Exploration of the Tensor Structure of the Higgs Boson Coupling to Weak Bosons in $e^+e^-$ Collisions

Probing signatures of anomalous interactions of the Higgs boson with pairs of weak vector bosons is an important goal of an $e^+e^-$ collider commissioned as a Higgs factory. We perform a detailed analysis of such potential of a collider operating at $250 - 300$ GeV. Mostly using higher dimensional operators in a gauge-invariant framework, we show that substantial information on anomalous couplings can be extracted from the total rates of $s$-and $t$-channel Higgs production. The most obvious kinematic distributions, based on angular dependence of matrix elements, are relatively less sensitive with moderate coefficients of anomalous couplings, unless one goes to higher centre-of-mass energies. Some important quantities to use here, apart from the total event rates, are the ratios of event rates at different energies, ratios of $s$ and $t$-channel rates at fixed energies, and under some fortunate circumstances, the correlated changes in the rates for $W$-boson pair-production. A general scheme of calculating rates with as many as four gauge-invariant operators is also outlined. At the end, we perform a likelihood analysis using phenomenological parametrization of anomalous $HWW$ interaction, and indicate their distinguishability for illustrative values of the strength of such interactions.


I. INTRODUCTION
Physicists are widely convinced now that they have discovered what closely resembles the Higgs boson [1,2] postulated in the standard electroweak model (SM) [3][4][5][6][7][8][9][10][11]. Along with widespread exhilaration, such a development brings in questions on whether this particle carries some signature of physics beyond the standard model. Many studies in this direction have appeared  in the context of the Large Hadron Collider (LHC) where the data available so far still allow some departure from SM behaviour. Even a finite invisible branching ratio (BR) for the Higgs cannot, at the moment, be ruled out [57,58]. The issue can be probed through careful measurements of the couplings of the Higgs (or Higgs-like scalar) to various pairs of SM particles. Among them, the couplings to pairs of vector bosons (HV V ) are measured in a relatively more reliable manner. This possibility has been explained in the context of an ep collider too [59,60].
In view of the cumulative demand for a closer probe on the HV V couplings (and of course the couplings to other SM particles), the most desirable endeavour, however, is to build an electron-positron collider which provides a clean environment for precise measurements of Higgs interaction strengths. The first step is of course to develop a Higgs factory (at √ s ≈ 250 -300 GeV). Such a machine will not only produce the Higgs boson copiously near resonance, but is also the first step before an e + e − machine at even higher energies is developed. In this paper, we incorporate some observations regarding the signatures of anomalous HV V couplings, manifest through higher dimensional operators (HDOs), at a Higgs factory. Other studies performed for an e + e − machine can be found in [61].
If the couplings arise through physics at a scale higher than that of electroweak symmetry breaking, then the resulting higher-dimensional effective interactions are expected to be gauge invariant. Such interactions have not only been identified, but constraints on their coefficients have also been obtained from the LHC data [52,[62][63][64][65][66]. In view of such analyses, the coefficients are often restricted to such values where many cherished kinematic distributions may fail to reveal their footprints. In the current study, we point out some features which influence the detectability (or otherwise) of the higher-dimensional couplings at a Higgs factory. At the same time, we emphasise some possible measurements that can elicit their signatures even for relatively small coefficients of such operators.
We concentrate on two Higgs production channels, namely, e + e − −→ ZH (the s-channel process) and e + e − −→ ννH (the t-channel process, which we separate with the help of a simple kinematic cut around the Higgs boson energy). In principle, the HDOs that will constitute our report can influence the rates in both channels. In contrast, the most obvious kinematic distributions, namely, those based on the angular dependence of matrix elements, drawn with moderate values of their coefficients do not show a perceptible difference with respect to the SM situation. Keeping this in view, we underscore the following points here: 1. The s-channel process has substantial rates at ≤ 300 GeV or thereabout. We show, through an analysis of the production amplitude squared, why one cannot expect significantly different angular distributions in this channel at such energies, if one uses moderate values of the operator coefficients.
