Constraints on the spin-parity and anomalous $HVV$ couplings of the Higgs boson in proton collisions at 7 and 8 TeV

The study of the spin-parity and tensor structure of the interactions of the recently discovered Higgs boson is performed using the $H\to ZZ,Z{\gamma }^{*},{\gamma }^{*}{\gamma }^{*}\to 4\ell$, $H\to WW\to \ell \nu \ell \nu$, and $H\to \gamma \gamma$ decay modes. The full data set recorded by the CMS experiment during the LHC run 1 is used, corresponding to an integrated luminosity of up to $5.1{\mathrm{fb}}^{-1}$ at a center-of-mass energy of 7 TeV and up to $19.7{\mathrm{fb}}^{-1}$ at 8 TeV. A wide range of spin-two models is excluded at a 99% confidence level or higher, or at a 99.87% confidence level for the minimal gravitylike couplings, regardless of whether assumptions are made on the production mechanism. Any mixed-parity spin-one state is excluded in the $ZZ$ and $WW$ modes at a greater than 99.999% confidence level. Under the hypothesis that the resonance is a spin-zero boson, the tensor structure of the interactions of the Higgs boson with two vector bosons $ZZ$, $Z\gamma$, $\gamma \gamma$, and $WW$ is investigated and limits on eleven anomalous contributions are set. Tighter constraints on anomalous $HVV$ interactions are obtained by combining the $HZZ$ and $HWW$ measurements. All observations are consistent with the expectations for the standard model Higgs boson with the quantum numbers $J^{\mathrm{PC}}=0^{++}$.

Figure 1

Illustration of the production of a system $X$ in a parton collision and its decay to two vector bosons $gg$ or $q\overline{q}\to X\to ZZ,WW,Z\gamma$, and $\gamma \gamma$ either with or without sequential decay of each vector boson to a fermion-antifermion pair [28, 59]. The two production angles ${\theta }^{*}$ and ${\mathrm{\Phi }}_1$ are shown in the $X$ rest frame, and the three decay angles ${\theta }_1$, ${\theta }_2$, and $\mathrm{\Phi }$ are shown in the $V$ rest frames. Here $X$ stands either for a Higgs boson, an exotic particle, or, in general, the genuine or misidentified $VV$ system, including background.

Figure 2

Distributions of the eight kinematic observables used in the $H\to VV\to 4\ell$ analysis: $m_{4\ell }$, $m_1$, $m_2$, $\cos {\theta }^{*}$, $\cos {\theta }_1$, $\cos {\theta }_2$, $\mathrm{\Phi }$, and ${\mathrm{\Phi }}_1$. The observed data (points with error bars), the expectations for the SM background (shaded areas), the SM Higgs boson signal (open areas under the solid histogram), and the alternative spin-zero resonances (open areas under the dashed histograms) are shown, as indicated in the legend. The mass of the resonance is taken to be 125.6 GeV and the SM cross section is used. All distributions, with the exception of $m_{4\ell }$, are presented with the requirement $121.5<m_{4\ell }<130.5\mathrm{GeV}$.

Figure 3

Distributions of $m_{\ell \ell }$ (left) and $m_{\mathrm{T}}$ (right) for events with 0 jets (upper row) and 1 jet (lower row) in the $H\to WW\to \ell \nu \ell \nu$ analysis. The observed data (points with error bars), the expectations for the SM background (shaded areas), the SM Higgs boson signal (open areas under the solid histogram), and the alternative spin-zero resonance (open areas under the dashed histograms) are shown, as indicated in the legend. The mass of the resonance is taken to be 125.6 GeV and the SM cross section is used.

Figure 4

Distributions of the kinematic discriminants for the observed data (points with error bars), the expectations for the SM background (shaded areas), the SM Higgs boson signal (open areas under the solid histogram), and the alternative spin-zero resonances (open areas under the dashed histograms) are shown, as indicated in the legend. The mass of the resonance is taken to be 125.6 GeV and the SM cross section is used. Top row from left to right: ${\mathcal{D}}_{\mathrm{bkg}}$, ${\mathcal{D}}_{0-}$, ${\mathcal{D}}_{CP}$; bottom row from left to right: ${\mathcal{D}}_{0h+}$, ${\mathcal{D}}_{\mathrm{int}}$, ${\mathcal{D}}_{\mathrm{\Lambda }1}$. All distributions, with the exception of ${\mathcal{D}}_{\mathrm{bkg}}$, are shown with the requirement ${\mathcal{D}}_{\mathrm{bkg}}>0.5$ to enhance signal purity.

