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.
JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci., 2012, Vol 57, No 7, pp 94-99 This paper is available online at http://stdb.hnue.edu.vn SQUARKS DECAY INTO QUARKS AND GLUINO IN THE MSSM Nguyen Chinh Cuong1 and Phung Van Hao2 Faculty of Physics, Hanoi National University of Education 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 Introduction The minimal 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): qL and qR , corresponding to the left and the right helicity states of a quark The mass matrix on the basis (qL , qR ) is given by [3] Mq2 = m2qL aq mq aq mq m2qR = Rq † mq21 0 mq22 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 Rq (1.2) Squarks decay into quarks and gluino in the MSSM According to Eq.(1.2), Mq2 is diagonalized by a unitary matrix Rq The weak eigenstates q1 and q2 are thus related to their mass eigenstates qL and qR by q1 q2 = Rq qL qR , where i Rq = i e φq cos θq e− φq sin θq i i −e− φq sin θq e φq 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 2.1 Content 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: √ q q a M (qi → q + g) = −u(k2 ) 2igs Trs (Ri1 PR − Ri2 PL )v(k3 ) (2.1) The tree - level decay width can be written as q q +q q Γ0 = β{(|Ri1 | + |Ri2 | )(mq2i − m2q − m2g ) + 4mq mg ℜ(Ri1 Ri2 )}, (2.2) 95 Nguyen Chinh Cuong and Phung Van Hao where k1 , k2 and k3 are the four - momenta of qi , q and g, respectively (Figure1.a), and β= εk(m2q , m2qi , m2g ) , 2πmq3i k (m2q , m2qi , m2g ) =m4q + mq4i + m4g − 2m2q mq2i − 2m2q m2g − 2.mq2i m2g , ε =4παs /3 Figure 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 b → d) result is: δΓ = δΓ(v) + δΓ(w) + δΓ(c) + δΓreal , (2.3) where δΓ (v) a gs2 Tss ′ fabc q q +q q = β [|Ri1 | + |Ri2 | − 2ℜ(Ri1 Ri2 )](mq2i − m2q − m2g − 2mq mg ) 8.π ℜ{B0 (m2q , m2g , m2qi ) + 2.(mq2i − m2q + m2g )C11 (m2g , m2q , m2qi , 0, m2g , mq2i ) + B0 (m2qi , 0, mq2i ) + 2.(m2g − m2qi − m2q )C12 (m2g , m2q , m2qi , 0, m2g , m2qi ) − B0 (m2g , 0, m2g ) + 2(m2g + m2qi − m2q )C0 (m2g , m2q , mq2i , 0, m2g , m2qi ) δΓ(w) I1 − B0 (m2q , 0, m2q ) + 2(m2qi − m2g − m2q + 2mq mg )C0 (m2g , mq2i , m2q , 0, m2g , m2q )}, −ε = Γ0 ℜ(I1 + I2 + I3 ), 4π = (1 + 2mq′ )[B0 (m2q , 0, m2q ) − B1 (m2q , 0, m2q )] − 0.5, I2 = (mq + 2m2q )B1 (m2q , m2g , m2qj ) + 2[m2g + (−1)j mg mqj Sin2θq ]B0 (m2q , m2g , m2qj ), I3 = (mg + 2m2g )B1 (m2g , m2qj , m2q ) + 2[m2qj + (−1)j mq mqj Sin2θq ]B0 (m2g , m2qj , m2q ), 96 Squarks decay into quarks and gluino in the MSSM q q δΓ(c) = β{(|Ri1 | + |Ri2 | ).