2. The t-channel process can have appreciable production rates at high energies (≈ a TeV), too. Because of the production of two neutrinos in the final state, this process provides limited phase-space for the exploration of the tensor structure of the HW W coupling. Here it is attempted to exploit the full kinematics of the Higgs boson by means of a correlated two-dimensional likelihood analysis. 3. We show that, given such impediment, it is possible to uncover signatures of the aforementioned BSM operators through measurements of rates at two different energies, which also cancels many systematic uncertainties. In general, the energy dependence of the rates can be sensitive to anomalous couplings. 4. The very fact that the additional operators should be electroweak gauge invariant imply not only higher-dimensional HV V interactions (V = W , Z , γ) but also anomalous W W V interactions (V = Z, γ) whose strengths are related to the former. We show that the concomitant variations in Higgs production and W-pair production at Higgs factories may elicit the presence of such BSM interactions. 5. We also show that if the centre-of-mass energy (CME) of the colliding particles is ≈ 500 GeV or more, then even moderate values of the operator coefficients can show some differences in the kinematic distributions. 6. Lastly, we perform the analysis in a framework that allows one to retain all the gaugeinvariant operators at the same time.
We summarise the gauge invariant couplings in the next section, and subsequently point out the 'phenomenological' anomalous couplings they lead to. In section III, we take up the s and t-channel Higgs production cross-sections in turn, and explain why one cannot expect too much out of kinematic distributions at Higgs factory energies, so long as the BSM coupling coefficients are subject to constraints imposed by the LHC data. Their detectable signatures through event ratios at two energies, and also via the simultaneous measurement of W -pair production are predicted in section III. A likelihood analysis and some related issues, mostly in terms of the phenomenological forms to which all new couplings reduce, are found in section IV. We summarise our conclusions in section V.

II. EFFECTIVE LAGRANGIAN FORMALISM
In this paper, we adopt two types of effective Lagrangian parametrizations which are commonly used in the literature to probe the anomalous HV V (where V = W, Z, γ) interactions.
In one parametrization, we take the most general set of dimension-6 gauge invariant operators which give rise to such anomalous HV V interactions. In the other one, we parametrize the HV V vertices with the most general Lorentz invariant structure. Although, this formalism is not the most transparent one from the viewpoint of the gauge structure of the theory, it is rather simple and more experiment-friendly. Both formalisms modify the HV V vertices by introducing non-standard momentum-dependent terms.
We assume that the SM is a low-energy effective theory of a more complete perturbation theory valid below a cut-off scale Λ. In the present study, we are concerned mainly with the Higgs sector. The first order corrections to the Higgs sector will come from gauge invariant dimension 6 operators as there is only one dimension-5 operator which contributes to the neutrino masses. The relevant additional Lorentz structures in HV V interactions are necessarily of dimensions higher than four. If they arise as a consequence of integrating out physics at a higher scale, all such operators will have to be invariant under SU (2) L × U (1) Y .
A general classification of such operators is found in the literature [67][68][69][70]. The lowest order CP-conserving operators which are relevant for Higgs phenomenology are • The operators containing the Higgs doublet Φ and its derivatives: • The operators containing the Higgs doublet Φ (or its derivatives) and bosonic field strengths : whereŴ µν = i g 2 σ a W a µν andB µν = i g 2 B µν and g, g are respectively the SU (2) L and The Higgs doublet is denoted by Φ and its covariant derivative is given as Following are the properties of the aforementioned HDOs: • O Φ,1 : Does not preserve custodial symmetry and is therefore severely constrained by the T -parameter (or equivalently the ρ parameter). It modifies the SM HZZ and HW W couplings by unequal multiplicative factors.
• O Φ,2 : Preserves custodial symmetry and modifies the SM HZZ and HW W couplings by multiplicative factors. This operator modifies the Higgs self-interaction as well.
• O GG : Introduces HGG coupling which is same in structure as the SM effective HGG coupling. Since our discussion is limited to the context of an e + e − collider and as we will also not consider the gluonic decay mode of the Higgs, we will not discuss this operator any further.
• O BW : Drives the tree-level Z ↔ γ mixing and is therefore highly constrained by the electroweak precision test (EWPT) data [62].
Modifies the HV V couplings by introducing new Lorentz structure in the Lagrangian. They are not severely constrained by the EWPT data [63,64].
