Higgs Physics at the LHC

Abstract

The search for Higgs bosons Higgs search is one of the main goals of the LHC physics program. In the Standard Model there is only one neutral Higgs boson predicted. Super-symmetric extensions of the Standard Model require at least two Higgs doublets which manifest as three neutral and two charged Higgs bosons. In the following chapter, mainly the search for the Standard Model Higgs boson and the possibilities to measure its properties at LHC are described. The latter subject is however also relevant for the possible discovery of more than one neutral Higgs particle, since these are expected to differ in mass, coupling structure and CP properties.

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The search for Higgs bosons Higgs search is one of the main goals of the LHC physics program. In the Standard Model there is only one neutral Higgs boson predicted. Super-symmetric extensions of the Standard Model require at least two Higgs doublets which manifest as three neutral and two charged Higgs bosons. In the following chapter, mainly the search for the Standard Model Higgs boson and the possibilities to measure its properties at LHC are described. The latter subject is however also relevant for the possible discovery of more than one neutral Higgs particle, since these are expected to differ in mass, coupling structure and CP properties.

7.1 Standard Model Higgs Searches with ATLAS and CMS

The search for SM Higgs bosons [1] is one of the primary goals of the LHC experiments. Present direct search results for the SM Higgs exclude masses, $M_\textrm{H}$, below 114.4 GeV [2] and $M_\textrm{H}$ values in the range 160–170 GeV at 95% C.L. [3]. Global Standard Model analyses of electroweak data prefer Higgs mass values below 186 GeV [4], as discussed in Chap. 4. In the given mass range, the SM Higgs is mainly produced through gluon–gluon fusion, $gg\to H$, where subsequent $H\to\gamma\gamma$, $H\to \textrm{ZZ}\to 4\ell\,(\ell=\textrm{e},\mu)$ and $H\to WW\to\ell \nu \ell \nu,{\textrm{q q} \ell \nu}$ decays are analysed. Figure 7.1 compiles the SM Higgs branching fractions and cross-sections for $\sqrt{s}=14 {\textrm TeV}$. They show that the two-photon decay can only be explored up to Higgs masses of about 120 GeV, while di-boson decays start to have a significant rate below the di-boson mass threshold. The t-channel vector-boson-fusion (VBF) process has a much lower production rate. However, it has experimentally useful signatures due to the event kinematics with two forward quark jets, suppressed central jet production and a central Higgs decay. In VBF, Higgs decays to WW and $\tau^{+}\tau^{-}$ are studied. In the very low Higgs mass region, $H\to {\textrm{b} \overline{\textrm{b}}}$ dominates, but is only detectable in Higgs-Strahlung from a ${\textrm{t} \overline{\textrm{t}}}$ pair, leading to a ${\textrm{t} \overline{\textrm{t}}} H\to{\textrm{t} \overline{\textrm{t}}} {\textrm{b} \overline{\textrm{b}}}$ final state.

Fig. 7.1

(a) SM Higgs branching fractions as a function of the Higgs mass, $M_\textrm{H}$ [5]. (b) Cross-sections of different SM Higgs production modes at the LHC using NLO calculations [5].

The search for $gg\to H\to\gamma\gamma$ is very challenging. The di-photon ${\textrm{q} \overline{\textrm{q}}}, \textrm{q}g\to \gamma\gamma+X$ and $gg\to\gamma\gamma$ backgrounds are irreducible and dominate the spectrum together with the QCD two-jet background with misidentified jets. Only a good mass resolution helps to identify the Higgs signal, as shown in Fig. 7.2 for simulated ATLAS data. Since there is a significant amount of material in the ATLAS detectors in front of the LAr calorimeter, reconstruction of photon conversions is necessary. Some 57% of the selected events have at least one conversion at a radius smaller than 80 cm from the interaction point. The single and double track conversion reconstruction recovers those photons with efficiencies between 40 and 90% depending on the conversion radius.

The photons are reconstructed in the electromagnetic calorimeter as clusters combining energy depositions in LAr cells in given $\varDelta\eta\times\varDelta\phi$ areas. In the LAr barrel region, $|\eta| <1.37$, the cluster sizes cover $3\times 7$ cells of the middle layer of the calorimeter in case the photons are tagged as converted, while unconverted photons are identified as $3\times 5$ clusters. In the LAr endcaps, $1.52<|\eta| <2.37$, a single cluster size of $5\times 5$ is chosen. Shower shape and and measurement of energy leakage in the first compartment of the Tile Calorimeter, as well as $\pi^0$ rejection in the LAr strips are used to reject hadronic background. The detection efficiency of photons with $p_T > 25 {\textrm GeV}$ is about 83% when low-luminosity pile-up at $10^{33} \textrm{cm}^{-2}s^{-1}$ is included . Furthermore, isolation from nearby tracks improves the rejection of hadronic jets to a factor of about 8,000 [5].

Also the primary vertex reconstruction is important since it influences the calculated Higgs mass. Due to the high granularity of the multi-layer calorimeter, the cluster bary-centres provide information about the direction of impact of the photons. Combined with a primary vertex constraint of the tracking system, the $H\to\gamma\gamma$ decay vertex can be measured with a resolution of about $0.1\,\upmu\textrm{m}$.

