Determination of the top-quark $\overline{\rm MS}$ running mass via its perturbative relation to the on-shell mass with the help of principle of maximum conformality
In the paper, we study the properties of the top-quark $\overline{\rm MS}$ running mass computed from its on-shell mass by using both the four-loop $\overline{\rm MS}$-on-shell relation and the principle of maximum conformality (PMC) scale-setting approach. The PMC adopts the renormalization group 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 $\overline{\rm MS}$-on-shell relation and taking the top-quark on-shell mass $M_t=172.9\pm0.4$ GeV as an input, we obtain the renormalization scale-invariant $\overline{\rm MS}$ running mass at the scale $m_t$, e.g., $m_t(m_t)\simeq 162.6\pm 0.4$ GeV, in which the error is the squared average of those from $Δα_s(M_Z)$, $ΔM_t$, and the approximate error from the uncalculated five-loop terms predicted by using the Padé approximation approach.
We present a comprehensive and self-consistent analysis for the thrust distribution by using the Principle of Maximum Conformality (PMC). By absorbing all nonconformal terms into the running coupling using PMC via renormalization group equation, the scale in the running coupling shows the correct physical behavior and the correct number of active flavors is determined. The resulting PMC predictions agree with the precise measurements for both the thrust differential distributions and the thrust mean values. Moreover, we provide a new remarkable way to determine the running of the coupling cons…
We present in detail a new systematic method which can be used to automatically eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions at all orders. We show that all of the nonconformal β-dependent terms in a QCD perturbative series can be readily identified by generalizing the conventional renormalization schemes based on dimensional regularization. We then demonstrate that the nonconformal series of pQCD at any order can be resummed systematically into the scale of the QCD coupling in a unique and unambiguous way due to a special degeneracy of the β-terms…
The principle of maximum conformality (PMC) provides a way to eliminate the conventional renormalization scale ambiguity in a systematic way. By applying the PMC scale setting, all non-conformal terms in perturbative series are summed into the running coupling, and one obtains a unique, scale-fixed prediction at any finite order. In the paper, we make a detailed PMC analysis for the spin-singlet heavy quarkoniums decay (into light hadrons) at the next-to-leading order. After applying the PMC scale setting, the decay widths for all those cases are almost independent of the initial renormalizati…
The conventional approach to fixed-order perturbative QCD predictions is based on an arbitrary choice of the renormalization scale, together with an arbitrary range. This {\it ad hoc} assignment of the renormalization scale causes the coefficients of the QCD running coupling at each perturbative order to be strongly dependent on the choice of both the renormalization scale and the renormalization scheme. However, such ambiguities are not necessary, since as a basic requirement of renormalization group invariance (RGI), any physical observable must be independent of the choice of both the renor…
In the paper, we analyze the $η_c$ decays into light hadrons at the next-to-leading order QCD corrections by applying the principle of maximum conformality (PMC). The relativistic correction at the ${\cal{O}}(α_s v^2)$-order level has been included in the discussion, which gives about $10\%$ contribution to the ratio $R$. The PMC, which satisfies the renormalization group invariance, is designed to obtain a scale-fixed and scheme-independent prediction at any fixed order. To avoid the confusion of treating $n_f$-terms, we transform the usual $\overline{\rm MS}$ pQCD series into the one under t…
We investigate the renormalization group (RG) running of the effective potential and the pole mass in the broken symmetry phase of the $R_{ξ}$ and the $\overline{R_{ξ}}$ gauges which have different RG running for the effective potential in the symmetric phase and show that if the vacuum expectation value (VEV) is expressed as a function of the other parameters of the theory by solving the minimization condition, then the effective potential in the broken symmetry phase in both gauges satisfies the same RG equation as one in the symmetric phase of the $\overline{R_{ξ}}$ gauge. The pole masses i…
A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale $μ$ of the running coupling $α_s(μ^2).$ The purpose of the running coupling in any gauge theory is to sum all terms involving the $β$ function; in fact, when the renormalization scale is set properly, all non-conformal $β\ne 0$ terms in a perturbative expansion arising from renormalization are summed into the running coupling. The remaining terms in the perturbative series are then identical to that of a conformal theory; i.e., the corresponding theory with $β=0$. The result…
