Top Quark Pole Mass: Definition & Precision

• Top Quark Pole Mass is defined by the location of the pole in the top quark propagator and is crucial for validating Standard Model predictions.
• Experimental methods extract mₜᴾᵒˡᵉ using inclusive and differential t-tbar cross section measurements compared with advanced NNLO+NNLL QCD calculations.
• Theoretical innovations, including renormalization scheme conversion and the Principle of Maximum Conformality, enhance precision despite intrinsic renormalon uncertainties.

The top quark pole mass, $m_t^{\rm pole}$, is a fundamental parameter of the Standard Model defined as the pole (on-shell) mass of the top quark propagator in perturbative Quantum Chromodynamics (QCD). As opposed to renormalization-scheme-dependent masses such as the $\overline{\text{MS}}$ (MS‐bar) mass, the pole mass is conceptually tied to the location of the complex pole in the renormalized two-point function of the top quark. Its precise determination is crucial for tests of the Standard Model, electroweak precision fits, and studies of vacuum stability. Current experimental strategies exploit precision measurements of inclusive and differential $t\bar{t}$ and $t\bar{t}+$jet cross sections as well as advanced theoretical calculations up to next-to-next-to-leading order (NNLO) with resummations to next-to-next-to-leading logarithms (NNLL).

1. Definition and Renormalization Scheme Ambiguities

The pole mass is defined by the position of the pole of the resummed quark propagator: $$S^{-1}(p) \Big|_{p^2 = (m_t^{\rm pole})^2 - i\,m_t^{\rm pole}\,\Gamma_t} = 0$$ where $\Gamma_t$ is the top quark total decay width. This definition is unambiguous order-by-order in perturbation theory in QCD. However, due to infrared (IR) renormalon ambiguities the pole mass exhibits an intrinsic non-perturbative uncertainty of order $\Lambda_{\rm QCD}$, quantified as $\sim110$ MeV for the top quark (Beneke et al., 2016). This limitation motivates the use of short-distance masses (e.g., $\overline{\text{MS}}$, MSR) in some contexts, but the pole mass remains the standard reference in perturbative heavy quark calculations, particularly those involving cross sections and threshold observables. The relation between the pole mass and the $\overline{\text{MS}}$ mass is known up to four loops and, for $N_L=5$,

$$m_t^{\rm pole} = m_t^{\overline{\text{MS}}}(m_t^{\overline{\text{MS}}})\,\left\{1 + \frac{4}{3}\frac{\bar\alpha_s}{\pi} + (-1.0414 N_L + 13.4434)\left(\frac{\bar\alpha_s}{\pi}\right)^2 + (0.6527N_L^2 -26.655N_L + 190.595)\left(\frac{\bar\alpha_s}{\pi}\right)^3\right\}$$

which implies the MS-bar mass is approximately 9.8–10 GeV lower than the pole mass for $m_t^{\rm pole} \sim 173$ GeV (Collaboration et al., 2011).

2. Experimental Determination via Differential and Inclusive Cross Sections

The top quark pole mass is extracted indirectly from $t\bar{t}$ and $t\bar{t}+$jet production cross sections measured in hadron collisions. The strong dependence of the cross section on $m_t^{\rm pole}$ enables its precise extraction when confronted with accurate theoretical predictions.

Key methods:

• Inclusive $\sigma_{t\bar{t}}$ method: The total $t\bar{t}$ production cross section is measured with high precision and compared to theoretical QCD predictions at NNLO+NNLL (including PDF and scale uncertainties) to extract $m_t^{\rm pole}$ by maximizing the likelihood

$$L(m_t) = \int f_{\rm exp}(\sigma | m_t)\, [f_{\rm scale}(\sigma| m_t) \otimes f_{\rm PDF}(\sigma | m_t)]\,d\sigma$$

where $f_{\rm exp}$ is a Gaussian encoding the measured cross section with uncertainties, $f_{\rm scale}$ accounts for renormalization/factorization scale uncertainty, and $f_{\rm PDF}$ captures PDF uncertainties (Collaboration et al., 2011, Brandt, 2017, Barillari, 2017).

