The 1-Jettiness DIS event shape: NNLL + NLO results

TL;DR

This work delivers a complete NNLL+NLO description of the 1-jettiness tau1 spectrum for DIS jet production, unifying resummation in the small-tau region with fixed-order perturbation theory at large tau1. It implements a SCET-based factorization for resummation, exact NLO fixed-order calculations with explicit jet algorithms (anti-$k_T$, $R=1.0$), and a smooth matching between regions, including non-perturbative soft radiation modeling and a jet-shape analysis to probe radiation patterns and jet energy loss. The paper validates infrared pole cancellation, demonstrates controlled scale uncertainties via profile functions, and demonstrates jet-algorithm independence in the resummation region while highlighting algorithm dependence in the fixed-order region. The framework is positioned for precision QCD studies and nuclear dynamics probes with HERA data and future EIC/LHeC measurements, with broad flexibility for jet algorithms and radiation-content variations inside jets.

Abstract

We present results for the complete NNLL+NLO (~ α_s) 1-jettiness (τ_1) event shape distribution for single jet (J) production in electron-nucleus (N_A) collisions e^- + N_A \to e^- + J + X, in the deep inelastic scattering (DIS) region where the hard scale is set by the jet transverse momentum P_{J_T}. These results cover the entire τ_1-spectrum including the resummation (τ_1<<P_{J_T}) and fixed-order (τ_1~ P_{J_T}) perturbative QCD regions. They incorporate non-perturbative soft radiation effects, the anti-k_T jet algorithm in the fixed-order calculation, and a smooth matching between the resummation and fixed-order perturbative QCD regions. The matching smoothly connects the spectrum in the resummation region, which can be computed without reference to an external jet algorithm, and the fixed-order region where an explicit jet algorithm must be specified. Our code, used for generating the numerical results, is flexible enough to incorporate different jet algorithms for the fixed-order calculation. We also perform a jet shape analysis, defined within the 1-jettiness framework, which allows one to control the amount of radiation included in the definition of the final state jet. This formalism can allow for detailed studies of jet energy-loss mechanisms and nuclear medium effects. The analysis presented here can be used for precision studies of QCD and as a probe of nuclear dynamics using data collected at HERA and in proposed future electron-ion colliders such as the EIC and the LHeC.

Figures (9)

• Figure 1: Schematic figure of the process $e^- + p \to J + X$ in the limit $\tau_1\ll P_{J_T}$. The restriction $\tau_1\ll P_{J_T}$ (left panel) allows only soft radiation between the beam and jet directions. In the region of large 1-jettiness $\tau_1 \sim P_{J_T}$ (right panel), additional hard radiation is allowed at wide angles from the leading jet and beam directions.
• Figure 2: Cancellation of $\epsilon^{-2}$ (left panel) and $\epsilon^{-1}$ (right panel) IR poles in Eqs. (\ref{['IRpole1']}) and (\ref{['IRpole2']}) with numerical errors, as a function of machine center of mass energy squared $s$. This serves a non-trivial check on the consistency of our results.
• Figure 3: In this plot, we compare the difference between the NLO QCD cross section $\sigma_{\rm QCD}^{\rm FO}$ and the expanded SCET ${\cal O}(\alpha_s)$ prediction $\sigma_{\rm SCET}^{\rm FO}$, weighted to $\sigma_{\rm QCD}^{\rm FO}$. The details on both cross sections are explained in the text. In the resummation region where $\tau_1 \ll p_{J_T}$, the difference between these two predictions scales as $\tau_1/p_{J_T} \log^2(\tau_1/p_{J_T})$. Therefore as $\tau_1 \to 0$, the difference tends to $0$, as one can see from this plot, which implies that SCET correctly reproduces the singular terms.
• Figure 4: We plot $|\mathrm{d}\sigma/\mathrm{d}\tau_1|$ for both full NLO QCD (red solid line) and the expanded SCET singular (blue dashed line) predictions as a function of $\tau_1$. We see that when $\tau_1 \sim 5 {\rm GeV}$, the singular contribution goes negative and the cross section is dominated by the nonsingular pieces which can not be predicted by SCET. This implies that the resummation should start to be turned off around this point and switched to the fixed-order QCD prediction smoothly. This figure justifies the parameters we choose for the profile-scales for matching as explained in the text.
• Figure 5: Profile functions for the SCET scales $\mu_{S,B,J}$ are chosen so that they converge to $\mu_{FO}$ in the large $\tau_1$ region where perturbarive QCD is appropriate. The left panel shows the collective scale variation defined through Eqs. (\ref{['profile1']}) and (\ref{['muFO']}). The right panel shows the SCET resummation scale variation defined through Eqs. (\ref{['profile1']}) and (\ref{['muSCET']}), corresponding to jet binning uncertainties. The total scale variation uncertainty is given by adding these two types of scale variation uncertainties in quadrature as in Eq.(\ref{['band']}).