Combined analysis of double Higgs production via gluon fusion at the HL-LHC in the effective field theory approach

We perform the combined analysis of the double Higgs production via gluon fusion in the $b\overline{b}\gamma \gamma$ and $b\overline{b}{\tau }^{+}{\tau }^{-}$ decay channels at the High-Luminosity LHC (HL-LHC). To validate our analysis, we reproduce the ATLAS result of the $b\overline{b}\gamma \gamma$ process including all contributions from fakes. For the $b\overline{b}{\tau }^{+}{\tau }^{-}$ decay channel, we perform the similar analysis to the CMS one. As an improvement, we also perform the multivariate analysis employing the boosted decision tree algorithm. Then, we derive 68% probability contours on anomalous Higgs couplings in the effective field theory (EFT) approach for various analyses. We find that the $b\overline{b}{\tau }^{+}{\tau }^{-}$ process outperforms the $b\overline{b}\gamma \gamma$ for the measurement of energy-growing operators, while adding the $b\overline{b}{\tau }^{+}{\tau }^{-}$ process is least beneficial for improving the precision of the Higgs self-coupling (mainly set by the $b\overline{b}\gamma \gamma$ process). We illustrate that the double Higgs production alone can be comparable to the single Higgs process in constraining the modification of the top Yukawa coupling in the positive direction. Focusing on the Higgs self-coupling as a special interest, we derive the precision as a function of various improvable parameters such as tag and mistag rates of tau leptons, heavy flavor jets, photon identification, and diphoton mass resolution to take into account future phenomenological studies. As an optimistic benchmark scenario, we illustrate that the 68% and 95% probability intervals of the Higgs self-coupling, ${\lambda }_3/{\lambda }_3^{\mathrm{SM}}$, at the HL-LHC can reach [0.2, 2.3] and $[-0.1,3.5]\cup [4.0,6.5]$, respectively, where the correlation among the EFT coefficients is taken into account.

Figure 1

Feyman diagrams contributing to double Higgs production via gluon fusion. The first three diagrams in the upper line are labeled as ${\mathcal{A}}_{\square }$, ${\mathcal{A}}_{△}$, and ${\mathcal{A}}_{△nl}$ and those in the bottom by ${\mathcal{A}}_3$ and ${\mathcal{A}}_4$ from the left in Sec. 2.

Figure 2

The number of signal ($N_S$), background ($N_B$), and significance after imposing some BDT cuts and the condition in Eq. (11). Dashed lines correspond to the result by the cut-and-count analysis. The thin vertical line denotes our BDT cut minimum.

Figure 3

The significance as a function of $m_{\tau }$ for the $b\overline{b}{\tau }^{+}{\tau }^{-}$ process against all the backgrounds (left) and $h\to {\tau }^{+}{\tau }^{-}$ against $Z\to {\tau }^{+}{\tau }^{-}$ (right). The significance was maximized for each $m_{\tau }$ value by adjusting a cut on the $m_{\tau \tau }^{\mathrm{reco}}$, while the other cuts remain the same.

Figure 4

The distributions of $m_{\tau \tau }^{\mathrm{reco}}$ for the signal and $Z({\tau }^{+}{\tau }^{-})+\text{jets}$ samples. The left and right figures are distributions for the fully hadronic and semileptonic channels.

Figure 5

Distributions of $M_{T2}$ for the fully hadronic and the semileptonic channels.

Figure 6

The significance as a function of $N_S$ (the number of signal events) for the fully hadronic $b\overline{b}{\tau }^{+}{\tau }^{-}$ mode (left) and the semileptonic $b\overline{b}{\tau }^{+}{\tau }^{-}$ mode (right).

