Broadening is a classic jet observable that probes the transverse momentum
structure of jets. Traditionally, broadening has been measured with respect
to the thrust axis, which is aligned along the (hemisphere) jet momentum
to minimize the vector sum of transverse momentum within a jet. In this
paper, we advocate measuring broadening with respect to the “broadening
axis”, which is the direction that minimizes the scalar sum of transverse
momentum within a jet. This approach eliminates many of the calculational
complexities arising from recoil of the leading parton, and observables like the jet
angularities become recoil-free when measured using the broadening axis. We
derive a simple factorization theorem for broadening-axis observables which
smoothly interpolates between the thrust-like and broadening-like regimes.
We argue that the same factorization theorem holds for two-point energy
correlation functions as well as for jet shapes based on a “winner-take-all
axis”. Using kinked broadening axes, we calculate event-wide angularities in
Article funded by SCOAP3
0$. The angular measures used in previous definitions, e.g.\ eq.~\eqref{eq:compare_new_old_ang}, are only monotonic in $\theta_i$ for $\beta\geq1$, which has the unfortunate consequence of making the same value of the angularity sensitive to both two and three jet configurations for $\beta < 1$. We choose our factor of 2 convention such that the angularities have a simple form in the collinear limit.} \begin{align} \label{eq:angularity} \ang{\beta} &=\frac{1}{E_\text{jet}}\sum_{i\in \text{jet}}E_i\left(2\frac{n\cdot p_i}{E_i}\right)^{\frac{\beta}{2}}\nonumber \\ &= \frac{1}{E_ \jet} \sum_{i\in \jet} E_i \left(2\sin \frac{\theta_{i, \hat{n}}}{2}\right)^\beta \nonumber \\ &\approx \frac{1}{E_ \jet}\sum_{i\in \jet} E_i \theta_{i, \hat{n}}^\beta \,, \end{align} where $p^\mu_i$ is the four-vector of a massless particle $i$, $E_i$ is its energy, and $\theta_{i, \hat{n}}$ is the angle between particle $i$ and axis $\hat{n}$.\footnote{In this paper, we will always treat hadrons as being effectively massless. Hadron masses are relevant for non-perturbative power corrections~\cite{Salam-ml-2001bd,Mateu-ml-2012nk}, which are beyond the scope of this paper.} For massless hadrons, the limit $\beta = 1$ corresponds to broadening and $\beta = 2$ to thrust. True to their names, the broadening axis minimizes $\ang{1}$ and the thrust axis minimizes $\ang{2}$. In terms of the exponent $a$ in the original angularities paper, we are using the notation $\beta = 2-a$, and $\beta>0$ ($a < 2$) is needed for infrared and collinear (IRC) safety. Crucially, in order for $\ang{\beta}$ to be recoil-free, we have to identify the jet region using a recoil-free jet algorithm such that the jet center coincides with the hard-collinear radiation and not the jet momentum axis. Remarkably, we find a simple factorization theorem for $\ang{\beta}$ about the broadening axis that has the same form for the entire range $0 < \beta < \infty$, including broadening itself with $\beta = 1$. This factorization theorem is derived using soft-collinear effective theory (SCET)~\cite{Bauer-ml-2000ew,Bauer-ml-2000yr,Bauer-ml-2001ct,Bauer-ml-2001yt,Bauer-ml-2002nz}. Our result is in contrast to the usual case of $\ang{\beta}$ about the thrust axis, where there is a factorization theorem for $1 < \beta < \infty$ that expands any recoil contributions as power suppressed~\cite{Berger-ml-2003iw,Hornig-ml-2009vb}, and a different factorization theorem that accounts for recoil at $\beta = 1$~\cite{Becher-ml-2011pf,Chiu-ml-2012ir}. Because broadening-axis observables do not require transverse recoil convolutions, our factorization theorem is $\beta$-independent, which is quite surprising given that the relative scaling of the collinear and soft modes depends on $\beta$. As a proof of concept, we resum the event-wide broadening-axis $\ang{\beta}$ distributions to next-to-leading logarithmic (NLL) accuracy for any value of $\beta$. We also apply our analysis to jet-based broadening-axis $\ang{\beta}$ distributions to next-to-leading logarithmic (NLL) accuracy, ignoring non-global logarithms~\cite{Dasgupta-ml-2001sh} at present. The behavior of our factorization theorem is particularly interesting in the vicinity of $\beta = 1$ (corresponding to broadening itself). Strictly speaking, rapidity factorization~\cite{Chiu-ml-2012ir} is needed at exactly $\beta = 1$, with or without recoil. However, ordinary soft/collinear factorization is valid for $\beta = 1 \pm \epsilon$ for non-zero $\epsilon$, and we can obtain the $\beta = 1$ distributions from the limits $\beta \to 1^+$ or $\beta \to 1^-$. By measuring angularities with respect to the broadening axis, we therefore achieve a smooth interpolation between the thrust-like ($\beta = 2$) and broadening-like ($\beta = 1$) regimes. Furthermore, we argue that our factorization theorem also holds for the two-point energy correlation function~\cite{Banfi-ml-2004yd,Jankowiak-ml-2011qa,Larkoski-ml-2012eh,Larkoski-ml-2013eya} as well as for angularities measured with respect to the ``winner-take-all axis'' (defined in section~\ref{subsec:WTA}, see also ref.