Weak scale as a trigger

Does the value of the Higgs mass parameter affect the expectation value of local operators in the Standard Model? For essentially all local operators the answer to this question is “no”, and this is one of the avatars of the hierarchy problem: Nothing is “triggered” when the Higgs mass parameter crosses zero. In this article, we explore settings in which Higgs mass parameters can act as a “trigger” for some local operators ${\mathcal{O}}_T$. In the Standard Model, this happens for ${\mathcal{O}}_T=\mathrm{Tr}(G\tilde{G})$. We also introduce a “type-0” two Higgs doublet model, with a $Z_4$ symmetry, for which ${\mathcal{O}}_T=H_1{H}_2$ is triggered by the Higgs masses, demanding the existence of new Higgs states necessarily comparable to or lighter than the weak scale, with no wiggle room to decouple them whatsoever. Surprisingly, this model is not yet entirely excluded by collider searches, and will be incisively probed by the high-luminosity run of the LHC, as well as future Higgs factories. We also discuss a possibility for using this trigger to explain the origin of the weak scale, invoking a landscape of extremely light, weakly interacting scalars ${\varphi }_i$, with a coupling to ${\mathcal{O}}_T$ needed to make it possible to find vacua with small enough cosmological constant. The weak scale trigger links the tuning of the Higgs mass to that of the cosmological constant, while coherent oscillations of the ${\varphi }_i$ can constitute dark matter.

Figure 1

Experimental constraints on the type-0 2HDM in the $m_{H^{\pm }}-m_H$ plane (masses of the charged and new $CP$-even Higgses). The three panels correspond to three different choices for the vev of the new Higgs doublet $v_1=⟨H_1^0⟩=(0.2,0.3,0.5)v$. The mass of the new $CP$-Odd Higgs $m_A$ is fixed to 160 GeV. In all three cases we allow a 10% tuning of quartics, i.e., we take ${\lambda }_3+{\lambda }_4+{\lambda }_5=0.1|{\lambda }_4+{\lambda }_5|=0.2(m_{H^{\pm }}/v{)}^2$. In red we show the bound from $H^{\pm }$ pair production at LEP [1, 2] and in yellow from $HZ$ associated production and decays to fermions [3]. In light blue we display the bound from ElectroWeak Precision Tests [4] on the $S$, $T$, and $U$ oblique parameters [5, 6]. In light green we show bounds from searches for $B\to X_s\gamma$ [7, 8]. Indirect constraints from Higgs coupling measurements set an upper bound on $\mathrm{BR}(h\to HH)$ (in blue). The impact of LHC searches for $t\to H^{+}b$ is shown in pink [9, 10, 11, 12]. Theoretical constraints (in gray) from low energy Landau poles and the SM Higgs mass are summarized at the beginning of Sec. 3b. At high masses there is no solution for the quartics which gives $m_h=125\mathrm{GeV}$.

Figure 2

Almost all gauge invariant local operators $\mathcal{O}$ in the SM have UV sensitive expectation values that are thus independent of $m_h^2$. We can probe their vevs by adding to the Lagrangian the parametrically weak interaction $\varphi \mathcal{O}$ and look at the effective action for $\varphi$. If we add, for example, $\varphi |h{|}^2$ or $\varphi qh{u}^c$ we can always close the loops in this figure and obtain a tadpole for $\varphi$ proportional to the cutoff. This happens because there are no global symmetries carried by $\mathcal{O}$, that are not already broken by the presence of $\mathcal{O}$ in the SM effective Lagrangian.

Figure 3

In a 2HDM with a $Z_4$ symmetry [Eqs. (3) and (5)] the $H_1{H}_2$ vev is a UV-insensitive, calculable function of the two Higgs masses. This can be seen by adding to the Lagrangian the parametrically weak interaction $\varphi H_1{H}_2$. We can only close the loop in this figure and generate a $\varphi$ tadpole independent of $⟨H_1{H}_2⟩$ with an insertion of $B\mu ,{\lambda }_6$ or ${\lambda }_7$ which break the $Z_4$ symmetry and are thus absent from our 2HDM potential in Eq. (5).

Figure 4

In the type-0 2HDM (Eq. (5), $⟨H_1{H}_2⟩$ is a UV-insensitive, calculable function of the masses of the two Higgses: $m_1^2$, $m_2^2$. In the left panel we show its classical value while in the right one we include quantum effects. $m_1^2$, $m_2^2$ in the figure are effective masses that include contributions from cross quartic couplings. ${\mathrm{\Lambda }}_{\mathrm{QCD}}$ is the QCD scale with all quark masses below ${\mathrm{\Lambda }}_{\mathrm{QCD}}$.

