{'fraction_non_alphanumeric': 0.047207664592204386, 'fraction_numerical': 0.016582050442188013, 'mean_word_length': 4.059030654515327, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Higgs Production: CP Studies in e-e- Collisions

We review the production of scalar Higgs-like particles in high-energy electron-electron collisions, via the fusion of electroweak gauge bosons. The emphasis is on how to distinguish a CP-even from a CP-odd Higgs particle. Among the more significant differences, we find that in the CP-odd case, the Higgs spectrum is much harder, and the dependence of the total cross section on the product of the polarizations of the two beams is much stronger, than in the CP-even case. We also briefly discuss parity violation, and the production of charged Higgs bosons.


Introduction
In the planning for a future linear collider [1,2] one has to explore not only the electronpositron mode and various photon modes, but also an electron-electron mode, in spite of concerns related to beam "disruption". One reason for an electron-electron collider to be interesting is that one may produce states not accessible in the annihilation channel, another is that a large electron polarization will be readily available. There is already a considerable literature on the electron-electron mode [3,4,5,6].
We here consider the production of Higgs particles 1 in electron-electron collisions. Apart from a precise determination of the Higgs mass, which will allow for certain consistency tests of the theory, one will want to determine its properties under the discrete symmetries, and the couplings to various other particles.
At high energies, the Higgs production at an electron-electron collider will proceed via gauge boson fusion [3,6], and thus not be suppressed by the s-channel annihilation mechanism [7]. Certain models also predict doubly charged Higgs particles [8], some of which can be produced more readily at an electron-electron collider. Scalar ("Higgs") particles, h, h − and h −− , are produced in the t-channel via Z-or W -exchange: as depicted in Fig. 1 for the case (1.1). (However, in some models, including the left-right symmetric model [9], the doubly-charged Higgs boson has practically no coupling to the ordinary, left-handed W bosons. They would not be produced by this mechanism.) It is well known that the CP property of the Higgs particle can be explored in the electron-positron annihilation mode from studies of angular and energy correlations [10,11,12]. In the present paper we analyze the corresponding situation for the t-channel, at an electron-electron collider, taking into account the effects of beam polarization.
We shall investigate to what extent various angular distributions and energy correlations are sensitive to whether the Higgs particle is even or odd under CP , in which case it will be denoted as H or A, respectively. It turns out that several of these distributions are quite sensitive to the CP property of the Higgs particle. Some of these results were presented elsewhere [13].
The ZZh coupling is taken to be [14] i2 5/4 G F m 2 Z g µν for h = H (CP even), η ǫ µνρσ k 1ρ k 2σ for h = A (CP odd), (1.4) where k 1 and k 2 are the momenta of the gauge bosons. Thus, we see immediately that near the forward direction, where k k k 1 and k k k 2 are antiparallel, the production of a CP -odd Higgs boson will be suppressed. In the MSSM, this ZZA coupling is absent at the tree level, but will be induced at the 1-loop level [15]. Our analysis is not restricted to any particular model. There could also be CP violation in the Higgs sector, in which case the Higgs bosons would not be CP eigenstates [16]. Such mixing could take place at the tree level [17], or it could be induced by radiative corrections. It has also been pointed out that such mixing might take place in the MSSM, and be resonant [18]. We shall discuss ways to look for CP violation in the effective ZZh coupling.
The focus will be on a light Higgs boson, as is favored by current LEP precision data [19], and the case of E cm = 500 GeV [2].
When one or both Z's are replaced by W 's (for the production of charged Higgs particles), we shall assume that the Lorentz structure of the coupling remains unchanged.
The paper is organized as follows. In Sect. 2 we give notations and discuss kinematics, in Sect. 3 we give various cross section formulas. In Sects. 4-6 we present a variety of numerical results: Section 4 is devoted to integrated cross sections and distributions where the final-state electrons are integrated over. These distributions would qualitatively be the same for the production of charged Higgs particles. In Sect. 5 we study correlations between the final-state electrons, and in Sect. 6 we consider parity violation. Sect. 7 is devoted to a brief, qualitative, discussion of charged Higgs particles, and in Sect. 8 we discuss statistics, with some concluding remarks in Sect. 9.