~\cite{Bertolini-ml-2013iqa}). This is true in a very strong sense: not only is the form of the factorization theorem identical, but the anomalous dimensions of all corresponding functions in each observable are identical for a given $\beta$. Only the finite terms of the functions can differ. Even though the observables are clearly distinct, the fact that even one of the soft or collinear limits agrees is sufficient to prove that they have the same large logarithmic behavior. The outline of the remainder of this paper is as follows. In section~\ref{sec:axisdef} we define the broadening axis, highlight the important recoil properties of broadening-axis observables, and show how to define jets using the broadening axis. We then introduce a factorization theorem for general angularities about the broadening axis in section~\ref{sec:factor}, calculating event-wide angularities in $e^+ e^-$ collisions to \nllp\ accuracy. We briefly discuss how to apply this factorization theorem to jet-based angularities in section~\ref{sec:shapefactor}. In section~\ref{sec:anom_dim_beta_1_limit}, we discuss the broadening limit ($\beta = 1$) and sketch how to obtain rapidity factorization from ordinary soft/collinear factorization. We compare our distributions to showering Monte Carlo programs and automated resummation tools in section~\ref{sec:mc}. We give a historical perspective on the broadening axis in section~\ref{sec:history}, and conclude in section~\ref{sec:conclude} with some potential applications for broadening-axis observables at the LHC\@. Various calculational details are left to the appendices. ]]>
E_2$}, \\ \hat{n}_2 & \text{if $E_2 > E_1$}, \end{cases} \end{align} where $\hat{n}_i = \vec{p}_i/|\vec{p}_i|$ are unit-normalized. This recombination scheme is (perhaps surprisingly) IRC safe, just like other weighted schemes like the $p_t^2$-scheme~\cite{Catani-ml-1993hr,Butterworth-ml-2002xg}, and it can be applied to any of the generalized $k_T$ algorithms including anti-$k_T$. Because the jet axis always aligns with the harder particle in a pair-wise recombination, soft radiation cannot change the jet axis, so the resulting jet axis is recoil-free. Note that the jet axis is only needed to determine the particles clustered into a given jet, but the actual jet four-vector can be defined by adding the jet's constituents (just as in the $E$-scheme, though here the jet momentum and jet axis will be offset because of recoil). Because finding the winner-take-all axis is computationally much faster than minimizing $\ang{\beta}$, we expect it will become the default way to define a recoil-free axis. We leave a more in depth study of the winner-take-all axis for future work. In section~\ref{subsec:pythia}, we will see that this winner-take-all axis yields nearly identical results to the broadening axis. This is a consequence of both being dominated by collinear dynamics. The difference between angularities measured with respect to the broadening axis versus the winner-take-all axis must be proportional to the typical collinear splitting angle. Since soft radiation cannot resolve such splittings, the two observables share the same soft function to leading power. Then to all orders in perturbation theory, the anomalous dimensions of all functions for either axis are identical; see section~\ref{subsec:anomdimrelatsion}. Indeed, to \nllp\ order, the two cross sections are identical. ]]>
0$, measured with respect to kinked broadening axes. We use this factorization theorem to calculate the broadening-axis angularities to \nllp\ order in section~\ref{subsec:NLLp_result}, including matching to the full $\mathcal{O}(\alpha_s)$ fixed-order result. ]]>
