CP violation in SUSY
Abstract
CP violation in supersymmetric models is reviewd with focus on explicit CP violation in the MSSM. The topics covered in particular are CPmixing in the Higgs sector and its measurement at the LHC, CPodd observables in the gaugino sector at the ILC, EDM constraints, and the neutralino relic density.
pacs:
12.60.JvSupersymmetric models and 11.30.ErCharge conjugation, parity, time reversal, and other discrete symmetries] ] ] ] ] ]
1 Introduction
Test of the discrete symmetries, charge conjugation C, parity P, and timereversal T, have played an important role in establishing the structure of Standard Model (SM). In particular, CP violation has been observed in the electroweak sector of the SM in the and systems. It is linked to a single phase in the unitary CabbiboKobayashiMaskawa (CKM) matrix describing transitions between the three generations of quarks; see e.g. Buras:2005xt () for a detailed review. It is important to note that this source of CP violation is strictly flavour nondiagonal.
The strong sector of the SM also allows for CP violation through a dimensionfour term , which is of topological origin. Such a term would lead to flavourdiagonal CP violation and hence to electric dipole moments (EDMs). The current experimental limits on the EDMs of atoms and neutrons Tl (); Hg (); n ()
(1) 
however constrain the strong CP phase to ! A comprehensive discussion of this issue can be found in Pospelov:2005pr (). While appears to be extremly tuned, the CKM contribution to the EDMs is several orders of magnitude below the experimntal bounds, e.g. . Therefore, while providing important constraints, the current EDM bonds still leave ample room for new sources of CP violation beyond the SM.
Such new sources of CP violation are indeed very interesting in point of view of the observed baryon asymmetry of the Universe
(2) 
with , and the number densities of baryons, antibaryons and photons, respectively; see Dine:2003ax (); Cline:2006ts () for recent reviews. The necessary ingredients for baryogenesis Sakharov:1967dj () i) baryon number violation, ii) C and CP violation and iii) departure from equilibrium are in principle present in the SM, however not with sufficient strength. In particular, the amount of CP violation is not enough. This provides a strong motivation to consider CP violation in extensions of the SM, as reviewed e.g. in Ibrahim:2007fb ().
In general, CP violation in extensions of the SM can be either explicit or spontaneous. Explicit CP violation occurs through phases in the Lagrangian, which cannot be rotated away by field redefinitions. This is the standard case in the MSSM, on which I will concentrate in the following. Spontaneous CP violation, on the other hand, occurs if an extra Higgs field develops a complex vacuum expectation value. This can lead to a vanishing term as well as to a complex CKM matrix. Spontaneous CP violation is a very interesting and elegant idea, but difficult to realize in SUSY and obviously not possible in the MSSM (where the Higgs potential conserves CP). There has, however, been very interesting new work on leftright symmetric models and SUSY GUTs. For instance, models based on supersymmetric SO(10) may provide a link with the neutrino seesaw and leptogenesis. I do not follow this further in this talk but refer to Ibrahim:2007fb () for a review.
2 CP violation in the MSSM
In the general MSSM, the gaugino mass parameters (), the higgsino mass parameter , and the trilinear couplings can be complex,
(3) 
(assuming to be real by convention) thus inducing explicit CP violation in the model. Not all of the phases in eq. (3) are, however, physical. The physical combinations indeed are and . They can

affect sparticle masses and couplings through their mixing,

induce CP mixing in the Higgs sector through radiative corrections,

influence CPeven observables like cross sections and branching ratios,

lead to interesting CPodd asymmetries at colliders.
Nontrivial phases, although constrained by EDMs, can hence significantly influence the collider phenomenology of Higgs and SUSY particles, and as we will see also the properties of neutralino dark matter.
