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Using Back-Scattered Laser Beams to Detect

arXiv:hep-ph/9206262v1 29 Jun 1992UCD-92-18June, 1992Using Back-Scattered Laser Beams to DetectCP Violation in the Neutral Higgs Sector⋆B. Grza¸dkowski† and J.F.

GunionDepartment of PhysicsUniversity of California, Davis, CA 95616AbstractWe demonstrate that the ability to polarize the photons produced by back-scattering laser beams at a TeV scale linear e+e−collider could make it possibleto determine whether or not a neutral Higgs boson produced in photon-photoncollisions is a CP eigenstate. The relative utility of different types of polarizationis discussed.

Asymmetries that are only non-zero if the Higgs boson is a CP mixtureare defined, and their magnitudes illustrated for a two-doublet Higgs model withCP-violating neutral sector.⋆Work supported, in part, by the Department of Energy.† On leave of absence from Institute for Theoretical Physics, University of Warsaw, WarsawPL-00-681, Poland.Address after October 1, 1992 : CERN, CH-1211 Geneve 23, Switzerland.

An important issue for understanding electroweak symmetry breaking andphysics beyond the Standard Model (SM) is whether or not there is CP viola-tion in the Higgs sector. For instance, both spontaneous and explicit CP violationare certainly possible in the context of a general two-Higgs-doublet model (2HDM),whereas CP violation is not possible at tree-level for the specific 2HDM of the Mini-mal Supersymmetric Model (MSSM) or its simplest extensions involving additionalsinglet scalar Higgs fields.

[1] The sensitivities of a variety of experimental observ-ables to CP violation in the neutral sector have been examined in the literature,ranging from neutron[2−4] and electron[8−11] electric dipole moments (for a generalreview and more complete references, see Ref. [12]) to asymmetries in top quarkdecay and production.

[13,14] However, CP violation in these situations arises via loopgraphs involving the neutral Higgs bosons. Numerically, current EDM experimentsare not sufficiently sensitive to constrain CP violation in the neutral Higgs sector,although future results could begin to impose restrictions.

Even if a non-zero EDMis experimentally observed its interpretation will be uncertain since other types ofnew physics could also be involved. Of course, detection of CP violation in topquark decays and production must await the large top production rates of the SSCand LHC.Even after a Higgs boson has been directly observed, it may be difficult todetermine whether or not it is a pure CP eigenstate.

In a CP-conserving 2HDM,there are three neutral Higgs bosons: two CP-even scalars, h0 and H0 (mh0 ≤mH0)and one CP-odd scalar, A0. If the 2HDM is CP-violating there will simply be threemixed states, φi=1,3.

In principle, the presence of CP violation can be detectedthrough the existence and/or strength of various couplings. For instance, in theCP-conserving case, at tree-level the A0 is predicted to have no WW, ZZ couplingswhile the h0 and H0 together should saturate the WW, ZZ couplings.

In the CP-violating case, all of the φi would have WW, ZZ couplings at tree-level. But, evenif three φi are observed to have WW, ZZ couplings, it would not be clear whetherthis was due to CP violation in a 2HDM or to the existence of more than twodoublets.

A better possibility is to note[15] that CP violation at tree-level would berequired if the couplings ZZφ1, ZZφ2 and Zφ1φ2 are all non-zero. To completelyavoid contamination from C-violating one-loop diagrams, three or more neutralHiggs bosons must be detected.Non-zero values for all three of the couplingsZφ1φ2, Zφ1φ3 and Zφ2φ3 are only possible if CP violation is present.

Finally, wenote that the use of correlations between the decay planes of the decay productsof the WW or ZZ vector boson pairs, in order to analyze the CP properties ofthe decaying Higgs boson,[16] will not be useful in the most probable case wherethe CP-even component of a mixed-CP φ state accounts for essentially all of theWW, ZZ coupling strength.We wish to contrast the above rather significant difficulties with the situationthat arises in collisions of polarized photons. High luminosity for such collisions ispossible using back-scattered laser beams at a TeV scale linear e+e−collider.

[17,18]Detection of the SM Higgs boson and of the neutral Higgs bosons of the MSSMusing back-scattered laser beam photons was first studied in Ref. [19].

More de-tailed Monte Carlo results have appeared in Ref. [20].

