PHYSICAL REVIEW D 101, 114024 (2020)

Determination of the top-quark MS running via its perturbative relation to the on-shell mass with the help of the principle of maximum conformality

† ‡ ∥ Xu-Dong Huang ,* Xing-Gang Wu , Jun Zeng, Qing Yu,§ Xu-Chang Zheng, and Shuai Xu¶ Department of Physics, Chongqing University, Chongqing 401331, People’s Republic of China

(Received 11 May 2020; accepted 8 June 2020; published 22 June 2020)

In the paper, we study the properties of the top-quark MS running mass computed from its on-shell mass by using both the four-loop MS-on-shell relation and the principle of maximum conformality (PMC) scale- setting approach. The PMC adopts the equation to set the correct magnitude of the strong running coupling of the perturbative series, its prediction avoids the conventional renormalization scale ambiguity, and thus a more precise pQCD prediction can be achieved. After applying the PMC to the four-loop MS-on-shell relation and taking the top-quark on-shell mass Mt ¼ 172.9 0.4 GeV as an input, we obtain the renormalization scale-invariant MS running mass at the scale mt, e.g., mtðmtÞ ≃ 162.6 0.4 GeV, in which the error is the squared average of those from ΔαsðMZÞ, ΔMt, and the approximate error from the uncalculated five-loop terms predicted by using the Pad´e approximation approach.

DOI: 10.1103/PhysRevD.101.114024

I. INTRODUCTION measurements at the Tevatron and the LHC gives the OS mass Mt ¼ 172.9 0.4 GeV. In quantum chromodynamics (QCD), the quark Practically, one usually adopts the modified minimal are elementary input parameters of the QCD Lagrangian. subtraction scheme (the MS scheme) to do the pQCD There are three light quarks (up, down, and strange) and three heavy ones (charm, bottom, and top). Comparing with other calculation, and the MS running quark mass is introduced. quarks, the top quark is special. It decays before hadroniza- As for the top-quark MS running mass, it can be related to the tion, which can be almost considered as a free quark. OS mass perturbatively which has been computed up to four- Therefore, the top-quark on-shell (OS) mass, or equivalently loop level [11–19]. Using this relation and the measured OS the pole mass, can be determined experimentally. The direct mass, we are facing the chance of determining a precise value measurements are based on analysis techniques which use for the top-quark MS running mass. In using the relation, an top-pair events provided by Monte Carlo (MC) simulation important thing is to determine the exact value of the strong for different assumed values of the top-quark mass. Applying (αs). The scale running behavior of αs is those techniques to data yields a mass quantity correspond- controlled by the renormalization group equation (RGE) or ing to the top-quark mass scheme implemented in the MC; the β function [20–23], which is now known up to five-loop thus, it is usually referred to as the “MC mass.” Since the top- level [24]. Using the Particle Data Group reference point quark MC mass is within ∼1 GeV of its OS mass [1], one can αsðMZÞ¼0.1181 0.0011 [10],wecanfixitsvalueatany treat the MC mass as the OS one [2–6]. Detailed discussions scale. And thus the remaining task for achieving the precise on the top-quark OS mass can be found in Refs. [7–9].As value of the perturbative series of the MS running mass over shown by the Particle Data Group [10], an average of various the OS mass is to determine the correct momentum flow and hence the correct αs value of the perturbative series. *[email protected] Conventionally, people uses the guessed renormalization † Corresponding author. scale as the momentum flow of the process and varies it [email protected] ‡ within an arbitrary range to estimate its uncertainty for the [email protected] § pQCD predictions. This naive treatment leads to the mis- [email protected] ∥ [email protected] matching of the strong coupling constant with its coeffi- ¶[email protected] cients, well breaking the renormalization group invariance [25–27] and leading to renormalization scale and scheme Published by the American Physical Society under the terms of ambiguities. And the effectiveness of this treatment depends the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to heavily on the perturbative convergence of the pQCD series. the author(s) and the published article’s title, journal citation, Sometimes, the scale is chosen so as to eliminate the large and DOI. Funded by SCOAP3. logarithmic terms or to minimize the contributions from