Let me note here that CP violation in the MSSM alone is a large field with a vast amount of literature; it is essentially impossible to give a complete review in 25 min. I will hence not try a tour de force but rather present some selected examples, and I apologize to those whose work is not mentioned here. This said, let us begin with the MSSM Higgs sector:
2.1 Higgssector CP mixing
The neutral Higgs sector of the MSSM consists in principle of two CPeven states, and , and one CPodd state, . Complex parameters, eq. (3), here have a dramatic effect, inducing a mixing between the three neutral states through loop corrections Pilaftsis:1998dd (); Demir:1999hj (); Pilaftsis:1999qt (). The resulting mass eigenstates (with by convention) are no longer eigenstates of CP. Owing to the large top Yukawa coupling, the largest effect comes from stop loops, with the size of the CP mixing proportional to Choi:2000wz ()
(4) 
CP mixing in the Higgs sector can change the collider phenomenology quite substantially. For example, it is possible for the lightest Higgs boson to develop a significant CPodd component such that its coupling to a pair of vector bosons becomes vanishingly small. This also considerably weakens the LEP bound on the lightest Higgs boson mass Abbiendi:2006cr (), as illustrated in Fig. 1, which shows the LEP exclusions at 95% CL (mediumgrey or lightgreen) and 99.7% CL (darkgrey or darkgreen) for the CPX scenario with maximal phases; the top mass is taken to be GeV. The CPX scenario Carena:2000ks () is the default benchmark scenario for studying CPviolating Higgsmixing phenomena. It is defined as
(5)  
The free parameters are , the charged Higgsboson pole mass , the common SUSY scale , and the CP phases. Typically one chooses , which leaves and as the relevant ones. The ATLAS discovery potential Schumacher:2004da () for Higgs bosons in the CPX scenario with is shown in Fig. 2. As can be seen, also here there remains an uncovered region at small and small Higgs masses, comparable to the holes at small in Fig. 1,
An overview of the implications for Higgs searches at different colliders is given in Godbole:2004xe (), and a review of MSSM Higgs physics at higher orders, for both CPconserving and CPviolating cases, in Heinemeyer:2004ms (). For an extensive discussion of Higgssector CP violation, see the CPNSH report Kraml:2006ga ().
A question that naturally arises is whether and how the CP properties of the Higgs boson(s) can be determined at the LHC. (At the ILC, which is a highprecision machine in particular for Higgs physics, this can be done quite well, see AguilarSaavedra:2001rg () and references therein). A very promising channel is leptons; cf. the contributions by Godbole et al., Buszello and Marquard, and Bluj in Kraml:2006ga (). Were here follow Godbole et al. Kraml:2006ga (); Godbole:2007cn (): The coupling can be written as in the general form
(6)  
up to a factor , where and the fourmomenta of the two bosons. The terms associated with and are CPeven, while that associated with is CPodd. is totally antisymmetric with . CP violation is be realized if at least one of the CPeven terms is present (i.e. either and/or ) and is nonzero. This can be tested through polar and azimuthal angular distributions in , c.f. Fig. 3. Denoting the polar angles of the fermions in the rest frames of the bosons by and , we have e.g.
(7) 
where are the threevectors of the corresponding fermions with and in their parent ’s rest frame but and in the Higgs rest frame, see Fig. 3. The angular distribution in () for a CPodd state is , corresponding to transversely polarized bosons, which is very distinct from the purely CPeven distribution proportional to for longitudinally polarized bosons in the large Higgs mass limit. will introduce a term linear in leading to a forwardbackward asymmetry. The distribution for is shown in Fig. 4 for a Higgs mass of GeV and a purely scalar, purely pseudoscalar and CPmixed scenario. The asymmetry is absent if CP is conserved (for both CPodd and CPeven states) but is nonzero if while simultaneously . Another probe of CP violation is the azimuthal angular distribution with the angle between the planes of the fermion pairs, see Fig. 3. For a detailed discussion of various distributions and asymmetries sensitive to CP violation in leptons, see Godbole:2007cn ().
