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Register a new userA new analysis of the pQCD contributions to the electroweak parameter $rho$ using the single-scale approach of principle of maximum conformality
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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 [Phys.Rev.D {bf 89},116001(2014)], we reanalyze the electroweak $rho$ parameter by using the PMC single-scale approach. Using the PMC conformal series and the Pad$acute{e}$ approximation approach, we observe that the residual scale dependence can be greatly suppressed and then a more precise pQCD prediction up to ${rm N^4LO}$-level can be achieved, e.g. $Deltarho|_{rm PMC}simeq(8.204pm0.012)times10^{-3}$, where the errors are squared averages of those from unknown higher-order terms and $Deltaalpha_s(M_Z)=pm 0.0010$. We then predict the magnitudes of the shifts of the $W$-boson mass and the effective leptonic weak-mixing angle: $delta M_{W}|_{rm N^4LO} =-0.26$ MeV and $delta sin^2{theta}_{rm eff}|_{rm N^4LO}=0.14times10^{-5}$, 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.
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The next-to-next-to-leading order (NNLO) pQCD prediction for the $gammagamma^* to eta_c$ form factor was evaluated in 2015 using nonrelativistic QCD (NRQCD). A strong discrepancy between the NRQCD prediction and the BaBar measurements was observed. Until now there has been no solution for this puzzle. In this paper, we present a NNLO analysis by applying the Principle of Maximum Conformality (PMC) to set the renormalization scale. By carefully dealing with the light-by-light diagrams at the NNLO level, the resulting high precision PMC prediction agrees with the BaBar measurements within errors, and the conventional renormalization scale uncertainty is eliminated. The PMC is consistent with all of the requirements of the renormalization group, including scheme-independence. The application of the PMC thus provides a rigorous solution for the $gammagamma^* to eta_c$ form factor puzzle, emphasizing the importance of correct renormalization scale-setting. The results also support the applicability of NRQCD to hard exclusive processes involving charmonium.
A key problem in making precise perturbative QCD predictions is the uncertainty in determining the renormalization scale $mu$ of the running coupling $alpha_s(mu^2).$ The purpose of the running coupling in any gauge theory is to sum all terms involving the $beta$ function; in fact, when the renormalization scale is set properly, all non-conformal $beta e 0$ 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 $beta=0$. The resulting scale-fixed predictions using the principle of maximum conformality (PMC) are independent of the choice of renormalization scheme -- a key requirement of renormalization group invariance. The results avoid renormalon resummation and agree with QED scale-setting in the Abelian limit. The PMC is also the theoretical principle underlying the BLM procedure, commensurate scale relations between observables, and the scale-setting method used in lattice gauge theory. The number of active flavors $n_f$ in the QCD $beta$ function is also correctly determined. We discuss several methods for determining the PMC scale for QCD processes. We show that a single global PMC scale, valid at leading order, can be derived from basic properties of the perturbative QCD cross section. The elimination of the renormalization scale ambiguity and the scheme dependence using the PMC will not only increase the precision of QCD tests, but it will also increase the sensitivity of collider experiments to new physics beyond the Standard Model.
In the paper, we study the properties of the $Z$-boson hadronic decay width by using the $mathcal{O}(alpha_s^4)$-order quantum chromodynamics (QCD) corrections with the help of the principle of maximum conformality (PMC). By using the PMC single-scale approach, we obtain an accurate renormalization scale-and-scheme independent perturbative QCD (pQCD) correction for the $Z$-boson hadronic decay width, which is independent to any choice of renormalization scale. After applying the PMC, a more convergent pQCD series has been obtained; and the contributions from the unknown $mathcal{O}(alpha_s^5)$-order terms are highly suppressed, e.g. conservatively, we have $Delta Gamma_{rm Z}^{rm had}|^{{cal O}(alpha_s^5)}_{rm PMC}simeq pm 0.004$ MeV. In combination with the known electro-weak (EW) corrections, QED corrections, EW-QCD mixed corrections, and QED-QCD mixed corrections, our final prediction of the hadronic $Z$ decay width is $Gamma_{rm Z}^{rm had}=1744.439^{+1.390}_{-1.433}$ MeV, which agrees with the PDG global fit of experimental measurements, $1744.4pm 2.0$ MeV.
In the paper, we study the $Upsilon(1S)$ leptonic decay width $Gamma(Upsilon(1S)to ell^+ell^-)$ by using the principle of maximum conformality (PMC) scale-setting approach. The PMC adopts the renormalization group equation to set the correct momentum flow of the process, whose value is independent to the choice of the renormalization scale and its prediction thus avoids the conventional renormalization scale ambiguities. Using the known next-to-next-to-next-to-leading order perturbative series together with the PMC single scale-setting approach, we do obtain a renormalization scale independent decay width, $Gamma_{Upsilon(1S) to e^+ e^-} = 1.262^{+0.195}_{-0.175}$ keV, where the error is squared average of those from $alpha_s(M_{Z})=0.1181pm0.0011$, $m_b=4.93pm0.03$ GeV and the choices of factorization scales within $pm 10%$ of their central values. To compare with the result under conventional scale-setting approach, this decay width agrees with the experimental value within errors, indicating the importance of a proper scale-setting approach.
We present a detailed study on the properties of the free energy density at the high temperature by applying the principle of maximum conformality (PMC) scale-setting method within the effective field theory. The PMC utilizes the renormalization group equation recursively to identify the occurrence and pattern of the non-conformal ${beta_i}$-terms, and determines the optimal renormalization scale at each order. Our analysis shows that a more accurate free energy density up to $g_s^5$-order level without renormalization scale dependence can be achieved by applying the PMC. We also observe that by using a smaller factorization scale around the effective parameter $m_E$, the PMC prediction shall be consistent with the Lattice QCD prediction derived at the low temperature.
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