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Single top quark production via SUSY-QCD FCNC couplings at the CERN LHC in the unconstrained MSSM
Nuclear Physics B 705 (2005) 3–32
Single top quark production via SUSY-QCD FCNC couplings at the CERN LHC in the unconstrained MSSM Jian Jun Liu a , Chong Sheng Li a , Li Lin Yang a , Li Gang Jin b a Department of Physics, Peking University, Beijing 100871, China b Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China
Received 13 April 2004; accepted 18 October 2004 Available online 13 November 2004
Abstract We evaluate the t c¯ and t u¯ productions at the LHC within the general unconstrained MSSM framework. We find that these single top quark productions induced by SUSY-QCD FCNC couplings have remarkable cross sections for favorable parameter values allowed by current low energy data, which can be as large as a few pb. Once large rates of the t c¯ and t u¯ productions are detected at the LHC, they may be induced by SUSY FCNC couplings. We show that the precise measurement of single top quark production cross sections at the LHC is a powerful probe for the details of the SUSY FCNC couplings. 2004 Elsevier B.V. All rights reserved. PACS: 14.65.Ha; 12.60.Jv; 11.30.Pb Keywords: Top quark; MSSM; FCNC
1. Introduction The flavor dynamics, such as the mixing of three generation fermions and their large mass differences observed, is still a great mystery in particle physics today. The heaviest E-mail address: [email protected] (C.S. Li). 0550-3213/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.10.048
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one of three generation fermions is the top quark with the mass close to the electroweak (EW) symmetry breaking scale. Therefore, it can play a role of a wonderful probe for the EW breaking mechanism and new physics beyond the standard model (SM) through its decays and productions. An important aspect of the top quark physics is to investigate anomalous flavor changing neutral current (FCNC) couplings. Within the SM, FCNC is forbidden at the tree-level and highly suppressed by the GIM mechanism [1] at one-loop level. All the precise measurements of various FCNC processes, for example, b → sγ [2], agree with the SM predictions. However, many new physics models allow the existence of the tree-level FCNC couplings, and may enhance some FCNC processes. In particular, the minimal supersymmetric standard model (MSSM) is one of the most popular new physics models. And it is hard to believe that this model will say no more about the flavor dynamics than the SM. The MSSM includes 105 free parameters beyond the SM: 5 real parameters and 3 CP-violating phases in the gaugino/higgsino sector, 21 squark and slepton masses, 36 new real mixing angles to define the squark and slepton mass eigenstates, and 40 new CP-violating phases that can appear in squark and slepton interactions [2], so in general, it can provide a new explanation of the source of FCNC couplings. Actually, many of the parameters are constrained by present experimental data, and some popular SUSY models, for example, the gravity-mediated supersymmetry broken model (SUGRA) [3], gauge mediated supersymmetry broken model (GMSB) [4], anomaly mediated supersymmetry broken model (AMSB) [5], gaugino mediated supersymmetry broken model (gMSB) [6] and Kaluza–Klein mediated supersymmetry broken model [7], make some ad hoc assumptions to reduce the number of independent parameters. Although the predictions of these models can agree with the present constraints from FCNC experiments, it should be noted that few experimental constraints on the top quark FCNC processes are available so far [2]. Since the CERN Large Hadron Collider (LHC) can produce abundant top events, the measurement of the top quark rare processes will become possible. Therefore, a good understanding of the theoretical predictions of these processes, especially within the general unconstrained MSSM framework, is important. And a careful investigation will provide some clues to the constraint relations among the supersymmetry (SUSY) parameters and more clear information about the SUSY breaking mechanisms. The top quark FCNC processes have been investigated in detail. There are two categories of these investigations, one of which is the top quark FCNC decays [8], especially for t → cg [9,10] and t → ch [11] in the MSSM. These results show that the branch ratio of t → cg can reach 10−4 [10], which enhances the one in the SM (∼ 10−11 ) significantly. The other category is the top quark productions through FCNC processes [12–17], ones of which induced by the anomalous top quark FCNC couplings at the colliders have been extensively discussed in a model independent way [12–16] and model dependent way [17], respectively. As we know, most of the works within SUSY models are limited in some constrained MSSM, where the top quark FCNC couplings were studied through some SUSY-CKM matrices or mass insertion approximation [18]. However, it is interesting to study the top quark FCNC productions using soft SUSY broken parameters directly, and then establish some relations between the production cross sections and soft SUSY broken parameters in the mass eigenstate formalism [19,20]. Actually, the general framework with SUSY FCNC mechanisms was presented several years ago [21], and there have been a lot of works studying SUSY FCNC processes within this framework [19,20]. But the top quark
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SUSY FCNC production processes have not been studied in above framework so far, and this means that the relations between the observables of the top quark production processes and the mixing among up squarks involving the third generation have not been established yet. In this paper, we will investigate the single top quark productions at the LHC induced by SUSY FCNC couplings in the general unconstrained MSSM, which include the top quark productions associated with the anti-charm quark and the anti-up-quark, and we believe that it is an important step towards revealing the top quark SUSY FCNC couplings and exploring some relations among SUSY broken parameters at the LHC. The paper is organized as follows. In Section 2 we describe the general framework of SUSY FCNC mechanisms. In Section 3 we evaluate the total production cross sections of the pp → t c( ¯ u) ¯ process induced by SUSY FCNC couplings, and the detail expressions for various couplings and form factors are given in Appendices A and B, respectively. In Section 4 we present our numerical calculations and discussions. Finally, Section 5 gives our conclusions.
2. The FCNC in the MSSM In order to make our paper self-contained, we start with a brief description of the FCNC mechanism in the MSSM, as shown in Ref. [21], and establish our notation conventions. The SUSY part of the MSSM Lagrangian is 1 a µν / ab λbG + Dµ φ † (Dµ φ) + i ψ¯ Dψ / LSUSY = − FG FGa µν + i λ¯ aGD 4 ˜ ˜ ˜ ∂W 1 ∂ 2W ∂W ψiT Cψj + h.c. − − ∂φi ∂φi 2 ∂φi ∂φj √ † a aT 1 2 † a † a − 2gG φ TG λG Cψ + h.c. − gG φ TG φ φ TG φ , 2 where ˜ = µH 1 H 2 + Y I J H 1 L˜ I E˜ J + Y I J H 1 Q˜ I D˜ J + YuI J H 2 Q˜ I U˜ J . W l R d R R
(1)
(2)
Here subscript G represents color, weak isospin and supercharge indices of the SM gauge group SU(3)C × SU(2)L × U (1)Y , respectively, a and b are the indices of adjoint representations of the non-abelian subgroups, and I, J = 1, 2, 3 are the generation indices. φ, ψ and λ represent scalar fields, matter fermion fields and gauginos, respectively. C is charge conjugation matrix. L˜ I (Q˜ I ) and E˜ RI (U˜ RI , D˜ RI ) represent slepton (squark) SU(2) left-hand doublets and right-hand singlets, respectively. H 1,2 represent two Higgs SU(2) doublets, and their vacuum expectation values are v1 v cos β √ √ 2 1 0 0 2 2 = = ≡ , H H (3) ≡ v√ sin β , v2 √ 0 0 2 2 √ where v = ( 2GF )−1/2 = 246 GeV, and the angle β is defined by tan β ≡ v2 /v1 , the ratio of the vacuum expectation values of the two Higgs doublets. µ is the Higgs mixing parameter.
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
The soft SUSY breaking Lagrangian is 1 M3 g˜ a T C g˜ a + M2 W˜ iT C W˜ i + M1 B˜ T C B˜ + h.c. 2 I J I J J 2 1† 1 2 2† 2 E˜ R H − MH H − L˜ I † ML2 L˜ J − E˜ RI † ME2 − MH 1H 2H IJ ˜ J 2 IJ ˜ J 2 IJ ˜ J ˜ I † MQ −Q Q − D˜ RI † MD DR − U˜ RI † MU2 UR IJ 1 I J + AE H L˜ E˜ R + AIDJ H 1 Q˜ I D˜ RJ + AIUJ H 2 Q˜ I U˜ RJ + BµH 1 H 2 + h.c. . (4)
Lsoft = −
2 , M2 Here MQ U,D and AU,D are the soft broken SU(2) doublet squark mass squared matrix, the SU(2) singlet squark mass squared matrix and the trilinear coupling matrix, respectively. E,U,D , the lepton and quark mass eigenstates are given Using 3 × 3 unitary matrices VL,R by
ν = VLE ΨL1 , l = VLE ΨL2 + VRE C Ψ¯ ET ,
u = VLU ΨQ1 + VRU C Ψ¯ UT , d = VLD ΨQ2 + VRD C Ψ¯ DT .
