Three-loop QCD+QED corrections to on-shell quark renormalization

1. Problem Statement

The precise determination of quark masses is fundamental to testing the Standard Model (SM) of particle physics. However, due to color confinement, quarks are not observed as free particles, making the definition of a "physical" mass non-trivial.

• The Pole Mass vs. $\overline{\text{MS}}$ Mass: The two most common definitions are the pole mass (on-shell mass) and the $\overline{\text{MS}}$ mass. While the pole mass is conceptually convenient for on-shell calculations, it suffers from an intrinsic non-perturbative ambiguity of order $\Lambda_{\text{QCD}}$ due to infrared (IR) renormalons. The $\overline{\text{MS}}$ mass is a short-distance mass free from these leading ambiguities.
• The Gap: While the relation between these masses and the associated renormalization constants have been computed up to three loops in pure QCD (and partially in mixed QCD+QED), the complete three-loop mixed QCD+QED corrections (terms of order $\mathcal{O}(\alpha_s^m \alpha_e^n)$ with $m+n=3$ and $m,n > 0$) to the on-shell quark mass and wave-function renormalization constants were previously unknown.
• Motivation: With the High-Luminosity LHC and future $e^+e^-$ colliders, experimental precision demands theoretical predictions that include both strong and electromagnetic interactions at high perturbative orders. Additionally, a new mass definition, the $\sigma$-mass (trace-anomaly subtracted), has been proposed to combine the benefits of the pole and $\overline{\text{MS}}$ schemes while avoiding IR renormalons; its extension to mixed QCD+QED was required.

2. Methodology

The authors performed a complete calculation of the heavy quark self-energy up to three loops in the presence of both QCD and QED interactions.

• Computational Setup:

  • Diagrams: Feynman diagrams were generated using DiaGen and cross-checked with FeynArts. The calculation includes a single massive quark flavor ($Q$) and $n_l$ massless quark flavors.
  • Gauge: Calculations were performed in a general Lorentz-covariant gauge ($\xi$) for both gluon and photon propagators to verify gauge independence of mass renormalization and track gauge dependence of wave-function renormalization.
  • Algebra: Color algebra (SU($N_c$)) and Dirac algebra were handled using FORM and CalcLoop.
  • Integral Reduction: Tensor amplitudes were reduced to scalar Feynman integrals. Integration-by-parts (IBP) identities were used to reduce these to a set of Master Integrals (MIs) using Kira, Blade, and IdSolver.
  • Evaluation: MIs were evaluated numerically at the on-shell kinematic point using AMFlow with high precision. The on-shell limit was approached via generalized power-logarithmic series expansions of differential equations to ensure reliability.
  • Reconstruction: Analytic expressions were reconstructed from numerical results using the PSLQ algorithm.

• Renormalization Scheme:

  • The calculation uses Dimensional Regularization (DR).
  • The on-shell renormalization constants ($Z_m^{\text{os}}$, $Z_Q^{\text{os}}$) are derived from the self-energy $\Sigma_B$.
  • The $\overline{\text{MS}}$ renormalization constants ($Z_m$, $Z_{\alpha_s}$, $Z_{\alpha_e}$) are extracted by requiring the cancellation of UV poles in the ratio between on-shell and $\overline{\text{MS}}$ schemes.

3. Key Contributions

The paper provides the first complete analytical results for the following quantities at three-loop order in mixed QCD+QED:

1. On-shell Mass Renormalization Constant ($Z_m^{\text{os}}$):

   • Derived up to $\mathcal{O}(\alpha_s^3, \alpha_s^2\alpha_e, \alpha_s\alpha_e^2, \alpha_e^3)$.
   • Includes contributions from massive quark loops ($n_h$) and distinct massless quark flavors ($n_q, n_{q'}$).
   • Explicitly handles the complex flavor structures arising in pure QED at three loops (quartic in flavor multiplicities).

2. On-shell Wave-function Renormalization Constant ($Z_Q^{\text{os}}$):

   • Derived for the heavy quark up to three loops.
   • Demonstrates the dependence on the gauge parameter $\xi$ starting at three loops in QCD (consistent with literature), while pure QED contributions remain $\xi$-independent.

3. Mass Relation ($Z_m^f$):

   • Established the finite conversion factor $Z_m^f = m_{\overline{\text{MS}}} / m_{\text{pole}}$ up to three loops.
   • This required deriving the three-loop $\overline{\text{MS}}$ mass renormalization constant $Z_m$ in the mixed QCD+QED scheme, which was previously unknown.

4. $\sigma$-Mass Relation:

   • Extended the definition of the trace-anomaly subtracted $\sigma$-mass ($m_\sigma$) to include mixed QCD+QED effects.
   • Provided the explicit conversion formula between $m_\sigma$ and the on-shell mass ($m_{\text{os}}$) up to three loops.

4. Key Results

• Analytic Expressions: The paper presents closed-form analytic expressions for all renormalization constants and mass relations. These are expressed in terms of:
  • Couplings: $\alpha_s$ and $\alpha_e$ (rescaled by $4\pi$).
  • Color factors: $C_F, C_A, N_c$.
  • Flavor multiplicities: $n_h$ (massive), $n_q, n_{q'}$ (massless).
  • Electric charges: $e_Q$ (heavy), $e_q, e_{q'}$ (light).
  • Logarithms: $L_\mu = \ln(\mu^2/m_{\text{os}}^2)$.
  • Transcendentals: $\pi$, $\zeta_3, \zeta_5$, $\text{Li}_4(1/2)$, and polylogarithms.
• Gauge Independence: The mass renormalization constant $Z_m^{\text{os}}$ is confirmed to be independent of the gauge parameter $\xi$, while $Z_Q^{\text{os}}$ exhibits $\xi$-dependence starting at three loops.
• Consistency Checks:
  • Pure QCD limits match existing results from Refs. [47, 48, 96].
  • Pure QED limits are consistent with known two-loop results and extend to three loops.
  • The $\sigma$-mass relation reduces to the pure QCD result in Ref. [62] when QED is turned off.
• Flavor Structures: A notable finding is the emergence of quartic flavor structures (e.g., $n_h^2, n_h n_q, n_q n_{q'}$) in the pure QED three-loop contributions, reflecting the complexity of multi-loop photon exchanges involving multiple fermion loops.

5. Significance

• Precision Physics: These results are essential for reducing theoretical uncertainties in the determination of heavy quark masses (specifically top and bottom) to match the precision of upcoming experimental data from the HL-LHC and future colliders.
• Renormalization Group Evolution: The derived anomalous dimensions and mass relations allow for more accurate running of quark masses between different energy scales, incorporating both electromagnetic and strong corrections simultaneously.
• New Mass Schemes: The extension of the $\sigma$-mass to mixed QCD+QED provides a robust, renormalon-free mass definition that is particularly useful for high-precision phenomenology where the pole mass ambiguity is a limiting factor.
• Theoretical Completeness: This work closes a significant gap in perturbative QCD+QED theory, providing the necessary ingredients for next-to-next-to-next-to-leading order (N$^3$LO) calculations involving heavy quarks in processes where both interactions are relevant.

The authors provide the results in a machine-readable format (supplemental file) to facilitate their use in global fits and phenomenological studies.