The mass resolution obtained after applying the full reconstruction of the $H\to\gamma\gamma$ decay is better than 1.5% which is illustrated in Fig. 7.2. The effect of the photon triggers is already included. These are efficient to more than 94% for di-photon p _T thresholds of 17 GeV. Here, the offline event selection is taken as a reference which requires that the photons are detected within the fiducial volume of the LAr calorimeter excluding the low-efficiency gap between barrel and endcap, and that the p _T of the two photons is larger than 25 and $40 {\textrm GeV}$, respectively.

Fig. 7.2

(a) Di-photon invariant mass spectrum after the application of cuts of the inclusive analysis, without additional requirements on jets, $E_{T}^\textrm{miss}$ or leptons [5]. (b) Invariant mass distribution for photon pairs from $H\to\gamma\gamma$ decays with $M_\textrm{H}=120 {\textrm GeV}$ after trigger and identification cuts [5]. Events with at least one converted photon are displayed as the shaded histogram

The expected performance of the inclusive Higgs boson detection is evaluated within a mass window of $\pm 1 \sigma$ around the peak value, where $\sigma$ is the mass resolution. For a Higgs mass of $120 {\textrm GeV}$, the accepted cross-section for the signal is 25.4 fb, combining gluon-fusion production, vector-boson fusion, ${\textrm{t} \overline{\textrm{t}}} H$ and $\textrm{W}H/\textrm{Z}H$ associated production. The background amounts to 950 fb, dominated by the irreducible QCD di-photon processes (53%) and by $\gamma$+jet production (33%). This shows that a very good performance and resolution of the LAr calorimeter will be important not to miss the $H\to\gamma\gamma$ signal.

The signal-to-background ratio can be further improved to about 1/10 and 1/2 by requiring one or two additional jets, respectively, because the leading jet in $gg\to H+\textrm {jet}$ and VBF production tends to be harder than for the di-photon background process. This reduces the accepted signal by factors 5 and 25, though. Other search strategies require $E_{T}^\textrm{miss}$ or $E_{T}^\textrm{miss}+\textrm {lepton}$ signatures, which enhances the associated production $\textrm{W}H$, $\textrm{Z}H$, ${\textrm{t} \overline{\textrm{t}}} H$ with respect to the background. The signal cross-sections are however below $0.01 \textrm{fb}$, which means that these channels become significant only if a larger amount of data, several hundred $fb^{-1}$, will be collected.

A similar performance is also achieved by the CMS experiment. The details of the measured photon shower shape in the CMS crystal calorimeter in different $\eta$ ranges is used to divide the data into performance categories. By optimising the analysis using cut-based and neural network criteria for photon selection and hadronic isolation, signal-to-background ratios close to 1/1 can be achieved in some of the event categories [6].

Another benchmark channel, where well performing calorimetry is essential, is the $H\to \textrm{ZZ}\to 4\ell$ channel. Here, high p _T electrons and muons are triggered on, and four leptons compatible with two Z decays, possibly off-shell, are selected. Electrons are reconstructed as clusters in the electromagnetic calorimeters matched to tracks in the inner detector. Shower shape and track quality criteria are applied. Isolation from hadronic activities is important to reject $Z{\textrm{b} \overline{\textrm{b}}}\to 4\ell+X$ and ${\textrm{t} \overline{\textrm{t}}}\to 4\ell+X$ production. Muons are detected in the muon spectrometer and a combined reconstruction with inner detector tracks is performed. Also here, isolation criteria rejecting muons with nearby tracks and clusters are used to reduce background. Both electrons and muons may be produced by B meson decays with vertices displaced from the primary vertex. To enhance the Higgs signal and reduce background with b quark content, cuts on the impact parameter are applied to muons and electrons. Eventually, lepton pairs of opposite charge, $\textrm {e}^{+}\textrm {e}^{-}$ and $\mu^+\mu^-$, are are combined and the $H\to ZZ\to 4\ell$ decay is fully reconstructed. The trigger efficiency for the four-lepton final states are typically above 98% compared to the offline event selection since either single-lepton triggers with p _T thresholds of about 20 GeV and isolation criteria or two-lepton signature with lower thresholds of 10–15 GeV with and without isolation are selecting the signal events.

The continuum $\textrm{ZZ}$ production represents the largest background in this channel. In the final step of the analysis, the invariant mass calculated from the four-momenta of the final state leptons is the main observable used to reject this background. The mass resolution is found to be below the 2% level in ATLAS for Higgs masses smaller than 200 GeV. Figure 7.3 shows the reconstructed mass peak in a data sample of $30 fb^{-1}$. At lower luminosities of around $1 fb^{-1}$, additional QCD jet, weak boson, and photon production backgrounds become important because dedicated studies are needed to control the systematics uncertainties related to those background sources. The reduction and control of the $\textrm{ZZ}$ continuum systematics is however primordial [7] and will require a good understanding of the detector performance and resolution with first data. The signal significances expected for the ATLAS analyses [5] in the $H\to \textrm{ZZ}$ channels are summarised in Fig. 7.3. The highest significances are obtained in the Higgs mass regions between 130 GeV and 160 GeV as well as between 200 and 500 GeV. The gap is mainly due to the rising $\textrm{ZZ}$ continuum cross-section at about twice the Z boson mass.

