On-Shell Scheme for Renormalized Mass

• The paper presents a complete calculation of four-loop QCD corrections that significantly reduce renormalization scale uncertainties in the OS-MS mass conversion.
• It employs the Principle of Maximum Conformality to absorb beta function terms into the running coupling, creating a conformal series free from arbitrary scale choices.
• A Padé Approximation is applied to estimate uncalculated higher-order effects, resulting in an unprecedented precision for the bottom-quark on-shell mass.

The on-shell scheme for renormalized mass defines particle masses such that they correspond to the physical poles of propagators, ensuring direct connection to measured quantities. In perturbative QCD, and specifically in the precision determination of heavy quark masses such as the bottom quark, this approach links the renormalized mass to observable parameters through high-order corrections and addresses the crucial issue of renormalization scale dependence. Recent progress has enabled the calculation of the bottom-quark on-shell (OS) mass, $M_b$, in terms of the running $\overline{\rm MS}$ mass, $\overline{m}_b$, up to four-loop order, incorporating advanced scale-setting methods to minimize theoretical uncertainties (Ma et al., 26 Jun 2024).

1. Four-Loop QCD Corrections and the $\overline{\rm MS}$–On-Shell Relation

The relation between the OS and $\overline{\rm MS}$ bottom-quark masses is given by the perturbative series: $$M_b = \overline{m}_b(\mu_r) \left\{ 1 + C_1(\mu_r)a_s(\mu_r) + C_2(\mu_r)a_s^2(\mu_r) + C_3(\mu_r)a_s^3(\mu_r) + C_4(\mu_r)a_s^4(\mu_r) + \mathcal{O}(a_s^5) \right\},$$ where $a_s = \alpha_s/4\pi$, and the coefficients $C_n(\mu_r)$ contain all explicit and implicit $\mu_r$–dependence, including logarithms of $\mu_r^2/\overline{m}_b^2$. This relation incorporates all known four-loop perturbative QCD corrections, substantially improving the theoretical precision over lower-order calculations and reducing the error due to truncating the perturbative expansion.

The explicit use of four-loop coefficients for the $\overline{\rm MS}$–OS mass conversion captures not only the bulk of QCD corrections but also higher-order matching effects and subleading renormalon contributions. With each loop order, the residual uncertainty in the relation decreases, yet even at four loops, a non-negligible (∼2%) scale-setting ambiguity remains if one relies on the conventional procedure of scale variation.

2. Renormalization Scale Uncertainty and the Principle of Maximum Conformality

In conventional fixed-order pQCD calculations, the choice of renormalization scale $\mu_r$ is ambiguous, leading to an artificial uncertainty. Typically, $\mu_r$ is varied around the central value $\mu_r = \overline{m}_b(\overline{m}_b)$ by a factor of two to estimate the scale uncertainty, resulting in an uncertainty of $2$–$2.2\%$ in the predicted $M_b$. The size and sign of individual loop corrections may vary significantly with $\mu_r$, further challenging the interpretation of the series.

The "Principle of Maximum Conformality" (PMC) resolves this by reorganizing the perturbative expansion to explicitly absorb all $\{\beta_i\}$-dependent terms—those governed by the QCD beta function—into the running coupling, thereby setting the effective scale $Q_*$ order by order: $$M_b|_{\rm PMC} = \overline{m}_b(Q_*) \bigg\{ 1 + r_{1,0} a_s(Q_*) + r_{2,0} a_s^2(Q_*) + r_{3,0} a_s^3(Q_*) + r_{4,0} a_s^4(Q_*) + \mathcal{O}(a_s^5) \bigg\}.$$ The scale $Q_*$ is fixed via a RG-invariant series—here, $Q_* \approx 1.917\,\rm GeV$ for the bottom quark mass—making the entire series insensitive to the arbitrary choice of $\mu_r$. This produces a "conformal series," free from renormalization scheme and scale uncertainties at fixed order.

This approach fundamentally eliminates the dominant source of theoretical uncertainty that has limited the utility of the OS mass in precision Standard Model tests and phenomenological applications.

