We consider the effective potential $V$ in the standard model with a single Higgs doublet in the limit that the only mass scale $\mu$ 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 $SU(3)$ gauge coupling $z$, and the $SU(2)\times U(1)$ 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 $N^{2}LL\dots N^{4}LL$ 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 Coleman-Weinberg scheme. The Higgs mass can then be determined from the effective potential: the $LL$ result is $m_H=219\mathrm{GeV}/c^2$ and decreases to $m_H=188\mathrm{GeV}/c^2$ at $N^{2}LL$ order and $m_H=163\mathrm{GeV}/c^2$ 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 $163\mathrm{GeV}/c^2$ is argued to be an upper bound on $m_H$.

• Received 19 December 2010

DOI:https://doi.org/10.1103/PhysRevD.83.105009