One CP--odd rephase-invariant phase appears in the chargino mass matrix in
the minimal Supersymmetric Standard Model. We investigate in detail the
phenomenological impacts of the CP-odd complex phase in the production of light
charginos in $e^+e^-$ annihilation. The values of the chargino masses and the
mixing angles, determining the size of the wino and higgsino components in the
chargino wave functions, are so sensitive to the CP-odd phase that the
constraints on the supersymmetric parameters based on the conventional
assumptions for the parameters are recommended to be re-evaluated including the
CP-odd phase.Comment: 9 pages, latex with 3 eps figur

Choi, S. Y.

Shim, J. S.

Song, H. S.

Song, W. Y.

English

arXiv

One CP--odd rephase-invariant phase appears in the chargino mass matrix in the minimal Supersymmetric Standard Model. We investigate in detail the phenomenological impacts of the CP-odd complex phase in the production of light charginos in $e^+e^-$ annihilation. The values of the chargino masses and the mixing angles, determining the size of the wino and higgsino components in the chargino wave functions, are so sensitive to the CP-odd phase that the constraints on the supersymmetric parameters based on the conventional assumptions for the parameters are recommended to be re-evaluated including the CP-odd phase

Choi, S Y

Shim, J S

Song, H S

Song, W Y

CERN Document Server

YUMS 98{12SNUTP 98{084August, 1998Phenomenological Impacts of the CP-odd Rephase-invariantPhase of the Chargino Mass Matrix in the Productionof Light Chargino-Pair in e+e− CollisionsS.Y. Choi1, J.S. Shim2y, H.S. Song3z and W.Y. Song3x1 Department of Physics, Yonsei University, Seoul 120{749, Korea2 Department of Physics, Myong Ji University, Yongin 449-728, Korea3 Center for Theoretical Physics and Department of Physics,Seoul National University, Seoul 151-742, KoreaAbstractOne CP{odd rephase-invariant phase appears in the chargino mass matrix in theminimal Supersymmetric Standard Model. We investigate in detail the phenomeno-logical impacts of the CP-odd complex phase in the production of light charginosin e+e− annihilation. The values of the chargino masses and the mixing angles,determining the size of the wino and higgsino components in the chargino wavefunctions, are so sensitive to the CP-odd phase that the constraints on the super-symmetric parameters based on the conventional assumptions for the parametersare recommended to be re-evaluated including the CP-odd phase.e-mail: sychoi@theory.yonsei.ac.krye-mail: jsshim@wh.myongji.ac.krze-mail: hssong@physs.snu.ac.krxe-mail: misson@re.snu.ac.kr1. IntroductionThe Minimal Supersymmetric Standard Model (MSSM) [1] is a well-dened quantumtheory of which the Lagrangian form is completely known, including the general R-paritypreserving, soft supersymmetry (SUSY) breaking terms.The full MSSM Lagrangian has 124 truly independent parameters - 79 real parametersand 45 CP-violating complex phases [2]. The number of parameters in MSSM is too largecompared to 19 in the Standard Model (SM). Therefore, many studies on possible directand indirect SUSY eects have been made by making several assumptions and investi-gating the variation of a few parameters [3, 4, 5]. Recently, it has, however, been shown[6] that limits on sparticle masses and couplings are very sensitive to the assumptionsand need to be re-evaluated without making any of the simplifying assumptions that havebeen standard.Despite the large number of phases in the model as a whole, just one CP-odd rephase-invariant phase [7], stemed from the chargino mass matrix, takes part in chargino pro-duction [8]. In light of this aspect, in the chargino