Fig. 7.3

(a) Reconstructed 4-lepton mass for SM Higgs signal and background, which is mainly from $\textrm{ZZ}$ continuum production [5]. (b) Expected signal significances in the $H\to \textrm{ZZ}$ channel, computed using Poisson statistics, for each of the three final states, and their combination [5]

In the mass range between 150 and $180 \textrm GeV$ the Higgs boson decays nearly exclusively to W-boson pairs. It is studied for both gluon-fusion, $H\to WW\to \textrm{e} \nu\mu \nu$ without additional jets, and vector-boson fusion production, $\textrm{qq}H\to \textrm{qq}WW\to {\textrm{q q} \textrm{e} \nu}\mu \nu$ and $\textrm{qq}H\to \textrm{qq}WW\to \textrm{qq}{\textrm{q q} \ell \nu}$ with additional jets in the forward region. The former is largely dominating but requires the rejection of final states with two leptons and missing energy, like W-pair, W+jets, ${\textrm{t} \overline{\textrm{t}}}$, $Z\to \tau^{+}\tau^{-}$ and heavy quark pair-production. For the latter, the same background sources are important, but the identification of two hadronic jets with a pseudo-rapidity gap of $|\varDelta\eta_{jj}|>2.5$ and a central jet veto further reduces those. This is due to the special VBF kinematics, as discussed in Chap. 1. To control the ${\textrm{t} \overline{\textrm{t}}}$ background a veto on b-jets is furthermore applied under the condition that the performance of the b-tagging algorithms is well understood.

Figure 7.4 summarises the ATLAS expectations [5] for the SM Higgs discovery potential with $10 fb^{-1}$ and the possible exclusion for $2 fb^{-1}$ of data. The most sensitive mass range is around 160 GeV. However, this mass range is currently already well tested by CDF and $D\emptyset$ [3]. An exclusion or first evidence for a SM Higgs signal will most probably be in experimental reach of the Tevatron experiments before ATLAS or CMS have collected enough data to become sensitive in this mass range.

Fig. 7.4

(a) The median discovery significance for the various Higgs decay channels and their combination for ${\cal L}=10 fb^{-1}$ in the low $M_\textrm{H}$ region [5]. Requiring significances above 5 translates into a possible discovery of the SM Higgs for $M_\textrm{H}>126 {\textrm GeV}$. (b) The median p-value obtained in a profile likelihood analysis [5] for excluding a Standard Model Higgs boson $(\mu=\sigma_\textrm{excluded}/\sigma_\textrm{SM}=1)$ for the various channels, as well as the combination for the lower mass range. Assuming ${\cal L}={\textit 2} fb^{-1}$ and asking for $p>0.95$, it will be possible to exclude the existence of the Standard Model Higgs boson in the mass range $115 {\textrm GeV} <M_\textrm{H}<480 {\textrm GeV}$

In the Higgs mass interval $114.4 {\textrm GeV} <M_\textrm{H}<120 {\textrm GeV}$, a discovery is most challenging and more luminosity may be needed. Higgs decays to ${\textrm{b} \overline{\textrm{b}}}$ dominate but are practically impossible to detect with sufficient significance in direct production due to overwhelming QCD background. Associated production of Higgs bosons in the ${\textrm{t} \overline{\textrm{t}}}$H, ZH and WH channels provides additional signatures to identify the signal process. Higgs production with top pairs is studied by selecting events with at least 4 b-jets, two from the Higgs decay and two from the t$\to$bW decays, and identified hadronic or leptonic W decays. The main background source is QCD and electroweak ${\textrm{t} \overline{\textrm{t}}}{\textrm{b} \overline{\textrm{b}}}$ production, but also ${\textrm{t} \overline{\textrm{t}}}$ events with additional hadronic jets not from B decays. They remain irreducible because requirements on mass windows for the reconstructed W and top decays only reduce other backgrounds efficiently. Eventually, signal significances in the order of $0.5-1 \sigma$ are obtained with $30 \textrm{fb}^{-1}$ of data [6,5], depending on the systematic uncertainties assumed. These are mainly driven by b-tag performance and b-jet misidentification rates, as well as energy scales for leptons, jets and missing transverse energy. Estimations of total background uncertainties, including predictions of the corresponding cross-sections, are in the order of 30%. Promising results [8] are obtained in the WH$\to\ell \nu+{\textrm{b} \overline{\textrm{b}}}$ and ZH$\to{\ell^+\ell^-}+{\textrm{b} \overline{\textrm{b}}},\nu\bar{\nu}+{\textrm{b} \overline{\textrm{b}}}$ channels using a dedicated identification method of the $H\to{\textrm{b} \overline{\textrm{b}}}$ system. If the Higgs boson is produced with large transverse momentum, $p_T>200 {\textrm GeV}$, and back-to-back to the gauge boson, background from ${\textrm{t} \overline{\textrm{t}}}$ and QCD events is suppressed. The boosted $H\to{\textrm{b} \overline{\textrm{b}}}$ decay is however difficult to detect since the b-jets are merged into a single jet if classical jet-cone algorithms are applied. The jet substructure can however be further resolved, for example, by applying the Cambridge/Aachen [9] clustering algorithm . It is based on angular distances in $\varDelta R=\sqrt{(\varDelta\eta)^2+(\varDelta\phi)^2}$ between two clusters and iteratively merges objects close in angular space until all $\varDelta R$ are larger than a given parameter $R_{\textrm{cut}}$. To resolve the jet substructure the last steps of the jet clustering are undone, until the jet that was formed last splits into two jets of nearly equal mass. The last condition should suppress soft gluon radiation. If both jets were tagged as a b-jet the event is considered to be a Higgs candidate. After having identified the ${\textrm{b} \overline{\textrm{b}}}$ system, the Cambridge/Aachen algorithm is repeated with a finer resolution parameter to resolve also a possible third jet from final state QCD radiation. The accompanying gauge boson decays are identified according to their two-lepton, one lepton and $E_{T}^\textrm{miss}$, or large $E_{T}^\textrm{miss}$ signatures. Events with other jets or leptons with large p _T in the central detector region, $|\eta| <2.5-3$, are rejected. Eventually, one obtains a sufficient Higgs mass resolution and background suppression with signal significances in the order of 4–6 standard deviations for Higgs masses of $M_\textrm{H}\approx 115 {\textrm GeV}$.