These studies show that2

expected luminosities are adequate to make large numbers of Higgs bosons viasuch γγ collisions, and that backgrounds to their detection in qq and ZZ decaychannels are not serious. In qq channels, γγ →qq production can be suppressedin the mHiggs ≫2mq limit by appropriate choices for the helicities of the incom-ing photons.

The ZZ channel in which one of the Z’s decays to l+l−is virtuallybackground free, and observation of even a handful of events would constitute anadequate signal. In this paper, we demonstrate that the ability to control the po-larizations of back-scattered photons provides a powerful means for exposing theCP properties of any single neutral Higgs boson that can be produced with rea-sonable rate.

In particular, we find that there are three polarization asymmetrieswhich are only non-zero if the Higgs boson is not a pure CP eigenstate, and whichcould well be large enough to be measurable. We will compute the maximal effectsachievable in a general 2HDM while maintaining a sizeable production rate.The basic physics behind our techniques is well-known.

[1] A CP-even scalarcouples to γ1γ2 via FµνF µν, yielding (in the center of mass of the two photons) acoupling strength proportional to e · e, while a CP-odd scalar couples via Fµν eF µν,implying a coupling proportional to (e × e)z. [Quantities without (with) a tildebelong to γ1 (γ2).] In the helicity basis, we employ conventions such that (for γ1moving in the +z direction and γ2 moving in the −z direction)e± = ∓1√2(0, 1, ±i, 0),ee± = ∓1√2(0, −1, ±i, 0) .

(1)For these choices we find (λ, eλ = ±1):e · e = −12(1 + λeλ) ,(e × e)z = 12iλ(1 + λeλ) . (2)We write the general amplitude for a mixed-CP state φ to couple to γγ as M =e · ee + (e × e)z o, where e (o) represents the CP-even (-odd) coupling strength.Then using Eq.

(2) the helicity amplitude squares and interferences of interest are:|M++|2 + |M−−|2 =2(|e|2 + |o|2) ,|M++|2 −|M−−|2 = −4Im (eo∗) ,2Re (M∗−−M++) =2(|e|2 −|o|2) ,2Im (M∗−−M++) = −4Re (eo∗) . (3)It is useful to define the three ratios:A1 ≡|M++|2 −|M−−|2|M++|2 + |M−−|2 ,A2 ≡2Im (M∗−−M++)|M++|2 + |M−−|2 ,A3 ≡2Re (M∗−−M++)|M++|2 + |M−−|2 .

(4)Note that A1 ̸= 0, A2 ̸= 0 , and |A3| < 1 only if both the even and odd CPcouplings e and o are present. For a CP-even (-odd) eigenstate A1 = A2 = 0 andA3 = +1 (−1).

In the following we demonstrate how to probe these three ratios,and we will compute their magnitudes in a general CP-violating 2HDM.3

The event rate for γγ production of any final state can be written in the form[18]dN = dLγγ3Xi,j=0⟨ζieζj⟩dσij ,(5)where dLγγ is the luminosity for two-photon collisions, the ζi (eζj) are the Stokespolarization parameters (with ζ0 = eζ0 ≡1) for γ1 (γ2), and σij are the correspond-ing cross sections. ζ2 and eζ2 are the mean helicities of the two photons, whilel =qζ21 + ζ23 and el =qeζ21 + eζ23 are their degrees of linear polarization.

dLγγ and⟨ζieζj⟩are obtained as functions of the γγ center-of-mass energy, W, by averagingover collisions, including a convolution over the energy spectra for the collidingphotons.⋆Expressing the σij in terms of the helicity amplitudes, we obtain in thecase of Higgs boson production:dN = dLγγdΓ14(|M++|2 + |M−−|2)(1 + ⟨ζ2eζ2⟩)+(⟨ζ2⟩+ ⟨eζ2⟩)A1 + (⟨ζ3eζ1⟩+ ⟨ζ1eζ3⟩)A2 + (⟨ζ3eζ3⟩−⟨ζ1eζ1⟩)A3,(6)where dΓ represents an appropriate element of the final state phase space, includingan initial state flux factor.The behaviors of dLγγ and the ⟨ζieζj⟩as functions of W are crucial to ourconsiderations.Using standard results for Compton scattering, it is shown inRef. [18] that both quantities depend sensitively upon the polarizations of theincoming electron and laser beams.In order to understand these convolution-weighted quantities, it will be useful to first discuss the energy spectrum and Stokesparameters for an individual back-scattered photon, obtained after integrating onlyover the azimuthal angles for its emission.

