Weight for random quark masses

In theories in which the parameters of the low energy theory are not unique, perhaps having different values in different domains of the universe as is possible in some inflationary models, the fermion masses would be distributed with respect to some weight. In such a situation the specifics of the fermion masses do not have a unique explanation, yet the weight provides the visible remnant of the structure of the underlying theory. This paper introduces this concept of a weight for the distribution of masses and provides a quantitative estimate of it from the observed quarks and leptons. The weight favors light quark masses and appears roughly scale invariant ( ρ ∼ 1/m). Some relevant issues, such as the running of the weight with scale and the possible effects of anthropic constraints, are also discussed. 1 Basic ideas Many of the parameters of the Standard Model, such as the quark masses and the weak mixing angles, appear without any obvious pattern. Perhaps the explanation for the specific values of these parameters is hidden in the physics at a deeper level. The goal of much of the work in particle physics has been to find the underlying theory, the golden Lagrangian, which by its structure explains the parameters of the Standard Model. If this is successful, it will indeed be satisfying. However, another possibility also exists that these parameters are in some sense random. This could potentially occur in various ways. For example, in some theories of inflation, different regions of the universe involve different parameters and perhaps even different low energy theories[1]. The dynamical fields which determine the properties of the low energy theory become fixed at different values in each domain, and subsequent inflation ensures that we live within only one of these domains. It is also conceivable in theories such as superstrings with different classically equivalent vacua, that different vacuum states could be selected in different regions of the universe. The moduli fields, whose vacuum expectation values determine the mass parameters of the low energy theory, are not fixed upon compactification[2]. Perhaps these fields are sampled in a random fashion rather than being determined uniquely by some mechanism. However, the general idea can at this time be considered distinct from the particular underlying theory. If some quasi-random mechanism is in fact at work, we should not expect to find a unique explanation for the values of the parameters in the Standard Model. Given our present incomplete knowledge, it is not any more scientific to assume that there is a unique explanation for a single set of parameters which holds throughout the universe than it is to consider the possibility that the parameters may vary in different domains. Indeed, the history of science has taught us that we do not occupy a privileged position. In this spirit, it is fair to explore the possibility that our domain of the Universe and our particular parameters are not unique. The quark masses do not appear strictly random either, as there are more light quarks than very massive ones. Indeed, we would not necessarily expect the masses to be equally distributed in such a multiple domain theory. The structure and dynamics of the fields which determine the low energy