Principle Of Maximum Conformality Research Articles

Precise determination of the top-quark on-shell mass via its scale- invariant perturbative relation to the top-quark mass * *Supported in part by the National Natural Science Foundation of China (12247129, 12175025, 12347101) and the Graduate Research and Innovation Foundation of Chongqing, China (ydstd1912)

The principle of maximum conformality (PMC) provides a systematic approach to solve the conventional renormalization scheme and scale ambiguities. Scale-fixed predictions of physical observables using the PMC are independent of the choice of renormalization scheme – a key requirement for renormalization group invariance. In this paper, we derive new degeneracy relations based on the renormalization group equations that involve both the usual β-function and the quark mass anomalous dimension -function. These new degeneracy relations enable improved PMC scale-setting procedures for correct magnitudes of the strong coupling constant and -running quark mass to be determined simultaneously. By using these improved PMC scale-setting procedures, the renormalization scale dependence of the -on-shell quark mass relation can be eliminated systematically. Consequently, the top-quark on-shell (or ) mass can be determined without conventional renormalization scale ambiguity. Taking the top-quark mass GeV as the input, we obtain GeV. Here, the uncertainties arise from errors combined with those from and the approximate uncertainty resulting from the uncalculated five-loop terms predicted through the Padé approximation approach.

Relation Between Pole and Running Masses of Heavy Quarks Using the Principle of Maximum Conformality

Abstract The relation of the pole and running heavy quark masses of the order of $\mathcal {O}\left(\alpha _s^4\right)$ in perturbative quantum chromodynamics (pQCD) can be obtained using the principle of maximum conformality (PMC), a formalism that provides a rigorous method for eliminating renormalization scale and scheme ambiguities for observables in pQCD. Using PMC, optimal renormalization scale for the heavy quark mass ratio is determined, independent of the renormalization scale and scheme up to order $\alpha _s^4$. Precise values are then obtained for the PMC pole masses of the heavy quarks $M_b^{\text{PMC}}=4.86^{+0.03}_{-0.02}$ GeV, $M_t^{\text{PMC}}=172.3\pm 0.6$ GeV, and the running mass $\overline{m}_t^{\text{PMC}}=162.6\pm 0.7$ GeV at the PMC scale.

Approximate N5LO Higgs Boson Decay Width Γ(H→γγ)

The precision and predictive power of perturbative QCD (pQCD) prediction depends on both a precise, convergent, fixed-order series and a reliable way of estimating the contributions of unknown higher-order (UHO) terms. It has been shown that by applying the principle of maximum conformality (PMC), which applies the renormalization group equation recursively to set the effective magnitude of αs of the process, the remaining conformal coefficients will be well matched with the corresponding αs at each order, leading to a scheme-and-scale invariant and more convergent perturbative series. The PMC series, being satisfied with the standard renormalization group invariance, has a rigorous foundation. Thus it not only can be widely applied to virtually all high-energy hadronic processes, but also can be a reliable platform for estimating UHO contributions. In this paper, by using the total decay width Γ(H→γγ) which has been calculated up to N4LO QCD corrections, we first derive its PMC series by using the PMC single-scale setting approach and then estimate its unknown N5LO contributions by using a Bayesian analysis. The newly suggested Bayesian-based approach estimates the magnitude of the UHO contributions based on an optimized analysis of the probability density distribution, and the predicted UHO contribution becomes more accurate when more loop terms have been known to tame the probability density function. Using the top-quark pole mass Mt = 172.69 GeV and the Higgs mass MH = 125.25 GeV as inputs, we obtain Γ(H→γγ)=9.56504keV, and the estimated N5LO contribution to the total decay width is ΔΓH=±1.65×10−4keV for the smallest credible interval of 95.5% degree of belief.

Revisiting the top-quark pair production at future e + e − colliders* *Supported in part by the Natural Science Foundation of China (12175025, 12147102, 12265011), by the Projects of Guizhou Provincial Department (YQK[2023]016, ZK[2023]141, [2020]1Y027, GZMUZK[2022]PT01)

In this study, we reanalyze the top-quark pair production at next-to-next-to-leading order (NNLO) in quantum chromodynamics (QCD) at future colliders using the Principle of Maximum Conformality (PMC) method. The PMC renormalization scales in are determined by absorbing the non-conformal β terms by recursively using the Renormalization Group Equation (RGE). Unlike the conventional scale-setting method of fixing the scale at the center-of-mass energy , the determined PMC scale is far smaller than the and increases with the , yielding the correct physical behavior for the top-quark pair production process. Moreover, the convergence of the pQCD series for the top-quark pair production is greatly improved owing to the elimination of the renormalon divergence. For a typical collision energy of GeV, the PMC scale is GeV; the QCD correction factor K for conventional results is , where the first error is caused by varying the scale and the second error is from the top-quark mass GeV. After applying the PMC, the renormalization scale uncertainty is eliminated, and the QCD correction factor K is improved to , where the error is from the top-quark mass GeV. The PMC improved predictions for the top-quark pair production are helpful for detailed studies of the properties of the top-quark at future colliders.

