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loop graph shown in Fig. 9.3.The scalar and fermion lines in the loopare messenger fields.Recall that the interaction vertices in Fig. 9.3are of gauge coupling strength even though they do not involve gaugebosons; compare Fig. 6.3g.In this way, gauge-mediation provides thatq,qmessenger loops give masses to the gluino and the bino, and,messenger loops give masses to the wino and bino fields. By computingthe 1-loop diagrams one finds that the resulting MSSM gaugino massesare given byMa=αa4πΛ,(a= 1,2,3),(9.40)(in the normalization discussed in section 8.4) where we have introduceda mass parameterΛ≡FS/ S.(9.41)
2229 Origins of supersymmetry breakingFig. 9.4.Contributions to MSSM scalar squared masses in gauge-mediatedsupersymmetry breaking models arise in leading order from these two-loopFeynman graphs.(Note that ifFSwere 0,then Λ=0 and the messenger scalarswould be degenerate with their fermionic superpartners and there wouldbe no contribution to the MSSM gaugino masses.)In contrast, thecorresponding MSSM gauge bosons cannot get a corresponding mass shift,since they are protected by gauge invariance. So supersymmetry breakinghas been successfully communicated to the MSSM (“visible sector”). Toa good approximation, eq. (9.40) holds for the running gaugino masses atan RG scaleQ0corresponding to the average characteristic mass of theheavy messenger particles, roughly of orderMmess∼yiS. The runningmass parameters can then be RG-evolved down to the electroweak scaleto predict the physical masses to be measured by future experiments.The scalars of the MSSM do not get any radiative corrections to theirmasses at one-loop order.The leading contribution to their massescomes from the two-loop graphs shown in Fig. 9.4, with the messengerfermions (heavy solid lines) and messenger scalars (heavy dashed lines)and ordinary gauge bosons and gauginos running around the loops. Bycomputing these graphs, one finds that each MSSM scalarφgets a (mass)2given by:m2φ= 2Λ2α34π2Cφ3+α24π2Cφ2+α14π2Cφ1.(9.42)HereCφaare the quadratic Casimir group theory invariants for the scalarφfor each gauge group. They are defined byCφaδji= (TaTa)jiwhere theTaare the group generators which act on the scalarφ. Explicitly, they
9.4 Gauge-mediated supersymmetry breaking models223are:Cφ3=⎧⎨⎩4/3 forφ=Qi,ui,di;0forφ=Li,ei, Hu, Hd(9.43)Cφ2=⎧⎨⎩3/4 forφ=Qi, Li, Hu, Hd;0forφ=ui,di,ei(9.44)Cφ1= 3Y2φ/5for eachφwith weak hyperchargeYφ.(9.45)The squared masses in eq. (9.42) are positive (fortunately!).The termsau,ad,aearise first at two-loop order, and are suppressedby an extra factor ofαa/(4π) compared to the gaugino masses. So, to avery good approximation one has, at the messenger scale,au=ad=ae= 0,(9.46)a significantly stronger condition than eq. (8.16). Again, eqs. (9.42) and(9.46) should be applied at an RG scale equal to the average mass ofthe messenger fields running in the loops.However, after evolving the