2470-0010=2020=101(11)=114024(9) 114024-1 Published by the American Physical Society HUANG, WU, ZENG, YU, ZHENG, and XU PHYS. REV. D 101, 114024 (2020)

R R high-order terms. And sometimes, the scale is so chosen to m0 ¼ Zmm ; ð1Þ directly achieve the prediction in agreement with the data. This kind of guessing work depresses the predicative power where R ¼ MS or OS. Under the MS scheme, one can MS of the pQCD theory, and sometimes is misleading, since there derive the expression of Zm by requiring the renormalized may be new physics beyond the standard model. to be finite, which has been calculated up to To eliminate the artificially introduced renormalization five-loop level [44–47]. Under the OS scheme, the expres- scale and scheme ambiguities, the principle of maximum OS sion of Zm can be obtained by requiring the quark two- conformality (PMC) scale-setting approach has been sug- point correlation function to vanish at the position of the OS gested [28–32]. The purpose of the PMC is to determine the α β mass, whose one-, two-, and three-loop QCD corrections effective s of a pQCD series by using the known -terms – α have been given in Refs. [11 15,48], and the electroweak of the pQCD series. The argument of the effective s is – called the PMC scale, which corresponds to the effective effects have also been considered in Refs. [49 58]. momentum flow of the process. It has been found that the Generally, the relation between the MS quark mass and magnitude of the determined effective αs is independent of OS quark mass can be written as any choice of renormalization scale; thus, the conventional m μ ZOS X renormalization scale ambiguity is eliminated by applying ð rÞ m ðnÞ n zmðμrÞ¼ ¼ ¼ zm ðμrÞas ðμrÞ; ð2Þ β M MS the PMC. The PMC shifts all nonconformal -terms into Zm n≥0 the strong coupling constant at all orders, and it reduces to the Gell-Mann and Low scale-setting approach [33] in the where asðμrÞ¼αsðμrÞ=4π, mðμrÞ is the MS running mass QED Abelian limit [34]. Furthermore, after adopting the with μr being the renormalization scale, and M is the OS PMC to fix the αs running behavior, the remaining ðnÞ perturbative coefficients of the resultant series match the quark mass. The perturbative coefficients zm have been series of conformal theory, leading to a renormalization known up to four-loop level [16,17], and the MS running scheme independent prediction. Using the PMC single- mass at the scale M takes the following perturbative form: scale approach [35], it has recently been demonstrated that the PMC prediction is scheme independent up to any fixed ð1Þ ð2Þ 2 mðMÞ¼Mf1 þ zm ðMÞasðMÞþzm ðMÞasðMÞ order [36]. The residual scale dependence due to the ð3Þ 3 ð4Þ 4 uncalculated higher-order terms is highly suppressed by þ zm ðMÞas ðMÞþzm ðMÞas ðMÞþg; ð3Þ the combined αs suppression and exponential suppression [37]. Because of the elimination of the divergent renorma- ðiÞ n n where the coefficients zm ðMÞ (i ¼ 1; …; 4) can be read lon terms like n!β αs [38–40], the convergence of the 0 from Ref. [17]. Using the displacement relation which pQCD series is naturally improved, which leads to a more relates the αs value at the scale μ1 with its value at any other accurate prediction. Moreover, the renormalization scale- μ and-scheme independent series is also helpful for estimat- scale 2, ing the contribution of the unknown higher orders, some – X∞ 1 ∂na μ examples can be found in Refs. [41 43]. a μ a μ sð rÞ −δ n; sð 1Þ¼ sð 2Þþ n! ∂ μ2 n ð Þ ð4Þ n¼1 ð ln 1Þ μr¼μ2 II. CALCULATION TECHNOLOGY 2 2 The renormalized mass under the MS scheme or the OS where δ ¼ ln μ2=μ1, and one can obtain the relation at any scheme can be related to the bare mass (m0)by renormalization scale μr, i.e., μ2 m M M 1 zð1Þ M a μ zð2Þ M β zð1Þ M r a2 μ zð3Þ M β zð1Þ M 2β zð2Þ M ð Þ¼ þ m ð Þ sð rÞþ m ð Þþ 0 m ð Þ ln M2 s ð rÞþ m ð Þþð 1 m ð Þþ 0 m ð ÞÞ μ2 μ2 μ2 r β2zð1Þ M 2 r a3 μ zð4Þ M β zð1Þ M 2β zð2Þ M 3β zð3Þ M r ×lnM2 þ 0 m ð Þln M2 sð rÞþ m ð Þþð 2 m ð Þþ 1 m ð Þþ 0 m ð ÞÞ ln M2 5 μ2 μ2 β β zð1Þ M 3β2zð2Þ M 2 r β3zð1Þ M 3 r a4 μ : þ 2 1 0 m ð Þþ 0 m ð Þ ln M2 þ 0 m ð Þln M2 s ð rÞþ ð5Þ

