Production of $HHH$ and $HHV(V=\gamma ,Z)$ at the hadron colliders

We consider the production of two Higgs bosons in association with a gauge boson or another Higgs boson at the hadron colliders. We compute the cross sections and distributions for the processes $pp\to HHH$ and $HHZ$ within the standard model. In particular, we compute the gluon-gluon fusion one-loop contributions mediated via heavy quarks in the loop. It is the leading order contribution to $pp\to HHH$ process. To the process $pp\to HHZ$, it is next-to-next-to-leading-order (NNLO) contribution in QCD coupling. We also compare this contribution to the next-to-leading-order (NLO) QCD contribution to this process. The NNLO contribution can be similar to NLO contribution at the Large Hadron Collider (LHC), and significantly more at higher center-of-mass energy machines. We also study new physics effects in these processes by considering $ttH$, $HHH$, $HHHH$, $HZZ$, and $HHZZ$ interactions as anomalous. The anomalous couplings can enhance the cross sections significantly. The $gg\to HHH$ process is specially sensitive to anomalous trilinear Higgs boson self-coupling. For the $gg\to HHZ$ process, there is some modest dependence on anomalous $HZZ$ couplings.

Figure 1

Different classes of diagrams contributing to $gg\to HHH$ process.

Figure 2

Different classes of diagrams contributing to $gg\to HHZ$ process.

Figure 3

SM contribution of pentagon (blue), box(green), and triangle (violet) diagrams to leading $p_T(H)$ distribution in $gg\to HHH$ at 13 TeV (left) and 100 TeV (right).

Figure 4

Kinematic distributions for $gg\to HHH$ in the SM at 13 TeV. These plots are obtained after $p_T$ ordering the Higgs bosons. $H_1$, $H_2$, and $H_3$ refer to the hardest, second hardest, and third hardest Higgs bosons in $p_T$ respectively.

Figure 5

$\frac{{\sigma }_{\mathrm{BSM}}}{{\sigma }_{\mathrm{SM}}}$ as function of various Higgs anomalous couplings affecting $gg\to HHH$ at 13 TeV.

Figure 6

Normalized leading $p_T(H)$ distribution in $gg\to HHH$ at 13 TeV for some benchmark values of anomalous trilinear Higgs self-coupling. In the lower panels $R$ is defined as the ratio of the distributions ($d\sigma /d{p}_T$) in BSM and in SM.

Figure 7

SM contribution of pentagon (blue), box (green) and triangle (violet) diagrams to leading $p_T(H)$ distribution in $gg\to HHZ$ at 13 TeV (left) and 100 TeV (right).

Figure 8

Kinematic distributions for $gg\to HHZ$ in the SM at 13 TeV. These plots are obtained after $p_T$ ordering the Higgs bosons. $H_1$ and $H_2$ refer to the hardest and second hardest Higgs bosons in $p_T$ respectively.

Figure 9

A comparison of normalized distributions for $p_T(H_1)$ and $p_T(Z)$ due to $gg\to HHZ$ and $qq\to HHZ$ in the SM at 13 TeV.

Figure 10

Combined $gg\to HHZ(\mathrm{LO})+qq\to HHZ(\mathrm{NLO})$ contribution to $p_T(H_1)$ and $p_T(Z)$ distributions in the SM at 13 TeV and 100 TeV. Lower panels show the ratio of $QQ(\mathrm{NLO})+GG(\mathrm{LO})$ and $QQ(\mathrm{NLO})$ for each of these distributions. The dashed straight line in the lower panel of each plot refers to the same quantity at inclusive or total cross section level (see $R_2$ in Table 3).

Figure 11

$\frac{{\sigma }_{\mathrm{BSM}}}{{\sigma }_{\mathrm{SM}}}$ as a function of anomalous couplings of the Higgs boson in $gg\to HHZ$ at 13 TeV.

Figure 12

Normalized leading $p_T(H)$ distribution in $gg\to HHZ$ at 13 TeV for some benchmark values of anomalous top Yukawa couplings.

Figure 13

Normalized leading $p_T(H)$ distribution in $gg\to HHZ$ at 13 TeV for some benchmark values of $HHH$ anomalous couplings.

Figure 14

Normalized leading $p_T(H)$ distribution in $gg\to HHZ$ at 13 TeV for some benchmark values of $HZZ$ and $HHZZ$ anomalous couplings.