(5)
Their corresponding diagonal 3 × 3 mass matrices are v cos β ml = − √ VRE YlT VLE† , 2 v cos β D T D† md = − √ VR Yd VL . 2
v sin β mu = √ VRU YuT VLU † , 2
As in the SM, the Cabibbo–Kobayashi–Maskawa (CKM) matrix is K = VLU VLD† . It is convenient to specify the squark mass matrices in the super-CKM basis, in which the mass matrices of the quark fields are diagonalized by rotating the superfields. The super-CKM basis U˜ 0 is defined as U VL U˜ L U˜ 0 = (6) . VRU U˜ R∗ And in this basis, the up squark mass matrix is 6 × 6 matrix, which has the form: 2 2 M ˜ LL + Fu LL + Du LL M ˜ LR + Fu LR U 2 † 2 U M2U˜ ≡ . M ˜ LR + (Fu LR )† M ˜ RR + Fu RR + Du RR U
(7)
U
The F terms and D terms are diagonal 3 × 3 submatrices, which are given by Fu LR = −µ(mu cot β)13 , Du LL = Du RR = cos 2β m2Z
Fu LL = Fu RR = m2u 13 , 1 2 2 − sin θW 13 , 2 3
(8) (9)
where θW is the Weinberg angle, 13 stands for the 3 ×3 unit matrix. And (M 2˜ )LL , (M 2˜ )RR
and (M 2˜ )LR contain the flavor-changing entries
U
MU2˜
LL
2 U† = VLU MQ VL ,
MU2˜
RR
T = VRU MU2 VRU † ,
U
U
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
MU2˜
LR
7
v sin β = − √ VLU A∗U VRU † , 2
(10)
which are directly related to the mechanism of SUSY breaking, and are in general not diagonal in the super-CKM basis. Furthermore, (M 2˜ )LR , arising from the trilinear terms U
in the soft potential, is not hermitian. The matrix M2˜ can further be diagonalized by an U additional 6 × 6 unitary matrix ZU to give the up squark mass eigenvalues
M2U˜
diag
† = ZU M2U˜ ZU .
(11)
As for the down squark mass matrix, we also can define M2˜ as the similar form D
of Eq. (7) with the replacement of (M 2˜ )I J (I, J = L, R) by (M 2˜ )I J . Note that since SU(2)L gauge invariance implies that
U (M 2˜ )LL U
D
= K(M 2˜ )LL K † , the matrices (M 2˜ )LL and D
U
(M 2˜ )LL are correlated and cannot be specified independently. D Thus, in the super-CKM basis, there are new potential sources of flavor-changing neutral current: neutralino–quark–squark coupling and gluino–quark–squark coupling, which arise from the off-diagonal elements of (M 2˜ )LL , (M 2˜ )LR and (M 2˜ )RR . As the former coupling U U U is in general much weaker than the latter one, in this paper, we only consider the SUSYQCD FCNC effects induced by the gluino–quark–squark coupling, which can be written as √
i 2gs T a −(ZU )I i PL + (ZU )(I +3)i PR , I, i = 1, 2, 3. (12) Here PL,R ≡ (1 ∓ γ5)/2 and T a is the SU(3) color matrix. Thus the flavor changing effects 2 , M 2 and A on the observables can be obtained through the of soft broken terms MQ U U matrix ZU . As mentioned in the introduction, most of previous literatures studied the FCNC processes in the MSSM by so-called mass insertion approximation [18]. However, when the off-diagonal elements in the up squark mass matrix become large, the mass insertion approximation is no longer valid [19,20]. Therefore, we use the general mass eigenstate formalism as described above, which has been adopted in the literatures recently [10,19, 20]. So any effects of SUSY FCNC couplings in loops on various observables will provide some information of the SUSY breaking mechanism. In practical calculations, because the squark mass matrix M2˜ is a 6 × 6 matrix, it is U very formidable to calculate the effects arising from the matrix on the observables of FCNC processes. Thus, as shown in Refs. [10,19,20], it is reasonable to calculate the observables by only switching on one off-diagonal element in (M 2˜ )LL , (M 2˜ )LR and (M 2˜ )RR at a U U U time. For the aim of this paper, the following strategy in the calculations of the process pp → t c( ¯ u) ¯ will be used: first we deal with the LL, LR, RL and RR block of the matrix M2˜ separately and in each block we only consider the effects of individual element on the U production cross sections, and then we investigate both the interference effects between different entries within one block and the interference effects between different blocks. To simplify the calculation we further assume that all diagonal entries in (M 2˜ )LL , (M 2˜ )LR , U
U
2 , and then we nor(M 2˜ )RL and (M 2˜ )RR are set to be equal to the common value MSUSY U
U
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
2 malize the off-diagonal elements to MSUSY [19,22],
ij δU LL
ij δU LR
= =
ij U , 2 MSUSY ij (M 2˜ )LR U , 2 MSUSY
(M 2˜ )LL
ij
(M 2˜ )RR ij U δU RR = , 2 MSUSY
ij δU RL
=
ij U 2 MSUSY
(M 2˜ )RL
Thus (M 2˜ )LL can be written as follows: U 12 1 δU LL 2 21 2 MU˜ LL = MSUSY δU LL 1 31 32 δU LL δU LL and analogously for all the other blocks.
(i = j, i, j = 1, 2, 3).
13 δ LL U 23 , δU LL 1
(13)
(14)
3. The process pp → t c( ¯ u) ¯ induced by SUSY FCNC couplings The process pp → t c( ¯ u) ¯ at the LHC can be induced through the SUSY-QCD FCNC ¯ s s¯ , cc, couplings with the initial partonic states uu, ¯ d d, ¯ uc, ¯ cu¯ and gg, as shown in Fig. 1. ¯ b s¯ and s b¯ initial states Our practical calculations show that the contributions from the bb, are negligibly small, so we do not discuss them below. As the t u¯ process is completely similar to the t c¯ one, here we only discuss the situations of t c¯ in detail. Neglecting the charm quark mass, the amplitude of the process gg → t c¯ can be written as gg
gg
gg
Mgg = Mself + Mvertex + Mbox, gg Mself ,
gg Mvertex
(15)
gg Mbox
and are the amplitudes of self-energy diagrams Fig. 1(a)–(e), where vertex diagrams Fig. 1(f), (g) and (h) and box diagrams Fig. 1(i)–(o), respectively. They can be further expressed as gg Mself
=
14
gg
fsi Mi ,
(16)
i=1 gg
Mvertex =
30
gg
fvi Mi ,
(17)
gg
(18)
i=1 gg
Mbox =
40
fbi Mi ,
i=1
respectively. Here fsi , fvi and fbi are form factors corresponding to the self-energy diagrams, vertex diagrams and box diagrams, respectively, and their expressions are given gg explicitly in Appendix B. Mi are the standard matrix elements, which are defined by gg
M1,2 = u(p ¯ t )/ε(k1 )k1 · ε(k2 )PL,R v(pc ), gg
M3,4 = u(p ¯ t )/ε(k2 )k2 · ε(k1 )PL,R v(pc ),
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
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Fig. 1. Gluon initial state subprocess Feynman diagrams for the top quark and anti-charm quark associated production. gg
M5,6 = u(p ¯ t )/ε(k2 )pt · ε(k1 )PL,R v(pc ), gg
M7,8 = u(p ¯ t )/k1 ε(k1 ) · ε(k2 )PL,R v(pc ), gg
¯ t )ε(k1 ) · ε(k2 )PL,R v(pc ), M9,10 = u(p gg
M11,12 = u(p ¯ t )/ε(k1 )/k1 ε/(k2 )PL,R v(pc ),
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32 gg
M13,14 = u(p ¯ t )/ε(k1 )/ε(k2 )PL,R v(pc ), gg
¯ t )/ε(k1 )/k1 k1 · ε(k2 )PL,R v(pc ), M15,16 = u(p gg
M17,18 = u(p ¯ t )/ε(k1 )pt · ε(k2 )PL,R v(pc ), gg
¯ t )/ε(k1 )/k1 pt · ε(k2 )PL,R v(pc ), M19,20 = u(p gg
M21,22 = u(p ¯ t )/ε(k2 )/k1 k2 · ε(k1 )PL,R v(pc ), gg
¯ t )/ε(k2 )/k1 pt · ε(k1 )PL,R v(pc ), M23,24 = u(p gg
M25,26 = u(p ¯ t )k1 · ε(k2 )pt · ε(k1 )PL,R v(pc ), gg
¯ t )pt · ε(k2 )k2 · ε(k1 )PL,R v(pc ), M27,28 = u(p gg
M29,30 = u(p ¯ t )pt · ε(k2 )pt · ε(k1 )PL,R v(pc ), gg
M31,32 = u(p ¯ t )/k1 k1 · ε(k2 )k2 · ε(k1 )PL,R v(pc ), gg
¯ t )/k1 k1 · ε(k2 )pt · ε(k1 )PL,R v(pc ), M33,34 = u(p gg
M35,36 = u(p ¯ t )k1 · ε(k2 )k2 · ε(k1 )PL,R v(pc ), gg
¯ t )/k1 pt · ε(k2 )k2 · ε(k1 )PL,R v(pc ), M37,38 = u(p gg
M39,40 = u(p ¯ t )/k1 pt · ε(k2 )pt · ε(k1 )PL,R v(pc ),
(19)
where k1,2 denote the momenta of incoming partons, while pt and pc are used for the outgoing top and anti-charm-quarks, respectively. For the quarks initiated subprocesses, we also define the standard matrix elements as q q¯
¯ 2 )Pα γµ u(k1 )u(p ¯ t )Pβ γ µ v(pc ), M1αβ = v(k q q¯
M2αβ = u(p ¯ t )Pα v(pc )v(k ¯ 2 )Pβp / t u(k1 ), q q¯
M3αβ = v(k ¯ 2 )Pα u(k1 )u(p ¯ t )Pβ v(pc ), q q¯
¯ 2 )Pα u(k1 )u(p ¯ t )Pβ k/1 v(pc ), M4αβ = v(k q q¯
M5αβ = v(k ¯ 2 )Pα v(pc )u(p ¯ t )Pβ u(k1 ), q q¯
M6αβ = v(k ¯ 2 )Pα v(pc )u(p ¯ t )Pβ k/2 u(k1 ), q q¯
M7αβ = u(p ¯ t )Pα u(k1 )v(k ¯ 2 )Pβ k/1 v(pc ), q q¯
¯ 2 )Pα k/1 v(pc )u(p ¯ t )Pβ k/2 u(k1 ), M8αβ = v(k q q¯
M9αβ = v(k ¯ 2 )Pαp / t u(k1 )u(p ¯ t )Pβ k/1 v(pc ), q q¯
M10αβ = v(k ¯ 2 )Pα γµ v(pc )u(p ¯ t )Pβ γ µ u(k1 ), q q¯
M11αβ = u(p ¯ t )Pα u(k2 )v(p ¯ c )Pβ v(k1 ), q q¯
M12αβ = u(p ¯ t )Pα u(k2 )v(p ¯ c )Pβ k/2 v(k1 ), q q¯
¯ t )Pβ k/1 u(k2 )v(p ¯ c )Pα v(k1 ), M13αβ = u(p
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
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Fig. 2. Quark initial state subprocess Feynman diagrams for the top quark and anti-charm quark associated production.
