Energy-energy correlators in deep inelastic scattering

The energy-energy correlator (EEC) is an event shape observable which probes the angular correlations of energy depositions in detectors at high energy collider facilities. It has been investigated extensively in the context of precision QCD. In this work, we introduce a novel definition of EEC adapted to the Breit frame in deep-inelastic scattering (DIS). In the back-to-back limit, the observable we propose is sensitive to the universal transverse momentum dependent (TMD) parton distribution functions and fragmentation functions, and it can be studied within the traditional TMD factorization formalism. We further show that the new observable is insensitive to experimental pseudorapidity cuts, often imposed in the laboratory frame due to detector acceptance limitations. In this work the singular distributions for the new observable are obtained in soft collinear effective theory up to $\mathcal{O}({\alpha }_s^3)$ and are verified by the full QCD calculations up to $\mathcal{O}({\alpha }_s^2)$. The resummation in the singular limit is performed up to next-to-next-to-next-to-leading logarithmic accuracy. After incorporating nonperturbative effects, we present a comparison of our predictions to pythia 8 simulations.

Figure 1

Illustration of the measurement of the transverse momentum ${\mathit{q}}_{\perp }$ of the hadron-$a$ with respect to the proton axis in the Breit frame.

Figure 2

${\mathrm{TEEC}}_{\text{Lab}}$ (left) and ${\mathrm{EEC}}_{\mathrm{DIS}}$ (right) distributions from pythia 8 with different rapidity cuts in the lab frame. For ${\mathrm{TEEC}}_{\text{Lab}}$ we consider a cut on the scattered lepton transverse momentum $p_T^e>20\mathrm{GeV}$ and for ${\mathrm{EEC}}_{\mathrm{DIS}}$ we consider cut on the photon virtuality $Q>20\mathrm{GeV}$. For both case we consider cut on $y=2P·q/s$: $0.01<y<0.95$.

Figure 3

Fixed-order $\ln \tau$ distributions in the $\tau \to 1$ limit. The solid lines represent the singular distribution predictions of SCET. The dashed lines are the fixed order QCD results. The dash-dotted lines are power corrections.

Figure 4

Resummed $\ln \tau$ distributions for EEC. The dark red, dark blue, and dark green bands correspond to NLL, NNLL, and NNNLL distribution. The dot-dashed lines are the LO nonsingular distributions (dark red) and the absolute value of NLO nonsingular ones (dark green).

Figure 5

Left: Resummed $\ln \tau$ distributions without (solid) and with nonperturbative factor. The color scheme is the same as in Fig. 4. Right: Comparing the effect of the hadronization effects and TMDPDF effects in the small transverse momentum region for $\mathrm{N}{}^3\mathrm{L}\mathrm{L}$ only.

Figure 6

Comparison of $\cos \theta$ distributions between the SCET predictions and pythia simulations. The dark red and dark green bands are the $\mathrm{NLL}+\mathrm{NLO}$ and $\mathrm{N}{}^3\mathrm{L}\mathrm{L}+\mathrm{NLO}$. The gray bands are from pythia 8 simulations with the default settings for uncertainty bands.