For the case of top-quark masses, schematically, we can rewrite the perturbative coefficients of the above equation as the fnfg-power series, 2 2 3 mtðMtÞ¼Mtf1 þ c1;0asðμrÞþðc2;0 þ c2;1nfÞasðμrÞþðc3;0 þ c3;1nf þ c3;2nfÞasðμrÞþðc4;0 2 3 4 þ c4;1nf þ c4;2nf þ c4;3nfÞas ðμrÞþg; ð6Þ where mt is the top-quark MS mass and Mt is the top-quark OS mass.

114024-2 DETERMINATION OF THE TOP-QUARK MS RUNNING … PHYS. REV. D 101, 114024 (2020)

To apply the PMC to fix the αs value with the help of and RGE, we first transform the nf series as the fβig series by 2 3β1ðrˆ − rˆ1;0rˆ3;2Þ using the degeneracy relations which are the general T 2;1 2 ¼ 2 properties of a non-Abelian gauge theory [59], 2rˆ1;0 2 2 4ðrˆ1;0rˆ2;0rˆ3;1 − rˆ2;0rˆ2;1Þþ3ðrˆ1;0rˆ2;1rˆ3;0 − rˆ1;0rˆ4;1Þ 2 mtðMtÞ¼Mt 1 þ r1;0asðμrÞþðr2;0 þ β0r2;1ÞasðμrÞ þ 3 rˆ1;0 2 3 β 4rˆ rˆ rˆ − 3rˆ rˆ2 2rˆ rˆ rˆ − 3rˆ rˆ2 þðr3;0 þ β1r2;1 þ 2β0r3;1 þ β0r3;2ÞasðμrÞ 0ð 2;1 3;1 1;0 4;2 1;0 þ 2;0 3;2 1;0 2;0 2;1Þ þ 3 5 rˆ1;0 þ r4;0 þ β2r2;1 þ 2β1r3;1 þ β1β0r3;2 2 3 2 2 β ð2rˆ1;0rˆ3;2rˆ2;1 − rˆ − rˆ rˆ4;3Þ 0 2;1 1;0 : þ rˆ3 ð13Þ 2 3 4 1;0 þ 3β0r4;1 þ 3β0r4;2 þ β0r4;3 asðμrÞþ : ð7Þ Using the present known four-loop relations, we can fix the Q The coefficients ri;j can be obtained from the known PMC scale up to NNLL accuracy. It can be found that is μ coefficients ci;j ði>j≥ 0Þ by applying the basic PMC independent of the choice of the renormalization scale r at formulas listed in Refs. [31,32]. The conformal coefficients any fixed order, and the conventional renormalization scale ambiguity is eliminated. This indicates that one can finish ri;0 are independent of μr, and the nonconformal coef- the fixed-order perturbative calculation by choosing any ficients ri;j (j ≠ 0) are functions of μr, i.e., renormalization scale as a starting point, and the PMC scale Q Xj and hence the PMC prediction shall be independent of k k 2 2 such choice. ri;j ¼ Cj rˆi−k;j−k ln ðμr=Mt Þ; ð8Þ k¼0 rˆ r III. NUMERICAL RESULTS where the reduced coefficients i;j ¼ i;jjμr¼Mt , the combi- nation coefficients Ck ¼ j!=½k!ðj − kÞ!, and i, j, k are the To do the numerical calculation, we adopt [10] j α M 0 1181 0 0011 M 172 9 0 4 polynomial coefficients. For convenience, we put the sð ZÞ¼ . . and t ¼ . . GeV. reduced coefficients rˆi;j in the Appendix. Applying the standard PMC single-scale approach [35], A. Properties of the top-quark MS running mass α Q the effective coupling sð Þ can be obtained by using all By setting all input parameters to be their central values the nonconformal terms and the perturbative series (7) into Eqs. (5) and (9), we present the top-quark MS running changes to the following conformal series: mass at the scale Mt under conventional and PMC scale- setting approaches in Fig. 1. It shows that the conventional m M M 1 rˆ a Q rˆ a2 Q tð tÞjPMC ¼ tf þ 1;0 sð Þþ 2;0 s ð Þ renormalization scale dependence becomes small when we rˆ a3 Q rˆ a4 Q ; have known more loop terms. Numerically, we obtain þ 3;0 s ð Þþ 4;0 s ð Þþg ð9Þ

