Renormalization Scale Uncertainty Research Articles

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.

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} .

Next-to-soft-virtual resummed prediction for pseudoscalar Higgs boson production at NNLO+NNLL¯

We present first results on the resummation of next-to-soft virtual (NSV) logarithms for the threshold production of pseudoscalar Higgs boson through gluon fusion at the LHC. These results are presented after resumming the NSV logarithms of the kind ${\mathrm{log}}^{i}(1\ensuremath{-}z)$ to $\overline{\mathrm{NNLL}}$ accuracy and matching them systematically to the fixed order NNLO cross sections. These results are obtained using collinear factorization, renormalization group invariance and recent developments in the NSV resummation techniques. The phenomenological implications of these NSV resummed results for 13 TeV LHC are studied and it is observed that these NSV logarithms are quite large. We also evaluate theory uncertainties and find that the renormalization scale uncertainties get reduced further with the inclusion of NSV corrections at various orders in QCD. We further study the impact of QCD corrections on mixed scalar-pseudoscalar states for different values of the mixing angle $\ensuremath{\alpha}$.

Gg→HH : Combined uncertainties

In this note we discuss the combination of the usual renormalization and factorization scale uncertainties of Higgs-pair production via gluon fusion with the novel uncertainties originating from the scheme and scale choice of the virtual top mass. Moreover, we address the uncertainties related to the top-mass definition for different values of the trilinear Higgs coupling and their combination with the other uncertainties.

Second-order QCD corrections to event shape distributions in deep inelastic scattering

We compute the next-to-next-to-leading order (NNLO) QCD corrections to event shape distributions and their mean values in deep inelastic lepton–nucleon scattering. The magnitude and shape of the corrections varies considerably between different variables. The corrections reduce the renormalization and factorization scale uncertainty of the predictions. Using a dispersive model to describe non-perturbative power corrections, we compare the NNLO QCD predictions with data from the H1 and ZEUS experiments. The newly derived corrections improve the theory description of the distributions and of their mean values.

Solution to the γγ*→ηc puzzle using the principle of maximum conformality

The next-to-next-to-leading order (NNLO) pQCD prediction for the $\gamma\gamma^* \to \eta_c$ 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 $\gamma\gamma^* \to \eta_c$ 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 precise determination of the top-quark pole mass

The Principle of Maximum Conformality (PMC) provides a systematic way to eliminate the renormalization scheme and renormalization scale uncertainties for high-energy processes. We have observed that by applying PMC scale setting, one obtains comprehensive and self-consistent pQCD predictions for the top-quark pair total cross section and the top-quark pair forward–backward asymmetry in agreement with the measurements at the Tevatron and LHC. As a step forward, in the present paper, we determine the top-quark pole mass via a detailed comparison of the top-quark pair cross section with the measurements at the Tevatron and LHC. The results for the top-quark pole mass are m_t=174.6^{+3.1}_{-3.2} GeV for the Tevatron with sqrt{S}=1.96 TeV, m_t=173.7pm 1.5 and 174.2pm 1.7 GeV for the LHC with sqrt{S} = 7 and 8 TeV, respectively. Those predictions agree with the average, 173.34pm 0.76 GeV, obtained from various collaborations via direct measurements. The consistency of the pQCD predictions using the PMC with all of the collider measurements at different energies provides an important verification of QCD.

Novel all-orders single-scale approach to QCD renormalization scale-setting

The Principle of Maximal Conformality (PMC) provides a rigorous method for eliminating renormalization scheme-and-scale ambiguities for perturbative QCD predictions. The PMC uses the renormalization group equation to fix the $\beta$-pattern of each order in an arbitrary pQCD approximant, and it then determines the optimal renormalization scale by absorbing all $\{\beta_{i}\}$ terms into the running coupling at each order. The resulting coefficients of the pQCD series match the scheme-independent conformal series with $\beta=0$. As in QED, different renormalization scales appear at each order; we call this the multi-scale approach. In this paper, we present a novel single-scale approach for the PMC, in which a single effective scale is constructed to eliminate all non-conformal $\beta$-terms up to a given order simultaneously. The PMC single-scale approach inherits the main features of the multi-scale approach; for example, its predictions are scheme-independent, and the pQCD convergence is greatly improved due to the elimination of divergent renormalon terms. As an application of the single-scale approach, we investigate the $e^+e^-$ annihilation cross-section ratio $R_{e^+e^-}$ and the Higgs decay-width $\Gamma(H \to b \bar{b})$, including four-loop QCD contributions. The resulting predictions are nearly identical to the multi-scale predictions for both the total and differential contributions. Thus in many cases, the PMC single-scale approach PMC-s, which requires a simpler analysis, could be adopted as a reliable substitution for the PMC multi-scale approach for setting the renormalization scale for high-energy processes, particularly when one does not need detailed information at each order. The elimination of the renormalization scale uncertainty increases the precision of tests of the Standard Model at the LHC.

