Inclusive jet spectrum for small-radius jets

TL;DR

The paper tackles the challenge of predicting the inclusive jet spectrum at small radius $R$, where terms of the form $\alpha_s^n \ln^n 1/R^2$ threaten fixed-order accuracy. It develops a systematic small-$R$ resummation framework ($LL_R$), devises a robust multiplicative matching to fixed-order results up to NNLO, and introduces a stand-in NNLO$_R$ to capture $R$-dependent NNLO effects in the absence of the full calculation. Hadronisation and underlying-event corrections are incorporated via Monte Carlo corrections, and predictions are confronted with ALICE and ATLAS data to test $R$-dependence, revealing improved agreement when both LL$_R$ and NNLO$_R$ contributions are included. The study also highlights sizable subleading $R$-enhanced NNLO terms and advocates uncorrelated scale variation to obtain realistic uncertainties, recommending measurements at three radii to disentangle perturbative, hadronisation, and UE effects. Overall, the work delivers a more reliable description of the inclusive jet spectrum across radii, informing precision jet phenomenology and future PDF and $\alpha_s$ determinations.

Abstract

Following on our earlier work on leading-logarithmic (LLR) resummations for the properties of jets with a small radius, R, we here examine the phenomenological considerations for the inclusive jet spectrum. We discuss how to match the NLO predictions with small-R resummation. As part of the study we propose a new, physically-inspired prescription for fixed-order predictions and their uncertainties. We investigate the R-dependent part of the next-to-next-to-leading order (NNLO) corrections, which is found to be substantial, and comment on the implications for scale choices in inclusive jet calculations. We also examine hadronisation corrections, identifying potential limitations of earlier analytical work with regards to their $p_t$-dependence. Finally we assemble these different elements in order to compare matched (N)NLO+LLR predictions to data from ALICE and ATLAS, finding improved consistency for the R-dependence of the results relative to NLO predictions.

Figures (24)

• Figure 1: Impact of $R$-dependent terms in the inclusive-jet spectrum, illustrated using the small-$R$ resummation factor obtained from the ratio of $\sigma^\text{LL$_R$}\xspace$ in Eq. (\ref{['eq:LLR-master']}) to the leading order inclusive jet spectrum $\sigma^\text{LO}\xspace$. It is shown as a function of the jet $p_t$ for different jet radius values. For each $R$ value, the plot illustrates the impact of two choices of $R_0$: $R_0= 1$ (our default) as solid lines and $R_0 = 1.5$ as dashed lines.
• Figure 2: Left: Comparison of the $R$ dependence in the exact and small-$R$ approximated NLO expansion, using Eq. (\ref{['eq:LLR-validation-ratio']}), shown as a function of jet transverse momentum $p_t$, for $\sqrt{s} = 7\,\mathrm{TeV}$ in the rapidity region $|y| < 0.5$. Right: comparison of $\Delta_{1+2}(p_t, R, R_\text{ref})$ and $\Delta_{1+2}^\text{LL$_R$}\xspace(p_t, R, R_\text{ref})$ (cf. Eq. (\ref{['eq:LLR-validation-ratio-Delta12']})). In both plots CT10 NLO PDFs Lai:2010vv are used, while the renormalisation and factorisation scales are set equal to the $p_t$ of the highest-$p_t$$R=1$ jet in the event (this same scale is used for all $R$ choices in the final jet finding).
• Figure 3: Left: size of the matching normalisation factor (left-hand factor of Eq. (\ref{['eq:multiplicative-matching-schemeB']}), normalised to LO), shown v. $p_t$ for various $R$ values and two $R_0$ choices. Right: size of the matched small-$R$ fragmentation factor (right-hand factor of Eq. (\ref{['eq:multiplicative-matching-schemeB']}); similar results are observed for the right-hand factor of Eq. (\ref{['eq:multiplicative-NLO']})). The results are shown for the scale choice $\mu_R = \mu_F = p_{t,\max}$, where $p_{t,\max}$ is the transverse momentum of the hardest jet in the event.
• Figure 4: Inclusive jet cross section for $p_t > 100\,\mathrm{GeV}$, as a function of $R$, normalised to the ($R$-independent) leading-order result. Left: the standard NLO result, compared to the "NLO-mult." result of Eq. (\ref{['eq:multiplicative-NLO']}) and the NLO+LL$_R$ matched result of Eq. (\ref{['eq:multiplicative-matching-schemeB']}). The scale uncertainty here has been obtained within a prescription in which the scale is varied simultaneously in the left and right-hand factors of Eqs. (\ref{['eq:multiplicative-matching-schemeB']}) and (\ref{['eq:multiplicative-NLO']}) ("correlated scale choice"). Right: the same plot, but with the scale uncertainties determined separately the left and right-hand factors of Eqs. (\ref{['eq:multiplicative-matching-schemeB']}) and (\ref{['eq:multiplicative-NLO']}), and then added in quadrature ("uncorrelated scale choice"). The plot also shows the NLO+LL$_R$ result for $R_0=1.5$ at our central scale choice.
• Figure 5: Left: comparison of the NLO, NNLO$_R$ and NNLO$_R$-mult. results for the inclusive jet cross section for $p_t > 100 \,\mathrm{GeV}$, as a function of $R$, normalised to the LO result. Right, corresponding comparison of NLO, NNLO$_R$ and NNLO$_R$+LL$_R$ together with the central curve for NNLO$_R$+LL$_R$ when $R_0$ is increased to $1.5$. In both plots, for the NNLO$_R$-mult. and NNLO$_R$+LL$_R$ results the scale-dependence has been evaluated separately in the normalisation and fragmentation contributions and added in quadrature to obtain the final uncertainty band.