q q¯
M14αβ = u(p ¯ t )Pα k/1 u(k2 )v(p ¯ c )Pβ k/2 v(k1 ), q q¯
M15αβ = u(p ¯ t )Pα γµ u(k2 )v(p ¯ c )Pβ γ µ v(k1 ),
(20)
where (α, β) = (L, L), (L, R), (R, L) and (R, R), respectively. And the amplitude for the quarks initiated subprocesses can be expressed as
Mq q¯ =
15
q q¯
g iαβ Miαβ ,
(21)
i=1 α,β=L,R
where the g iαβ are form factors which can be fixed by the straightforward calculations of the relevant diagrams. However, the Feynman diagrams contributing to the subprocesses with the different initial states can be different. We first describe the amplitude of the subprocess cc¯ → t c, ¯ which has the largest set of Feynman diagrams, i.e., all diagrams in Fig. 2, and the explicit expressions of the non-zero g iαβ of each diagram can be found in Appendix B. In Table 1 we list all other channels that can contribute to t c( ¯ u) ¯ production and their related Feynman diagrams, and their corresponding form factors can be obtained from ones of cc¯ → t c¯ by modifying some couplings as shown in the table.
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Table 1 The form factors of other channels obtained from ones of cc¯ → t c. ¯ Here, ViL(R) are defined in the Appendix A. The subscript ρ represents the corresponding diagram Channel
Related diagrams in Fig. 2
Form factors
uu¯ → t u¯
b1 , b2 , b3 , b4 , v1 , v2 , v3 , v4 , s1 , s2 , s3 , s4
gρ (V5L(R) ↔ V4L(R) )
iαβ iαβ
gρ (V5L(R) ↔ V4L(R) )
uu¯ → t c¯ cc¯ → t u¯
b1 , b2 , b3 , b4 , v1 , v2 , s1 , s2
iαβ
gρ (V5L(R) ↔ V4L(R) ) iαβ
gρ (V5L(R) ↔ V4L(R) )
uc¯ → t c¯ cu¯ → t u¯
b1 , b2 , b3 , b4 , v3 , v4 , s3 , s4
iαβ
gρ (V5L(R) ↔ V4L(R) ) iαβ
gρ (V5L(R) ↔ V4L(R) )
cu¯ → t c¯ uc¯ → t u¯
iαβ
gρ (V5L(R) ↔ V4L(R) )
b1 , b2 , b3 , b4
iαβ
gρ (V6L(R) ↔ V5L(R) )
s s¯ → t c( ¯ u) ¯ d d¯ → t c( ¯ u) ¯
iαβ
gρ (V6L(R) ↔ V4L(R) )
b2 , b3 , v1 , v2 , s1 , s2
The partonic level cross section for κλ → t c¯ is tˆ+ σˆ
κλ
= tˆ−
1 κλ 2 M d tˆ 16π sˆ 2
(22)
with m2 + m2t − sˆ 1 tˆ± = c (23) ± (ˆs − (mc + mt )2 )(ˆs − (mc − mt )2 ), 2 2 √ ¯ states. The total hadronic cross section where sˆ is the c.m. energy of the κλ (gg or q q) for pp → κλ → t c¯ can be written in the form σ (s) =
κ,λ
Here as
1
dz √ (mc +mt )/ s
dL κλ σˆ dz
(κλ → t c¯ at sˆ = z2 s).
(24)
√ s is the c.m. energy of the pp states, and dL/dz is the parton luminosity, defined dL = 2z dz
1
dx fκ/p (x, µ)fλ/p z2 /x, µ , x
(25)
z2
where fκ/p (x, µ) and fλ/p (z2 /x, µ) are the κ and λ parton distribution functions, respectively.
4. Numerical calculation and discussion In the following we present some numerical results for the total cross section of the single top quark production induced by SUSY-QCD FCNC couplings at the LHC.
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
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Fig. 3. The total cross sections for the pp → t u¯ as a function of the mass of gluino.
In our numerical calculations the SM parameters were taken to be mt = 174.3 GeV, MW = 80.423 GeV, MZ = 91.1876 GeV, mc = 1.2 GeV, sin2 (θW ) = 0.23113 and αs (MZ ) = 0.1172 [2]. And we used the CTEQ6L PDF [23] and took the factorization scale and the renormalization scale as µf = µr = mt /2. The relevant SUSY parameters are µ, tan β, MSUSY and mg˜ , which are unrelated to flavor changing mechanism, and may be fixed from flavor conserving observables at the future colliders. And they are chosen as follows: MSUSY = 400, 1000 GeV, tan β = 3, 30, mg˜ = 200, 300 GeV and µ = 200 GeV. ij As for the range of the flavor mixing parameters, (δU )LL are constrained by correspondij 12 ) ¯ mixing [26], and ing (δD )LL [19,22,24,25], in which (δU LL also is constrained by K–K 31 12 12 12 D0 –D¯ 0 mixing makes constraints on (δU )LL , (δU )LR and (δU )RL [27]. And (δU )LL , 32 31 32 ¯ (δU )LL , (δU )RL and (δU )RL are constrained by the chargino contributions to Bd –Bd mixing [24]. Finally, there also are constraints on the up squark mass matrix from the chargino contributions to b → sγ [19,28]. Taking into account above constraints, in our numerical calculations, we use the following limits: 12 ) , (δ 12 ) 12 (i) (δU LL U LR and (δU )RL are less than 0.08MSUSY /(1 TeV); 13 12 (ii) (δU )RR and (δU )LL are limited below 0.2MSUSY /(1 TeV); 23 23 23 23 13 13 13 (iii) (δU )LL , (δU )LR , (δU )RL , (δU )RR , (δU )LR , (δU )RL and (δU )RR vary from 0 to 1.
Our results are shown in Figs. 3–10. There are two common features of these curves: one is that the cross sections increase rapidly with the mixing parameters increasing, and the other is that the cross sections depend strongly on the gluino mass mg˜ , but weakly depend on tan β (so we just discuss the results for tan β = 30 below). Comparing with above cases, the dependence on MSUSY is medium.
