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\setcopyright{Copyright 2025 CERN for the benefit of the ATLAS Collaboration. CC-BY-4.0 license.}
\title{Towards a new definition of jet flavour}
%\shorttitle{B-jet charge identification with GNN\lowercase{s} for \(A^b_{FB}\) studies}
\author{Alberto L.~Rescia\thanks{alberto.rescia@ge.infn.it}, on behalf of the ATLAS Collaboration}
\instlist{\item Dipartimento di Fisica, Universit\`{a} di Genova and INFN, Sezione di Genova, \\Via Dodecaneso 33, 16146, Italy}
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\maketitle

\begin{abstract}
This contribution presents new flavour jet algorithms which provide an infrared and collinear safe jet flavour definition at NNLO and beyond. In addition, a measurement of $Z$ boson production in association with \mbox{$b/c$-jets} at \mbox{$\sqrt{s} = 13$ TeV} by the ATLAS experiment at the LHC is presented. Lastly, a first comparison of fixed-order predictions made with the GHS algorithm to data is made.
\end{abstract}

\section{Introduction}

The identification of final states containing heavy flavour jets is key to many physics goals at the Large Hadron Collider (LHC). Experimental collaborations at the LHC rely on flavour tagging algorithms to identify these final states containing \mbox{$b$-jets} and \mbox{$c$-jets}. Observables measured from these final states are then compared to truth-level observables calculated from Monte Carlo predictions on which analogous fiducial selections have been placed. As higher-order predictions become more readily available, it is important to guarantee homogeneity between the definitions of heavy flavour jets at both truth and reconstruction level. The assignment of a flavour label to truth-level jets will henceforth be referred to as \emph{flavour labelling}.

These proceedings outline problems which flavour labelling strategies employed by experimental collaboration currently face and highlight some state-of-the-art algorithms which may serve as a solution to these problems~\cite{Behring:2025ilo}. In addition, a measurement of $Z$ boson production in association with \mbox{$b/c$-jets} carried out by the ATLAS experiment is presented~\cite{ATLAS:2024tnr}. Predictions obtained with a new flavour labelling strategy are compared to this measurement.

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\section{Flavour labelling}

\subsection{Current strategies}

Particle-level jets obtained in predictions by experimental collaborations, specifically ATLAS and CMS \cite{ATLAS:2008xda, CMS:2008xjf}, are usually labelled in accordance with one of two strategies:

\begin{enumerate}
    \item \textbf{Ghost labelling:} Jets are clustered with all stable, visible particles in the final state using the anti-$k_t$ algorithm. The undecayed heavy hadrons in the final state are identified and their four-momenta are suppressed to by some small factor to be nearly zero. The jet clustering algorithm is run with the final state particles described above together with these ``ghost'' hadrons. If a hadron is clustered within a certain jet, that jet is ascribed the flavour of that hadron. $b$-flavour is given precedence over $c$-flavour, and jets receive the same label regardless of the number of hadrons matched to them~\cite{CMS:2012feb}. 

    \item \textbf{Cone labelling:} Jets are clustered with all stable, visible particles in the final state using the anti-$k_t$ algorithm. The undecayed heavy hadrons in the final state with $p_\mathrm{T} > 5$ GeV are identified. If the distance in the rapidity-azimuthal angle plane ($y-\phi$ plane) between the jet and heavy hadron is $\Delta R(\text{jet, hadron)} < 0.3$, the jet is ascribed the flavour of the hadron. Also in this case, $b$-flavour has priority over $c$-flavour, and jets receive the same label independent of the number of hadrons matched to them~\cite{ATLAS:2015thz}. 
\end{enumerate}

\subsection{Problems at NNLO}

At next-to-next-to-leading order (NNLO) there are three critical faults with the labelling strategies currently in use.

\begin{enumerate}
    \item Soft gluons radiated in the parton shower can split collinearly to a heavy flavour pair, i.e. $g\rightarrow b\bar{b}$. This splitting can lead to the ``contamination'' of a jet's flavour, i.e. the assignment of $b$-flavour when in reality the jet originates from a matrix element containing no $b$-flavour.
    \item Soft gluons can undergo a wide-angle splitting to a heavy flavour pair. In this case, the flavour of up to two jets may be ``contaminated'' by the spurious flavour that is generated by the gluon splitting. This scenario is illustrated in Figure \ref{gbb splitting}.
    \item The radiation of a gluon off a heavy-flavour quark can cause the quark's $p_\mathrm{T}$ to fall below the threshold for the cone algorithm. This can persist after hadronisation.
\end{enumerate}

\begin{figure}
    \centering
    \includegraphics[width=0.32\linewidth]{z+jet_gbb.eps}
    \caption{A wide-angle gluon splitting (1, 2) in a \mbox{$Z$+jet} event, leading to the contamination of the flavour of jet 3 by the quark 2~\cite{Caola:2023wpj}.}
    \label{gbb splitting}
\end{figure}

\subsection{Proposed solutions}
The solution to this problem is to cluster jets while simultaneously tracking flavour throughout the entire event, in order to identify heavy-flavour pairs originating from gluon splittings. The first such algorithm to do so was the \mbox{\emph{flavour-$k_t$ algorithm}}, published in 2007~\cite{Banfi:2007gu}. However, this never caught on as the jets are clustered with $k_t$-like kinematics, rather than the more widely-used anti-$k_t$ kinematics.

