CP-Violation in the \(ZZZ\) and \(ZWW\) vertices at \(e^+e^-\) colliders in Two-Higgs-Doublet Models

JHEP05(2016)025

Published for SISSA by Springer

Received: March 10, 2016 Accepted: April 13, 2016 Published: May 4, 2016

CP-violation in the ZZZ and ZW W vertices at e

e

colliders in Two-Higgs-Doublet Models

B. Grzadkowski,^a O.M. Ogreid^b and P. Osland^c

aFaculty of Physics, University of Warsaw, Pastura 5, 02-093 Warsaw, Poland

bBergen University College,

Postboks 7030, N-5020 Bergen, Norway

cDepartment of Physics, University of Bergen, Postboks 7803, N-5020 Bergen, Norway

E-mail: [email protected],[email protected], [email protected]

Abstract: We discuss possibilities of measuring CP violation in the Two-Higgs-Doublet Model by studying effects of one-loop generatedZZZ andZW W vertices. We discuss a set of CP-sensitive asymmetries forZZ and W⁺W⁻ production at lineare⁺e⁻-colliders, that directly depends on the weak-basis invariant ImJ₂ that parametrises the strength of CP violation. Given the restrictions on this model that follow from the LHC measurements, the predicted effects are small. Pursuing such measurements is however very important, as an observed signal might point to a richer scalar sector.

Keywords: Beyond Standard Model, CP violation, Higgs Physics ArXiv ePrint: 1603.01388

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Contents

1 Introduction 1

2 The model 2

3 The ZZZ vertex 3

3.1 Lorentz structure 3

3.2 Results 4

4 The ZW⁺W⁻ vertex 6

4.1 Lorentz structure 6

4.2 Results 7

5 Asymmetries 7

5.1 e⁺e⁻→ZZ 8

5.2 e⁺e⁻→W⁺W⁻ 11

6 Discussion 15

A The ZZZ vertex 16

A.1 TheHHH triangle diagram 16

A.2 TheHHGtriangle diagrams 17

A.3 TheHHZ triangle diagrams 18

A.4 Bubble diagrams 19

A.5 Tadpole diagrams 19

B The ZW⁺W⁻ vertex 19

C Extracting ImJ2 — a case study 20

D Some asymmetry prefactors F 21

D.1 The prefactorF1(β,Θ) ofA^ZZ1 21

D.2 The prefactorF(s₁,Θ) ofA^ud 22

D.3 The prefactorsF^{W W} and ˜F^{W W} of A^{W W} and ˜A^{W W} 23

1 Introduction

Anomalous contributions to trilinear electroweak vector boson couplings have been thor- oughly studied [1–5] and searched for, at LEP [6], at Fermilab [7–10] and at the LHC [11–

21]. Experimentally, the

V W⁺W⁻, V =γ, Z (1.1)

couplings are considered the more accessible, whereas the

V ZZ, V =γ, Z (1.2)

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couplings are considered more challenging. Both classes may have a CP-violating, as well as a CP-conserving part.

In the Standard Model (SM), at the tree level, only the γW W and ZW W couplings are non-zero, whereas all four receive contributions at the one-loop level. In the SM, CP- violating effects can only be induced via the CKM matrix. However, at one-loop order, there is no such contribution, since there might be only two relatively complex-conjugated

¯

qq⁰W vertices, hence CP-violating phases of the CKM matrix would cancel. An extended Higgs sector may naturally modify this at the one-loop level, since new sources of CP violation could enter in a non-trivial way.

As is well known, the Two-Higgs-Doublet Model allows for CP violation, either explicit or spontaneous [22]. Early work on CP violation in the Higgs sector related it to the couplings of neutral scalars to the electroweak gauge bosons, as well as to the charged scalars [23,24]. The conditions for having CP violation in the model can be expressed in terms of three invariants, in ref. [25] denoted ImJ₁, ImJ₂ and ImJ₃. If any one of them is non-zero, then CP is violated [25] (see also ref. [26]). Further criteria would allow to distinguish spontaneous and explicit CP violation [25,27].

Standard-model contributions to theZZZandZW W vertices have been studied in [28]

and [29], respectively. Since there is some scope for further constraining or even measuring CP violation in these couplings, we present an updated review of these observables, and also propose some new ones.

The paper is organized as follows. After a brief review of the model and the basic CP- violating invariants in section 2, we discuss one-loop contributions to theZZZ andZW W vertices in sections3 and4. Selected CP-violating asymmetries that could be measured in e⁺e⁻ collisions are discussed in section 5, and concluding remarks are given in section 6.