Hence for the Higgs sector, we will choose our basis as In the presence of the above operators, the Lagrangian is parametrised as where κ is the scale factor of the SM-like coupling, something which needs to be accounted for when considering BSM physics. f i is a dimensionless coefficient which denotes the strength of the i th operator and Λ is the cut-off scale above which new physics must appear. We keep κ to be the same for the HW W and HZZ couplings so that there is no unacceptable contribution to the ρ-parameter. Another operator considered in this work is . This only affects the triple gauge boson couplings and does not affect the Higgs sector.
The effective Lagrangian which affects the Higgs sector is HW W HW + µν W −µν HZZ HZ µν Z µν HZγ HA µν Z µν + g Hγγ HA µν A µν , where g (1) where g W W γ = g s, g W W Z = g c, κ V = 1 + ∆κ V and g Z 1 = 1 + ∆g Z 1 with The limits on these operators have been derived in many references. The most comprehensive of these are listed in references [52,[62][63][64][65]. These operators, even within their current limits, have been shown to modify the efficiencies of the various selection cuts for the relevant final states in the context of the LHC [66].
All of the aforementioned HDOs lead essentially to one effective coupling (each for HW W and HZZ), when CP -violation is neglected. These can be alternatively used in a phenomenological way for example, the H(k)W + µ (p)W − ν (q) vertex can be parametrised as [71]: where deviations from the SM form of Γ µν SM (p, q) = −gM W g µν would indicate the presence of BSM physics. These BSM deviations, including CP -violating ones (not considered among the gauge invariant operators), can be specified as where λ and λ are the effective strengths for the anomalous CP-conserving and CP-violating operators respectively.
Precise identification of the non-vanishing nature of λ, λ is a challenging task. If ever  We concentrate on two main Higgs production mechanisms viz. e + e − → ZH and e + e − → ννH, at an e + e − collider with energies ranging from 250 GeV to 500 GeV. The e + e − → ZH channel includes only the s-channel processes -e + e − → Z * /γ * → ZH (shown in Fig. 1(a)).
The s and t-channel processes have different kinematics and hence are affected differently by the inclusion of the HDOs. Moreover, the t-channel process allows us to explore the tensor structure of the HW W vertex alone, free from any contamination from the HZZ and HZγ vertices. On the other hand, the s-channel process is free from any contamination due to the HW W vertex. Hence, the measurement of the s-channel contribution will shed light on the tensorial nature of the HZZ and HZγ vertices. We, therefore, analyse the s and t-channel processes separately to shed more light on the anomalous behaviour of the HV V vertices. We separate the s-channel (t-channel) contribution from the e + e − → ννH events by applying a simple kinematic cut on the Higgs energy (E H ) as follows: where √ s is the CME of the two colliding e + e − beams and ∆ is an energy-window around Here, E c H -cut is complementary to the E H -cut. We use ∆ = 5 GeV throughout our analysis 1 . We must mention here that for the rest of this paper the s-channel process will be studied at the ZH level without any cuts, unless otherwise specified. One can easily get an estimate of the cross-section for any decay modes of Z by multiplying the appropriate BR. This is because for the e + e − → l + l − H channel, a simple invariant mass cut on the two leptons about the Z boson mass will separate the s-channel to a very high degree. For e + e − → ννH, on the other hand, the cut on E H separates the s and t-channels. The schannel contribution surviving the cut is found to be very close to what one would have found from the rate for l + l − H, through a scaling of BRs. One is thus confident that the E H -cut is effective in minimising mutual contamination of the s and t-channel contributions.
It should also be mentioned here that the effects of beam energy spread are not taken into account in Eq. 10 for simplification. While we present the basic ideas of distinguishing anomalous interactions of the Higgs, the relevant energy window for precision studies has to factor in the effects of bremsstrahlung as well as beamstrahlung (depending on whether the Higgs factory is a circular or a linear collider).