3. Estimation of Uncalculated Higher-Order Corrections via the Padé Approximation

Even after including four-loop terms, residual uncertainties due to uncalculated higher-order (UHO) corrections remain. The Padé Approximation Approach (PAA) is used to estimate these terms by approximating the truncated series with a rational function: $$\rho^{N/M}(Q) = a_s \frac{b_0 + b_1 a_s + \ldots + b_N a_s^N}{1 + c_1 a_s + \ldots + c_M a_s^M},$$ where the coefficients $b_i$, $c_j$ are determined by matching the known perturbative coefficients. In practice, [0/($n-1$)] Padé approximants are recommended for conformal series.

For the PMC-resummed conformal series, the UHO uncertainty becomes very small (e.g., $\pm0.017$ GeV), while in the conventional scale-setting approach, the UHO uncertainty remains large (e.g., $\pm0.524$ GeV). This striking difference is a direct reflection of the theoretical improvement enabled by PMC.

4. Numerical Determination of the Bottom-Quark On-Shell Mass

The computation uses input values:

• $\overline{m}_b(\overline{m}_b) = 4.183 \pm 0.007$ GeV,
• $\alpha_s(M_Z) = 0.1180 \pm 0.0009$.

The conventional method, with scale variation and Padé-based UHO estimation, yields: $$M_b|_{\rm Conv.} = 5.062^{+0.533}_{-0.525}~\rm GeV.$$ With PMC, the breakdown of individual orders is

$$5.100 + 0.667 - 0.439 - 0.153 + 0.197 = 5.372~\rm GeV,$$

and the total uncertainty (including $\Delta\alpha_s$, $\Delta\overline{m}_b$, UHO) is reduced to: $M_b|_{\rm PMC} = 5.372^{+0.091}_{-0.075}~\rm GeV.$ The uncertainty is the squared average of the errors from $\Delta\alpha_s(M_Z)$, $\Delta\overline{m}_b(\overline{m}_b)$, and the Padé-based UHO estimate.

5. Relevant Formal Properties and LaTeX Formulas

The OS mass in terms of the $\overline{\rm MS}$ mass is represented via: $$\frac{M_b}{\overline{m}_b(\mu_r)} = Z_m^{\overline{\rm MS}} / Z_m^{\rm OS} = \sum_{n=0}^\infty a_s^n(\mu_r) c_m^{(n)}(\mu_r),\qquad c_m^{(0)} = 1.$$ For fixed-order expansion: $$M_b|_{\rm conv.} = \overline{m}_b(\mu_r) \left\{ 1 + \mathcal{C}_1(\mu_r)a_s(\mu_r) + \ldots + \mathcal{C}_4(\mu_r)a_s^4(\mu_r) \right\}.$$ PMC reorganizes this as

$$M_b|_{\rm PMC} = \overline{m}_b(Q_*) \left( 1 + r_{1,0}a_s(Q_*) + r_{2,0}a_s^2(Q_*) + r_{3,0}a_s^3(Q_*) + r_{4,0}a_s^4(Q_*) \right),$$

with $Q_*$ determined by RG invariance via

$$\ln \frac{Q_*^2}{\overline{m}_b^2(Q_*)} = S_0 + S_1 a_s(Q_*) + S_2 a_s(Q_*)^2 + \ldots$$

All explicit scale dependence is thus absorbed into $Q_*$.

6. Phenomenological Implications and Outlook

The implementation of high-order corrections, PMC scale setting, and the Padé Approximation yields a bottom-quark OS mass with unprecedented precision and theoretical control:

• The lattice of uncertainties—scale, parametric, uncalculated higher orders—collapses to a small window, enhancing predictive accuracy for observables sensitive to $M_b$.
• The improved $M_b$ is essential for processes with $b$-quark kinematics dominated by energy scales close to the pole mass (e.g., B-meson decay, threshold production).
• This methodological framework can be immediately generalized to other heavy quark masses (e.g., top) and matched with experimental determinations in future precision collider studies.

The combination of complete four-loop corrections, PMC scale setting, and explicit estimation of higher-order effects via PAA constitutes the current state-of-the-art in rigorous heavy-flavor mass renormalization methodology (Ma et al., 26 Jun 2024).