Figure 7

Left: 68% probability contours of the likelihoods in ($c_{2t}$, $c_3$) plane using the cut-based analysis: the inclusive analysis of the double Higgs production in the $b\overline{b}\gamma \gamma$ (dashed red), the exclusive analysis of the $b\overline{b}\gamma \gamma$ (solid red), inclusive analysis of the $b\overline{b}{\tau }^{+}{\tau }^{-}$ (solid blue), and combined analysis (solid black). The combined analysis combines the exclusive analysis of the $b\overline{b}\gamma \gamma$ channel and the inclusive one of the $b\overline{b}{\tau }^{+}{\tau }^{-}$ channel. Right: similarly, 68% probability contours of the likelihood in ($c_{2t}$, $c_{2g}$) plane with the same color/line codes as the left plot. The $c_g=c_{2g}=0$, $c_t=1$ in the left plot ($c_g=0$, $c_3=c_t=1$ in the right plot) was chosen.

Figure 8

Left: 68% probability contours of the likelihoods in ($c_{2t}$, $c_3$) plane using the multivariate analysis: the exclusive analysis of the double Higgs production in the $b\overline{b}\gamma \gamma$ (red), inclusive analysis of the $b\overline{b}{\tau }^{+}{\tau }^{-}$ (blue), and the combined analysis (black). Right: similarly, 68% probability contours of the likelihood in ($c_{2t}$, $c_{2g}$) plane with the same color/line codes as the left plot. The $c_g=c_{2g}=0$, $c_t=1$ in the left plot ($c_g=0$, $c_3=c_t=1$ in the right plot) was chosen.

Figure 9

Left: 68% probability contours of the likelihoods of the double Higgs production in the $b\overline{b}\gamma \gamma$ decay channel in the ($c_{2t}$, $c_3$) plane (left) and ($c_{2t}$, $c_{2g}$) plane (right): with ATLAS cuts (dashed), with cuts in [11] except the modification mentioned before (dotted), with multivariate analysis (solid). The $c_g=c_{2g}=0$, $c_t=1$ in the left plot ($c_g=0$, $c_3=c_t=1$ in the right plot) was chosen.

Figure 10

Left: 68% probability contours of the likelihoods in (${\overline{c}}_u$, ${\overline{c}}_6$) plane using the cut-based analysis of the double Higgs production process. The combined analysis corresponds to the exclusive $b\overline{b}\gamma \gamma$ (solid red) plus the inclusive $b\overline{b}{\tau }^{+}{\tau }^{-}$ (dashed red). Right: 68% probability contours of the likelihood of the exclusive $b\overline{b}\gamma \gamma$ plus single Higgs processes (dashed black). The $b\overline{b}{\tau }^{+}{\tau }^{-}$ is further combined (solid black). ${\overline{c}}_H={\overline{c}}_g={\overline{c}}_d=0$ was set in all contours except the solid red line in the right plot where the likelihood was marginalized over ${\overline{c}}_H$, ${\overline{c}}_g$, and ${\overline{c}}_d$.

Figure 11

Left: 68% probability contours of the likelihoods in (${\overline{c}}_u$, ${\overline{c}}_6$) plane using the multivariate analysis of the double Higgs production process. The combined analysis corresponds to the exclusive $b\overline{b}\gamma \gamma$ (solid red) plus the inclusive $b\overline{b}{\tau }^{+}{\tau }^{-}$ (dashed red). Right: 68% probability contours of the likelihood of the exclusive $b\overline{b}\gamma \gamma$ plus single Higgs processes (dashed black). The $b\overline{b}{\tau }^{+}{\tau }^{-}$ is further combined (solid black). ${\overline{c}}_H={\overline{c}}_g={\overline{c}}_d=0$ was set in all contours except the solid red line in the right plot where the likelihood of the combined analysis (exclusive $b\overline{b}\gamma \gamma$, inclusive $b\overline{b}{\tau }^{+}{\tau }^{-}$, and single Higgs) was marginalized over ${\overline{c}}_H$, ${\overline{c}}_g$, and ${\overline{c}}_d$.