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
Fig. 4. The total cross sections for the pp → t c( ¯ u) ¯ with LL off-diagonal elements. Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
13 ) Fig. 5. The total cross sections for the pp → t c( ¯ u) ¯ with RR off-diagonal elements (δU RR . Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
15
23 ) Fig. 6. The total cross sections for the pp → t c( ¯ u) ¯ with RR off-diagonal elements (δU RR . Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
13 ) Fig. 7. The total cross sections for the pp → t c( ¯ u) ¯ with LR off-diagonal elements (δU LR . Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
16
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
23 ) Fig. 8. The total cross sections for the pp → t c( ¯ u) ¯ with LR off-diagonal elements (δU LR . Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
Fig. 3 shows that in general the production rates of gluon fusion processes are several times larger than ones of the q q¯ annihilation, and these rates all depend strongly on the 13 ) gluino mass mg˜ . For example, assuming (δU LR = 0.7 and tan β = 30, the production rates of gg → t u¯ decrease from 700 fb at mg˜ = 200 GeV to 70 fb at mg˜ = 500 GeV. 23 ) in the LL block. For Fig. 4 shows the dependence of the cross sections on the (δU 23 ) tan β = 30 the cross section of t c¯ production can reach around 12 fb when (δU LL = 0.7, MSUSY = 400 GeV, and mg˜ = 200 GeV, as shown in Fig. 4(b), but for t u¯ production it 13 ) is one order of magnitude smaller. The production cross sections arising from (δU LL are 13 less than 1 fb as (δU )LL is limited below 0.2MSUSY/(1 TeV), so we do not show the corresponding curves here. 13 ) and (δ 23 ) in the RR block, Figs. 5 and 6 give the cross sections as the functions of (δU U respectively. We can see that the cross sections of t u¯ production are larger than ones of 13 ) t c¯ production for the case where (δU RR = 0 and all the other mixing parameters are set to be 0, and in contrast with this case, the cross sections of t c¯ production are larger than 23 ones of t u¯ production for the case of (δU )RR = 0. For example, assuming tan β = 30, MSUSY = 400 GeV, and mg˜ = 200 GeV, the cross sections of t c¯ and t u¯ productions are 23 ) 13 12 fb and 1 fb for (δU RR = 0.7, respectively, but 2 fb and 13 fb for (δU )RR = 0.7, respectively. 13 ) and (δ 23 ) in the LR Figs. 7 and 8 show the dependence of the cross sections on (δU U block, respectively. We can see that the cross sections are much larger than ones arising from the mixing in the LL and RR blocks, which indicates that the SUSY FCNC effects
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
17
Fig. 9. Typical interference effects between different matrix elements within one block in (a) and (b), and between different blocks in (c) and (d). Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
induced by the mixing in the LR block can enhance the cross sections significantly. For example, for tan β = 30, MSUSY = 400 GeV, and mg˜ = 200 GeV, the cross section can reach 23 ) 13 643 fb for (δU LR = 0.7, and 75 fb for (δU )LR = 0.7, as shown in Figs. 7(b) and 8(b), respectively. Here we see again that the production cross sections of t c¯ and t u¯ depend 23 ) 13 strongly on (δU LR and (δU )LR , respectively. The cases of the RL block are similar to ones of the LR block, so we do not show them here. From Figs. 7 and 8 we can find that the production rates may be as large as a few pb for favorable parameter values. In the above discussion for Figs. 3–8, we only show the results for mg˜ = 200 GeV and MSUSY = 400 GeV, and the corresponding results for other cases (mg˜ = 200 GeV and MSUSY = 1000 GeV, mg˜ = 300 GeV and MSUSY = 400 or 1000 GeV) decrease significantly, but they are still remarkable and vary from a few fb to hundreds of fb. In Fig. 9, we give the results to uncover the interference effects between different entries within one block and between different blocks for t c¯ production. Fig. 9(a) and (b) show 13 ) 23 the typical interference effects between (δU LR and (δU )LR within the LR block and 13 23 between (δU )RR and (δU )RR within the RR block, respectively. Fig. 9(c) and (d) show the interference effects between the LL and LR block and between the LL and RR block for the typical parameters, respectively. In general, those interference effects enhance the cross sections. And the interference effects for t u¯ production are very similar to the case of t c¯
18
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
Fig. 10. Similar as Fig. 9, but for t u. ¯ Table 2 Cross sections in fb for the t c¯ and t u¯ productions. Here tan β = 30, MSUSY = 400 GeV and mg˜ = 200 GeV, the off-diagonal element in the table equals to 0.7 and others are set to zero Subprocess
23 ) (δU LL
13 ) (δU RR
23 ) (δU RR
13 ) (δU LR
23 ) (δU LR
gg → t u¯ q q¯ → t u¯ gg → t c¯ q q¯ → t c¯
0 0.2 11.0 1.6
14.4 4.7 0 1.8
0 0.2 10.5 1.5
650.4 124.1 0 74.5
0 12.6 633.0 10.0
Total
12.8
20.9
12.2
849.0
655.6
production and are shown in Fig. 10. Although the interference effects do not change the cross sections of single top quark production at the LHC significantly, the consideration of them is of importance to reveal the details of flavor changing mechanism. Finally, for convenience, we list some typical results in Table 2.
5. Conclusions We have evaluated the t c¯ and t u¯ productions at the LHC within the general unconstrained MSSM framework. These single top quark productions are induced by SUSY-
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
19
QCD FCNC couplings and have remarkable cross sections for favorable parameter values allowed by current low energy data, which can be as large as a few pb. It has been pointed out [13] that the top quark FCNC production signals are more accessible than the top quark FCNC decay signals. And according to the model independent analysis in Ref. [14], it is possible to observe FCNC effects through single top quark productions at the LHC. Thus, our above results show that once large rates of the t c¯ and t u¯ productions are detected at the LHC, they may be induced by SUSY FCNC couplings, and such flavor changing must come from the LR or RL block, which is related to the soft trilinear couplings AU . Therefore, we believe that the precise measurement of single top quark production cross sections at the LHC is a powerful probe for the details of the SUSY FCNC couplings.
Acknowledgements This work was supported in part by the National Natural Science Foundation of China and Specialized Research Fund for the Doctoral Program of Higher Education.
Appendix A. Couplings Here we list the relevant couplings in the amplitudes. i, j stand for generation indices and r, s stand for color indices. V1 = −igs T a , V2 = −gs fabc , V3 = −igs fabc , √ √ q a q V4R = i 2gs Trsa Z6i , V4L = −i 2gs Trs Z3i , √ √ q q V5R = i 2gs Trsa Z5i , V5L = −i 2gs Trsa Z2i , √ √ q q V6R = i 2gs Trsa Z4i , V6L = −i 2gs Trsa Z1i , i 1 V7 = gs2 δab + dabc T c . 2 3
(A.1) (A.2) (A.3) (A.4) (A.5)
