Determination of αs from static QCD potential: OPE with renormalon subtraction and lattice QCD

Published for SISSA by Springer Received: December 8, 2018 Revised: March 17, 2019 Accepted: April 15, 2019 Published: April 29, 2019 Hiromasa Takaura,a,1 Takashi Kaneko,b Yuichiro Kiyoc and Yukinari Suminod a Department of Physics, Kyushu University, Fukuoka, 819-0395 Japan b Theory Center, KEK, Tsukuba, Ibaraki, 305-0801 Japan c Department of Physics, Juntendo University, Inzai, 270-1695, Japan d Department of Physics, Tohoku University, Sendai, 980-8578 Japan E-mail: , , , Abstract: We determine the strong coupling constant αs from the static QCD potential by matching a theoretical calculation with a lattice QCD computation. We employ a new theoretical formulation based on the operator product expansion, in which renormalons are subtracted from the leading Wilson coefficient. We remove not only the leading renormalon uncertainty of O(ΛQCD ) but also the first r-dependent uncertainty of O(Λ3QCD r2 ). The theoretical prediction for the potential turns out to be valid at the static color charge distance ΛMS r . 0.8 (r . 0.4 fm), which is significantly larger than ordinary perturbation theory. With lattice data down to ΛMS r ∼ 0.09 (r ∼ 0.05 fm), we perform the matching in a wide region of r, which has been difficult in previous determinations of αs from the potential. Our final result is αs (MZ2 ) = 0.1179+0.0015 −0.0014 with 1.3% accuracy. The dominant uncertainty comes from higher order corrections to the perturbative prediction and can be straightforwardly reduced by simulating finer lattices. Keywords: QCD Phenomenology, Lattice field theory simulation ArXiv ePrint: 1808.01643 1 Corresponding author. Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2019)155 JHEP04(2019)155 Determination of αs from static QCD potential: OPE with renormalon subtraction and lattice QCD Contents 1 2 Theoretical framework 2.1 Formula to subtract renormalons 2.2 Treatment of US scale 2.3 Higher order perturbative uncertainty 4 5 8 10 3 αs determination 3.1 Lattice simulations 3.2 Analysis (I): two-step analysis 3.2.1 Continuum extrapolation 3.2.2 Consistency checks and comparison with conventional methods 3.2.3 αs determination: matching between OPE and lattice result 3.3 Analysis (II): global fit 3.4 Summary of results 11 11 12 12 15 20 23 28 4 Conclusions and discussion 29 A Coefficients of perturbative calculation 30 B Formulation to extract VSRF (r) from VS (r) 31 C Definition of ΛMS 33 D χ2 and covariance matrix 33 E Case including data at r = a in Analysis (I) 35 F Additional analyses on systematic errors 36 1 Introduction Today, facing frontier experiments of particle physics, such as the ones at LHC and Super B Factory (Belle II), there exist increasing demands for more accurate theoretical predictions based on QCD on various phenomena of the strong interaction. Precise determination of the strong coupling constant αs , which is a fundamental parameter of QCD, sets a benchmark for such predictions. In fact, many theoretical developments are required for improving accuracy of αs determination, and once αs is determined, it serves as an input parameter for various predictions. For instance, a precise value of αs will play crucial roles –1– JHEP04(2019)155 1 Introduction in measurements of Higgs boson properties, in searches for new physics, or in high-precision flavor physics. It is also demanded in the context of precise determination of the top quark mass, predicting running of the Higgs quartic coupling, etc. Let us quote the current value of αs , given as the world-combined result by the Particle Data Group (PDG), αs (MZ2 ) = 0.1181 ± 0.0011 [1]. Dominant contributions to this value come from determinations by lattice QCD, which have smaller errors than other determinations using more direct experimental inputs. The Flavor Lattice Averaging Group (FLAG) reports an average of lattice determinations as αs (MZ2 ) = 0.1182 ± 0.0012 [2] based on the studies in refs. [3–7]. The relative accuracies of these current values are 0.9–1.0 per cent. The method of finite volume scheme combined with step-scaling [8–11] can resolve this problem even at currently available lattice cutoffs. In this method, discretization and finite volume effects are kept under control by a finite volume scheme, while lattice data after the step-scaling running can be matched with perturbation theory at sufficiently high scale. As a result, matching with perturbative prediction can be performed at 10–100 GeV. A recent determination based on this method gives αs (MZ2 ) with 0.7 per cent relative accuracy [12] (not yet included in the above average values). In this paper, we determine αs by taking an alternative approach to the window problem: we enlarge the validity range of a theoretical calculation to lower energy where lattice calculations are accurate due to Q  a−1 . For this purpose, we use the operator product expansion (OPE) as a theoretical framework. Its difference from perturbative calculations can be stated as follows. Perturbative predictions have an inevitable uncertainty known as renormalon uncertainty, which stems from a certain divergent behavior of perturbative series at large orders. (See ref. [13] for a review of renormalon.) For a dimensionless observable R(Q) with typical energy scale Q, a renormalon uncertainty is estimated as O((ΛQCD /Q)n ) with a positive integer n (dependent on the observable). In the context of the OPE of the same observable, given by R(Q) = C1 (Q) + CO1 (Q) h0|O1 |0i + ... , Qn (1.1) the perturbative result is encoded in the leading Wilson coefficient C1 . In fact, the renormalon uncertainty of C1 generally has the same order of magnitude as the leading nonperturbative effect (the second term), which corresponds to dim[O1 ] = n [14]. It is expected that the renormalon uncertainty in the leading Wilson coefficient gets canceled when the –2– JHEP04(2019)155 In determinations of αs by lattice QCD, we need to pay attention to the so-called “window problem,” as pointed out in the FLAG report [2]. This is a problem that it is difficult to find a wide enough region where both lattice QCD and perturbative QCD predictions are accurate. A lattice simulation is carried out with a finite lattice spacing a, whose inverse plays the role of an ultraviolet (UV) cutoff scale. Hence, the lattice results are accurate in the energy region Q  a−1 . On the other hand, perturbative calculations are accurate at Q & 1 GeV( ΛQCD ∼ 300 MeV). Determinations of αs are performed by matching of both results. It turns out that, for currently available lattice cutoff scales, the energy window 1 GeV . Q  a−1 cannot be taken widely. –3– JHEP04(2019)155 nonperturbative matrix element is added. Hence, the OPE may realize a wider validity range due to the absence of the renormalon uncertainty, in particular at lower energy side. However, the OPE cannot be made a maximal use as long as we naively calculate C1 in the o (...truncated)