Squarks decay into quarks and gluino in the MSSM

JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol. 57, No. 7, pp. 94-99 This paper is available online at SQUARKS DECAY INTO QUARKS AND GLUINO IN THE MSSM Nguyen Chinh Cuong1 and Phung Van Hao2 1Faculty of Physics, Hanoi National University of Education 2 Son Tay High School, Hanoi Abstract. We present a phenomenological study of the decay of squarks (top and bottom) into quarks (top and bottom) and gluino in the Minimal Supersymmetric Standard Model (MSSM). The formulae for the decay width and numerical results are obtained. We have calculated the conclusion of the one-loop vertex correction, wave-function correction and renormalization of the bare couplings to the decay width. We revealed that the effect of the complex parameters At and Ab could be quite significant in a large region of the MSSM parameter space. Keywords: MSSM, squark, CP violation. 1. Introduction Theminimal supersymmetric standard model (MSSM) is one of the most promising extensions of the standard model (SM) [1]. Only three terms in the supersymmetric Lagrangian can give rise to CP violating phases. The superpotential contains a complex coefficient µ in the bilinear term of the Higgs superfield. There are two complex terms in the soft supersymmetry (SUSY) breaking part: the gaugino mass M˜ and the left and right handed squark mixing term Aq [2]. µ = |µ|eiφµ = |µ|eiφ1, Aq = |Aq|eiφq˜ = |Aq|eiφ2 , M˜ = |M˜ |eiφM˜ = |M˜ |eiφ3. (1.1) In the MSSM, there are two types of scalar quarks (squarks): q˜L and q˜R, corresponding to the left and the right helicity states of a quark. The mass matrix on the basis (q˜L, q˜R) is given by [3] M2q˜ = ( m2q˜L aqmq aqmq m 2 q˜R ) = ( Rq˜ )†( m2q˜1 0 0 m2q˜2 ) Rq˜. (1.2) Received April 16, 2012. Accepted October 20, 2012. Physics Subject Classification: 60 44 01. Contact Phung Van Hao, e-mail address: k20ch.phunghao@yahoo.com.vn 94 Squarks decay into quarks and gluino in the MSSM According to Eq.(1.2), M2q˜ is diagonalized by a unitary matrix R q˜. The weak eigenstates q˜1 and q˜2 are thus related to their mass eigenstates q˜L and q˜R by( q˜1 q˜2 ) = Rq˜ ( q˜L q˜R ) , where Rq˜ = ( e i 2 φq˜ cos θq˜ e − i 2 φq˜ sin θq˜ −e− i2φq˜ sin θq˜ e i2φq˜ cos θq˜ ) . (1.3) As known, CP violation arises naturally in three generations of SM and it can appear only through the phase in the CKM - matrix. In the MSSM with complex parameters, additional complex couplings can lead to CP violation within one generation at one - loop level [2]. Recently, the gauge boson in the MSSM with explicit CP violation has been studied [4] and CP violation as a probe of flavor origin in supersymmetry has been discussed [5]. To discover new particles in the MSSM, some collider problems have been studied [6, 7]. The CP violation has been considered and the one-loop correction has been caculated in these problems. Similarly, some decays of squarks have been studied when calculating the one-loop correction and evaluating the effects of CP violation on decay width. The particular researches are: squark decays into Higgs bosons and squark [8], squark decays into Charginos (or Neutralinos) and quark [9], Squark decays into Gauge bosons and squark [10]. Since the decays of squarks into quarks and gluino have not been calculated in detail, in this article, we study these problems in the MSSM with complex parameters Aq. Not only the analytic results but also the numerical results and the comparative graphs are given. The one-loop vertex correction, wave-function correction and renormalization of the bare couplings to the decay width have been caculated. 