Appendix B. Form factors This appendix lists all the form factors in the amplitude of various subprocess of pp → t c, ¯ in terms of 2-, 3- and 4-points one-loop integrals [29]. For convenience, we define the abbreviations of one-loop integrals as following: B a = B s, m2g˜ , m2g˜ , B d = B t, m2g˜ , m2g˜ , B t = B 0, m2g˜ , m2q˜ , B q = B m2t , m2g˜ , m2q˜ , B g = B t, m2g˜ , m2q˜ , B m = B u, m2g˜ , m2q˜ , C = C m2t , s, 0, m2q˜ , m2g˜ , m2g˜ , C d = C m2t , s, 0, m2g˜ , m2q˜ , m2q˜ , C t = C m2t , t, 0, m2q˜ , m2g˜ , m2g˜ , C q = C m2t , t, 0, m2g˜ , m2q˜ , m2q˜ , C g = C 0, m2t , s, m2g˜ , m2q˜ , m2g˜ , C f = C m2t , 0, s, m2g˜ , m2q˜ , m2g˜ ,
20
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
C n = C 0, 0, t, m2g˜ , m2g˜ , m2q˜ , C m = C 0, 0, t, m2q˜ , m2q˜ , m2g˜ , C l = C 0, t, 0, m2g˜ , m2g˜ , m2q˜ , C o = C 0, t, 0, m2g˜ , mq2˜ , m2g˜ , C r = C 0, u, 0, m2q˜ , m2g˜ , m2q˜ , C p = C 0, u, 0, m2g˜ , m2q˜ , m2g˜ , C x = C 0, u, m2t , m2g˜ , m2g˜ , m2q˜ , C y = C 0, u, m2t , m2q˜ , m2q˜ , m2g˜ , C w = C m2t , t, 0, m2q˜ , m2g˜ , mq2˜ , C z = C m2t , t, 0, m2g˜ , m2q˜ , m2g˜ , D = D 0, 0, m2t , 0, t, s, m2g˜ , m2q˜ , m2g˜ , m2q˜ , D v = D 0, 0, m2t , 0, t, s, m2q˜ , m2g˜ , m2q˜ , m2g˜ , D t = D 0, m2t , 0, 0, u, s, m2q˜ , m2g˜ , m2q˜ , m2g˜ , D q = D m2t , 0, 0, 0, u, t, m2q˜ , m2g˜ , m2q˜ , m2q˜ , D g = D 0, 0, m2t , 0, t, s, m2g˜ , m2g˜ , m2q˜ , m2g˜ , D f = D 0, 0, m2t , 0, t, s, m2q˜ , mq2˜ , m2g˜ , m2q˜ , D s = D 0, m2t , 0, 0, u, s, m2g˜ , m2g˜ , mq2˜ , m2g˜ , D m = D 0, m2t , 0, 0, u, s, m2q˜ , m2q˜ , m2g˜ , mq2˜ , D n = D m2t , 0, 0, 0, u, t, m2g˜ , m2q˜ , m2q˜ , m2g˜ , D l = D m2t , 0, 0, 0, u, s, m2q˜ , m2g˜ , m2g˜ , m2q˜ . Below we display the non-zero form factors for the gluon fusion subprocess (we only display the odd number form factors, and the even number ones can be obtained by the replacement of V4L ↔ V4R and V5L ↔ V5R , correspondingly). For self-energy diagram (see Fig. 1(a)–(e)), fsi are:
1 V mg˜ m2t tB0t V1 V5L sV1 + t − m2t V3 fs1 = 3 4L 2 2 mt π s(mt − t)t g + mt V1 −B0 mg˜ m3t sV1 V5L
q q g , + m2t − t mg˜ mt B0 V5R − B1 m2t V5L (sV2 + tV3 ) + m2t stV1 V5R B1 fs3
1
=
(B.1)
q V4L −mt m2t − u V1 mg˜ mt B0 V5R
m3t π 2 s(m2t − t)t
q − B1 m2t V5L (uV3 − sV1 ) − m2t V1 mg˜ B0t uV5L u − m2t V3 − sV2 + mt sV1 B0m mg˜ mt V5L − uV5R B1m , mg˜ B0t V1 V5L mg˜ B0t V2 V5L 1 fs5 = V V + 1 4L 8π 2 mt t − m3t mt u − m3t q
+
q
V1 (B1 m2t V5L − mg˜ mt B0 V5R ) m2t u
q
+
q
V2 (B1 m2t V5L − mg˜ mt B0 V5R ) m2t t
(B.2)
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32 g
g
V1 (tV5R B1 − B0 mg˜ mt V5L )
V1 (uV5R B1m − B0m mg˜ mt V5L )
21
+ , t (t − m2t ) u(u − m2t ) 1 q −mt m2t − u V1 mg˜ mt B0 V5L V fs7 = 3 4R 2 2 mt π s(mt − t)t
q − B1 m2t V5R (uV3 − sV1 ) − m2t V1 mg˜ B0t uV5R u − m2t V3 − sV2 + mt sV1 B0m mg˜ mt V5R − uV5L B1m , 1 fs9 = V1 V4R 2mg˜ m2t s B0m V1 − B0t V2 V5R 2 2 16mt π 2 s(mt − u) q q − m2t − u B1 V5R m3t − mg˜ m2t B0 V5L + mg˜ m2t B0t V5R V3 − 2m3t sV1 V5L B1m ,
(B.4)
1 fs11 = fs5 , 2
(B.6)
+
fs13 =
1
16m2t π 2 t (t − m2t )(m2t − u) g + m2t V1 mg˜ B0 m2t − u (s
V4R
(B.3)
(B.5)
q q m2t − u V1 V2 B1 m3t V5R − mg˜ m2t B0 V5L
+ u)V1 + m2t − t t B0m V1 − B0t V2 V5R . + mt V1 V5L t t − m2t B1m (B.7)
For vertex diagram Fig. 1(f), (g) and (h), the non-zero form factors fti are: l 1 fv1 = V 4L stV1 C0 mg˜ mt V2 V5L 2 2 8π st (t − mt )
g l l o o − B0 V2 − 2C00 V2 + 2C00 V1 − (s + u) C12 V2 − C12 V1 V5R
d + m2t − t V5L t −C0 m2g˜ + C0 m2q + B0a − 2C00 + C1 m2t V2 − 2C00 V1 V3 g
z w + sV1 V2 B0 − 2C00 + m2t C1z − 2V1 C00 − mg˜ mt V2 V5R C0 tV3 + sV1 C0z (B.8) , 1 2 V m fv3 = − u mg˜ mt V2 V5R C0 uV3 − sV1 C0x 4L t 8π 2 su(u − m2t ) d
V1 − −C0 m2g˜ + C0 m2q + B0a − 2C00 + C1 m2t V2 V3 + V5L u 2C00 y x + sV1 V2 B0m − 2C00 + m2t C2x − 2V1 C00 p − suV1 mg˜ mt C1 V2 + C1r V1 V5L
p p r r + B0m − 2C00 + C12 u V2 + 2C00 V1 − C12 uV1 V5R (B.9) , o g l V ) 2C00 V1 V5R V2 (C0l mg˜ mt V5L − B0 V5R + 2C00 1 5R V + V fv5 = 1 4L 2 2 8π 2 t − mt mt − t r V r V1 (−mg˜ mt V5LC1r − 2C00 5R + C12 V5R ) + m2t − u
22
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32 p
− −
p
p
V2 (C1 mg˜ mt V5L + (B0m − 2C00 + C00 u)V5R ) 1 t
m2t − t
w w w w V1 mg˜ mt V5R C1w + V5L 2C00 − tC12 + m2t C1w + C11 + C12 y
x + m2 C x ) − m m V C x ) 2V1 V5L C00 V2 (V5L (B0m − 2C00 g˜ t 5R 0 t 2 − + u u 1 g z z z z + V2 mg˜ mt V5R C1z + V5L B0 − 2C00 + tC12 − m2t C11 + C12 , t (B.10) 1 p V4L suV1 mg˜ mt C0 V2 V5L fv7 = 8π 2 su(u − m2t )
p r − B0m V2 − 2C00 + 2C00 V1 V5R + m2t − u mg˜ mt V2 V5R C0 uV3 − sV1 C0x d
V1 − −C0 m2g˜ + C0 m2q + B0a − 2C00 + C1 m2t V2 V3 + V5L u 2C00 y x + sV1 V2 B0m − 2C00 + m2t C2x − 2V1 C00 (B.11) , 1 1 p 2sV1 mt −B0m V2 + 2C00 V2 V fv9 = 4R 2 2 16π s(mt − u)
p r − 2C00 V1 V5L + mg˜ C0 uV2 V5R + m2t − u mg˜ C0 m2t V2 + C2 (t − u)V2 + (t − u) C1 V2 − C0d + C1d + C2d V1 V5L + mt −B0a V2 + 2C00 + C0 (mg˜ − mq )(mg˜ + mq ) + (C11 + C12 )(t − u)
d d d (t − u)V1 V5R V3 V1 + C1d + C11 + C12 − C1 (s + 2u) V2 + 2C00 2V1 V2 (mt uV5R C2x − mg˜ uV5LC0x ) , + (B.12) u 2 1 1 11 o r V1 2 − C00 fv = mt + C00 V 4L 2 2 2 16π (mt − u)(mt − t) p
r o + C00 t + C00 u V1 V5R + V2 mg˜ mt C0 m2t − t + C0l m2t − u V5L 2
p g l mt − u + B0 u − m2t V5R + − B0m − 2C00 m2t − t + 2C00
g
1 z x V1 V2 V5L t V0m − 2C00 + m2t uC1z + m2t C2x + u B0 − 2C00 tu w y − mg˜ mt V5R tC0x + uC0z − 2V1 V5L uC00 (B.13) + tC00 , +
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
23
r V V 2mt C00 C0l mg˜ (s + u)V2 V5R 1 1 5L x + m = V V C V − V 1 4R 2 5L g ˜ 0 16π 2 m2t − u m2t − t
1 m p p + 2 V2 mt B0 − 2C00 V5L − mg˜ C0 uV5R mt − u w 2mt V1 V5R C00 + − mt V2 V5R C2x t g 1 z + V2 mg˜ (s + u)V5L C0z − mt V5R B0 − 2C00 + (s + u)C1z , (B.14) t l 1 o V1 V4R C12 mt V2 V5L − mt C12 V1 V5L fv15 = − 2 2 8π (mt − t)
+ mg˜ C0l + C2l V2 + C2o V1 V5R , (B.15)
fv13
1 l o V2 − C12 V1 V4L (s + u)V1 C12 2 8π (m2t − t) y x + m2t − t V1 V5L V2 C12 + V1 C12 ,
fv17 = −
fv19 =
fv21 =
fv23 =
fv25 =
V5R
l 1 o V 4R uV1 C12 mt V2 V5L − mt C12 V1 V5L 2 2 8π (mt − t)u
+ mg˜ C0l + C2l V2 + C2o V1 V5R
y + m2t − t V1 mg˜ V5L V1 C2 − V2 C0x + C2x x y y y x + mt V5R V2 C12 + C2x + C22 + V1 C12 + C2 + C22 , p 1 r V1 V4R mt C12 V2 − C12 V1 V5L 2 2 8π (mt − u) p
p + mg˜ C0 + C1 V2 + C1r V1 V5R ,
1 V m2t − u V1 mg˜ V5L V2 C0z + C1z − V1 C1w 4R 2 2 8π t (mt − u) z z w w − mt V5R V1 C1w + C11 + V2 C1z + C11 + C12 + C12 p r − tV1 mt C12 V2 − C12 V1 V5L p
p , + mg˜ C0 + C1 V2 + C1r V1 V5R l 1 o V4R stV1 C12 mt V2 V5L − mt C12 V1 V5L 2 4π 2 s(mt − t)t
+ mg˜ C0l + C2l V2 + C2o V1 V5R
+ m2t − t t mg˜ (C1 + C2 )V2 − C0d + C1d + C2d V1 V5L
d d V1 V5R V3 + mt (C1 + C11 + C12 )V2 + C1d + C11 + C12
+ sV1 mg˜ V5L V1 C1w − V2 C0z + C1z z z w w + C12 + C12 + mt V5R V1 C1w + C11 + V2 C1z + C11 ,
(B.16)
(B.17)
(B.18)
(B.19)
(B.20)
24
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
1 y sV1 mg˜ V5L V1 C2 − V2 C0x + C2x V 4R 2 4π su y y y x x + V1 C12 + C2 + C22 + C2x + C22 + mt V5R V2 C12
− u mg˜ (C1 + C2 )V2 − C0d + C1d + C2d V1 V5L
d d V1 V5R V3 , + mt (C1 + C11 + C12 )V2 + C1d + C11 + C12 (B.21) 1 y
V4R m2t − t V1 mg˜ V5L V2 C0x + C2x − V1 C2 fv29 = 2 2 4π (mt − t)u x y y y x + V1 C12 + C2 + C22 + C2x + C22 − mt V5R V2 C12 l
o mt V2 V5L − mt C12 V1 V5L + mg˜ C0l + C2l V2 + C2o V1 V5R − uV1 C12 .