Technical details are relegated to appendices.

2 The model

We adopt a standard parametrization for the scalar potential of the 2HDM (see, for exam- ple, [30]) with

Φi = ϕ⁺_i

(vi+ηi+iχi)/√ 2

!

, i= 1,2. (2.1)

In the general CP-violating case, the model contains three neutral scalars, which are linear compositions of the ηi and χi:





 H₁ H₂ H3





=R





 η₁ η₂ η3





, (2.2)

with η₃ a linear combination of the χ_i that is orthogonal to the Goldstone field G₀. Fur- thermore, the 3×3 rotation matrix R satisfies

RM²R^T=M²diag = diag(M₁², M₂², M₃²), (2.3) whereM² is the neutral-sector mass-squared matrix, and with M₁ ≤M₂ ≤M₃.

The weak-basis invariants revealing CP violation were originally expressed by Lavoura, Silva and Botella [23,24], in terms of couplings and rotation-matrix elements. The notation

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Z₁^µ

Z₃^β Z₂^α

= ieΓ ^µαβ

−→

ր

ց p1

p2

p3

Figure 1. The generalZZZ vertex.

ImJ_i, where the invariants were expressed in terms of potential parameters was introduced by Gunion and Haber [25]. It was recently discussed in more detail by the present au- thors [30] (where also ImJ3 was replaced by another related invariant which we named ImJ₃₀). The invariant ImJ₂, which represents CP violation in the mass matrix, can be written as

ImJ₂ = 2e1e2e3

v⁹ (M₁²−M₂²)(M₂²−M₃²)(M₃²−M₁²)

= 2e1e2e3

v⁹ X

i,j,k

_ijkM_i⁴M_k², (2.4)

whereMiare the neutral Higgs masses, andei ≡v1Ri1+v2Ri2 represents their couplings to aZ or aW (for a full dictionary of couplings determined bye_i, see appendix B of ref. [30]).

We shall in this paper focus on processes in which ImJ2 is responsible for the CP violation. This invariant is the only one which does not involve charged scalars. Charged scalars are involved in processes for which ImJ₁ and/or ImJ₃₀ are responsible for the CP violation. For the explicit form of these invariants and processes to which they contribute, we refer to ref. [30].

3 The ZZZ vertex

One of the simplest vertex functions to which ImJ2 contributes, is the effectiveZZZvertex discussed in appendix A. Since each ZH_iH_j vertex contains a factor _ijk (see appendix B of ref. [30]), it follows that i, j, k must be some permutation of 1,2,3 and thus an over-all factor of e₁e₂e₃ will emerge.

CP-violating form factors for triple gauge boson couplings have previously been studied in the 2HDM in refs. [31–33].

3.1 Lorentz structure

Phenomenological discussions [2–5] of the ZZZ vertex have presented its most general Lorentz structure. In ref. [4] the CP-violating vertex is analyzed, with all Z₁, Z₂, Z₃ off- shell. A total of 14 Lorentz structures are identified, all preserving parity. Some of these vanish when one or more Z is on-shell. (For a detailed discussion of this structure, see ref. [34].) We characterize them by momenta and Lorentz indices (p₁, µ), (p₂, α) and

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(p₃, β), and let Z₁ be off-shell while Z₂ and Z₃ are on-shell. In addition, we assume that Z1 couples to a pair of leptons e⁺e⁻ and neglect terms proportional to the lepton mass.

Then according to [3] the structure reduces to the form.¹ eΓ^αβµ_ZZZ =iep²₁−M_Z²

M_Z²

hf₄^Z(p^α₁g^µβ+p^β₁g^µα) +f₅^Zµαβρ`_ρi

, (3.1)

where

`≡p2−p3≡2p2−p1 (3.2)

withebeing the proton charge, the momenta (p1incoming andp2, p3outgoing) and Lorentz indices as defined in figure 1. The dimensionless form factor f₄^Z violates CP while f₅^Z conserves CP.

Our aim is to determine the CP-violating contributions to the ZZZ vertex, hence the contributions to f₄^Z. Let us here make some qualitative comments. Summing over i, j, k (see figure 12 in appendix A) one might think that contributions to the triangle diagram would pairwise cancel because of the factor _ijk. Indeed, the scalar triangle diagrams do sum to zero, but there are non-vanishing tensor contributions, due to the momentum factors at the ZHiHj vertices.

Three classes of Feynman diagrams give contributions to the effective CP-violating ZZZ vertex, all proportional to ImJ₂. They are triangle diagrams withH_iH_jH_kalong the internal lines, as well as diagrams where one neutral Higgs boson is replaced by a neutral GoldstoneG₀ field, or a Z,

f₄^Z =f₄^Z,HHH +f₄^Z,HHG+f₄^Z,HHZ. (3.3) These three contributions are calculated in appendix A.