Under the conventional scale setting, the renormalization scale uncertainty usually constitutes a systematic error for a fixed-order perturbative QCD estimation. The recently suggested principle of maximum conformality (PMC) provides a principle to eliminate such scale ambiguity in a step-by-step way. Using the PMC, all non-conformal terms in perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. In the paper, we make a detailed PMC analysis for both the polarized and the unpolarized cross se…
A key problem in making precise perturbative QCD predictions is to set the proper renormalization scale of the running coupling. The extended renormalization group equations, which express the invariance of physical observables under both the renormalization scale- and scheme-parameter transformations, provide a convenient way for estimating the scale- and scheme-dependence of the physical process. In this paper, we present a solution for the scale-equation of the extended renormalization group equations at the four-loop level. Using the principle of maximum conformality (PMC) / Brodsky-Lepage…
The {\it Principle of Maximum Conformality} (PMC), which generalizes the conventional Gell-Mann-Low method for scale-setting in perturbative QED to non-Abelian QCD, provides a rigorous method for achieving unambiguous scheme-independent, fixed-order predictions for physical observables consistent with the principles of the renormalization group. In addition to the original multi-scale-setting approach (PMCm), two variations of the PMC have been proposed to deal with ambiguities associated with the uncalculated higher order terms in the pQCD series, i.e. the single-scale-setting approach (PMCs)…
Single-differential cross section predictions for top quark pair production are presented at next-to-leading order, using running top quark mass renormalization schemes. The evolution of the mass of the top quark is performed in the MSR scheme $m_{\mathrm{t}}^{\textrm{MSR}}(μ)$ for renormalization scales $μ$ below the $\overline{\textrm{MS}}$ top quark mass $\overline{m}_{\mathrm{t}}(\overline{m}_{\mathrm{t}})$, and in the $\overline{\textrm{MS}}$ scheme $\overline{m}_{\mathrm{t}}(μ)$ for scales above. In particular, the implementation of a mass renormalization scale independent of the strong …
Using the newly available $α_s^6$-order QCD correction to the Higgs decay channel $H\to gg$, we make a detailed discussion on the perturbative properties of the decay width $Γ(H\to gg)$ by using the principle of maximum conformality (PMC). The PMC provides a way to eliminate the conventional renormalization scheme-and-scale ambiguities, which uses the renormalization group equation to determine the optimal running behavior of the strong coupling constant at each order via a recursive way. Even though there is no ambiguity for setting the renormalization scale, there is residual scale dependenc…
Scheme-independent determination of the QCD running coupling at all scales from jet observables using the principle of maximum conformality and infinite-order scale setting
Precise determination of the top-quark on-shell mass $M_t$ via its scale-invariant perturbative relation to the top-quark $\overline{\rm MS}$ mass ${\overline m}_t({\overline m}_t)$
The measured $B_d \to π^0π^0$ branching fraction deviates significantly from conventional QCD predictions, a puzzle which has persisted for more than 10 years. This may be a hint of new physics beyond the Standard Model; however, as we shall show in this paper, the pQCD prediction is highly sensitive to the choice of the renormalization scales which enter the decay amplitude. In the present paper, we show that the renormalization scale uncertainties for $B\to ππ$ can be greatly reduced by applying the Principle of Maximum Conformality (PMC), and more precise predictions for CP-averaged branchi…
The QCD coupling $α_s$ is the most important parameter for achieving precise QCD predictions. By using the well measured effective coupling $α^{g_1}_{s}(Q)$ defined from the Bjorken sum rules as a basis, we suggest a novel and self-consistency way to fix the $α_s$ at all scales: The QCD light-front holographic model is adopted for its infrared behavior, and the fixed-order pQCD prediction under the principle of maximum conformality (PMC) is used for its high-energy behavior. Using the PMC scheme-and-scale independent perturbative series, and by transforming it into the one under the physical $…
The next-to-next-to-leading order (NNLO) pQCD correction to the inclusive decays of the heavy quarkonium $η_Q$ ($Q$ being $c$ or $b$) has been done in the literature within the framework of nonrelativistic QCD. One may observe that the NNLO decay width still has large conventional renormalization scale dependence due to its weaker pQCD convergence, e.g. about $(^{+4\%}_{-34\%})$ for $η_c$ and $(^{+0.0}_{-9\%})$ for $η_b$, by varying the scale within the range of $[m_Q, 4m_Q]$. The principle of maximum conformality (PMC) provides a systematic way to fix the $α_s$-running behavior of the process…