These are determined by four functions.In particular, the energy spectrum for γ1 is directly proportional to a function C00,while its Stokes parameters take the form ζi = Ci0/C00, whereC00 =11 −y + 1 −y −4r(1 −r) −2λePcrx(2r −1)(2 −y)C20 =2λerx[1 + (1 −y)(2r −1)2] −Pc(2r −1)11 −y + 1 −yC10 =2r2Pt sin 2κ ,C30 = 2r2Pt cos 2κ . (7)In Eq.

(7) Pc (Pt) is the degree of circular (transverse) polarization of the initial⋆We note that the definition of ⟨. .

.⟩employed here is not the same as that of Ref. [18],Eq.

(29), which does not include a convolution at fixed W. Our definition of ⟨. .

.⟩is thatimplicitly employed in Figs. 13-15 of Ref.

[20].4

laser photon (P 2c +P 2t ≤1), κ is the azimuthal angle of the direction of its maximumlinear polarization, and λe is the mean helicity of the electron offof which thephoton is scattered; note that any sensitivity to the transverse polarization of theinitial electron has vanished after the azimuthal emission angle integration. Thequantities r, x and y are defined byr =yx(1 −y) ,y = ωE ,x = 4Eω0m2e≃15.3 ETeV ω0eV,(8)where ω (ω0) is the final (initial) photon energy, and E is the initial electron energy.The maximum value of y is ymax = x/(1 + x), in which limit r = 1.

All these samequantities as related to the second back-scattered photon will be denoted with atilde.Given that the mass of the Higgs boson is not likely to be near to the e+e−center-of-mass energy, a flat luminosity spectrum as a function of W is best forHiggs boson searches. This means that a flat energy spectrum as a function ofy is preferred for the individual photons, and to study the CP properties of theHiggs boson large values of the ζi (i = 1, 2, 3) are required.

To simultaneouslyachieve both, it turns out that the two extreme choices of full circular and fulltransverse polarization for the initial laser photon are most useful. Consider first2λePc = ±1, i.e.

full circular polarization for the initial laser photon and maximalaverage helicity for the incoming electron. For 2λePc = −1, C00 (and, consequently,the photon spectrum) is peaked as a function of y just below ymax, whereas for2λePc = +1 one finds a rather flat (and, hence, more desirable) spectrum over abroad range of y falling sharply to 0 as y →ymax.

The behaviors of the ζi followfrom Eq. (7).

For Pc = ±1, Pt must be zero and only ζ2 (and ζ0 ≡1) can be non-zero. The choice of |2λePc| = 1 allows ζ2 to be maximal; one finds ζ2 →+Pc, −Pcfor y →0, ymax, respectively.In the case of 2λePc = +1 (preferred for Higgsstudies), ζ2 ∼+Pc over almost the entire y range; only very near to y = ymax doesζ2 change sign and approach −Pc.

This is highly desirable behavior for isolatingA1.In contrast, in the case of 2λePc = −1, associated with a peaked energyspectrum, ζ2 slowly switches sign in the middle of the allowed y range. Thus, wehave a very fortunate conspiracy in which the 2λePc = +1 choice which yields thebest photon energy spectrum for study of a Higgs boson, also yields nearly 100%circular polarization for the back-scattered photon’s polarization for most y values.The other extreme choice for the incoming laser beam polarization is to takePc = 0, Pt = 1.

In this case, C00 is independent of λe and varies slowly as a functionof y over the entire range y = 0 to y = ymax. Eqs.

(6) and (7) make it apparentthat large l will be required in order to measure A2 and A3. For Pt = 1, the linearpolarization, l =qζ21 + ζ23, vanishes at y = 0 (l ≃y2/x2) and is rather small untily >∼ymax/2; as y →ymax, l approaches a maximum of lmax = 2/[(1+x)+(1+x)−1].Obviously, to maximize l it would be highly advantageous to have a machine designwith as small a value of x as possible.

It is also useful to note that if |2λe| = 1then ζ2 goes from 0 at y = 0 to a maximum of |ζ2|max =√1 −l2 at y = ymax.5

For typical values of x (e.g. of order 2 −4) |ζ2|max can be sufficiently large that itwould be useful in suppressing qq backgrounds.Of course, to isolate A1, A2 and A3, it is necessary to consider the polarizationsof both of the back-scattered photons.