Figure 5

Experimental constraints on the $CP$-even Higgs $H$ for $m_{H^{\pm }}=m_A$ and different values of ${\lambda }_{345}$. From top to bottom we increase $m_{H^{\pm }}$. From left to right we move from 1% tuning (${\lambda }_{345}=0.01|{\lambda }_4+{\lambda }_5|$) to natural values of the quartics (${\lambda }_{345}=|{\lambda }_4+{\lambda }_5|$). In red we show the bound from $e^{+}{e}^{-}\to Z\to AH$ at LEP [3] and in yellow from $HZ$ associated production [3] followed by decays to fermions. In light blue we display the current sensitivity of $H\to \gamma \gamma$ at LEP and the LHC [20, 21, 22] and a projection for the HL-LHC obtained rescaling [22]. In light green we show bounds from searches for $B\to K^{(*)}H\to K^{(*)}\mu \mu$ at LHCb [23, 24]. Indirect constraints from Higgs coupling measurements (purple and blue) are discussed in Sec. 3b2. The pink shaded area shows the strongest bound point-by-point between searches for flavor changing processes, mainly $b\to s\gamma$ [7, 8], and LHC searches for $t\to H^{+}b$ [9, 10, 11, 12]. Theoretical constraints (in gray) from low energy Landau poles and the SM Higgs mass are summarized at the beginning of Sec. 3b.

Figure 6

The landscape contains a UV sector and an IR sector (in this figure). The high energy sector is generated by fields of mass close to the cutoff $m_{\mathrm{\Phi }}\sim M_{*}$ and does not have enough vacua to scan the CC down to ${\mathrm{\Lambda }}_{\mathrm{obs}}\simeq {\mathrm{meV}}^4$, but can scan the Higgs mass(es) $m_H^2$ down to the weak scale. The low energy sector is generated by fields of mass $m_{\varphi }\sim v^2/M_{*}$ and has a number of nondegenerate minima dependent on the Higgs vev. When $⟨h⟩\simeq v$ we can scan the CC down to its observed value.

Figure 7

Values of the Cosmological Constant in our two-sectors landscape. When $⟨{\mathcal{O}}_T⟩=0$ the UV landscape does not have enough minima to scan the CC from $M_{*}^4$ down to ${\mathrm{\Lambda }}_{\mathrm{obs}}\simeq {\mathrm{meV}}^4$. The minimal value of the CC in the landscape is $M_{*}^4/N_{\mathrm{UV}}\gg {\mathrm{meV}}^4$. When $⟨{\mathcal{O}}_T⟩\ne 0$ the degeneracy in the vacua of the low energy landscape in Fig. 6 is broken and if ${\mu }_S^{{\mathrm{\Delta }}_T}\lesssim ⟨{\mathcal{O}}_T⟩\lesssim {\mu }_B^{{\mathrm{\Delta }}_T}$ we can harness the full potential of its $2^{n_{\varphi }}$ vacua and scan the CC down to ${\mathrm{meV}}^4$. If $⟨{\mathcal{O}}_T⟩\gg {\mu }_B^{{\mathrm{\Delta }}_T}$ the low energy landscape loses all its minima but one and the minimal CC in the landscape is again $M_{*}^4/N_{\mathrm{UV}}\gg {\mathrm{meV}}^4$.

Figure 8

The smallest CC in the landscape as a function of ${\mu }^2\equiv ⟨H_1{H}_2⟩$. In the light blue area the CC is smaller than its observed value, while for ${\mu }^2>{\mu }_B^2$ or ${\mu }_S<{\mu }_S^2$ it is much larger, $M_{*}^4/N_{\mathrm{UV}}\gg {\mathrm{meV}}^4$.

Figure 9

Laboratory and astrophysical constraints on scalars coupled to the Higgs boson via the trilinear interaction $\kappa m_{\varphi }{\sum }_{i=1}^{n_{\varphi }}{\varphi }_i|H{|}^2/\sqrt{n_{\varphi }}$ [we neglect unimportant $\mathcal{O}(1)$ factors introduced by the mixing of the two Higgses]. The bounds include tests of the equivalence principle [142, 143, 144, 145], tests of the Newtonian and Casimir potentials (5th force) [146, 147, 148, 149, 150, 151, 152, 153, 154], stellar cooling [155], and black hole super-radiance [156, 157]. The pink solid line shows the target given by the scalars being dark matter. We shaded in gray the region where $\kappa >1$ (i.e., ${\mathrm{\Lambda }}_H\lesssim \mathrm{TeV}$).