system the analyses with the generalparameter set are not so much more dicult than those with parameters assumed real.The CP-odd phase may be constrained indirectly by the electron or neutron electric dipolemoment [7] and may be small, but the constraints on its actual size depends strongly onthe assumptions in those analyses. As a matter of fact, the analysis of the electric dipolemoments has not been done with a general phase structure, which will certainly giveweaker constraints on the CP-odd phase in the chargino mass matrix. So, unless thereexist any concrete demonstrations for a small value of the phase, it will be more reasonableto take the CP-odd phase as a free parameter.Charginos are produced in e+e− collisions, either in diagonal or in mixed pairs. How-ever, the second chargino ~2 is generally expected to be signicantly heavier than therst state. At LEP2 [9], and potentially even in the rst phase of e+e− linear colliders(see e.g. Ref. [10]), the chargino ~1 may be, for some time, the only chargino state thatcan be studied experimentally in detail. In the present note, we will focus on the diagonalpair production of the light chargino ~1 in e+e− collisions,e+e− ! ~+1 ~−1and investigate in detail the phenomenological impacts of the CP-odd complex phase inthe determinations of the relevant SUSY parameters in the production process.The production of the light chargino-pair is completely described by the chargino mass~1 , two mixing angles determining the size of the wino and higgsino components in thecharginos, and the sneutrino mass. So, rst of all, in Section 2, we briefly recapitulatethe elements of the mixing formalism and quantitatively discuss the dependence of thechargino masses and the mixing angles on the CP-odd phase. In Section 3 the crosssection for chargino production along with the light chargino mass is mapped over theparameter space, especially for the gaugino mass M2 and the higgsino mass parameterjj, by varing the CP-odd phase. Then, we examine the phenomenological impacts of theCP-odd phase in constraining the parameter space. Conclusions are given in Section 4.12. Chargino Masses and Mixing AnglesIn the MSSM, the spin{1/2 partners of the W boson and charged Higgs boson, ~W and~H, mix to form chargino mass eigenstates ~1;2. The mass eigenvalues m~1;2 and themixing angles and phases are determined by the elements of the chargino mass matrix inthe ( ~W−; ~H−) basis [1]MC = M2p2mW cosp2mW sin  !; (1)which is built up by the fundamental supersymmetric parameters; the gaugino mass M2,the Higgs mass parameter , and the ratio tan = v2=v1 of the vacuum expectation valuesof the two neutral Higgs elds which break the electroweak symmetry. In CP{noninvarianttheories, the gaugino mass M2 and the Higgs mass parameter  can be complex. However,by reparametrizations of the elds, M2 can be assumed real and positive without loss ofgenerality [7] so that the only non{trivial invariant phase is attributed to : = jjei: (2)In these theories the complex chargino mass matrix (1) is diagonalized by two unitarymatrices UL and UR, which can be parameterized in the following way:UL = cosL e−iL sinL−eiL sinL cosL!;UR = eiγ1 00 eiγ2! cosR e−iR sin R−eiR sin R cos R!; (3)and which render URMCUyL diagonal. The two chargino mass eigenvalues are given bym2~1;2=12M22 + jj2 + 2m2W C; (4)withC =q(M22 − jj2)2 + 4m4W cos2 2 + 4m2W (M22 + jj2 + 2M2jj sin 2 cos ): (5)The mixing angles L;R are given by the relationscos 2L = −M22 − jj2 − 2m2W cos 2C;sin 2L = −2mWqM22 + j2j+ (M22 − jj2) cos 2 + 2M2jj sin 2 cos C; (6)cos 2R = −M22 − jj2 + 2m2W cos 2C;sin 2R = −2mWqM22 + j2j − (M22 − jj2) cos 2 + 2M2jj sin 2 cos C; (7)2The four nontrivial phase