When good circular polarization for bothlaser beams is available, A1 would be most easily isolated by making the wide-spectrum choices of 2λePc = +1 and 2eλe ePc = +1. From Eq.

(6), we see that theterm proportional to A1 changes sign if we reverse the sign of all of the helicitiesof the incoming electrons and laser beams — λe, eλe, Pc and ePc — thereby keeping2λePc and 2eλe ePc fixed at +1 while reversing the sign of both ζ2 and eζ2.Todetermine A2 and A3 we would take Pt = 1 and ePt = 1 and note that the coefficientof A2 (A3) is proportional to ⟨lel⟩sin 2(κ + eκ) (⟨lel⟩cos 2(κ + eκ)). A2 could thus beisolated by taking the difference of cross sections for κ + eκ = +π/4 and −π/4,while the difference of cross sections for κ + eκ = 0 and π/2 would determine A3.A convenient explicit form for the number of Higgs boson events is obtainedby normalizing to the two-photon decay width of the Higgs boson obtained aftersumming over final state photon polarizations.

From Refs. [19] and [20] the numberof Higgs bosons, Nφ, produced after averaging over colliding photon polarizationsis:Nφ = dLγγdWW =mφ4π2Γ(φ →γγ)m2φ≃1.54 × 104 Leefb−1 EeeTeV−1 Γ(φ →γγ)KeV mφGeV−2F(mφ) ,(9)where F(W) = (Eee/Lee)dLγγ/dW is a slowly varying function whose value de-pends upon details of the machine design, but is O(1).

In Eq. (9), Eee and Lee arethe e+e−machine energy and integrated luminosity.

For the case of interest, wheresome of the Stokes parameters have non-zero average values, Eq. (9) is modifiedby the curly bracket appearing in Eq.

(6), with ζi and eζi replaced by ⟨ζi⟩and ⟨eζi⟩,etc. All such averages depend upon the γγ invariant mass, W. For instance, forA1 = A2 = 0, and Pt = ePt = 0, the expressions for Nφ in Eq.

(9) are multipliedby (1 + ⟨ζ2eζ2⟩(mφ))Typical behaviors of F(W) and ⟨ζ2eζ2⟩(W) are amply illustrated in Ref. [20]for the case of 2λePc ≃+1 (i.e.

Pt = 0) upon which we shall focus. The mostimportant points to note are the following.

(a) A broad spectrum for F(W), advantageous for Higgs studies, can be achievedusing 2λePc and 2eλe ePc both as close to +1 as possible. (b) For 2λePc ∼+1, 2eλe ePc ∼+1 and Pc ePc ∼+1, ⟨ζ2eζ2⟩(W) is near to +1 forW up to 50%–70% of Eee.

This means that for mφ <∼70%Eee, Higgs boson6

production, proportional to 1+⟨ζ2eζ2⟩(mφ) (see Eq. (5)), will be enhanced sig-nificantly relative to qq backgrounds which, for small mq, will be suppressedby the factor 1 −⟨ζ2eζ2⟩(mφ), as we outline shortly.

(c) For Pc ≃ePc ∼±1, ⟨ζ2⟩∼⟨eζ2⟩∼±1 in this same range of W = mφ, so thatA1 can be easily isolated by simultaneously changing the signs of Pc and ePcfor the incoming laser beams (keeping 2λePc and 2eλe ePc fixed near +1).We now turn to estimating the observability of the Higgs boson and the as-sociated polarization asymmetries. In our normalization conventions we computeΓ(φ →γγ) as:Γ(φ →γγ) =α2g21024π3m3φm2W|e|2 + |o|2,(10)wheree =XiSiφ ,o =XiP iφ ,(11)and Siφ, P iφ represent CP-even, CP-odd triangle contributions of type i.

The onlynon-zero P iφ’s derive (at the one-loop triangle diagram level) from i =chargedfermion. Although our computations include fermion loops from all quarks andleptons, we illustrate by displaying the expressions for e and o keeping only b, t,W, and H+ triangles.

In this case, we havee =Nce2t sttF s1/2(τt) + Nce2bsbbF s1/2(τb) + sW +W −F1(τW +) + sH+H−F0(τH+) ,o =Nce2t pttF p1/2(τt) + Nce2bpbbF p1/2(τb) ,(12)where Nc = 3 for quarks, et = 2/3 and eb = −1/3 are the t and b fractionalcharges, and τi ≡4m2i /m2φ. The Fi’s are those defined in Appendix C of Ref.