Q 163 where is the PMC scale, which corresponds to the effective momentum flow of the process and is determined by requiring all the nonconformal terms vanish. The PMC Q Q2=M2 162.5 scale ,orln t , can be expanded as a perturbative series, and up to next-to-next-to-leading log (NNLL) accuracy, we have 162 Q2 T T a M T a2 M O a3 ; ln 2 ¼ 0 þ 1 sð tÞþ 2 s ð tÞþ ð sÞ ð10Þ Mt 161.5 where the coefficients are 161 100 150 200 250 300 rˆ2;1 T0 ¼ − ; ð11Þ rˆ1;0 FIG. 1. The top-quark MS running mass, mtðMtÞ,upto 2 four-loop QCD corrections under the conventional (Conv.) β0ðrˆ − rˆ1;0rˆ3;2Þ 2 rˆ rˆ − rˆ rˆ T 2;1 ð 2;0 2;1 1;0 3;1Þ 1 ¼ 2 þ 2 ð12Þ and PMC scale-setting approaches. The renormalization scale rˆ1;0 rˆ1;0 1 μr ∈ ½2 Mt; 2Mt.

114024-3 HUANG, WU, ZENG, YU, ZHENG, and XU PHYS. REV. D 101, 114024 (2020) m M 162 170; 162 462 μ ∈ 1 M ; tð tÞjConv ¼½ . . GeV for r ½2 t TABLE I. The PAA predictions of the magnitudes of the four- 2M m M 162 052; 162 522 loop and five-loop terms (in unit: GeV) using the conventional t GeV, and tð tÞjConv ¼½ . . GeV 1 series. The central value is for renormalization scale μr ¼ Mt, and for μr ∈ ½3 Mt; 3Mt; e.g., the net scale errors are only the errors are for μr ∈ ½Mt=2; 2Mt. ∼0.2% and ∼0.3%, respectively. We should point out that such small net scale dependence for the four-loop pre- N4LO N5LO diction is due to the well convergent behavior of the −0 208þ0.063 EC . −0.083 perturbative series, e.g., the relative magnitudes of LO: 0 067 0 024 2 3 4 PAA 1=1 ; −0.169þ . ½1=2; −0.087þ . NLO: N LO: N LO: N LO ¼ 1: 4.6%: 1%: 0.3%: 0.1% for ½ −0.086 −0.038 2=1 −0 077þ0.022 the case of μr ¼ Mt, and also due to the cancellation of the ½ ; . −0.038 scale dependence among different orders. The scale errors for each loop term are unchanged and large. For example, the mtðMtÞ has the following perturbative series up to four- that the conventional pQCD series which has a weaker loop level: convergence due to renormalon divergence, the diagonal- type PAA series is preferable [63,64]; for the present case, m M 172 9 − 7 903−0.834 − 1 854þ0.391 1=1 1=2 2=1 tð tÞjConv ¼ . . þ0.624 . −0.276 the ½ -type and the ½ -type or ½ -type are the 0 175 0 063 preferable PAA types to predict the magnitudes of the − 0.560þ . − 0.208þ . ðGeVÞ −0.178 −0.083 N4LO- and the N5LO-terms, respectively. And for the more 162 375−0.205 ; ¼ . þ0.087 ðGeVÞ ð14Þ convergent PMC conformal series, the preferred PAA type is consistent with that of the Gell-Mann and Low method where the central values are for μr ¼ Mt, and the upper and [41] and that of the generalized Crewther relation [65]; e.g., lower errors are for μr ¼ Mt=2 and μr ¼ 2Mt, respectively. for the present case, the ½0=2-type and the ½0=3-type are It shows that the absolute scale errors are 18%, 36%, 63%, the preferable