In collider physics, jet algorithms are a ubiquitous tool for clustering
particles into discrete jet objects. Event shapes offer an alternative way to
characterize jets, and one can define a jet multiplicity event shape, which can
take on fractional values, using the framework of “jets without jets”. In this
paper, we perform the first analytic studies of fractional jet multiplicity
Article funded by SCOAP3
0$ is dominated by logarithms of $\dN{n\pm}$. We will later show that the logarithms of $\dN{n\pm}$ arise from soft emissions, and the most important terms scale single logarithmically as $(\as \ln \dN{n\pm})^k$. These logarithms combine at all orders to suppress the cross section as $\dN{n\pm} \to 0$, leading to the disjoint behavior at integer $\Njet$. Note that single-logarithmic suppression is not as strong as the more familiar double-logarithmic suppression, so there are no Sudakov peaks in the $\dN{n\pm}$ distributions. ]]>
R \,, \qquad \theta_{13}, \theta_{23} < R, \ee or permutations of the parton labels. If all $E_i > \Ecut$, then the contribution to the jet multiplicity is \be \Njet = \frac{E_1}{E_1 + E_3} + \frac{E_2}{E_2 + E_3} + \frac{E_3}{E_1 + E_2 + E_3} \in (1,2) \,. \ee In terms of energy fractions \be \label{zi} z_i \equiv \frac{E_i}{E_1 + E_2 + E_3} \,, \qquad z_1 + z_2 + z_3 = 1 \,, \qquad \zcut \equiv \frac{\Ecut}{E_1 + E_2 + E_3} \,, \ee which removes the overall energy scale of the jet as a degree of freedom, the observable takes the value \be \Njet = \frac{z_1}{z_1 + z_3} + \frac{z_2}{z_2 + z_3} + z_3 \,. \ee For this merged triplet, near-integer behavior is obtained when one or two of the partons goes soft, as shown in \Fig{fig:phasespace}. In these soft limits, the observable may depend on a single soft momentum or a product of soft momenta. For example, if particle 1 is soft, \be \label{Njet1soft} \Njet \xrightarrow[\; z_1 \, \ll \, 1 \;]{} \; 1 + z_1 \frac{z_2 (2-z_2)}{1-z_2} + \ord{z_1^2} \,. \ee Or, if particles 1 and 2 are both soft, \be \label{Njet12soft} \Njet \xrightarrow[\; z_{1,2} \, \ll \, 1 \;]{} \; 1 + 2 z_1 z_2 + \ord{z_{1,2}^3} \,. \ee The first case is typical of event shape observables, as the near-integer behavior is linear in the soft particle's energy. The second case, however, is unusual --- it goes as the \emph{product} of soft momenta, another demonstration of the non-additive nature of $\Njet$. This feature creates a novel structure in the perturbative series, reminiscent of rapidity divergences in SCET$_{\rm II}$~\cite{Chiu-ml-2009yx,Chiu-ml-2011qc}. We discuss this further in \Sec{rapidity}, using a toy observable $\Delta = z_1 z_2$ that exhibits the same analytic features. With more emissions, near-integer behavior requires the soft limit of one or more particles.\footnote{We are neglecting special configurations of energetic particles where the value of $\Njet$ happens to be near-integer, since those regions of phase space are power suppressed.} As mentioned above, an arbitrary collection of energetic particles will yield an integer $\Njet$ if the particles can be grouped into merged clusters where each particle in the cluster is within $R$ of all other particles in the cluster (such that each cluster has $\Njet = 1$). Non-integer values are obtained when this is not satisfied (as in the merged triplet example above), though generically the resulting values are far from integers. Near-integer values are obtained by starting from an integer $\Njet$ configuration and then adding any number of soft particles which are not within the mutual radius $R$ of the cluster. The contribution of these soft particles to $\Njet$ will be suppressed by \be \frac{E_{\rm soft}}{E_{{\rm soft},R}} \,, \ee and the deviation of $\Njet$ from integer values is similarly suppressed by