1.Institute of High Energy Physics, CAS, Beijing 100049, China 2.School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China Received Date:2019-05-28 Available Online:2019-10-01 Abstract:We study the sensitivity of constraining the model independent HZZ coupling based on the effective theory up to dimension-6 operators at a future Higgs factory. Using the current conceptual design parameters of the Circular Electron Positron Collider, we give the experimental limits for the model independent operators given by the total Higgsstrahlung cross-section and the angular distribution of Z boson decays. In particular, we give the very small sensitivity limit for the CP violation parameter $ \tilde g$, which will be a clear window to test the Standard Model and look for new physics signals.
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2.New physics effects on HZZ couplingThe generic effective Hamiltonian in the $ HZZ $ sector is written as [27]
where $ Z_{\mu\nu} = \partial_{\mu}Z_{\nu}-\partial_{\nu}Z_{\mu} $ and $ {\widetilde{Z}}_{\mu\nu} = \frac{1}{2}\epsilon_{\mu\nu\rho\sigma}Z^{\rho\sigma} $. The effective Feynman rule can be derived from Eq (1) as
In this parameterization, $ g_0 = e M_Z /(c_w s_w ) $ is the HZZ coupling in the Standard Model. Taking the convention of [28], $ g_3 $ is a small number in units of $ g_0 $, while $ g_1, g_2, {\tilde g} $ are small numbers in units of $ e^2 /( g_0 s^2_w c^4_w ) $ , so that the interaction is consistent with the dimension of mass. The new type of couplings $ g_1, g_2, {\tilde g} $ should be smaller than one in SM, since most experimental data are consistent with SM. The number of free parameters in new physics is then reduced from 12 to only 4 [28], while keeping a sufficiently general structure for the interaction between the Higgs and vector bosons. In contrast to the $ \kappa_Z $ parametrization [29], which has only one parameter, these four parameters are effective for revealing the details of potential new physics. We will focus only on the real Z that is produced in association with the Higgs boson. It decays into a pair of leptons, either $ e^- e^+ $ or $ \mu^-\mu^+ $, since they are the particles with the highest detection efficiency and carry the polarization message of a Z boson. Even in lepton colliders it will be hard to tag the electric charge of jets, and we have to choose between the electron or muon as the spin analyzer at a price of reduced statistics. The kinematics of this process is illustrated in Fig. 1. Obviously, the above mentioned new physics coupling of $ HZZ $ beyond SM may make the $ e^+ e^- \to Z^* \to HZ $ cross-section different from SM. Furthermore, the complicated new physics structure in Eq. (2) may also change the polarization fraction of the Z boson, making the angular distribution of the final lepton pairs different from SM. Figure1. (color online) Kinematics of $ e^+e^- \to H Z(l^+l^-) $.
In the Standard Model, the off-shell photon can also contribute to the strahlung production via the for a given design of a lepton collider. In this regard, we do not need to worry about the constraint of the $ \gamma^* ZH $ coupling on the electric dipole moment (EDM) of electron [30, 31]. The momenta and helicities of the incoming (anti-) electron and outgoing bosons are defined as:
where $ \sigma_{1,2} = +\displaystyle\frac{1}{2}, -\displaystyle\frac{1}{2} $ and $ \lambda = -1, 0, +1 $. The invariant amplitude for the Higgs production is
$\begin{split} {\cal M}^\lambda =& {\bar v}(p_1) ( v_e I + a_e \gamma_5 ) \gamma_{\tau} u( p_2 )\\&\times P^{\tau\mu} V_{\mu\nu} ( k + q , k ) \epsilon^{\lambda,\nu} , \end{split}$
(4)
where $ P^{\tau\mu} $ is the propagator of the virtual Z boson in unitary gauge, and the polarization vector $ \epsilon^\lambda (k) $ of the real Z is
where $ \tau $ is the helicity of the spin analyzer in the Z decay, and $ d^\tau_\lambda ( \vartheta, \varphi) $ is the usual $ \displaystyle\frac{1}{2}- $representation of the rotation group. There is also a Breit-Wigner factor, but it is left out as it is an overall factor. The scatting angle $ {\hat \vartheta} $, polarization angle $ \vartheta $ and azimuthal angle $ \varphi $ are defined in Fig. 1. 22.1.Total cross-section for Higgsstrahlung
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2.1.Total cross-section for Higgsstrahlung