‘Maximal conformality’ does not work

The so-called “principle of maximal conformality” is ineffective and does nothing to resolve the renormalization-scheme-dependence problem. Some essential facts about that problem are summarized. It is stressed that RG invariance is a symmetry and that any viable method for resolving the scheme-dependence problem should be formulatable in terms of the invariants of that symmetry.

Novel method to reliably determine the QCD coupling from Ruds measurements and its effects to muon g − 2 and alpha left({M}_Z^2right) within the tau-charm energy region

We present a novel method for precisely determining the QCD running coupling from Ruds measurements in electron-positron annihilation. When calculating the fixed-order perturbative QCD (pQCD) approximant of Ruds, its effective coupling constant {alpha}_sleft({Q}_{ast}^2right) is determined by using the principle of maximum conformality, a systematic scale-setting method for gauge theories, whose resultant pQCD series satisfies all the requirements of renormalization group. Contribution due to the uncalculated higher-order (UHO) terms is estimated by using the Bayesian analysis. Using Ruds data measured by the KEDR detector at 22 centre-of-mass energies between 1.84 GeV and 3.72 GeV, we obtain {alpha}_sleft({M}_Z^2right) = {0.1227}_{-0.0132}^{+0.0117}left(exp .right)pm 0.0016left(textrm{the}.right) , where the theoretical uncertainty (the.) is negligible compared to the experimental one (exp.). Numerical analyses confirm that the new method for calculating Ruds removes conventional renormalization scale ambiguity, and the residual scale dependence due to the UHO-terms will also be highly suppressed due to a more convergent pQCD series. This leads to a significant stabilization of the perturbative series, and a significant reduction of theoretical uncertainty. It thus provides a reliable theoretical basis for precise determination of the QCD running coupling from Ruds measurements at future Tau-Charm Facility. It can also be applied for the precise determination of the hadronic contributions to muon g − 2 and QED coupling alpha left({M}_Z^2right) within the tau-charm energy range.

Extending the predictive power of perturbative QCD using the principle of maximum conformality and the Bayesian analysis

In addition to the evaluation of high-order loop contributions, the precision and predictive power of perturbative QCD (pQCD) predictions depends on two important issues: (1) how to achieve a reliable, convergent fixed-order series, and (2) how to reliably estimate the contributions of unknown higher-order terms. The recursive use of renormalization group equation, together with the Principle of Maximum Conformality (PMC), eliminates the renormalization scheme-and-scale ambiguities of the conventional pQCD series. The result is a conformal, scale-invariant series of finite order which also satisfies all of the principles of the renormalization group. In this paper we propose a novel Bayesian-based approach to estimate the size of the unknown higher order contributions based on an optimized analysis of probability distributions. We show that by using the PMC conformal series, in combination with the Bayesian analysis, one can consistently achieve high degree of reliability estimates for the unknown high order terms. Thus the predictive power of pQCD can be greatly improved. We illustrate this procedure for two pQCD observables: R_{e^+e^-} and R_tau , which are each known up to four loops in pQCD. Numerical analyses confirm that by using the scale-independent and more convergent PMC conformal series, one can achieve reliable Bayesian probability estimates for the unknown higher-order contributions.

