CP Violation in Decays at Colliders^{1}^{1}1Supported in part by the Committee for Scientific Research under grant 2 0165 91 01, Poland.
Bohdan Grza̧dkowski^{2}^{2}2Email:.
Institute for Theoretical Physics
University of Warsaw
Hoża 69, PL00681 Warsaw, Poland
Abstract
A helicity asymmetry for top quarks originating from Higgs boson decays is investigated within the 2Higgs Doublet Model. The asymmetry is sensitive to CP violation in the scalar sector of the model, whereas it vanishes in the Standard Model. It has been checked that without any fine tuning of parameters the asymmetry can reach . Standard decay patterns and are utilized as spin analyzers to measure the asymmetry. The Higgs production mechanism considered here is . It has been shown that signal from the asymmetry can easy overcome the noise for Higgs bosons produced in future linear colliders at energy operating with integrated luminosity .
1 Introduction
In spite of spectacular successes of experimental highenergy physics (e.g. precision tests of the Standard Model) the origin of CP violation is still a mystery from both experimental and theoretical point of view. As it is known very well there is only one solid experimental signal of CP violation, namely decay [1]. The classical method for incorporating CP violation into a model of electroweak interactions has been proposed two decades ago by Kobayashi and Maskawa [2], it relays simply on explicit CP violation through complex Yukawa couplings. This is the mechanism adopted by the Standard Model (SM) of electroweak interactions. It is obvious from the above that our present experimental knowledge is very limited and theoretical tools adopted as modest as possible. However, fortunately the simplest and most attractive extension of the SM, 2Higgs Doublet Model (2HDM) provides much richer middles to describe CP violation, namely spontaneous and/or explicite CP violation in the scalar sector of the model [3]. As will be seen in the next section both scenarios result in mixing between imaginary and real parts of complex Higgs fields, and a consequence of this is that mass eigenstates do not possess defined CP properties : CP must be violated! This paper is devoted to investigate possibilities for a detection of such sources of CP violation.
Linear highenergy collider (NLC) can prove to be very useful laboratories to study the physics of the Higgs boson. One of the many interesting issues one can investigate there is, for instance, whether or not CP is violated in decay. In this paper, some kind of helicity asymmetries in the final state will be discussed. In general, because of hadronization effects such observables might be contaminated by large theoretical uncertainties. The top quark however, because of its huge mass offers few relevant advantages for the study of CP violation:

If GeV, it would decay before it can form a bound state [4]; therefore the perturbative description is much more reliable.

For the same reason, the spin information of the top quark would not be diluted by hadronization, therefore helicity asymmetries can provide a very useful tool in searching for CP violation.

Again, because of the large mass of the top quark, its properties are sensitive to interactions mediated by Higgs bosons.

The KobayashiMaskawa [2] mechanism of CP violation is strongly suppressed for majority of topquark interactions; therefore it is sensitive to nonconventional sources of CP violation.
There exists in the literature a number of papers [5, 6] investigating CP violation within 2HDM in production, Higgs boson decay and the top quark decay. Here we will consider Higgs boson production in future colliders through the classical Bjorken mechanism followed by the decay where decays in a standard pattern: . The aim of this paper was not only to include a realistic production mechanism into account, but first of all to establish a method of measurement the asymmetry defined for the decay at the level of decay products. We emphasize that the method adopted here controls possible CPviolating effects appearing in the process of Higgs boson production and also in decays. Since the asymmetry will be measured at the level of those effects may mix with CP violation in and influence an interpretation of measurements. We shell prove that the asymmetry discussed here is free of those effects and could be efficiently measured at future linear colliders.
2 The Model
We will consider here 2HDM of electroweak interactions. The model is defined by Yukawa couplings and the Higgs potential. It is known [7] that if two independent vacuum expectation values contributed to quark mass matrices a model would predict treelevel flavour changing neutral currents coupled to Higgs bosons. A standard solution to eliminate them is to impose a discrete symmetry D:
(1) 
Invariant quark Yukawa interactions read:
(2) 
where are generation indices and is defined as .