fv27 =
(B.22) And for box diagrams Fig. 1(i)–(o), all form factors
fbi
are non-zero:
f s 1 g g g g s s V4L C2 + C0 − 2D00 + 2D002 + 2D003 − 2D003 u + D13 + D23 2 8π g
g g g g g + D12 + D13 + D22 (s + u) + D23 (s + u) + D2 m2t V5L − D0 mg˜ mt V5R V22 l n n + 2D003 + C2m − 2D003 + D23 u V1 V5L V2 f 2 f m + 2 D002 + D003 − D003 (B.23) V1 V5L , 1 f g g g g s C0 − C0 − C1 − C2 − 2D001 − 2D00 fb3 = − 2 V4L 8π s s s s s s s s u + −D12 u + 2 D001 + D002 − D13 + D23 + D13 − D22 + D23 g g g − D13 (s + u) + D12 (t − s) V5L + mg˜ mt −D1 + D1s + D2s V5R V22
l l n l + V1 −2D002 + D2l m2q − m2g˜ + D12 m2t + C1n + 2D002 + D22 u V5L 2 f m m V1 V5L , − D2l mg˜ mt V5R V2 + 2 −D001 + D001 + D002 (B.24) f 1 f g g g g s fb5 = − 2 V4L C1 + C2 + C1 + C2 + 2D00 − 2D002 + 2D00 8π g
g g s s s − 2D002 + D23 u + D22 u + D23 t + D12 t V5L − mg˜ mt D2 + D2s V5R V22 l l 2 l l + V1 2D001 + 2D002 + D1l + D2l m2q − m2g˜ − D11 + D12 mt
n n l l − D12 u V5L + C0n + C2n − 2 D001 + D002 + D22 l 2 f l m + D1 + D2 mg˜ mt V5R V2 − 2 D002 + D002 V1 V5L , (B.25) f 1 g g g V mg˜ mt D0s V5R − C0 + 2D001 + 2D002 + 2D003 fb7 = 4L 8π 2 s
s s s + D2s u V5L V22 − 2 2D00 + D001 + D002 + D003 l l n n n + 2 −D002 − D003 + D00 + D002 + D003 V1 V5L V2 f 2
f f f m m m m − 2 D00 + D001 + D002 + D003 − D00 − D001 − D002 − D003 V1 V5L , fb1 = −
(B.26)
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
fb9 =
25
f g
1 s V mg˜ V5L 2 C0 − 2 D00 + D00 − D0s u V22 4R 2 16π 2 l f n m V1 V2 + 4 D00 + D00 V1 + C0d V7 − 4 D00 + D00 f f g g s + mt V5R −2 − D2s + D3s m2t + C1 + C2 − 2D001 2D002 + 2D00 l s l n V1 V2 + D001 + m2t D2s + 2D3s − D2s u − D3s m2t V22 − 4 D00 − D001 − D001 2
f f f m d , + 4 D00 + D001 + D002 − D001 V3 − C1 V7 (B.27)
fb11 =
fb13 =
fb15 =
fb17 =
fb19 =
fb21 =
fb23 =
f 1 g g g s s V + C − 4D − 4D + D u + D t V5L V V C 2 4L 2 00 2 0 0 00 2 16π 2 g
l n − mg˜ mt D0 + D0s V5R − 2 D00 + D00 (B.28) V3 V5L , f 1 f g g s V2 V4R mt C1 + C2 + C2 + 2D00 + 2D00 − D2s u V2 16π 2
l + 2D00 V1 V5R f
g g g g − mg˜ C0 − C0 − D2 s − D0 s − D0 + D0s u V2 V5L , (B.29) 1 n V2 V4R −mg˜ D3n V1 V5L + mt D13 V1 V5R 2 8π g g g + V2 mg˜ D0 + D2 + D3 − D3s V5L g
g g g g s V5R , − mt D12 + D13 + D2 + D22 + D23 + D13 (B.30) s 1 f g g g s s s C + C2 + 2D00 + 2D002 + 2 D00 + mt2 D23 + D002 − D23 (s + t) 8π 2 2 s g g g s s − D12 + D2s + D22 + D23 u + D12 + D2 + D22 (s + u) l l m n n n − 2D001 − 2D002 − 2D001 + 2D002 − C2 − 2D00
2 n f n m u V1 V2 + 2 D002 + D002 V3 V4L V5L , + D12 + D2n + D22 (B.31) g
1 g s n n V2 + D1n + D11 V1 V5R + D12 V2 V4R mt D12 + D2G + D22 − D12 2 8π g
g − mg˜ D0 + D2 + D0s + D2s V2 + D0n + D1n + D2n V1 V5L , (B.32) g
1 V2 V4R mg˜ D1 V2 − D0s + D1s + D2s V2 + D2l V1 V5L 2 8π g s s s s V2 + mt D13 + D12 + D13 + D2s + D22 + D23 l
l l l + D12 + D2 + D22 + D23 V1 V5R , (B.33) g
1 g V2 V4R mg˜ D0 + D2 + D0s + D2s V2 − D1n + D2l V1 V5L 2 8π g s s V2 − mt −D23 + D2s + D22 + D23 l
l l l l l + D1 + D11 + 2D12 + D13 + D2l + D22 + D23 (B.34) V1 V5R ,
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
g 2 1 g g s V4R mg˜ D0 − D22 − D23 + D23 V2 2 4π l f 2
f l n n m + D13 + D23 + D23 + D13 V1 V2 + D22 + D23 − D23 V1 V5L g 2 g g g g g m + mt D122 + D123 + D22 + D222 + D223 + D23 − D23 V2 l l n n V1 V2 + D133 + D123 − D133 − D123 2
f f f f f f m V3 V5R , + D122 + D123 + D22 + D222 + D223 + D23 + D123 (B.35) 1 g g s s V5L + D2s + D22 V4R mg˜ −D1 − D12 + D1s + D12 fb27 = 2 4π g
g g s s + mt D112 + D122 + D12 + D112 + D122 V5R V22 l l n n + V1 mg˜ D12 + D22 + D12 + D2n + D22 V5L l
l l n n n V5R V2 + mt D112 + D12 + D122 − D112 − D12 − D122 f m m mg˜ V5L + V12 D12 − D12 − D22
f f f m m mt V5R , + D112 + D12 + D122 + D112 + D122 (B.36) 1 g g s fb29 = − 2 V4R mg˜ (D2 + D22 + D2s + D22 )V5L 4π g
g g s V5R V22 − mt D122 + D22 + D222 − D122 l l l n n n V5L + V1 mg˜ D11 + 2D12 + D22 + D1n + D11 + 2D12 + D2n + D22 l l l l l n n n + mt D11 + D111 + 2D112 + D12 + D122 − D11 − D111 − 2D112
f n n m V5R V2 − V12 D22 + D22 mg˜ V5L − D12 − D122
f f f f m mt V5R , + D122 + D22 + D222 + D122 − D122 (B.37)
fb25 = −
1 g g g g g g g D112 + D113 + D12 + D122 + 2D123 + D13 + D133 4π 2 2 s s s s s s s − D113 − 2D123 − D13 − D133 − D23 − D223 − D233 V2 l l l n n n + D223 + D23 + D233 − D23 − D223 − D233 V1 V2 f f f f f f f m m + D112 + D113 + D12 + D122 + 2D123 + D13 + D133 − D113 − 2D123 2
m m m m m V1 V4L V5L , − D13 − D133 − D223 − D23 − D233 (B.38) 1 g g g g g g g g fb33 = D12 + D122 + D123 + D13 + D2 + 2D22 + D222 + 2D223 4π 2 2 g g s s s s s + 2D23 + D233 + D123 + D13 + D223 + D23 + D233 V2 l l l l l l n n n − D123 + D13 + D133 + D223 + D23 + D233 − D123 − D223 − D133 f f f f f f n n V1 V2 + D122 + D123 + D22 + D222 + 2D223 + D23 − D23 − D233 2
f m m m m V1 V4L V5L , + D233 + D123 + D223 + D23 + D233 (B.39)
fb31 =
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
fb35 =
27
g 1 g g g s s V mg˜ D12 + D13 + D13 V5L − mt D112 + D113 + D23 4R 2 4π
g g g g s s V5R V22 + D12 + D122 + D123 + D13 − D113 − D123 l l
n n + V1 mg˜ D23 + D23 − D123 V5L + mt D123 V5R V2 f f m m mg˜ V5L − V12 D12 + D13 + D13 + D23
f f f f f f m m mt V5R , − D123 + D112 + D113 + D12 + D122 + D123 + D13 − D113
(B.40) 1 g g g g g s = − 2 D112 + D12 + D122 + D123 + D13 + D112 4π 2 s s s s s s s s V2 + 2D12 + 2D122 + D123 + D13 + D2s + 2D22 + D222 + D223 + D23 l l l l l n n + −D122 − D123 − D22 − D222 − D223 + D12 + D122 n n n n n V1 V2 + D123 + D2n + 2D22 + D222 + D223 + D23 f f f f m m + D112 + D123 + D12 + D122 + D112 + D12
m m m m m V12 V4L V5L , + 2D122 + D123 + D22 + D222 + D223 (B.41) 1 g g g g g g fb39 = −D12 − D122 − D2 − 2D22 − D222 − D223 4π 2 2 g s s s s s s V2 − D23 + D12 + D122 + D2s + 2D22 + D222 + D223 + D23 l l l l l l l l + −D112 − D113 − D12 − 2D122 − 2D123 − D22 − D222 − D223
fb37
n n n n n n n + D112 + D113 + 2D12 + 2D122 + 2D123 + D13 + D2n + 2D22 f f f f n n n + D222 + D223 + D23 V1 V2 + −D122 − D22 − D222 − D223 2
m m m m + D122 + D22 + D222 + D223 V1 V4L V5L .