(2mq δmq + 2mg δmg − δm2qi ) δmq δmg δm2qi +q q − 4(mq δmg + mg δmq )ℜ(Ri1 Ri2 )}, −ε = ℜ{mq [B0 (m2q , 0, m2q ) − B1 (m2q , 0, m2q ) − 0.5] 2π + 2[A0 (mq2j ) + m2q B1 (m2q , m2g , mq2j ) + (m2g + (−1)j mg mqj Sin2θq )B0 (m2q , m2g , mq2j )]}, −ε ℜ{A0 (m2q ) + m2g B1 (m2g , m2qj , m2q ) = π2 + (m2qj + (−1)j mq mqj Sin2θq )B0 (m2g , m2qj , m2q )}, −ε ℜ{m2g [2B0 (mq2i , 0, mq2i ) − B1 (m2qi , 0, m2qi )] − Sij Sji A0 (m2qi ) = 4.π + 4[A0 (m2q ) + m2qi B1 (m2qi , m2g , m2q ) + (m2g + (−1)i mg mq Sin2θq )B0 (mq2i , m2g , m2q )]} d4 q , A0 (m ) = 2 iπ (q − m2 + iε) 1; qµ d4 q , B0;µ (p2 , m21 , m22 ) = 2 iπ (q − m1 + iε)[(q + p)2 − m22 + iε] Bµ (p2 , m21 , m22 ) = pµ B1 (p2 , m21 , m22 ), C0;µ;µν Cµ Cµν ≡ C0;µ;µν (p2 , k , (p + k)2 , m21 , m22 , m23 ) d4 q 1; qµ ; qµν = , 2 2 iπ (q − m1 + iε)[(q + p) − m22 + iε][(q + p + k)2 − m23 + iε] = pµ C11 + kµ C12 , = pµ pν C21 + kµ kν C22 + {pk}µν C23 + δµν C24 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(qi → g + q + g) (2.4) And the decay width can be written as Γ = Γ0 + δΓ 2.2 (2.5) 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, mt2 = 650GeV, mt1 = 350GeV, mb2 = 520GeV, mb1 = 170GeV, |µ| = 300GeV, mg = 500GeV, cosθt = - 0.5 and cosθb = - 0.9 We first discuss the decays b2 → b + g Figure shows the dependence of the ratios 0 ΓR /Γ and ΓR /Γ on φ2 in the above case In the decay b2 → b + g, φ2 can contribute ≈ −1.4% → 0% to the Γ0 and contribute ≈ −4.6% → 0% to the Γ Figure The dependence of Γ0R /Γ0 and ΓR /Γ on φ2 in the decays b2 → b + g for mg = 500GeV, mb2 = 520GeV, mb1 = 170GeV and cosθb = - 0.9 We turn to the decays t2 → t + g The dependence of Γ0R /Γ0 and ΓR /Γ on φ2 is shown in Figure 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 b2 → b + g In the decay t2 → t + g, φ2 can contribute ≈ −0.8% → 0% to the Γ0 and contribute ≈ −9.5% → 0% to the Γ Figure The dependence of Γ0R /Γ0 and ΓR /Γ on φ2 in the decays t2 → t + g for mg = 500GeV, mt2 = 650GeV, mt1 = 350GeV and cosθt = - 0.5 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 [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 [7] Nguyen Thi Thu Huong, Ha Huy Bang, Nguyen Chinh Cuong and Dao Thi Le Thuy, 2007 International Journal of Theoretical Physics 46, pp 39-48 [8] Nguyen Chinh Cuong, Dao Thi Le Thuy and Ha Huy Bang, 2003 Communications in Physics, 14, pp 27-33 [9] N.C.Cuong and H.H.Bang, 2004 Communications in Physics 14, pp 23-30 [10] Dao Thi Le Thuy, Nguyen Chinh Cuong and Ha Huy Bang, 2004 Communications in Physics, 14, pp 157-164 [11] A Barlt, and et al., 1998 Phys Lett B, 419, p 243 99 ... 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... 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.. .Squarks decay into quarks and gluino in the MSSM According to Eq.(1.2), Mq2 is diagonalized by a unitary matrix Rq The weak eigenstates q1 and q2 are thus related to their mass eigenstates