Theory Uncertainties in the Extraction of $α_s$ from Drell-Yan at Small Transverse Momentum

TL;DR

The paper tackles the challenge of extracting $\alpha_s(m_Z)$ from the Drell–Yan $q_T$ spectrum at small $q_T$ by emphasizing the importance of theory correlations across bins. It argues that traditional scale variations fail to capture these correlations and advocates for a theory-nuisance-parameter (TNP) framework, implemented in an Asimov pseudodata setup to study perturbative and nonperturbative uncertainties. The study demonstrates that profiling TNPs against high-precision data can substantially reduce the perturbative uncertainty on $\alpha_s(m_Z)$ (to around $4.5\times 10^{-4}$ in the presented tests), while nonperturbative uncertainties can be constrained with lattice QCD inputs, yielding a total uncertainty compatible with competitive high-precision determinations. The results underscore the viability of a precise $\alpha_s(m_Z)$ extraction from the $Z$-boson $q_T$ spectrum when both the perturbative correlations and nonperturbative physics are treated with robust, data-driven and lattice-informed methods.

Abstract

We perform a detailed pseudodata study to estimate the expected theory uncertainty in the extraction of the strong coupling constant, $α_s(m_Z)$, from a fit to the measured Drell-Yan transverse momentum ($q_T$) spectrum at small $q_T \ll m_Z$. We consider two approaches to estimate the dominant perturbative uncertainties. We first discuss that the traditional approach based on varying unphysical scales is insufficient here because it cannot correctly account for bin-by-bin theory correlations in the $q_T$ spectrum, which are critically important in this case. We then use this case as a nontrivial application of a new approach based on theory nuisance parameters (TNPs), which encodes the correct theory correlations by construction. Moreover, the TNPs can be profiled in the fit thereby allowing the data to constrain the theory uncertainties in a consistent manner. We furthermore discuss the interplay with nonperturbative effects in the peak region $q_T \lesssim 10$ GeV, from where most of the $α_s$ sensitivity originates. The associated nonperturbative uncertainties on $α_s$ when fitting only the $q_T$ spectrum are large. They can in principle be reduced by including additional constraints on the nonperturbative Collins-Soper kernel from lattice QCD calculations. We find that these improvements in the treatment of perturbative and nonperturbative uncertainties and their correlations will enable a competitive $α_s$ extraction from Drell-Yan data at small $q_T$. We also discuss the implications of our findings, calling into question a recent $α_s$ extraction from the $Z$ $q_T$ spectrum by the ATLAS experiment.

Figures (13)

• Figure 1: Interpreting a differential spectrum by scanning scale variations. Left: The theory prediction (black line) with a theory uncertainty band (blue) from the envelope of scale variations compared to measurements (red) at three points $x_{1,2,3}$. Middle: The normalized uncertainty band (blue) with three possible scale variations (orange, green, violet) filling out the band. Right: The assumed correlations $\rho_{ij}$ between the uncertainties at the points $x_i$ and $x_j$ resulting from a given scale variation. Note that for $\rho_{ij} = 0$ the uncertainty at either $x_i$ or $x_j$ must vanish.
• Figure 2: Left panel: Relative differences in the $q_T$ spectrum of individual scale variations at N$^{4}$LL. Different colored lines show different classes of variations as defined in \ref{['sec:profile_scale_variations']}. The red dashed lines show the relative difference from varying $\alpha_s(m_Z)$ for comparison. Right panel: The resulting $\alpha_s(m_Z)$ values when performing the Asimov fit to the central N$^4$LL prediction for each scale variation. Different colors correspond to the same classes of variations as on the left. The grey band shows the fit uncertainty for comparison.
• Figure 3: Left panel: Relative uncertainties in the $q_T$ spectrum with TNPs at N$^{3+1}$LL. The different lines show the impact of varying the corresponding TNP by $+1$ or $-1$, corresponding to 68% theory CL. The yellow band shows their sum in quadrature. Right panel: Corresponding uncertainty on $\alpha_s(m_Z)$ from performing the Asimov fit to the central N$^{3+1}$LL prediction for each TNP variation. The grey band shows the fit uncertainty for comparison.
• Figure 4: Left panel: Relative uncertainties in the $q_T$ spectrum at N$^{3+1}$LL before and after profiling the TNPs. The different lines show the post-fit relative impact of each TNP and the orange band the total post-fit uncertainty. The yellow band shows the pre-fit uncertainty corresponding to the yellow band in \ref{['fig:N3p1LL_scanning']}. Right panel: Post-fit constraints on the TNPs (error bars) and their impact on $\alpha_s(m_Z)$, with the solid (dashed) grey band showing the impact of the post-fit downward (upward) TNP variations.
• Figure 5: Left panel: Uncertainties in the $q_T$ spectrum at N$^{2+1}$LL relative to the N$^4$LL, before (yellow band) and after (orange band) profiling the TNPs. The different lines show the post-fit relative impact of each TNP. Right panel: Post-fit constraints on the TNPs (error bars) and their impact on $\alpha_s(m_Z)$, with the solid (dashed) grey band showing the impact of the post-fit downward (upward) TNP variations. The stars indicate the true values of the TNPs.