The Higgs Boson Mass from Three-loop Effective Potential of Massless Standard Model

The effective potential of massless standard model (SM) is calculated up to three-loop order. The stability of the effective potential and the Higgs boson mass are investigated up to three-loop order. We found that, Higgs boson mass $m_{H}$ of one-loop order is large. The two-loop and three-loop results are not appreciably different from each other. The two-loop and three-loop radiative corrections have led to an improvement of Higgs boson mass and the value of the scalar coupling. For the value $m_{t}=170$ GeV at the energy scale $\mu\approx 5.68\times10^2$GeV, we get $m_{H2-loop}\approx 125.4$ GeV. At the energy scale $\mu\geq28\times10^2$, the scalar coupling $\lambda$ at two-loop becomes negative and leads to metastable vacuum while the three-loop level is stable even at high $\mu$ $\approx 10^{19}$ GeV. For the larger $\mu$-range $(3 \times 10^{3} \text{GeV}\leq \mu \leq 20 \times 10^{3} \text{GeV})$ spontaneous symmetry breaking for one-loop and three-loop occurs at approximately the same scalar coupling values. We get $m_{H1-loop}\approx 126.14$ GeV at $\mu \approx 6.5 \times 10^{3}$ GeV while for three-loop $m_{H3-loop}\approx 126$ GeV at $\mu \approx 20 \times 10^{3}$ GeV.