(B.42)
For the quarks initiated subprocesses, we list all form factors of cc¯ initial state: = gs14LR = gs6LR 4 4
mg˜ B0t V12 V4R V5R , 8mt π 2 t
gs6RL = gs14LL = gs6LR (V4L ↔ V4R , V5L ↔ V5R ), 4 4 4 gs6LR 3 gs6RL 3 gs5LR 2 gs5RL 2
q V12 V4R (B1 m2t V5R − mg˜ mt B0 V5L ) , = gs14LR = 3 8m2t π 2 t = gs14LL = gs6LR (V4L ↔ V4R , V5L ↔ V5R ), 3 3 q q 2 V1 V4R (mg˜ mt B0 V5L − B1 m2t V5R , = gs14RR = 2 8m2t π 2 s = gs14RL = gs5LR (V4L ↔ V4R , V5L ↔ V5R ), 2 2
= gs14RR =− gs5LR 1 1
mg˜ B0t V12 V4R V5R , 8mt π 2 s
gs5RL = gs14RL = gs5LR (V4L ↔ V4R , V5L ↔ V5R ), 1 1 1
(B.43) (B.44) (B.45) (B.46) (B.47) (B.48) (B.49) (B.50)
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
gv2LL = gv2LR = 4 4
q 1 q q V1 V1 V4R mt C1 + C11 + C12 V5R 2 8π t
q q q − mg˜ (C0 + C1 + C2 )V5L ,
= gv2LL (V4L ↔ V4R , V5L 4 q C V1 V1 V4R V5R , = gv14LR = 00 gv6LR 4 4 4π 2 t
gv2RL 4
= gv2RR 4
(B.51)
↔ V5R ),
(B.52) (B.53)
gv6RL = gv14LL = gv6LR (V4L ↔ V4R , V5L ↔ V5R ), 4 4 4 1 = gv2LR = − 2 V1 V2 V4R mg˜ C1t + C2t V5L gv2LL 3 3 8π t
t t V5R , + mt C1t + C11 + C12
(B.55)
↔ V4R , V5L ↔ V5R ), 1 = gv14LR = − 2 V1 V2 V4R C0t V5R m2g˜ + mt C0t V5Lmg˜ gv6LR 3 3 8π t
t − C0t m2q + B0d − 2C00 + m2t C1t V5R ,
gv2RL 3
= gv2RR 3
(B.54)
= gv2LL (V4L 3
= gv6LR (V4L ↔ V4R , V5L 3 d C V1 V1 V4R V5R , gv5LR = gv14RR = 00 2 2 4π 2 s
gv6RL 3
= gv14LL 3
↔ V5R ),
= gv8RR 2
(B.59) (B.60)
(B.61)
= gv8LL (V4L 2
↔ V4R , V5L ↔ V5R ), 1 V1 V2 V4R C0 mg˜ mt V5L = gv14RR = gv5LR 1 1 8π 2 s
− −C0 m2g˜ + C0 m2q + B0a − 2C00 + C1 mt2 V5R ,
(B.62)
(B.63)
(B.64) ↔ V4R , V5L ↔ V5R ), V1 V2 V4R ((C1 + C2 )mg˜ V5L + (C1 + C11 + C12 )mt V5R ) , (B.65) = gv8LR = gv8LL 1 1 8π 2 s gv8RL (B.66) = gv8RR = gv8LL (V4L ↔ V4R , V5L ↔ V5R ), 1 1 1
gv5RL 1
= gv14RL 1
(B.57) (B.58)
gv5RL = gv14RL = gv5LR (V4L ↔ V4R , V5L ↔ V5R ), 2 2 2 1 d d gv8LL mt V5R = gv8LR = + C12 V1 V1 V4R C1d + C11 2 2 2 8π s
d − C0 + C1d + C2d mg˜ V5L , gv8RL 2
(B.56)
= gv5LR (V4L 1
q
3 V mg˜ D3 V4R 5L , 16π 2 gb1RR = gb1LL (V4L ↔ V4R , V5L ↔ V5R ), 4 4
= gb1LL 4
= gb1LR 4
(B.67) (B.68)
q 2 V4L V5R mg˜ D3 V4R , 16π 2
(B.69)
gb1RL = gb1LR (V4L ↔ V4R , V5L ↔ V5R ), 4 4 = gb4LL 4
2 (m D q V mg˜ V4L V4R g˜ 0 5L
q + mt (D0 16π 2
(B.70) q + D1
q + D2 )V5R )
,
(B.71)
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
29
gb4RR = gb4LL (V4L ↔ V4R , V5L ↔ V5R ), 4 4 = gb4LR 4
3 (m D q V mg˜ V4L g˜ 0 5L
q + mt (D0 16π 2
q + D1
(B.72) q + D2 )V5R )
,
(B.73)
= gb4LR (V4L ↔ V4R , V5L ↔ V5R ), gb4RL 4 4 gb9LL = gb9LR = 4 4
(B.74)
q 2 V D13 V4L V4R 5L , 16π 2
(B.75)
= gb9RR = gb9LL (V4L ↔ V4R , V5L ↔ V5R ), gb9RL 4 4 4 gb11LL = gb11LR = 4 4
2 V (m D q V V4L 4R g˜ 1 5L
q + mt (D1 16π 2
(B.76)
q + D11
= gb11RR = gb11LL (V4L ↔ V4R , V5L ↔ V5R ), gb11RL 4 4 4 gb13LL = gb13RL = 4 4
q + D12 )V5R )
,
(B.77) (B.78)
q 2 V D00 V4L V4R 5L , 8π 2
(B.79)
= gb13RR = gb13LL (V4L ↔ V4R , V5L ↔ V5R ), gb13LR 4 4 4
(B.80)
3 V mg˜ D3t V4R 5L gb1LL = − , 3 16π 2
(B.81)
= gb1LL (V4L ↔ V4R , V5L ↔ V5R ), gb1RR 3 3
(B.82)
2 V V mg˜ D3t V4R 4L 5R , 16π 2 = gb1LR (V4L ↔ V4R , V5L ↔ V5R ), gb1RL 3 3
gb1LR =− 3
= gb4LL 3
2 (m D t V mg˜ V4L V4R g˜ 0 5L 16π 2
− mt D2t V5R )
(B.83) (B.84) ,
(B.85)
= gb4LL (V4L ↔ V4R , V5L ↔ V5R ), gb4RR 3 3 gb4LR = 3
(B.86)
3 (m D t V t mg˜ V4L g˜ 0 5L − mt D2 V5R ) , 16π 2
(B.87)
= gb4LR (V4L ↔ V4R , V5L ↔ V5R ), gb4RL 3 3
(B.88)
t 2 V D13 V4L V4R 5L , 16π 2 = gb9RR = gb9LL (V4L ↔ V4R , V5L ↔ V5R ), gb9RL 3 3 3
gb9LL = gb9LR = 3 3
= gb11LR = gb11LL 3 3
2 V (m D t V V4L 4R t 12 5R 16π 2
− mg˜ D1t V5L )
(B.89) (B.90) ,
(B.91)
= gb11RR = gb11LL (V4L ↔ V4R , V5L ↔ V5R ), gb11RL 3 3 3
(B.92)
t 2 V D00 V4LV4R 5L 13RL = g = − , gb13LL b3 3 2 8π
(B.93)
= gb13RR = gb13LL (V4L ↔ V4R , V5L ↔ V5R ), gb13LR 3 3 3
(B.94)
gb2LL = 2
2 V (m V D v − m V D v ) V4L 4R t 5R 23 g˜ 5L 3 , 16π 2
(B.95)
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
gb2RR = gb2LL (V4L ↔ V4R , V5L ↔ V5R ), 2 2 = gb2LR 2
(B.96)
v ) − mt V5R D23 , 16π 2
3 (m V D v V4R g˜ 5L 3
(B.97)
= gb2LR (V4L ↔ V4R , V5L ↔ V5R ), gb2RL 2 2 gb5RR = gb5RL = 2 2
2 V (m V D v mg˜ V4L 4R g˜ 5R 0 16π 2
− mt V5L D2v )
(B.98) ,
= gb5LR = gb5RR (V4L ↔ V4R , V5L ↔ V5R ), gb5LL 2 2 2 2 V V Dv V4L 4R 5R 00 , 8π 2 = gb6LR (V4L ↔ V4R , V5L ↔ V5R ), gb6RL 2 2
gb6LR = 2
(B.99) (B.100) (B.101) (B.102)
3 V Dv V4L 5L 13 gb10LL = − , 2 16π 2
(B.103)
= gb10LL (V4L ↔ V4R , V5L ↔ V5R ), gb10RR 2 2
(B.104)
2 V V Dv V4L 4R 5L 13 = − , gb10LR 2 16π 2
(B.105)
= gb10LR (V4L ↔ V4R , V5L ↔ V5R ), gb10RL 2 2
(B.106)
2 V V Dv mg˜ V4L 4R 5L 1 , 16π 2 = gb12RR = gb12LL (V4L ↔ V4R , V5L ↔ V5R ), gb12RL 2 2 2
= gb12LR =− gb12LL 2 2
= gb14LL 2
3 V Dv V4L 5L 00 , 8π 2
= gb14LL (V4L ↔ V4R , V5L ↔ V5R ), gb14LR 2 2 2 V V (D0 + D1 + D2 + D3 )mg˜ V4L 4R 5L , 2 16π = gb3RR = gb3LL (V4L ↔ V4R , V5L ↔ V5R ), gb3LR 1 1 1
gb3LL = gb3RL = 1 1
2 V V D00 V4L 4R 5R , 8π 2 = gb5LR (V4L ↔ V4R , V5L ↔ V5R ), gb5RL 1 1
gb5LR =− 1
+ (D0 + D1 + D2 )mt V5R ) , 16π 2 = gb6RR = gb6LL (V4L ↔ V4R , V5L ↔ V5R ), gb6LR 1 1 1 = gb6RL =− gb6LL 1 1
2 (D m V mg˜ V4L V4R 0 g˜ 5L
3 V (D12 + D2 + D22 + D23 )V4L 5L , 2 16π gb7RR = gb7LL (V4L ↔ V4R , V5L ↔ V5R ), 1 1
= gb7LL 1
= gb7LR 1
2 V V (D12 + D2 + D22 + D23 )V4L 4R 5R , 16π 2
(B.107) (B.108) (B.109) (B.110) (B.111) (B.112) (B.113) (B.114) (B.115) (B.116) (B.117) (B.118) (B.119)
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
gb7RL = gb7LR (V4L ↔ V4R , V5L ↔ V5R ), 1 1 + (D12 + D2 + D22 )mt V5R ) , 16π 2 = gb8LL (V4L ↔ V4R , V5L ↔ V5R ), gb8RR 1 1 =− gb8LL 1
2 V (D m V V4L 4R 2 g˜ 5L
+ (D12 + D2 + D22 )mt V5R ) , 16π 2 gb8RL = gb8LR (V4L ↔ V4R , V5L ↔ V5R ), 1 1 gb8LR =− 1
3 (D m V V4R 2 g˜ 5L
3 V5L D00 V4L , 8π 2 gb14RR = gb14RL (V4L ↔ V4R , V5L ↔ V5R ). 1 1
=− gb14RL 1
31
(B.120) (B.121) (B.122) (B.123) (B.124) (B.125) (B.126)
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Single top quark production via SUSY-QCD FCNC couplings at the CERN LHC in the unconstrained MSSM Jian Jun Liu a , Chong Sheng Li a , Li Lin Yang a , Li Gang Jin b a Department of Physics, Peking University, Beijing 100871, China b Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China
Received 13 April 2004; accepted 18 October 2004 Available online 13 November 2004
Abstract We evaluate the t c¯ and t u¯ productions at the LHC within the general unconstrained MSSM framework. We find that these single top quark productions induced by SUSY-QCD FCNC couplings have remarkable cross sections for favorable parameter values allowed by current low energy data, which can be as large as a few pb. Once large rates of the t c¯ and t u¯ productions are detected at the LHC, they may be induced by SUSY FCNC couplings. We show that the precise measurement of single top quark production cross sections at the LHC is a powerful probe for the details of the SUSY FCNC couplings. 2004 Elsevier B.V. All rights reserved. PACS: 14.65.Ha; 12.60.Jv; 11.30.Pb Keywords: Top quark; MSSM; FCNC
