rm ren}$. The kinematic variables $\tau_{1,2}$ are the “massive” Mandelstam variables,

[TABLE]

denoting the momenta of the incoming partons and outgoing heavy quark by $p_{1,2}$ and $p_{3}$, respectively. The partonic cross sections scale as $d\hat{\sigma}^{ij\rightarrow Q+X}\sim\tau_{1,2}^{-n}$, and, thanks to the heavy-quark mass, remain finite even at $p_{\rm T}=0$. The heavy-quark cross sections can be turned into, say D0-meson production cross sections by folding with $Q\rightarrow\mathrm{D^{0}}$ fragmentation functions (FFs) ${D_{Q\rightarrow\mathrm{D}^{0}}(z)}$. The fragmentation variable $z$ is not unique when the masses of the heavy quark and D0 meson are kept non zero. For simplicity, we define $z\equiv{E_{\rm D^{0}}}/{E_{Q}}$, where $E_{\rm D^{0}}$ and $E_{Q}$ are the energies of the D0 meson and heavy quark in the center-of-mass frame of the p-p collision. Assuming that the fragmentation is collinear we get,

[TABLE]

where the (lower case) partonic and (upper case) hadronic variables are related by

[TABLE]

and $M_{\rm T}=\sqrt{P_{\rm T}^{2}+M_{Q}^{2}}$ marks the hadronic transverse mass. While this framework appears to work well at low $P_{\rm T}$ (see e.g. [7]), the description seems to deteriorate towards high $P_{\rm T}$. Presumably, this can be attributed to the $\log(p_{\rm T}^{2}/m^{2})$ behaviour of the partonic cross sections which begin to dominate and should be resummed, as we will come to conlcude.

3 D-meson production in general-mass variable-flavour-number scheme

The GM-VFNS description can be obtained from FFNS by resumming the $\log(p_{\rm T}^{2}/m^{2})$ terms that appear in FFNS. As an example, in Figure 1 the diagram (a) gives rise to such logarithmic behaviour when the initial-state gluon splits into a $Q\overline{Q}$ pair. This is only the first of the whole series of diagrams that are in GM-VFNS summed into the heavy-quark PDF $f_{Q}^{p}$.

Effectively, this summation can be realized by including the heavy-quark initiated contribution (c) and a subtraction term (b) that avoids the double counting between diagrams (a) and (c). The contribution from $Qg\rightarrow Q+X$ channel, represented here by the diagram (c), can be written as

[TABLE]

The subtraction term (b) is obtained from this same expression by replacing the heavy-quark PDF with its perturbative expression,

[TABLE]

where $\alpha_{s}$ is the QCD coupling, and $P_{qg}$ is the usual gluon-to-quark splitting function. However, the exact form of $d\hat{\sigma}^{Qg\rightarrow Q+X}(\tau_{1},\tau_{2})$ in the above expressions is not unique [8]. In practice, we can only require that the zero-mass $\overline{\rm MS}$ expressions are recovered at high $p_{\rm T}$,

[TABLE]

The easiest option is to define ${d\hat{\sigma}^{Qg\rightarrow Q+X}(\tau_{1},\tau_{2})}\equiv{d\hat{\sigma}^{qg\rightarrow q+X}(\tau_{1}^{0},\tau_{2}^{0})}$, i.e. use the zero-mass expressions to begin with. This is known as the SACOT scheme [9]. The problem of this choice is that it leads to infinite cross sections towards $P_{\rm T}\rightarrow 0$ due to the ${d\hat{\sigma}^{qg\rightarrow q+X}}/d^{3}p\sim(\tau_{1,2}^{0})^{-n}$ behaviour of the partonic cross sections. In the so-called FONLL scheme [4] this is avoided by multiplying the partonic cross section by an ad-hoc damping factor $p_{\rm T}^{2}/(p_{\rm T}^{2}+c^{2}m^{2})$ with $c\sim 5$, which serves to tame the unphysical behaviour at small $p_{\rm T}$. Alternatively, the problematic behaviour can be avoided simply by retaining the kinematics of the $Q\overline{Q}$-pair production which, deep down, is the underlying process we describe. With this physical picture in mind, we define ${d\hat{\sigma}^{Qg\rightarrow Q+X}(\tau_{1},\tau_{2})}\equiv{d\hat{\sigma}^{qg\rightarrow q+X}(\tau_{1},\tau_{2})}$ taking $\tau_{1,2}={m_{\rm T}e^{\mp y}}/({\sqrt{s}x_{2,1}})$ as in FFNS. This leads to well-behaved cross sections in the $P_{\rm T}\rightarrow 0$ limit. We call this the SACOT-$m_{\rm T}$ scheme, as it shares the same underlying idea as the SACOT-$\chi$ scheme in deeply-inelastic scattering [10].