In Table I We also present the s (σ s,ac ννH ) and t-channel (σ t,ac ννH ) crosssections separated from the ννH events after applying the cut defined in Eq. 10. The superscript ac means after cut.
channel contribution but the E c H -cut cuts out a small portion around E H from the t-channel contribution. Therefore, the s-channel cross-sections after this cut increase slightly from their without-cut values due to this small t-channel contamination. On the other hand, the t-channel cross-sections after cut decrease slightly from their without-cut values. We also estimate the interference between the s and t-channel diagrams and present the numbers in Table I. Interference contribution is expected to be tiny in the √ s region sufficiently away from the s-channel threshold energy (M H + M Z ) ≈ 226 GeV. We find that the interference contribution is only ∼ 3.5% of the total cross-section for √ s = 300 GeV, in the SM. This reaffirms the statement at the end of the previous paragraph. We also note that the inclusion of HDOs with moderate values of coefficients does not affect this contribution much. Hence, by neglecting the interference term, we approximate the total ννH cross-section as where σ ZH is the s-channel cross section and BR Z→νν is the invisible branching fraction (≈ 20%) of the Z-boson. and also the s and t channels separately. In an inset plot we show the distribution due to this . The red, green, blue histograms are for the total (s + t + interf erence), s and t channels respectively. The inset (orange) plot shows the interference (total − s − t) contribution.
interference. This clearly shows that it is negligible when compared to the s and t channel contributions. This nature generally holds for the parameter space under consideration.

B. A general expression for the cross-sections
In this analysis, we keep κ, f W W /TeV 2 , f W /TeV 2 , f BB /TeV 2 and f B /TeV 2 as free parameters. The HW W vertex depends on three parameters (κ, f W W and f W ) whereas the HZZ and the HZγ vertices depend on five parameters (κ, The κ dependence enters the HZγ vertex through the W -loop in the effective HZγ vertex. The amplitude for the process e + e − → ZH/ννH is a linear combination of at a CME of √ s. Hence, the cross-section can be written in the following closed form where The matrices of coefficients for the e + e − → ZH process at √ s = 250 GeV and 300 GeV Similar matrices for the t-channel process (after the E c H -cut) for the channel e + e − → ννH at √ s = 250 GeV and 300 GeV are We must mention here that the matrices in Eq. 14 are three-dimensional compared to the five-dimensional matrices in Eq. 13 because the t-channel only involves the HW W vertex which is not affected by the operators O BB and O B (Eqs. 4,5). We also observe that in Eq. 13, the coefficients of the matrix related to either f BB or f B are much less pronounced compared to the coefficients involving the other three parameters, viz. κ, f W W and f W . Also from Eq. 14 we see that barring the (1,1) entry in the matrices, all the other coefficients are small implying that the HDOs will have small but non-negligible effects on the t-channel cross-sections for energies at the Higgs factories.
An explanation of relatively less dependence of the t-channel cross-section compared to the s-channel on the anomalous operators can also be understood from Fig. 3. The plots reveal that, for the former process (essentially a vector boson fusion channel), the Higgs emerges with much smaller energy. The higher-dimensional couplings, on the other hand, contain derivatives which translate into a direct dependence on the energy of the Higgs, thus putting the t-channel process at a relative disadvantage. The Higgs energy distribution shows a longer tail for higher centre-of-mass energies, thus offering a partial recompense to the t-channel process for an energy as high as a TeV.
In this study we also consider the process e + e − → W + W − which involves the triple-gauge As we can see above, all the C ij s are very small when compared to C 11 , which gives us the SM cross-section. We will discuss this channel in more details later in this paper.

C. Energy dependence of s and t-channel cross-sections
It is well-known that in SM, the cross-section for the s-channel falls with the CME as 1/S and that for the t-channel, rises as nS [77]. However, for sets of values of our parameters, different from the SM, the nature of the s-channel curve can be completely different from its SM-counterpart. The t-channel cross-section however is not affected so significantly on the introduction of HDOs as has been discussed in detail in the previous sub-section. We show the variation of the s and t-channel processes for √ s ranging from 250 GeV to 900 GeV. In contrast to the SM nature of a fall in the s-channel cross-section with energy, the introduction of HDOs does in no way ensure such a nature which can be seen in Fig.4  superscript ac denotes the after cut scenario. 2 The visible rise with √ s (in Fig.4(a)

D. More information from the total rates
The total rates and their ratios at different CMEs can be important probes to identify the tensor structure of the HV V couplings. We show how the total rates for the s and t-channel processes are affected on the introduction of the effective operators ( Eqs. 13 and 14).