this ratio as long as $E_{{\rm soft},R} > E_\cut$. In this way, the near-integer behavior is determined by soft emissions in the vicinity of hard clusters, and soft emissions will generate logarithms of $\dN{n\pm}$. By contrast, collinear splittings cannot generate near-integer behavior of $\Njet$ and hence do not generate logarithms of $\dN{n\pm}$. For collinear splittings of angle $R_c$, the only effect on $\Njet$ comes from particles who are within $R$ of either of the two daughters or the parent, but not within $R$ of all three. For small $R_c$, this is a power-suppressed region of phase space and not logarithmically enhanced. Said another way, the effect on $\Njet$ from a small-angle splitting is not smooth, as it either preserves the value of $\Njet$ or discretely changes it by including or excluding particles from the various $E_{i,R}$ terms; this behavior cannot generate logarithms of $\dN{n\pm}$. So unlike standard jet shapes (like jet mass) which depend on both the softness and collinearity of a splitting, the near-integer $\Njet$ value depends only on the softness of emissions. ]]>
{\centering\arraybackslash}m{0.15\textwidth} | >{\centering\arraybackslash}m{0.2\textwidth} | >{\centering\arraybackslash}m{0.2\textwidth} | >{\centering\arraybackslash}m{0.314\textwidth}|} \hline Observable & $\cR_A$ & $\cR_B$ & $\cR_C$\\ \hline $\dNm$ &\; \includegraphics[width=0.15\textwidth]{dNmA.pdf} &\; \includegraphics[width=0.15\textwidth]{dNmB.pdf} &\; --\\ \hline $\dNp$ &\; \includegraphics[width=0.15\textwidth]{dNpA.pdf} &\; \includegraphics[width=0.15\textwidth]{dNpB.pdf} &\; \includegraphics[width=0.15\textwidth]{dNpCsc.pdf} \includegraphics[width=0.15\textwidth]{dNpCcs.pdf} \includegraphics[width=0.15\textwidth]{dNpCss.pdf}\\ \hline $\dNtm$ &\; \includegraphics[width=0.15\textwidth]{dN3A.pdf} &\; \includegraphics[width=0.15\textwidth]{dN3B.pdf} &\; --\\ \hline \end{tabularx} ]]>
R) \, \Theta(\angle_{23}
2.5) \biggr] \,. $$ } Alternatively, we could define the nonsingular term by removing \emph{both} sets of singular terms over the entire spectrum.\footnote{This would correspond to the nonsingular definition $$ \frac{\df\sigma_{\rm ns}}{\df\Njet} (2 < \Njet < 3)= \frac{\df\sigma_{\rm full}}{\df\Njet} - \biggl[ \frac{\df\sigma_{\rm sing}}{\df\dNp} + \frac{\df\sigma_{\rm sing}}{\df\dNtm}\biggr] \,. $$ } This is a valid approach as well, since the singular $\dNp$ and $\dNtm$ distributions are governed by different soft limits of phase space, hence there is no double-counting by including both singularities. Since we are mainly interested in describing the behavior in the vicinity of \mbox{$\Njet = 2$}, though, for the rest of the paper we will simply adopt the nonsingular definition in \Eq{eq:nonsingdefinition}, even in the vicinity of $\Njet = 3$: \be \label{eq:def2p_nonsingular} \frac{\df\sigma_{\rm ns}}{\df\Njet} (2 < \Njet < 3) = \frac{\df\sigma_{\rm ns}}{\df \dNp} = \frac{\df\sigma_{\rm full}}{\df\Njet} - \frac{\df\sigma_{\rm sing}}{\df\dNp} \,. \ee That said, this non-singular term turns out to be dominated by the singular $\dNtm$ piece. Writing the nonsingular term as \be \label{eq:def2p_residual} \frac{\df\sigma_{\rm ns}}{\df \dNp} = \frac{\df\sigma_{\rm sing}}{\df\dNtm} + \frac{\df\sigma_{\rm res}}{\df\dNp}, \ee the residual term is quite small, even in the vicinity of $\Njet = 2.5$. This is shown in \Fig{fig:singularVSnonsingular}, where the full QCD result between $2 < \Njet < 3$ is decomposed into the $\dNp$ singular, $\dNtm$ singular, and residual terms for $R = 0.4$ and $\zcut = 0.2$. The fact that the $\dNp$ nonsingular term is nearly saturated by the $\dNtm$ singular term suggests that higher-order logarithmic terms can play an important role in determining the shape of the $\Njet$ distribution, even for large deviations from integer values of the observable. \begin{figure} \centering \includegraphics[width=0.6\textwidth]{FO_ns2to3.pdf} ]]>