[1].We remind the reader that for large τ, F1(τ) →7, F0(τ) →−1/3, F s1/2(τ) →−4/3,and F p1/2(τ) →−2; note in particular the small size of F0(τ) in this limit. Thereduced CP-even (scalar, s) and CP-odd (pseudoscalar, p) couplings are given by⋆stt =u2sin β ,ptt = −u3 cot β ,sbb =u1cos β ,pbb = −u3 tan β ,sW +W −= u2 sin β + u1 cos β .

(13)Reduced couplings for the charged leptons follow those for bb, while in Eq. (12) one⋆To get the correct relative sign between the contributions of a fermion, f, to e and o, it iscrucial to note that (in our convention) its reduced s (p) coupling to φ must be defined asthe coefficient of −gmf/(2mW ) times +1 (+iγ5).7

would have Nc = 1 and charge −1. In the above, the ui specify the eigenstate φ inthe Φi basis of Ref.

[3] (see Ref. [13] for more details).

In a 2HDM, Pi u2i = 1, butthey are otherwise unconstrained. Results for the SM Higgs boson correspond totaking u1 = cos β, u2 = sin β, and u3 = 0.

More generally, for a CP-even eigenstatewe would have u3 = 0, while for a CP-odd eigenstate |u3| = 1. We note that thebranching ratios for the φ to decay to bb, tt, W +W −, ZZ.

etc. are determined usingthese same reduced couplings by appropriately weighting the results for CP-evenand CP-odd scalars as given in Appendix B of Ref.

[1]. In Eq.

(13) we have notgiven an expression for sH+H−. The most general possibility is rather complicated,and there is a great deal of freedom in its magnitude.

However, it is proportionalto m2W/m2H+ and will not be large if mH+ is large, unless the mass of one of theother neutral Higgs bosons is much larger than mH+. Because of the very uncertainvalue of sH+H−, the reasonable probability that it will be small, and the fact thatit enters with a relatively small coefficient from F0(τH), we neglect the chargedHiggs loop in the numerical estimates to be given later.Of course, it is also necessary to understand the structure of possible back-grounds to our Higgs boson signal.

Consider the case of the bb background withmφ ≫2mb. In this limit, the amplitudes M++ and M−−for γγ →bb may beneglected.

Further, parity invariance implies that M−+ = M+−. We then obtainas the form of the background corresponding to that of Eq.

(6) for the Higgs boson:dN = dLγγdΓ12 |M+−|2 n1 −⟨ζ2eζ2⟩+ ⟨ζ3eζ3⟩+ ⟨ζ1eζ1⟩o,(14)Note that if the colliding photons have perfect circular polarization with ζ2 =eζ2 = ±1 (in which case ζ1 = ζ3 = eζ1 = eζ3 = 0), then this background from bbproduction can be essentially eliminated.† Most importantly for determining theCP properties of the φ, we note that in Eq. (14) there are no terms containing thesame ζ/eζ-dependent factors that multiply A1, A2 and A3 in Eq.

(6). This remainstrue even if M++ and M−−cannot be neglected (as, in particular, in the case ofthe tt background).

Thus, for example, the background cancels when isolating A1by simultaneously changing the sign of both ζ2 and eζ2.We will not attempt to study the backgrounds in detail here. Instead, we willuse the results of Refs.

[19] and [20] to estimate that a Higgs boson could be seen ifits polarization-averaged event rate is such that there are at least 80 bb decays, or 80tt decays, or 20 clean ZZ (with one Z →l+l−) decays. Of course, a more detailedMonte Carlo would be required to determine accurately the minimal signal as afunction of mφ that would be required in the various channels.

For instance, forbb invariant masses below about 50 GeV, 80 Higgs events in the bb channel would† Even if perfect polarization cannot be achieved, γγ →bb can be dramatically suppressedby requiring that the b and b not emerge close to the beam direction.8

not be adequate. In addition, the number of background events depends stronglyon the polarization mode employed.

As we have already noted, the bb backgroundcan be greatly suppressed if 2λePc = 2eλe ePc ∼+1 and mφ is below about 70% ofEee (so that ⟨ζ2eζ2⟩(mφ) ∼1). This is certainly the appropriate configuration formeasuring A1.