Another possibility to test Higgs CP mixing at the LHC are correlations arising in the production process. Here the azimuthal angle correlations between the two additional jets in events have emerged as a promising tool Plehn:2001nj (). Higgs boson production in association with two tagging jets, analysed in detail in DelDuca:2006hk (), is mediated by electroweak vector boson fusion and by gluon fusion. The latter proceeds through topquark loops, which induce an effective vertex. Writing the Yukawa coupling as , where and denote scalar and pseudoscalar Higgs fields, the tensor structure of the effective vertex has the form Hankele:2006ma (); Klamke:2007cu ()
(8) 
with
(9) 
The azimuthal angle correlation of the two jets is hence sensitive to the CPnature of the Yukawa coupling. To resolve interference effects between the CPeven coupling and the CPodd coupling it is, however, important to measure the sign of . This can be done by defining as the azimuthal angle of the “away” jet minus the azimuthal angle of the “toward” jet with respect to the beam direction Hankele:2006ma (). The corresponding distributions, for two jets with GeV, , and , are shown in Fig. 5 for three scenarios of CPeven and CPodd Higgs couplings Klamke:2007cu (). All three cases are well distinguishable, with the maxima in the distributions directly connected to the size of the scalar and pseudoscalar contributions, and .
2.2 Gauginos and sfermions
The CPviolating phases in (3) directly enter the neutralino, chargino, and sfermion mass matrices, hence having an important impact on the masses and couplings of these particles. This is particularly interesting for the precision measurements envisaged at the ILC. The effects of CP phases in measurements of neutralinos, charginos, and sfermions at the ILC have been studied in great detail by various groups; see below as well as references in Ibrahim:2007fb (); MoortgatPick:2005cw (). They fall into two different classes. On the one hand, there are CPeven observables: spartice masses, cross sections, branching ratios, etc.. If measured precisely enough, they allow for a parameter determination, either analytically Choi:2000ta (); Kneur:1998gy () or through a global fit Bartl:2003pd (). Beam polarization MoortgatPick:2005cw () is essential, but some ambiguities in the phases always remain. We do not discuss this in more detail here. On the other hand, there are CPodd (or Todd) observables, e.g. rate asymmetries or tripleproduct asymmetries, which are a direct signal of CP violation. Indeed the measurement of CPodd effects is necessary to prove that CP is violated, and to determine the model parameters, including phases, in an unambiguous way.
An expample for a rate asymmetry is the chargino decay into a neutralino and a boson, . Here, nonzero phases can induce an asymmetry between the decay rates of and ,
(10) 
through absorptive parts in the oneloop corrections Eberl:2005ay (). Figure 6 shows the dependence of on for GeV, GeV, GeV, GeV, and various . has its maximum at and is larger at large negative values of the phase of . The obvious advantage of such a rate asymmetry is that it can be measured in a ‘simple’ counting experiment. Analogous asymmetries have been computed for in Christova:2002ke (); Christova:2006fb (); Frank:2007ca (). Ref. Christova:2006fb () also discusses CPviolating forwardbackward asymmetries.
Tripleproduct asymmetries rely on spin correlations between sparticle production and decay processes. They have been computed for neutralino Kizukuri:1990iy (); Choi:1999cc (); Barger:2001nu (); Bartl:2003tr (); Choi:2004rf (); AguilarSaavedra:2004dz (); Bartl:2004jj (); Choi:2005gt () and chargino Bartl:2004vi (); Kittel:2004kd (); Bartl:2006yv () production in followed by two or threebody decays. Let me take the most recent work Bartl:2006yv () on charginopair production with subsequent threebody decay as an illustrative example. The processes considered are () at a linear collider with longitudinal beam polarizations, followed by threebody decays of the ,
(11) 
where . It is assumed that the momenta , , and of the associated particles can be measured or reconstructed. The relevant triple products are:
(12)  
(13) 
Note that , relates momenta of initial, intermediate and final particles, whereas , uses only momenta from the initial and final states. Therefore, both triple products depend in a different way on the production and decay processes. From one can define Todd asymmetries
(14) 
where is the number of events for which . A genuine signal of CP violation is obtained by combining with the corresponding asymmetry for the chargeconjugated processes:
(15) 
Figure 7 shows the phase dependence of for followed by for GeV and polarized beams. The authors conclude that can be probed at the level in a large region of the MSSM parameter space, while has a somewhat lower sensitivity.
