DSpace at VNU: QCD corrections to squark production in e(+)e(-) annihilation in the MSSM with complex parameters tài liệ...
International Journal of Theoretical Physics, Vol 46, No 1, January 2007 ( C 2006) DOI: 10.1007/s10773-006-9190-4 QCD Corrections to Squark Production in e+ e− Annihilation in the MSSM with Complex Parameters Nguyen Thi Thu Huong,1,4,5 Ha Huy Bang,2,4 Nguyen Chinh Cuong,3 and Dao Thi Le Thuy3 Received September 29, 2005; accepted March 13, 2006 Published Online: January 3, 2007 We discuss the pair production of scalar quarks in e+ e− annihilation within the MSSM with complex parameters We calculate the SUSY-QCD corrections to the cross section e+ e− → q˜i q¯˜ j (i, j = 1, 2) and show that the effect of the CP phases of these complex parameters on the cross section can be quite strong in a large region of the MSSM parameter space This could have important implications for squarks searches and the MSSM parameter determination in future collider experiments KEY WORDS: MSSM; CP violation PACS number(s): 14.80 Ly, 12.60 Jv, 13.10 + q, 13.88 + e INTRODUCTION Supersymmetry is the currently best motivated extension of the Standard Model (SM) of particle physics which allows to stabilize the gauge hierarchy without getting into conflict with electroweak precision data Among all possible supersymmetric theories, the Minimal Supersymmetric Standard Model (MSSM) occupies a special position It is not only the simplest, i.e most economical, potentially realistic supersymmetric field theory, but it also has just the right particle content to allow for the unification of all gauge interactions The MSSM predicts the existence of scalar partners to all known quarks and leptons Each fermions has two spin zero partners called sfermions f˜L and f˜R , one for each chirality eigenstate: the mixing between f˜L and f˜R is proportional to the corresponding fermion The Abdus Salam International Centre for Theoretical Physics, P.O Box 586, 34100 Trieste, Italy de Physique Th´eorique LAPTH, Chemin de Bellevue, B.P 110, F-74941 Annecy-leVieux, Cedex, France Hanoi University of Education, 136 Xuan Thuy Str., Cau Giay, Hanoi Permanent address: Department of Physics, Vietnam National University, 334 Nguyen Trai, Hanoi, Vietnam To whom correspondence should be addressed; e-mail: nguyenhuong1982@yahoo.com Laboratoire 41 0020-7748/07/0100-0041/0 C 2006 Springer Science+Business Media, Inc 42 Huong, Bang, Cuong, and Thuy mass, and so negligible except for the third generation In particular, this model gives the possibility for one of the scalar partners of the top quark (t˜1 ) to be lighter than other scalar quarks and also than the top quark (Ellis and Rudaz, 1983) So far most phenomenological studies on supersymmetric (SUSY) particle searches have been performed in the MSSM with real SUSY parameters Studies of the 3rd generation sfermions are particularly interesting because of the effects of the large Yukawa couplings The lighter sfermion mass eigenstates may be relatively light and they could be thoroughly studied at an e− e+ linear collider (Accomando et al., 1998) An analysis of the QCD corrections to scalar quark pair production in e+ e− annihilation in the MSSM with real parameters was performed in (Arhrib et al., 1995; Eberl et al., 1996) The assumption that all SUSY parameters are real, however, may be too restrictive The higgsino mass parameter µ, the gaugino masses M˜ i and the trilinear scalar coupling parameters Af of the sfermions f˜ may be complex In the MSSM the complex parameters provide the CP violating phases Recently, a phenomenological study of τ -sleptons τ˜1,2 