SUSY Thresholds at a Muon Collider

One of the useful features of muon colliders is the naturally narrow spread in beam energies. Measurements of threshold cross sections then become a prime candidate for precision measurements of particle masses, widths, and couplings as well as determining particle spin. We describe the potential for measuring cross sections near threshold in supersymmetric theories. INTRODUCTION Muon colliders have negligible bremstrahlung and the beam energy can be measured very accurately by muon precession in the collider ring. Momentum spreads as low as ∆P/P = 0.003% are thought to be achievable for a lowenergy collider [1,2]. The beam energy could be determined with a precision of ∆E/E = 10−6 by measuring the time-dependent decay asymmetry resulting from the naturally polarized muons [3]. In addition, initial state radiation is smaller than at a electron-positron machine. These features make muon colliders especially useful for studying narrow resonances [4], and for measuring production cross sections near threshold where they change very rapidly [5–7]. However the cross sections are smaller near threshold, so high luminosity is required. An important feature of cross sections near threshold is that they isolate the effects of the width and spin of the produced particle. The threshold cross section is a function of the particle’s mass, decay width, spin, and coupling strength(s), and it (largely) factorizes near the threshold into an overall normalization and an energy-dependent part (profile). The energy profile depends on the particle mass which dictates where in energy the cross section begins to rise; we say the cross section “turns on” near 2m. The particle width and spin govern how fast the cross section rises with energy. The width is important because, on the one hand, we can think 1) To appear in the Proceedings of the 5th International Conference on Physics Potential and Development of μμ Colliders (MuMu99), San Francisco, CA, 15-17 December 1999. of the particle being produced and then subsequently decaying (the narrow width approximation); but on the other hand, one can more accurately view the process as one in which the decay products are the final state and the decaying particle is a virtual particle. The rate the cross section rises depends then on the required “off-shellness” of this virtual particle. As an example consider the cross section for μ+μ− → W+W− followed by the decay W → ff ′ is given by [8] σ(s) = Bf1f2Bf3f4 ∫ s 0 ds1ρ(s1) ∫ ( √ s−s1)2 0 ds2ρ(s2)σ(s, s1, s2) , (1) where σ(s, s1, s2) is the cross section for producing two virtural W bosons with invariant masses-squared of s1 and s2, and Bf1f2 is the branching ratio for the decay W → f1f2. The weighting factor ρ(s) arises from the W boson propagator and controls the rise of the cross section, ρ(s) = 1 π √ sΓ(s) (s − M W ) 2 + M W Γ(s) 2 . (2) The cross section σ(s, s1, s2) is controlled by kinematics and angular momentum conservation for the region that ρ(s1) and ρ(s2) have significant support near s1, s2 ≈ MW . The couplings and radiative corrections are approximately energy-independent over this narrow range, and do not impact on the energy profile, but do affect the overall number of events expected. CHARGINO PRODUCTION Supersymmetric charginos can be pair produced at a muon collider. The two contributing Feynman diagrams for μ+μ− → χ̃1 χ̃ − 1 is shown in Fig. 1. These two diagrams interfere desctructively over most of parameter space. The threshold measurement of the process μ+μ− → χ̃1 χ̃ − 1 yields the following: a measurement of the χ̃1 mass, an indirect measurement of the ν̃μ mass, and a measurement of the overall normalization of the cross section.