We must make a statement about the values of the coefficients, f i /Λ 2 (i is the index of the operator under consideration) chosen in the rest of the paper. In most cases, f i /Λ 2 is allowed to vary in the range [−20, 20] TeV −2 . Now, a reasonable criterion for the validity of the effective field theory [78] for the operators under study and E is the scale of the process. For the production case, it is the centre of mass energy of the e + e − colliding beams, which is 250 − 300 GeV, while for decays, it is the mass of the Higgs boson. For the production case, we perform a rough calculation taking g ≈ 0.65, g ≈ 0.74 and the cut-off scale Λ = 1 TeV. Hence, for the   3.5 It should however be mentioned that the actual presence of anomalous couplings in e + e − → W + W − is best reflected in a detailed study of various kinematic regions [79].
Such a study, however is not the subject of the present paper.
The main conclusion emerging from Figs. 5, 6 and 7 are as follows : • In Figs. 5(a) and 6(a), for the process e + e − → ZH, we find that the operator O W W changes the cross section from its SM expectation by ∼ 30% even in the range −5 < f W W < 5. The major contribution to the cross section modification comes from the    • We see that in Fig. 7, the cross-sections do not vary significantly with the operator coefficients. This is because the e + e − → W + W − channel has a strong ν e mediated t-channel contribution which does not involve the triple-gauge boson vertex. This has a significant interference with the s-channel. In order to bring out the feature of the triple gauge boson vertices, we need to devise some strategy which will tame down the t-channel effect, such as using right-polarised electrons if one uses a linear collider.

Two parameters at the same time
In Figs. 8 and 9, we show some fixed cross-section contours in the planes of two parameters varied at the same time. In Figs.8 and 9, all the parameters apart from the ones shown in the axes, are kept fixed. In each of these figures, we have marked regions in brown where the cross-section is σ(SM ) ± 10% × σ(SM ). Hence, we see that for each of these plots, some regions even with large values of the parameters can closely mimic the SM cross-section. The above statement for the ranges of the coefficients of the HDOs will be somewhat modified if we consider the Higgs decays. This is because then we will have branching ratios depending on the effects of the HDOs. Even for fermionic decays of the Higgs, which are independent of the operators under study, the BR will have non-trivial effects on the operator couplings through the total decay width. But, we must mention here that unless we go to very high values of the operator coefficients, the total decay width remains close to the SM expectation and hence fermionic decay channels would show similar features as these plots. Of course, when we study the effects of all the operators in the basis that we have considered by considering every possible decay mode of the Higgs, then the higher-dimensional operators will come to play at the HV V decay vertices also. Hence, we will get modified bounds on the operator coefficients from a similar approach. We should mention that these operators are also constrained by the electroweak precision observables, v iz. S, T and U parameters.
An important observation which is carried forward from Fig. 5 (a) is that the HZZ and

All parameters at the same time
The most general case will be to vary all the parameters simultaneously to obtain the most realistic parameter space. Here, we demonstrate this scenario for the cut-applied t-channel cross section in the e + e − → ννH channel. In Figs.10 (a), (b) and (c) we present three slices of the 3-dimensional hyper-surface. For each of these plots, there is a third parameter which has been varied. We see that a very large parameter space is allowed which can mimic the SM cross section within its 10% value. Of course these plots are for illustrative purposes only. In Fig. 10 (d), we have shown one such slice of the five-dimensional hyper-surface in the space of (κ, f W W , f W , f BB and f B ) for the s-channel process. processes for √ s = 300 GeV and 500 GeV respectively. We find that the angular dependence for the s-channel is very sensitive in some regions of the parameter space allowed by the EWPT constraints and the LHC data. We also find the cos θ dependence can be completely opposite as we increase the CME. This can be seen in Figs. 11(a) and 11(b), if we compare the curves for BP1. In contrast, the t-channel is not significantly affected by the inclusion of HDOs. The angular dependence of the differential cross-sections can be expressed as It is found that, between coefficients a and b above, a is more affected by the anomalous couplings rather than b, unless √ s is 500 GeV or well above that. As a result, angular distributions are insensitive to the new interactions at the proposed energy scale of a Higgs factory.