Elimination of QCD Renormalization Scale and Scheme Ambiguities

The setting of the renormalization scale (μr) in the perturbative QCD (pQCD) is one of the crucial problems for achieving precise fixed-order pQCD predictions. The conventional prescription is to take its value as the typical momentum transfer Q in a given process, and theoretical uncertainties are then evaluated by varying it over an arbitrary range. The conventional scale-setting procedure introduces arbitrary scheme-and-scale ambiguities in fixed-order pQCD predictions. The principle of maximum conformality (PMC) provides a systematic way to eliminate the renormalization scheme-and-scale ambiguities. The PMC method has rigorous theoretical foundations; it satisfies the renormalization group invariance (RGI) and all of the self-consistency conditions derived from the renormalization group. The PMC has now been successfully applied to many physical processes. In this paper, we summarize recent PMC applications, including event shape observables and heavy quark pair production near the threshold region in e+e− annihilation and top-quark decay at hadronic colliders. In addition, estimating the contributions related to the uncalculated higher-order terms is also summarized. These results show that the major theoretical uncertainties caused by different choices of μr are eliminated, and the improved pQCD predictions are thus obtained, demonstrating the generality and applicability of the PMC.

Detailed comparison of renormalization scale-setting procedures based on the principle of maximum conformality

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) and the procedures based on ``intrinsic conformality" (PMC$_\infty$). In this paper, we will give a detailed comparison of these PMC approaches by comparing their predictions for three important quantities $R_{e^+e^-}$, $R_{\tau}$, and $\Gamma(H \to b \bar{b})$ up to four-loop pQCD corrections. The PMCs approach determines an overall effective running coupling $\alpha_s(Q)$ by the recursive use of the renormalization group equation, whose argument $Q$ represents the actual momentum flow of the process. Our numerical results show that the PMCs method, which involves a somewhat simpler analysis, can serve as a reliable substitute for the full multi-scale PMCm method, and that it leads to more precise pQCD predictions with small residual scale dependence.

Precise perturbative predictions from fixed-order calculations

The intrinsic conformality (iCF) is a general property of the renormalizable gauge theory, which ensures the scale invariance of a fixed-order series at each perturbative order. Following the idea of iCF, we suggest a novel single-scale setting approach under the principle of maximum conformality (PMC) with the purpose of removing the conventional renormalization scheme and scale ambiguities. We call this newly suggested single-scale procedure the PMC∞-s approach, in which an overall effective α s , and hence an overall effective scale is achieved by identifying the {β 0} terms at each order. Its resultant conformal series is scale-invariant and satisfies all renormalization group requirements. The PMC∞-s approach is applicable to any perturbatively calculable observable, and its resultant perturbative series provides an accurate basis for estimating the contribution from the unknown higher-order (UHO) terms. Using the Higgs decay into two gluons up to five-loop quantum chromondynamics (QCD) corrections as an example, we show how the PMC∞-s works, and we obtain and . Here the errors are squared averages of those mentioned in the body of the text. The approximation approach (PAA) and the Bayesian approach (BA) have been adopted to estimate the contributions from the UHO terms. We also demonstrate that the PMC∞-s approach is equivalent to our previously suggested single-scale setting approach (PMCs), which also follows from the PMC but treats the {β i } terms from a different point of view. Thus, proper use of the renormalization group equation can provide a solid way to solve the scale-setting problem.

QCD improved top-quark decay at next-to-next-to-leading order

We analyse the top-quark decay at the next-to-next-to-leading order (NNLO) in QCD by using the Principle of Maximum Conformality (PMC) which provides a systematic way to eliminate renormalization scheme and scale ambiguities in perturbative QCD predictions. The PMC renormalization scales of the coupling constant alpha _s are determined by absorbing the non-conformal beta terms that govern the behavior of the running coupling by using the Renormalization Group Equation (RGE). We obtain the PMC scale Q_star =15.5 GeV for the top-quark decay, which is an order of magnitude smaller than the conventional choice mu _r=m_t, reflecting the small virtuality of the QCD dynamics of the top-quark decay process. Moreover, due to the non-conformal beta terms disappear in the pQCD series, there is no renormalon divergence and the NLO QCD correction term is greatly increased while the NNLO QCD correction term is suppressed compared to the conventional results obtained at mu _r=m_t. By further including the next-to-leading (NLO) electroweak corrections, the finite W boson width and the finite bottom quark mass, we obtain the top-quark total decay width Gamma ^{textrm{tot}}_t=1.3112^{+0.0190}_{-0.0189} GeV, where the error is the squared averages of the top-quark mass Delta m_t=pm 0.7 GeV, the coupling constant Delta alpha _s(M_Z)=pm 0.0009 and the estimation of unknown higher-order terms using the PAA method with [N/M]=[1/1]. The PMC improved predictions for the top-quark decay are complementary to the previous PMC calculations for top-quark pair production and helpful for detailed studies of properties of the top-quark.

