Measuring the ratio of $HWW$ and $HZZ$ couplings through $W^+W^-H$ production

For a generic Higgs boson, measuring the relative sign and magnitude of its couplings with the $W$ and $Z$ bosons is essential in determining its origin. Such a test is also indispensable for the 125-GeV Higgs boson. We propose that the ratio of the $HWW$ and $HZZ$ couplings $\lambda_{WZ}$ can be directly determined through the $W^+W^-H$ production, where $H$ denotes a generic Higgs boson, owing to the tree-level interference effect. While this is impractical at the LHC due to the limited sensitivity, it can be done at future $e^+e^-$ colliders, such as a 500-GeV ILC with the beam polarization $P(e^-,e^+)=(-0.8,+0.3)$ in the $jj\ell^{\pm}bb$ and $\ell^{\pm}\ell^{\pm}\ell^{\mp}jj$ channels. The discovery potential of a general ratio and the power to discriminate it from the SM value are studied in detail. Combining the cross section of $e^+e^-\to W^+ W^- H$ with the measurements of $HZZ$ coupling at the HL-LHC, one can further improve the sensitivity of $\lambda_{WZ}$.


I. INTRODUCTION
After the discovery of the 125-GeV Higgs boson [1,2], it is important to measure all its couplings with other particles to verify whether it is the Standard Model (SM) Higgs boson. In particular, the measurements of HV V couplings are crucial to testing exactly how electroweak symmetry breaking (EWSB) occurs. In the SM, there is a residual global SU (2) symmetry known as the custodial symmetry [3] in the Higgs and gauge sectors after the EWSB to guarantee that the electroweak ρ parameter at tree level, where θ W is the weak mixing angle, and m Z and m W are the Z and W boson masses, respectively. It is also found that the the tree-level HW W and HZZ couplings satisfy [4] g SM
For a neutral Higgs boson H that may have an origin beyond the SM, we write its couplings with the W and Z bosons as where the coefficients κ W and κ Z are the scale factors [5,6]. Note that H can generally be any neutral Higgs boson, though our following discussions will focus on the 125-GeV Higgs boson for definiteness and obvious interest. If H is the SM Higgs boson, then κ W = κ Z = 1. However, κ W,Z can have generally different values for H with origin from an extended Higgs sector. It is therefore useful to consider the ratio In Higgs-extended models, even if ρ = 1 is imposed at tree level, the ratio λ W Z can be different from 1 and even have a negative sign if at least two distinct non-trivial Higgs representations are introduced [7][8][9]. Ref. [9] has shown that λ W Z is generally arbitrary when the Higgs sector of the SM is extended with a complex triplet and a real triplet while keeping ρ = 1. It is therefore important to be able to determine the magnitude and sign of λ W Z for any neutral Higgs boson, including the 125-GeV Higgs boson.
Experimentally, it is difficult to directly determine the sign of λ W Z . Existing measurements [10,11] of the HW W and HZZ couplings, depending on the production or decay rates, are more sensitive to the magnitude rather than the sign of λ W Z . From the combined measurements in Large Hadron Collider (LHC) Run I [10], the 2σ confidence level (C.L.) interval of λ W Z is determined to be It is clear that there is a discrete two-fold ambiguity in the sign of λ W Z that cannot be resolved in such experiments.
At the High-Luminosity LHC (HL-LHC) with an integrated luminosity of 3 ab −1 , the HW W and HZZ couplings are expected to be measured with the precision [14] δκ W /κ W ≤ 5%, δκ Z /κ Z ≤ 4%, assuming that the central values are the ones predicted by the SM [15]. The corresponding 1σ uncertainty on λ W Z is then It was investigated in Ref. [8] that the magnitude and sign of λ W Z could be measured in the differential distribution of H → ZZ * → 4 due to the interference between amplitudes at tree and one-loop levels, which are proportional to the HZZ and HW W couplings, respectively. However, more diagrams should be considered at the one-loop level [16,17] to maintain gauge invariance and consistency [6]. In this work, we show that a desirable interference occurs among tree-level amplitudes in the W + W − H production process. We therefore propose to use this channel to experimentally fix λ W Z .
This paper is arranged as follows. In Section II, we discuss the current constraints on the ratio λ W Z of the HW W and HZZ couplings and the projected sensitivity at the HL-LHC. We argue why it is impractical to employ the proposed method at the LHC.