Notation and kinematics
The eeZ vector and axial vector couplings are denoted g V and g A , as defined by the interaction ψ(x)γ µ (g V − g A γ 5 )ψ(x)Z µ (x). As a parameterization of their ratio, we define the angle χ by (2.2) and g the SU(2) electroweak coupling constant. In the present work, the only reference to this angle χ is through sin 2χ. In the case of the eeZ coupling, we have sin 2χ ≃ 0.1393, whereas for the eνW coupling, which is purely left-handed, we have sin 2χ = 1. The momenta of the final-state leptons will be referred to by polar angles θ 1 and θ 2 [see Eq. (1.1)], and that of the Higgs particle by the polar angle θ h : For "forward" production, we will thus have cos θ 1 ≃ 1, cos θ 2 ≃ −1. Furthermore, an azimuthal angle φ will refer to the relative orientation of the two planes formed by the final and initial-state leptons (in ref. [13] the definition used was cos φ → − cos φ), The two beams will be taken to be longitudinally polarized, with degrees of polarizations given by P 1 and P 2 , respectively (P i > 0 for a right-handed polarization). We shall express the cross sections in terms of the variables 5) where (neglecting the electron mass) For the two final-state electrons, we distinguish p ′ 1 and p ′ 2 , according to which has the higher energy, 3 The e − e − → h e − e − cross section For Higgs production from an electron-electron initial state, via the so-called fusion mechanism (with Z exchange), there are two diagrams, because of the symmetry of the two electrons in the final state. The corresponding two amplitudes differ by the substitutions p ′ 1 ↔ p ′ 2 , corresponding to (t 1 , t 2 ) ↔ (u 1 , u 2 ), and by an over-all sign. We present the differential cross section in two different forms. Both are useful, according to which distribution we want to study. For the study of distributions of the final-state electrons, we express the differential cross section as where F (t 1 , t 2 ) is a propagator factor, (3. 2) The over-all constant is given as 2 with the Jacobian (3.4) and ε half the energy difference between the two electrons, Since the two final-state electrons are indistinguishable, we shall identify the momenta such that E ′ 1 ≥ E ′ 2 ; thus, ǫ ≥ 0. The maximum value is given by the beam energy and the Higgs mass as For the purpose of studying distributions in cos θ h and E h , it is more convenient to express the cross section as 3 where the over-all constant is given as (3.8) andJ is the Jacobian,J The dynamics is given by X (h) and Z (h) . We shall below consider three cases: (1) The CP -even case, (2) the CP -odd case, and (3) the case of CP violation.

The CP -even case
For the CP -even case, we find

The CP -odd case
For the CP -odd case, we find (3.12) and For comparison, we give in Appendix A the corresponding results, including beam polarization effects, as well as the t-channel contribution, for the more familiar case of (3.14)

CP violation
To allow for the possibility of CP violation in the interaction between electroweak gauge bosons and the Higgs, we introduce a mixing angle α as follows: Thus, for α = 0 or π/2, the Higgs has even or odd CP , respectively, whereas for sin 2α = 0 the production mechanism violates CP . This amounts to allowing for both terms, and their interference. The suitably averaged square of the amplitude will then take the form In the notation of Eqs. (3.1) and (3.7), we get 17) where the amount of CP violation is given by sin 2α, with The CP -even terms X (H) and Z (H) are given by Eq. (3.10), and the CP -odd ones, X (A) and Z (A) , by (3.11)-(3.13).
There is no CP -violating contribution to the interference between the t-and u-channel terms. For the t-and u-channel CP -violating terms, which are proportional to we findX A quantitative study of this case of CP violation is presented in Sec. 6.

Gross features of the cross section
The Z propagators will favor production at small momentum transfers, i.e., with the final-state electrons close to the beam directions. This is indeed how the CP -even Higgs particle is produced. However, a finite momentum transfer is required to produce a CPodd particle, as is seen from the coupling (1.4) and also from the explicit expressions (3.11)- (3.13). This statement will be illustrated quantitatively in the following.

Total cross section
For a collider at √ s = 500 GeV, the cross section for producing a Standard Model Higgs with a mass of 100 GeV, is 9 fb, and falling steeply with mass, as illustrated in Fig. 2 (denoted "even"). The corresponding Bjorken cross section is around 60 fb [2]. We also compare with the cross section for producing a CP -odd Higgs boson, taking the coupling strength η such that the two cross sections coincide at m h = 100 GeV. 4 Since it may be difficult to observe electrons at small angles, and in order to reduce certain backgrounds, we also study the effect of a cut, with respect to the beam, on the polar angles of the final-state electrons. Three sets of curves are given in Fig. 2, the upper ones are for no cut, whereas the lower ones correspond to cuts at 5 • (as suggested by Minkowski [6]) and at 15 • . 5 Similar results are given by Hikasa [3] at higher energies 6 , and in [6].
The energy dependence is illustrated in Fig. 3. As the energy increases, the cross section grows. This is characteristic of the t-channel fusion mechanism, and rather different from the case of the Bjorken mechanism. However, an angular cut will temper this growth with energy; for the CP -even case, the cross section may even decrease with energy (see also [6]).
Polarization-dependent total cross sections will be discussed in Sec. 4.4 below.