1$) or below ($\beta < 1$) the soft scale. For the special case $\beta = 1$ (discussed more in section~\ref{sec:anom_dim_beta_1_limit}), the jet and soft scales coincide. The relative positions of all modes in the theory are illustrated in figure~\ref{fig:invariant_mass}. From the point of view of the factorization theorem, the relative hierarchy between $\mu_J$ and $\mu_S$ will be largely irreverent, though we will have to choose appropriate scales in appendix~\ref{sec:scalechoice}.\footnote{The relative hierarchy is interesting from the point of view of non-perturbative physics. For $\beta<1$, the leading non-perturbative power correction will be controlled by the collinear modes, not the soft modes as it is for $\beta > 1$ observables like thrust~\cite{Lee-ml-2006nr}. Non-perturbative corrections to the jet function are not well-understood, so we will not discuss this issue further in this paper.} \begin{figure}[t] \centering \subfloat[]{ \includegraphics[trim = 0 0 0 0, clip, width=6.1cm]{figures/JetsAboveSoft.pdf} } $\qquad\qquad$ \subfloat[]{ \includegraphics[width=6.1cm]{figures/JetsBelowSoft.pdf} } ]]>
1$ (left) and $\beta < 1$ (right). Shown is the light-cone momentum plane $(n\cdot p, \bar{n}\cdot p)$ and the dots correspond to the hard ($H$), left-collinear ($J$), right-collinear ($\bar{J}$), and soft ($S$) modes. The curves indicate the natural invariant mass scale for the hard ($\mu_H$), jet ($\mu_J$), and soft ($\mu_S$) functions. For $\beta > 1$, the soft modes have smaller invariant mass than the collinear modes, but this hierarchy is inverted for $\beta < 1$. The arrows indicate the direction of renormalization group evolution if modes are factorized at the jet scale (as we will do in this paper). ]]>
0$, recoil is a power-suppressed correction for all of the broadening-axis angularities. It is instructive to compare this behavior to the thrust-axis angularities, where only for $\beta > 1$ is recoil power-suppressed~\cite{Bauer-ml-2008dt}. For the thrust-axis angularities, one can perform a multipole expansion~\cite{Grinstein-ml-1997gv,Bauer-ml-2000yr,Beneke-ml-2002ni} of the jet function in $\vec{k}_\perp$ to account for the displacement of the collinear direction due to soft recoil. This displacement is power suppressed for $\beta>1$ since the $p_\perp$ of a soft mode is a factor of $\lambda^{(\beta - 1)/\beta}$ smaller than the $p_\perp$ of a collinear mode, so the effect of recoil can be formally ignored to leading power. By contrast, the broadening axis lies in the collinear direction for all $\beta>0$, independent of the soft emissions in the jet, so the multipole expansion is \emph{irrelevant}. There are still power corrections to eq.~\eqref{recoil_power_corrections_for_jet_function} for the broadening axis, but they are generated by hard perturbative emissions, the finite size of the collinear region, and similar effects. \sloppy Using eq.~\eqref{recoil_power_corrections_for_jet_function}, we can trivially perform the transverse momentum integrals in eq.~\eqref{eq:factorized-X-sec-start} to achieve the simplified factorization theorem for broadening-axis observables: \begin{align} \label{eq:factorized-X-sec-fin} \frac{1}{\sigma_0}\frac{d^2\sigma}{d\ang{\beta}_Ld\ang{\beta}_R} & = H\left(Q^2,\mu\right) \int de_n \, de_{\bar n}\, {\cal J}_{n}(Q,e_n) {\cal J}_{\bar n}\left(Q,e_{\bar n}\right) {\cal S}\left(\ang{\beta}_L-e_n,\ang{\beta}_R-e_{\bar{n}}\right), \end{align} where the jet and soft functions are defined as \begin{align}\label{eq:jet-and-soft-functions-final-form} {\cal J}_{n}(Q,e_n) &= \frac{(2\pi)^3}{N_c}\tr \langle 0 \vert \frac{\nbarslash}{2} \chi_{n} \delta\left( \bar n \cdot \hat P - Q \right) \delta\left(e_n -\hat e_{\hat b_L}\right) \delta^{(2)}\left(\hat P_{\perp}\right) \bar{\chi}_{n} \vert 0\rangle\,,\nonumber \\ {\cal J}_{\bar{n}}(Q,e_{\bar{n}})&= \frac{(2\pi)^3}{N_c} \langle 0 \vert \bar{\chi}_{\bar n} \delta\left( n \cdot \hat P - Q \right) \delta\left(e_{\bar n} -\hat e_{\hat b_R}\right) \delta^{(2)}\left(\hat P_{\perp}\right)\frac{\nslash}{2} \chi_{\bar n} \vert 0\rangle\,,\nonumber \\ {\cal S}(e_L,e_R)& = \frac{1}{N_c}\tr \langle 0 \vert S^{\dagger}_{\bar n} S_{n} \delta\left(e_L-\hat e_{\hat b_L}\right)\delta\left(e_R-\hat e_{\hat b_R}\right) S^{\dagger}_{n} S_{\bar n} \vert 0\rangle \ . \end{align} This factorization theorem will be the basis for our subsequent analysis. The event-wide observable from eq.~\eqref{eq:eventwideangularity} is obtained by summing over the left and right hemispheres \begin{equation} \frac{d\sigma}{d\ang{\beta}_{\rm event}} = \int d \ang{\beta}_L \, d \ang{\beta}_R \, \frac{d^2\sigma}{d\ang{\beta}_Ld\ang{\beta}_R} \, \delta\left(\ang{\beta}_{\rm event} - \ang{\beta}_L - \ang{\beta}_R\right), \end{equation} and the procedure for calculating the jet-based observables is given in section~\ref{sec:shapefactor}. \fussy \pagebreak The key ingredient needed to derive this factorization theorem was eq.