2.3 EDM constraints
Let us next discuss the EDM constraints in some more detail. The constraints eq. (1), especially the one on , translate into a tight bound on the electron EDM,
(16) 
Setting all soft breaking parameters in the selectron and gaugino sector equal to , one can derive a simplified formula for the oneloop contributions Falk:1999tm ()
(17)  
where , and by convention. Note the enhancement of the first term. It is the main reason why the phase of is more severely constrained than the phases of the parameters. The phases of the third generation, , only enter the EDMs at the twoloop level. However, there can be a similar enhancement for these twoloop contributions Pilaftsis:1999bq (), so they have to be taken into account as well.
Indeed, the EDM constraints pose a serious problem in the general MSSM: for GeV masses and phases, the EDMs are typically three(!) orders of magnitude too large ellis (); buch (); pol (); dug (). Some efficient suppression mechanism is needed to satisfy the experimental bounds. The possibilities include

small phases,

heavy sparticles nath (); ko (); ckn (); Falk:1995fk (),

flavour offdiagonal CP phases Abel:2001vy (),

lepton flavour violation Bartl:2003ju ().
Detailed analyses of the EDM constraints have recently been performed e.g. in Pospelov:2005pr (); Olive:2005ru (); Ayazi:2007kd (). As example that large phases can be in agreement with the current EDM limits, Fig. 8 shows the results for the CMSSM benchmark point D of Olive:2005ru (), which has . The strongest constraint comes from the EDM of Tl; that of Hg is not shown because it is satisfied over the whole plane. As can be seen, for this benchmark point there is no limit to , while .
2.4 Neutralino relic density
If the is the lightest supersymmetric particle (LSP) and stable, it is a very good cold dark matter candidate. In the framework of thermal freezeout, its relic density is , where is the thermally averaged annihilation cross section summed over all contributing channels. These channels are: annihilation of a bino LSP into fermion pairs through channel sfermion exchange in case of very light sparticles; annihilation of a mixed binoHiggsino or binowino LSP into gauge boson pairs through channel chargino and neutralino exchange, and into topquark pairs through channel exchange; and annihilation near a Higgs resonance (the socalled Higgs funnel); and finally coannihilation processes with sparticles that are close in mass with the LSP. Since the neutralino couplings to other (s)particles sensitively depend on CP phases, the same can be expected for and hence .
The effect of CP phases on the neutralino relic density was considered in Falk:1995fk (); Gondolo:1999gu (); Nihei:2004bc (); Argyrou:2004cs (); Balazs:2004ae (); Gomez:2005nr (); Choi:2006hi (); Cirigliano:2006dg (); Lee:2007ai (), although only for specific cases. The first general analysis, (i) including all annihilation and coannihilation processes and (ii) separating the phase dependece of the couplings from pure kinematic effects, was done in Belanger:2006qa ().
It was found that modifications in the couplings due to nontrivial CP phases can lead to variations in the neutralino relic density of up to an order of magnitude. This is true not only for the Higgs funnel but also for other scenarios, like for instance the case of a mixed binohiggsino LSP. Even in scenarios which feature a modest phase dependence once the kinematic effects are singled out, the variations in are comparable to (and often much larger than) the range in of the WMAP bound. Therefore, when aiming at a precise prediction of the neutralino relic density from collider measurements, it is clear that one does not only need precise sparticle spectroscopy but one also has to precisely measure the relevant couplings, including possible CP phases.
This is illustrated in Fig. 9, which shows the regions where the relic density is in agreement with the WMAP bound, , for the case of a mixed binohiggsino LSP. When all phases are zero, only the narrow blue (dark grey) band is allowed. When allowing all phases to vary arbitrarily, while still satisfying the EDM constraints, the allowed band increases to the the green (light grey) region. In the – plane (left panel), the allowed range for increases roughly from GeV to GeV for a given . In terms of relative mass differences (right panel) this means that in the CPviolating case much smaller – mass differences can be in agreement with the WMAP bound than in the CPconserving case.