In Section III, we study λ W Z in the e + e − → W + W − H process at a future e + e − collider with the colliding energy of 500 GeV. We separately consider two kinds of final states, jj ± bb and ± ± ∓ jj , and study their reaches. Section IV summarizes our results.
FIG. 1: The representative Feynman diagrams of aā → W + W − H, where a(ā) denotes initial-state q(q) or e − (e + ). The blue and red vertices represent the interactions of HW W and HZZ, respectively. The squared amplitude of (a) and (b) with the SM HW W coupling is denoted by |M W | 2 in Eq. (7), and the squared amplitude of (c) with the SM HZZ coupling is denoted by |M Z | 2 , while the SM interference of (a) and (b) Here we have neglected the diagrams with small Yukawa couplings of the Higgs boson to light fermions.
While current measurements cannot determine the sign of λ W Z , we propose that both its sign and magnitude can be directly measured through the tree-level interference in W + W − H production. To show this, we first make a general discussion about the W + W − H production at colliders. At the parton level, the aā → W + W − H process involves three types of Feynman diagrams shown in Fig. 1, where a(ā) denotes q(q) and e − (e + ) at hadron colliders and e + e − colliders, respectively. The squared amplitude of this process can be organized as: where M W involves only the HW W coupling, M Z only the HZZ coupling, and M W Z the interference. The total cross section of the production process can then be expressed In what follows, we will only make a brief comment on the W + W − H production at the LHC because we find our proposed channel to be impractical in this case. It was claimed in Ref. [18] that the pp → W + W − H process in the SM had the best sensitivity in the ± ± ∓ jj channel and a 3.0σ sensitivity can be reached at the HL-LHC with an integrated luminosity of 3 ab −1 . However, we find that the sensitivity is actually lower when other omitted factors are taken into account. First, Ref. [18] considered the backgrounds of W W Z, jjW Z with Z → τ + τ − , where the τ lepton could further decay into e or µ, but ignored the backgrounds of W W Z, jjW Z with Z → + − ( = e, µ) under the assumption that they could be completely removed using the invariant mass cut of the same-flavor and opposite-sign (SFOS) lepton pairs. Nevertheless, such a selection cut would also reduce the signal and other background events, with a net result of lowering the signal significance as we have checked. Second, to reconstruct the W boson, the cut |m jj − m W | < 5 GeV [18] was imposed, which was the most important cut to suppress the backgrounds of jjW W W , jjW Z, ttW , etc. However, the efficiency of such a cut was somewhat overestimated [19]. Finally, jet matching was not considered in Ref. [18]. This might notably increase the backgrounds of jjW W W , jjW Z, etc. at hadron colliders.
Due to the above-mentioned low sensitivity of the pp → W + W − H process in the SM at the LHC, we reckon that it is not practical to use it to study the ratio λ W Z . This leads us to the next section where we consider probing λ W Z through the e + e − → W + W − H channel at future e + e − colliders. also be studied at other future e + e − colliders, provided the colliding energy is sufficiently high [26,27].
To study the process e + e − → W + W − H with general κ W and κ Z , we first consider the following benchmark scenarios: The corresponding total cross sections are σ BP1 = 12 fb, σ BP2 = 17.11 fb, σ BP3 = 13.54 fb.
Thus we obtain each term in Eq. (8) in this case as follows: The diagrams involving the HW W coupling dominate the one involving the HZZ coupling by about one order of magnitude, and the interference between the two types of diagrams is destructive.
Two decay channels H → bb and H → W W * are considered with the corresponding final states being jj ± bb and ± ± ∓ jj , respectively. Let us now define BP1 , BP2 and BP3 to be the efficiencies of the signals under a fixed set of cuts for the BP1, BP2, and BP3 scenarios, respectively. The signal cross section with arbitrary HW W and HZZ couplings after the selection cuts can now be written as where B = B(κ W , κ Z ) denotes the combined branching ratios of the W and H decays in the final state and depends on the coefficients κ W and κ Z , and W , Z and W Z denote 1 In Ref. [24], as an example at the ILC, the integrated luminosities of 4 ab −1 at √ s = 500 GeV is shared equally by the beam polarization choices of P (e − , e + ) = (−0.8, +0.3) and P (e − , e + ) = (+0.8, −0.3). But it is also stressed that the polarization scheme should be modified depending on future experimental results [24]. For other studies with P (e − , e + ) = (−0.8, +0.3) and the maximal integrated luminosity of 4 ab −1 at the 500-GeV ILC, see for example Ref. [25]. the cut efficiencies of the associated parts. One can then solve to obtain Therefore, through a simulation for the three benchmark scenarios, we are able to extract W , Z and W Z for general analyses.