Figure 12

Left: 68% probability contours in the (${\overline{c}}_u$, ${\overline{c}}_g$) plane using the result by the multivariate analysis of the double Higgs production process. The combined analysis (solid black) corresponds to the exclusive analysis of the $b\overline{b}\gamma \gamma$ (dashed red) plus the inclusive analysis of the $b\overline{b}{\tau }^{+}{\tau }^{-}$ (dashed blue). Right: 68% probability contours of the likelihood of the various combinations. The black dashed contour is obtained by the double Higgs production in the $b\overline{b}\gamma \gamma$ (further combining with the inclusive $b\overline{b}{\tau }^{+}{\tau }^{-}$ gives a solid black contour). The single Higgs processes are split into three categories: all single Higgs processes except $t\overline{t}h$ (dashed blue), $t\overline{t}h$ alone (dotted black), and all single Higgs processes (dashed red). The double Higgs production in the $b\overline{b}\gamma \gamma$ plus single Higgs fit corresponds to the orange region. Combining all double Higgs production channels plus all single Higgs processes appears in the light blue region. ${\overline{c}}_H={\overline{c}}_6={\overline{c}}_d=0$ was chosen.

Figure 13

The surrounding region of the first island of the single Higgs process in Fig. 12 is zoomed in. The line/color coding is the same as Fig. 12 on the right panel.

Figure 14

Likelihood distributions as a function of ${\overline{c}}_6$ using the cut-based analysis (left) and the multivariate analysis (right). The combined analysis corresponds to the exclusive analysis of the $b\overline{b}\gamma \gamma$ plus the inclusive analysis of the $b\overline{b}{\tau }^{+}{\tau }^{-}$. All the other parameters were set to the SM values (no marginalization).

Figure 15

Likelihood distributions as a function of ${\overline{c}}_6$ using the cut-based analysis (left) and the multivariate analysis (right). The combined analysis combines the exclusive $b\overline{b}\gamma \gamma$, the inclusive $b\overline{b}{\tau }^{+}{\tau }^{-}$, and single Higgs processes. The marginalization was performed over ${\overline{c}}_g$ and ${\overline{c}}_u$ with the priors from the single Higgs data.

Figure 16

The upper value of the 68% probability interval of ${\overline{c}}_6$ ($=c_3-1$) around the SM point, ${\overline{c}}_{6+}$, as a function of $\tau$-tag rate (left) and $b$-tag rate (right) at the HL-LHC. The red-cross line (in an arbitrary size) corresponds to the HL-LHC.

Figure 17

The upper value of the 68% probability interval of ${\overline{c}}_6$ ($=c_3-1$) around the SM point, ${\overline{c}}_{6+}$, as a function of fake rate at the HL-LHC. The red-cross line (in an arbitrary size) corresponds to the HL-LHC.

Figure 18

Distribution of $m_{\gamma \gamma }$ from ATLAS [32], CMS [8], and our simulation.

Figure 19

Likelihood distributions as a function of ${\overline{c}}_6$ using the multivariate analysis at the HL-LHC (dashed) and OPT-HL-LHC (solid). The combined analysis combines the exclusive $b\overline{b}\gamma \gamma$, the inclusive $b\overline{b}{\tau }^{+}{\tau }^{-}$, and single Higgs processes. The marginalization was performed over ${\overline{c}}_g$ and ${\overline{c}}_u$ with the priors from the single Higgs data.

Figure 20

The upper value of the 68% probability interval of ${\overline{c}}_6$ ($=c_3-1$) around the SM point, ${\overline{c}}_{6+}$, as a function of $f_{\mathrm{res}}$ at the OPT-HL-LHC. The $f_{\mathrm{res}}=1$ corresponds to the OPT-HL-LHC.

Figure 21

The $m_{\tau \tau }^{\mathrm{reco}}$ and $m_{bb}^{\mathrm{reco}}$ distributions for $f_{\mathrm{res}}=1$ (default) and 0.5 (50% improved).

Figure 22

Similar plots to Fig. 2 for more BDT analyses with the smaller set of variables.

Figure 23

Similar plots to Fig. 6 for more BDT analyses with the smaller set of variables.