Validity of Padé approximations in vacuum polarization at three- and four-loop order

Abstract

The heavy-quark contribution to the polarization function Π at higher perturbative orders is presently only known approximately. We scrutinize the accuracy of state-of-the-art approximations at three-and fourloop order. At three loops, we present for the first time a result with arbitrary numerical precision for general kinematics and compare to the best Padé estimate. At four loops, we calculate the fourth (inverse) moment of the nonsinglet heavy-quark vacuum polarization in order to test the prediction for this moment based on the Padé approximation.

Top-quark physics at the CLIC electron-positron linear collider

“…An accurate prediction of the √ s distribution requires a matched calculation that includes the enhancement of the cross section at the tt production threshold from boundstate effects and remains valid at centre-of-mass energies well above threshold. The theoretical predictions used in this study are based on the NNLL renormalization group improved threshold cross section of [23], and O(α 3 s ) predictions for the continuum production [102,103], which have been smoothly matched together [104]. The cross section for e + e − → tt + X + γ ISR factorises into the ISR photon emission from the incoming leptons and the e + e − → tt + X inclusive production.…”

Precise determination of αs from relativistic quarkonium sum rules

“…The O(α 2 s ) contribution to the first n = 30 moments has been computed in [15][16][17][18][19]. 2 At O(α 3 s ), analytic computations exist only for n = 1 [21][22][23], n = 2, n = 3, and n = 4 [20,24,25]. At this order, values for n > 4 have been estimated using semi-analytical procedures [26][27][28][29].…”

“…In Ref [20]. the three-loop vector correlator has been obtained numerically for any value of s/m 2 to arbitrary precision 3.…”

Charm- and Bottom-Quark Masses: Recent Sum Rule Results

“…Originally the idea has been suggested for the analysis of the charm quark by the ITEP group [1] and, several years later, developed further and applied to the bottom quark [2]. An analysis based on low moments has been performed originally in three- [3,4] and subsequently four-loop [5,6,7,8,9,10,11,12] approximation. Recently new experimental results for the electronic widths of J/Ψ and Ψ in combination with a more precise determination of α s have led to a further shrinking of the uncertainty in the charm quark mass [13].…”

“…[14,15,16] and, subsequently, [17,18] even up to n = 30. The four-loop contributions up to n = 3 were calculated in [5,6,8,10], recently even the moment with n = 4 was evaluated [12]. Equating theory with the experimentally measured moments…”