~\eqref{recoil_power_corrections_for_jet_function}, so one can really think of eq.~\eqref{recoil_power_corrections_for_jet_function} as \emph{defining} what it means to be a recoil-free observable. Indeed, the energy correlation functions (section~\ref{sec:comparison}) and angularities with respect to the winner-take-all axis (section~\ref{subsec:WTA}) satisfy the same property, so the factorization theorem above applies equally well to those observables. Of course, even though the jet function is insensitive to soft recoil effects, the jet function does depend on the relative angles between collinear radiation. Consequently, different recoil-free observables will have different jet (and soft) functions, even though they share the same factorization theorem. As we discuss in detail in section~\ref{subsec:anomdimrelatsion}, recoil-free observables with the same behavior in the soft limit will necessarily have the same jet and soft anomalous dimensions to all orders, a fact which is generically not true for recoil-sensitive observables. ]]>
1$.} The global component of the cross section factorizes when the jets in the event are well-separated in angle compared to the jet radius $R_0$. Factorization-violating non-global~\cite{Dasgupta-ml-2001sh} or clustering~\cite{Banfi-ml-2005gj} logarithms are beyond the scope of this paper, but we do not expect the broadening axis to present any additional complications with respect to those issues. We now sketch a derivation of the factorization theorem for broadening-axis angularities measured on a single jet. The key point is that in the absence of recoil, it is possible to extend the factorization proof for thrust-axis angularities given in ref.~\cite{Ellis-ml-2010rwa} to the broadening-axis case. What ref.~\cite{Ellis-ml-2010rwa} showed is that even in the presence of a jet algorithm, the jet-based angularities factorize into contributions from soft and collinear modes within the jet (where the soft function itself depends on the light-like directions of unmeasured jets). The proof of ref.~\cite{Ellis-ml-2010rwa} is only valid for $\beta>1$, though, since only then is the jet function recoil-free (i.e.\ independent of the injected transverse momentum to leading power). However, apart from recoil, nothing in the proof of ref.~\cite{Ellis-ml-2010rwa} precluded an extension to $\beta < 1$, since the same power counting and identification of modes holds equally well for all $\beta > 0$. For the broadening-axis angularities, eq.~\eqref{recoil_power_corrections_for_jet_function} guarantees that there are no recoil effects for all $\beta > 0$. Thus, in the absence of recoil, we can simply recycle the $\beta > 1$ factorization theorem from ref.~\cite{Ellis-ml-2010rwa} and apply it to the jet-based broadening-axis angularities for all $\beta > 0$. The only subtlety in the above argument is that for $\beta < 1$, one has to use a recoil-free jet algorithm (see section~\ref{subsec:WTA}), such that the jet region is determined solely by the location of the collinear modes and is independent of soft recoil. Having dealt with that subtlety, then the jet-based factorization theorem is analogous to the event-wide factorization theorem from section~\ref{sec:eventfactor} in the sense that one can smoothly continue the result for $\beta > 1$ thrust-axis angularities to the $\beta>0$ broadening-axis angularities. In general, we would have to use the full machinery of ref.~\cite{Ellis-ml-2010rwa} to calculate the (recoil-free) jet-based angularities, including a careful treatment of the color structure of unmeasured jets. Since we will only ever work to \nllp\ accuracy, though, there is a simple procedure to determine the global structure of the cross section for the jet-based broadening-axis angularity from the corresponding event-wide result. Recall that our factorization theorem for the event-wide angularities in eq.~\eqref{eq:factorized-X-sec-fin} divided the event into two hemispheres, with each hemisphere containing a single collinear direction. As we define a jet by a single collinear direction, we can integrate over the unmeasured hemisphere to compute the cross section of the angularity in the measured hemisphere: \begin{equation} \frac{1}{\sigma_0}\frac{d\sigma}{d\ang{\beta}_{\rm jet}} = \int d\ang{\beta}_L\, \left. \frac{1}{\sigma_0}\frac{d^2\sigma}{d\ang{\beta}_Ld\ang{\beta}_R} \right|_{\ang{\beta}_R = \ang{\beta}_{\rm jet}} \ . \end{equation} At this point, we have made no approximations, but we are only describing hemisphere jets. To \nllp\ accuracy, the jet radius can be incorporated by boosting the hemisphere into a region of radius $R_0$ about the collinear direction. As shown in ref.