The most general potential for the model is the following:
(3)  
The first term is symmetric under D symmetry. It has been noticed [8] long time ago that if one being orthodox and naïve, restricts himself to absolutely symmetric Lagrangian (in order to keep a model renormalizable), then the potential is just . However, a symmetric potential does not allow neither for explicite (in the potential) nor spontaneous CP violation. That was the reason why one had to introduce a third Higgs doublet in order to remove FCNC and preserve spontaneous or explicite CP violation in a Higgs sector. However, as showed by Symanzik [9] soft breaking of a symmetry (by operators dimension 3 and less) preserves renormalizability, therefore it is allowed to add to the potential and as we shell see this is exactly what is needed to break CP in the potential. Terms contained in break the D symmetry hard and therefore can not by accepted. ^{3}^{3}3 In this case one would have to introduce all possible interactions of dimension 4 to keep the model renormalizable. That means that FCNC would be necessary.
It is not difficult to find a condition on CP conservation; one can show that if
(4) 
then there exist phases such that the following CP transformation is a symmetry of the model:
(5) 
Therefore, now we know how to break CP explicitly in the potential. It is easy to see that “spontaneous” (by noninvarainat vacuum expectation values) CP breaking is also possible. Let us first choose a phase of field such that is real and a phase of such that coupling is real, then one can expect a complex vacuum expectation value for the second doublet . Solving minimum condition for
(6) 
we get [10]
(7) 
So, we are able to conclude that in the 2HDM both explicit and spontaneous CP violation is possible keeping the model renormalizable and avoiding FCNC at the tree level. Two above scenarios lead to a mixing between real and imaginary parts of Higgs fields in the mass matrix, a consequence of this is that Yukawa interactions can not be invariant under CP:
(8) 
where is a physical Higgs boson and , are functions of mixing angles which diagonalize the mass matrix. In a basis, where longitudinal components of decouple, the rotation matrix is given by ^{4}^{4}4Notice a misprint in the formula (21) of ref. [11], the element (1,2) should read ””.
(9) 
where and . Parameters of the Yukawa coupling (8) are given [11] in terms of the elements of matrix:
(10)  
where , , and .
3 The Asymmetry
In this section we will calculate the following CP violating helicity asymmetry:
(11) 
where by we understand a decay width of the lightest Higgs boson into pair with indicated helicities. Since under CP: , nonzero would be a signal of CP violation. Let us define an effective interaction of the lightest Higgs () with a pair by:
(12) 
and are coefficients calculable order by order within a given model. can be expressed in terms of and , at oneloop approximation we get:
(13) 
Hereafter, for a particle of mass , we adopt a notation . Let us define partial contributions to through the formula:
(14) 
where summation runs over all contributing diagrams shown in fig. 1.
Let us list all partial contributions to the asymmetry. Photon and gluon exchange give:
(15) 
where i=QED, QCD, and . QCD contribution is usually of the order of , whereas the QED one is much smaller, at the level of .
The single exchange contribution can be written as:
(16) 
where and . Single correction is always small, of the order of the QED one.
A Higgs boson exchange gives the following contribution:
(17) 
where corresponds to two lightest Higgs bosons considered here. Numerically, both and are very small, at the level of and , respectively (we use ).
The exchange contribution is the following:
where the function is defined as:
(19) 
is the gauge coupling constant and .
The exchange diagram gives:
(20)  
where
(21) 
Both and diagram gives numerically relevant contributions of the order of few per cent. The above contributions to the asymmetry has been obtained before [6], we agree with their result, however a sign convention adopted here is opposite.