Figure 7.5 shows the reconstructed $H\to{\textrm{b} \overline{\textrm{b}}}(g)$ mass spectrum for all WH and ZH channels combined. B-tag efficiencies of 60% and a mistag probability of 2% are furthermore assumed. The dependence of the final significance on these parameters, as well as on the minimal p _T of the Higgs is also indicated in Fig. 7.5. The significance decreases with increasing Higgs masses, increasing Higgs p _T and increasing b-mistag rate. The performance is however better than expected for the ${\textrm{t} \overline{\textrm{t}}}$H channel and very promising in the low Higgs mass region, which is preferred by fits to electroweak data.

Fig. 7.5

(a) Reconstructed $H\to{\textrm{b} \overline{\textrm{b}}}(g)$ mass spectrum in the combined WH and ZH search channels, expected for an integrated luminosity of $30 fb^{-1}$ and a LHC centre-of-mass energy of $14 {textrm TeV}$ [8]. (b) Combined WH and ZH signal significance as a function of Higgs mass, of the minimal p _T of the $H\to{\textrm{b} \overline{\textrm{b}}}(g)$ system, and of the b-tagging efficiency and mistag rate [8]

In general, with the full data set of ATLAS, the SM Higgs can not escape detection. The performance of CMS is quite similar, as shown in Fig. 7.6 and both LHC experiments will cross-check each others findings. The LHC data will eventually give the definitive verdict about the SM Higgs boson.

Fig. 7.6

Signal significance as a function of the Higgs boson mass for an integrated luminosity of $30 fb^{-1}$ for different Higgs boson production and decay channels as expected for the CMS experiment [6]

7.2 Determination of Higgs Boson Properties at the LHC

Once a Higgs boson has been discovered a determination of the particle properties will be needed to verify their consistency with the Standard Model predictions. By analysing the mass spectrum, the cross-sections and branching fractions, the Higgs mass, width and the Higgs couplings can be measured. The detailed production and decay kinematics will give a handle to determine the spin of the discovered particle and its CP properties.

7.2.1 Measurement of Mass, Width and Couplings

The first parameter which will be know at the moment of the discovery of the Higgs boson is its mass. However, the corresponding precision varies with Higgs mass and accessible decay channels. As can be seen in Fig. 7.7, the combined $H\to\gamma\gamma$ and $H\to \textrm{ZZ}$ channels cover the mass range above $\approx 115 {\textrm GeV}$. Other final states, like $H\to WW$, suffer from missing energy in the final state and a direct determination of the Higgs mass is not easily possible. The relative mass resolution in the $\gamma\gamma$ final state is about 1% [5,6] for both ATLAS and CMS. Applying a likelihood method, the Higgs mass can be determined in this channel with a statistical precision below 0.2% using $30 fb^{-1}$ of data [6].

Fig. 7.7

(a) The determination of the Higgs mass is best in the $H\to \gamma\gamma$ and $H\to \textrm{ZZ}$ decay channels since the full final state can be reconstructed and the decay spectrum is measured with good resolution. (b) The measurement of the Higgs width is also derived from the reconstructed Higgs decay spectrum. Because of the small physical Higgs width the experimental data will only be able to measure $\Gamma_H$ for Higgs masses above $\approx 200 {\textrm GeV}$. The corresponding simulated performances are shown for the CMS experiment [6]

To estimate the precision on the $M_\textrm{H}$ determination in the $\textrm{ZZ}$ final state, a simple Gaussian shape of the peak is assumed and the relative statistical uncertainties are calculated as $\sigma_{M_\textrm{H}}=\sigma_{\textrm{gauss}}/\sqrt{N_S}$ and $\sigma_{\Gamma_H}=\sigma_{\textrm{gauss}}/\sqrt{2N_S}$, with the number of Higgs signal events, N _S, in the mass peak interval [6]. This results in uncertainties below 1% on $M_\textrm{H}$ for Higgs boson masses up to 500 GeV, and 5–50% on $\Gamma_H$, again for $30 fb^{-1}$. Such a measurement will put the Standard Model to a very important test. A result which is not in the favourable low-mass range would imply a break-down of the theory at higher energy scales (see Chap. 1).

In an ATLAS study [10] also a possible scenario for the measurement of the Higgs couplings to fermions and bosons has been evaluated. According to Eq. (1.35), they are given by

$$g_W=\frac{M_\textrm{W}}{2v}\;,\;\;g_Z=\frac{M_\textrm{Z}}{2v}\;;\;\;g_d=\frac{\sqrt{2}m_f}{v}\;;\;\;g_u=-\frac{\sqrt{2}m_f}{v}\;,$$

((7.1))

for the W and Z bosons, and up- and down-type fermions. Starting from the measured product of cross-section and branching fraction in the different channels, shown in Fig. 7.8, one can derive eventually also absolute values of the couplings. This requires however a series of additional assumptions. Under the condition that there is only one CP-even and spin-0 Higgs boson, only Standard Model couplings, and visible branching ratios which follow the Standard Model predictions, the absolute measurement is possible. The result obtained in this restricted scenario is shown in Fig. 7.8. Relaxing the last requirement on the branching fractions, only ratios of Higgs boson couplings are accessible.