Higgs energy distributions
An interesting observable to consider, is the Higgs energy distribution We show in Fig. 4 such distributions, for √ s = 500 GeV, and for two Higgs masses, m h = 120 GeV and 150 GeV. In the CP -even case, the Higgs particle is rather low-energetic, whereas in the CP -odd case, the spectrum is much harder, as discussed previously. When one imposes a cut on the opening angle of the final-state electron momenta w.r.t. the beam, the CP -even spectrum becomes harder, whereas the CP -odd one is practically unchanged. Curves are shown (dashed and dotted) in Fig. 4, corresponding to cuts at opening angles of 5 and 10 degrees. (No cut is imposed on the Higgs particle.) However, even with such a cut, there is a clear distinction between the two cases. We shall return to these distributions in Sec. 8.

Higgs polar-angle distributions
Next, we consider the Higgs polar-angle distribution In the absence of any cut, there is a clear distinction between the two cases of CP , the CP -even distribution being much more peaked along the beam direction. When cuts are imposed on the opening angles of the final-state electrons, this difference is reduced, but not seriously. The dashed and dotted curves in Fig. 5 show the effects of imposing a cut on the opening angle of the final-state electrons, with respect to the beam, of 5 • and 10 • , respectively. The dependence of these distributions on the Higgs mass, is very weak.
Such polar-angle distributions may therefore be valuable, in particular if one can get data near the beam directions.

Polarization-dependent correlations
The dependence on longitudinal beam polarization enters in the following way 1 ) [20]. These quantities A (h) 1 and A (h) 2 might be useful in distinguishing the even and odd case, since the unknown coupling strength η cancels.
The quantity A 1 is most easily extracted if both beams have equal and opposite polarizations. For the integrated cross section, A 1 is shown in Fig. 6. There is a very strong discrimination between the two CP cases. For the CP -odd case, the cross section is much reduced if the two beams have large and opposite polarizations, whereas in the even case, there is only a small reduction. The large value of A 1 in the CP -odd case implies that the cross section is very much reduced if both beams are longitudinally polarized. This suppression is due to the fact that in the CP -odd case, the two intermediate Zs must have orthogonal polarizations. 7 In the extraction of A 1 from data, there will be a contamination from the A 2 term in (4.3) when P 1 + P 2 = 0. For the parameters given in Fig. 6, A 2 ranges from -16% to -18% and from -21% to -30% for CP even and odd, respectively.

Final-state electron-electron correlations
In the electron-electron mode, the angular distributions are more complicated than in the positron-electron mode, due to the fact that the propagators will depend on the angles of interest, through t j and u j .

Azimuthal correlations
We first consider distributions in the azimuthal angle φ defined in (2.4). These are obtained by integrating the differential cross section, Eq. (3.1), over the energy difference, given by ε, up to ε max , as well as over the polar angles, θ j , for 0 ≤ | cos θ j | ≤ cos θ c : In this case, there is no particular need to use the events where the final-state electrons are close to the beam direction, so we impose a stronger cut, cos θ c = 0.9. (It may even be difficult to determine the azimuthal angles for electrons which are close to the beam direction.) Results are shown in Fig. 7, for the unpolarized case. We consider two c.m. energies, and two Higgs masses. The distributions generally favor the region around φ ∼ π, i.e., when the two final-state electrons have non-vanishing and opposite transverse momenta (w.r.t. the beam), as opposed to φ ∼ 0, when they are more parallel. This broad feature is purely kinematic, more energy is available to create a Higgs particle if the two virtual Zs have opposite transverse momenta. On top of this broad feature, there is in the CP -odd case a dip around φ = π, if the Higgs momentum is sufficiently high (i.e., at low mass).
If the two beams have opposite polarizations, the difference between the two cases can be quite spectacular, as is illustrated in Fig. 8 for CP = 1 and CP = −1.