The differential cross-section for Higgs production in the Born approximation reads
where $ \beta = {( 1 + m^4_Z/s^2 + m^4_H/s^2 - 2m^2_Z/s - 2m^2_H/s -} $${ 2 m^2_Z m^2_H/s^2 ) }^{1/2} $, with s the center-of-mass energy squared, and $ | { p}_l | $ the momentum of the lepton which is the Z spin analyzer. The definition of spin matrix $ \rho^{ \lambda ' \lambda } $ respects the fact that no beam polarization is expected from the incoming leptons in CEPC. In the case of longitudinal beam polarizations in a linear collider like ILC, a detailed study was published in [17, 32]. After integration in the phase space, the total cross-section is:
The SM coupling $ C_{l{\bar l}} = \left( a^2_e + v^2_e \right) \left(a^2_f+ v^2_f \right) $ for the leptonic final states of Z will be defined by the experiment. Since new physics couplings are a small perturbation of the SM couplings, we only keep the leading order linear terms. It is interesting that the anomalous couplings appear as a combination:
$ g '_3 = 2 g_2 ( s + m^2_Z ) + g_3 + g_1 \sqrt{s} E_Z. $
(11)
This relation further reduces the number of free parameters to three, $ g_1 $, $ g '_3 $ and $ {\tilde g} $. We would also like to point out that this relation takes place at the level of the amplitude of ZH associated production, so it can be regarded as a new parametrization for analyzing the Higgsstrahlung channel. To isolate the $ g_2 $ contribution, one has to investigate the decay channel Higgs into Z pair, whose yield seems relatively small, and is an independent issue beyond the scope of this work. 22.2.Polarization in Z boson decays -->
2.2.Polarization in Z boson decays
Although there are only three effective couplings left, one can not distinguish their contribution by the total cross-section measurement only. Different kinds of new physics will give more information on the angular distributions of the decay products ofthe Z boson, which characterize its polarization fractions. The polar angle distribution of the outgoing lepton is derived as
The fraction of each spin polarization, characterized by the distribution of the polarization angle $ \vartheta $ , is obtained by integrating out the scatting angle $ {\hat \vartheta} $. In principle, the final lepton coupling $ a_f, v_f $ may receive extra contributions if anomalous $ Z l \bar{l} $ interaction is included. Since CEPC proposes a better option, a Z-factory run to explore this possibility, we keep in this work $ a_f, v_f $ as in SM . It is interesting to note that the fraction of transverse polarization can be increased if the integration in scattering angle is performed in a reduced interval, for example, in a forward region defined by $ | \cos {\hat \vartheta} | > \cos \displaystyle\frac{\pi}{4} $,
It is obvious that the contribution from $ \Gamma^\pm (\vartheta) $ is enhanced by a factor of 3.3 in the forward region. In the experiments, this polarization distribution, together with the total cross-section, will be used to fit the parameters $ g_1 $ and $ g '_3 $. 22.3.Azimuthal angle distribution for CP violation -->
2.3.Azimuthal angle distribution for CP violation
Up to now, all the analyses are independent of the CP violation term $ {\widetilde{g}} $ in the effective Hamiltonian of Eq. (1), which characterizes the CP violating effects in new physics beyond the Standard Model. We need to study the azimuthal angle $ \varphi $ dependence of the Z boson decays in order to study CP violation effects:
Here, the first two terms are background from SM, and CP violation appears in the third term, with the $ \sin2\varphi $ dependence of the signal against the background of $ \cos 2\varphi $. There is no $ \sin\varphi $ term in the above equation. However, it can be recovered by breaking the symmetry in decay angle $ \vartheta $ integration, $ 0\to \pi/2 $ or $ \pi/2 \to \pi $, at a price of $ \cos\varphi $ background in SM
One can see from Fig. 2 that the distribution with $ \sin\varphi $ reveals the CP violation as breaking of the height equality of the two peaks, while the $ \sin 2\varphi $ term makes a phase shift with respect to CP conserving SM background. Figure2. (color online) Differential cross-section of the Higgs production as a function of azimuthal angle $ \varphi $ , Eq.(17). The black line is for SM, while the blue line corresponds to possible new physics beyond SM.