Fig. 7.8

(a) Expected precision, including systematic uncertainties, on the measurement of the product $\sigma\times BR$ for the different Higgs decay channels. (b) From these, the absolute values of the couplings of the Higgs to fermions and bosons can be derived [10]

7.2.2 Measurement of Spin and CP Quantum Numbers

Most interesting will be the study of the intrinsic properties of the Higgs boson, like its spin and the CP properties. The Standard Model Higgs boson is a scalar particle with positive parity and charge conjugation eigenvalues, P and C, shortly noted as $J^{PC}=0^{++}$. These parameters can be measured in Higgs production [11] and decay [12] separately. For a quantitative estimate of the sensitivity, the Standard Model Lagrangian is usually extended to include anomalous contributions . They can be from CP-even and CP-odd couplings, CP-violating mixtures of these [13] , or from couplings with a Higgs spin structure different from $J=0$. CP violation in the Higgs sector would be a striking sign for physics beyond the Standard Model. Indeed, the strength of CP violation of the SM, observed only in the $K^0-\bar{k}^0$ and $B^0-\bar{B}^0$ systems to date and described by the CKM mixing matrix [14], is not sufficient to explain the baryon/anti-baryon asymmetry in the universe [15]. An additional source of CP violation beyond that of the SM may be needed. An extended Higgs sector together with CP-violating super-symmetry (SUSY) is one possible option of physics beyond the SM that may explain the baryon asymmetry [16]. The measurement of the properties of the Higgs sector and any possible CP violation in Higgs production or decay will be an important question which can be answered once a Higgs boson is discovered [17].

To include different CP structures, the Standard Model HVV vertex function which couples the Higgs field to the vector boson fields, $T^{\mu \nu}=(2M_V/v) g^{\mu \nu}$, is generalised to the tensor [18,12]:

$$\begin{aligned}T^{\mu\nu}&{}=a_1(q_1,q_2) g^{\mu\nu} +a_2(q_1,q_2) \left[q_1\cdot q_2 g^{\mu\nu} - q_2^\mu q_2^\nu\right]\nonumber\\ &{}+a_3(q_1,q_2)\varepsilon^{\mu\nu\rho\sigma}q_{1\rho}q_{2\sigma}\end{aligned}$$

((7.2))

The Lorentz-invariant form factors, a _i, depend on the four-momenta of the weak bosons, q ₁ and q ₂. The factor a ₁ of the first CP-even term appears already in the SM at tree level. The CP-even coupling a ₂ and the CP-odd coupling a ₃ are from higher dimension operators [19,20,18]. They appear first at dimension-5 level and the effective Lagrangian may be written as [18]:

$$\begin{aligned}{\cal L}_5&{}= \frac{g_{5e}^{HWW}}{\varLambda_{5e}} H W^+_{\mu\nu} W^{-\mu\nu}+\frac{g_{5o}^{HWW}}{2 \varLambda_{5o}} H \varepsilon_{\mu\nu\rho\sigma} W^{+\rho\sigma} W^{-\mu\nu}\nonumber\\ &{}+\frac{g_{5e}^{HZZ}}{2 \varLambda_{5e}} H Z_{\mu\nu} Z^{\mu\nu}+\frac{g_{5o}^{HZZ}}{4 \varLambda_{5o}} H \varepsilon_{\mu\nu\rho\sigma} Z^{\rho\sigma} Z^{\mu\nu}\;,\end{aligned}$$

((7.3))

where the $\Lambda_{5e/o}$ define the energy scale of the effective theory. The anomalous couplings are then given by

$$a_2(q_1,q_2)=-\frac{2}{\varLambda_{5e}}g_{5e}^{HWW}\;;\;\;a_3(q_1,q_2)=-\frac{2}{\varLambda_{5o}}g_{5o}^{HWW}$$

((7.4))

and

$$a_2(q_1,q_2)=-\frac{2}{\varLambda_{5e}}g_{5e}^{HWW}\;;\;\;a_3(q_1,q_2)=-\frac{2}{\varLambda_{5o}}g_{5o}^{HWW}$$

((7.5))

for the HWW and HZZ vertex. If one assumes the relative strength of W and Z contributions to behave like in the Standard Model, one can additionally require $g_{5e/o}^{HZZ}\!=\!\cos^2{\theta_\textrm{w}} g_{5e/o}^{HWW}$. Setting $a_1\!=\!0$ and assuming a Higgs mass of 120 GeV, Higgs production rates similar to those in the SM are obtained with $\Lambda_{5e/o}\approx 480 {\textrm GeV}$ if only CP-even or only CP-odd couplings are active $\left(g_{5e/o}^{HWW}\!\!=\!1,g_{5o/e}^{HWW}\!=0\right)$.