In Figs. 11(e) and 11(f), we show the normalised dσ/dp T,h and dσ/dy h distributions respectively for the t-channel where p T,h is the transverse momentum of the Higgs and y h is its rapidity. We want to emphasise that it is very difficult to see any significant differences in the various kinematic distributions in most of the parameter space allowed by the LHC and EWPT constraints while performing experiments with smaller CME. In both the channels, we do not consider the final decay products of the Higgs. If we consider the Higgs boson decaying to fermionic final states, then the HDOs under consideration will not affect these decay vertices and the above normalised distributions will remain intact. However, if we consider the bosonic decay modes of the Higgs, then the HDOs will affect these distributions non-trivially.
We end this subsection with the following admission. Various kinematical distributions are canonically emphasized as the best places to find the signature of non-standard Lorentz structures in interaction terms. While this expectation is not completely belied in the present case as well, we note that the anomalous couplings are reflected in distributions at relatively high CMEs. The reason behind this has already been explained above. While this prospect is encouraging, electron-positron colliders, especially those designed as Higgs factories, are likely to start operating at energies as low as 250 − 300 GeV. Our observation is that the imprint of anomalous couplings can be found even at such low energies at the level of total rates and their ratios. A detailed study involving all possible decay products and their various correlations can in principle go further in revealing traces of anomalous couplings.
We will take up such a study in a subsequent work.

F. Discussion on relevant backgrounds
We wish to see the effects of anomalous HV V couplings on the Higgs production alone.
Therefore, we do not look at bosonic decay modes of Higgs and limit our discussion only to Finally we estimate two of the aforementioned backgrounds by applying the cuts below: • Non-Higgs e + e − → bb We demand the two b's to fall within the Higgs-mass window and the two 's to fall within the Z-mass window as follows: Finally the total background cross-section for the bb final state is defined as, B bb = η 2 b σ bb where η b is the b-tagging efficiency which we take as 0.6 for our analysis. The signal is also scaled by the same factor, η 2 b .
• Non-Higgs e + e − → bb + E We demand the two b's to fall within the Higgs-mass window, |M (bb) − M h | < 10 GeV. Here the background is The signal 3 has also been scaled by    In Table II, we show the cross-sections for both the signal and background scenarios. For the signal we have considered two benchmark points, viz. SM and BP1 (x i ∈ {1, −3, 8, −4, 3})). We show the cross-sections once after applying just the trigger cuts (designated with the subscript tc) and next by applying the channel-specific selection cuts (written with a subscript ac) along with the basic trigger cuts. All the numbers have been multiplied by η 2 b . We see that the effects of the invariant mass selection cuts on the signal cross-sections are negligible whereas these are very effective in reducing the backgrounds almost completely.
The study performed here is at parton level. Shower, hadronization and detector effects are expected to have an impact on the effective cross-sections reported in Table II. That said, these effects will not change the conclusions of the paper.

IV. LIKELIHOOD ANALYSIS FOR t-CHANNEL
The kinematics of the final state associated to the s-channel production has been studied extensively in the past. As pointed out in section I, the t-channel production provides The primary intent of this section is to shed light on the relative improvement of this twodimensional approach, rather than determining absolute sensitivity to the size of anomalous couplings. The latter requires a detailed study that carefully incorporates experimental effects. This is beyond the scope of this paper.
We use a test-statistic (TS) to distinguish the BSM hypothesis from its SM counterpart by defining the logarithm of a profile likelihood ratio (q ij = ln λ ij ) for two different hypotheses i and j defined as where λ ij is the ratio of two likelihood functions L(P i |D i ) and L(P j |D i ) describing two different hypotheses 4 , D i is the data set used and P i,j are the probability density functions.
Due to the discrete nature of the probabilities in this analysis, the likelihood functions are defined as products of binned Poisson probabilities over all channels and bins [1]. From the TS, a p-value can be calculated to quantify the extent to which a hypothesis can be rejected.
In general, a p-value is a portion of the area under a normalised TS which, after calculation, is the percentage confidence level (CL) by which a hypothesis can be rejected.