3.2 Results

The total one-loop contribution to f₄^Z for the ZZZ vertex calculated in appendix A is given by a linear combination of the three-point tensor coefficient functions C001 and C1

(we adopt the LoopTools notation [35]) of various arguments, f₄^Z(p²₁) = 2α

πsin³(2θ_W) M_Z² p²₁−M_Z²

e1e2e3

v³

×X

i,j,k

_ijk

C001(p²₁, M_Z², M_Z², M_i², M_j², M_Z²) +C001(p²₁, M_Z², M_Z², M_Z², M_j², M_k²) +C₀₀₁(p²₁, M_Z², M_Z², M_i², M_Z², M_k²)−C₀₀₁(p²₁, M_Z², M_Z², M_i², M_j², M_k²)

+M_Z²C₁(p²₁, M_Z², M_Z², M_i², M_Z², M_k²)

. (3.4)

This structure was identified 20 years ago by Chang, Keung and Pal [32], who studied the set of diagrams presented in appendices A.1 and A.2. We find numerically that our

1Here, we follow the convention of Hagiwara et al. [2], which we also adopt in section4for theZW W vertex by putting 0123 = −⁰¹²³ = +1, whereas Gounaris et al. [3] have chosen the convention where ⁰¹²³= +1.

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5 10 15 20 25 30 35 40

−2

−1.5

−1

−0.5 0 0.5 1 1.5

2 f^Z₄ [10^-4]v³/e₁e₂e³

2

/ MZ

s1

200

250

300 350

Figure 2. Real (solid lines) and imaginary (dashed) part of the form factor f4^Z (divided by e1e2e3/v³) as a function ofp²₁/M_Z², forp²₂=p²₃=M_Z² and four values of neutral-Higgs masses M2

of eq. (3.5), as indicated (in GeV). Below threshold,s1=p²₁= 4M_Z², the function is not defined.

result for the sum of these diagrams is identical to their result. There are, however, also diagrams with an internal Z line, arising from the ZZHi vertex which was not included in their study. These contributions are calculated in appendix A.3, and numerical studies show that these are actually the dominant contributions.

For the neutral-Higgs masses

M1= 125 GeV, M2 = (200,250,300,350) GeV, M3 = 400 GeV, (3.5) we show in figure 2 the value of f₄^Z(p²₁)v³/(e1e2e3) as a function of p²₁/M_Z². The normal- ization factor,e1e2e3/v³, is typically of O(0.1) (only small regions of the parameter space are compatible with theoretical and experimental constraints [36, 37]). Defining δ as a measure of deviation of the H1V V coupling from its SM strength, e1 = v(1−δ), and using e²₂+e²₃ =v²−e²₁, one can easily find [30] that for small δ, (e1e2e3)/v³ < δ, so it is suppressed by the H₁V V coupling approaching the SM limit.

The form factor f₄^Z has been constrained by experiments at LEP, Fermilab and the LHC. Recently, CMS [20] has presented an impressive bound on f₄^Z (assumed real):

−0.0022 < f₄^Z < 0.0026. This result is obtained in the 2`2ν channel from the 7 and 8 TeV data sets. It is still two orders of magnitude above what is generated in the 2HDM by a non-zero ImJ₂.

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Z^µ

(W⁺)^β (W⁻)^α

= − ie cot θ _W Γ ^µαβ

−→

ր

ց p1

p2

p3

Figure 3. The generalZW W vertex.

4 The ZW⁺W⁻ vertex

Contrary to the ZZZ vertex, the ZW W vertex is present at the tree level, with a well- known, CP-conserving structure:

igZW WΓ^αβµ_tree =−igcosθW[g^αβ(p2−p3)^µ+g^βµ(p1+p3)^α−g^µα(p1+p2)^β] (4.1a)

=−igcosθ_W

"

g^αβ`^µ+g^βµ

−1 2`+3

2p₁ α

−g^µα 1

2`+3 2p₁

β#

, (4.1b) whereg_{ZW W} =−ecotθ_W,p₁ is incoming whilep₂ and p₃ are outgoing, and in the second line, we make use of`=p2−p3.

Triangle diagrams discussed in appendix B contribute to the CP-violating ZW⁺W⁻ vertex. In fact, they give a contribution proportional to the invariant ImJ₂, which is one measure of CP violation in the Two-Higgs-Doublet model [25] (referred to asJ1 in earlier work by Lavoura, Silva and Botella [23,24]).