PAA types to predict the magnitudes of the and 70% for the NLO-terms, N2LO-terms, N3LO-terms, N4LO- and the N5LO-terms, respectively. More explicitly, and N4LO-terms, respectively. following the procedures described in detail in On the other hand, Fig. 1 shows that after applying the Refs. [37,41], we give the PAA predictions of the uncalcu- PMC, the relative magnitudes of LO: NLO: N2LO: N3LO: lated higher-order pQCD contributions in Table I and N4LO of the pQCD series changes to 1: 7.2%: 0.5%: 0.3%: Table II for conventional and PMC scale-setting < 0.1%. And there is no renormalization scale dependence approaches, respectively. In those two tables, “EC” stands for mtðMtÞ at any fixed order, for the exact results from the known perturbative series, and “PAA” stands for the PAA prediction by using the known m M 4 tð tÞjPMC perturbative series; e.g., the N LO PAA prediction is obtained by using the known N3LO series, etc. ¼ 172.9 − 12.497 þ 0.919 þ 0.551 − 0.095 ðGeVÞ The effectiveness of the PAA approach depends heavily ¼ 161.778 ðGeVÞ; ð15Þ on how well we know the perturbative series and the accuracy of the known perturbative series. Generally, which is unchanged for any choice of renormalization because of large scale dependence for each order terms, scale. The PMC scale or, equivalently, the effective the PAA predictions based on the conventional series is not Q 12 30 momentum flow of the process is ¼ . GeV, which reliable. For the present case, the PAA prediction is is fixed up to NNLL accuracy, acceptable due to the fact that (1) the perturbative series has a good convergence, (2) the large cancellation of the Q2 −4 686 − 51 890a M − 2126 558a2 M scale dependence among different orders, and (3) the scale ln 2 ¼ . . sð tÞ . sð tÞ Mt dependence of the first several dominant terms are small. More explicitly, Table I shows that by using the conven- ¼ −4.686 − 0.445 − 0.156: ð16Þ tional pQCD series under the choices of μr ∈ ½Mt=2; 2Mt, the PAA predicted N4LO-term is about 70%–88% of the The relative magnitudes of each loop terms are 1∶9%∶3%, which shows a good convergence. As a conservative estimation, if using the last known term as the magnitude TABLE II. The PAA predictions of the magnitudes of the four- of its unknown NNNLL term, the change of momentum loop and five-loop terms (in unit: GeV) using the PMC conformal ΔQ ≃ þ1.00 series, which is independent of any choice of renormalization flow is small, ð−0.92 Þ GeV. One usually wants to know the magnitude of the scale. “ ” 4 5 unknown high-order pQCD corrections. We adopt the N LO N LO Pad´e approximation approach (PAA) [60–62], which pro- −0 095 vides a practical way for promoting a finite series to an EC . PAA ½0=2; −0.086 ½0=3; −0.020 analytic function, to do such a prediction. It has been found