~\cite{Ellis-ml-2010rwa}, this can be accomplished by appropriately rescaling the canonical soft scale by the jet radius, \begin{equation} \label{eq:softrescalingtojet} \mu_S \to \frac{\ang{\beta}Q_J}{2^{\beta-1}\sin^{\beta - 1} \frac{R_0}{2}} \,, \end{equation} where $Q_J$ is the energy of the jet and the jet scale is unchanged. This holds for both cone and $k_T$-type algorithms to \nllp, but higher order corrections will in general differ between the two algorithms. We will only consider the recoil-free jet algorithms presented in section~\ref{subsec:WTA} in the rest of this paper for which these results hold at \nllp\ accuracy. ]]>
0$, including broadening itself at $\beta = 1$. While previous studies of thrust-axis angularities required a different treatment of $\beta = 1$ compared to $\beta \ge 2$, our \nllp\ resummation of the broadening-axis observables achieves a smooth interpolation between the \scetii\ and \sceti\ regimes. Here, we used the broadening axis to define recoil-free jet observables, but one could envision other applications of the broadening axis, particularly at the LHC\@. By definition, a recoil-free axis is insensitive to perturbative soft radiation, but this same requirement means that it is insensitive to other kinds of soft jet contamination, including initial state radiation, underlying event, pileup contamination, detector noise, and even the QCD fireball in heavy-ion collisions. The advantage of defining jets with respect to the broadening axis is that the jet center will always be aligned along the hard collinear modes, though of course the jet energy will still be impacted by soft effects. In practice, we suspect that the winner-take-all axis (instead of the broadening axis) will become the preferred recoil-free axis for future jet studies, since it can be easily implemented in existing jet clustering algorithms like anti-$k_T$. In the context of jet substructure studies, the broadening axis can be used to measure the degree of recoil within a jet. Because a jet from a boosted object with $N$ hard prongs is unlikely to have one of its prongs aligned along the jet momentum axis, one can use the angle between the thrust axis and the broadening axis as a boosted object discriminant.\footnote{A similar observation was made in ref.~\cite{tmb}. Amusingly, for a single emission within a jet, the thrust-broadening angle is identical to recoil-free broadening, thus the two observables have the same resummation to NLL order, though not the same factorization theorem.} This kind of logic was exploited in ref.~\cite{Curtin-ml-2012rm} to define ``axis pull'' and ``axis contraction'' variables based on $N$-subjettiness. Similarly, measuring the thrust-broadening angle may help in discriminating pileup jets from QCD jets~\cite{CMS-ml-2013wea}. Finally, our focus has been on perturbative aspects of the broadening axis, but it would be interesting to study non-perturbative power corrections to $\ang{\beta}$, especially for $0 < \beta < 1$. The simple form of the broadening-axis factorization theorem suggests that it should be straightforward to include appropriate non-perturbative shape functions~\cite{Korchemsky-ml-1999kt,Korchemsky-ml-2000kp}. We expect that the form of broadening-axis power correction should differ from the thrust-axis case~\cite{Dokshitzer-ml-1998qp}, and it may be that the broadening-axis case more resembles the earlier broadening power correction study in ref.~\cite{Dokshitzer-ml-1998pt}. Of particular interest would be to compare the non-perturbative corrections between the angularities $\ang{\beta}$ and the energy correlation functions $\C{1}{\beta}$ for $0 < \beta < 1$. While both are recoil-free, $\ang{\beta}$ is sensitive to $\mathcal{O}(\Lambda_{\rm QCD}/Q)$ angles between individual collinear modes and the broadening axis while $\C{1}{\beta}$ is sensitive to $\mathcal{O}(\Lambda_{\rm QCD}/Q)$ angles between all pairs of collinear modes. Hopefully, such a study would shed light on the comparative advantages of axis-based versus axis-free jet observables. ]]>
