Q-balls generically exist in the supersymmetric extensions of the standard
model. Taking into account the additional sources of CP violation, which are
naturally accomodated by the supersymmetric models, it is shown that the Q-ball
matter depends additively on individual CP phases, whereas mass per unit charge
in the Q-ball depends only on the relative phases. There are regions of the
parameter space where there is no stable Q-ball solution in the CP-conserving
limit whereas finite CP phases induce a stable Q-ball.Comment: 6 p

Boz, M.

Demir, D. A.

Pak, N. K.

English

arXiv

Q-balls generically exist in the supersymmetric extensions of the standard model. Taking into account the additional sources of CP violation, which are naturally accommodated by the supersymmetric models, it is shown that the Q-ball matter depends additionally on individual CP phases, whereas mass per unit charge in the Q-ball depends only on the relative phases. There are regions of the parameter space where there is no stable Q-ball solution in the CP-conserving limit whereas finite CP phases induce a stable Q-ball

Boz, M

Demir, DA

Pak, NK

Middle East Technical University Research Information System

Q-ball formation in the MSSM with explicit CP violation

Pak, Namık Kemal

OpenMETU (Middle East Technical University)

arXiv:hep-ph/0004070v1  7 Apr 2000Modern Physics Letters A,❢c World Scientific Publishing CompanyIC/00/45Q–ball formation in the MSSM with explicit CP violationM. BozPhysics Department, Hacettepe UniversityAnkara, 06533, TurkeyD. A. DemirThe Abdus Salam International Center For Theoretical PhysicsTrieste, I-34100, ItalyN. K. PakPhysics Department, Middle East Technical UniversityAnkara, 06531, TurkeyAbstractQ–balls generically exist in the supersymmetric extensions of the standardmodel. Taking into account the additional sources of CP violation, whichare naturally accomodated by the supersymmetric models, it is shown thatthe Q–ball matter depends additively on individual CP phases, whereas massper unit charge in the Q–ball depends only on the relative phases. There areregions of the parameter space where there is no stable Q–ball solution in theCP–conserving limit whereas finite CP phases induce a stable Q ball.1. IntroductionNon-topological solitons, in particular Q–balls, are extended objects with finitemass and spatial extension, and arise in scalar field theories when there is an exactcontinious symmetry and some kind of attractive interaction, as already classifiedby Coleman 1.As first pointed out by Kusenko 2, Q-balls naturally exist in supersymmetrictheories thanks to global baryon (B) and lepton (L) number symmetries. Besides,theories with an extended scalar sector can support non–baryonic Q–balls 3. Q–balls have found applications in modelling several physical processes ranging fromleptoquarks to dark matter 4,5.It is well known that the MSSM has unremovable physical phases which can beidentified with the phases of µ parameter and A terms 6. According to the vacuumstability arguments these phases relax to CP conserving points 7. However, thesame arguments are not sufficient to relax the phases in the minimal extensions 8.Hence, it is plausable to take these phases finite and look for their effects in lowenergy processes.1As summarized above, the MSSM predicts the existence of both non–topologicalsolitons and finite CP violation. However, so far the investigations on the Q–ballformation in the MSSM have not dealt with the effects of the CP violation. In thisshort note we investigate the effects of explicit CP violation in the MSSM on theQ–ball formation. In the next section we discuss this issue in detail using the MSSMscalar potential. We particularly analyze the effects of the phases in the trilinearcouplings and the µ parameter. In the last section we summarize the main findings.2. MSSM with explicit CP violation and Q-ballsThe MSSM scalar sector contains two Higgs doublets (with opposite hyper-charge) and scalar partners of quarks and leptons. The supersymmetry and gaugesymmetry are broken by the soft supersymmetry breaking terms which introduce anumber of mass parameters to the potential. Both the Higgsino Dirac mass termµ and the trilinear couplings in soft terms are complex, and they lead to CP vi-olation beyond the CKM matrix already present in the SM. Denoting the neutralcomponents of the Higgs doublets as φ1 and φ2, the MSSM scalar potential for onegeneration of sfermions reads as:VMSSM =[(huAuφ2 − huµ∗φ∗1)ũLũ∗R − (hdAdφ1 − hdµ∗φ∗2)d̃Ld̃∗R−(heAeφ1 − heµ∗φ∗2)ẽLẽ∗R + h.c.]