3 Conclusions
The observed baryon asymmetry of the Universe necessiates new sources of CP violation beyond those of the SM. I this talk, I have discussed effects of such new CP phases, focussing on the case of the MSSM. The topics covered include CPmixing in the Higgs sector and its measurement at the LHC, CPodd observables in the gaugino sector at the ILC, EDM constraints, and the neutralino relic density. Each topic was discussed by means of some recent example(s) from the literature. For a more extensive discussion, in particular of topice that could not be covered here, I refer the reader to the recent review by Ibrahim and Nath Ibrahim:2007fb ().
References
 (1) A. J. Buras, arXiv:hepph/0505175.
 (2) B. C. Regan et al., Phys. Rev. Lett. 88, 071805 (2002).
 (3) M. V. Romalis, W. C. Griffith and E. N. Fortson, Phys. Rev. Lett. 86, 2505 (2001).
 (4) P. G. Harris et al., Phys. Rev. Lett. 82, 904 (1999).
 (5) M. Pospelov and A. Ritz, Annals Phys. 318 (2005) 119 [arXiv:hepph/0504231].
 (6) M. Dine and A. Kusenko, Rev. Mod. Phys. 76 (2004) 1 [arXiv:hepph/0303065].
 (7) J. M. Cline, arXiv:hepph/0609145.
 (8) A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5 (1967) 32 [JETP Lett. 5 (1967 SOPUA,34,392393.1991 UFNAA,161,6164.1991) 24].
 (9) T. Ibrahim and P. Nath, arXiv:0705.2008 [hepph].
 (10) A. Pilaftsis, Phys. Lett. B435, 88 (1998), [arXiv:hepph/9805373].
 (11) D. A. Demir, Phys. Rev. D60, 055006 (1999), [arXiv:hepph/9901389].
 (12) A. Pilaftsis and C. E. M. Wagner, Nucl. Phys. B553, 3 (1999), [arXiv:hepph/9902371].
 (13) S. Y. Choi, M. Drees and J. S. Lee, Phys. Lett. B481, 57 (2000), [arXiv:hepph/0002287].
 (14) The ALEPH, DELPHI, L3 and OPAL Collaborations, and the LEP Working Group for Higgs Boson Searches, G. Abbiendi et al., Eur. Phys. J. C 47 (2006) 547 [arXiv:hepex/0602042].
 (15) M. S. Carena, J. R. Ellis, A. Pilaftsis and C. E. M. Wagner, Phys. Lett. B 495 (2000) 155 [arXiv:hepph/0009212].
 (16) M. Schumacher, arXiv:hepph/0410112.
 (17) R. M. Godbole et al., arXiv:hepph/0404024.
 (18) S. Heinemeyer, Int. J. Mod. Phys. A21, 2659 (2006), [arXiv:hepph/0407244].
 (19) S. Kraml et al. (eds.), Workshop on CP studies and nonstandard Higgs physics, CERN2006009, arXiv:hepph/0608079.
 (20) J. A. AguilarSaavedra et al. [ECFA/DESY LC Physics Working Group], TESLA Technical Design Report Part III: Physics at an Linear Collider, arXiv:hepph/0106315.
 (21) R. M. Godbole, D. J. Miller and M. M. Muhlleitner, arXiv:0708.0458 [hepph].
 (22) T. Plehn, D. L. Rainwater and D. Zeppenfeld, Phys. Rev. Lett. 88, 051801 (2002) [arXiv:hepph/0105325].
 (23) V. Del Duca et al., JHEP 0610 (2006) 016 [arXiv:hepph/0608158].
 (24) V. Hankele et al., Phys. Rev. D 74 (2006) 095001 [arXiv:hepph/0609075].
 (25) G. Klamke and D. Zeppenfeld, JHEP 0704 (2007) 052 [arXiv:hepph/0703202].
 (26) G. A. MoortgatPick et al., arXiv:hepph/0507011.
 (27) S. Y. Choi, A. Djouadi, M. Guchait, J. Kalinowski, H. S. Song and P. M. Zerwas, Eur. Phys. J. C 14 (2000) 535 [arXiv:hepph/0002033].