In the past few years, new algorithms which solve this problem have been published. These so-called flavour algorithms are Interleaved flavour neutralisation~\cite{Caola:2023wpj}, Soft-drop flavour~\cite{Caletti:2022hnc}, CMP~\cite{Czakon:2022wam}, and Flavour dressing (GHS)~\cite{Gauld:2022lem}. Typically, at hadron level, these algorithms all consider jets clustered with final states where heavy hadrons decays have been inhibited. A detailed study comparing the performance of these algorithms has recently been published \cite{Behring:2025ilo}.

\subsection{Flavour dressing}

To give an example of a flavour algorithm, let us describe the GHS algorithm. The algorithm is based on a new distance measure used to determine whether flavoured particles originate from a gluon splitting. We refer the reader to reference~\cite{Gauld:2022lem} for the distance measures involved. 

Jets are first clustered with an algorithm of our choosing, for instance anti-$k_t$. We then consider a set containing all jets and individual particles in the final state. We must then build a set of distance measures. If a particle $p_i$ is a constituent of the jet $j_k$, we populate the set of distances with $d(p_i, j_k)$. We add to this set the distance $d(p_i, p_j)$ if both particles $p_i$ and $p_j$ are heavy-flavour particles. In the case in which either particle is not a heavy flavour particle, but they are both associated to the same jet $j_k$, we still add the distance measure to the set.

We then compare the distances in this set. If $d(p_i, p_j) < d(p_i, j_k)$, the four-momenta of the particles are combined and they are replaced in the set with the result $k_{ij}$. If instead $d(p_i, j_k) < d(p_i, p_j)$, $p_i$ is assigned to the jet $j_k$ and all other elements containing $p_i$ are removed. Lastly, if the distance between the particle $p_i$ and the beam $B$ $d(p_i, B)$ is less than all other distances involving $p_i$, all elements containing $p_i$ are removed from the set. 

When particles are combined, so is their flavour. This can be done either based on the net-flavour recombination scheme, where the flavour $f(k_{ij}) = f(p_i) + f(p_j).$ Here, heavy flavour particles are assigned flavour $+1$ while anti-particles are assigned flavour $-1$. Alternatively, flavour can be combined with the mod-2 flavour recombination scheme, where all (anti-)particles are assigned flavour $+1$ and the sum is carried out modulo 2. Jets are assigned flavour based on the resulting sum of all flavour of the particles associated to them.

\section{ATLAS $Z + b/c$-jet production}
As mentioned, NLO and NNLO predictions for \mbox{$Z+b/c$-jet} production obtained with GHS were compared to data. These data originate from a measurement carried out by the ATLAS experiment~\cite{ATLAS:2024tnr}. 

The measurement was carried out on final states composed of $Z$ bosons decaying leptonically to either electrons or muons in association with heavy-flavour jets utilising \mbox{140 fb$^{-1}$} of data collected at \mbox{$\sqrt{s} = 13$ TeV}.

Events must contain same-flavour opposite-sign electrons or muons, and each individual lepton is required to have \mbox{$p_\mathrm{T} >$ 27 GeV}. The dilepton pair invariant mass must be compatible with the mass of the $Z$ boson \mbox{$76 < m_{\ell\ell}  < 106$ GeV}. 
Jets are required to have transverse momentum \mbox{$p_\mathrm{T} > 20$ GeV} and pseudorapidity \mbox{$|\eta| < 2.5$}. Jets are tagged as $b$ ($c$) using the ATLAS DL1r tagger with 85\% (30\%) efficiency~\cite{ATLAS:2022qxm}. 
To suppress the $t\bar{t}$ background, a missing transverse energy selection \mbox{$p_\mathrm{T}^{miss} < 60$ GeV} requirement is also considered for events with \mbox{$p_\mathrm{T}^{\ell\ell} < 150$ GeV}.

Matrix elements are calculated at NLO using \mbox{\code{MadGraph5\_aMC}@NLO 2.6.5}~\cite{Alwall:2014hca} and showered with \mbox{\code{Pythia} 8.245}~\cite{Sjostrand:2014zea} with the FxFx merging procedure (\mbox{\code{MGaMC+Py8}})~\cite{Frederix:2012ps}. The matrix element calculation utilises the NNPDF3.1$_{\text{LUX}}$QED PDF set~\cite{NNPDF:2017mvq} with the 5 flavour scheme (5FS). Alternative NLO predictions are generated with Sherpa 2.2.11~\cite{Sherpa:2019gpd} with the ATLAS configuration. Again a 5FS is considered in the matrix element.

