Four-loop QCD Research Articles

On the flavour dependence of the $O(\alpha_{s}^{4})$ correction to the relation between running and pole heavy quark masses

Recently the four-loop perturbative QCD contributions to the relations between pole and running masses of charm, bottom and top quarks were evaluated in the $ \overline{MS}$ scheme with identical numerical error bars. In this work the flavour dependence of the $O(\alpha_{s}^{4})$ correction to these asymptotic series is obtained in the semi-analytical form with the help of the least squares method. The numerical structure of the corresponding asymptotic perturbative relations between pole and running c -, b - and t -quark masses is considered and the theoretical errors of the $O(\alpha_{s}^{4})$ contributions are discussed. The explicit dependence for these relations on the renormalization scale $ \mu^{2}$ and the flavour number nl is presented.

Top-quark pole mass in the tadpole-freeMS¯scheme

The complex pole mass of the top quark is presented at full two-loop order in the Standard Model, augmenting the known four-loop QCD contributions. The input parameters are the MS-bar Yukawa and gauge couplings, the Higgs self-coupling, and the Higgs vacuum expectation value (VEV). Here, the VEV is defined as the minimum of the full effective potential in Landau gauge, so that tadpoles vanish. This is an alternative to earlier results that instead minimize the tree-level potential, resulting in a VEV that is gauge-fixing independent but accompanied by negative powers of the Higgs self-coupling in perturbative expansions. The effects of non-zero Goldstone boson mass are eliminated by resummation. I also study the renormalization scale dependence of the calculated pole mass.

General properties on applying the principle of minimum sensitivity to high-order perturbative QCD predictions

As one of the key components of perturbative QCD theory, it is helpful to find a systematic and reliable way to set the renormalization scale for a high-energy process. The conventional treatment is to take a typical momentum as the renormalization scale, which assigns an arbitrary range and an arbitrary systematic error to pQCD predictions, leading to the well-known renormalization scheme and scale ambiguities. As a practical solution for such scale setting problem, the "Principle of Minimum Sensitivity" (PMS), has been proposed in the literature. The PMS suggests to determine an optimal scale for the pQCD approximant of an observable by requiring its slope over the scheme and scale changes to vanish. In the paper, we present a detailed discussion on general properties of PMS by utilizing three quantities $R_{e^+e^-}$, $R_\tau$ and $\Gamma(H\rightarrow b\bar{b})$ up to four-loop QCD corrections. After applying the PMS, the accuracy of pQCD prediction, the pQCD convergence, the pQCD predictive power and etc., have been discussed. Furthermore, we compare PMS with another fundamental scale setting approach, i.e. the Principle of Maximum Conformality (PMC)... Our results show that PMS does provide a practical way to set the effective scale for high-energy process, and the PMS prediction agrees with the PMC one by including enough high-order QCD corrections, both of which shall be more accurate than the prediction under the conventional scale setting. However, the PMS pQCD convergence is an accidental, which usually fails to achieve a correct prediction of unknown high-order contributions with next-to-leading order QCD correction only, i.e. it is always far from the "true" values predicted by including more high-order contributions.

Four-loop corrections with two closed fermion loops to fermion self energies and the lepton anomalous magnetic moment

We compute the eighth-order fermionic corrections involving two and three closed massless fermion loops to the anomalous magnetic moment of the muon. The required four-loop on-shell integrals are classified and explicit analytical results for the master integrals are presented. As further applications we compute the corresponding four-loop QCD corrections to the mass and wave function renormalization constants for a massive quark in the on-shell scheme.

Four-loop QCD analysis of the Bjorken sum rule

We study the polarized Bjorken sum rule (BSR) at low momentum transfers in the range 0.22<Q<1.73 GeV with the four-loop N3LO expression for the coefficient function CBj(αs) in the framework of the common QCD perturbation theory (PT) and the singularity-free analytic perturbation theory (APT). The analysis of the PT series for CBj(αs) gives a hint to its asymptotic nature manifesting itself in the region Q<1 GeV. It relates to the observation that the accuracy of both the three- and four-loop PT predictions happens to be at the same 10% level. On the other hand, the usage of the two-loop APT allows one to describe the precise low energy JLab data down to Q∼300 MeV and gives a possibility for reliable extraction of the higher twist (HT) corrections. At the same time, above Q∼700 MeV the APT two-loop order with HT is equivalent to the four-loop PT with HT compatible to zero and is adequate to current accuracy of the data.

Standard and ϵ-finite master integrals for the ρ-parameter

We have constructed an ϵ-finite basis of master integrals for all new types of one-scale tadpoles which appear in the calculation of the four-loop QCD corrections to the electroweak ρ-parameter. Using transformation rules from the ϵ-finite basis to the standard “minimal-number-of-lines” basis, we obtain as a by-product analytical expressions for few leading terms of the ϵ-expansion of all members of the standard basis. The new master integrals have been computed with the help of the Padé method and by use of difference equations independently.

Four-Loop QCD Corrections to the ElectroweakρParameter

The four-loop QCD corrections to the electroweak rho parameter arising from top and bottom quark loops are computed. Specifically we evaluate the missing "nonsinglet" piece. Using algebraic methods the amplitude is reduced to a set of around 50 new master integrals which are calculated with various analytical and numerical methods. The inclusion of the newly completed term halves the final value of the four-loop correction for the minimally renormalized top-quark mass. The predictions for the shift of the weak mixing angle and the W-boson mass are thus stabilized.

Four-loop singlet contribution to the electroweak ρ parameter

We compute the four-loop QCD contribution to the electroweak ρ parameter induced by the singlet diagrams of the Z-boson self-energy. The numerical impact on the weak mixing angle and the W-boson mass is small.

Automatic reduction of four-loop bubbles

We give technical details about the computational strategy employed in a recently completed investigation of the four-loop QCD free energy. In particular, the reduction step from generic vacuum bubbles to master integrals is described from a practical viewpoint, for fully massive as well as QED-type integrals.

Order-α s3 determination of the strange quark mass

We present a QCD sum rule calculation of the strange-quark mass including four-loop QCD corrections to the correlator of scalar currents. We obtain m̄ s (1GeV) = 205.5 ± 19.1 MeV.

The four-loop β-function in quantum chromodynamics

We present the analytical calculation of the four-loop QCD β-function within the minimal subtraction scheme.