angles fL; R; γ1; γ2g also depend on the invariant angle ; fortheir expressions we refer to the appendix of the work [4].Note that cos  in eqs. (4), (6) and (7) for the chargino masses and the mixing anglesappears along with a unique combination factor:M2jj sin 2 = M2jj2 tan1 + tan2 :This is a reflection of the fact that the CP-odd phase angle  can be absorbed by eldre{denitions if at least one of the chargino mass matrix elements vanishes. In particular,when tan is very small or very large, i.e. one of the two Higgs vacuum expectation valuesv1 and v2 is relatively very smalla, the eects of the phase angle  diminish. Keeping inmind that the CP-odd phase eects are very small for large tan, we present numericalanalyses for a xed value of tan = 2 in the following.The light and heavy chargino masses are presented in Figs. 1(a) and (b) as a functionof the cosine of the CP-violating phase angle  for a representative set of parameters.The parameters are chosen in the higgsino region M2  jj, the gaugino region M2  jjand in the mixed region M2  jj for tan = 2 asgaugino region : (M2; jj) = (80 GeV; 200 GeV);higgsino region : (M2; jj) = (210 GeV; 70 GeV);mixed region : (M2; jj) = (90 GeV; 90 GeV):(8)The two masses are very sensitive to the phase angle  in all scenarios; the sensitivityis more prominent in the light chargino mass than in the heavy chargino mass, and it ismost prominent in the mixed scenario. Figs. 1(c) and (d) exhibit cos 2L and cos 2R asa function of cos b. Similarly, both of them depend more strongly on the CP-violatingphase angle  in the mixed scenario than in the gaugino and higgsino scenarios.3. Production Cross SectionThe process e+e− ! ~+1 ~−1 is generated by the three mechanisms: s{channel γ andZ exchanges, and t{channel ~ exchange. The transition matrix element, after a Fierztransformation of the ~{exchange amplitude,Te+e− ! ~+1 ~−1=e2sQhv(e+)γPu(e−)i hu(~−1 )γPv(~+1 )i; (9)can be expressed in terms of four bilinear charges, classied according to the chiralities;  = L;R of the associated lepton and chargino currentsQLL = 1 +DZs2W c2W(s2W −12)s2W −34−14cos 2L;aTwo vacuum expectation values v1 and v2 can not be larger than v =pv21 + v22  250 GeV.bThe two sines, fsin 2L,sin 2Rg, and the four nontrivial phase angles fL;R; γ1;2g are not numericallypresented in the present work because they are not involved in the diagonal pair production of the lightor heavy charginos as shown in the next Section.3QLR = 1 +DZs2W c2W(s2W −12)s2W −34−14cos 2R+D~4s2W(1 + cos 2R);QRL = 1 +DZc2Ws2W −34−14cos 2L;QRR = 1 +DZc2Ws2W −34−14cos 2R: (10)The rst index in Q refers to the chirality of the e current, the second index to thechirality of the ~1 current. The ~ exchange aects only the LR chirality charge while allother amplitudes are built up by γ and Z exchanges. D~ denotes the sneutrino propagatorD~ = s=(t −m2~), and DZ the Z propagator DZ = s=(s −m2Z + imZΓZ); the non{zerowidth can in general be neglected for the energies considered in the present analysis sothat the charges are real.The bilinear charges fQRR; QRLg (fQLR; QLLg) with right (left) electron chirality de-pend only on cos 2R (cos 2L) so that the dependence of the production amplitude on theCP-odd phase angle  is easily readable. Fig. 2 shows the total production cross sectionas a function of (a) cos  for a xed sneutrino mass of 200 GeV and the parameter ineq. (8), and (b) the sneutrino mass in the mixed scenario of the parameter set (8) at ac.m. energy of 200 GeV. Interestingly, while the chargino masses and the mixing anglesare most sensitive to the CP-violating phase angle in the mixed scenario, the productioncross section itself is most sensitive to the phase in the