 (28) J. L. Kneur and G. Moultaka, Phys. Rev. D 59 (1999) 015005 [arXiv:hepph/9807336]; Phys. Rev. D 61 (2000) 095003 [arXiv:hepph/9907360].
 (29) A. Bartl, S. Hesselbach, K. Hidaka, T. Kernreiter and W. Porod, Phys. Rev. D 70 (2004) 035003 [arXiv:hepph/0311338].
 (30) H. Eberl, T. Gajdosik, W. Majerotto and B. Schrausser, Phys. Lett. B 618 (2005) 171 [arXiv:hepph/0502112].
 (31) E. Christova, H. Eberl, W. Majerotto and S. Kraml, Nucl. Phys. B 639 (2002) 263 [Erratumibid. B 647 (2002) 359] [arXiv:hepph/0205227]; JHEP 0212 (2002) 021 [arXiv:hepph/0211063].
 (32) E. Christova, H. Eberl, E. Ginina and W. Majerotto, JHEP 0702 (2007) 075 [arXiv:hepph/0612088].
 (33) M. Frank and I. Turan, Phys. Rev. D 76 (2007) 016001 [arXiv:hepph/0703184]; arXiv:0708.0026 [hepph].
 (34) Y. Kizukuri and N. Oshimo, Phys. Lett. B 249 (1990) 449.
 (35) S. Y. Choi, H. S. Song and W. Y. Song, Phys. Rev. D 61 (2000) 075004 [arXiv:hepph/9907474].
 (36) V. D. Barger, T. Falk, T. Han, J. Jiang, T. Li and T. Plehn, Phys. Rev. D 64 (2001) 056007 [arXiv:hepph/0101106].
 (37) A. Bartl, H. Fraas, O. Kittel and W. Majerotto, Phys. Rev. D 69 (2004) 035007 [arXiv:hepph/0308141]; Eur. Phys. J. C 36 (2004) 233 [arXiv:hepph/0402016].
 (38) S. Y. Choi, M. Drees and B. Gaissmaier, Phys. Rev. D 70 (2004) 014010 [arXiv:hepph/0403054].
 (39) J. A. AguilarSaavedra, Nucl. Phys. B 697 (2004) 207 [arXiv:hepph/0404104].
 (40) A. Bartl, H. Fraas, S. Hesselbach, K. HohenwarterSodek and G. A. MoortgatPick, JHEP 0408 (2004) 038 [arXiv:hepph/0406190].
 (41) S. Y. Choi, B. C. Chung, J. Kalinowski, Y. G. Kim and K. Rolbiecki, Eur. Phys. J. C 46 (2006) 511 [arXiv:hepph/0504122].
 (42) A. Bartl, H. Fraas, O. Kittel and W. Majerotto, Phys. Lett. B 598 (2004) 76 [arXiv:hepph/0406309].
 (43) O. Kittel, A. Bartl, H. Fraas and W. Majerotto, Phys. Rev. D 70 (2004) 115005 [arXiv:hepph/0410054].
 (44) A. Bartl, H. Fraas, S. Hesselbach, K. HohenwarterSodek, T. Kernreiter and G. MoortgatPick, Eur. Phys. J. C 51 (2007) 149 [arXiv:hepph/0608065].
 (45) T. Falk, K. A. Olive, M. Pospelov and R. Roiban, Nucl. Phys. B 560, 3 (1999) [arXiv:hepph/9904393].
 (46) A. Pilaftsis, Phys. Rev. D 62, 016007 (2000) [arXiv:hepph/9912253].
 (47) J. R. Ellis, S. Ferrara and D. V. Nanopoulos, Phys. Lett. B 114 (1982) 231.
 (48) W. Buchmuller and D. Wyler, Phys. Lett. B 121 (1983) 321.
 (49) J. Polchinski and M. B. Wise, Phys. Lett. B 125 (1983) 393.
 (50) M. Dugan, B. Grinstein and L. J. Hall, Nucl. Phys. B 255 (1985) 413.
 (51) P. Nath, Phys. Rev. Lett. 66 (1991) 2565.