In this work, we perform a parton-level simulation with the signal and background events generated using MG5 aMC@NLO v2.4.3 [28]. In order to mimic detector resolution effects, particle four-momenta are smeared with a Gaussian distribution. The jet energy resolution and lepton momentum resolution are approximately described by [29] 2 respectively.

A. The H → bb case
We first consider the signal process e + e − → W + W − H in the jj ± bb channel, where j denotes a light-flavored quark jet, H → bb, and one of the W bosons decays hadronically while the other decays leptonically. The main backgrounds include tt and W W Z with Z → bb, jj. The following basic cuts are imposed at the generator level: where the angular distance in the and φ i are the pseudorapidity and azimuthal angle of particle i, respectively.
2 It should be noted that although the energy of an electron can be measured in the electromagnetic calorimeter, the measurement is not statistically independent of the tracking determination of its momentum so that these two measurements cannot be combined [30]. In practice, we smear the electron momentum with track system performance in Eq. (15) as in Refs. [31,32]. We assume that the b-tagging efficiency, the rates of misidentifying the c-jet and the light-quark jet as a b-jet are 0.8, 0.08 and 0.01, respectively [33]. To reconstruct the W and H bosons, we require that the invariant mass of the light jet pair and the b-jet pair where the Higgs boson mass m H = 125 GeV here. To suppress the tt background, we further require p T > 30 GeV. The cut flow of the signals in scenarios (BP1, BP2, BP3) and backgrounds are summarized in Table I. It is observed that the m bb cut can effectively reduce both backgrounds.
Here the combined decay branching ratio where B 0 = 16.4% is the corresponding value for the SM Higgs boson [36] and the total width of H is 4 with the SM decay branching ratios and total width of the Higgs boson given by B 0 HW W = 0.214, B 0 HZZ = 2.62% and Γ 0 H = 4.07 MeV [36]. We have assumed that the couplings of H to the SM fermions have the same values as in the SM. 3 The asymmetric cut on the invariant mass of bb is required to take into account the effects of photons from initial-state radiation (ISR) [34] or unmeasured neutrinos from semi-leptonic b-decays [35]. 4 The modification of the partial width H → γγ from κ W is much smaller than that of the partial width H → W + W − and thus neglected in Eq. (19).
The cut efficiencies are the ratios of the cross sections in the last column in Table I to the corresponding cross sections with no cut. For the three benchmark scenarios, they are found to be BP1 = 0.296 BP2 = 0.308 BP3 = 0.300.
Therefore, we obtain the cut efficiency for each of the contributions in Eq. (12) as We then evaluate the discovery significance using [37] S D = 2 (n s + n b )ln where the number of signal and background events after selection cuts are with L being the integrated luminosity, and σ tt , σ W W Z the corresponding cross sections of the backgrounds of tt, W W Z with Z → bb, jj after the selection cuts. we define the test ratio as [37] 5 , where the likelihood function L(λ W Z ) is defined as with the Poisson distribution function given by We can calculate the p-value of the non-SM hypothesis using the likelihood ratio in Eq. (24) by assuming that the actual observation is taken to be the median of the Q distribution under the SM hypothesis [37], i.e., The non-SM hypothesis λ W Z = λ 0 is then rejected at the Z-sigma level with Z = Φ(1 − p) −1 , where Φ is the cumulative distribution of the standard Gaussian [37]. Explicitly, In Fig. 4, we show the power to discriminate λ W Z from its SM value. The left plot shows that the (BP2, BP3) scenarios can be rejected from the SM (BP1) at 95% C.L. (i.e., Z ≥ 1.64) when the integrated luminosity goes above 290 fb −1 and 2.2 ab −1 , respectively.
The right plot, on the other hand, shows the region in the |κ W |-λ W Z plane that satisfies Z ≤ 1.64 with an integrated luminosity of 4 ab −1 . We thus find that if |κ W | 0.9 as determined by other measurements, a good portion of negative λ W Z region can be completely excluded at 95% C.L. through the e + e − → W + W − H process in the jj ± bb channel.
