One-loop Weak Dipole Form Factors and Weak Dipole Moments of Heavy Fermions

The one–loop weak–magnetic and weak–electric dipole form factors of heavy fermions in a generic model are derived. Numerical predictions for the τ lepton and b quark Weak Anomalous Magnetic and Electric Dipole Moments (AWMDM and WEDM) in the SM and MSSM are reviewed. The MSSM contribution to the τ (b) AWMDM could be, in the high tan β scenario, four (thirty) times larger than the Electroweak SM one, but still a factor five below the QCD contribution (in the b case). More interesting is the CP–odd sector where the contribution to the τ (b) WEDM in the MSSM could be up to twelve orders of magnitude larger than in the SM. Talk given at the XXXIIIrd Rencontres de Moriond Work partially supported by the EC under contract ERBFMBICT972474 V ff effective vertex for on–shell fermions The most general V ff effective vertex describing the interaction between a neutral vector boson and two on–shell fermions can be conventionally written in terms of six independent form factors as: Γ ff μ (s) = i { γμ [ F V V − F V A γ5 ] − (q + q̄)μ [ iF V S + F V P γ5 ] + σμν(q + q̄) ν [ iF V M + F V E γ5 ]} . (1) Here q and q̄ are respectively the outgoing momenta of the fermion and the antifermion and s is the square of the total momentum p = q + q̄. The form factors F V i are, in general, functions of the total energy s and of all the other possible kinematic invariants of the process. F V V and F V A are the usual vector and axial–vector form factors. Being related to D = 4 operators they are the only terms that can appear, at tree–level, in the Lagrangian of a renormalizable theory. F V S (s) and F V P (s) are the so–called scalar and pseudo–scalar form factors. They are usually negligible. Finally F V M(s) and F V E (s) are known as magnetic and electric form factors. The Anomalous Magnetic Dipole Moment (AMDM) and the Electric Dipole Moment (EDM), associated to a neutral vector boson V , are defined as: aVf = 2mf e F V M(s = M 2 V ) and d V f = −F V E (s = M 2 V ). (2) Here e is the electron charge, mf and MV are the fermion and boson masses. If V = γ (Mγ = 0) Eq. 2 reproduces the usual definitions of the photon AMDM and EDM. For V = Z Eq. 2 defines the Anomalous Weak Magnetic Dipole Moment (AWMDM = af ) and the Weak Electric Dipole Moment (WEDM = df ). Although the formulation could be completely general in the following we concentrate our analysis on the Weak Dipole Form Factors (WDFFs). One–loop generic expressions of the Weak Dipole Form Factors All the possible one–loop contributions to the af (s) and d Z f (s) form factors can be classified in terms of the six classes of triangle diagrams depicted in Fig. (1). The vertices are labelled by generic couplings, according to the following interaction Lagrangian, for vector bosons V (k) μ = Aμ, Zμ, Wμ, W † μ, fermions Ψk and scalar bosons Φk: L = ieJ(W † μνW Z −W W † μZν + Z W † μWν) + eV (k) μ Ψ̄jγ (V (k) jl − A (k) jl γ5)Ψl + { eΨ̄f (Sjk − Pjkγ5)ΨkΦj + eKjkZ V (k) μ Φj + h.c. } + ieGjkZ Φ†j ↔ ∂μ Φk. (3) Every class of diagrams is calculated analytically and expressed in terms of the couplings introduced in (3) and the one–loop three–point integrals. The computation is done in the ’t 1 Class I: k j l f f Class II: k j l f f Class III: k j l f f Class IV: k j l f f Class V: k j l f f Class VI: j k l f f Figure 1: One-loop topologies for a general V ff effective vertex. Hooft-Feynman gauge. Similar expressions are also derived in Ref.[1,2,3]. – [Class I]: vector boson exchange contribution. af 2mf (I) = α 4π {