+m2Q̃|Q̃|2 +m2ũ|ũR|2 +m2d̃R|d̃R|2 +m2L̃|L̃|2 +m2ẽR |ẽR|2+m21|φ1|2 +m22|φ2|2 + |µ|2|φ2|2 + |µ|2|φ1|2 + h2e|L̃2||ẽR|2+h2d|Q̃2||d̃R|2 + h2e|L̃2||ẽR|2 + h2u|Q̃2||ũR|2 + h2e|φ1|2|ẽR|2+h2d|φ1|2|d̃R|2 + h2e|φ1ẽL|2 + h2u|φ2ũL|2 + h2d|φ1d̃L|2 (1)plus the D term contributions which will not be shown explicitely. This scalarpotential has two global symmetries: U(1)B and U(1)L corresponding to baryonnumber and lepton number symmetries, respectively. These are the exact symme-tries of the theory and their breaking (spontaenous and otherwise) lead to B– andL– violating processes. As was emphasized in Refs. 1 and 2 it is mainly the cubiccouplings that generate B–ball or L–ball type solitonic solutions. It is appearentthat slepton doublet L̃ and right–handed slepton ẽR contribute to L–balls whereasone needs the squark doublet Q̃ and right–handed squarks ũR and d̃R to form B–balls. In both cases Higgs fields are necessary. In this form the scalar potentialinvolves several scalar fields, and a true analysis of the Q–ball formation requires aminimization of the multi–field quantity 1m2eff (φ1, · · · , ẽR) ≡ 2VMSSM/∑Ballcharge× |field|2 (2)that guarantees the stability of the Q–ball against decaying into its constitutents.In this formula charge and field denote the baryon number (lepton number) andsquark fields (slepton fields). Instead of dealing with coupled equations of motion forfields contributing to a particular Q–ball, practically one can describe the nature ofthe Q–ball by using a single scalar degree of freedom 2. This approximation is quiteaccurate especially forD– and F–flat potentials as the degrees of freedom orthogonalto flat directions will be much more massive 4,5. For this purpose it is convenientto introduce a scalar field ϕ representing the Q-ball matter, and decompose thecomponent fields in terms of ϕ using the dimensionless parameters ξi, with ξ < 1and∑i ξ2i = 1, as follows:ẽL,R = ξeL,Rϕ, ũL,R = ξuL,Rϕ, d̃L,R = ξdL,Rϕ, φ1,2 = ξ1,2ϕ . (3)2Using this decomposition, the scalar potential VMSSM takes the formVϕ = M2ϕ|ϕ|2 +McReϕ|ϕ|2 −MsImϕ|ϕ|2 + λ|ϕ|4 (4)for both L– and B–balls. Here M2ϕ is the linear combination of the scalar quadraticmass parameters in VMSSM . The quartic coupling λ is a linear combination ofYukawa couplings hu,d,e and gauge couplings g3,2,1 following, respectively, from theF–term and D–term contributions.As the general analyses of Q–ball formation Refs. 1, 2, 3 show explicitly, thecrucial parameters in Eq. (4) are the trilinear mass parameters Mc and Ms. ForL–balls one hasMc = 2[− cos(φAe)Ãe + cos(φµ)µ̃](5)Ms = 2[− sin(φAe)Ãe + sin(φµ)µ̃](6)whereÃe = he|Ae|ξ1ξeLξeRµ̃ = he|µ|ξ2ξeLξeR (7)where φAe and φµ are the phases of Ae and µ parameter, respectively.For B–balls, however, Mc and Ms have the following expressionsMc = 2[cos(φAu)Ãu − cos(φAd)Ãd + cos(φµ)µ̃](8)Ms = 2[sin(φAu)Ãu − sin(φAd)Ãd + sin(φµ)µ̃](9)whereÃu = hu|Au|ξ2ξuLξuRÃd = hd|Ad|ξ1ξdLξdRµ̃ = |µ|(hdξ2ξdLξdR − huξ1ξuLξuR) . (10)The form of the scalar potential Eq. (4) is such that U(1)B or U(1)L symmetriesare not manifest at all. In particular, Imϕ and Reϕ refer to Higgs fields φ1,2which do not contribute to the charge of the Q–ball. For instance, in thin–wallapproximation and in the notation of Ref. 2, the total charge and the effective massof the B–ball are respectively given byB = 2aω|ϕ|2Vm2eff =1a[M2ϕ +McReϕ−MsImϕ+ λ{(Reϕ)2 + (Imϕ)2}]where a =13[ξ2uL + ξ2uR+ ξ2dL + ξ2dR](11)Thus, the total charge vanishes if squarks are absent; that is, the Higgs fields do notplay a role in charge accumulation. However, there is no stable Q–ball if the Higgsfields are absent, as can be seen from vanishing of the trilinear couplings Mc andMs. In this sense, Re and Im parts of ϕ in Eq. (4) refer to the time–independentphase of ϕ generated by the non–trivial phases in µ and