Part of the $\log(p_{\rm T}^{2}/m^{2})$ terms in FFNS come also from final-state splittings. As an example, Figure 2 shows a diagram in which an outgoing gluon splits into a $Q\overline{Q}$ pair.

The resulting $\log(p_{\rm T}^{2}/m^{2})$ terms are absorbed into the fragmentation-scale $(\mu_{\mathrm{frag}})$ dependent gluon FFs, $D_{g\rightarrow\mathrm{D}^{0}}(z,\mu^{2}_{\rm frag})$, giving rise to gluon-fragmentation contributions,

[TABLE]

The subtraction term avoiding the double counting is again the same expression, but replacing the gluon FF by its perturbative form,

[TABLE]

In line with our scheme choice, the $\overline{\rm MS}$ zero-mass matrix elements for $d\hat{\sigma}^{gg\rightarrow g+X}(\tau_{1},\tau_{2})$ with the massive expressions for $\tau_{1,2}$, are adopted. Even if the heavy quarks do not explicitly appear in the $gg\rightarrow g+X$ process, the underlying process is also here the $Q\overline{Q}$-pair production.

With this schematic justification, our “master formula” within GM-VFNS is,

[TABLE]

Unlike in FFNS, all partonic subprocesses are included and the FFs are scale dependent. In the limit $p_{\rm T}\rightarrow 0$, the partonic cross sections reduce to FFNS, but towards $p_{\rm T}\rightarrow\infty$ they tend to the zero-mass $\overline{\rm MS}$ expressions. Our numerical realization of SACOT-$m_{\rm T}$ scheme is crafted around the Mangano-Nason-Ridolfi code [3] for heavy quarks, and the incnlo code [11] for zero-mass partons. All terms up to $\mathcal{O}(\alpha_{s}^{3})$ are included.

4 Description of the LHCb D0 data

In Figure 3, we compare the LHCb 13 TeV D0 data [12] with our GM-VFNS theory calculation. The darker bands show the NNPDF3.1 (pch) [13] PDF uncertainty, and the lighter bands combine the scale-variation and PDF errors. We have used here the KKKS08 FFs [14]. The calculation agrees very well with the data though the scale variation leads to a significant uncertainty at small $P_{\rm T}$. We have found that, the contribution from the three FFNS processes, including the subtraction terms, is less than 10% for $P_{\rm T}\gtrsim 3\,{\rm GeV}$ and only around 3% for $P_{\rm T}\gtrsim 10\,{\rm GeV}$. This demonstrates that the $\log(p_{\rm T}^{2}/m^{2})$ terms in FFNS become quickly the dominant ones and must be resummed as done in GM-VFNS. A comparison using FFNS + parton-shower approach (with the same PDFs) is also presented. Here, we have used the Powheg-Box event generator [5] matched to the Pythia 8 [15] parton shower. We see that the Powheg+Pythia setup has a tendency to underpredict the experimental spectrum, and the ratio between two $\sqrt{s}$ is clearly flatter than the GM-VFNS prediction. We deduce that the leading reason is that by starting only with $\mathrm{c\overline{c}}$ pairs (generated by Powheg) one neglects the contributions in which the parton shower excites the $\mathrm{c\overline{c}}$ pair only later on. In GM-VFNS these are resummed to the scale-dependent FFs and, indeed, e.g. the gluon-to-D contributions can be around 50% of the total budget. The gluon fragmentation also significantly changes the $x$ regions in which the PDFs are sampled. Thus, the use of FFNS-based computation when using D-meson data to fit GM-VFNS PDFs would inflict a potential bias.

Acknowledgments

The support by the Academy of Finland Projects 297058 & 308301, the Carl Zeiss Foundation, and the state of Baden-Württemberg through bwHPC, are acknowledged. We have used the computing facilities of the Finnish IT Center for Science (CSC) in our work.