R)\Theta( \theta_{\bq g} > R) \Theta (z_g < \zcut) \,. \ee At higher orders, the phase space restrictions differ from discrete jet algorithms (which also differ among themselves), and the $\Njet = 2$ cross section has unique features. Consider an $\ord{\as^2}$ configuration contributing to the near-integer behavior, as discussed in the \Sec{singular}. A concrete example is the $\dNp$-$\cR_C$ configuration in \Tab{tab:PhaseSpace}, which has a collinear $q\to qg$ splitting with another soft gluon (labeled $s$) emitted such that \be \label{configscc} \theta_{qg} < R \,, \quad \theta_{qs} > R \,, \quad \theta_{gs} < R \,. \ee This event will give a non-integer contribution to $\Njet$, and its contribution to the cross section is of the form \be \label{N2contribnear} \sim \as^2 \biggl[ \frac{1}{\epsilon^2} \delta(\Njet - 2) + \zeta \delta(\Njet - 2) + \kappa_0 \, \cL_0 (\dNp) + \kappa_1 \, \cL_1 (\dNp) \biggr] \,. \ee The plus distribution terms were already calculated in \Eq{eq:ab2}. The $\delta(\dNp)$ terms were suppressed in our previous discussion (since we were focused on the non-integer behavior), but both the $1/\epsilon^2$ term and the $\zeta$ term follow directly from the calculations in \Sec{singular}. The structure of \Eq{N2contribnear} reflects the hybrid jet algorithm/event shape behavior of $\Njet$. The plus distribution terms have no support at $\Njet = 2$, since the plus function prescription removes any divergence there.\footnote{To see this, we can use the definition of the plus function in \Eq{eq:plus_definition}. Integrating $[q(x)]_+$ against $\delta(x)$ gives the value at 0, \be [q(0)]_+ = \int \df x \, [q(x)]_+ \, \delta(x) = 0 \,. \ee } This feature is similar to other event shape variables with a singular limit, such as thrust, where there is zero cross section in the singular limit at higher orders. By contrast, the delta function terms behave more like a jet algorithm, as anticipated in \Eq{eq:deltafunctioninteger}, with a finite cross section at exactly $\Njet = 2$. The IR pole from the real radiation at $\Njet = 2$ will cancel divergences from virtual matrix elements, such that the $\Njet = 2$ cross section is IR finite.\footnote{It is straightforward to see that at $\ord{\as^2}$ the cross section at $\Njet = 2$ must be IR finite. Since non-integer values of $\Njet$, as well as $\Njet = 3$ and 4, have IR finite cross sections, and since the inclusive cross section is IR finite, the remaining piece, the cross section at $\Njet = 2$, must be as well.} The additional finite $\zeta$ term at $\Njet = 2$ implies that the non-global structure in the near-integer part of the cross section is also contributing at $\Njet = 2$. ]]>
2$. This indicates that we have made a reasonable estimate of the higher-order corrections not included in our calculation. Because we normalize to the window $\dNp \in (10^{-4}, 10^{-2})$, it is not surprising that there is a normalization discrepancy in $\dNm$, though one can see that the shape agreement for $\Njet < 2$ is excellent. We suspect that the normalization issues for $\Njet < 2$ may be due to the treatment of correlated soft radiation (and angular ordering) in the Monte Carlo generators, since the $\dNm$ distribution is dominated by double-soft emissions with a $C_F C_A$ color structure. The overall shape agreement between \pythia and \herwig is quite surprising, given that they exhibit a factor of 3 difference (not shown) in their predicted value of $\sigma(\Njet = 2)$, suggesting that the shape of the non-integer distributions is more robust than the cross section at integer values. ]]>
0$, and appear in many applications. ]]>