4.1 Lorentz structure

Phenomenological discussions [2] of the ZW W vertex have presented its most general Lorentz structure. We let Z be off-shell while both W^± are on-shell, again assuming that Z couples to a pair of leptonse⁺e⁻so that we may neglect terms proportional to the lepton mass. Then according to [2] the structure reads

Γ^αβµ_{ZW W} =f₁^Z`^µg^αβ − f₂^Z

M_W² `^µp^α₁p^β₁ +f₃^Z

p^α₁g^µβ−p^β₁g^µα +if₄^Z

p^α₁g^µβ+p^β₁g^µα

+if₅^Zµαβρ`_ρ

−f₆^Zµαβρp1ρ− f₇^Z

M_W² `<sup>µαβρσ</sup>p1ρ`σ. (4.2) The tree-level vertex contributes tof₁ and f₃:

f₁^tree = 1, f₃^tree= 2. (4.3)

The dimensionless form factors f₄^Z, f₆^Z and f₇^Z violate CP while the others conserve CP.

Recent LHC experiments [17,19,21] have constrained the CP-conserving anomalous cou- plings, but not the CP-violatingf₄^Z.

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10 20 30 40 50 60 70 80 90 100

−3

−2

−1 0

1 2

3 f^Z₄ [10^-4]v³/e₁e₂e³

2

/ MW

s1

200

250

300 350

Figure 4. Real (solid lines) and imaginary (dashed) part of the form factor f4^Z (divided by e1e2e3/v³) as a function ofs1/M_W² , forp²₂=p²₃=M_W² and four values of neutral-HiggsM2 masses of eq. (3.5), as indicated (in GeV).

Our aim is to determine the CP violating contributions to the ZW W vertex, hence the contributions to f₄^Z.

4.2 Results

The total one-loop contribution to f₄^Z for the ZW W vertex calculated in appendix B is given by a linear combination of the three-point tensor coefficient functionsC001of various arguments,

f₄^Z p²₁

= −α

πsin²(2θ_W) e1e2e3

v³ X

i,j,k

_ijk

C001(p²₁, M_W² , M_W² , M_i², M_j², M_W² )

−C001(p²₁, M_W² , M_W² , M_i², M_j², M_H²±)

. (4.4)

This quantity was also studied by He, Ma and McKellar [31]. Assuming that they have used the (−i) prescription in their eq. (5), we find numerical agreement apart from an overall sign. Furthermore, the result for the imaginary part given in their eq. (6) is twice as large as the one in eq. (5).

For the neutral-Higgs masses given by equation (3.5), we show in figure4 the value of f₄^Z(p²₁)v³/(e₁e₂e₃) as a function of s₁/M_W² .

5 Asymmetries

We are going to discuss the possibility of testing CP violation at futuree⁺e⁻ colliders [38, 39]. It is assumed that polarizations of the final-state vector bosons could be determined

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experimentally.² We adopt CP-sensitive observables defined forW⁺W⁻andZZin [33,40], and [32], respectively. Below, we present some predictions for those and other asymmetries for the 2HDM.

5.1 e⁺e⁻ → ZZ

Helicities of the ZZ (and W⁺W⁻) pairs can be measured statistically by studying decay products of the final vector bosons. Therefore, we will define a number of differential asymmetries assuming that both the momenta and helicities of the ZZ pair could be determined. Since our goal is to measure the CP-violating form factorf₄^Z, our asymmetries will (to leading order) be proportional tof₄^Z. Let us first start by considering

A^ZZ₁ ≡ σ+,0−σ0,−

σ+,0+σ0,−

, (5.1)

A^ZZ₂ ≡ σ_0,+−σ−,0

σ0,++σ−,0

, (5.2)

whereσ_λ,¯λ are unpolarized-beam cross sections for the production of ZZ with helicitiesλ and ¯λ, respectively. The cross sections can be expressed through the helicity amplitudes fore⁺(σ)e⁻(¯σ)→Z(λ)Z(¯λ) as follows

σ_λ,¯λ=X

σ,¯σ

Mσ,¯σ;λ¯λ(Θ)M^?_σ,¯σ;λλ¯(Θ), (5.3) where σ and ¯σ are the helicities of e⁻ and e⁺, respectively. Expressions for these cross sections can readily be written out using the results from Chang, Keung and Pal [32].