Figure 5

(left) Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ for the $J^P=1^{+}$ hypothesis of $q\overline{q}\to X(1^{+})\to ZZ$ tested against the SM Higgs boson hypothesis ($0^{+}$). The expectation for the SM Higgs boson is represented by the yellow histogram on the right and the alternative $J^P$ hypothesis by the blue histogram on the left. The red arrow indicates the observed $q$ value. (right) Observed value of $-2\mathrm{\Delta }\ln \mathcal{L}$ as a function of $f(J^P)$ and the expectation in the SM for the $q\overline{q}\to X(1^{+})\to ZZ$ alternative $J^P$ model.

Figure 6

Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ as a function of $f_{b2}$ for the spin-one $J^P$ models tested against the SM Higgs boson hypothesis in the $q\overline{q}\to X\to ZZ$ (left) and decay-only $X\to ZZ$ (right) analyses. The median expectation for the SM Higgs boson is represented by the red squares with the green (68% C.L.) and yellow (95% C.L.) solid color regions and for the alternative $J^P$ hypotheses by the blue triangles with the red (68% C.L.) and blue (95% C.L.) hatched regions. The observed values are indicated by the black dots.

Figure 7

Summary of the $f(J^P)$ constraints as a function of $f_{b2}$ from Table 7, where the decay-only measurements are performed using the efficiency of the $q\overline{q}\to X\to ZZ$ selection. The expected 68% and 95% C.L. regions are shown as green and yellow bands. The observed constraints at 68% and 95% C.L. are shown as the points with error bars and the excluded hatched regions.

Figure 8

(left) Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ for the $J^P=2_{h2}^{+}$ hypothesis of $gg\to X(2_{h2}^{+})\to ZZ$ tested against the SM Higgs boson hypothesis ($0^{+}$). The expectation for the SM Higgs boson is represented by the yellow histogram on the right and the alternative $J^P$ hypothesis by the blue histogram on the left. The red arrow indicates the observed $q$ value. (right) Observed value of $-2\mathrm{\Delta }\ln \mathcal{L}$ as a function of $f(J^P)$ and the expectation in the SM for the $gg\to X(2_{h2}^{+})\to ZZ$ alternative $J^P$ model.

Figure 9

Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ for the spin-two $J^P$ models tested against the SM Higgs boson hypothesis in the $X\to ZZ$ analyses. The expected median and the 68.3%, 95.4%, and 99.7% C.L. regions for the SM Higgs boson (orange, the left for each model) and for the alternative $J^P$ hypotheses (blue, right) are shown. The observed $q$ values are indicated by the black dots.

Figure 10

Summary of the $f(J^P)$ constraints for the spin-two models from Table 8, where the decay-only measurements are performed using the efficiency of the $gg\to X\to ZZ$ selection. The expected 68% and 95% C.L. regions are shown as the green and yellow bands. The observed constraints at 68% and 95% C.L. are shown as the points with error bars and the excluded hatched regions.

Figure 11

(left) Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ for the $J^P=1^{+}$ hypothesis of $q\overline{q}\to X(1^{+})\to WW$ against the SM Higgs boson hypothesis ($0^{+}$). The expectation for the SM Higgs boson is represented by the yellow histogram on the right and the alternative $J^P$ hypothesis by the blue histogram on the left. The red arrow indicates the observed $q$ value. (right) Observed value of $-2\mathrm{\Delta }\ln \mathcal{L}$ as a function of $f(J^P)$ and the expectation in the SM for the $q\overline{q}\to X(1^{+})\to WW$ alternative $J^P$ model.

Figure 12

(left) Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ for the $J^P=2_{h2}^{+}$ hypothesis of $gg\to X(2_{h2}^{+})\to WW$ against the SM Higgs boson hypothesis ($0^{+}$). The expectation for the SM Higgs boson is represented by the yellow histogram on the right and the alternative $J^P$ hypothesis by the blue histogram on the left. The red arrow indicates the observed $q$ value. (right) Observed value of $-2\mathrm{\Delta }\ln \mathcal{L}$ as a function of $f(J^P)$ and the expectation in the SM for the $gg\to X(2_{h2}^{+})\to WW$ alternative $J^P$ model.