Till now, in the existing literature [6], Higgs selfenergy initial state diagrams have been omitted. Usually their contributions are numerically small and sometimes could be neglected. However, there is a potentially large resonance effect possible, namely if two lightest Higgs bosons have similar masses then mixed selfenergy diagram gives a very large contribution, even if we stay not too close to the resonance : , where is a width of the corresponding Higgs boson. The Higgs boson selfenergy contribution to the asymmetry could be written as:
(22) 
where contributions from , , and to are listed below
(23)  
where . The selfenergy gives the following contribution:
(24) 
After the asymmetry has been properly calculated we are ready to rise a question:
4 How to Measure the Asymmetry?
Linear highenergy colliders provide a cleanest environment for the Higgs boson production. For mass ranges we are considering here the dominant production mechanism is the Bjorken process: . ^{5}^{5}5 The production through the or fusion is less relevant for at . After the Higgs boson is produced it will rapidly decay and we concentrate on mode here since we would like to measure the asymmetry . However, pair would again decay and the only way to measure is to look for some observables defined at the level of decay products which would be sensitive to the asymmetry. Following ideas from our previous paper [12] we shell consider here decays and and look for final states with defined helicities.
Let us consider top quark decays. At the tree level in the highenergy limit the quark always has helicity , therefore helicity conservation tells us that with helicity coming from the top of would like to go forward in the direction of flying top, whereas with will go mainly in the opposite direction. emerging from the top of with the same helicities as above would obviously go in the opposite direction. An analogous picture holds for the decays. It should be noticed that the allowed helicities of , at the tree level and for massless quarks, are , and that helicity conservation never permits to produce together with , respectively. Therefore, it seems natural to consider the following asymmetries:
(25)  
(26)  
(27) 
where denote polar angles of measured in the rest frames with respect to directions seen from the Higgs boson rest frame, respectively. stands for the crosssection obtained by integrating over the full production phase space and over azimuthal angles.
In order to calculate we will need helicity amplitudes for :
(28)  
where denotes an amplitude for a given helicity and is the top quark energy in the rest frame.
For top quark decays, , the following parameterization could be adopted:
(29) 
where are projection operators, is the momentum and is the Kobayashi–Maskawa matrix. Because is on shell, two additional form factors do not contribute. At the tree level and .
Amplitudes for production with helicities and could be written in terms of amplitudes for and for and decays, as follows:
(30) 
Since we are interested in the leading contribution to the asymmetries (which is an interference between oneloop (see fig. 1) and treelevel diagrams) all amplitudes providing higher order corrections, have been neglected above.
A direct calculation at the lowest order leads to the following result:
(31) 
It is very relevant to notice that, in the leading order, there is no contributions from CP violating effects in decays ( decays enter at the tree level; ), therefore the asymmetries measure directly CP violation in the decay . It is worth to emphasize that possible CP violating interactions in the production process are irrelevant since we observe inclusively all Higgs bosons produced and therefore there is no possibility to memorize anything concerning the production mechanism, including possible CP violation. We can therefore conclude that are sensitive purely to CP violation in the Higgs boson decay . There is a close analogy between asymmetries considered here and those we have discussed in ref. [12] for . However, there, we were also able to find an asymmetry which was a good measure of CP violation in decays. It was generated by and helicity configurations for . However, to generate those configurations one needs nonzero amplitudes or for the pair what is, of course, impossible in our case, for pairs produced in the decay .
It is very important to notice another consequence of vanishing and amplitudes for , namely, in the case of Higgs decay we do not need to identify helicities of produced in the decay process. Following ref. [12] it is useful to define the the asymmetry , where and helicity states for are summed:
(32) 
where are defined in the formula (27). In the case of the above asymmetry has been introduced in order to increase the statistics, however here, since the asymmetry counts all possible helicity configurations for , and therefore we do not need to measure .
It is easy to check that is directly related to our initial asymmetry :
(33) 
The above result makes the experimental determination of the asymmetry much easier.^{6}^{6}6We thank W.Y. Keung for bringing this point to our attention.