Au,d,e parameters.3As is seen from Eq. (4), the main effect of complex µ and Au,d,e is to introduceMs 6= 0 which is proportional to Imϕ. There are three distinct limits in which theQ–ball matter gains different CP characteristics depending on the values of Mc andMs.So far discussions of Q–balls have been based on purely real µ and Au,d,e inwhich case Ms ≡ 0. Then only Reϕ has a trilinear coupling and stable Q–matteris thus composed of Reϕ which is purely CP even.In the opposite limit, that is, for purely imaginary µ and Au,d,e one has Mc ≡ 0.Then only Imϕ has a trilinear coupling, and thus, the resulting Q–ball is composedof Imϕ, and Q–matter is purely CP odd.In the general case, where µ and Au,d,e are complex parameters with nonvan-ishing real and imaginary parts, both Imϕ and Reϕ contribute to the Q–matter.Namely, m2eff is minimized for Reϕ = −Mc/2λ and Imϕ = Ms/2λ, so that massper unit charge for B–ball reads asm2eff (B − ball) =1a[M2ϕ −14λ(M2c +M2s )]. (12)According to Coleman’s theorem 1, if 0 < m2eff < M2ϕ/a then the resulting B-ball isstable. One also notices that m2eff depends explicitly on the relative phase betweenany pair of Ãd, Ãu and µ̃, using Eq.(8):m2eff (B − ball) =1a[M2ϕ −1λ(Ãu2+ Ãd2+ µ̃2 − 2ÃuÃd cos(φAu − φAd)+2Ãuµ̃ cos(φAu − φµ)− 2Ãdµ̃ cos(φAd − φµ))](13)Depending on the values of these CP phases m2eff takes on a range of values.Therefore, mass per unit charge in the B–ball varies with the CP violating phases.To illustrate this, one can consider the simple case of ũLũR B–ball, that is, Ad = 0.In this case m2eff varies from its minimum value (φAu − φµ = 0)[m2eff (B − ball)]min=1a[M2ϕ −1λ(Ãu + µ̃)2 ](14)to the maximal value (φAu − φµ = π)[m2eff (B − ball)]max=1a[M2ϕ −1λ(Ãu − µ̃)2 ](15)with the mean (φAu − φµ = π/2)[m2eff (B − ball)]mean=1a[M2ϕ −1λ(Ãu2+ µ̃2) ](16)taking µ̃ positive. One notices that none of the conditions above implies a specificvalue for the phases φAu and φµ. Indeed the mass parameters Mc and Ms take onthe following valuesMc = 2(Ãu + µ̃)cosφAu , Ms = 2(Ãu + µ̃)sinφAuMc = 2(Ãu − µ̃)cosφAu , Ms = 2(Ãu − µ̃)sinφAu (17)Mc = 2(Ãu cosφAu + µ̃ sinφAu), Ms = 2(Ãu sinφAu − µ̃ cosφAu)4for φAu − φµ = 0, π and π/2, respectively. Hence, despite the phase independenceof mass per unit charge (14)–(16), the value of the condensate, determined byMc and Ms, is an explicit function of the CP phases. This follows from the factthat the Q–matter ϕ depends on the individual phases additively whereas m2effdepends only on the relative phases. This particular pattern of phase structure canbe important for Q–ball formation. In the purely CP–conserving limit, one wouldhave m2eff =[m2eff (B − ball)]mintogether with Mc = 2(Ãu + µ̃)and Ms = 0,correspondig to the first line of (17) with φAu = 0. However, as the Eqs. (14)–(16)and (17) suggest clearly the nonvanishing CP phases offer more alternatives.To see the implications of such a phase dependence, one can consider the specialcase of Ãu = µ̃ and µ̃2/λ = M2ϕ/4. Then, for φAu − φµ = 0, one has m2eff = 0 withnonvanishing Mc and Ms. Therefore, for this parameter set one has a massless Q–ball, or equivalently, a Q–matter distribution over entire space. In this case, at leastin the thin–wall approximation, there is no macroscopic structure with finite size.On the other hand, for φAu − φµ = π, m2eff = M2ϕ/a, and this corresponds to thecritical value ofm2eff below which there would be a stable Q–ball solution. A carefullook at Mc and Ms shows that they vanish identically and Q–ball formation withoutthe trilinear couplings is already out of question. Therefore this possibility leavesno room for Q–ball formation. Finally, for φAu − φµ = π/2, however, one obtainesm2eff = M2ϕ/2a leaving both Mc and Ms nonvanishing. This is a regular Q–ballsolution, and it corresponds to maximal CP–violation, for instance, in the Higgssector 6. Indeed, Mc and Ms can vary with φAu further and this does not affect theQ–ball structure obtained for[m2eff (B − ball)]mean. Therefore, depending on theamount of CP violation, there may be regions of the parameter space that supporta stable Q–ball solution though the strictly CP–conserving MSSM does not.Another implication of these phases