The features of the ordinary least squares method, which gives a possible way to a solution of the overdetermined systems of algebraic equations and allows to estimate the uncertainties of the obtained solutions, are considered. As the important physical example we define four-loop QCD coefficients in the dependence of the relation between pole and running heavy quarks masses on the number of light flavors, using the existing results of numerical supercomputer based calculations of the corresponding four-loop contributions at different fixed numbers of light flavors. Stability of the found sol…
The D0 collaboration at FermiLab has recently measured the top-quark pair forward-backward asymmetry in $\bar p p \to t \bar{t} X$ reactions as a function of the $t\bar{t} $ invariant mass $M_{t\bar{t}}$. The D0 result for $A_{\rm FB}(M_{t\bar{t}}>650\, {\rm GeV})$ is smaller than $A_{\rm FB}(M_{t\bar{t}})$ obtained for small values of $M_{t\bar{t}}$, which may indicate an "increasing-decreasing" behavior for $A_{\rm FB}(M_{t\bar{t}}>M_{\rm cut})$. This behavior is not explained using conventional renormalization scale-setting, even by a next-to-next-to-leading order (N$^2$LO) QCD calculation …
It is conventional to choose a typical momentum transfer of the process as the renormalization scale and take an arbitrary range to estimate the uncertainty in the QCD prediction. However, predictions using this procedure depend on the renormalization scheme, leave a non-convergent renormalon perturbative series, and moreover, one obtains incorrect results when applied to QED processes. In contrast, if one fixes the renormalization scale using the Principle of Maximum Conformality (PMC), all non-conformal $\{β_i\}$-terms in the perturbative expansion series are summed into the running coupling…
This contribution attempts to determine the $b$-quark pole mass $M_b$ and \MSbar running mass $\overline{m}_b$ with two different approaches at the next-to-next-to-leading order (NNLO) corrections. At the first approach, we derive a relation between the $b$-quark pole mass $M_b$ and its \MSbar running mass $\overline{m}_b$ at the NNLO corrections based on the perturbative Quantum Chromo Dynamics (pQCD) predictions. At the second approach, we extract numerical values of the $b$-quark pole and \MSbar running masses based on the NNLO phenomenology of H1 and ZEUS Collaborations combined beauty ver…
A key problem in making precise perturbative QCD (pQCD) predictions is how to set the renormalization scale of the running coupling unambiguously at each finite order. The elimination of the uncertainty in setting the renormalization scale in pQCD will greatly increase the precision of collider tests of the Standard Model and the sensitivity to new phenomena. Renormalization group invariance requires that predictions for observables must also be independent on the choice of the renormalization scheme. The well-known Brodsky-Lepage-Mackenzie (BLM) approach cannot be easily extended beyond next-…
We present results concerning the non-perturbative evaluation of the ghost-gluon running QCD coupling constant from $N_f=2$ twisted-mass lattice calculations. A novel method for calibrating the lattice spacing, independent of the string tension and hadron spectrum is presented with results in agreement with previous estimates. The value of $Λ_{\overline{MS}}$ is computed from the running of the QCD coupling only after extrapolating to zero dynamical quark mass and after removing a non-perturbative OPE contribution that is assumed to be dominated by the dimension-two $\VEV{A^2}$ gluon condensat…
We study the relationship between the $\overline{\mathrm{MS}}$ Yukawa coupling and the pole mass for the bottom and top quarks at the two-loop electroweak order ${\cal O}(α^2)$ in the gaugeless limit of the standard model. We also consider the $\overline{\mathrm{MS}}$ to pole mass relationships at this order, which include tadpole contributions to ensure the gauge independence of the $\overline{\mathrm{MS}}$ masses. In order to suppress numerically large tadpole contributions, we propose a redefinition of the running heavy-quark mass in terms of the $\overline{\mathrm{MS}}$ Yukawa coupling. We…
The Effects of the Massless O(alpha_s^2), O(αα_s), O(α^2) QCD and QED Corrections and of the Massive Contributions to Gamma(H^0\rightarrow b\overline{b})
We present a novel method for precisely determining the running QCD coupling constant $α_s(Q^2)$ over a wide range of $Q^2$ from event shapes for electron-positron annihilation measured at a single annihilation energy $\sqrt{s}$. The renormalization scale $Q^2$ of the running coupling depends dynamically on the virtuality of the underlying quark and gluon subprocess and thus the specific kinematics of each event. The determination of the renormalization scale for event shape distributions is obtained by using the Principle of Maximum Conformality (PMC), a rigorous scale-setting method for gaug…
