Quark Mass Relations to Four-Loop Order in Perturbative QCD

Abstract

We present results for the relation between a heavy quark mass defined in the on-shell and minimal subtraction (MS) scheme to four-loop order. The method to compute the four-loop on-shell integral is briefly described and the new results are used to establish relations between various short-distance masses and the MS quark mass to next-to-next-to-next-to-leading order accuracy. These relations play an important role in the accurate determination of the MS heavy quark masses. DOI: 10.1103/PhysRevLett.114.142002 PACS numbers: 12.38.Bx, 12.38.Cy, 14.65.Fy, 14.65.Ha The precise knowledge of quark masses plays an important role in many phenomenological applications. This is in particular true for the heavy top, bottom, and charm quarks. For example, the top quark mass enters as a crucial parameter the combined electroweak fits which have been used to obtain indirect information about the Higgs boson mass, and nowadays serve as consistency checks for the standard model, see, e.g., Refs. [1,2]. The uncertainty in the top quark mass is also dominant in the analyses of the stability of the electroweak vacuum [3][4][5]. A prominent example where a precise bottom quark mass is required are B-meson decays which are often proportional to the fifth power of m b . Precise charm and bottom quark masses are important to obtain accurate predictions for the Higgs boson decays into the respective quark flavors. Also in the context of top and bottom Yukawa coupling unification, precise mass values are indispensable since they serve as boundary conditions at low energies. Last but not least, quark masses enter the Lagrange density of the standard model as fundamental parameters. Thus, it is mandatory to obtain precise numerical values by comparing high-order theoretical predictions with precise experimental data.At lowest order in perturbation theory there is no need to fix the renormalization scheme for the quark masses. However, after including quantum corrections it is necessary to apply renormalization conditions which fix the renormalization scheme. A natural scheme for heavy quark masses, i.e., the charm, bottom, and top quark masses, is the on-shell (OS) scheme where one requires that the inverse heavy quark propagator with momentum q has a zero at the position of the on-shell mass, M, i.e., for q 2 ¼ M 2. It is well known that perturbation theory has a bad convergence behavior in case the on-shell quark mass is used as a parameter. Another widely used renormalization scheme is based on minimal subtraction. This means that the mass parameter entering the quark propagator is defined in such a way that just divergent terms (and no finite contributions) are absorbed such that the quark propagator is finite (after wave function renormalization). In this Letter we consider four-loop corrections to the relation between the on-shell and the minimal subtraction (MS) definition of a heavy quark mass that allows for a precise conversion from one renormalization scheme into the other.For the various heavy quarks, different methods re...

On the light massive flavor dependence of the large order asymptotic behavior and the ambiguity of the pole mass

“…[1][2][3][4][5][6], and a 4 was determined numerically in refs. [7,8], where the quoted numerical uncertainties have been taken from ref. [8].…”

“…In refs. [1][2][3][4][5][6][7][8] where the terms show the series in powers of the strong coupling α s (m Q ) in the scheme that includes Q as a dynamical flavor. The fourth order coefficient displays the numerical uncertainties from [8], which are, however, much smaller than other types of uncertainties considered in this paper.…”

Indirect determinations of the top quark mass

“…Since the SM predictions for flavour observables are often expressed in terms of the running MS top quark mass m t (m t ), it is useful to give here the relation between the pole top mass M t and m t (m t ). Accounting for QCD corrections only 4 we find [20] …”

“…On the theoretical side, since BR(K L → π 0 νν) is fully dominated by short-distance effects, the main sources of uncertainties arise from |V cb |, β and γ, see eq.s (3.15), (3.41). In particular, the current error budget for BR( We estimate that the top mass uncertainty will be 20) where δK νν is the experimental error on From figure 3 we also learn that ∆m Bs and B s → µ + µ − are the most accurate M t discriminators, while other observables like ∆m B d , K and K → πνν play a sub-leading role. The latter point is also illustrated by figure 5, which shows how experimental improvements in each flavour observable affect the uncertainty on M t , assuming present (left) and future (right) theory uncertainties.…”

“…One interesting possibility is to identify observables that can be computed in QCD beyond the leading order in terms of the running top mass evaluated at a sufficiently high-energy scale, so that the perturbative expansion is completely reliable. The running top mass is then translated into the pole mass by means of a relation now known at four-loops in QCD [20]. This programme has been applied to the total inclusive tt cross section [21][22][23][24][25], from which it was possible to extract the following values of the pole top mass: (M t ) σ tt = 172.9 ± 2.6 GeV ATLAS [26] 176.7 ± 2.9 GeV CMS [27] .…”