5 Results
According to most popular proposals for NLC we will consider a machine operating at . Since we are interested in decays, mass of the lighter Higgs boson will be varied in the range , such that top quarks with mass between and could be produced. Since mass adopted for the next heavier Higgs boson is we are never closer to the propagator pole (see selfenergy graphs) as . A width of the lightest Higgs boson varies in the range (for ), therefore we never approach the pole at by more than . We assume here, that the charged Higgs boson is heavy: . From mixing and measurements there exists a lower bound on [13] as a function of the charged Higgs boson mass and , however for heavy charged Higgs boson that bound is not very restrictive and therefore we can safely consider .
In fig. 2 we present results for obtained for fixed values of the Higgsboson mixing angles: , and . Hereafter we use for QCD corrections running evaluated at lighter Higgs mass. As seen from the figure the asymmetry could be substantial, at the level of . However, it is obvious that a big asymmetry is only a necessary condition to observe it. Sufficiently large number of Higgs bosons must be produced in order to see a signal. A necessary and sufficient condition which must be satisfied to observe at the level is that the asymmetry should be greater then statistical fluctuations of the signal:
(34) 
where is the number of pairs produced.
To calculate we assume the Bjorken mechanism for the Higgs boson production calculated consistently within 2HDM we are considering. An integrated luminosity we are adopting is . In order to find a maximal signal we scan over mixing angles for fixed value of , Higgs masses , and the top quark mass. Results for maximal are presented in fig. 3. It is seen that the is substantial and the observation should be possible. In fig. 4 we present values of the asymmetry corresponding to maximal . For is at the level of . Notice that since for increasing we suppress coupling to , therefore for the maximal asymmetry is smaller.
The asymmetry is a ratio of an appropriate rate difference to the total width for the decay , therefore the maximalization procedure is a competition between the minimal width and the maximal rate difference. In order to make sure that we are not supperssing the width, we control the total number of pairs produced and in the maximalization procedure we always demand that at least 35 pairs must be produced.
In order to illustrate the behavior of major effects we present in fig. 5 separated contributions() from gluon, and exchange and mixed self energy for , again the numbers presented correspond to maximal . As it could have been anticipated the gluonic contributions are decreasing while selfenergy ones are increasing with Higgs masses since we are approaching the pole at .
Although, any detailed study of the background is beyond the scope of this work, a few remarks are in order here. For the process there is of course a potential background coming from direct production, however since we assume the Higgs mass known the invariant mass cut should provide an effective way to remove balk of the background assuming sufficient mass resolution. The other useful tool is certainly a presence of a monochromatic boson in the final state. One should also have in mind that we do not take into account any experimental cuts and, of course, some number of events must be lost because of nonperfect efficiency. The results that we have obtained here presumed the narrow width approximation, where all possible interference effects between a production and decays are neglected. In order to justify this we must in addition assume that the final mass resolution is sufficiently good to be sure that and are coming from onshell top quarks.
6 Summary
A helicity asymmetry (see eq.( 11)) for top quarks originating from Higgs boson decay has been investigated within the 2HDM. The asymmetry is sensitive to CP violation in the Higgs sector of the 2HDM, whereas it vanishes in the SM. We have checked that the asymmetry can even reach . is defined for the process , however, since top quarks decay very fast the asymmetry can only be measured through decay products.
Standard decay patterns and has been utilized ^{7}^{7}7Since we assume that the charged Higgs is very heavy, the decay is kinematicaly forbidden. as spin analyzers. We have showed that an angular helicity asymmetries (see eq. (26)) defined in terms of quantum numbers of decay products are simply proportional to the for the decay . It was important to notice that the above proportionality holds even if we assume the most general patterns for and , what means that decays enter only at the tree level and therefore can not provide any extra source of CP noninvariance. We have checked that, in fact, one do not need to measure helicities of and therefore the angular asymmetry is a direct measure of CP violation in .
The Higgs production mechanism considered here was . We have proved that signal from the asymmetry can easy overcome the noise for Higgs bosons produced in future linear colliders at energy operating with integrated luminosity . An experimental evidence for the asymmetry discussed in this letter would be a direct signal of CP violation beyond the SM.
Acknowledgments
We thank J. Abraham for useful discussions.
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