Letting Θ be the angle between the e⁻ beam direction and the Z whose helicity is given by the first index λ, and defining γ = √s₁/(2M_Z) and β² = 1−γ⁻², we find to lowest order in f₄^Z

A^ZZ₁ =−4βγ⁴

(1 +β²)²−(2βcos Θ)²

F¹(β,Θ) Imf₄^Z, (5.4) withF1(β,Θ) given in appendix D.

In the low-energy limit (β →0) this simplifies to A^ZZ₁ = −4β

ξ1−3ξ1cos²Θ + 2 (ξ1−ξ2) cos³Θ Imf₄^Z

(ξ₃+ξ₄) + 2ξ₃cos Θ−3 (ξ₃+ξ₄) cos²Θ−4ξ₃cos³Θ + 4 (ξ₃+ξ₄) cos⁴Θ, (5.5) where the ξi are given in appendix D. Furthermore, we find

A^ZZ₂ =A^ZZ₁ (cos Θ→ −cos Θ). (5.6) These asymmetries are both shown in figure 5. The sharp peaks near the forward and backward directions are due to an interplay of three factors: (1) the near-divergence of the t-channel propagator, (2) the factor [∆σ∆λ(1 +β²)−2 cos Θ] of the amplitude (see eq. (5) in ref. [32]) and (3) the Wigner functions proportional to 1±cos Θ.

2Investigating angular distributions of the vector boson decay products one can indeed measure their polarizations.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−10 8

− 6

−

−4 2

− 0 2 4 6 8

10 A^ZZ₁ /(γ⁴ Im f^Z₄)

/π Θ 500 150

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−10 8

− 6

−

−4 2

− 0 2 4 6 8

10 A^ZZ₂/(γ⁴ Im f^Z₄)

/π Θ

150 500

1500

Figure 5. The asymmetriesA^ZZ₁ (Θ) of eq. (5.4) andA^ZZ₂ (Θ) of eq. (5.6)(both divided byγ⁴Imf₄^Z) as functions of Θ for three beam energiesE as indicated (in GeV).

Introducing the abbreviations

ξ = 2 sinθ_Wcosθ_W(1−6 sin²θ_W + 12 sin⁴θ_W)

1−8 sin²θ_W + 24 sin⁴θ_W −32 sin⁶θ_W + 32 sin⁸θ_W '1.65, (5.7) ξ˜= −4 sinθWcosθW 1−6 sin²θW + 12 sin⁴θW −16 sin⁶θW

1−8 sin²θW + 24 sin⁴θW −32 sin⁶θW + 32 sin⁸θW ' −0.78, (5.8) the following asymmetries can be defined and calculated to leading order in f₄^Z:

A^ZZ ≡ σ+,0+σ0,+−σ0,−−σ−,0

σ_+,0+σ_0,++σ0,−+σ−,0

= −2βγ⁴[(1 +β²)²−(2βcos Θ)²][1 +β²−(3−β²) cos²Θ]ξImf₄^Z

(1 +β²)²−(3 + 6β²−β⁴) cos²Θ + 4 cos⁴Θ , (5.9) A˜^ZZ ≡ σ_+,0−σ_0,+−σ0,−+σ−,0

σ_+,0+σ_0,++σ0,−+σ−,0

= −2βγ⁴cos Θ[(1 +β²)²−(2βcos Θ)²] β²−cos²ΘξIm˜ f₄^Z

(1 +β²)²−(3 + 6β²−β⁴) cos²Θ + 4 cos⁴Θ . (5.10) The asymmetriesA^ZZ and ˜A^ZZ are both shown in figure 6 for three values of the energy.

Since the former is defined symmetrically with respect to the two Z bosons, the expres- sion is forward-backward symmetric. At high energies and intermediate angles, it is well approximated byA^ZZ ' −4γ⁴ξImf₄^Z.

In the low-energy limit, these become

A^ZZ → −2β(1−3 cos²Θ)ξImf₄^Z

1−3 cos²Θ + 4 cos⁴Θ , (5.11) A˜^ZZ → 2βcos³Θ ˜ξImf₄^Z

1−3 cos²Θ + 4 cos⁴Θ. (5.12)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−5

−4

−3

−2

−1 0 1 2 3 4

5 ^Z)

Im f4 4ξ γ β

ZZ/(

A

π Θ/ 150

500 1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

−1.5

−1

−0.5 0 0.5 1 1.5

2 ₄~ξ

γ β

ZZ/(

A~

Z) Im f4

π Θ/ 150

500

1500

Figure 6. The asymmetriesA^ZZ(Θ) and ˜A^ZZ(Θ) of eqs. (5.9) and (5.10) (divided byβγ⁴ξImf₄^Z andβγ⁴ξ˜Imf4^Z) and as functions of Θ for three beam energiesE as indicated (in GeV).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2 ^Z/π)

Re f4 2ξ γ β A''/(

π Θ/ 150

500 1500

Figure 7. The asymmetryA⁰⁰(Θ) of eq. (5.15) (divided by βγ²ξRef₄^Z/π) as a function of Θ for three beam energiesE as indicated (in GeV).

