## Abstract

The membership of a list of `fundamental' constants necessarily depends on who is compiling the list. A hydrodynamicist might reasonably include the density and viscosity of water, while an atomic physicist would doubtless include the proton mass and electronic charge. This talk deals with a different sort of list: a list of the constants that appear in the laws of nature at the deepest level that we yet understand, constants whose value we cannot calculate with precision in terms of more fundamental constants, not just because the calculation is too complicated (as for the viscosity of water or the mass of the proton) but because we do not know of anything more fundamental. The membership of such a list of fundamental constants thus reflects our present understanding of fundamental physics. Also, each constant on the list is a challenge for future work, to try to explain its value. The parameters that appear at the most fundamental level in our present theories of elementary particles are (1) the electroweak and strong gauge couplings g$_{1}$, g$_{2}$, g$_{3}$, and (2) the masses and self couplings of the `Higgs' scalars, and (3) the coupling constants for the interaction of the scalars to quarks and leptons. The gauge couplings themselves determine the observables e$^{2}$ = g$_{1}^{2}$g$_{2}^{2}$/(g$_{1}$+g$_{2}^{2}$) and sin$^{2}\theta $ = g$_{2}^{2}$/(g$_{1}^{2}$+g$_{2}^{2}$), which are experimentally known to be 4$\pi $/137 and 0.22. The scalar masses and self-couplings determine the scalar vacuum expectation values, and hence the Fermi coupling G$_{\text{F}}$ = $\langle \phi ^{0}\rangle ^{-2}$/$\surd $2, which is experimentally known to be (293 GeV)$^{-2}$. Combined with the gauge couplings, these vacuum expectation values also determine the W and Z masses, with predicted values of about 83 and 93 GeV, which are now happily in agreement with experiment. Finally, the scalar vacuum expectation values and scalar-fermion couplings determine the quark and lepton masses, which experimentally range over more than three orders of magnitude. From these experimental masses we deduce that the scalar-fermion coupling constants are much smaller than the gauge couplings and very different from each other, but we have no clear idea of why this should be so. It is somewhat misleading to list the gauge couplings as fundamental parameters. We know in the case of quantum chromodynamics that the QCD coupling is not a constant; rather g$_{3}^{2}$ varies with the energy E as 24$\pi ^{2}$/25 ln (E/$\Lambda _{3}$), (for E below the bottom mass). The constant $\Lambda _{3}$ determines the general scale of strong interaction physics, including the current-algebra constant F$_{\pi}$ and the `constituent quark' mass m$_{\text{q}}$. Experimentally $\Lambda _{3}\sim $ 150 MeV, F$_{\pi}\sim $ 190 MeV, and m$_{\text{q}}\approx {\textstyle\frac{1}{3}}$m$_{\text{N}}$ = 310 MeV. We do not yet know how to calculate F$_{\pi}$ and m$_{\text{q}}$ from $\Lambda _{3}$. The only place the Planck scale of 1.7 $\times $ 10$^{18}$ GeV, and this will involve new dimensionless ratios of coupling parameters. I think it is likely that all these apparently fundamental constants will ultimately be determined by a condition of consistency of quantum mechanics with relativity, a condition that requires that all couplings have values that place the theory on a trajectory that is attracted to an ultraviolet fixed point.