Figure 13

(left) Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ as a function of $f_{b2}^{WW}$ for the hypothesis of the spin-one $J^P$ models against the SM Higgs boson hypothesis in the $q\overline{q}\to X\to WW$ analysis. (right) Distribution of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ as a function of $f(q\overline{q})$ for the $2_{h2}^{+}$ hypothesis against the SM Higgs boson hypothesis in the $H\to WW$ analysis. The median expectation for the SM Higgs boson is represented by the red squares with the green (68% C.L.) and yellow (95% C.L.) solid color regions and for the alternative $J^P$ hypotheses by the blue triangles with the red (68% C.L.) and blue (95% C.L.) hatched regions. The observed values are indicated by the black dots.

Figure 14

Distribution of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ for the spin-two $J^P$ models tested against the SM Higgs boson hypothesis in the $X\to WW$ analyses. The expected median and the 68.3%, 95.4%, and 99.7% C.L. regions for the SM Higgs boson (orange, the left for each model) and for the alternative $J^P$ hypotheses (blue, right) are shown. The observed $q$ values are indicated by the black dots.

Figure 15

Summary of the $f(J^P)$ constraints for the spin-one and spin-two models from Tables 9 and 10 in the $X\to WW$ analyses. The expected 68% and 95% C.L. regions are shown as the green and yellow bands. The observed constraints at 68% and 95% C.L. are shown as the points with error bars and the excluded hatched regions.

Figure 16

Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ in the combination of the $X\to ZZ$ and $WW$ channels for the hypotheses of $q\overline{q}\to X(1^{+})$ (top) and $gg\to X(2_{h2}^{+})$ (bottom) tested against the SM Higgs boson hypothesis. The expectation for the SM Higgs boson is represented by the yellow histogram on the bottom of each plot and the alternative $J^P$ hypothesis by the blue histogram on the top. The red arrow indicates the observed $q$ value.

Figure 17

(left) Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ in the combination of the $X\to ZZ$ and $WW$ channels for the hypothesis of $gg\to X(2_m^{+})$ tested against the SM Higgs boson hypothesis. The expectation for the SM Higgs boson is represented by the yellow histogram on the right and the alternative $J^P$ hypothesis by the blue histogram on the left. The red arrow indicates the observed $q$ value. (right) Distribution of $q$ as a function of $f(q\overline{q})$ for the $2_m^{+}$ hypothesis against the SM Higgs boson hypothesis in the $X\to ZZ$ and $WW$ channels. The median expectation for the SM Higgs boson is represented with the solid green (68% C.L.) and yellow (95% C.L.) regions. The alternative $2_m^{+}$ hypotheses are represented by the blue triangles with the red (68% C.L.) and blue (95% C.L.) hatched regions. The observed values are indicated by the black dots.

Figure 18

Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ for the spin-one and spin-two $J^P$ models tested against the SM Higgs boson hypothesis in the combined $X\to ZZ$ and $WW$ analyses. The expected median and the 68.3%, 95.4%, and 99.7% C.L. regions for the SM Higgs boson (orange, the left for each model) and for the alternative $J^P$ hypotheses (blue, right) are shown. The observed $q$ values are indicated by the black dots.

Figure 19

(left) Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ in the combination of the $X\to ZZ,WW$, and $\gamma \gamma$ channels for the hypothesis of $gg\to X(2_m^{+})$ tested against the SM Higgs boson hypothesis ($0^{+}$). The expectation for the SM Higgs boson is represented by the yellow histogram on the right and the alternative $J^P$ hypothesis by the blue histogram on the left. The red arrow indicates the observed $q$ value. (right) Distributions of the test statistic $q=-2\ln ({\mathcal{L}}_{J^P}/{\mathcal{L}}_{0^{+}})$ as a function of $f(q\overline{q})$ for the hypotheses of the $2_m^{+}$ model tested against the SM Higgs boson hypothesis in the $X\to ZZ,WW$, and $\gamma \gamma$ channels. The median expectation for the SM Higgs boson is represented with the solid green (68% C.L.) and yellow (95% C.L.) regions. The alternative $2_m^{+}$ hypotheses are represented by the blue triangles with the red (68% C.L.) and blue (95% C.L.) hatched regions. The observed values are indicated by the black dots.