Other possibilities of testing CP violation in e⁺e⁻ → ZZ have been investigated by Chang, Keung and Pal [32], who note that the angular distribution of `⁻ from a Z decay is determined by the spin-density matrix of the Z (see eq. (10) of ref. [32]):

ρ(Θ)_λ1λ2 =N⁻¹(Θ) X

σ,¯σ,λ¯

Mσ,¯σ,λ1,¯λ(Θ)M^∗_σ,¯σ,λ2,λ¯(Θ). (5.13) where again, σ and ¯σ are helicities of e⁻ and e⁺, respectively, and the λ and ¯λ refer to the two Z helicities. They advocate a certain difference of cross sections, integrated over

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azimuthal quadrants of the final-state leptons, which is not suppressed by the approximate C-symmetry. They have thus defined such a “folded” asymmetry A⁰⁰(Θ) in their eq. (15), and shown that it equals

A⁰⁰(Θ) =−1

π [Imρ(Θ)+,−−Imρ(π−Θ)−,+]. (5.14) To lowest order inf₄^Z, this quantity is proportional to Ref₄^Z:

A⁰⁰(Θ) = β(1 +β²)γ²[(1 +β²)²−(2βcos Θ)²] sin²ΘξRef₄^Z

π[2 + 3β²−β⁶−β²(9−10β²+β⁴) cos²Θ−4β⁴cos⁴Θ]. (5.15) This asymmetry is shown in figure 7 for three values of the energy. Superficially, it looks like this asymmetry might be unbounded at high energies. This is not the case, since at high energies (see appendix C) f₄^Z falls off like (1/γ⁶) logγ.

In the low-energy limit (β →0), it simplifies:

A⁰⁰(Θ)→ βsin²ΘξRef₄^Z

2π . (5.16)

5.2 e⁺e⁻ → W⁺W⁻

Let us follow the same approach as fore⁺e⁻ →ZZ in thee⁺e⁻→W⁺W⁻case by forming the asymmetries [33]:

A^{W W}₁ ≡ σ+,0−σ0,−

σ_+,0+σ0,−

, (5.17)

A^{W W}₂ ≡ σ_0,+−σ−,0

σ0,++σ−,0

, (5.18)

where σ_λ,¯λ are unpolarized-beam cross sections for the production of W⁻ and W⁺ with helicities λ and ¯λ, respectively. The cross sections can be expressed through the helicity amplitudes for e⁺(σ)e⁻(¯σ) → W⁻(λ)W⁺(¯λ) like in eq. (5.3), where σ and ¯σ are the helicities of e⁻ and e⁺, respectively. The amplitudes Mσ,¯σ;λ¯λ(Θ) were first calculated in [2]. Here, Θ is the angle between thee⁻ and the W⁻ momenta.

Following the notation of [33], we find for the case of polarized initial beams (σ,σ),¯ and to lowest order in f₄^Z:

(σ,σ) = (+¯ −) : A^{W W}₁ = s₁

M_Z²Imf₄^Z, (5.19a)

(σ,σ) = (¯ −+) : A^{W W}₁ = −β²(1−2 sin²θ_W)s1

β²(2 sin²θ_WM_Z² −s₁) + (s₁−M_Z²)Y Imf₄^Z, (5.19b) where

Y ≡1− (1 +β)

γ²(1 +β²−2βcos Θ). (5.20) withγ =√s1/(2MW) and β²= 1−γ⁻².

For the unpolarized case, we find (still to lowest order in f₄^Z):

A^{W W}₁ = N₁^(a)(1−cos Θ)²+N₁^(b)(1 + cos²Θ)

D^(a)₁ (1−cos Θ)²+D^(b)₁ (1 + cos²Θ) βs₁Imf₄^Z (5.21)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5

− 4

− 3

−

−2 1

− 0 1 2 3 4

5 A^WW₁ /(γ² Im f^Z₄)

/π Θ 500 150

1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5

− 4

− 3

−

−2 1

− 0

1 2 3 4

5 A^WW₂ /(γ² Im f^Z₄)

/π Θ 150

150 500

1500

Figure 8. The asymmetriesA^{W W}₁ andA^{W W}₂ vs Θ for three values of the beam (orW) energy E, 150 GeV, 500 GeV and 1500 GeV, as indicated.