0: \qquad \left| p_{T1} \right|+\left| p_{T2} \right|+\left| p_{Ts} \right| =\left(\frac{1}{2}-\epsilon\right)\theta_C+\delta\phi + 2\epsilon \gamma, \end{equation} which is minimized for $\gamma = 0$. That is, the broadening axis is unchanged under soft, wide angle emission from hard particles. Because of this, the change in the value of any angularities $\ang{\beta}$ measured about the broadening axis before and after the soft emission is suppressed by the energy $\delta$ of the soft emission. Now consider the case where the soft emission lies between particles 1 and 2. In this case, eq.~\eqref{eq:scalarsumthreeparticle} becomes \begin{equation} -\theta_C < \phi < 0: \qquad \left| p_{T1} \right|+\left| p_{T2} \right|+\left| p_{Ts} \right| =\left( \frac{1}{2}-\epsilon \right) \theta_C+\delta\phi + 2(\epsilon-\delta) \gamma \,, \end{equation} and the location of the broadening angle $\gamma$ depends on the precise relationship between $\delta$ and $\epsilon$. For $\delta > \epsilon$, the sum is minimized when the broadening axis lies along the direction of the soft emission. While this might seem like it results in the broadening axis being IRC unsafe, the broadening axis discontinuously moves back to particle 1 once $\delta$ decreases below $\epsilon$ (in particular, in the soft limit $\delta \to 0$). This discontinuous change should not be cause for concern, though, since the phase space for the soft emission to lie in the plane between particles 1 and 2 has zero measure and is not enhanced by any singularities. If instead the soft emission lies out of the plane of the two hard particles, then the broadening axis returns to the hardest particle continuously as the energy of the soft emission becomes small. Of course, for this to be consistent with the behavior when the soft emission is in the plane, the rate of return increases without bound as the soft emission approaches the plane of the hard particles. Thus, for sufficiently soft emissions, the broadening axis is unchanged, explicitly illustrating its IRC safety. ]]>
0$, the quark jet function can be computed in this frame from \begin{align} J^{(1)}_q\left(\ang{\beta}\right)&=g^2 \bar{\mu}^{2\epsilon} \nu^\eta C_F \int \frac{dl^+}{2\pi} \frac{1}{(l^+)^2}\int \frac{d^d q}{(2\pi)^d}\, \left(q^-\right)^{-\eta} \, \left( 4 \frac{l^+}{q^-}+(d-2)\frac{l^+-q^+}{Q-q^-} \right)\, \nonumber \\ &\quad \times 2\pi\delta\left(q^+q^--q_\perp^2\right)\Theta\left(q^+\right)\Theta\left(q^-\right)\nonumber \\ &\quad \times \Theta\left(Q-q^-\right) \Theta\left(l^+-q^+\right) 2\pi\delta\left( l^+-q^+ -\frac{q_\perp^2}{Q-q^-} \right) \nonumber \\ &\quad \times \left\{ \Theta\left( \frac{Q}{2}-q^- \right) \delta\left( \ang{\beta} - (2Q)^{\beta-1}\left(Q-q^-\right)^{-\beta} \left(q^-\right)^{1-\beta}q_\perp^\beta \right) \right. \nonumber\\ &\left. \qquad \quad+\ \Theta\left( q^- -\frac{Q}{2} \right)\delta\left( \ang{\beta} - (2Q)^{\beta-1}\left(Q-q^-\right)^{1-\beta} \left(q^-\right)^{-\beta}q_\perp^\beta \right) \right\} \,, \end{align} in $d=4-2\epsilon$ dimensions with $\overline{\text{MS}}$ scale $\bar{\mu}$ defined as \begin{equation} \bar{\mu}^2 = \mu^2 \frac{e^{\gamma_E}}{4\pi} \,, \end{equation} where $\gamma_E$ is the Euler-Maschroni constant. We have also introduced $\eta$ which regulates rapidity divergences for $\beta=1$ and has a corresponding scale $\nu$. Similarly, the gluon jet function can be computed from \begin{align} J^{(1)}_g\left(\ang{\beta}\right)&=2g^2 \bar{\mu}^{2\epsilon}\nu^\eta \int \frac{dl^+}{2\pi} \frac{1}{l^+}\int \frac{d^d q}{(2\pi)^d}\, (q^-)^{-\eta} \,\frac{1}{Q-q^-} \, 2\pi\delta\left(q^+q^--q_\perp^2\right)\nonumber \\ &\quad \times\Theta\left(q^+\right)\Theta\left(q^-\right) \Theta\left(Q-q^-\right) \Theta(l^+-q^+) 2\pi\delta\left( l^+-q^+ -\frac{q_\perp^2}{Q-q^-} \right) \nonumber \\ &\quad \times \left[ n_f T_R \left(1-\frac{2}{1-\epsilon}\frac{q^+q^-}{Q l^+}\right) - C_A \left( 2-\frac{Q}{q^-} - \frac{Q}{Q-q^-}-\frac{q^+q^-}{Q l^+} \right) \right] \nonumber \\ &\quad \times \left\{ \Theta\left( \frac{Q}{2}-q^- \right) \delta\left( \ang{\beta} - (2Q)^{\beta-1}\left(Q-q^-\right)^{-\beta} \left(q^-\right)^{1-\beta}q_\perp^\beta \right) \right. \nonumber\\ &\left. \quad\qquad+\ \Theta\left( q^- -\frac{Q}{2} \right)\delta\left( \ang{\beta} - (2Q)^{\beta-1}\left(Q-q^-\right)^{1-\beta} \left(q^-\right)^{-\beta}q_\perp^\beta \right) \right\} \ . \end{align} Performing the integrals, the quark jet function is \begin{align}\label{bare_quark_jet_function} J_q^{(1)}\left(\ang{\beta}\right)&=\frac{\alpha_s}{\pi}\frac{C_F}{\beta}4^{\frac{\beta-1}{\beta}\epsilon} \frac{e^{\gamma_E \epsilon}}{\Gamma(1-\epsilon)}{\ang{\beta}}^{-1-\frac{2\epsilon}{\beta}}\left(\frac{\mu^2}{Q^2}\right)^\epsilon\left(\frac{\nu}{Q}\right)^\eta\left\{\frac{2\beta}{2(\beta-1)\epsilon+\beta\eta}+\frac{3}{2}\right.\nonumber\\ &\quad\left.-\ \epsilon\left(-\frac{13}{2}+\frac{2\pi^2}{3}+\frac{3}{\beta}-\frac{\pi^2}{3\beta}+\frac{3\log2}{\beta}\right)+{\cal O}\left(\epsilon^2\right)\right\} \,, \end{align} and the gluon jet function is \begin{align}\label{bare_gluon_jet_function} J_g^{(1)}\left(\ang{\beta}\right)&=\frac{\alpha_s}{\pi}\frac{C_A}{\beta}4^{\frac{\beta-1}{\beta}\epsilon}\frac{e^{\gamma_E \epsilon}}{\Gamma(1\!