On the other hand, the maximal achievable ⟨ζ2eζ2⟩for Pc = ePc = 0(so as to maximize Pt and ePt) would not be very near 1 and full suppression of thebb background would not be possible when probing A2 and A3.For our illustrative results, we will adopt a machine energy of Eee = 0.5 TeV,an integrated luminosity of Lee = 20 fb−1, and use the rough value of F(W) ≃1in Eq. (9) in computing event rates.

We will search for the values of |A1|, |A2|,and |A3| that maximize the observability of CP violation, subject to the minimumevent number requirements stated above. This search is performed separately foreach of the observables by randomly scanning over all allowed ui values.

We willpresent our results as functions of mφ for various values of mt and tan β. We willnot consider mφ smaller than 60 GeV.

For such low mφ, not only would the bbbackground be large (as noted above), but also it turns out that existing experi-mental constraints from Z decays at LEP require that the φW +W −coupling mustbe so suppressed that a significant number of φ cannot be made in γγ collisions.‡Regarding the statistical significance, NSD, of event number differences deriv-ing from the A1,2,3, it will be useful to define two standardized scenarios, the firstappropriate for measuring A1 and the second for measuring A2 and A3. Supposethe number of events of a given type for unpolarized photons is N. For measuringA1 we take ⟨ζ2⟩≃⟨eζ2⟩= ±1 (this requires Pc ePc ≃1 and mφ <∼70%Eee) andfind 2N(1 ± A1) events.

The statistical significance of the asymmetry fluctuationis then N1SD ≡√2N|A1|. For measuring A2 and A3, we take Pt ≃ePt ≃1 andadopt the “typical” values of ⟨l⟩≃⟨el⟩≃0.4 (not attained until mφ >∼60%Eeeaccording to our estimates) and ⟨ζ2eζ2⟩= 0.5 (we assume that 2λe ≃2eλe ≃±1 sothat this latter average is large, in order to suppress qq backgrounds, when l and elare large — see earlier discussion).

The number of events obtained (after averag-ing over 2λe = 2eλe = ±1) for κ + eκ = +π/4, −π/4 (for A2) or for κ + eκ = 0, π/2(for A3) is N[(1 + ⟨ζ2eζ2⟩) ± ⟨lel⟩A2,3]. Using the typical values stated above weobtain an approximate statistical significance for the event number fluctuation ofN2SD ≡0.13√N|A2|, in the case of A2.

For A3, recall that the signal for CPviolation is |A3| < 1. Thus, what matters in this case is whether the magnitudeof fluctuation predicted for |A3| = 1 could be distinguished from that associatedwith some smaller value of |A3|.

The statistical significance that we shall associatewith this difference will be N3SD ≡0.13√N(1 −|A3|).‡ Recall that the W loop is the most important contributor to the φγγ coupling.9

Figure 1: The values for |A1| (———) and |A2| (−−−−) and (1 −|A3|) (· · · · · · ) whichyield the largest standard scenario statistical significances, N 1SD, N 2SD, and N 3SD, respectively(see text), as a function of mφ. We have taken tan β = 2 and mt = 150 GeV.

Extrema areobtained for 150,000 random choices of the ui subject to the requirement that there be at least80 events in the bb decay channel of the φ, or 20 events in the ZZ (one Z →l+l−) channel, or80 events in the tt channel when the colliding photon polarizations are averaged over.Results for tan β = 2 and mt = 150 GeV are presented in Figs. 1 and 2.

InFig. 1 we plot the values of |A1,2| and 1 −|A3| which yield the maximal statis-tical significance (N1SD, N2SD, or N3SD, as defined above) for detection of each ofthese three observables.

In all cases, we impose the minimal event number require-ments stated earlier. In Fig.

2 we plot the corresponding NSD values themselves.The best statistical significances (as plotted) are achieved using the bb channel formφ < 2mZ, using the ZZ channel for 2mZ < mφ < 2mt, and using the tt channelfor mφ > 2mt. In all cases, the best ui choices are such that u3 is large, therebyguaranteeing a significant contribution to the CP-odd amplitude, o.

Also, as dis-cussed in more detail below, in any particular region of mφ, the ui that maximizethe NSD’s have certain preferred relative signs. Otherwise, the ui choices whichyield the extrema plotted are unnoteworthy.

Fine tuning is not required to obtainNSD values close to those illustrated. We have also verified that fine-tuning ofthe parameters of the 2HDM potential (see Ref.

[3], Eq. (55)) is not necessary toattain these ui choices.