would be on the scattering of fermionsfrom the Q–ball. Indeed, a typical cross section has the form σ ∼ 4πR2 whereR2 ∼ 1/m2eff . Therefore, the supersymmetric CP phases influence the formation aswell as interactions of the Q–balls with surrounding plasma. In fact, this expectationis confirmed by the recent analysis of the scattering of the dark matter particlesfrom the nucleons 10.So far we have discussed Q–ball formation without imposing any D– and/or F–flatness. However, the scalar potential of the low–energy supersymmetric theorieshas many flat directions. Such flat potentials are phenomenologically relevant as itis possible to produce large enough Q–balls that can resist the evaporation on timescales of the order of the age of the universe 4,5. As has been listed in Ref. [9], thereare slepton as well as squark flat directions with corresponding Q–balls. In fact,radiatively corrected flat directions in the MSSM induce a potential of the formVϕ = M2ϕ|ϕ|2 + λ2|ϕ|2(d−1)M2(d−3)Pl+(λAϕdM(d−3)Pl+ h.c.)(18)where d is the mass dimension of the nonrenormalizable operator in the potential,and A is a typical trilinear coupling in the soft supersymmetry breaking part. Forthe purpose of this work the essential piece in this formula is A dependent part. Asdiscussed above, if A–terms are complex parameters, in general, the Reϕ as well asImϕ can develop nonvanishing vacuum expectation values leading to CP violatingQ–matter.3. Discussions5In this work we have discussed Q-ball formation in the MSSM with explicit CPviolation. It is seen that the complex µ parameter and A–terms can affect the Q-ballformation process. In particular, it is shown that the scalar vacuum expectationvalue in the Q–ball depends additively on the soft CP phases whereas the mass perunit charge of the Q–ball depends only on the relative phases. There are regionsof the parameter space where finite CP phases induce a stable Q–ball where thestrictly CP–conserving does not.Once the Q–ball is formed, as long as the Q–matter is CP violating one, oneexpects that the scatterings of the light fermions from the Q–ball as well as itsdecay to fermions (neutralino LSP, for example) can affect their phenomenology.Particularly interesting is the Q–matter contributing to dark matter where thedetection rates will change with the CP violating phases considerably 10.It is worthy of reemhasizing that:1. The vacuum expectation values of the fields, that is, the Q–ball matter ϕdepends on the individual relative CP phases additively; however,2. The mass per unit charge in the Q-ball depends only on the relative phases.Hence, there are regions of the parameter space where the Q–matter is nonvanishing;however, the Q–ball solutions are not stable. Moreover, there are regions of theparameter space where finite CP phases suport a stable Q–ball whereas the CPconserving MSSM does not. These discussions equally apply to L–balls as well, andone may obtain new kinds of Q–ball solutions by including appropriate soft termssuch as R–parity violating ones.AcknowledgmentsD. A. D. thanks Alexander Kusenko for useful conversations.1. S. Coleman, Nucl. Phys. B262 (1985) 263.2. A. Kusenko, Phys. Lett. 405B (1997)108; Phys. Lett. 404B 285.3. D. A. Demir, Phys. Lett. 450B (1999)215.4. A. Kusenko, M. Shaposhnikov, P.G. Tinyakov, Pisma Zh. Eksp. Teor. Fiz. 67 (1998)229 [JETP Lett. 67 (1998) 247]; A. Kusenko, V. Kuzmin, M. Shaposhnikov, P.G.Tinyakov, Phys. Rev. Lett. 80 (1998) 3185; A. Kusenko, M. Shaposhnikov, Phys. Lett.418B (1998) 46; G. Dvali, A. Kusenko, M. Shaposhnikov, Phys. Lett. 417B (1998)99.5. K. Enqvist, J. McDonald, Phys. Lett. 440B (1998) 59; Phys. Rev. Lett. 81 (1998)3071; Nucl. Phys. B538 (1998) 321; Phys. Lett. 425B (1998) 309.6. D. A. Demir, Phys. Rev. D60 (1999)055006; D60 (1999) 095007; A. Pilaftsis, C. E.M. Wagner, Nucl. Phys. B553 (1999) 3.7. S. Dimopoulos, S. Thomas, Nucl. Phys. B465 (1996) 23.8. D. A. Demir, hep-ph/9911435.9. See the last reference in [5].10. T. Falk, A. Ferstl, K. A. Olive, hep-ph/9908311.6

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Q-ball formation in the MSSM with explicit CP violation

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