New determination of |Vcb| using the three-loop QCD corrections for the B → D⁎ semi-leptonic decays

We present a new determination of the Cabibbo-Kobayashi-Maskawa matrix element |Vcb| by using the three-loop perturbative QCD corrections for the B→D⁎ semi-leptonic decay. The decay width of B→D⁎ semi-leptonic decay can be factorized as perturbatively calculable short-distance part and the non-perturbative but universal long-distance part. We adopt the principle of maximum conformality (PMC) single-scale setting approach to deal with the perturbative series so as to achieve a precise fixed-order prediction for the short-distance parameter ηA. By applying the PMC, an overall effective αs value is achieved by recursively using the renormalization group equation, which inversely results in a precise scale-invariant pQCD series. Such scale-invariant series also provides a reliable basis for predicting the contributions from uncalculated perturbative terms. We then obtain ηA=0.9225−0.0168+0.0117, where the error is the squared average of those from Δαs(MZ)=±0.0010 and the uncertainties caused by the uncalculated higher-order perturbative terms. By using the data of B→D⁎ℓν¯ℓ, we finally obtain |Vcb|PMC=(40.60−0.57+0.53)×10−3, which is consistent with the PDG value within errors.

Total decay width of using the infinite-order scale-setting approach based on intrinsic conformality* *Supported by the Chongqing Graduate Research and Innovation Foundation (CYB21045, ydstd1912), the Natural Science Foundation of China (12175025, 12147102), and the Fundamental Research Funds for the Central Universities (2020CQJQY-Z003, 2021CDJZYJH-003)

We conducted a detailed study on the properties of the total decay width of the Higgs decay channel up to -order QCD corrections by using the newly suggested infinite-order scale-setting approach, which is based on both the principle of maximum conformality and intrinsic conformality. This approach is called PMC. By using the PMC approach, we observed that the conventional renormalization scale ambiguity in perturbative QCD calculation is eliminated, and the residual scale dependence due to unknown higher-order terms can also be highly suppressed. We then obtained an accurate perturbative QCD prediction on the total decay width, e.g., keV, where the errors are squared averages of those from all the mentioned error sources.

New analyses of event shape observables in electron-positron annihilation and the determination of αs running behavior in perturbative domain

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 uncertainty and underestimate the experimental data. Moreover, since the renormalization scale is simply fixed to the center-of-mass energy sqrt{s} , only one value of the coupling αs at the single scale sqrt{s} can be extracted. In contrast, the PMC renormalization scales are determined by absorbing the nonconformal β contributions that govern the behavior of the running coupling via the Renormalization Group Equation (RGE). The resulting PMC scales change with event shape kinematics, reflecting the virtuality of the underlying quark and gluon subprocess. The PMC scales thus yield the correct physical behavior of the scale and the PMC predictions agree with precise event shape distributions measured at the LEP experiment. More importantly, we can precisely determine the running of the QCD coupling constant αs(Q2) over a wide range of Q2 in perturbative domain from event shape distributions measured at a single center-of-mass energy sqrt{s} .

PMC$_\infty$: Infinite-Order Scale-Setting method using the Principle of Maximum Conformality and preserving the Intrinsic Conformality

We show results for Thrust and C-parameter in e^+ e^-e+e− annihilation to 3 jets obtained using the recently developed new method for eliminating the scale ambiguity and the scheme dependence in pQCD namely the Infinite-Order Scale-Setting method using the Principle of Maximum Conformality (PMC_\infty∞). This method preserves an important underlying property of gauge theories: intrinsic Conformality (iCF). It leads to a remarkably efficient method to eliminate the conventional renormalization scale ambiguity at any order in pQCD. A comparison with Conventional Scale Setting method (CSS) is also shown.

Novel and Self-Consistency Analysis of the QCD Running Coupling α s(Q) in Both the Perturbative and Nonperturbative Domains

The quantum chromodynamics (QCD) coupling α s is the most important parameter for achieving precise QCD predictions. By using the well measured effective coupling defined from the Bjorken sum rules as a basis, we suggest a novel 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 V scheme, we observe that a precise α s running behavior in both the perturbative and nonperturbative domains with a smooth transition from small to large scales can be achieved.