114024-4 DETERMINATION OF THE TOP-QUARK MS RUNNING … PHYS. REV. D 101, 114024 (2020) exact N4LO-term, and the PAA predicted N5LO is about 42%–43% of the exact N4LO-term. On the other hand, the PAA predictions with the help of the renormalization scheme and scale-invariant PMC conformal series is much more reliable. Table II shows that by using the PMC conformal series, the PAA predicted N4LO-term is about 91% of the exact N4LO-term, and the PAA predicted N5LO is about 21% of the exact N4LO-term (showing better convergence). Thus, the approximate top-quark MS mass up to N5LO level becomes

m M 162 288−0.181 1=2 tð tÞjConv ¼ . þ0.049 ðGeVÞ½ -type ð17Þ

−0.183 ¼ 162.298 0 049 ðGeVÞ½2=1-type ð18Þ þ . FIG. 2. The value of the top-quark MS running mass mtðmtÞ versus its OS mass Mt ¼ 172.9 0.4 GeV under conventional m M 161 758 : tð tÞjPMC ¼ . ðGeVÞ ð19Þ (Conv.) and PMC scale-setting approaches. The solid line is the PMC prediction, which is independent of the choice of μr. As a final remark, the top-quark MS running mass at two The dashed line is the prediction of the conventional scale- scales μ1 and μ2 can be related via the following equation setting approach, and the error band shows its errors for μ ∈ M =2; 2M μ M =2 [45]: r ½ t t, whose lower edge is for r ¼ t and upper edge is for μr ¼ 2Mt.

ctð4asðμ1ÞÞ mtðμ1Þ¼mtðμ2Þ ; ð20Þ ctð4asðμ2ÞÞ Eqs. (23), (24) show that the PMC prediction is more sensitive to the value of ΔαsðMZÞ. This is reasonable since 4 2 α where the function ctðxÞ¼x7ð1 þ 1.3980x þ 1.7935x − the purpose of PMC is to achieve an accurate s value of the 0.6834x3 − 3.5356x4Þ. Using Eqs. (14), (15), and (20),we process, and inversely, a slight change of its running obtain the top-quark MS running mass at the scale mt: behavior derived from RGE may lead to sizable alterations. Numerically, the determined effective momentum flow 0 081 Q ≃ 12 m m 163 182þ . ; GeV is much smaller than the guessed momentum tð tÞjConv ¼ . −0.190 ðGeVÞ ð21Þ flow OðMtÞ, and the strong coupling constant is more Λ sensitive to the variation of QCD. mtðmtÞj ¼ 162.629 ðGeVÞ: ð22Þ PMC As for the error from the choice of the top-quark OS mass ΔMt ¼0.4 GeV, we obtain