gaugino scenario. This implies thatthe sensitivities to the phase angle  is strongly correlated with those to the sneutrinomass involved in the t-channel snuetrino exchange amplitude. Fig. 2(b) clearly showsthat the total cross section grows up very sharply with cos  and the sneutrino mass forlargest destructive interference in the production amplitude shifts to a lower value. Prioror simultaneous determination of m~ will be therefore necessary to determine the phaseand the other SUSY parameters.The maximal (M2; jj) parameter space, probed in the production of light charginosin e+e− collisions with a c.m. energy of 200 GeV, is presented in Fig. 3(a) for threedierent values of cos  with tan = 2. As cos  increases, the covered parameter spaceis enlarged. However, the maximal parameter space can not be fully covered in actualexperiments, but its searchable regime relies on the number of produced charginos and thereconstruction eciency of the chargino signals determined from their decay patterns [11].In the case that the light chargino decays into the lightest neutralino, [usually consideredto be the lightest supersymmetric particle (LSP)], a fermion and an anti-fermion, thedierence between the chargino mass and the LSP mass plays a crucial role. Since theLSP mass also depends strongly on the CP-odd phase angle  as well as an extra phaseangle [6], the determination of the upper limits of the chargino cross section, which canbe excluded at high energy colliders such as LEPII, will be rather involved, but of coursedoable. In the present work, we simply take tot = 2 pb as a reference value for the limitof the production cross section while its estimates based on the detailed investigations ofthe chargino decays will be touched on in our future work. Fig. 3(b) shows the contoursfor the total cross section tot = 2 with m~ = 200 GeV at a c.m. energy of 200 GeV forthree dierent values of cos . It is clear that the excluded region of the parameter space4depend strongly on the phase angle  and its dependence is most prominent in the mixedscenario.4. ConclusionsWe have analyzed how the parameters of the chargino system, the masses of the charginos~ and the size of the wino and higgsino components in the chargino wave functions,parametrized in terms of the two angles L and R, are aected by the CP-violatingrephase-invariant angle  in the chargino mass matrix. In addition, we have studied thedependence of the production cross section of the light chargino-pair in e+e− collisions onthe angle  in detail.The chargino masses and the production cross section of the light chargino-pair ine+e− collisions are very sensitive to the CP-violating angle . The sensitivities are mostprominent around tan  = 1 and in the mixed scenario with M2  jj, where the limitson the light chargino mass might be much weaker than has been reported [11] and need tobe re-evaluated. It is then essential to probe the parameter space including the possiblecomplex phase angle  in the search of the SUSY particles at high energy colliders.AcknowledgmentsThis work was supported by Korean Science and Engineering Foundation in part throughlarge collaboration project, Project No. 96-0702-01-01-2 and in part through the Centerfor Theoretical physics, Seoul National University.References[1] For reviews of supersymmetry and the Minimal Supersymmetric Standard Model,see H. Nilles, Phys. Rep. 110 (1984) 1; H.E. Haber and G.L. Kane, Phys. Rep. 117(1985) 75.[2] S. Dimopoulos and D. Sutter, Nucl. Phys. B 452 (1995) 496; H. Haber, Proceedingsof the 5th International Conference on Supersymmetries in Physics (SUSY’97), May1997, ed. M. Cvetic and P. Langacker, hep-ph/9709450.