Squarks decay into quarks and gluino in the 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.

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

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 Atand Ab could be

quite significant in a large region of the MSSM parameter space

Keywords: MSSM, squark, CP violation

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 coefficientµ in the bilinear term of the Higgs superfield There are two complex terms in the soft supersymmetry (SUSY) breaking part: the gaugino mass fM and the left and right handed squark mixing term Aq[2]

µ = |µ|eiφµ = |µ|eiφ1, Aq = |Aq|eiφe q = |Aq|eiφ2, fM = |fM |eiφM f = |fM |eiφ3 (1.1)

In the MSSM, there are two types of scalar quarks (squarks): qeL and qeR, corresponding to the left and the right helicity states of a quark The mass matrix on the basis (qeL,eqR) is given by [3]

Mqe2 =



m2 e

qL aqmq

aqmq m2

e qR



= Rqe
†

m2 e q1 0

0 m2 e q2



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

According to Eq.(1.2), M2

e

q is diagonalized by a unitary matrix Rq e The weak eigenstatesqe1 andqe2are thus related to their mass eigenstatesqeLandqeRby

 e

q1 e

q2



= Re q

 e

qL e

qR

 , where

Re q = e

i

2 φeqcos θqe e− 2iφeqsin θqe

−e− i

2 φeqsin θqe e2iφ qecos θqe

!

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 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(eqi → q + eg) = −u(k2)√

2igsTa

rs(Ri1eqPR− Rqi2ePL)v(k3) (2.1) The tree - level decay width can be written as

Γ0 = β{(|Ri1eq|2+ |Rqi2e|2)(m2qie − m2q− m2e g) + 4mqmegℜ(R+ei1qReqi2)}, (2.2)

wherek1,k2andk3are the four - momenta ofeqi, q andeg, respectively (Figure1.a), and

β =εk(m

2

q, m2 e

qi, m2 e

g) 2πm3 e qi ,

k2(m2q, m2qie, m2eg) =m4q+ m4eqi+ m4eg− 2m2q.m2eqi − 2m2qm2eg − 2.m2e qim2eg,

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) = βg

2

sTa

ss ′fabc 8.π2 [|Rqi1e|2+ |Ri2eq|2− 2ℜ(R+ei1qRqi2e)](m2qie − m2q− m2e g− 2mqmeg) ℜ{B0(m2q, m2eg, m2eqi) + 2.(m2eqi− m2q+ m2eg)C11(m2eg, m2q, m2qie, 0, m2eg, m2eqi)

+ B0(m2eqi, 0, m2eqi) + 2.(m2eg− m2eqi − m2q)C12(m2eg, m2q, m2qie, 0, m2eg, m2eqi)

− B0(m2eg, 0, m2eg) + 2(m2eg + m2eqi − m2q)C0(m2eg, m2q, m2eqi, 0, m2eg, m2qie)

− B0(m2q, 0, m2q) + 2(m2qie − m2e g− m2q+ 2mqmeg)C0(m2eg, m2eqi, m2q, 0, m2eg, m2q)},

δΓ(w) = Γ0−ε

4π2ℜ(I1+ I2+ I3),

I1 = (1 + 2mq ′)[B0(m2q, 0, m2q) − B1(m2q, 0, m2q)] − 0.5,

I2 = (mq+ 2m2q)B1(m2q, m2eg, m2qje) + 2[m2eg+ (−1)jmegmeqjSin2θqe]B0(m2q, m2eg, m2qje),

I3 = (meg + 2m2eg)B1(m2eg, m2qje, m2q) + 2[m2qje + (−1)jmqmeqjSin2θqe]B0(m2eg, m2qje, m2q),

δΓ(c) = β{(|Ri1eq|2+ |Rqi2e|2).(2mqδmq+ 2megδmeg− δm2

e

qi)

− 4(mqδmeg+ megδmq)ℜ(R+ei1qRqi2e)},

δmq = −ε

2π2.ℜ{mq[B0(m2q, 0, m2q) − B1(m2q, 0, m2q) − 0.5]

+ 2[A0(m2

e

qj) + m2

qB1(m2

q, m2 e

g, m2 e

qj) + (m2

e

g+ (−1)jmegmeqjSin2θeq)B0(m2

q, m2 e

g, m2 e

qj)]},

δmeg = −ε

π2ℜ{A0(m2q) + m2egB1(m2eg, m2qje, m2q)

+ (m2qje + (−1)jmqmeqjSin2θqe)B0(m2eg, m2qje, m2q)},

δm2qie = −ε

4.π2.ℜ{m2e g.[2B0(m2eqi, 0, m2eqi) − B1(m2qie, 0, m2qie)] − SijSjiA0(m2eqi)

+ 4[A0(m2q) + m2qieB1(m2qie, m2eg, m2q) + (m2eg+ (−1)imegmqSin2θeq)B0(m2qie, m2eg, m2q)]}

A0(m2) =

Z

d4q

iπ2

1 (q2− m2+ iε),

B0;µ(p2, m2

1, m2

2) =

Z d4q

iπ2

1; qµ (q2− m2

1+ iε)[(q + p)2− m2

2+ iε],

Bµ(p2, m2

1, m2

2) = pµB1(p2, m21, m22),

C0;µ;µν ≡ C0;µ;µν(p2, k2, (p + k)2, m21, m22, m23)

=

Z d4q

iπ2

1; qµ; qµν (q2− m2

1+ iε)[(q + p)2− m2

2+ iε][(q + p + k)2− m2

3+ iε],

Cµ = pµC11+ kµC12,

Cµν = pµpνC21+ kµkνC22+ {pk}µνC23+ δµνC24

The total vitualδΓ(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(eqi → g + q + eg) (2.4) And the decay width can be written as

2.2 Numerical results

Let us now turn to the numerical analysis Squark masses and mixing angles are fixed by the assumptionsMDe = 1.12MQe 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

chosen tanβ = 3, met2 = 650GeV,met1 = 350GeV,meb2 = 520GeV, meb1 = 170GeV,|µ| = 300GeV,meg = 500GeV, cosθet= - 0.5 and cosθeb= - 0.9

We first discuss the decays eb2 → b + eg Figure 2 shows the dependence of the ratios

Γ0

R/Γ0 and ΓR/Γ on φ2 in the above case In the decay eb2 → b + eg, φ2 can contribute

≈ −1.4% → 0% to the Γ0 and contribute≈ −4.6% → 0% to the Γ

Figure 2 The dependence ofΓ0

R/Γ0andΓR/ Γ on φ2 in the decays eb2 → b + eg

formeg= 500GeV,meb2 = 520GeV,meb1 = 170GeV and cosθeb= - 0.9

We turn to the decays et2 → t + eg 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 ofφ2 In this case, the effect of φ2 on the decay width is stronger than that of the decays eb2 → b + eg In the decay et2 → t + eg, φ2 can contribute

≈ −0.8% → 0% to the Γ0 and contribute≈ −9.5% → 0% to the Γ

Figure 3 The dependence ofΓ0

R/Γ0andΓR/ Γ on φ2 in the decayset2 → t + eg

formeg = 500GeV,met2 = 650GeV,met1 = 350GeV and cosθet= - 0.5

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

ofΓR/Γ on φ2 differs in each situation, and normally the effect ofφ2 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

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