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3.1.Limits from the total cross-section
Using the above mentioned cuts, we scanned the new physics parameters $ ( g_1 , g'_3 ) $ simultaneously. Their sensitivity limits occur when $ \Delta \sigma / \sigma \geqslant \sqrt{2}/ \sqrt{N_{evt}} $, where $ N_{evt} $ is the observed number of events (signal) in $ ZH \rightarrow l^+ l^- b {\bar b} $, as shown in Fig. 3. The new physics parameters $ ( g_1 , g'_3 ) $ inside the parallelogram will be difficult to distinguish from SM within experimental errors. It can be seen that CEPC can set limits at $ | g'_3| \leqslant 0.015 $ and $ |g_1| \leqslant 0.035 $. Figure3. New physics sensitivity limits from the total cross-section measurements. The parameters inside the parallelogram are difficult to distinguish from the Standard Model within experimental errors.
One may also set lower limits $ | g'_3| \leqslant 0.005 $ and $ |g_1| \leqslant 0.015 $ , by discarding systematical errors and relaxing event selection. On the other hand, the reconstruction of the recoil Z boson leads to an inclusive analysis with Higgs decaying to anything other than the $ b {\bar b} $ final states. In this case, a tighter limit can be set with about three times larger statistics. In such a reconstruction, it is possible to reduce the dependence on invisible Higgs decays, but the details are beyond the scope of this paper. 23.2.Limits from Z polarization -->
3.2.Limits from Z polarization
With the planned luminosity and the analysis outlined at the beginning of this section, we study the sensitivity to new physics using the polarization angle distribution, given in Eq. (12) and (15). The expected event number distribution with the polarization angle $ \vartheta $ is shown in Fig. 4, with blue points for Eq. (12) and black points for Eq. (15). The polarization angle distribution will be different from the black plot in Fig. 4 if only the forward region of the decay angle is taken into account, as in Eq. (15). Since it comes with lower statistics (only half of the number of events), it is omitted in the present analysis until there is better input for experimental systematics. Figure4. (color online) Expected event number distribution with the polarization angle $ \vartheta $, blue for Eq. (12) and black for Eq. (15).
We show the experimental limits from polarization angle distributions for new physics parameters $ g_1 $ and $ g'_3 $ in Fig. 5. The parameter region inside the blue lines is indistinguishable from the Standard Model. We also plot the limits from the cross-section study in Fig. 3. Fig. 5 shows that the two limiting regions overlap, which means that the sensitivity limits are further restricted to the meshed region. The polarization angle distribution will anyway be helpful since it will constrain new physics from a different direction than the cross-section. It is also worth to point out that in Eq. (12) the polarization angle distribution is normalized by the cross-section. This means that there is less dependence on the uncertainties from Higgs production or decay, because the actual analysis is done solely by fitting the shape. As a result, the HZZ couplings are better determined. Figure5. (color online) Limits from the polarization angle, with no sensitivity to new physics in the region delimited by the blue line. The region delimited by the black line is from Fig. 3. The overlap of two regions is the meshed region.
23.3.Limits for CP Violation parameter $ \tilde g $ -->
3.3.Limits for CP Violation parameter $ \tilde g $
Using Eq. (16), we show the expected event number distribution with the azimuthal angle $ \varphi $ in Fig. 6, where CP violation effect may appear. Without loss of generality, only one of the new physics parameters $ g_1 $ or $ g'_3 $ is scanned at a time together with the CP violation parameter $ \tilde g $. Figure6. (color online) Expected event number distribution with the azimuthal angle $ \varphi $.
Again, the forward region defined in Eq. (17) is omitted in the present analysis due to its lower statistics. After a careful study of the background, we derive the experimental limit for $ \tilde g $ in correlation with $ g'_3 $ , shown in Fig. 7. The correlation sensitivity of $ \tilde g $ and $ g_1 $ is shown in Fig. 8. These figures indicate that the experimental sensitivity limit for $ \tilde g $ is $ -0.04 \sim 0.01 $. Figure7. Experimental limits for $ \tilde g $ from the azimuthal angle distribution, correlated with $ g'_3 $ , with no sensitivity in the inner region.
Figure8. Experimental limits for $ \tilde g $ from the azimuthal angle distribution, correlated with $ g_1 $ , with no sensitivity in the inner region.