Constraints on these couplings were already set at LEP by the L3 experiment [21]. In the LEP analysis a different effective theory based on dimension-6 operators was used [22,23,24], and CP-odd terms were neglected:

$$\begin{aligned}{\cal L}_{\textrm{eff}}&{}= \frac{g_2}{2 {M_\mathrm{W}}} \left( d\sin^2{\theta_\textrm{w}} + d_B \cos^2{\theta_\textrm{w}} \right) {\textrm H} {\textrm A}_{\mu\nu} {\textrm A}^{\mu\nu}\\ &{}+\frac{g_2}{{M_\mathrm{W}}} \left(\varDelta g^{{\rm Z}}_{1} \sin 2{\theta_\textrm{w}} - \varDelta \kappa_{\gamma}\!\cot{\theta_\textrm{w}} \right) {\textrm A}_{\mu\nu} {\textrm Z}^{\mu} \partial^{\nu} {\textrm H}\\ &{}+\frac{g_2}{2 {M_\mathrm{W}}} \sin 2{\theta_\textrm{w}} \left( d - d_B \right) {\textrm H} {\textrm A}_{\mu\nu} {\textrm Z}^{\mu\nu}\\ &{}+\frac{g_2}{{M_\mathrm{W}}} \left(\varDelta g^{{\rm Z}}_{1} \cos 2{\theta_\textrm{w}} + \varDelta \kappa_{\gamma}\!\tan^2{\theta_\textrm{w}} \right) {\textrm Z}_{\mu\nu} {\textrm Z}^{\mu} \partial^{\nu} {\textrm H}\\ &{}+\frac{g_2}{2 {M_\mathrm{W}}} \left(d\cos^2{\theta_\textrm{w}} +d_B \sin^2{\theta_\textrm{w}} \right) {\textrm H} {\textrm Z}_{\mu\nu} {\textrm Z}^{\mu\nu}\\ &{}+\frac{g_2 {M_\mathrm{W}}}{2 \cos^2{\theta_\textrm{w}}} \delta_{\textrm Z} {\textrm H} {\textrm Z}_\mu {\textrm Z}^\mu\\ &{}+\frac{g_2 {M_\mathrm{W}}}{\textrm M}_Z^2 \Delta g_1^Z \left({\textrm W}^{+}_{\mu\nu} {\textrm W}_{-}^\mu \partial^{\nu} {\textrm H} + h.c.\right)\\ &{}+\frac{g_2}{{M_\mathrm{W}}} \frac{d}{\cos 2{\theta_\textrm{w}}} {\textrm H} {\textrm W}^{+}_{\mu\nu} {\textrm W}_{-}^{\mu\nu},\end{aligned}$$

((7.6))

where g ₂ is the $SU(2)_L$ coupling constant. One can identify both sets of CP-even couplings using the relations:

$$g_{5e}^{HZZ} = \Lambda _{5e} \frac{{g_2 }}{{M_W }}\left( {d\,\cos ^2 \theta _W + d_B \,\sin ^2 \theta _W + \Delta g_1^Z \,\cos 2\theta _W + \Delta k_\gamma \tan ^2 \theta _W } \right)$$

((7.7))

$$g_{5e}^{HZZ} = \Lambda _{5e} \frac{{g_2 }}{{M_W }}\left( {d + \frac{{M_W^2 }}{{M_Z^2 }}\Delta g_1^Z } \right)$$

((7.7))

The anomalous couplings $\varDelta g^{{\rm Z}}_{1}$ and $\varDelta \kappa_{\gamma}$ have already been studied in the analysis of W-pair production at LEP since they also describe possible deviations of the triple-gauge-boson couplings of W bosons with photons and Z bosons [23]. Their measurement in $\textrm {e}^+\textrm {e}^{-}\to{\textrm WW}$ events was presented in Chap. 3 and they were found to be in very good agreement with the SM expectations. In the general HVV vertex, the additional couplings d and $d_{b}$ [22] appear. Here, their effects on the Higgs sector shall be further discussed.

The existence of $H\gamma \gamma$ and $HZ\gamma$ couplings would actually lead to large $H \to \gamma \gamma$ and $H \to Z\gamma$ branching fractions, which, at tree level, are zero in the Standard Model. These decay modes have complementary sensitivities to the different couplings. In addition, the decay $H \to WW^{\left( * \right)}$ would be enhanced in the presence of anomalous HWW couplings. This was studied by L3 [21] by interpreting the Higgs search results in the $\textrm {e}^{+}\textrm {e}^{-}\to HZ$ channels in $H\to\textrm{f}\bar{f}$ and $H\to\gamma\gamma$ final states and by extending the signatures to $\textrm {e}^{+}\textrm {e}^{-}\to H\gamma$ with $H\to \gamma\gamma$, $H\to Z\gamma$, and $H\to{\textrm WW}^{(*)}$, as well as to the boson-fusion channel $\textrm {e}^{+}\textrm {e}^{-}\to\textrm {e}^{+}\textrm {e}^{-} H\to\textrm {e}^{+}\textrm {e}^{-}\gamma\gamma$. The results are shown in Fig. 7.9. For Higgs masses up to about 100 GeV any anomalous contributions to the HVV vertex are excluded, also for the $\Delta g_1^Z$ and $\Delta k_\gamma$ parameters not shown in the plot. For masses between 100 and 170 GeV, d and $d_{B}$ couplings larger than 0.5 are found to be incompatible with data at 95% C.L. The ($M_\textrm{H}$ dependent) limits on $\Delta g_1^Z$ and $\Delta k_\gamma$ are less stringent than those from the TGC measurements , which are an order of magnitude more sensitive.

