Unveiling Mass Effects: How the Finite Mass of Gauge Bosons Shapes PDFs for Muon Colliders
Executive Summary
In muon colliders, the presence of finite masses for gauge bosons notably modifies parton distribution functions in the leading-log regime. Unlike the massless gauge limit, where collinear singularities generate large logarithms of the type ln Q2 over a cutoff scale, the boson mass acts as a physical regulator. This alters off-shell propagators and splitting functions, reducing boson radiation below the boson scale and changing the structure of dominant logs to terms dependent on ln Q2 over MV2 and corrections proportional to MV2 over Q2 at high scales.
Numerical Impact and Low Scales
Numerically, the effect is especially relevant at low scales Q near or below the boson mass. Muon distributions undergo a suppression of collinear radiation that can translate into typical corrections of several percent to tens of percent depending on the energy range and the applied angular cut. For Q close to MV, the splitting functions show a smooth transition between the massive and asymptotic regimes, requiring explicit matching between massive and massless schemes. Phase-space integrals are bounded by the mass, which reduces the probability of soft boson emission and modifies the effective DGLAP evolution in the leading log.
Off-Shell Propagators and Splitting Function Modifications
Off-shell propagators incorporate terms of the form 1 over p2 minus MV2 that introduce thresholds and attenuation when the virtuality p2 approaches MV2. In practice, this means that splitting kernels receive MV2-dependent corrections that suppress regions of extreme collinearity. Effective splitting functions cease to be purely universal in the massless regime and acquire mass dependencies that must be included in the calculation of initial PDFs for muon colliders.
Consequences for Simulation and Experimental Design
For simulators and sensitivity analyses in muon colliders, it is crucial to implement schemes that account for boson masses in shower generation and PDF evolution. Ignoring these effects can lead to overestimating rates of processes with boson emission and biasing theoretical uncertainty estimates. We recommend employing matching with massive log resummations and validating against fixed higher-order calculations in transition regions.
Practical Recommendations
1 Implement splitting kernels with mass regularization and include logarithmic terms ln Q2 over MV2 in the evolution. 2 Use matching between massive and massless schemes around Q comparable to MV. 3 Evaluate numerical corrections in initial PDF files and provide uncertainties associated with the mass. 4 Validate through Monte Carlo simulations with showers that respect massive kinematics.
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