In Monte Carlo (MC) studies, these TSs emerge as binned peaks which show up on running pseudo-experiments, each of which returns a value for the TS based on a randomly generated set of pseudo-data. The number of pseudo-data points generated is fixed by the cross-section of the process being studied. The TSs concerned in this analysis are always produced in pairs, in order to discriminate between the SM and BSM hypotheses. This pair of TSs is represented as The q U TS tends to have a more positive value due to its ordering, and we refer to it as the upper TS for our purposes, while we refer to q L as the lower TS. A hypothesis can be rejected by calculating the associated p-value as follows where m q U is the median of the upper TS, q U . The confidence by which a hypothesis can be rejected, can alternatively be quantified by knowing the significance of the separation between the two TSs. The median-significance, Z med , is defined as the number of standard deviations between the median of q L and the left edge of the p-value area, that is, the median of q U . As stated above, we focus on the t-channel process (in e + e − → ννH) which has not been studied as extensively as the s-channel. The s-channel (t-channel) contributions can be separated out from the ννH events by applying the E H -cut (E c H -cut) in Eq. 10. For this purpose, we work with the phenomenological parametrization of anomalous HW W interaction characterised by λ and λ , as defined in Eq. 9.
In our analysis, the vertices for the Lagrangians in the SM and in BSM with spin-0 bosons are calculated in FeynRules [72] and passed to the event-generator MadGraph [74], which is used for the generation of the matrix elements for Higgs production in the t-and s-channels. MC samples are produced at parton level. Effects related to detector resolution are taken into account when defining requirements to suppress the contamination from the s-channel process (see Eq. 10).
We set the stage for the likelihood analysis by showing some plots for distributions in terms of λ and λ . In Figs. 13(a) and (b), we show the p H (Higgs momentum) and θ H (the angle of the Higgs with the beam-axis) distributions respectively for the t-channel at √ s = 250 GeV. We see that significant deviations from the SM can be seen. This is in contrast to what was shown for the gauge invariant formulation (in Fig. 11) because there we stick to moderate values of the parameter coefficients, whereas for example, here,  The lower plots display the corresponding results for admixtures with the CP-odd term. In this case the sensitivity of the polar angle is similar to that of the momentum. As a result, the improvement from the 2D analysis is significant, to the extent that the sensitivity can be enhanced by about a factor of two. The sensitivity of the angular variable grows with the CME.
The results provide a good motivation for the role of an electron positron collider in understanding the nature of the HV V couplings. The plots in Fig. 15 show the utility in using two dimensional distributions in discerning the rejection of hypotheses. That is, using the same accrued data from two separate one dimensional distributions, one can enhance the confidence in rejecting hypotheses. The correlation of the two dimensional distributions thus carries vital information about the dynamics of the processes which are studied in e + e − collisions.

V. SUMMARY AND CONCLUSIONS
We have attempted to demonstrate the efficacy as well as limitations of an e + e − Higgs factory operating at 250 − 300 GeV in probing anomalous, higher-dimensional couplings of a Higgs to W -and Z-pairs, suppressed by a scale O(TeV). For this purpose, we have mostly adhered to the set of gauge-invariant operators that can lead to such interactions, since it is such terms that are expected to emerge on integrating out physics above the electroweak symmetry breaking scale. We have utilised the consequent correlation of the anomalous HW W , HZZ and HZγ couplings, and also the concomitant effect on ZW W/γW W interactions, as reflected in gauge boson pair-production rates.
The general conclusion reached by this study is that the total rates can be quite useful as probes of higher-dimensional operators. Based on this, we have performed a detailed analysis of the cross-sections for s-and t-channel Higgs production, specifying event selection criteria for minimising their mutual contamination. A general scheme of computing the rates with more than one gauge-invariant operators has been outlined. Based on such an analysis, we conclude that, even with the additional operators well within the erstwhile experimental bounds (including those form the LHC), a number of observations can probe them at a Higgs factory. These include not only the individual total cross-sections but also their ratios at different values of √ s and also the ratio of the s-and the t-channel Higgs production rates at fixed energies. We also indicate the correlated variation of W -pair production rates. The Higgs production rate contours with more than one type of anomalous gaugeinvariant operators are also presented. Finally, using some illustrative values of anomalous HW W couplings in a more phenomenological parametrization, we indicate the viability of a correlated two-dimensional likelihood analysis to fully exploit the kinematics of the Higgs boson. The latter is particularly relevant to disentangle the SM from CP-violating admixtures. On the whole, we thus conclude that a Higgs factory can considerably improve our understanding of whether the recently discovered scalar is the SM Higgs or not, as evinced from its interactions with a pair of weak gauge bosons.