B. Theoretical uncertainties m m 163 182þ0.380 ; tð tÞjConv ¼ . −0.381 ðGeVÞ ð25Þ After eliminating the renormalization scale uncertainty via using the PMC approach, there are still several error sources, α Δα M m m 162 629þ0.379 : such as the s fixed-point error sð ZÞ, the error of top- tð tÞjPMC ¼ . −0.381 ðGeVÞ ð26Þ quark OS mass ΔMt, the unknown contributions from six- loop and higher-order terms, etc. The uncertainty of the four- Figure 2 shows that the top-quark MS running mass mtðmtÞ ð4Þ loop coefficient zm ðMÞ has been discussed in Ref. [17], depends almost linearly on its OS mass, whose error is at 1 whose magnitude ∼0.0004 GeV is negligibly small. For the same order of OðΔMtÞ. convenience, when discussing one uncertainty, the other In the above subsection, we have predicted the magni- input parameters shall be set as their central values. tude of the uncalculated N5LO-terms. If treating the As for the αs fixed-point error, by using ΔαsðMZÞ¼ absolute value of the PAA predicted N5LO magnitude as 0.0011 together with the four-loop αs-running behavior, a conservative estimation of the error of present N4LO Λ 209 5þ13.2 Λ we obtain QCD;nf¼5 ¼ . −12.6 MeV and QCD;nf¼6 ¼ prediction, we shall have an extra error from the unknown 88 3þ6.2 perturbative terms, e.g., . −5.9 MeV. Then we obtain the top-quark MS running mass at the scale mt 1As an addendum, if taking the small difference of the OS m m 163 182þ0.103 ; tð tÞjConv ¼ . −0 103 ðGeVÞ ð23Þ mass and the Monte-Carlo mass into consideration [66], . MC Mt ¼ Mt − ½0.29 GeV; 0.85 GeV, then the above central m m 0 11; −1 19 m m 162 629þ0.118 : values of tð tÞ shall be altered by about ½ . . GeV tð tÞjPMC ¼ . −0.119 ðGeVÞ ð24Þ for both conventional and PMC scale-setting approaches.

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Δm m 0 080 ; 1=2 tð tÞjConv ¼ . ðGeVÞ ½ -type ð27Þ achieved. Thus, we think the PMC is an important approach for achieving precise pQCD predictions, since its predic- ¼0.072 ðGeVÞ; ½2=1-type ð28Þ tion is independent of the choice of renormalization scale. It should be extremely important for lower fixed-order pQCD predictions, when there are not enough terms to suppress ΔmtðmtÞj ¼0.018 ðGeVÞ: ½0=3-type: ð29Þ PMC the large scale uncertainty of each loop term.

IV. SUMMARY ACKNOWLEDGMENTS In the present paper, we have presented a more accurate This work is partly supported by the National Natural prediction of the top-quark MS running mass from the Science Foundation of China under Grants No. 11625520, experimentally measured top-quark OS mass by applying No. 11947406 and No. 11905056, the Project Supported by the PMC to eliminate the conventional renormalization Graduate Research and Innovation Foundation of Chongqing, scale ambiguity. As a combination, we obtain China (Grant No. CYB19065), the China Postdoctoral Science Foundation under Grant No. 2019M663432, and m m 163 182þ0.410 ; 1=2 tð tÞjConv ¼ . −0.445 ðGeVÞ ½ -type ð30Þ by the Chongqing Special Postdoctoral Science Foundation under Grant No. XmT2019055. 163 182þ0.408 ; 2=1 ¼ . −0.444 ðGeVÞ ½ -type ð31Þ

þ0.397 APPENDIX: THE PMC REDUCED mtðmtÞj ¼ 162.629−0 400 ðGeVÞ; ½0=3-type ð32Þ PMC . PERTURBATIVE COEFFICIENTS rˆi; j where the errors are squared averages of those from In this Appendix, we give the required PMC reduced 5 ΔαsðMZÞ, ΔMt, and the uncalculated N LO-terms pre- coefficients rˆi;j for the perturbative series of the top-quark dicted by using the PAA. Among the errors, the one caused MS running mass over its OS mass up to four-loop level, i.e., by ΔMt is dominant, and we need more accurate data to suppress this uncertainty. The conventional predictions rˆ1;0 ¼ −4CF; ðA1Þ have also the renormalization scale uncertainty by varying 55 1 2 2 μ ∈ M ; 2M rˆ2;0 ¼ CACF 6ζ3 þ 5π − − 4π ln 2 r ½2 t t, even though its magnitude is small due to 4 the cancellation of scale errors among different orders. Up 4 2 7 2 2 to the present known N LO level, the predictions under the þ CF − 12ζ3 − 5π þ 8π ln 2 PMC and conventional scale-setting approaches are con- 8 2 sistent with each order. However, it has been found that þð12 − 4π ÞCFTF; ðA2Þ after applying the PMC, a scale-invariant and more con- 71 2 vergent pQCD series, and a more reliable prediction of rˆ2;1 ¼ − − π CF; ðA3Þ contribution from unknown higher-order terms can be 8