The Higgs-boson decay up to -order under the minimal momentum space subtraction scheme

We study the Higgs-boson decay width up to order under the minimal momentum space subtraction (mMOM) scheme. A major uncertainty of a finite-order perturbative quantum chromodymaics (pQCD) prediction is the perceived ambiguity in setting the renormalization scale. In the present paper, to achieve a precise pQCD prediction without renormalization scale uncertainty, we adopt the principle of maximum conformality (PMC) to set the renormalization scale of the process. The PMC has a solid theoretical foundation, which is based on renormalization group invariance and utilizes the renormalization group equation to fix the renormalization scale of the process. The key point of the application of the PMC is how to correctly set the {βi} terms of the process to achieve the correct αs-running behavior at each perturbative order. It is found that the ambiguities in dealing with the {βi} terms of the decay width under the scheme can be avoided by using the physical mMOM scheme. For this purpose, for the first time we provide the PMC scale-setting formulas within the mMOM scheme up to a four-loop level. By using the PMC, it is found that a more reliable pQCD prediction on can indeed be achieved under the mMOM scheme. As a byproduct, the convergence of the resultant pQCD series has been greatly improved due to the elimination of renormalon terms. By taking the newly measured Higgs mass, MH = 125.09 ± 0.21 ± 0.11 GeV, our PMC prediction of the decay width is, keV, in which the first error is from the Higgs mass uncertainty and the second error is the residual renormalization scale dependence by varying the initial renormalization scale .

Renormalization-scale uncertainty in the decay rate of false vacuum

We study radiative corrections to the decay rate of false vacua, paying particular attention to the renormalization-scale dependence of the decay rate. The decay rate exponentially depends on the bounce action. The bounce action itself is renormalization scale dependent. To make the decay rate scale-independent, radiative corrections, which are due to the field fluctuations around the bounce, have to be included. We show quantitatively that the inclusion of the fluctuations suppresses the scale dependence, and hence is important for the precise calculation of the decay rate. We also apply our analysis to a supersymmetric model and show that the radiative corrections are important for the Higgs-stau system with charge breaking minima.

A possible solution to the [formula omitted] puzzle using the principle of maximum conformality

The measured Bd→π0π0 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→ππ can be greatly reduced by applying the Principle of Maximum Conformality (PMC), and more precise predictions for CP-averaged branching ratios B(B→ππ) can be achieved. Combining the errors in quadrature, we obtain B(Bd→π0π0)|PMC=(0.98−0.31+0.44)×10−6 by using the light-front holographic low-energy model for the running coupling. All of the CP-averaged B→ππ branching fractions predicted by the PMC are consistent with the Particle Data Group average values and the recent Belle data. Thus, the PMC provides a possible solution for the Bd→π0π0 puzzle.

Finite quark-mass effects in the NNLOPS POWHEG+MiNLO Higgs generator

We include finite top- and bottom-mass effects in the next-to-next-to-leading order parton shower (NNLOPS) event generator for inclusive Higgs boson production in gluon fusion based upon the POWHEG+MiNLO approach. Since fixed-order results for quark-mass effects only reach NLO accuracy, we add them to the NNLOPS generator at that accuracy. We explore uncertainties related to the unknown all-order logarithmic structure of bottom-mass effects by comparing the assumption of full exponentiation to no exponentiation at all. Phenomenological results showing the effects of finite quark-masses in the NNLOPS simulation are presented. These suggest that the aforementioned uncertainty is well contained within the envelope of plain renormalization and factorization scale uncertainties.

