Higgs couplings and naturalness

Many extensions of the standard model postulate the existence of new weakly coupled particles, the top partners, at or below the TeV scale. The role of the top partners is to cancel the quadratic divergence in the Higgs mass parameter due to top loops. We point out the generic correlation between naturalness (the degree of fine-tuning required to obtain the observed electroweak scale), and the size of top partner loop contributions to Higgs couplings to photons and gluons. If the fine-tuning is required to be at or below a certain level, a model-independent lower bound on the deviations of these Higgs couplings from the standard model can be placed (assuming no cancellations between contributions from various sources). Conversely, if a precise measurement of the Higgs couplings shows no deviation from the standard model, a certain amount of fine-tuning would be required. We quantify this connection, and argue that a measurement of the Higgs couplings at the per cent level would provide a serious and robust test of naturalness.

Figure 1

Fractional deviation of the Higgs coupling to gluons (left panel) and photons (right panel) from the SM value, as a function of the top partner mass. Top row: Spin-0 top partner. Bottom row: Spin-$1/2$ top partner. Regions currently allowed by the LHC and Tevatron data are shown in dark-gray/green (68% c.l.) and light-gray/yellow (95% c.l.).

Figure 2

Regions allowed by the LHC and Tevatron measurements of the Higgs rates in the $R_{\gamma }-R_g$ plane, at the 68% c.l. (dark-gray/green) and 95% c.l. (light-gray/yellow). The spin-0 top partner model predicts deviations along the solid/blue line, while the spin-$1/2$ top partner induces deviations along the dashed/red line. The points on both lines correspond to the partner masses of 350, 500, 650, and 800 GeV. For comparison, projected constraints from the LHC-14 [1] are shown by dashed/purple ellipses.

Figure 3

Regions allowed by the LHC and Tevatron data in the $\mathrm{\Delta }-\log \mathrm{\Lambda }$ plane, at the 68% c.l. (dark-gray/green) and 95% c.l. (light-gray/yellow). Here, $\mathrm{\Lambda }$ is the scale (in GeV) where the logarithmic divergence in the Higgs mass renormalization is cut off. Left panel: Spin-0 top partner. Right panel: Spin-$1/2$ top partner.

Figure 4

Fine-tuning as a function of the fractional deviation of the Higgs coupling to gluons (left panel) and photons (right panel) from the SM value, and the energy scale $\mathrm{\Lambda }$ (in GeV) where the logarithmic divergence in the Higgs mass renormalization is cut off. Top row: Spin-0 top partner. Bottom row: Spin-$1/2$ top partner. Regions currently allowed by the LHC and Tevatron data are shown in dark-gray/green (68% c.l.) and light-gray/yellow (95% c.l.).

Figure 5

Contours of fine tuning (solid/blue) and $R_g$ (dashed/red) for fixed values of $\theta$, with $\mathrm{\Lambda }=20\mathrm{TeV}$. The shaded regions correspond to points where $|R_g-1|<0.01$ but for which the amount of fine tuning is less than what is predicted for a one scalar partner model with $R_g=0.01$. The cross-hatched regions corresponds to points where $|c_i{v}^2/m_{0,i}^2|>1$. The top partner mass $m_1$ is in units of GeV.

Figure 6

Fine tuning (thick lines) as a function of $\theta$ for fixed values of $\mu$ and $R_g$, with $\mathrm{\Lambda }=20\mathrm{TeV}$. The regions shaded in black indicate values of $\theta$ where $|c_i{v}^2/m_{0,i}^2|>1$; the regions shaded in red/dark-gray are unphysical due to $m_1^2<0$. Light-gray/green regions indicate values of $\theta$ for which the fine-tuning is less than what is predicted for a one scalar partner model with $R_g=0.01$.