Unsafe but Calculable: Ratios of Angularities in Perturbative QCD

TL;DR

The paper shows that ratios of IRC-safe jet angularities, though not IRC safe themselves, can be computed in perturbative QCD using resummation due to Sudakov safety. By deriving the leading-log double differential distribution d^2σ/(de_α de_β) and marginalizing to obtain dσ/dr, it reveals a unique sqrt(α_s) scaling and demonstrates reliable predictions via Monte Carlo showers that include multiple emissions. Higher-order refinements (MLL, LO matching) and non-perturbative analyses through shape functions confirm that NP effects are power-suppressed at high jet energies, validating perturbative control over ratio observables. The results generalize to other ratio-type observables and underscore the broader landscape of calculable quantities beyond IRC-safe constraints in perturbative QCD.

Abstract

Infrared- and collinear-safe (IRC-safe) observables have finite cross sections to each fixed-order in perturbative QCD. Generically, ratios of IRC-safe observables are themselves not IRC safe and do not have a valid fixed-order expansion. Nevertheless, in this paper we present an explicit method to calculate the cross section for a ratio observable in perturbative QCD with the help of resummation. We take the IRC-safe jet angularities as an example and consider the ratio formed from two angularities with different angular exponents. While the ratio observable is not IRC safe, it is "Sudakov safe", meaning that the perturbative Sudakov factor exponentially suppresses the singular region of phase space. At leading logarithmic (LL) order, the distribution is finite but has a peculiar expansion in the square root of the strong coupling constant, a consequence of IRC unsafety. The accuracy of the LL distribution can be further improved with higher-order resummation and fixed-order matching. Non-perturbative effects can sometimes give rise to order one changes in the distribution, but at sufficiently high energies Q, Sudakov safety leads to non-perturbative corrections that scale like a (fractional) power of 1/Q, as is familiar for IRC-safe observables. We demonstrate that Monte Carlo parton showers give reliable predictions for the ratio observable, and we discuss the prospects for computing other ratio observables using our method.

Figures (14)

• Figure 1: Summary of results from this paper. Plotted is the differential cross section for the angularity ratio $r_{2,1} \equiv e_2 / e_1$. The curves appear in increasing order of theoretical accuracy, with each subsequent curve including all previous effects. "LL" is the baseline leading logarithmic result (with the scaling $\alpha_s \log^2 r \simeq 1$) that demonstrates Sudakov safety. "LL+LO" includes $\mathcal{O}(\alpha_s)$ fixed-order matching in the Log-R scheme. "MLL+LO" includes modified leading log resummation which has running $\alpha_s$. "MLL+MC" is Monte Carlo resummation which includes multiple emissions. Finally, "MLL+MC+$\delta_{\rm NP}$" includes an estimate of non-perturbative corrections through a simple shape function. We note that most parton shower Monte Carlo programs include all of these effects.
• Figure 2: Jets and angles defined with respect to the broadening axis $\hat{b}$ defined in Eq. (\ref{['eq:broadening_axis']}). For two particles, the broadening axis coincides with the direction of the most energetic particle. Throughout this paper, we focus on jets initiated by an energetic quark, and take the jet radius to be $R_0 = 1$.
• Figure 3: Allowed phase space region for the double differential cross section in the $(e_\alpha,e_\beta)$ plane. The allowed phase space is gray, lines of constant ratio $r_{\alpha,\beta} \equiv e_\alpha/e_\beta$ are illustrated by dashed lines, and the forbidden regions are white. The boundary of the lower forbidden region has zero slope only at $e_\beta=0$, such that $r_{\alpha,\beta} = 0$ is only possible at the origin.
• Figure 4: Phase space for strongly-ordered emissions in the $\left(\log1/\theta,\log1/z\right)$ plane. Left: a single emission dominates the value of both $e_\alpha$ and $e_\beta$. Right: one emission dominates the value of $e_\alpha$ in blue, while a second emission dominates the value of $e_\beta$ in red. In both cases, further emissions are forbidden in the gray region below the solid black line, and the area of the gray forbidden region determines the Sudakov factor. The singular region of phase space is up and to the right.
• Figure 5: LL differential cross section for the ratio observable $r_{\alpha,\beta} \equiv e_\alpha/e_\beta$ from Eq. (\ref{['eq:resumrLL']}). Left: numerator fixed to $\alpha = 2$ (thrust measure), and denominator sweeping over $\beta$. Right: $\alpha = 1$ (broadening measure), sweeping $\beta$. In both cases, the $\beta = 0$ curves give the LL differential cross section for the angularity with the corresponding $\alpha$, i.e. $r_{\alpha,0} = e_\alpha$.