Abstract
In grand unified theories higher order contributions to the effective potential contain terms of the type (λ log υ 1 2 υ 2 2 ) n where λ is some Higgs coupling and υ 1, υ 2 are the vacuum expectation values of certain scalar fields whose ratio we require phenomenologically to be extremely large ( υ 1 2 υ 2 2 ≈ 10 24 or so). We study these terms in a toy model and find taht they can be absorbed into suitably defined coupling constant whose evolution equations (in this leading logarithm approximation) seem to indicate that no natural large hierachy is possible. For consistency of the calculation, however, certain “non-leading” terms must also be included and then it is found that the evolution equations have to be reinterpreted, suggesting a very natural explanation of the large ratio of vacuum expectation values.
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