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I believe that mass is property of every particle,as well as spin etc.Now I'm interested in SUSY particles in cMSSM model.Can it be,that mass of a SUSY particle (at one point in five parameter space) depend on process it is participating in,or parents particles? I doubt it.

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As long as one picks a fixed definition of the mass - there are various different "types of masses" like the running mass at some scale, and so on - then it is unique for a particular particle. There's lot of freedom in SUSY and other BSM models and if masses of everything were also process-dependent, it would be fuzzy, indeed. –  LuboŇ° Motl Dec 16 '12 at 9:09

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