Figure 20

Summary of allowed confidence level intervals on anomalous coupling parameters in $HVV$ interactions under the assumption that all the coupling ratios are real (${\varphi }_{ai}^{VV}=0$ or $\pi$). The expected 68% and 95% C.L. regions are shown as the green and yellow bands. The observed constraints at 68% and 95% C.L. are shown as the points with errors and the excluded hatched regions. In the case of the $f_{\mathrm{\Lambda }1}^{Z\gamma }$ measurement, there are two minima and two 68% C.L. intervals, while only one global minimum is indicated with a point. The combination of the $HZZ$ and $HWW$ measurements is presented, assuming the symmetry $a_i=a_i^{WW}$, including $R_{ai}=0.5$.

Figure 21

Expected (dashed line) and observed (solid line) likelihood scans using the template method for the effective fractions $f_{\mathrm{\Lambda }1}$, $f_{a2}$, $f_{a3}$ (from top to bottom) describing $HZZ$ interactions. Plots on the left show the results when the couplings studied are constrained to be real and all other couplings are fixed to the SM predictions. The $\cos {\varphi }_{ai}$ term allows a signed quantity where $\cos {\varphi }_{ai}=-1$ or $+1$. Plots on the right show the results where the phases of the anomalous couplings and additional $HZZ$ couplings are left unconstrained, as indicated in the legend. The $f_{a3}$ result with ${\varphi }_{a3}$ unconstrained (in the bottom-right plot) is from Ref. [12].

Figure 22

Observed likelihood scans using the template method for pairs of effective fractions $f_{\mathrm{\Lambda }1}$ vs. $f_{a2}$, $f_{\mathrm{\Lambda }1}$ vs. $f_{a3}$, and $f_{a2}$ vs. $f_{a3}$ (from top to bottom) describing $HZZ$ interactions. Plots on the left show the results when the couplings studied are constrained to be real and all other couplings are fixed to the SM predictions. Plots on the right show the results when the phases of the anomalous couplings are left unconstrained. The SM expectations correspond to points (0,0) and the best fit values are shown with the crosses. The confidence level intervals are indicated by the corresponding $-2\mathrm{\Delta }\ln \mathcal{L}$ contours.

Figure 23

Expected (dashed line) and observed (solid line) likelihood scans for $f_{a2}$ (top left) and $f_{a3}$ (top right), and observed likelihood scan for the $f_{a2}$ vs. $f_{a3}$ fractions (bottom), obtained using the template method (3D, black line) and the multidimensional distribution method (8D, red line) in the study of anomalous $HZZ$ interactions. The couplings are constrained to be real.

Figure 24

Expected (dashed line) and observed (solid line) likelihood scans using the template method for the effective fractions $f_{\mathrm{\Lambda }1}^{Z\gamma }$ (top), $f_{a2}^{Z\gamma }$ (middle left), $f_{a3}^{Z\gamma }$ (middle right), $f_{a2}^{\gamma \gamma }$ (bottom left), and $f_{a3}^{\gamma \gamma }$ (bottom right). The couplings studied are constrained to be real and all other couplings are fixed to the SM predictions. The $\cos {\varphi }_{ai}^{VV}$ term allows a signed quantity where $\cos {\varphi }_{ai}^{VV}=-1$ or $+1$.

Figure 25

Expected (dashed line) and observed (solid line) likelihood scans for effective fractions $f_{\mathrm{\Lambda }1}$ (top), $f_{a2}$ (middle), $f_{a3}$ (bottom). The couplings studied are constrained to be real and all other anomalous couplings are fixed to the SM predictions. The $\cos {\varphi }_{ai}$ term allows a signed quantity where $\cos {\varphi }_{ai}=-1$ or $+1$. Plots on the left show the results of the $H\to WW\to \ell \nu \ell \nu$ analysis expressed in terms of the $HWW$ couplings. Plots on the right show the combined $H\to WW$ and $H\to ZZ$ result in terms of the $HZZ$ couplings for $R_{ai}=0.5$. Measurements are shown for each channel separately, and two types of combinations are present: using $a_1^{WW}=a_1$ (red line) and without such a constraint (magenta line).

Figure 26

Observed conditional likelihood scans of $f_{\mathrm{\Lambda }1}$ (top), $f_{a2}$ (middle), $f_{a3}$ (bottom) for a given $R_{ai}$ value from the combined analysis of the $H\to WW$ and $H\to ZZ$ channels using the template method. The results are shown with custodial symmetry $a_1=a_1^{WW}$ (left) and without such an assumption (right). Each cross indicates the minimum value of $-2\mathrm{\Delta }\ln \mathcal{L}$ and the contours indicate the one-parameter confidence intervals of $f_{ai}$ for a given value of $R_{ai}$.