This remains true even if we demand that φ is the Higgseigenstate of lowest mass and that the contribution of the H+ loop to the γγφcoupling is negligible. Of course, if φ is not the lightest eigenstate, then decays10

Figure 2: The maximum statistical significances N 1SD, N 2SD and N 3SD for observing |A1| (———),|A2| (−−−−), and (1 −|A3|) (· · · · · · ), respectively (for the standard scenarios definedin the text), as a function of mφ. We have taken tan β = 2 and mt = 150 GeV.

Extrema areobtained for 150,000 random choices of the ui subject to the requirement that there be at least80 events in the bb decay channel of the φ, or 20 events in the ZZ (one Z →l+l−) channel, or80 events in the tt channel when the colliding photon polarizations are averaged over.such as φ →φ′φ′ ′, φ →φ′Z, etc. become possible, and can easily be dominant.Since the CP asymmetries we consider are defined by the initial photon polariza-tions, presumably these alternative final states would also allow measurement ofthe associated production rate differences.According to Fig.

2, at least one of the asymmetries could be observablethroughout almost the entire mφ range studied. However, we must keep in mindthat the NSD values presented, based on the polarization averages assumed forthe standard scenarios, are overestimated in certain mass regions.

In particular,in the case of A1, for mφ > 2mt ⟨ζ2⟩≃⟨eζ2⟩= ±1 may not be achievable forEee = 0.5 TeV, since mφ > 70%Eee for much of this range. In the case of A2 andA3, the large linear polarizations characterizing our standard A2, A3 scenario canonly be realized for mφ >∼60%Eee.

However, in practice this does not appear to bean important restriction, since viable statistical significances for detection of CPviolation via these latter two asymmetries are mostly attained for mφ > 2mt, inany case. Thus, determination of the CP properties of the φ via A2 and A3 willbe confined to the mφ > 2mt region.Let us now discuss the asymmetry observables corresponding to these maxi-11

mal NSD values, see Fig. 1.

We focus first on A1. |A1| is generally small anddecreases with increasing mφ for mφ < 2mW.

This is because the only significantimaginary contribution to the sum of loop amplitudes comes from the b loop, andthis imaginary part declines in magnitude as τb ln(4/τb). In this region, maximalstatistical significance is achieved for values of u3 >∼0.6, and if u1 and u2 have thesame sign.

The latter implies that the φWW coupling is not small, thereby keepingthe basic production rate significant (recall that the W loop generally dominates).Once mφ is above the W +W −threshold a much larger imaginary part for theCP-even amplitude, e, is possible. In order to maximize interference, preferredvalues of u3 remain large.

Further, u1 and u2 continue to have the same sign;this allows for a large W loop imaginary part, large basic production cross section,and large φ →ZZ branching ratio. Not surprisingly, the best signal is in the ZZchannel for 2mZ < mφ < 2mt.

As mφ passes beyond 2mt, for this case wheremt is significantly larger than mW it is easier to achieve a large t-loop imaginarypart than W-loop imaginary part. The best statistical significance is achieved ifu1 and u2 have opposite signs, so as to suppress the φWW coupling.

This has atwofold effect: the imaginary part of the t loop is enhanced relative to the overallamplitude magnitude (enhancing the level of interference) and φ →tt decays canbe dominant, thereby allowing for the most significant signal to be found in the ttchannel.For this same case of mt = 150 GeV and tan β = 2, A2 and A3 are lesspromising for CP studies, and the reasons behind their behaviors are more complex.We discuss only the mφ > 2mt region, where reasonably large linear polarization ismost likely to be achieved. Here, the best N2SD and N3SD values are attained usingthe tt decay channel.

u1 and u2 are chosen to have appropriately balanced oppositesigns (see the form for sW +W −in Eq. (13)) such as to (simultaneously) severelysuppress the W triangle loop contribution to e and the decays φ →WW, ZZ.

Formoderate values of u3, the CP-even and CP-odd amplitudes can then be madecomparable because of the dominance of the t loop and |A2| and 1 −|A3| take onvalues near unity. Values of N2SD and N3SD above 1 are generally possible.

Overall,however, the suppression arising from the small value of ⟨lel⟩∼0.16 that is presentab initio implies that large statistical significances for these asymmetries wouldrequire much larger luminosity. For this reason, the remainder of our discussionwill focus on A1.The achievable statistical significances for A1 (as well as the other asymmetries)depend upon tan β and mt.