and τ -sneutrinos ν˜ τ has been presented (Bartl et al., 2002), and the effect of the CP phases on the t˜1,2 and b˜1,2 decays has been found (Bartl et al., 2003) in the MSSM with complex parameters Next, in (Thuy et al., 2004) the one loop vertex correction to the decay width of squark decays into W and Z bosons within the MSSM with complex parameters has been calculated and the numerical results are also performed In this article we study the effects of the phases of the complex parameters Aq on the cross sections of the process: e+ e− → q˜i q˜¯j We point out that these effects can be quite strong in a large region of the MSSM parameter space This could have an important impact on the search for squarks and the determination of the MSSM parameters at future colliders Our present study is an extension of the corresponding one in the MSSM with real parameters in (Arhrib et al., 1995; Eberl et al., 1996) DIAGONALIZATION OF MASS MATRICES We neglect generation mixing As pointed out in (Dugan et al., 1985; Christora and Fabbrichesi, 1993) only three terms in the supersymmetric Lagrangian can give rise to CP-violating phases which cannot be rotated away: The superpotential contains a complex coefficient µ in the term bilinear in the Higgs superfields The soft supersymmetry breaking operators introduce two further complex terms, the gaugino masses M˜ i and the left- and right-handed squark mixing term Aq In the MSSM one has two types of scalar quarks (squarks), q˜L and q˜R , corresponding to the left and right helicity states of a quark The mass matrix in the basis (q˜L , q˜R ) is given by Ellis and Rudaz (1983) M2q˜ = m2q˜L aq mq aq mq m2q˜R = (Rq˜ )+ m2q˜1 0 m2q˜2 Rq˜ , (1) QCD Corrections to Squark Production 43 with q + m2q , m2q˜L = M2Q˜ + m2Z cos 2β I3L − eq sW (2) 2 m2q˜R = M2{u, ˜ + eq mZ cos 2βsW + mq , ˜ D} (3) aq = Aq − µ{cot β, tan β}, (4) q for {up, down} type squarks, respectively eq and I3 are the electric charge and ˜ and mq is the mass of the third component of the weak isospin of the squark q, the partner quark MQ˜ , Mu˜ , and MD˜ are soft SUSY breaking masses, and Aq are trilinear couplings According to eq (1) M2q˜ is diagonalized by a unitary matrix Rq˜ The weak eigenstates q˜L and q˜R are thus related to their mass eigenstates q˜1 and q˜2 by q˜1 q˜2 q˜L , q˜R = Rq˜ (5) e− ϕq sin θq˜ i e− ϕq cos θq˜ i Rq˜ = i e ϕq cos θq˜ i −e ϕq sin θq˜ (6) , with θq˜ is the squark mixing angle and ϕq = arg(Aq ) The mass eigenvalues are given by m2q˜1,2 = m2q˜L + m2q˜R ∓ m2q˜L − m2q˜R + 4aq2 m2q (7) By convention, we choose q˜1 to be the lighter mass eigenstate For the mixing angle θq˜ we require ≤ θq˜ ≤ π We thus have cos θq˜ = −|aq |mq m2q˜L − m2q˜1 m2q˜L − m2q˜1 sin θq˜ = , + aq2 m2q m2q˜L − m2q˜1 (8) + aq2 m2q ˜ Qγ ˜ AND Q ˜Q ˜ Z COUPLINGS THE Q From the matrix we can find the interaction of a neutral gauge boson V = γ , Z with squarks in the general forms ↔ ↔ ˜L∗ ∂ µ q˜L + q˜R∗ ∂ µ q˜R ) Lq˜ qγ ˜ = ieeq Aµ (q ↔ = ieeq Aµ (Ri1 Rj + Ri2 Rj )q˜j∗ ∂ µ q˜i q˜ ↔ q˜ ≡ ieeq δij q˜j∗ ∂ µ q˜i , q˜ q˜ (9) 44 Huong, Bang, Cuong, and Thuy where δ= cos ϕq + i sin ϕq cos 2θq˜ −i sin ϕq sin 2θq˜ −i sin ϕq sin 2θq˜ cos ϕq − i sin ϕq cos 2θq˜ , (10) and Lq˜ qZ ˜ = ≡ ↔ ↔ ig Zµ cqL q˜L∗ ∂ µ q˜L + cqR q˜R∗ ∂ µ q˜R cw ↔ ig Zµ Cij q˜j∗ ∂ µ q˜i , cw (11) where C= ⎛ q I3L cos2 θq˜ − eq sW cos ϕq ⎜+i I q cos2 θ − e s cos 2θ sin ϕ ⎜ q W q q˜ q˜ 3L ⎜ ⎜ q ⎝− I3L sin 2θq˜ cos ϕq q ⎞ q − 12 I3L sin 2θq˜ cos ϕq ⎟ ⎟ ⎟ ⎟ ⎠ q +i eq sW − 12 I3L sin 2θq˜ sin ϕq q I3L sin2 θq˜ − eq sW cos ϕq q +i eq sW − 12 I3L sin 2θq˜ sin ϕq +i I3L sin2 θq˜ + eq sW cos 2θq˜ sin ϕq (12) We obtain the corresponding Feynman