[3] A. Leike, Int. J. Mod. Phys. A3 (1988) 2895; M.A. Diaz and S.F. King,Phys. Lett. B349 (1995) 105; B373 (1996) 100; J.L. Feng and M.J. Strassler,Phys. Rev. D51 (1995) 4461 and D55 (1997) 1326; G. Moortgat-Pick and H. Fraas,hep-ph/9708481.[4] S.Y. Choi et al., hep-ph/9806279.[5] G. Moortgat-Pick, H. Fraas, A. Bartl and, and W. Majerotto, hep-ph/9804306.[6] M. Brhlik and G.L. Kane, hep-ph/9803391.5[7] Y. Kizukuri and N. Oshimo, Proceedings of the Workshop on e+e− Collisions at 500GeV: The Physics Potential, Munich-Annecy-Hamburg 1991/93, DES 92-123A+B,93-123C, ed. P. Zerwas; T. Ibrahim and P. Nath, Phys. Lett. B418 (1998) 98;Phys. Rev. D57 (1998) 478; hep-ph/9807591.[8] J. Ellis, J. Hagelin, D. Nanopoulos and M. Srednicki, Phys. Lett. 127B (1983) 233;V. Barger, R.W. Robinett, W.Y. Keung and R.J.N. Phillips, Phys. Lett. B131 (1983)372; D. Dicuss, S. Nandi, W. Repko and X. Tata, Phys. Rev. Lett. 51 (1983) 1030;S. Dawson, E. Eichten and C. Quigg, Phys. Rev. D31 (1985) 1581; A. Bartl andH. Fraas and W. Majerotto, Z. Phys. C30 (1986) 441.[9] Proceedings of the Workshop on Physics at LEP II, Report No. CERN-96-01, eds.G. Altarelli, T. Sjo¨strand, and F. Zwirner.[10] E. Accomando et al., Phys. Rept. 299 (1998) 1.[11] See, for example, Particle Data Group, Eur. Phys. J. C3 (1998) 1.6−1 −0.5 0 0.5 1cosθµ−1−0.500.51cos2φ L(c)−1 −0.5 0 0.5 1cosθµ050100Light chargino mass [GeV] (a)−1 −0.5 0 0.5 1cosθµ−1−0.500.51cos2φ R(d)−1 −0.5 0 0.5 1cosθµ100200300Heavy chargino mass [GeV] (b)Figure 1: (a) the light chargino mass, (b) the heavy chargino mass, (c) cos 2L, and (d)cos 2R as a function of the cosine of the CP-violating phase angle  for the representativeset of SUSY parameters in eq. (8): solid line for the gaugino case, dashed line for thehiggsino case, and dot-dashed line for the mixed case.7−1 −0.5 0 0.5 1cosθµ01234σtot [pb](a)0 100 200 300Sneutrino mass [GeV]01234σtot [pb](b)Figure 2: The cross section for the production of light charginos (a) as a function ofcos  with m~ = 200 GeV for the parameters set in eq. (8), and (b) as a function ofthe sneutrino mass in the mixed scenario atps = 200 GeV; in (a) the solid line for thegaugino case, the dashed line for the higgsino case, and the dot-dashed line for the mixedcase, and in (b) the solid line for cos  = −1, the dashed line for cos  = 0, and thedot-dashed line for cos  = 1.80 100 200 300|µ| [GeV]0100200300M2 [GeV](a) Light Chargino Mass = 100 GeVcosθµ=−1cosθµ= 0cosθµ=+10 100 200 300|µ| [GeV]0100200300M2 [GeV](b) Total Cross Section = 2 pbcosθµ=−1cosθµ= 0cosθµ=+1Figure 3: Contours for (a) the light chargino mass m~1 = 100 GeV and (b) the total crosssection tot = 2 pb with m~ = 200 GeV atps = 200 GeV; the line with circle symbols forcos  = −1, the line with plus symbols for cos  = 0 and the line with star symbols forcos  = +1.9

Phenomenological Impacts of the CP-odd Rephase-Invariant Phase of the Chargino Mass Matrix in the Production of Light Chargino-Pair in $e^+e^-$ Collisions

Phenomenological Impacts of the CP-odd Rephase-Invariant Phase of the
  Chargino Mass Matrix in the Production of Light Chargino-Pair in $e^+e^-$
  Collisions

Phenomenological Impacts of the CP-odd Rephase-Invariant Phase of the Chargino Mass Matrix in the Production of Light Chargino-Pair in $e^+e^-$ Collisions

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Phenomenological Impacts of the CP-odd Rephase-Invariant Phase of the Chargino Mass Matrix in the Production of Light Chargino-Pair in e+e−e^+e^-e+e− Collisions

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Phenomenological Impacts of the CP-odd Rephase-Invariant Phase of the Chargino Mass Matrix in the Production of Light Chargino-Pair in $e^+e^-$ Collisions