The unfolded results are compared to various theoretical predictions, obtained by varying FS. In particular, for \mbox{$Z+b$-jet} events, 4 and 5FS predictions are obtained again using \mbox{\code{MGaMC+Py8}}. Likewise, 3 and 4 FS predictions are obtained with \mbox{\code{MGaMC+Py8}} for \mbox{$Z+c$-jet} events. 

In addition to this, NLO and NNLO fixed-order 5FS predictions are compared to the data. To do so, two correction factors are applied to the unfolded data. The first accounts for hadronisation and multi-parton interactions (MPI). It is obtained through the ratio of simulated distributions at hadron-level with MPI enabled and the same distributions at parton-level, using the constituent definitions for the GHS algorithm.  The other correction accounts for the difference in constituents between the cone and GHS algorithm at hadron level, namely the presence of decayed or undecayed heavy hadrons in the final state. It is derived by looking at the ratio of hadron-level distributions obtained with the cone algorithm and the GHS algorithm. 

Results for \mbox{$Z+b$-jets} are shown in Figure \ref{b-jet data}. Here the $Z$ boson $p_\mathrm{T}$ ($p_\mathrm{T}(Z)$), leading \mbox{$b$-jet} $p_\mathrm{T}$ and distance between the $Z$ boson and leading \mbox{$b$-jet} $\Delta\tilde{R}_\text{Zb}$ are shown. For the first observable, the NLO fixed-order predictions yield a softer spectrum. The discrepancy is reduced when considering the NNLO predictions, though it does not fully disappear. There is better agreement between the two fixed-order predictions and data for the leading \mbox{$b$-jet} $p_\mathrm{T}$. They are able to describe most of the spectrum well, though they underpredict the data above $p_\mathrm{T} > 500$~GeV. For $\Delta\tilde{R}_\text{Zb}$ there is excellent agreement with data, except in a region where divergences occur.

\begin{figure}
    \centering
    \includegraphics[width=0.38\linewidth]{ptzb}
    \includegraphics[width=0.38\linewidth]{ptb}
    \includegraphics[width=0.38\linewidth]{drzb}
    \caption{Cross sections for \mbox{$Z+b$-jet} production as a function of $p_\mathrm{T}(Z)$ (top left), leading \mbox{$b$-jet} $p_\mathrm{T}$ (top right) and  $\Delta\tilde{R}_\text{Zb}$ (bottom). Comparisons are made to predictions obtained with \mbox{\code{MGaMC+Py8}}, Sherpa 2.2.11 and fixed-order predictions at NLO and NNLO with jets clustered with the GHS algorithm~\cite{ATLAS:2024tnr}.}
    \label{b-jet data}
\end{figure}

Results for \mbox{$Z+c$-jets} are shown in Figure~\ref{c-jet data}. The $Z$ boson $p_\mathrm{T}$ and leading \mbox{$c$-jet} $p_\mathrm{T}$, though this time they are accompanied by the leading \mbox{$c$-jet} \mbox{Feynman-x} variable \mbox{$x_\text{F} = 2p_z(c)/\sqrt{s}$}. This observable is proportional to the fraction of energy produced in the forward $z$ direction, and is particularly sensitive to differences in PDF models~\cite{Feynman:1969ej}. 

NLO fixed-order predictions once again result in softer spectra for $p_\mathrm{T}(Z)$ and the leading \mbox{$c$-jet} $p_\mathrm{T}$. There is little to no improvement this time when considering NNLO predictions. For the $x_\text{F}$ distribution, the fixed-order predictions describe the shape of the data well, but systematically underestimate the normalisation.

\begin{figure}
    \centering
    \includegraphics[width=0.38\linewidth]{ptzc}
    \includegraphics[width=0.38\linewidth]{ptc}
    \includegraphics[width=0.38\linewidth]{xf}

    \caption{Cross sections for \mbox{$Z+c$-jet} production as a function of $p_\mathrm{T}(Z)$ (top left), leading $c$-jet $p_\mathrm{T}$ (top right) and  $x_\text{F}$ (bottom). Comparisons are made to predictions obtained with \mbox{\code{MGaMC+Py8}}, Sherpa 2.2.11 and fixed-order predictions at NLO and NNLO with jets clustered with the GHS algorithm~\cite{ATLAS:2024tnr}.}
    \label{c-jet data}
\end{figure}

\section{Conclusions}

Problems with current flavour labelling strategies and novel algorithms which offer a solution have been presented together with an ATLAS measurement on \mbox{$Z+b/c$-jet} production. Fixed order \mbox{$Z$+jet} predictions obtained with the GHS algorithm have been compared to data. Results generally show good agreement for distributions such as the leading \mbox{$b$-jet} $p_\mathrm{T}$ and $\Delta\tilde{R}_\text{Zb}$. Good shape agreement is found for $x_\text{F}$, though normalisation is off. Lastly, the $p_\mathrm{T}(Z)$ and leading \mbox{$c$-jet} $p_\mathrm{T}$ spectra are predicted to be softer than what is observed. This analysis constitutes the first comparison of predictions made with novel flavour jet algorithms to data.
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