Fig. 7.9

Regions excluded by L3 [21] at 95% C.L. as a function of the Higgs mass for the anomalous couplings: (a) d, (b) $d_{B}$, always setting the other respective couplings to zero

Eventually, one can convert the L3 measurements into constraints on the HVV couplings, which yields [11]:

$$\begin{aligned}&{}g_{5e}^{HWW}\in [-0.78,0.73]\;;\;\;g_{5e}^{HZZ}\in [-0.63,0.55]\;\;\textrm {for}\;M_H=120 {\textrm GeV}\\ &{}g_{5e}^{HZZ}\in [-2.0,1.5]\;;\phantom{00}\;;g_{5e}^{HZZ}\in [-1.6,1.3]\phantom{00}\;\;\textrm {for}\;M_H=160 {\textrm GeV}\end{aligned}$$

using Eq. (7.7), assuming classic error propagation and neglecting correlations. ATLAS studied the measurement of $g_{5e/o}^{HZZ}$ in the VBF channel $\textrm{qq}H\to \textrm{qq}\tau^{+}\tau^{-}$ with fully leptonic and lepton-hadron decay mode at $M_\textrm{H}=120 {\textrm GeV}$ as well as in the VBF channel $\textrm{qq}H\to \textrm{qq}{\textrm WW}\to \textrm{qq}\ell \nu \ell\nu$ at $M_\textrm{H}=160 {\textrm GeV}$. The distribution of the reconstructed angle between the two forward-tag jets, $\varDelta \phi_\textrm{jj}$, was used to evaluate the sensitivity to the anomalous couplings. Figure 7.10 shows the distributions in the three channels for the Standard Model (SM, $a_1=1,a_2=0,a_3=0$), for a scenario with only anomalous CP-even coupling turned on (CPE, $a_1=0,a_2=1,a_3=0$), and finally with only an anomalous CP-odd coupling (CPO, $a_1=0,a_2=0,a_3=1$). It could be shown that from a $\chi^2$ analysis of the $\varDelta\phi_{\textrm{jj}}$ distributions, the CPE and CPO scenarios can be excluded at 95% C.L. with $10 fb^{-1}$ of data, using only the $H\to WW$ channel. The low-mass $H\to\tau^{+}\tau^{-}$ channel only becomes sensitive beyond $30 fb^{-1}$.

Fig. 7.10

Angle between the forward-tag jets in a high statistics sample of simulated VBF Higgs events for different HVV couplings: the Standard Model (SM) case, and pure anomalous CP-even (CPE) and CP-odd (CPO) couplings [11]

The Higgs coupling structure is also easily accessible in the $H\to \textrm{ZZ}\to \ell ^+\ell ^- \ell \prime^{+}\ell \prime^{-}$ decay channel, where the lepton pairs, $\ell ^+\ell ^-$ and $\ell \prime^{+}\ell \prime^{-}$, are muon or electron pairs. From distributions of the Z decay plane correlations and the lepton angles in the Z rest frames, the helicity structure of the HZZ vertex can be determined. The observables that are most sensitive to the CP properties of the Higgs are

• the angle between the oriented Z decay decay planes, $\phi$, in the Higgs rest frame, and

• the cosine of the polar angle of the fermion, $\cos{\theta^*}$, in the Z rest frame.

Their theoretical distribution expected in the Standard Model and for an anomalous CP-even and CP-odd HZZ coupling are shown in Fig. 7.11.

Fig. 7.11

Distribution of the fermion polar angle, $\theta^*$, in the Z rest frame (left) and the angle $\phi$ between the decay planes of the Z bosons (right) for the Standard Model (SM), and an anomalous CP-even (CPE) and CP-odd (CPO) vertex. The distributions are shown at Monte Carlo generator level [25]

The ZZ final state was studied by ATLAS [26,25] and CMS [6,27]. The theoretical frameworks used in these analyses were based on a coupling structure:

$$T^{\mu\nu}_{HZZ}=\frac{ig M_Z}{\cos\theta_\textrm{W}}\left\{ A g^{\mu\nu}+\frac{B}{M_Z^2}p^\mu p^\nu+\frac{C}{M_Z^2} \varepsilon^{\mu\nu\rho\sigma}p_\rho k_\sigma\right\}\;,$$

((7.9))

with the four-momenta of the Z bosons, q ₁ and q ₂, their sum, $p=q_1+q_2$, and their difference, $k=q_1-q_2$. The parameters A, B, and C describe the SM couplings strength and anomalous CP-even and CP-odd contributions, very similar to Eq. (7.2). An alternative model [28] with a different parameterisation of the HZZ vertex is also used in the analysis. This model is integrated in the Pythia [29] event generator. Purely CP-even, CP-odd, and mixed scenarios are available. In the mixed scenario, a parameter $\eta$ is introduced, and the CP-odd term is varied proportional to $\eta^2$, while the interference term scales with $\eta$. The $\eta$ parameter is not directly equivalent to C and contains a small admixture of the CP-even B-term. The limit $\eta\to 0$ corresponds to the Standard Model and $|\eta|\to\infty$ to a pure CP-odd coupling. A mapping $\xi=\textrm {atan}(\eta)$ projects the full coupling range to the interval $[-\phi,\phi]$.