The recent determination of the $β$--function of the QCD running coupling $α_{\overline{MS}}(Q^2)$ to 5-loops{\color{blue},} provides a verification of the convergence of a novel method for determining the fundamental QCD parameter $Λ_s$ based on the Light-Front Holographic approach to nonperturbative QCD. The new 5-loop analysis, together with improvements in determining the holographic QCD nonperturbative scale parameter $κ$ from hadronic spectroscopy, leads to an improved precision of the value of $Λ_s$ in the ${\overline{MS}}$ scheme close to a factor of two; we find $Λ^{(3)}_{\overline{MS…
The basis of the $\{β\}$-expansion for the perturbative series evaluated in the $\overline{MS}$ scheme for the renormalization group invariant quantities is summarized.Comparison with a similar representation,used within the BLM-motivated Principle of Maximal Conformality,is discussed.We stress that the original $\{β\}$-expansion contains a completed list of terms rather than its PMC analog. The arguments in favour of the complete $\{β\}$-expansion are presented. They are based on the relations which follow from the power $β$-function generalization of the Crewther relation for the nonsinglet …
The precise values of the running quark and lepton masses $m^{}_f(μ)$, which are defined in the modified minimal subtraction scheme ($\overline{\rm MS}$) with $μ$ being the renormalization scale and the subscript $f$ referring to all the charged fermions in the Standard Model (SM), are very useful for the model building of fermion masses and flavor mixing and for the precision calculations in the SM or its new-physics extensions. In this paper, we calculate the running fermion masses by taking account of the up-to-date experimental results collected by Particle Data Group and the latest theore…
As an alternative to directly integrating their defining equations to find the running coupling $a(μ)$ and the running mass $m(μ)$, we expand these quantities in powers of $\ln\left(\fracμ{μ^\prime}\right)$ and their boundary values $a(μ^\prime)$ and $m(μ^\prime)$. Renormalization group summation is used to partially sum these logarithms. We consider this approach using both the $\overline{MS}$ and 't Hooft renormalization schemes. We also show how the couplings and masses in any two mass independent renormalization schemes are related.
We present a lattice determination of the $Λ$ parameter in three-flavor QCD and the strong coupling at the Z pole mass. Computing the nonperturbative running of the coupling in the range from $0.2\,$GeV to $70\,$GeV, and using experimental input values for the masses and decay constants of the pion and the kaon, we obtain $Λ_{\overline{\rm MS}}^{(3)}=341(12)\,$MeV. The nonperturbative running up to very high energies guarantees that systematic effects associated with perturbation theory are well under control. Using the four-loop prediction for $Λ_{\overline{\rm MS}}^{(5)}/Λ_{\overline{\rm MS}…
We consider electroweak corrections to the relation between the running $\overline{\mathrm{MS}}$ mass $m_b$ of the $b$ quark in the five-flavor QCD$\times$QED effective theory and its counterpart in the Standard Model (SM). As a bridge between the two parameters, we use the pole mass $M_b$ of the $b$ quark, which can be calculated in both models. The running mass is not a fundamental parameter of the SM Lagrangian, but the product of the running Yukawa coupling $y_b$ and the Higgs vacuum expectation value. Since there exist different prescriptions to define the latter, the relations considered…
In this paper, we give comprehensive analyses for event shape observables in electron-positron annihilation by using the Principle of Maximum Conformality (PMC) which is a rigorous scale-setting method to eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions. Conventionally the renormalization scale and theoretical uncertainties in event shape observables are often evaluated by setting the scale to the center-of-mass energy $\sqrt{s}$. The event shape distributions using this conventional scale setting are plagued by the large renormalization scale uncertai…
In this paper various theoretical approaches are used to define the dependence of the estimated $\mathcal{O}(α^5_s)$ and $\mathcal{O}(α^6_s)$-corrections to the QCD relation between pole and $\rm{\overline{MS}}$ running masses of heavy quarks on the number of light flavours. It is found that recently studied asymptotic formula for the coefficients of this relation, based on the infared-renormalon method, does not reproduce sign-alternating structure in the flavour-dependence of the five and six-loop corrections, which holds in three other used by us approaches.
