Tests of the triple Higgs boson form factor in ${\mu }^{-}{\mu }^{+}\to HH$

We study the sensitivity of the process ${\mu }^{-}{\mu }^{+}\to HH$ to the $q^2$ dependence of the $HHH$ form factor, which can reflect the Higgs boson structure, especially in the case of compositeness. We compute the Born and one-loop SM contribution to this process. We then show how the ${\mu }^{-}{\mu }^{+}\to HH$ polarized and unpolarized cross sections are modified by the presence of various types of anomalous contributions to the $HHH$ form factor, in particular Higgs constituents in the case of compositeness.

Figure 1

The complete SM one-loop HC amplitudes (upper panels) and the corresponding HV amplitudes (lower panels). Left panels present the energy dependencies at $\theta =60°$, while right panels present the angular dependencies at $\sqrt{s}=0.5\mathrm{TeV}$.

Figure 2

SM one-loop differential cross sections as defined in Eqs. (13), (14). Left panels present the energy dependencies at $\theta =60°$, while right panels present the angular ones at $\sqrt{s}=0.5\mathrm{TeV}$.

Figure 3

The $s$ dependence of SM form factor $T_{HHH}^{\mathrm{SM}}$ defined in Eq. (7), together with the new physics contributions to it from $A(s{)}_{FFF}$ of Eq. (21) (upper panels), $A(s{)}_{XXX}$ of Eq. (20) (middle panels), and $A(s{)}_{\text{Res}}$ of Eq. (22) (lower panels). Left and right panels present real and imaginary parts, respectively. T refers to the SM contribution, and A refers to the new physics contributions, with the following parameters $m_X=0.5\mathrm{TeV}$ and $g_{HXX}=-10\mathrm{TeV}$, $m_F=0.5\mathrm{TeV}$ and $g_{HFF}=-4$, $M_R=1\mathrm{TeV}$, ${\mathrm{\Gamma }}_R=0.3\mathrm{TeV}$, and $g_{HR}{g}_{RHH}=0.5\mathrm{TeV}$. The SRS predictions given in Eqs. (a1) and (a2) are denoted T(sim).

Figure 4

The new physics contributions to the HV amplitudes induced by the $HHH$ form factors of Fig. 3. The HC amplitudes and the imaginary parts in the “new fermion” and “new scalar” HV amplitudes are vanishing, and they are not shown. The left panel presents the energy dependencies, while the right panel presents the angular ones, as in Fig. 1.

Figure 5

Dependencies of $[d\sigma (\mathrm{SM}+\mathrm{new})/d\cos \vartheta -d\sigma (\mathrm{SM})/d\cos \vartheta ]/[d\sigma (\mathrm{SM})/d\cos \vartheta ]$, on the “new fermion” (upper), “new scalar” (middle), and the “new resonance” (lower panel) contributions for the total HV case in Eq. (13) and for the unpolarized case.