Usually large data statistics, in the order of $50-100 fb^{-1}$, is necessary to measure the anomalous couplings in $H\to \textrm{ZZ}$ decays. CMS simulated signal and background events, which are practically only from ZZ continuum, for a luminosity of $60 fb^{-1}$. The likelihood analysis of the shape of the angular distributions and the reconstructed Higgs mass spectrum yields a sensitivity in the order of $\varDelta\xi=0.2$ for $M_\textrm{H}=200 {\textrm GeV}$, shown in Fig. 7.12, improving with mass and signal statistics to $\varDelta\xi=0.14$ for $M_\textrm{H}=300 {\textrm GeV}$. This result is in good agreement with expectations for the ATLAS experiment [25], where it could also be shown that systematic effects from resolution and modelling of signal and background can eventually be controlled.

Fig. 7.12

(a) Estimated sensitivity to the anomalous HZZ coupling $\xi=\textrm {atan}(\eta)$ of the Pythia model as a function of a possible signal enhancement factor C for a Higgs mass of $200 {\textrm GeV}$ and an integrated luminosity of $60 fb^{-1}$ [27]. The enhancement factor assumes that the Higgs production cross-section scales proportional to C, where for $C=1$ the SM expectation is obtained. (b) Expected significance for excluding spin 1 and anomalous CP-even and CP-odd scenarios for $100 fb^{-1}$ of ATLAS data [26]

In addition to an anomalous CP structure, the effect of a Higgs-like particle with different spin $J\ne 0$ was evaluated. Clearly, this is mainly to reject this possibility once the Higgs is discovered in the $H\to \textrm{ZZ}$ channel. The main sensitivity to models with $J>0$ and anomalous HZZ couplings at high Higgs mass is obtained from the analysis of the fermion decay angle, $\cos\theta^*$. This sensitivity can be enhanced by measuring the Z polarisation, assuming a distribution of the decay angle according to [30]:

$$f(\theta)=T(1+\cos^2\theta^*)+L\sin^2\theta^* \;,$$

((7.10))

where T and L are the fraction of transverse and longitudinal polarisation of the Z bosons. A similar method was successfully applied by the KTeV Collaboration who measured the parity of the neutral pion in the four-electron decay, $\phi^0\to \gamma^*\gamma^*\to \textrm {e}^{+}\textrm {e}^{-}\textrm {e}^{+}\textrm {e}^{-}$ [31]. The variable

$$R=\frac{L-T}{L+T}$$

((7.11))

quantifies the relative difference of longitudinal and transverse polarisation fractions, L and T. For a pure CP-even HZZ coupling a value of $R=+1$ is expected, corresponding to a 100% longitudinal Z polarisation, while $R=-1$ for a pure CP-odd coupling, corresponding to a 100% transverse Z polarisation. In the Standard Model, R is around 0.5, varying with the Higgs mass. Since the background is flat in $\cos\theta^*$ the normalised probability density as a function of $\theta^*$ is given by:

$$P(\theta^*)=\frac{\frac{3}{4}N_s\left(\frac{1-R}{3-R}(1+\cos^2\theta^*)+\frac{1+R}{3-R}\sin^2\theta^*\right) + \frac{1}{2}N_b}{N_s+N_b} \;.$$

((7.12))

If the number of background events, N _b, is assumed to be known, the number of signal events is obtained from data as $N_s=N_\textrm{data}-N_b$. An effective value of R is determined from the reconstructed fermion angles, $\theta^*_i$, of each event i, maximising the log-likelihood function

$$\log L=\sum_{i=1}^{N_\textrm{data}} \log P\left(\theta^*_i\right)\;.$$

((7.13))

Additional sensitivity provides the polar angle distribution which follows

$$F(\phi)=1+\alpha\cos(\phi)+\beta\cos(2\phi)$$

((7.14))

for the HZZ signal (see Fig. 7.11). A likelihood analysis, including also the background shape, of both polar angle and decay plane angle was performed by ATLAS [26], and the result is shown in Fig. 7.12. A CP-odd scalar boson can be well excluded with $100 fb^{-1}$ of data, while an exclusion of a vector-like particle with spin $J=1$ is only possible for Higgs masses above $200 {\textrm GeV}$. This is again in good agreement with [25].

In the very low-mass region, where $H\to\gamma\gamma$ may be the first discovery channel, only the spin can be constrained using the well-known Landau–Yang theorem [32]. In case this decay mode is observed at the LHC, the Higgs boson must necessarily be of spin-0 nature, or a multiple of $J=2$.

The measurement of the CP structure in the $H\to\tau^{+}\tau^{-}$ decay, which may well be a preferred decay scenario of a low-mass super-symmetric Higgs boson, is possible but challenging. In principle, the polarisation of the $\tau$ leptons, measured by analysing the $\tau$ decay kinematics, may be used to infer on the Higgs CP and spin structure. But in contrary to the $H\to ZZ\to 4\ell$ channel, the centre-of-mass frame can not be directly reconstructed, which drastically reduces the sensitivity of the decay angular distributions [33]. The Higgs spin and CP structure will therefore be mainly determined in the VBF channels $H\to\tau^{+}\tau^{-}$ and $H\to WW^{(*)}$, as well as in $H\to \textrm{ZZ}$ decays.

In summary, one can conclude that the Standard Model Higgs parameters, like mass, width, branching fractions, as well as spin and CP structure can be probed at the LHC with data samples that correspond to integrated luminosities in the order of $30 fb^{-1}$. For further interesting analyses, like the measurement of the Higgs trilinear self-coupling in $H\to HH\to \textrm{WWWW, WWZZ, ZZZZ}$ decays, two orders of magnitude more data is needed [34], which may be delivered by an upgraded super-LHC [35] with instantaneous luminosities of $10^{35} \textrm{cm}^{-2}s^{-1}$.