181π2 53π4 19027 16 1 rˆ C2 C 51π2ζ 219ζ − 130ζ − − − π2 2 C C2 384 − 76π2ζ 3;0 ¼ A F 3 þ 3 5 6 30 216 þ 3 ln þ A F Li4 2 3 518π2 5731 π4 728 − 112ζ 180ζ − 16 42 − 16π2 22 − π2 2 T C C 8π2ζ 3 þ 5 þ 3 þ 12 15 þ ln ln 3 ln þ F A F 3 28π4 4372π2 32 1696 56π4 2608π2 88ζ − 40ζ − − 144 π2 22 π2 2 C2 − 288ζ − þ 3 5 9 27 þ þ 3 ln þ 9 ln þ F 9 3 27 64 1216 1 4π4 613π2 2969 − 24 − π2 22 π2 2 C3 40ζ − 768 − 4π2ζ − 324ζ − − − 3 ln þ 9 ln þ F 5 Li4 2 3 3 3 3 12 608 − 32 42 32π2 22 464π2 2 − π2C T2 ; ln þ ln þ ln 45 F F ðA4Þ 1 73ζ 26π2 19π4 20335 4 8 44 119π4 rˆ C C 32 3 − − 42 π2 22 − π2 2 C2 3;1 ¼ A F Li4 2 þ 2 þ 3 90 432 þ 3 ln þ 3 ln 3 ln þ F 90 1 95π2 1927 8 16 88 26π2 − 64 − 55ζ − − − 42 − π2 22 π2 2 C T 22 − 24ζ − ; Li4 2 3 6 48 3 ln 3 ln þ 3 ln þ F F 3 3 ðA5Þ

114024-6 DETERMINATION OF THE TOP-QUARK MS RUNNING … PHYS. REV. D 101, 114024 (2020) 13π2 2353 rˆ C −14ζ − − ; 3;2 ¼ F 3 3 216 ðA6Þ 3 2 2 2 2 3 3 rˆ4;0 ¼ −947.046CACF − 1269.84CACF þ TFð3216.18CACF þ 568.364CACF − 335.759CFÞþ3671.8CACF abcd abcd 2 2 219.883CAdF dF 4 3 abcd abcd þ TFð587.571CF − 2497.16CACFÞ − − 1787.65CF − 1050.64CFTF þ 33.28dF dA TF abcd abcd − 254.891dF dF ; ðA7Þ

abcd abcd 2 2 2 3 2 19.9893dF dF rˆ4;1 ¼ 932.846CACF þ TFð81.3012CF − 834.037CACFÞ− 473.22CACF − 21.945CF − 780.54CFTF þ ; TF ðA8Þ

2 rˆ4;2 ¼ −112.694CACF þ 85.1974CF − 590.868CFTF; ðA9Þ

rˆ4;3 ¼ −439.436CF; ðA10Þ where

2 1 NC − 1 NC ¼ 3;TF ¼ ;CA ¼ NC;CF ¼ ; 2 2NC N2 − 1 N4 − 6N2 18 N N2 − 1 N2 6 dabcddabcd ð C Þð C C þ Þ ;dabcddabcd Cð C Þð C þ Þ : F F ¼ 2 F A ¼ ðA11Þ 96NC 48

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