-\!\epsilon)}{\ang{\beta}}^{-1-\frac{2\epsilon}{\beta}}\left(\frac{\mu^2}{Q^2}\right)^\epsilon\left(\frac{\nu}{Q}\right)^\eta\left\{\frac{2\beta}{2(\beta\!-\!1)\epsilon\!+\!\beta\eta}\!+\!\frac{11}{6}-\frac{1}{3}\frac{n_f T_F}{C_A}\right.\nonumber\\ &\quad\left.- \ \epsilon\left(-\frac{67}{9}+\frac{2\pi^2}{3}+\frac{137}{36\beta}-\frac{\pi^2}{3\beta}+\frac{11\log2}{3\beta}+\frac{n_fT_R}{C_A}\left(\frac{23}{18}-\frac{29}{36\beta}-\frac{2\log2}{3\beta}\right)\right)\right. \nonumber \\ &\left.\quad +\ {\cal O}\left(\epsilon^2\right) \vphantom{\frac12} \right\} \ . \end{align} For consistency of the factorization, the limit $\eta\to 0$ must be taken first, and then the limit $\epsilon\to 0$ can be taken. The rapidity regulator $\eta$ is therefore only active if $\beta=1$ and does not appear in the jet function if $\beta \neq 1$. Expanding in $\eta$ and $\epsilon$ isolates the singularities of the jet function which allows for renormalization. \boldmath ]]>
k^+$, then the soft emission is closer to the $\bar{n}$ axis and its angle with respect to the $\bar{n}$ axis is \begin{equation} \sin^2\frac{\theta}{2} = \frac{k^+}{k^- + k^+} \ . \end{equation} The energy of a soft emission is \begin{equation} E=\frac{k^- + k^+}{2} \,, \end{equation} and so the value of the angularity when $k^->k^+$ is \begin{equation} \ang{\beta} = 2^{\beta-1}Q^{-1}\left(k^+\right)^{\beta/2}\left(k^- + k^+\right)^{1-\beta/2}\ . \end{equation} When $k^+>k^-$, the expression for the angularity is the same, with $k^+\leftrightarrow k^-$. Then, at one-loop for dijet events, the soft function can be computed from \begin{align} S^{(1)}(\ang{\beta})&=4g^2\bar{\mu}^{2\epsilon}\nu^\eta C_i\int\frac{d^dk}{(2\pi)^d} \frac{\left|k^-- k^+\right|^{-\eta} }{k^+ k^-}\,2\pi \delta\left(k^2\right)\Theta\left(k^0\right) \nonumber \\ & \quad \times \left\{ \Theta\left(k^- - k^+\right) \delta\left( \ang{\beta}-2^{\beta-1}Q^{-1}\left(k^+\right)^{\beta/2}\left(k^- + k^+\right)^{1-\beta/2} \right)\right.\nonumber \\ &\quad\qquad\left.+\ \Theta\left(k^+-k^-\right) \delta\left( \ang{\beta}-2^{\beta-1}Q^{-1}\left(k^-\right)^{\beta/2}\left(k^++k^-\right)^{1-\beta/2} \right) \right\} . \end{align} $C_i$ is the total color of the jets in each hemisphere and $\eta$ is the rapidity regulator. Evaluating this expression, we find \begin{align}\label{eq:softfunc} S^{(1)}(\ang{\beta})&= \frac{\alpha_s}{\pi} C_i \frac{2^{1-(1-\beta)(\eta+2\epsilon)}e^{\gamma_E\epsilon}}{\Gamma(1-\epsilon)}{\ang{\beta}}^{-1-2\epsilon-\eta}\left(\frac{ \mu^2}{Q^2} \right)^\epsilon \left( \frac{\nu}{Q} \right)^\eta \\ & \quad \times \frac{\Gamma(1-\eta)\Gamma\left(\frac{\beta}{2}\eta+(1-\beta)\epsilon\right)}{\Gamma\left(1-(1-\beta)\epsilon-\frac{1}{2}(2-\beta)\eta\right)}\nonumber\\ &\quad \times \,_{2}F_{1}\left(-(1-\beta)\epsilon+\frac{\beta}{2}\eta,\frac{1}{2}(\beta-2)(2\epsilon+\eta);1-(1-\beta)\epsilon+\frac{1}{2}(\beta-2)\eta;-1\right), \nonumber \end{align} where $_2F_1(a,b;c;z)$ is the hypergeometric function. As for the jet function, we must first take $\eta\to0$ and then $\epsilon\to 0$ and so the rapidity regulator is only relevant for $\beta = 1$. \boldmath ]]>
0$ is then given by: \begin{align}\label{eq:fo_exp} \frac{1}{\sigma_0}\frac{d\sigma}{d\ang{\beta}}&= \frac{\alpha_s C_F}{\pi}\int_0^{1}dx\int_x^{1}dy\, \Theta(1-x-2y)\, \delta\left(\ang{\beta}-2^{\beta-1}x^{\frac{\beta}{2}}(1-y)^{-\frac{\beta}{2}}(x+y)^{1-\frac{\beta}{2}}\right)\nonumber\\ &\quad\times\left\{\frac{2-3(1+x)(1-y)+3\left(1+x^2\right)(1-y)^2+3x(1+x)(1-y)^3}{x (1-y)(1-(1+x)(1-y))}\right\}, \end{align} where within a sector, before relabeling, $x$ is the invariant mass between second-most and least energetic partons, normalized to $Q^2$. These particles form the broadening axis and the measured parton. Then $y$ is the invariant mass between the least energetic parton and the most energetic parton. If $\beta=2$, all integrals can be performed analytically, and the result is: \begin{align} \frac{1}{\sigma_0}\frac{d\sigma}{d\ang{2}} &=\frac{\alpha_s C_F}{\pi}\left\{6\frac{\left(1-\ang{2}\right)\left(-32-4\ang{2}+6\left(\ang{2}\right)^2-4\left(\ang{2}\right)^3+\left(\ang{2}\right)^4\right)}{\ang{2}\left(2-\ang{2}\right)\left(4-\ang{2}\right)^2\left(2+\ang{2}\right)}\right.\nonumber\\ &\quad\left.