Polar-angle correlations
Next we consider distributions in the polar angles of the electrons, Such distributions are shown in Figs. 9, for CP even and odd, and for the case of no polarization. There is a rather strong difference between the two cases, the cross section being much more peaked for electrons emitted close to the forward direction in the CP even case. To produce an odd parity state, angular momentum has to be transferred, and the electrons must therefore undergo a more violent scattering. A less differential distribution can be obtained as follows. Let cos Θ = 1 2 (cos θ 1 − cos θ 2 ), (5.4) or cos θ 1 = cos Θ + 1 2 w, cos θ 2 = − cos Θ + 1 2 w, (5.5) and consider We show in Fig. 10 such distributions, for the even and odd cases. For CP = 1, the cross section is much more peaked towards the "forward" direction, cos Θ = ±1, consistent with Fig. 9.

Energy correlations
Finally, we consider the distribution in relative electron-energy difference. Introducing the scaled energy difference as x = ε/ε max [see Eqs. (3.5) and (3.6)], we will consider Such distributions are shown in Fig. 11. For the CP -odd case, this distribution is "harder", it falls off less rapidly for large energy differences x.
In electron-positron annihilation, with Higgs production via the Bjorken process, analogous distributions also exhibit a considerable sensitivity to whether the Higgs particle is even or odd under CP [11].

CP violation
As discussed in the Introduction, there could also be CP -violation in the Higgs sector, in which case the Higgs particles would not be eigenstates of CP .
While the presence of both even (X (H) and Z (H) ) and odd (X (A) and Z (A) ) terms in the cross section reflect parity violation, only the termsX t andX u explicitly violate parity. For these to be observed, one has to assign a value to p p p ′ 1 ×p p p ′ 2 [cf. Eq. (3.19)], i.e., one needs to distinguish the final-state electrons.
We show in Fig. 12 azimuthal distributions of the kind shown in Fig. 7, allowing for CP violation. Since these involve a symmetrical integration over both hemispheres, − cos θ c ≤ cos θ 1,2 ≤ cos θ c , the parity-violating termsX t andX u [cf. Eq. (3.20)] cancel. However, the parity violation leads to a superposition of the two cases, CP = +1 and CP = −1. Such distributions may suffice to provide evidence of parity violation.
One way to access the parity-violating termsX t andX u , is to introduce the weight factor cos θ 1 to distinguish the two hemispheres. Thus, we consider (the two electrons are here distinguished by E ′ 1 > E ′ 2 ) the asymmetry This quantity is shown in Fig. 13 for the same parameters and cuts as were used in Fig. 12.
To lowest order, the effect is linear in η. Thus, given enough data, the effect can be sizable. Other ways to search for CP violation in the ZZ-Higgs coupling are discussed in [10,11,14] 7 Charged Higgs Production If the produced Higgs is charged, there will be one or two final-state neutrinos. These cannot be detected, so distributions of the kind discussed in section 5 are not available. One may instead consider distributions of the charged Higgs particles themselves.

Singly-charged Higgs production
Singly-charged (negative) Higgs particles, which are expected in certain models [15], can be produced in e − e − collisions through the exchange of one Z and one W − boson. The cross section would be given by formulas analogous to those presented in Sec. 3, where the numerical coefficients involving the polarizations P i and the relative strength of the axial coupling, sin 2χ, would be replaced as follows for the t-channel terms (with accompanying changes in the propagator masses): (7.1) and similarly for the u-channel terms, with P 1 and P 2 interchanged. The interference terms would have the coefficient substitution where sin 2χ ≃ 0.1393 refers to the eeZ coupling. These substitutions would only change quantitative aspects of the cross sections. Thus, we expect all qualitative features discussed in Sec. 4 to remain valid.

Doubly-charged Higgs production
Doubly-charged Higgs particles, h −− , which are expected in the left-right-symmetric [9] and other models [8], can be produced in electron-electron collisions, not only in the schannel, but also via W W exchange. This mechanism does not require lepton-number violation, but the W W h −− coupling is absent in certain models [5]. Apart from an overall, model-dependent constant, the cross section would be given by the formulas of Sec. 3, with sin 2χ = 1. Distributions of the kinds given in Figs. 4 and 6 would readily reveal whether such a particle was even or odd under CP .