2. Content 2.1. Tree level results and vertex corrections Our terminologies and notations are in ref. [11]. At tree level, the amplitude of squark decay into quarks and gluino has the general form: M0(q˜i → q + g˜) = −u(k2) √ 2igsT a rs(R q˜ i1PR − Rq˜i2PL)v(k3). (2.1) The tree - level decay width can be written as Γ0 = β{(|Rq˜i1|2 + |Rq˜i2|2)(m2q˜i −m2q −m2g˜) + 4mqmg˜ℜ(R+q˜i1 Rq˜i2)}, (2.2) 95 Nguyen Chinh Cuong and Phung Van Hao where k1, k2 and k3 are the four - momenta of q˜i, q and g˜, respectively (Figure1.a), and β = εk(m2q , m 2 q˜i , m2g˜) 2πm3q˜i , k2(m2q, m 2 q˜i , m2g˜) =m 4 q +m 4 q˜i +m4g˜ − 2m2q.m2q˜i − 2m2qm2g˜ − 2.m2q˜im2g˜, ε =4παs/3. Figure 1. Feynman diagrams for the O(αs) SUSY - QCD corrections to squark decay into quarks and gluino: (a) tree level, (b) and (c) vertex corrections and (d) real gluon emission The one loop vertex correction (Figure 1 b→ d) result is: δΓ = δΓ(v) + δΓ(w) + δΓ(c) + δΓreal, (2.3) where δΓ(v) = β g2sT a ss′fabc 8.π2 [|Rq˜i1|2 + |Rq˜i2|2 − 2ℜ(R+q˜i1 Rq˜i2)](m2q˜i −m2q −m2g˜ − 2mqmg˜) ℜ{B0(m2q, m2g˜, m2q˜i) + 2.(m2q˜i −m2q +m2g˜)C11(m2g˜, m2q, m2q˜i, 0, m2g˜, m2q˜i) + B0(m 2 q˜i , 0, m2q˜i) + 2.(m 2 g˜ −m2q˜i −m2q)C12(m2g˜, m2q , m2q˜i, 0, m2g˜, m2q˜i) − B0(m2g˜, 0, m2g˜) + 2(m2g˜ +m2q˜i −m2q)C0(m2g˜, m2q, m2q˜i, 0, m2g˜, m2q˜i) − B0(m2q , 0, m2q) + 2(m2q˜i −m2g˜ −m2q + 2mqmg˜)C0(m2g˜, m2q˜i, m2q, 0, m2g˜, m2q)}, δΓ(w) = Γ0 −ε 4π2 ℜ(I1 + I2 + I3), I1 = (1 + 2mq′)[B0(m 2 q , 0, m 2 q)− B1(m2q , 0, m2q)]− 0.5, I2 = (mq + 2m 2 q)B1(m 2 q , m 2 g˜, m 2 q˜j ) + 2[m2g˜ + (−1)jmg˜mq˜jSin2θq˜]B0(m2q , m2g˜, m2q˜j), I3 = (mg˜ + 2m 2 g˜)B1(m 2 g˜, m 2 q˜j , m2q) + 2[m 2 q˜j + (−1)jmqmq˜jSin2θq˜]B0(m2g˜, m2q˜j , m2q), 96 Squarks decay into quarks and gluino in the MSSM δΓ(c) = β{(|Rq˜i1|2 + |Rq˜i2|2).(2mqδmq + 2mg˜δmg˜ − δm2q˜i) − 4(mqδmg˜ +mg˜δmq)ℜ(R+q˜i1 Rq˜i2)}, δmq = −ε 2π2 .ℜ{mq[B0(m2q , 0, m2q)− B1(m2q , 0, m2q)− 0.5] + 2[A0(m 2 q˜j ) +m2qB1(m 2 q, m 2 g˜, m 2 q˜j ) + (m2g˜ + (−1)jmg˜mq˜jSin2θq˜)B0(m2q , m2g˜, m2q˜j)]}, δmg˜ = −ε π2 ℜ{A0(m2q) +m2g˜B1(m2g˜, m2q˜j , m2q) + (m2q˜j + (−1)jmqmq˜jSin2θq˜)B0(m2g˜, m2q˜j , m2q)}, δm2q˜i = −ε 4.π2 .ℜ{m2g˜.[2B0(m2q˜i , 0, m2q˜i)− B1(m2q˜i , 0, m2q˜i)]− SijSjiA0(m2q˜i) + 4[A0(m 2 q) +m 2 q˜i B1(m 2 q˜i , m2g˜, m 2 q) + (m 2 g˜ + (−1)img˜mqSin2θq˜)B0(m2q˜i , m2g˜, m2q)]}. A0(m 2) = ∫ d4q iπ2 1 (q2 −m2 + iε) , B0;µ(p 2, m21, m 2 2) = ∫ d4q iπ2 1; qµ (q2 −m21 + iε)[(q + p)2 −m22 + iε] , Bµ(p 2, m21, m 2 2) = pµB1(p 2, m21, m 2 2), C0;µ;µν ≡ C0;µ;µν(p2, k2, (p+ k)2, m21, m22, m23) = ∫ d4q iπ2 1; qµ; qµν (q2 −m21 + iε)[(q + p)2 −m22 + iε][(q + p+ k)2 −m23 + iε] , Cµ = pµC11 + kµC12, Cµν = pµpνC21 + kµkνC22 + {pk}µνC23 + δµνC24. The total vitual δΓ(v)+δΓ(w)+δΓ(c) is utraviolet (UV) finite. In order to cancel the infrared (IR) divergence we include the emission of real (hard and soft) gluons, see Figure 1d, δΓreal ≡ Γ(q˜i → g + q + g˜). (2.4) And the decay width can be written as Γ = Γ0 + δΓ. (2.5) 2.2. Numerical results Let us now turn to the numerical analysis. Squark masses and mixing angles are fixed by the assumptions MD˜ = 1.12MQ˜ and |At| = |Ab| = 300GeV. In order to study the dependence of the ratio of the two decay widths ΓR and Γ on φ2 (for simplicity of notation, we abbreviate ΓR to the decay width in the case of real parameters), we have 97 Nguyen Chinh Cuong and Phung Van Hao chosen tanβ = 3, mt˜2 = 650GeV, mt˜1 = 350GeV, mb˜2 = 520GeV, mb˜1 = 170GeV, |µ| = 300GeV,mg˜ = 500GeV, cosθt˜ = - 0.5 and cosθb˜ = - 0.9. We first discuss the decays b˜2 → b+ g˜. Figure 2 shows the dependence of the ratios Γ0R/Γ 0 and ΓR/Γ on φ2 in the above case. In the decay b˜2 → b + g˜, φ2 can contribute ≈ −1.4%→ 0% to the Γ0 and contribute ≈ −4.6%→ 0% to the Γ. Figure 2 The dependence of Γ0R/Γ 0 and ΓR/Γ on φ2 in the decays b˜2 → b+ g˜ for mg˜ = 500GeV, mb˜2 = 520GeV, mb˜1 = 170GeV and cosθb˜ = - 0.9 We turn to the decays t˜2 → t + g˜. The dependence of Γ0R/Γ0 and ΓR/Γ on φ2 is shown in Figure 3. We can see from the graphs that the decay width changes significantly in accordance with the raising of φ2. In this case, the effect of φ2 on the decay width is stronger than that of the decays b˜2 → b + g˜. In the decay t˜2 → t + g˜, φ2 can contribute ≈ −0.8%→ 0% to the Γ0 and contribute ≈ −9.5%→ 0% to the Γ. Figure 3. The dependence of Γ0R/Γ 0 and ΓR/Γ on φ2 in the decays t˜2 → t + g˜ for mg˜ = 500GeV, mt˜2 = 650GeV, mt˜1 = 350GeV and cosθt˜ = - 0.5 3. Conclusion From the above studies, we come to some conclusions concerning squark decay into quarks and gluino. First, the effect of CP violation on the decay width is relatively large and it needs to be paid attention to when studying this problem. Second, the dependence 98 Squarks decay into quarks and gluino in the MSSM of ΓR/Γ on φ2 differs in each situation, and normally the effect of φ2 on the decay width of the stop decays is stronger than that of the sbottom decays [6, 9, 10]. Our results have the same significance as the results obtained from other such collisions and decays which are related to new articles in the MSSM [7, 8, 9, 10], and contribute to new physics. Evaluating the effect of CP violation on the decay width is expected to give useful results to experimental research and the discovery of new particles in the MSSM. Acknowledgements. This research is supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam. Grant number: 103.03-2012.80. REFERENCES [1] H. E. Haber and G. L. Kane., 1985. Phys. Rep., 177 75. [2] W. Bernrenther and M. Suzuki, 1991. Rev. Mod. Phys., 63, pp. 3-13. [3] J. Ellis and S. Rudaz, 1993. Phys. Lett. B, 128 248. [4] A. Pilaftsis and Calos E. M. Wagner, 1999. Phys. Lett. B, 553 3. [5] D. A. Demir, A. Masiero and O. Vives, hep-ph/9911337. [6] N.T.T.Huong, N.C.Cuong, H.H.Bang and D.T.L.Thuy, 2010. International Journal of Theoretical Physics 49, pp. 1457-1464. 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