-\frac{4}{\ang{2}\left(2-\ang{2}\right)}\text{ln}\left[\frac{\ang{2}}{2-\ang{2}}\right]+\frac{3}{2}\text{ln}\left[\frac{\ang{2}\left(4-\ang{2}\right)}{4-\left(\ang{2}\right)^2}\right] \vphantom{\frac{\left(1-\ang{2}\right)\left(-32-4\ang{2}+6(\ang{2})^2-4\left(\ang{2}\right)^3+\left(\ang{2}\right)^4\right)}{\ang{2}\left(2-\ang{2}\right)\left(4-\ang{2}\right)^2\left(2+\ang{2}\right)}}\right\}. \end{align} When $\beta\neq 2$, one of the integrals can be done via the delta function constraint after the change of variables: \begin{align} y&\rightarrow u-(1-u)v \,,\nonumber \\ x&\rightarrow (1-u)v \ . \end{align} The remaining integral can be computed numerically. We can split the fixed-order cross section at ${\cal O}(\alpha_s)$ into two contributions. First, there is the singular contribution that can be computed in the effective theory. This is defined as the part of the cross section in eq.~\eqref{eq:totangexp} that remains when $\ang{\beta}\to 0$: \begin{align}\label{eq:sing_distribution_event} \frac{\ang{\beta}}{\sigma_0} \frac{d\sigma^\text{sing}}{d\ang{\beta}} &= \lim_{\ang{\beta}\to 0}\frac{\ang{\beta}}{\sigma_0} \frac{d\sigma}{d\ang{\beta}}\nonumber \\ &=-\frac{\alpha_s C_F}{\pi\beta}\left[4\log \left(2^{1-\beta}\ang{\beta}\right)+3\right] \ . \end{align} After subtracting off the singular contribution, the rest defines the non-singular contribution. The non-singular contribution is beyond the scope of the effective theory and is necessary for matching the resummed cross section to achieve formal next-to-leading order (NLO) accuracy. There are two non-trivial checks of the fixed-order result in eq.~\eqref{eq:fo_exp}. The first is that it must reproduce the singular behavior in eq.~\eqref{eq:sing_distribution_event}. The second is that the cross section must vanish at \begin{equation}\label{eq:max_ang} \ang{\beta}_{\max} =3^{\frac{\beta}{2}-1}. \end{equation} This is because the maximum value of the angularity at $\mathcal{O}(\alpha_s)$ is given by the ``Mercedes-Benz'' configuration when all three partons have equal energy. In eq.~\eqref{eq:fo_exp}, this corresponds to $x$ and $y$ in the integrand both equaling $1/3$, at which point $\ang{\beta} = \ang{\beta}_{\max}$. In figure~\ref{fig:non-sing}, we plot singular and non-singular contributions to the cross section at ${\cal O}(\alpha_s)$ for $\beta = 0.5,1,2$. The two above cross checks are satisfied because the non-singular piece goes to zero as $\ang{\beta} \to 0$ and $\sigma^\text{sing} + \sigma^\text{non-sing} = 0$ at $\ang{\beta} = \ang{\beta}_{\max}$. The resummation is important in the region where the singular contribution to the cross section dominates and the fixed order result is important where the non-singular contribution dominates. As we discuss in appendix~\ref{sec:scalechoice}, we will turn off the resummation when $\sigma^\text{sing}$ and $\sigma^\text{non-sing}$ \linebreak are~equal. \begin{figure}[t] \centering \includegraphics[trim = 0 0 0 0, clip, width=10cm]{figures/sing_V_non_sing.pdf} ]]>
1,\\ \label{def_can_region_jet}\text{Jet: }\frac{\alpha_s \left(\ang{\beta}_cQ\right)}{\beta}C_i\log \ang{\beta}_c\sim 1\,\text{if } \beta<1, \end{align} where $C_i$ is the appropriate color factor for the jet. The resummation should be completely turned off when the perturbative non-singular corrections become as important as the singular pieces, so \begin{align} \frac{d\sigma^\text{sing}}{d\ang{\beta}}\left(\ang{\beta}_e\right)\sim\frac{d\sigma^\text{non-sing}}{d\ang{\beta}}\left(\ang{\beta}_e\right). \end{align} Together, the tail and far-tail regions is defined by $\ang{\beta}_c<\ang{\beta}<\ang{\beta}_e$, and the boundary between them is somewhat ambiguous. For concreteness, we set the tail/far-tail boundary by $\ang{\beta}_t\sim\ang{\beta}_e/3$. A peculiar feature of these profiles is that the $\ang{\beta}_c$ and $\ang{\beta}_e$ boundaries start to coincide as $\beta \to 0$. This can be understood because $\ang{\beta}_c$ grows as $\beta\rightarrow 0$ but the position where the singular corrections vanish remains fixed. Hence the tail plus far-tail region necessarily shrinks. Thus, unlike the case for thrust, as $\beta \to 0$ one can expect both singular and non-singular corrections to be substantial in the peak/tail regions of the cross section. We saw this in the plots in section~\ref{subsec:NLLprime}, where there are large differences between the NLL and \nllp+NLO calculations even in the peak region. \begin{figure}[t] \centering \subfloat[]{\label{fig:prof_2} \includegraphics[trim = 0 0 0 0, clip, width=4.4cm]{figures/ProfilePlotBeta2.pdf} } $\quad$ \subfloat[]{\label{fig:prof_1} \includegraphics[width=4.4cm]{figures/ProfilePlotBeta1.pdf} } $\quad$ \subfloat[]{\label{fig:prof_05} \includegraphics[trim = 0 0 0 0, clip, width=4.5cm]{figures/ProfilePlotBetap5.pdf} } ]]>