Pole MassEdit

Pole mass

Pole mass is a fundamental parameter used in quantum field theory to describe the mass of a fermion, most prominently the quarks of quantum chromodynamics Quantum Chromodynamics. It is defined in relation to the quark propagator: the pole mass m_p is the value for which the full propagator has a pole, i.e., where its inverse vanishes in the complex energy plane. In practice this means locating the energy at which the propagator describes a particle with on-shell kinematics. For a colored particle such as a quark, this definition is clean in perturbation theory but becomes subtle once nonperturbative effects of confinement are taken into account. Nevertheless, the pole mass remains a widely used parameter in the phenomenology of heavy quarks, especially the top Top quark and bottom quarks, where it serves as a convenient reference scale for short-distance processes.

Introductory overview - In the Standard Model and its natural extensions, quark masses arise from Yukawa couplings to the Higgs field, and their numerical values play a central role in predicting collider outcomes and precision tests of the theory. The pole mass captures, in a gauge-invariant sense at perturbative order, the on-shell energy scale associated with a quark if one could observe an isolated pole in the propagator. - Because quarks in QCD are confined, no free quark is directly observed. This makes the pole mass a parameter that is meaningful within perturbation theory and in high-energy processes where short-distance physics dominates, but it is not an observable in isolation. The distinction between a quark’s pole mass and its long-distance behavior is one of the key themes in modern QCD.

Definition and formalism

The pole mass m_p is defined as the location of the pole of the full quark propagator S(p) in the complex energy plane, satisfying S(p)^{-1} = 0 at p^2 = m_p^2 in perturbation theory. This ties the mass to the on-shell condition for a colored fermion within a calculational scheme that treats interactions with gluons.
In practice, the pole mass is extracted within a renormalization framework, where masses are defined through a chosen renormalization scheme. The most common comparison is with the MSbar (modified minimal subtraction) mass m_MSbar, a short-distance mass defined in a particular renormalization scale scheme. The relationship between m_p and m_MSbar is known perturbatively and can be computed to high orders in the strong coupling α_s.
The exact numerical relation between m_p and m_MSbar depends on the renormalization scale and on higher-order radiative corrections. This relation is the subject of careful perturbative analysis, including the computation of radiative coefficients to several loop orders, and it is important for translating between different mass definitions used in theory and experiment.

Relationship to other mass definitions

Short-distance masses, such as the MSbar mass, are defined to be insensitive to long-distance, nonperturbative QCD effects. They are particularly useful for high-precision tests and for maintaining controlled perturbative expansions in calculations of cross sections and decay rates.
The pole mass and MSbar mass are related by a perturbative series in α_s, with coefficients that have been computed to multiple loops. In addition to MSbar, other short-distance schemes have been proposed and used in phenomenology, such as the 1S mass and RS mass, each designed to minimize certain theoretical uncertainties in specific calculations.
In the heavy-quark sector, the choice of mass definition affects the convergence of the perturbative series and the stability of predictions for observables like production cross sections and decay spectra. The pole mass remains natural in some contexts, but its intrinsic limitations have prompted widespread use of alternative schemes for precision work.

Infrared issues and renormalon ambiguity

A central theoretical point is that the pole mass in QCD is not free from nonperturbative ambiguities. Infrared renormalons—a manifestation of the asymptotic nature of the perturbative expansion in QCD—induce an intrinsic uncertainty in the pole mass of order Λ_QCD, the characteristic nonperturbative scale of QCD. This means that, beyond a certain precision, the pole mass cannot be defined unambiguously within perturbation theory.
The practical upshot is that while the pole mass is a useful and intuitive concept for certain high-energy applications, there is a fundamental limit to how precisely it can be determined. For many purposes, especially where nonperturbative effects are relevant, short-distance mass definitions provide cleaner and more stable predictions.
This tension has sparked systematic work to understand the size of the renormalon effects and to develop mass definitions that suppress or remove the associated ambiguities, which in turn improves the reliability of precision QCD tests.

Practical implications in experiments and phenomenology

In collider physics, the mass of the top quark is a primary example where the pole mass has been historically used as a convenient reference. Direct measurements of the top mass through kinematic reconstruction and others methods are often interpreted in terms of a pole-mass-like parameter, though the mapping to a strict pole mass is nuanced because experimental mass parameters are extracted using Monte Carlo event generators with their own definitions and modeling assumptions.
Cross-section-based extractions and global fits frequently prefer short-distance masses (e.g., MSbar) due to their better perturbative behavior. The translation between m_pole and m_MSbar is essential for combining different kinds of measurements and for comparisons with theory predictions across energy scales.
For bottom and charm quarks, nonperturbative effects are more pronounced, making the pole mass even more delicate as a phenomenological quantity. In these cases, alternative mass definitions and careful treatment of nonperturbative corrections are standard practice.

Controversies and debates

The central debate centers on how best to parameterize quark masses for precision predictions. Some researchers advocate continuing use of the pole mass in contexts where it provides a transparent physical picture and where calculations are formulated in schemes that effectively isolate short-distance physics. Others argue that the intrinsic renormalon ambiguity limits the pole mass’s usefulness and that short-distance masses should be the default for high-precision work, with explicit conversions when needed.
A related issue is what experimental mass measurements actually correspond to. Direct mass determinations often rely on Monte Carlo simulations whose mass parameter does not perfectly coincide with the pole mass, MSbar mass, or any single field-theory definition. This has driven efforts to design observables and analysis strategies that map more cleanly onto well-defined mass schemes.
The preference for short-distance masses aligns with a broader philosophy in high-energy physics: place emphasis on quantities that admit stable, well-behaved perturbative expansions and that minimize nonperturbative uncertainties. In this view, pole mass is acknowledged as a historically important and practically convenient reference, but not the ultimate arbiter of quark mass values in the era of precision QCD.

Measurement and determination

The extraction of a pole-mass-like parameter from data involves comparing measurements with theoretical predictions computed in a chosen mass scheme and then translating between schemes using perturbation theory.
For the top quark, measurements from hadron colliders and from future lepton colliders contribute to a consistent picture when interpreted through the lens of short-distance mass definitions, with careful accounting of experimental systematics and theoretical uncertainties.
The ongoing program in lattice QCD, effective field theories, and global fits continues to refine the precise relationships among different mass definitions and to reduce ambiguities arising from nonperturbative QCD effects.