 (52) Y. Kizukuri and N. Oshimo, Phys. Rev. D 45 (1992) 1806; Phys. Rev. D 46 (1992) 3025.
 (53) A. G. Cohen, D. B. Kaplan and A. E. Nelson, Phys. Lett. B 388 (1996) 588 [arXiv:hepph/9607394].
 (54) T. Falk, K. A. Olive and M. Srednicki, Phys. Lett. B354, 99 (1995), [arXiv:hepph/9502401].
 (55) T. Falk and K. A. Olive, Phys. Lett. B 375 (1996) 196 [arXiv:hepph/9602299]; Phys. Lett. B 439 (1998) 71 [arXiv:hepph/9806236].
 (56) T. Ibrahim and P. Nath, Phys. Lett. B 418 (1998) 98 [arXiv:hepph/9707409]; Phys. Rev. D 57 (1998) 478 [Erratumibid. D 58 (1998) 019901, D 60 (1999) 079903] [arXiv:hepph/9708456]; Phys. Rev. D 58 (1998) 111301 [Erratumibid. D 60 (1999) 099902] [arXiv:hepph/9807501].
 (57) M. Brhlik, G. J. Good and G. L. Kane, Phys. Rev. D 59 (1999) 115004 [arXiv:hepph/9810457].
 (58) M. Brhlik, L. L. Everett, G. L. Kane and J. Lykken, Phys. Rev. Lett. 83 (1999) 2124 [arXiv:hepph/9905215].
 (59) A. Bartl, T. Gajdosik, W. Porod, P. Stockinger and H. Stremnitzer, Phys. Rev. D 60 (1999) 073003 [arXiv:hepph/9903402].
 (60) S. Pokorski, J. Rosiek and C. A. Savoy, Nucl. Phys. B 570 (2000) 81 [arXiv:hepph/9906206].
 (61) R. Arnowitt, B. Dutta and Y. Santoso, Phys. Rev. D 64 (2001) 113010 [arXiv:hepph/0106089].
 (62) S. Y. Ayazi and Y. Farzan, JHEP 0706 (2007) 013 [arXiv:hepph/0702149].
 (63) S. Abel, S. Khalil and O. Lebedev, Nucl. Phys. B 606 (2001) 151 [arXiv:hepph/0103320].
 (64) A. Bartl, W. Majerotto, W. Porod and D. Wyler, Phys. Rev. D 68 (2003) 053005 [arXiv:hepph/0306050].
 (65) K. A. Olive, M. Pospelov, A. Ritz and Y. Santoso, Phys. Rev. D 72 (2005) 075001 [arXiv:hepph/0506106].
 (66) P. Gondolo and K. Freese, JHEP 07, 052 (2002), [arXiv:hepph/9908390].
 (67) T. Nihei and M. Sasagawa, Phys. Rev. D70, 055011 (2004), [arXiv:hepph/0404100].
 (68) M. Argyrou, A. B. Lahanas, D. V. Nanopoulos and V. C. Spanos, Phys. Rev. D70, 095008 (2004), [arXiv:hepph/0404286].
 (69) C. Balazs, M. Carena, A. Menon, D. E. Morrissey and C. E. M. Wagner, Phys. Rev. D71, 075002 (2005), [arXiv:hepph/0412264].
 (70) M. E. Gomez, T. Ibrahim, P. Nath and S. Skadhauge, Phys. Rev. D72, 095008 (2005), [arXiv:hepph/0506243].
 (71) S. Y. Choi and Y. G. Kim, Phys. Lett. B 637, 27 (2006) [arXiv:hepph/0602109].
 (72) V. Cirigliano, S. Profumo and M. J. RamseyMusolf, JHEP 0607 (2006) 002 [arXiv:hepph/0603246].
 (73) J. S. Lee and S. Scopel, Phys. Rev. D 75 (2007) 075001 [arXiv:hepph/0701221].
 (74) G. Belanger, F. Boudjema, S. Kraml, A. Pukhov and A. Semenov, Phys. Rev. D 73 (2006) 115007 [arXiv:hepph/0604150].