Thrust distribution for 3-jet production from e+e− annihilation within the QCD conformal window and in QED

We investigate the theoretical predictions for thrust distribution in the electron positron annihilation to three-jets process at NNLO for different values of the number of flavors, Nf. To determine the distribution along the entire renormalization group flow from the highest energies to zero energy we consider the number of flavors near the upper boundary of the conformal window. In this regime of number of flavors the theory develops a perturbative infrared interacting fixed point. We then consider also the QED thrust obtained as the limit Nc→0 of the number of colors. In this case the low energy limit is governed by an infrared free theory. Using these quantum field theories limits as theoretical laboratories we arrive at an interesting comparison between the Conventional Scale Setting - (CSS) and the Principle of Maximum Conformality (PMC∞) methods. We show that within the perturbative regime of the conformal window and also out of the conformal window the PMC∞ leads to a higher precision, and that reducing the number of flavors, from the upper boundary to the lower boundary, through the phase transition the curves given by the PMC∞ method preserve with continuity the position of the peak, showing perfect agreement with the experimental data already at NNLO.

Reanalysis of the top-quark pair hadroproduction and a precise determination of the top-quark pole mass at the LHC * *Supported in part by the Natural Science Foundation of China (11625520, 11705033, 11905056, 11947406), the Project of Guizhou Provincial Department (KY[2021]003) and the Department of Energy Contract (DE-AC02-76SF00515. SLAC-PUB-17567)

In this study, we calculate the pQCD production cross-section at the NNLO and determine the top-quark pole mass from recent measurements at the LHC at the center-of-mass energy TeV to a high precision by applying the principle of maximum conformality (PMC). The PMC provides a systematic method that rigorously eliminates QCD renormalization scale ambiguities by summing the nonconformal contributions into the QCD coupling constant. The PMC predictions satisfy the requirements of renormalization group invariance, including renormalization scheme independence, and the PMC scales accurately reflect the virtuality of the underlying production subprocesses. By using the PMC, an improved prediction for the production cross-section is obtained without scale ambiguities, which in turn provides a precise value for the top-quark pole mass. Moreover, the prediction of PMC calculations that the magnitudes of higher-order PMC predictions are well within the error bars predicted from the known lower-order has been demonstrated for the top-quark pair production. The resulting determination of the top-quark pole mass, GeV, from the LHC measurement at TeV agrees with the current world average cited by the Particle Data Group (PDG). The PMC prediction provides an important high-precision test of the consistency of pQCD and the SM at TeV with previous LHC measurements at lower CM energies.

Generalized Crewther relation and a novel demonstration of the scheme independence of commensurate scale relations up to all orders * *Supported by the Chongqing Graduate Research and Innovation Foundation (ydstd1912, CYB21045), the National Natural Science Foundation of China (11625520, 12047564) and the Fundamental Research Funds for the Central Universities (2020CQJQY-Z003)

In this study, we investigate in detail the generalized Crewther Relation (GCR) between the Adler function (D) and the Gross-Llewellyn Smith sum rules coefficient ( ) using the newly proposed single-scale approach of the principle of maximum conformality (PMC). The resultant GCR is scheme-independent, with the residual scale dependence due to unknown higher-order terms highly suppressed. Thus, a precise test of QCD theory without renormalization schemes and scale ambiguities can be achieved by comparing with data. Moreover, a demonstration of the scheme independence of the commensurate scale relation up to all orders is presented. Additionally, for the first time, the Pade approximation approach has been adopted for estimating the unknown -loop contributions from the known four-loop perturbative series.

A new analysis of the pQCD contributions to the electroweak parameter ρ using the single-scale approach of principle of maximum conformality

It has been observed that conventional renormalization scheme and scale ambiguities for the pQCD predictions can be eliminated by using the principle of maximum conformality (PMC). However, being the intrinsic nature of any perturbative theory, there are still two types of residual scale dependences due to uncalculated higher-order terms. In the paper, as a step forward of our previous work Wang et al. (2014) [16], we reanalyze the electroweak ρ parameter by using the PMC single-scale approach. Using the PMC conformal series and the Padé approximation approach, we observe that the residual scale dependence can be greatly suppressed and then a more precise pQCD prediction up to N4LO-level can be achieved, e.g. Δρ|PMC≃(8.204±0.012)×10−3, where the errors are squared averages of those from unknown higher-order terms and Δαs(MZ)=±0.0010. We then predict the magnitudes of the uncertainties of the W-boson mass and the effective leptonic weak-mixing angle: δMW|N4LO=−0.26 MeV and δsin2⁡θeff|N4LO=0.14×10−5, which are well below the precision anticipated for the future electron-position colliders such as FCC, CEPC and ILC. Thus by measuring those parameters, it is possible to test SM with high precision.