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We introduce mu-scalar curvature for a K"ahler metric with a moment map muand start up a study on constant mu-scalar curvature K"ahler metric as ageneralization of both cscK metric and K"ahler-Ricci soliton and as acontinuity path to extremal metric. We study some fundamental constraints tothe existence of constant mu-scalar curvature K"ahler metric by investigating avolume functional as a generalization of Tian-Zhu's work, which is closelyrelated to Perelman's W-functional. A new K-energy is studied as an approach tothe uniqueness problem of constant mu-scalar curvature and as a prelude to newK-stability concept.

Constant $\mu$-scalar curvature K\"ahler metric -- formulation and foundational results