Reanalysis of the higher order perturbative QCD corrections to hadronicZdecays using the principle of maximum conformality

A complete calculation of the ${\cal O}(\alpha_s^4)$ perturbative QCD corrections to the hadronic decay width of the $Z$-boson has recently been performed by Baikov et al.[1]. In their analysis, Baikov et al. relied on the conventional practice of simply guessing the renormalization scale and taking an arbitrary range to estimate the pQCD uncertainties. This procedure inevitably introduces an arbitrary, scheme-dependent theoretical systematic error in the predictions. In this paper, we show that the renormalization scale uncertainties for hadronic $Z$ decays can be greatly reduced by applying the principle of maximum conformality (PMC), a rigorous extension of the BLM method. The PMC prediction is independent of the choice of renormalization scheme; i.e., it respects renormalization group invariance, and thus it provides an optimal and theoretically rigorous method for setting the renormalization scale. We show that the convergence of the pQCD prediction for the $Z$ hadronic width is greatly improved using the PMC since the divergent renormalon series does not appear. The magnitude of the high-order corrections quickly approach a steady point. The PMC predictions also have the property that any residual dependence on the choice of initial scale is highly suppressed, even for low-order predictions. Thus, one obtains optimal fixed-order predictions for the $Z$-boson hadronic decay rates thus enabling high precision tests of the Standard Model.

The Higgs boson inclusive decay channels $$H \rightarrow b\bar{b}$$ H → b b ¯ and $$H \rightarrow gg$$ H → g g up to four-loop level

The principle of maximum conformality (PMC) has been suggested to eliminate the renormalization scheme and renormalization scale uncertainties, which are unavoidable for the conventional scale setting and are usually important errors for theoretical estimations. In this paper, by applying PMC scale setting, we analyze two important inclusive Standard Model Higgs decay channels, $H\rightarrow b\bar{b}$ and $H\rightarrow gg$, up to four-loop and three-loop levels accordingly. After PMC scale setting, it is found that the conventional scale uncertainty for these two channels can be eliminated to a high degree. There is small residual initial scale dependence for the Higgs decay widths due to unknown higher-order $\{\beta_i\}$-terms. Up to four-loop level, we obtain $\Gamma(H\rightarrow b\bar{b}) = 2.389\pm0.073 \pm0.041$ MeV and up to three-loop level, we obtain $\Gamma(H\rightarrow gg) = 0.373\pm0.030$ MeV, where the first error is caused by varying $M_H=126\pm4$ GeV and the second error for $H\to b\bar{b}$ is caused by varying the $\overline{\rm MS}$-running mass $m_b(m_b)=4.18\pm0.03$ GeV. Taking $H\to b\bar{b}$ as an example, we present a comparison of three BLM-based scale setting approaches, e.g. the PMC-I approach based on the PMC-BLM correspondence, the $R_\delta$-scheme and the seBLM approach, all of which are designed to provide effective ways to identify non-conformal $\{\beta_i\}$-series at each perturbative order. At four-loop level, all those approaches lead to good pQCD convergence, they have almost the same pQCD series, and their predictions are almost independent on the initial renormalization scale. In this sense, those approaches are equivalent to each other.

Formula omitted] production at the B factories under the principle of maximum conformality

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 sections for the double charmonium production process, e++e−→J/ψ(ψ′)+χcJ with (J=0,1,2). The running behavior for the coupling constant, governed by the PMC scales, are determined exactly for the specific processes. We compare our predictions with the measurements at the B factories, BaBar and Belle, and the theoretical estimations obtained in the literature. Because the non-conformal terms are different for various polarized and unpolarized cross sections, the PMC scales of these cross sections are different in principle. It is found that all the PMC scales are almost independent of the initial choice of renormalization scale. Thus, the large renormalization scale uncertainty usually adopted in the literature up to ∼40% at the NLO level, obtained from the conventional scale setting, for both the polarized and the unpolarized cross sections are greatly suppressed. It is found that the charmonium production is dominated by J=0 channel. After PMC scale setting, we obtain σ(J/ψ+χc0)=12.25−3.13+3.70 fb and σ(ψ′+χc0)=5.23−1.32+1.56 fb, where the squared average errors are caused by bound state parameters as mc, |RJ/ψ(0)| and |RχcJ′(0)|, which are non-perturbative error sources in different to the QCD scale setting problem. In comparison to the experimental data, a more accurate theoretical estimation shall be helpful for a precise testing of QCD and for determining whether there is new physics beyond the Standard Model.