• Differential cross section in $t\bar{t} +$1 jet events: The normalized differential cross section as a function of $1/m_{t\bar{t}+{\rm jet}}$ or a related threshold-sensitive variable ($\rho = 340~{\rm GeV}/m_{t\bar{t}+{\rm jet}}$ or $\rho_s = (2 m_0)/\sqrt{s_{t\bar{t}+1~{\rm jet}}}$) is measured. This observable is particularly sensitive to $m_t^{\rm pole}$ near threshold; its shape is fitted to NLO or NNLO predictions to extract the mass. The use of normalized distributions reduces experimental and theoretical uncertainties (Collaboration, 2015, Collaboration, 2022, 2207.13413, Collaboration, 3 Jul 2025).

Recent ATLAS and CMS measurements in dileptonic final states employ advanced unfolding techniques (Iterative Bayesian Unfolding, maximum likelihood or profiled likelihood fits) and machine learning regression to reconstruct $m_{t\bar{t}+{\rm jet}}$ with improved resolution before comparison to theory (Collaboration, 2022, 2207.13413, Collaboration, 3 Jul 2025). Systematic uncertainties (jet energy scale, b-tagging, background modeling, PDF, and $\alpha_s$) are incorporated as nuisance parameters in the fit.

3. Theoretical Calculations and Mass Scheme Conversion

The QCD predictions employed in pole mass extractions include:

• NLO, NLO+PS, NLO+NLL(NNLL), approximate NNLO, and full NNLO (when available), always with the pole mass in the on-shell scheme as the argument.
• Cross-checks: For specific final states, two theoretical descriptions are compared—one treating top quarks as stable (on-shell, $2 \to 3$) and another including full decays and off-shell effects ($2 \to 7$) (Collaboration, 3 Jul 2025). Consistency within uncertainties between these approaches validates the robustness of the extraction and the parton-level modeling.

Conversion to the $\overline{\text{MS}}$ mass uses the three- or four-loop mass relation (shown above). The measured cross section (parameterized, e.g., as

$$\sigma_{t\bar{t}}(m_t^{\rm MC}) = \frac{1}{(m_t^{\rm MC})^4}[a + b(m_t^{\rm MC} - m_0) + c(m_t^{\rm MC} - m_0)^2 + d(m_t^{\rm MC} - m_0)^3]$$

) is cross-correlated with theory $\sigma_{t\bar{t}}(m_t^{\rm pole})$ via likelihood fits to extract the preferred pole mass (Collaboration et al., 2011).

4. Precision, Systematic Uncertainties, and Recent Measurements

Recent measurements at the LHC using $\sqrt{s} = 13$ TeV $pp$ data and in $t\bar{t}+$jet events reach sub-GeV to 1.5 GeV total uncertainties:

• ATLAS ($2 \to 3$ description): $$m_t^{\rm pole} = 170.7 \pm 0.3~{\rm(stat.)} \pm 1.4~{\rm(syst.)} \pm 0.3~{\rm(scale)} \pm 0.2~({\rm PDF} \oplus \alpha_s)$$ GeV (Collaboration, 3 Jul 2025).
• CMS: $m_t^{\rm pole} = 172.93 \pm 1.36$ GeV (ABMP16NLO PDF), $m_t^{\rm pole} = 172.94 \pm 1.37$ GeV (Collaboration, 2022, 2207.13413).

Uncertainties on $m_t^{\rm pole}$ measurements arise from:

• Statistical: Limited by dataset size.
• Experimental/systematic: Jet energy scale, background modeling, lepton corrections, and b-tagging efficiency dominate (1–1.4 GeV).
• Theoretical: Scale variation (0.3–1.0 GeV), PDF uncertainties (0.2–0.5 GeV), and $\alpha_s$.
• Renormalon ambiguity: Sets the ultimate floor on the pole mass definition at $\sim$110 MeV, as shown by combining the known asymptotic behavior of the QCD series and exact four-loop coefficients (Beneke et al., 2016). The uncertainty due to higher unknown orders (five loops and beyond) is estimated at $\sim$300 MeV.

