Quark mass relations to four-loop order (1502.01030v2)

Abstract: We present results for the relation between a heavy quark mass defined in the on-shell and $\bar{\rm MS}$ scheme to four-loop order. The method to compute the four-loop on-shell integral is briefly described and the new results are used to establish relations between various short-distance masses and the $\bar{\rm MS}$ quark mass to next-to-next-to-next-to-leading order accuracy. These relations play an important role in the accurate determination of the $\bar{\rm MS}$ heavy quark masses.

• The paper presents a four-loop analysis converting on-shell to MS quark masses, achieving N³LO precision.
• It employs advanced methods like the Laporta algorithm and integration by parts to tackle complex integrals.
• Numerical findings reduce mass-renormalization uncertainties, enhancing precision in heavy quark determinations.

Quark Mass Relations to Four-Loop Order in Perturbative QCD

The paper presents a systematic and comprehensive four-loop analysis of quark mass relations within perturbative Quantum Chromodynamics (QCD), focusing on the conversion between on-shell (OS) and Modified Minimal Subtraction ($\overline{\mathrm{MS}}$) schemes. Accurately determining the masses of heavy quarks such as the top, bottom, and charm quarks is crucial for various theoretical and experimental endeavors, making these calculations significant for the field.

The authors employ sophisticated computational techniques to derive these relations and culminate in the expression of the four-loop corrections necessary for achieving Next-to-Next-to-Next-to-Leading Order (N$^3$LO) precision in renormalization. A salient aspect of this work is the evaluation and manipulation of a substantial number of complex integrals, executed using automation tools and simplifying algorithms such as the Laporta algorithm and integration by parts identities.

Numerical and Analytical Results

The paper provides crucial numerical results for the four-loop coefficients, particularly for $z_m^{(n)}$, which encapsulate the relationship between the pole mass and the $\overline{\mathrm{MS}}$ mass. For instance, the findings such as $z_m^{(4)}$ for $n$ ranging from three to five loop orders, illustrate how these coefficients contribute to reducing the mass-renormalization uncertainty.

Moreover, the authors offer numerical conversion examples between threshold masses such as Potential Subtracted (PS), 1S, and Renormalon Subtracted (RS) masses, and the $\overline{\mathrm{MS}}$ mass at varying orders of perturbation. Their tables reveal details of these conversions, showing noteworthy improvements when four-loop corrections are incorporated, which substantially diminish the theoretical uncertainties associated with threshold mass definitions.

Implications

This work underscores the importance of renormalization scheme conversion at high precision levels, which is vital for aligning theoretical predictions with experimental measurements. The ability to convert into $\overline{\mathrm{MS}}$ masses with enhanced accuracy directly impacts electroweak precision fits, B-meson decay spectroscopy, and Yukawa coupling unification studies.

Future Prospects

Beyond serving as a robust reference for current standard model analyses, these results lay the groundwork for even higher-order corrections as computational power and techniques improve. Future advancements may focus on enhancing numeric accuracy, deriving analytical formulations for still untackled contributions, and possibly extending similar precision-level corrections to other QCD-renormalized quantities.

This paper represents a substantial contribution to the ongoing improvements in precision perturbative QCD and holds significant promise for enhancing the accuracy of particle physics predictions, with practical applications increasingly relevant in global fits and searches for physics beyond the Standard Model.