Application of the principle of maximum conformality to the top-quark forward-backward asymmetry at the Tevatron

The renormalization scale uncertainty can be eliminated by the Principle of Maximum Conformality (PMC) in a systematic scheme-independent way. Applying the PMC for the $t\bar{t}$-pair hadroproduction at the NNLO level, we have found that the total cross-sections $\sigma_{t\bar{t}}$ at both the Tevatron and LHC remain almost unchanged when taking very disparate initial scales $\mu^{\rm init}_R$ equal to $m_t$, $10\,m_t$, $20\,m_t$ and $\sqrt{s}$, which is consistent with renormalization group invariance. As an important new application, we apply PMC scale-setting to study the top-quark forward-backward asymmetry. We observe that the more convergent perturbative series after PMC scale-setting leads to a more accurate top-quark forward-backward asymmetry. The resulting PMC prediction on the asymmetry is also free from the initial renormalization scale-dependence. Because the NLO PMC scale has a dip behavior for the $(q\bar{q})$-channel at small subprocess collision energies, the importance of this channel to the asymmetry is increased. We observe that the asymmetries $A_{FB}^{t\bar{t}}$ and $A_{FB}^{p\bar{p}}$ at the Tevatron will be increased by 42% in comparison to the previous estimates obtained by using conventional scale-setting; i.e. we obtain $A_{FB}^{t\bar{t},{\rm PMC}} \simeq 12.5%$ and $A_{FB}^{p\bar{p},{\rm PMC}} \simeq 8.28%$. Moreover, we obtain $A_{FB}^{t\bar{t},{\rm PMC}}(M_{t\bar{t}}>450 \;{\rm GeV}) \simeq 35.0%$. These predictions have a $1\sigma$-deviation from the present CDF and D0 measurements; the large discrepancies of the top-quark forward-backward asymmetry between the Standard Model estimate and the CDF and D0 data are thus greatly reduced.

Azimuthal decorrelation of Mueller-Navelet jets at the Tevatron and the LHC

We study the production of Mueller-Navelet jets at hadron colliders in the Balitsky-Fadin-Kuraev-Lipatov framework. We show that a measurement of the relative azimuthal angle {delta}{phi} between the jets can provide a good testing ground for corrections due to next-leading logarithms (NLL). Besides the well-known azimuthal decorrelation with increasing rapidity interval {delta}{eta} between the jets, we propose to also measure this effect as a function of R=k{sub 2}/k{sub 1}, the ratio between the jet transverse momenta. Using renormalization-group improved NLL kernel, we obtain predictions for d{sigma}/d{delta}{eta}dRd{delta}{phi}. We analyze NLL-scheme and renormalization-scale uncertainties, and energy-momentum conservation effects, in order to motivate a measurement at the Tevatron and the LHC.

Analytic calculation of two-loop QCD corrections tob→sℓ+ℓ−in the highq2region

We present our results for the NNLL virtual corrections to the matrix elements of the operators O1 and O2 for the inclusive process b → sl+l− in the kinematical region q2 > 4mc2, where q2 is the invariant mass squared of the lepton-pair. This is the first analytic two-loop calculation of these matrix elements in the high q2 region. We give the matrix elements as an expansion in mc/mb and keep the full analytic dependence on q2. Making extensive use of differential equation techniques, we fully automatize the expanding of the Feynman integrals in mc/mb. In coincidence with an earlier work where the master integrals were obtained numerically [1], we find that in the high q2 region the αs corrections to the matrix elements sl+l−|O1,2|b calculated in the present paper lead to a decrease of the perturbative part of the q2-spectrum by 10%−15% relative to the NNLL result in which these contributions are put to zero and reduce the renormalization scale uncertainty to ~ 2%.

Second-Order QCD Corrections to the Thrust Distribution in Electron-Positron Annihilation

We compute the next-to-next-to-leading-order (NNLO) QCD corrections to the thrust distribution in electron-positron annihilation. The corrections turn out to be sizable, enhancing the previously known next-to-leading-order prediction by about 15%. Inclusion of the NNLO corrections significantly reduces the theoretical renormalization scale uncertainty on the prediction of the thrust distribution.

NLO QCD corrections to Higgs production with associated heavy-quark pairs at hadron colliders

The reactions p p pp → t tH + X represent important channels in the search for the Standard Model Higgs boson at the Tevatron and the LHC. In the leading perturbative order the cross sections suffer from severe (renormalization and factorization) scale uncertainties. The next-to-leading order QCD corrections stabilize the cross sections considerably. The calculation of these corrections is briefly reviewed, and a few numerical results are discussed.