with the following abbreviations:

N₁^(a)= (1 +β²−2βcos Θ){X1−2 sin²θ_W[(1−β²)(1−β+ 2 cos Θ)s1

−(1−3β−β²+ 2β²cos Θ−β³+ 2 cos Θ)M_Z²]}, (5.22a) N₁^(b)= 8 sin⁴θ_Wβ(1 +β²−2βcos Θ)²M_Z², (5.22b) D^(a)₁ =X₁²−4 sin²θ_Wβ(1 +β²−2βcos Θ)X₁M_Z², (5.22c) D^(b)₁ = 8 sin⁴θWβ²(1 +β²−2βcos Θ)²M_Z⁴, (5.22d) X1 = (1−β²)(1−β+ 2 cos Θ)s1−(1−2β−β²+ 2 cos Θ)M_Z². (5.22e) In the low-energy limit (β →0), this simplifies:

A^{W W}₁ →







4M_W² M_Z²

2M_W² −M_Z²

(4M_W² −M_Z²)(1+2 cos Θ)βImf₄^Z, β <∼ |1 + 2 cos Θ|, β1,

−^2MW2 (^16MW⁴ −5M_W² M_Z²−2M_Z⁴)

M_Z²(^10MW⁴ −2M_W² M_Z²+M_Z⁴) Imf₄^Z, |1 + 2 cos Θ|<∼β 1, (5.23) where we have also substituted the tree-level relation sin²θ_W = 1−M_W² /M_Z².

Furthermore, we find

A^{W W}₂ =−A^{W W}₁ (cos Θ→ −cos Θ;β → −β). (5.24) We display these asymmetries A^{W W}₁ and A^{W W}₂ in figure8. An overall factorγ²Imf₄^Z is factored out, and hence for A^{W W}₁ , the graphs for 500 GeV and 1500 GeV are practically indistinguishable. The main structure is due to the first term in the numerator of eq. (5.17) passing through zero close to a minimum of the denominator.

We may also combine these two asymmetries into one, either by addition or subtraction.

Again calculating to lowest order in f₄^Z:

A^{W W} ≡ σ+,0+σ0,+−σ0,−−σ−,0

σ+,0+σ0,++σ0,−+σ−,0

=β 1 +β²−2βcos Θ

F^{W W}Imf₄^Z, (5.25)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−2

−1.5

−1 0.5

− 0 0.5

1 A^WW/(γ² Im f^Z₄)

/π Θ 150

500 1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6

0.8 A~^WW/(γ² Im f^Z₄)

/π Θ 150

500

1500

Figure 9. The asymmetries A^{W W} and ˜A^{W W} (divided by γ²Imf₄^Z) vs Θ for three values of the beam (orW) energyE, 150 GeV, 500 GeV and 1500 GeV, as indicated.

A˜^{W W} ≡ σ_+,0−σ_0,++σ0,−−σ−,0

σ+,0+σ0,++σ0,−+σ−,0

=β 1 +β²−2βcos ΘF˜^{W W}Imf₄^Z, (5.26) where the functions F^{W W} and ˜F^{W W}, given in appendix D, can be expressed as ratios of polynomials in cos Θ.

A further possibility of testing CP violation in e⁺e⁻ → W W has been investigated in [33]. Adopting the helicity amplitudes obtained there, they have defined the up-down asymmetry A^ud(Θ) in their eq. (32), and shown that it equals

A^ud(Θ) = 3 8

√2 [Imρ(Θ)_+,0−Im ¯ρ(Θ)−,0−Imρ(Θ)−,0+ Im ¯ρ(Θ)_+,0], (5.27) with ρ(Θ) the spin-density matrix of the W⁻ boson and ¯ρ(Θ) the spin-density matrix of theW⁺boson, as defined by their eqs. (26) and (28). To lowest order in f₄^Z, this quantity is proportional to Ref₄^Z. It is a rather complicated function, depending on the W velocity β, the angle Θ, the ratio M_Z²/s₁, as well as sin²θ_W. We focus on the angular dependence, and write it as

A^ud= 3βp

1−β²(1 +β²−2βcos Θ) sin ΘF(s₁,Θ) Ref₄^Z, (5.28) with

F(s1,Θ)≡ N₀^ud+N₁^udcos Θ +N₂^udcos²Θ

D^ud₀ +D^ud₁ cos Θ +D₂^udcos²Θ +D₃^udcos³Θ +D₄^udcos⁴Θ (5.29) given in appendix D. The angular dependence of this asymmetry is shown in figure 10.