In Fig. 3 results for the maximal achievable statisticalsignificance (subject to minimum event number requirements) for observation ofA1 are given for a variety of other tan β, mt choices.

The corresponding values of|A1| are given in Fig. 4.

We briefly explain the principle features of these plots.First, consider keeping mt fixed at 150 GeV and increasing tan β to 10. Recall thatlarge tan β suppresses the tt coupling and enhances the bb coupling (see Eq.

(13)).For small mφ, |A1| is enhanced compared to small tan β since the (only) imaginarycomponents of e and o come from the b loop and are now much larger. Meanwhile, alarge event rate can be maintained through the W loop contribution to e. However,for 2mt >∼mφ >∼2mW, even though Im e can be large due to the W loop, o came12

Figure 3: The maximum statistical significances for observing |A1|, N 1SD (as defined in the text),as a function of mφ. Various values for mt and tan β are considered: tan β = 2, mt = 200 GeV(———), tan β = 2, mt = 100 GeV (−−−−), tan β = 10, mt = 200 GeV (· · −· · −),tan β = 10, mt = 150 GeV (· −· −), and tan β = 10, mt = 100 GeV (· · · · · · ).

Extrema areobtained as described for Fig 2. Curves terminate when the minimal event number requirementscan no longer be met.mainly from the t loop at moderate tan β, which loop is now severely suppressedby the large tan β value.

For mφ > 2mt, τW is getting small and the W loopcontribution to event rate cannot be kept large.Since the t loop contributionto the φ production rate and B(φ →tt) are also both suppressed, it becomesimpossible to maintain minimal event rates in either the ZZ or tt channels. Thus,detection of A1 for mφ >∼2mW becomes difficult at large tan β.Keeping tan β = 2 and increasing mt from 150 GeV to 200 GeV simply movesthe mφ = 2mt threshold (beyond which the t loop and tt channel dominate A1 andφ decays, respectively) to a higher value.

The systematics behind the behaviors ofN1SD and |A1| are very much as described above for tan β = 2 and mt = 150 GeV.Lowering mt to 100 GeV at tan β = 2 means that for mφ >∼200 GeV the e ando amplitudes can both have large imaginary parts.Thus, for mφ ∼200 GeVboth N1SD and the corresponding |A1| are much larger than for mt = 150 GeV.However, since both τW and τt become small as mφ increases further, the W- and t-loop contributions to e and o (and event rates) fall, and both N1SD and |A1| declinerapidly. Beyond a certain point (mφ ∼275 GeV) there are simply not enough ZZ13

Figure 4: The values for |A1| which yield the largest N 1SD values as a function of mφ. Variousvalues of mt and mφ are illustrated: tan β = 2, mt = 200 GeV (———), tan β = 2, mt = 100 GeV(−−−−), tan β = 10, mt = 200 GeV (· · −· · −), tan β = 10, mt = 150 GeV (· −· −), andtan β = 10, mt = 100 GeV (· · · · · · ).

Extrema are obtained as described for Fig. 2.

Curvesterminate when the minimal event number requirements can no longer be met.events to satisfy our minimal requirement.In summary, of the three possible CP-sensitive polarization asymmetries, A1provides the best opportunities for studying the CP properties of a neutral Higgsboson produced in γγ collisions of polarized back-scattered laser photons. A non-zero value for A1 requires that the φγγ coupling have an imaginary part, as wellas both CP-even and CP-odd contributions.

For a mixed-CP Higgs boson withmφ <∼2mW , measurement of A1 will be easiest if tan β is large since the b loop,which makes the only large contribution to the imaginary part for such mφ values,will be enhanced. For mφ > 2mW, the required imaginary part is dominated bythe W loop (or t-loop if mφ is also > 2mt).

Large tan β makes detection of A1in this region more difficult since the dominant CP-odd contribution derives fromthe t loop, which will be suppressed. Nonetheless, it is clear from our analysisthat collisions of polarized back-scattered laser photons will provide a significantopportunity for determining the CP properties of any neutral Higgs boson thatcan be produced with reasonable event rate.

Certainly, a substantial effort shouldbe made to design a machine with maximal polarization for the incoming electronsand laser beams and the highest possible luminosity.14

AcknowledgementsWe are grateful to D. Borden, D. Caldwell, and T. Barklow for discussionsregarding the degree of polarization achievable for the backscattered photons. Oneof us (JFG) would like to thank the CERN theory group for support during theinitial stages of this work.

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