We consider the effective potential V in the standard model with a single Higgs doublet in the limit that the only mass scale present is radiatively generated. Using a technique that has been shown to determine V completely in terms of the renormalization group (RG) functions when using the Coleman-Weinberg renormalization scheme, we first sum leading-log (LL) contributions to V using the one loop RG functions, associated with five couplings (the top quark Yukawa coupling x, the quartic coupling of the Higgs field y, the SUð3Þ gauge coupling z, and the SUð2Þ Â Uð1Þ couplings r and s). We then employ the two loop RG functions with the three couplings x, y, z to sum the next-to-leading-log (NLL) contributions to V and then the three to five loop RG functions with one coupling y to sum all the N 2 LL . . . N 4 LL contributions to V. In order to compute these sums, it is necessary to convert those RG functions that have been originally computed explicitly in the minimal subtraction scheme to their form in the ColemanWeinberg scheme. The Higgs mass can then be determined from the effective potential: the LL result is m H ¼ 219 GeV=c 2 and decreases to m H ¼ 188 GeV=c 2 at N 2 LL order and m H ¼ 163 GeV=c 2 at N 4 LL order. No reasonable estimate of m H can be made at orders V NLL or V N 3 LL since the method employed gives either negative or imaginary values for the quartic scalar coupling. The fact that we get reasonable values for m H from the LL, N 2 LL, and N 4 LL approximations is taken to be an indication that this mechanism for spontaneous symmetry breaking is in fact viable, though one in which there is slow convergence towards the actual value of m H . The mass 163 GeV=c 2 is argued to be an upper bound on m H .
The perturbative effective potential V in the massless 4 model with a global ON symmetry is uniquely determined to all orders by the renormalization group functions alone when the ColemanWeinberg renormalization condition d 4 V d 4 j is used, where represents the renormalization scale. Systematic methods are developed to express the n-loop effective potential in the Coleman-Weinberg scheme in terms of the known n-loop minimal-subtraction (MS) renormalization group functions. Moreover, it also proves possible to sum the leading-and subsequent-to-leading-logarithm contributions to V. An essential element of this analysis is a conversion of the renormalization group functions in the Coleman-Weinberg scheme to the renormalization group functions in the MS scheme. As an example, the explicit five-loop effective potential is obtained from the known five-loop MS renormalization group functions and we explicitly sum the leading-logarithm, next-to-leading-logarithm, and further subleadinglogarithm contributions to V. Extensions of these results to massless scalar QED are also presented. Because massless scalar QED has two couplings, conversion of the renormalization group functions from the MS scheme to the Coleman-Weinberg scheme requires the use of multiscale renormalization group methods.
When one uses the Coleman-Weinberg renormalization condition, the effective potential V in the massless φ 4 4 theory with O(N) symmetry is completely determined by the renormalization group functions. It has been shown how the (p + 1) order renormalization group function determine the sum of all the N p LL order contribution to V to all orders in the loop expansion. We discuss here how, in addition to fixing the N p LL contribution to V , the (p + 1) order renormalization group functions also can be used to determine portions of the N p+n LL contributions to V . When these contributions are summed to all orders, the singularity structure of V is altered. An alternate rearrangement of the contributions to V in powers of ln φ, when the extremum condition V ′ (φ = v) = 0 is combined with the renormalization group equation, show that either v = 0 or V is independent of φ. This conclusion is supported by showing the LL, · · · , N 4 LL contributions to V become progressively less dependent on φ.
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