Statistical considerations
It is of interest to estimate how many events are needed to determine the CP from distributions of the kinds presented here. One of the most promising ones appears to be the Higgs energy distribution, shown in Fig. 4. We will assume that CP is conserved, so that the problem can be formulated in terms of statistical hypothesis testing as H 0 : CP = 1 and H 1 : CP = −1. Information would be gained if H 0 were rejected.
We denote the CP = +1 distribution by f (x) and the CP = −1 distribution by g(x). The problem is well suited for the Neyman-Pearson test [21], and following this approach, we will reject H 0 if, for n events, the likelihood ratio  Table 1: Recognition probabilities. Here, n is the number of events, α is the level of significance of the test and 1 − β the probability that one can recognize a CP = −1 distribution in Higgs energy data, for E cm = 500 GeV and at three Higgs masses, m h = 100, 120 and 150 GeV. See the text for further details.
where x i denote observed values of E h , and k is a critical constant to be determined. The constant k determines the level of "significance", α, of the test, i.e., the probability that we reject a correct hypothesis. An estimate for k can be obtained by Monte Carlo simulations. By drawing s samples of n x-values from the f (x) distribution, the ratios L 1 , L 2 , . . . L s can be calculated by applying Eq. (8.1). The empirical (1 − α) · 100% percentile in the simulated L distribution can be used as an estimate of k.
The "power", 1 − β (the probability of rejecting H 0 when H 0 is false), of this resulting test can also be estimated by Monte Carlo simulations. Samples should then be drawn from the g distribution, and the proportion of samples that are rejected in the test estimates 1 − β. Results are given in table 1 for E cm = 500 GeV and three values of the Higgs mass, m h = 100, 120 and 150 GeV. For each mass value, two cases are considered: (i) No cut on the final-state electron momenta, (ii) the electron momenta have to satisfy θ ≥ 10 • . (With a cut at 5 • , these probabilities are practically the same as without any cut.) We see that at a mass of 120 GeV, already 10 events suffice to reveal a CP = −1 distribution, at the level of 95-97%, with a risk of falsely rejecting the correct hypothesis, α, of only 5%. The discrimination is easier if one can get data near the beam direction, and if the Higgs particle is light, as is also seen from Fig. 4.

Concluding remarks
We have studied the production of generic Higgs particles in e − e − collisions, focusing on distributions which might be useful in distinguishing a CP -even from a CP -odd particle. Longitudinal beam polarization effects are taken into account.
We have not discussed backgrounds. These would depend on how the Higgs boson is detected. A light Higgs would dominantly decay to b quarks, and the background would not be severe, mostly from single Z and W production and the two-photon process [6]. A heavier Higgs would decay to W and Z bosons, and the background would be a problem [3,22].
In the CP -even case, the Higgs particle tends to be softer, and events are more aligned with the beam direction than in the CP -odd case. In fact, the Higgs energy distribution may be one of the better observables for discriminating the two cases.
Furthermore, the dependence on the product of the two beam polarizations is much larger in the CP -odd case. This dependence, which is represented by an observable A 1 , becomes a better "discriminator" for increasing Higgs masses, when the Higgs momentum decreases, and other methods may tend to become less efficient.
If the two final-state electrons are observed, a certain azimuthal distribution, as well as the electron polar-angle distributions, will also be useful for discriminating the two cases.
Finally, we suggest ways to search for possible parity-violating effects in the ZZ-Higgs coupling.
Appendix A. The e + e − cross section For the positron-electron case, there is, in addition to the familiar Bjorken diagram, also a t-channel diagram. The cross section can be expressed as with F (t 1 , t 2 ) defined by Eq. (3.2). The amplitude for the t-channel diagram is related to the corresponding one for the electron-electron case by the spinor substitutionsv(p 1 ) →ū(p ′ 1 ), and v(p ′ 1 ) → u(p 1 ), which amount to (s 1 , s 2 ) ↔ (−u 2 , −u 1 ). Also, the convention for the positron polarization is different, such that P 1 → −P 1 .
Furthermore, the (s-channel) Bjorken diagram is related to the t-channel diagram in a way similar to what is the case for the electron-electron diagrams. Thus, the unpolarized cross section for the Bjorken diagram is related to that of the t-channel diagram by (s 1 , s 2 ) ↔ (−t 1 , −t 2 ). However, the positron polarization for the Bjorken diagram would correspond to a final-state polarization in the t-channel diagram (which we sum over). Thus, the polarization-dependent parts of these cross sections are not related in this simple way.