Higgs boson mass in the standard model at two-loop order and beyond

We calculate the mass of the Higgs boson in the standard model in terms of the underlying Lagrangian parameters at complete 2-loop order with leading 3-loop corrections. A computer program implementing the results is provided. The program also computes and minimizes the standard model effective potential in Landau gauge at 2-loop order with leading 3-loop corrections.

Figure 1

Topologies for the one- and two-loop vacuum and self-energy scalar basis integrals used in this paper and defined in Refs. [38, 39]. The dot in the $T$ topology stands for a derivative with respect to the squared mass $x$.

Figure 2

The standard model Higgs VEV, $v(Q)$, obtained from minimization of the effective potential is shown in the left panel as a function of the renormalization scale $Q$. The dashed line shows the results for the 2-loop minimization condition of Eqs. (4.18)–(4.20) of Ref. [22], while the solid line is the 2-loop plus leading 3-loop result obtained by including also Eq. (4.21) of the same reference. The input parameters $m^2,\lambda ,y_t,g_3,g,g^{\prime }$ are obtained at the scale $Q$ by 3-loop renormalization group running starting from Eqs. (4.1)–(4.6). The right panel shows the ratio of $v(Q)$ to the value $v_{\text{run}}(Q)$ obtained from directly running it using its renormalization group equation and input value Eq. (4.7).

Figure 3

The standard model Lagrangian Higgs squared mass parameter, obtained from minimization of the effective potential is shown in the left panel as $\sqrt{-m^2}$ as a function of the renormalization scale $Q$. The dashed line shows the results for the 2-loop minimization condition of Eqs. (4.18)–(4.20) of Ref. [22], while the solid line is the 2-loop plus leading 3-loop result obtained by including also Eq. (4.21) of the same reference. The input parameters $v,\lambda ,y_t,g_3,g,g^{\prime }$ are obtained at the scale $Q$ by 3-loop renormalization group running starting from Eqs. (4.2)–(4.7). The right panel shows the ratio of $m^2(Q)$ to the value $m_{\text{run}}^2(Q)$ obtained from directly running it using its renormalization group equation and input value Eq. (4.1).

Figure 4

The calculated Higgs pole mass $M_h$ as a function of the renormalization scale $Q$, in various approximations. The input data at $Q$ are obtained from 3-loop renormalization group running of $\lambda ,y_t,g,g^{\prime },g_3,v$ starting from Eqs. (4.2)–(4.7). The dotted (green) line is the tree-level approximation $\sqrt{2\lambda }v$. The short-dashed (orange) line is the 1-loop approximation obtained from Eqs. (2.43) and (2.46). The long-dashed (red) line is the 1-loop approximation with the 2-loop QCD corrections from Eq. (2.47). The lower solid (blue) line is the 2-loop $M_h$ as calculated from Eqs. (2.43) and (2.46)–(2.48), while the upper solid (black) line also includes the leading 3-loop corrections of Eqs. (3.2)–(3.4).

Figure 5

A close-up of the dependence of the calculated $M_h$ on $Q$, as in Fig. 4. The lower (blue) line is the full 2-loop $M_h$ as calculated from Eqs. (2.43) and (2.46)–(2.48). The upper (magenta) line is the full 2-loop plus the 3-loop QCD contribution of Eqs. (3.2)–(3.3), not shown in Fig. 4. The middle (black) line is the full 2-loop plus the 3-loop corrections of Eqs. (3.2)–(3.4), with the left dot marking the case $s=h$ and the right dot marking the case $\overline{\ln }(t)=0$.

Figure 6

The Higgs self-coupling parameter, ${\lambda }_{M_h}(Q)$ as calculated from a fixed pole mass $M_h=125.818\mathrm{GeV}$ using Eqs. (2.43) and (2.46)–(2.48) and Eqs. (3.2)–(3.4), with $y_t,g,g^{\prime },g_3,v$ obtained at the scale $Q$ by 3-loop renormalization group running starting from Eqs. (4.3)–(4.7). The right panel shows the ratio of ${\lambda }_{M_h}(Q)$ to the value ${\lambda }_{\text{run}}(Q)$ obtained from directly running it using its renormalization group equation and input value Eq. (4.2).