In the low-energy limit, β→0, this reduces to A^ud→ −3

4β(1−2 sin²θW)M_W²

4M_W² −M_Z² sin Θ Ref₄^Z. (5.30)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.14

− 0.12

− 0.1

− 0.08

− 0.06

− 0.04

− 0.02

−

0 A^ud/(γ² Re f^Z₄)

/π Θ 150

500 1500

Figure 10. The asymmetry A^ud of eq. (5.28) (divided by γ²Ref₄^Z) vs Θ for three values of the beam (orW) energyE, 150 GeV, 500 GeV and 1500 GeV, as indicated.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.08 0.07

− 0.06

− 0.05

−

−0.04

−0.03

−0.02 0.01

−

0 A'^ud/(γ² Re f^Z₄)

/π Θ 150

500 1500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.14 0.12

−

−0.1

−0.08 0.06

−

−0.04

−0.02 0

A'' ^Z

/ Re f4 ud

/π Θ 150

500 1500

Figure 11. The asymmetriesA^0ud andA^00uddivided byγ²Ref4^Z and Ref4^Z, respectively, vs Θ for three values of the beam (or W) energy E, 150 GeV, 500 GeV and 1500 GeV, as indicated. Here, E0=¹₄√s1has been used.

On the other hand, at high energies, the prefactor F(s₁,Θ) grows as γ² (see ap- pendix D), but this is tempered by the high-energy fall-off of f₄^Z.

Chang, Keung and Phillips [33] have also defined an asymmetry A^0ud(Θ) in their eq. (34), and shown that it equals

A^0ud = 3√ 2

4π {[a(E0)−b(E0)] [Imρ(Θ)+,0−Im ¯ρ(Θ)−,0]

−[a(E0) +b(E0)] [Imρ(Θ)−,0−Im ¯ρ(Θ)+,0]}, (5.31)

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witha(E₀) andb(E₀) defined in [33] following their eq. (34). To the lowest order inf₄^Z we find that

A^0ud= 3βp

1−β²(1 +β²−2βcos Θ) sin Θs₁Ref₄^Z

π D^ud₀ +D₁^udcos Θ +D₂^udcos²Θ +D^ud₃ cos³Θ +D^ud₄ cos⁴Θ

× {[a(E0)−b(E0)]N(β,cos Θ) + [a(E0) +b(E0)]N(−β,−cos Θ)}, (5.32) where

N(β,cos Θ) =N₀^0ud+N₁^0udcos Θ +N₂^0udcos²Θ (5.33) and

N₀^0ud = 1−2β²−β³

(1−β)² 1−2 sin²θ_W s1

− 1−4β−β²+ 2β³−2 1−6β−β²+β³+β⁵

sin²θW

m²_Z, (5.34) N₁^0ud = 1 + 3β+ 2β²

(1−β)² 1−2 sin²θW

s1

−

(1 +β)²−2 1 + 3β+ 5β²+β³−2β⁴

sin²θW

+8β 1 +β²

sin⁴θ_W m²_Z, N₂^0ud = 2β² 1−4 sin²θ_W + 8 sin⁴θ_W

m²_Z. (5.35)

Finally, they have also defined A^00ud=−1

π (Imρ(Θ)+,−−Im ¯ρ(Θ)−,+), (5.36) which to the lowest order in f₄^Z equals

A^00ud = 4β 1−β²2

1−2 sin²θW

1 +β²−2βcos Θ

sin²Θ s1−m²_Z

s1Ref₄^Z π D^ud₀ +D₁^udcos Θ +D₂^udcos²Θ +D^ud₃ cos³Θ +D^ud₄ cos⁴Θ .

(5.37) In the low-energy limit these become:

A^0ud = −3βm²_W 2m²_W −m²_Z

sin ΘRef₄^Z 4πm²_Z 4m²_W −m²_Z

× {[a(E0)−b(E0)] (1 + cos Θ) + [a(E0) +b(E0)] (1−cos Θ)}, (5.38) A^00ud = −βm²_W 2m²_W −m²_Z

sin²ΘRef₄^Z

πm²_Z 4m²_W −m²_Z . (5.39)

The asymmetriesA^0ud and A^00ud are shown in figure 11. LikeA^ud, they vary rapidly near the backward direction.

6 Discussion

The mixing of CP-even and odd components of the scalar fields lead to couplings among all pairs of neutral mass eigenstates and the gauge particles, which in turn lead to loop-induced trilinear couplings among the electroweak gauge particles,W andZ. The CP-violating part of these couplings, which we have discussed here, are all proportional to the quantity ImJ₂.