5. Cross-Checks and Consistency with Other Methods

Results from $t\bar{t}$ cross section–based pole mass determinations are found to be consistent with direct kinematic reconstruction and template-based extractions of the so-called “Monte Carlo mass” (which is interpreted as $m_t^{\rm pole}$ to an accuracy of $\sim1$ GeV). Recent ATLAS and CMS “direct” measurements in $t\bar{t}$ and single top events occupy the $m_t^{\rm MC} = 172$–173 GeV range with total uncertainties approaching a few hundred MeV (Collaboration, 2015, Collaboration, 2021, Nisius, 2017, Pearson, 2017). The agreement with pole mass results supports the interpretation that direct-reconstruction techniques are effectively measuring the pole mass (Collaboration et al., 2011).

Monte Carlo mass calibration studies reveal the MC parameter is close to the MSR mass at $R=1$ GeV and can differ from the pole mass by up to 0.6 GeV, reinforcing the need to state the renormalization scheme of the extracted parameter (1803.02321).

6. Theoretical Innovations: Renormalization Scale Setting

The Principle of Maximum Conformality (PMC) has been applied to remove renormalization scale uncertainty in NNLO $t\bar{t}$ cross-section predictions by systematically absorbing $\beta$-dependent terms into the running coupling and setting optimized order-by-order scales. Extractions using the PMC yield

$m_t^{\rm pole} = 172.5 \pm 1.4~{\rm GeV},$

in very good agreement with other determinations and with dramatically reduced theoretical scale uncertainty (Wang et al., 2020, Wang et al., 2017).

7. Future Directions and Precision Limits

Upcoming improvements are expected from reductions in PDF uncertainties achieved by global fits to differential distributions in $t\bar{t}$ and $t\bar{t}j$ production, including rapidity and longitudinal momentum. Such PDF updates, via tools like ePump, can yield a 12–20% reduction in top mass uncertainty in high-statistics environments (HL-LHC and future 100 TeV machines) (Gombas et al., 2023).

The irreducible uncertainty in the pole mass definition (~110 MeV from renormalons) defines the ultimate precision limit for this scheme, motivating the parallel reporting of short-distance masses for future high-precision theoretical uses (Beneke et al., 2016, 1803.02321).

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Summary Table: Representative Recent Measurements of $m_t^{\rm pole}$ via $t\bar{t}$ and $t\bar{t}+$jet Cross Sections

Experiment/Method	$m_t^{\rm pole}$ [GeV]	Total Uncertainty (GeV)
ATLAS $t\bar{t}+$jet, $2\to3$ (2025)	$$170.7 \pm 0.3 (\rm stat) \pm 1.4 (\rm syst) \pm 0.3 (\rm scale) \pm 0.2 (\rm PDF)$$	$\sim$1.5 (Collaboration, 3 Jul 2025)
CMS $t\bar{t}+$jet (2022, ABMP16NLO)	$172.93 \pm 1.36$	1.36 (Collaboration, 2022)
CMS $t\bar{t}+$jet (2022, ABMP16NLO)	$172.94 \pm 1.37$	1.37 (2207.13413)
LHC Tevatron avg (direct)	$172$–$173$	$<1$ (Brandt, 2017)
PMC NNLO Theory ($\sqrt{s}=13$ TeV)	$172.5 \pm 1.4$	1.4 (Wang et al., 2020)

These results illustrate the robust consistency and increasing precision of $m_t^{\rm pole}$ determinations through multiple experimental and theoretical strategies, all ultimately bounded in precision by the intrinsic limitations of the pole mass concept itself.