  The quite low Higgs particle mass makes it natural for the next high energy
facility to be a circular e+e- Higgs factory and, after that, a next-generation
p,p collider in the same tunnel. Surveying the luminosity-limiting phenomena of
synchrotron radiation power loss, beam-beam interaction limitations, and
beamstrahlung, scaling laws are established that fix all parameters of the
Higgs factory, as functions of assumed radius $r$, and RF power $P$. at least
to a first approximation. Historically the accelerator formalisms of electron
and hadron rings have been distinguished largely by the importance of
synchrotron radiation for electrons, and its unimportance for protons. While
electron beams equilibrate within seconds, proton beam distributions have
survived largely intact for extended periods. For future hadron colliders, this
distinction will no longer be valid. This will have a large impact on the
design of the future FCC-pp proton collider whose parameters can be
extrapolated using formulas previously applicable only to electron rings. The
most important scaling law has the luminosity depending on $r$ and $P_{\rm rf}$
only as a function of their product $rP$. This relation simplifies choosing a
tunnel radius that is optimal for both Higgs factory and p,p collider while
(almost) minimizing the initial cost---by increasing $r$ and decreasing $P$
proportionally, the tunnel circumference can be increased (as required for p,p)
without significant increase in Higgs factory cost.

Accelerator Physics