rules from ↔ q˜j∗ ∂ µ q˜i = i(ki + kj )µ , where ki and kj are the four-momenta of q˜i and q˜j in direction of the charge flow The coupling between a gauge boson V and two squarks q˜i and q˜j with i, j = 1, is given by (the directions of the momenta are shown in Fig 1): µ q˜i q˜j γ = −ieeq (k¯ + k)µ δij , Fig The V q˜i q˜j vertex (i, j = 1, 2) (13) QCD Corrections to Squark Production 45 Fig Feynman diagrams for the process e+ e− → q˜i q¯˜ j (i, j = 1, 2) µ q˜i q˜j Z =− ig ¯ (k + k)µ cij cW (14) TREE LEVEL FORMULAE In this section we discuss the pair production of squarks in e+ e− collisions The process e+ e− → q˜i q¯˜ j proceeds via γ and Z exchange, see Fig Using the results just obtained we get the cross section at tree level for unpolarized beams: σ0 = √ 3/2 2π α λij s eq2 |δij |2 − eq ve ve2 + ae2 + + (c δ + c δ ) · D + |cij |2 DZZ , ij ij γ Z ij ij 2 4 4cW sW 16cW sW (15) where s is the c.m energy squared, λij = [(s − − − charge of the squarks (et = 2/3, eb = −1/3) in the units of e(= −1, ve = −1 + 4sw2 (with sW ≡ sin θW , cW ≡ cos θW ), and m2i Dγ Z = DZZ = m2j )2 s s − MZ2 s − m2Z + m2Z Z + m2Z Z s2 s − m2Z 4m2i m2j ], eq √ is the 4π α), ae = , (16) (17) SUSY—QCD CORRECTIONS The SUSY QCD corrected cross section, corresponding to Fig can be written as σ = σ + δσ (18) 46 Huong, Bang, Cuong, and Thuy Fig Feynman diagrams for the lowest order SUSY—QCD corrections to e+ e− → q˜i q˜ j Here 3/2 δσ = π α λij αs Re{δA1 + δA2 + δA3 + δA4 + δA5 }, s 3π with δA1 = iδii δjj B1 m2i , m2g , m2i + B0 m2i , m2g , m2i + B1 m2j , m2g , m2j +B0 m2j , m2g , m2j − 2s − 2m2i − 2m2j − m2g C11 + C0 + 8δii B1 m2i , m2g , m2i + 2B0 m2i , m2g , m2i − iδii δii A m2i − m2g B0 m2i , m2g , m2i (19) QCD Corrections to Squark Production 47 A m2q − m2g B0 m2i , m2g˜ , m2q δki + 4i k=1 − mg˜ δki − mq mg˜ Cki B0 m2i , m2g˜ , m2q + 2Sik Sik A m2k eq2 |δij |2 + ve2 + ae2 |Cij |2 4 16CW SW DZZ − eq ve Cij δij+ + Cij+ δij 2 4CW SW + · Dγ Z , (20) δA2 = δij 2m2g˜ + m2i + m2j + m2q α + m2q β C11 + C0 − B0 m2i , m2g˜ , m2q β − B0 m2i , m2g˜ , m2q α + m2q α + m2q β + 0.5s + 0.5m2i + 0.5m2j + 2mq α mq β · C11 − 2mg˜ mq β + mq α · Eij C11 + C0 + eq2 δij , (21) δA3 = − δij RL 2m2g˜ + m2i + m2j + m2q α + m2q β C11 + C0 − B0 m2i , m2g˜ , m2q β − B0 m2i , m2g˜ , m2q α + m2q α + m2q β + 0.5s + 0.5m2i + 0.5m2j · C11 + 2mg˜ mq β Cij RL + mq α Cij LR C11 + C0 − 2mq α mq β Cij C11 + eq ve δij Dγ Z , 2 2CW SW (22) δA4 = 2Eij δij (2m2g˜ + m2i + m2j + m2q α + m2q β C11 + C0 − B0 m2i , m2g˜ , m2q β − B0 m2i , m2g˜ , m2q α + m2q α + m2q β + 0.5s + 0.5m2i + 0.5m2j + 2mq α mq β · C11 − 2mg˜ mq β + mq α · Eij C11 + C0 ) + eq ve δij Dγ Z , 2 2CW SW (23) δA5 = 2Eij δij RL 2m2g˜ + m2i + m2j + m2q α + m2q β C11 + C0 − B0 m2i , m2g˜ , m2q β − B0 m2i , m2g˜ , m2q α + m2q α + m2q β + 0.5s + 0.5m2i + 0.5m2j · C11 + 2mg˜ mq β Cij RL + mq α Cij LR C11 + C0 − 2mq α mq β Cij C11 + ve2 + ae2 DZZ 4 16CW SW (24) 48 Huong, Bang, Cuong, and Thuy and q˜ q˜ q˜ q˜ q˜ q˜ q˜ q˜ q˜ q˜ q˜ q˜ δij RL = CqR Ri1 Rj + CqL Ri2 Rj , Cij LR = CqL Ri1 Rj + CqR Ri2 Rj , Cij RL = CqR Ri1 Rj + CqL Ri2 Rj , q˜ q˜ q˜ q˜ q˜ q˜ q˜ q˜ Eij = Ri1 Rj + Ri2 Rj , Sij = Ri1 Rj − Ri2 Rj , C0 = C0 m2i , s, m2j , m2g , m2i , m2j , C11 = C11 m2i , s, m2j , m2g , m2i , m2j , C0 = C0 m2i , s, m2j , m2g˜ , m2q α , m2q β , C11 = C11 m2i , s, m2j , m2g˜ , m2q α , m2q β We now turn to the numerical analysis of the SUSY—QCD corrections In Fig we show the φ ≡ ϕq dependence of the σR0 /σC0 , δσR0 /δσC0 and the cases of real and complex paδσC /σC0 , with R and C indices corresponding to √ rameters respectively The input parameters are: s = 1000 GeV, mt˜1 = 400 GeV, mt˜2 = 600 GeV, mb˜1 = 400 GeV, mb˜2 = 450 GeV, mg˜ = 600 GeV, | cos θt˜| = | cos θb˜ | = 0, Here we concentrate on the range of the complex phase ϕq of the Aq -parameter, it must be less than of order 10−2 –10−3 to avoid generating electric dipole moments for the neutron, electron, and atom in conflict with observed data (Review of Particle Physics, 1998) In the range of φ shown, σR0 /σC0 Fig Cross sections and their corrections of (a) e+ e− → t˜1 t¯˜1 , (b) e+ e− → t˜2 t¯˜2 , (c) e+ e− → b˜1 b¯˜ , and (d) e+ e− → b˜2 b¯˜ as functions of , for cos θt = √ cos θb = 0.5; s = 1000 GeV, mt˜1 = mb˜1 = 400 GeV, mt˜2 = mg˜ = 600 GeV, mb˜2 = 450 GeV QCD Corrections to Squark Production Fig Continued 49 50 Huong, Bang, Cuong, and Thuy varies from 100% to 99% in cases of t˜1 t¯˜1 or t˜2 t¯˜2 productions (Fig 4(a) and (b)) and is about from 100% to 99.5% and 100% for b˜1 b¯˜ and b˜2 b¯˜ productions (Fig 4(c) and (d)) The corrections δσC0 /σC0 are from −28.4% to −25%, from −38.8% to −36.5%, and from −45.5% to −42.5% for the t˜1 t¯˜1 , t˜2 t¯˜2 , b˜1 b¯˜ and b˜2 b¯˜ productions, respectively We have also computed δσR0 /δσC0 It is about from 99.5% to 96.5%, from 99.5% to 93%, from 100% to 99.5%, and from 100% to 93%, for the above mentioned processes, respectively CONCLUSIONS In this paper, we have calculated the QCD radiative corrections to the production of squarks in e+ e− collisions within the MSSM with complex parameters In particular, we focus on the CP phase dependence of the production cross sections for e+ e− → q˜i q¯˜ j From numerical results we have found that the effects of the phases on the cross sections can be quite significant in a large region of the MSSM parameter space This could have important implications for t˜i and b˜i searches at future colliders and the determination of the underlying MSSM parameters ACKNOWLEDGMENTS We are grateful to Prof G Belanger for suggesting the problem and for her valuable comments H H Bang wishes to thank Prof P Aurenche for his help and encouragement N T T Huong would like to thank the International Atomic Energy Agency and UNESCO for hospitality at the Abdus Salam International Center for Theoretical Physics, Trieste, Italy This work was supported in part by Special Project on Natural Sciences of the Vietnam National University under the Grant number QG 04 03 REFERENCES Accomando, E., et al (1998) Phys Rept 229, 1–78 Arhrib, A., Capdequi-Peyranere, M., and Djouadi, A (1995) Phys Rev D 52, 1404–1417 Bartl, A., Hidaka, K., Kernreiter, T., and Porod, W (2002) hep-th/0207186 Bartl, A., Hesselbach, S., Hidaka, K., Kernreiter, T., and Porod, W (2004) Phys Rev D 70, 035003 hep-th/0311338 Christora, E and Fabbrichesi, M (1993) Phys Lett B315, 338 Dugan, M., Grinstein, B., and Hall, L (1985) Nucl Phys B255, 413 Eberl, H., Bartl, A., and Majerotto, W (1996) Nucl Phys B 472, 481 Ellis, J and Rudaz, S (1983) Phys Lett 128B, 248 Kraml, S (1999) hep-ph/9903257 Review of Particle Physics (1998) Eur Phys J C3, 1–794 Thuy, D T., Cuong, N C., and Bang, H H (2004) Comm in Phys 14(3), 76 ... CONCLUSIONS In this paper, we have calculated the QCD radiative corrections to the production of squarks in e+ e− collisions within the MSSM with complex parameters In particular, we focus on the CP... higgsino mass parameter µ, the gaugino masses M˜ i and the trilinear scalar coupling parameters Af of the sfermions f˜ may be complex In the MSSM the complex parameters provide the CP violating... in the term bilinear in the Higgs superfields The soft supersymmetry breaking operators introduce two further complex terms, the gaugino masses M˜ i and the left- and right-handed squark mixing