The nonleptonic decays $B_{d,s} \to K^{*0} {\overline K^{*0}}$ are reanalyzed in perturbative QCD approach, which is based on the $k_{\rm T}$ factorization. In the standard model, the calculated branching fraction and longitudinal polarization fraction of $B_{d} \to K^{*0} {\overline K^{*0}}$ are in agreement with experimental measurements, while the predictions of $B_{s} \to K^{*0} {\overline K^{*0}}$ cannot agree with data simultaneously. The parameter that combines of longitudinal polarization fractions and branching fractions evaluated to be $L_{K^*\overline{K}^{*0}}^{\rm PQCD}= 12.7^{+5.6…
We report on the first computation of the strong running coupling at the physical point (physical pion mass) from the ghost-gluon vertex, computed from lattice simulations with three flavors of Domain Wall fermions. We find $α_{\overline{\rm MS}}(m_Z^2)=0.1172(11)$, in remarkably good agreement with the world-wide average. Our computational bridge to this value is the Taylor-scheme strong coupling, which has been revealed of great interest by itself because it can be directly related to the quark-gluon interaction kernel in continuum approaches to the QCD bound-state problem.
Motivated by the LHCb-group discovery of exotic hadrons in the range (6.2 $\sim$ 6.9) GeV, we present new results for the masses and couplings of $0^{++}$ fully heavy $(\bar{Q}Q)(Q\bar{Q})$ molecules and $(QQ)(\overline{QQ})$ tetraquaks states from relativistic QCD Laplace Sum Rule (LSR) within stability criteria where Next-to-Leading Order (NLO) Factorized (F) Perturbative (PT) corrections is included. As the Operator Product Expansion (OPE) usually converges for $d\leqslant 6-8$, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to $\langle G^3 \rangle$. We also e…
We present the C++ program library mr that allows us to reliably calculate the values of the running parameters in the Standard Model at high energy scales. The initial conditions are obtained by relating the running parameters in the $\overline{\mathrm{MS}}$ renormalization scheme to observables at lower energies with full two-loop precision. The evolution is then performed in accordance with the renormalization group equations with full three-loop precision. Pure QCD corrections to the matching and running are included through four loops. We also provide a Mathematica interface for this prog…
We examine the perturbative calculation of the inclusive semi-leptonic decay rate $Γ$ for the $b$-quark, using mass-independent renormalization. To finite order of perturbation theory the series for $Γ$ will depend on the unphysical renormalization scale parameter $μ$ and on the particular choice of mass-independent renormalization scheme; these dependencies will only be removed after summing the series to all orders. In this paper we show that all explicit $μ$-dependence of $Γ$, through powers of ln$(μ)$, can be summed by using the renormalization group equation. We then find that this explic…
The complete next-to-next-to-next-to-leading order short-distance and bound-state QCD corrections to $Υ(1S)$ leptonic decay rate $Γ(Υ(1S)\to \ell^+\ell^-)$ has been finished by Beneke {\it et al.} \cite{Beneke:2014qea}. Based on those improvements, we present a renormalization group (RG) improved pQCD prediction for $Γ(Υ(1S)\to \ell^+\ell^-)$ by applying the principle of maximum conformality (PMC). The PMC is based on RG-invariance and is designed to solve the pQCD renormalization scheme and scale ambiguities. After applying the PMC, all known-type of $β$-terms at all orders, which are control…
In the present paper, we first give a detailed study on the pQCD corrections to the leading-twist part of BSR. Previous pQCD corrections to the leading-twist part derived under conventional scale-setting approach up to ${\cal O}(α_s^4)$-level still show strong renormalization scale dependence. The principle of maximum conformality (PMC) provides a systematic way to eliminate conventional renormalization scale-setting ambiguity by determining the accurate $α_s$-running behavior of the process with the help of renormalization group equation. Our calculation confirms the PMC prediction satisfies …
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 $\rm{\overline{MS}}$-scheme with identical numerical error bars. In this work the flavour dependence of the $\mathcal{O}(α_s^4)$ 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 $\mathcal{O…