As the central value of observed e + e − → W + W − H cross section may be different from the SM value, we consider three different cases for measured central value:  (30) in the jj ± bb channel with an integrated luminosity of 4 ab −1 . We note that for |κ W | = 1, the predicted signal cross section cannot be lower than about 0.97σ SM S . Fig. 5 shows allowed parameter regions using the above assumed experimental results. Depending on the measurement outcome and the information of |κ W | and/or |κ Z | from other experiments, one can readily fix the value, particularly the sign, of λ W Z . For example, if |κ W | is determined to be around 1 and σ central S = σ SM S , then a positive λ W Z is favored. Since the signal cross section depends on κ W (κ Z ) and λ W Z , we can obtain the relative error on λ W Z if that on κ W (κ Z ) is known. As discussed in Section II, the projected 1σ relative errors on κ W and κ Z at the HL-LHC are δκ W /κ W < 0.05 and δκ Z /κ Z < 0.04. 6 The definition of 1σ statistical error using the log likelihood ratio can be found in Ref. [41].

Moreover,
where δσ S is equal to ∆σ SM S , which is defined in Eq. (30). Therefore, we obtain δλ W Z /λ W Z < 0.063 for the SM scenario (BP1) with the combination of the e + e − → W + W − H process and the |κ Z | measurement at the HL-LHC, which is even better than that with the combination of |κ W | and |κ Z | measurements at the HL-LHC as shown in Eq. (6). Alternatively, we would have δλ W Z /λ W Z < 1.15 if the |κ W | measurement is used instead. This is because ∂σ S /∂λ W Z with σ S in terms of |κ Z | and λ W Z is larger than that in terms of |κ W | and λ W Z .
In this subsection, we consider the signal process e + e − → W + W − H in the ± ± ∓ jj channel. For the SM Higgs boson, the sensitivity of this channel is not competitive with that of the jj ± bb channel. However, for an exotic Higgs boson with mass close to the SM Higgs boson and not coupling to the SM fermions at tree level, such as the fiveplet H 0 5 in the Georgi-Machacek (GM) model [42][43][44][45][46], this channel provides a unique signature. Main backgrounds here include 4W , W W Z with Z → + − , τ + τ − , where the τ lepton can further decay into e or µ. We impose the same basic cuts as those in Eq. (16). In order to reconstruct the W boson and suppress the background with Z boson decays, we require that [19,34] The cut flow of the signals in scenarios (BP1, BP2, BP3) and backgrounds are summarized in Table II. The m SFOS + − cut is seen to be effective in reducing the dominant background of W W Z( + − ). Suppose the total width of H is denoted by Γ H and, again, the SM Higgs boson width is denoted by Γ 0 H . The combined decay branching ratio in either scheme is given by: similar to that in the H → bb decay channel. The SM value of the combined branching ratio is B 0 = 0.578% [36]. The total width of H 7 Through simulations, the cut efficiencies of the benchmark scenarios are found to be Hence, we obtain the cut efficiencies of each contribution in Eq. (12) as Similar to the jj ± bb channel, the ± ± ∓ jj channel is also more sensitive to negative λ W Z as compared to positive λ W Z . Again, the λ W Z 0.46 region has a better sensitivity, and the asymmetric shape in the contours enables us to solve the ±λ W Z ambiguity.
However, since the discovery potential of the ± ± ∓ jj is partially determined by λ W Z as In Fig. 7, we show the ability to discriminate λ W Z . The left plot shows that the region in the plotted area can be completely excluded at 95% C.L. through the e + e − → W + W − H process in the ± ± ∓ jj channel. Fig. 8 plots the 1σ region allowed by the assumed measurements given in Eq. (29). As compared to the jj ± bb channel, the allowed regions in the ± ± ∓ jj channel are slightly more symmetric with respect to λ W Z = 0 and thus less sensitive to the sign of λ W Z .
We can also obtain the relative error on λ W Z by combining the e + e − → W + W − H process in the ± ± ∓ jj channel and the measurement of |κ Z | at the HL-LHC, which gives δλ W Z /λ W Z < 0.071.

Scheme B
For scheme B where H is fermiophobic, the ± ± ∓ jj channel provides a unique signature. Such a scheme happens to the fiveplet Higgs boson H 0 5 in the GM model [42][43][44][45][46], where λ W Z = −1/2. For this, we introduce two more benchmark scenarios: For concreteness, we keep m H = 125 GeV in the following numerical analysis, noting that it is conceptually analogous to apply the method on an exotic Higgs boson of a different mass. Fig. 9 shows the discovery potential of the e + e − → W + W − H process in the ± ± ∓ jj channel for scheme B with five benchmark scenarios of the couplings (BP1, BP2, BP3, GM1, GM2) assuming that H is fermiophobic. Compared to scheme A, the sensitivity is notably improved due to an increase in the branching ratio of H → W + W − . According to the left plot, λ W Z = −1/2 can be confirmed with an integrated luminosity of about 90 fb −1 and 870 fb −1 for GM1 and GM2, respectively. Note that the sensitivities of BP2 and GM1 are close to each other since for |κ W | = 1 a change of λ W Z from −1 to −1/2 can increase the production cross section and reduce the decay branching ratio at the same time. The right plot shows that the e + e − → W + W − H process can be discovered with an integrated luminosity of 4 ab −1 provided |κ W | 0.35.
For this scheme, we are mostly interested in discriminating λ W Z = −1/2 from the SM value of λ W Z = 1. In Fig. 10, we show the discriminating power for the two GM scenarios from the SM Higgs boson. From the left plot, we find that H in the GM1 and GM2 scenarios can be distinguished from the SM case with an integrated luminosity of about 30 fb −1 and 1.14 ab −1 , respectively. As a comparison, it is claimed that the λ W Z = 1 and −1/2 cases can be discriminated at 95% C.L. at the LHC through H → ZZ * → 4 with an integrated luminosity of 2 ab −1 [8]. The right plot shows the Z ≤ 1.64σ region within which the λ W Z = −1/2 case cannot be excluded at 95% C.L. Left: discriminating the GM1 and GM2 scenarios from the SM. Right: the assumption of λ W Z = −1/2 can be distinguished from the SM assumption of λ W Z = 1 at 95% C.L.
for a varying |κ W | out of the blue region.

IV. SUMMARY AND CONCLUSIONS
In examining properties of a Higgs boson, the discovered 125-GeV SM-like and any new Higgs bosons alike, it is important to determine separately its couplings to the W and Z bosons, including their relative sign and the magnitudes of these couplings. In this work, we have proposed to use the W + W − H production process to determine the sign and magnitude of the ratio of the HW W and HZZ couplings at future colliders. This process includes Feynman diagrams involving both couplings at tree level. We found that the sensitivity of this process at the LHC is low. We therefore focused our study on the e + e − → W + W − H process at a future e + e − collider. For concreteness, we took the 125-GeV Higgs boson as an explicitly example. In this case, we have found that a 500-GeV e + e − collider with appropriately polarized beams is suitable for such an analysis.
We have considered two decay channels of the Higgs boson and thus divided our discussions according to two kinds of final states jj ± bb and ± ± ∓ jj . In the latter case, we further examined the cases when H is SM-like or fermiophobic in its couplings to the SM fermions. We analyzed a few benchmark coupling scenarios in terms of the scale factors κ W and κ Z . For all the scenarios considered, we obtained the discovery potential for determining the ratio of HW W and HZZ couplings, λ W Z . The results were shown in the plane of |κ W | and λ W Z . Due to the destructive interference between the diagrams involving the HW W coupling and that involving the HZZ coupling, we found that the e + e − → W + W − H process had different sensitivities for different values of λ W Z .
In particular, the negative λ W Z scenarios generally require less luminosity to reach a 5σ discovery.
To discriminate a particular λ W Z scenario from the SM case, we further made use of the log likelihood ratio. By evaluating such a quantity, we have obtained the required integrated luminosities to discriminate two theory assumptions at 95% C.L. as well as the region in the plane of |κ W | and λ W Z that satisfied Z ≤ 1.64 with an integrated luminosity of 4 ab −1 . We have also investigated the impact of three possible signal cross section outcomes and plotted the 1σ region in the |κ W |-λ W Z plane. It was found that for a smaller (larger) cross section, the e + e − → W + W − H process is less (more) sensitive to the sign of λ W Z . By combining the cross section measurement of the proposed process and the measurement of |κ Z | at the HL-LHC, it will be straightforward to determine λ W Z at a high precision, as compared to purely from the measurements of |κ W | and |κ Z | at the HL-LHC.
Finally, we would like to emphasize that our method of determining λ W Z for the 125-GeV Higgs boson can be readily applied to another new Higgs boson that couples to both W and Z bosons. We also note that a similar tree-level interference effect in the pp → jjW ± H and pp → jjZH processes can be used to determine λ W Z , the result of which will be given in a separate work [47].