1. Introduction The flavor dynamics, such as the mixing of three generation fermions and their large mass differences observed, is still a great mystery in particle physics today. The heaviest E-mail address: [email protected] (C.S. Li). 0550-3213/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2004.10.048
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
one of three generation fermions is the top quark with the mass close to the electroweak (EW) symmetry breaking scale. Therefore, it can play a role of a wonderful probe for the EW breaking mechanism and new physics beyond the standard model (SM) through its decays and productions. An important aspect of the top quark physics is to investigate anomalous flavor changing neutral current (FCNC) couplings. Within the SM, FCNC is forbidden at the tree-level and highly suppressed by the GIM mechanism [1] at one-loop level. All the precise measurements of various FCNC processes, for example, b → sγ [2], agree with the SM predictions. However, many new physics models allow the existence of the tree-level FCNC couplings, and may enhance some FCNC processes. In particular, the minimal supersymmetric standard model (MSSM) is one of the most popular new physics models. And it is hard to believe that this model will say no more about the flavor dynamics than the SM. The MSSM includes 105 free parameters beyond the SM: 5 real parameters and 3 CP-violating phases in the gaugino/higgsino sector, 21 squark and slepton masses, 36 new real mixing angles to define the squark and slepton mass eigenstates, and 40 new CP-violating phases that can appear in squark and slepton interactions [2], so in general, it can provide a new explanation of the source of FCNC couplings. Actually, many of the parameters are constrained by present experimental data, and some popular SUSY models, for example, the gravity-mediated supersymmetry broken model (SUGRA) [3], gauge mediated supersymmetry broken model (GMSB) [4], anomaly mediated supersymmetry broken model (AMSB) [5], gaugino mediated supersymmetry broken model (gMSB) [6] and Kaluza–Klein mediated supersymmetry broken model [7], make some ad hoc assumptions to reduce the number of independent parameters. Although the predictions of these models can agree with the present constraints from FCNC experiments, it should be noted that few experimental constraints on the top quark FCNC processes are available so far [2]. Since the CERN Large Hadron Collider (LHC) can produce abundant top events, the measurement of the top quark rare processes will become possible. Therefore, a good understanding of the theoretical predictions of these processes, especially within the general unconstrained MSSM framework, is important. And a careful investigation will provide some clues to the constraint relations among the supersymmetry (SUSY) parameters and more clear information about the SUSY breaking mechanisms. The top quark FCNC processes have been investigated in detail. There are two categories of these investigations, one of which is the top quark FCNC decays [8], especially for t → cg [9,10] and t → ch [11] in the MSSM. These results show that the branch ratio of t → cg can reach 10−4 [10], which enhances the one in the SM (∼ 10−11 ) significantly. The other category is the top quark productions through FCNC processes [12–17], ones of which induced by the anomalous top quark FCNC couplings at the colliders have been extensively discussed in a model independent way [12–16] and model dependent way [17], respectively. As we know, most of the works within SUSY models are limited in some constrained MSSM, where the top quark FCNC couplings were studied through some SUSY-CKM matrices or mass insertion approximation [18]. However, it is interesting to study the top quark FCNC productions using soft SUSY broken parameters directly, and then establish some relations between the production cross sections and soft SUSY broken parameters in the mass eigenstate formalism [19,20]. Actually, the general framework with SUSY FCNC mechanisms was presented several years ago [21], and there have been a lot of works studying SUSY FCNC processes within this framework [19,20]. But the top quark
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
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SUSY FCNC production processes have not been studied in above framework so far, and this means that the relations between the observables of the top quark production processes and the mixing among up squarks involving the third generation have not been established yet. In this paper, we will investigate the single top quark productions at the LHC induced by SUSY FCNC couplings in the general unconstrained MSSM, which include the top quark productions associated with the anti-charm quark and the anti-up-quark, and we believe that it is an important step towards revealing the top quark SUSY FCNC couplings and exploring some relations among SUSY broken parameters at the LHC. The paper is organized as follows. In Section 2 we describe the general framework of SUSY FCNC mechanisms. In Section 3 we evaluate the total production cross sections of the pp → t c( ¯ u) ¯ process induced by SUSY FCNC couplings, and the detail expressions for various couplings and form factors are given in Appendices A and B, respectively. In Section 4 we present our numerical calculations and discussions. Finally, Section 5 gives our conclusions.
2. The FCNC in the MSSM In order to make our paper self-contained, we start with a brief description of the FCNC mechanism in the MSSM, as shown in Ref. [21], and establish our notation conventions. The SUSY part of the MSSM Lagrangian is 1 a µν / ab λbG + Dµ φ † (Dµ φ) + i ψ¯ Dψ / LSUSY = − FG FGa µν + i λ¯ aGD 4 ˜ ˜ ˜ ∂W 1 ∂ 2W ∂W ψiT Cψj + h.c. − − ∂φi ∂φi 2 ∂φi ∂φj √ † a aT 1 2 † a † a − 2gG φ TG λG Cψ + h.c. − gG φ TG φ φ TG φ , 2 where ˜ = µH 1 H 2 + Y I J H 1 L˜ I E˜ J + Y I J H 1 Q˜ I D˜ J + YuI J H 2 Q˜ I U˜ J . W l R d R R
(1)
(2)
Here subscript G represents color, weak isospin and supercharge indices of the SM gauge group SU(3)C × SU(2)L × U (1)Y , respectively, a and b are the indices of adjoint representations of the non-abelian subgroups, and I, J = 1, 2, 3 are the generation indices. φ, ψ and λ represent scalar fields, matter fermion fields and gauginos, respectively. C is charge conjugation matrix. L˜ I (Q˜ I ) and E˜ RI (U˜ RI , D˜ RI ) represent slepton (squark) SU(2) left-hand doublets and right-hand singlets, respectively. H 1,2 represent two Higgs SU(2) doublets, and their vacuum expectation values are v1 v cos β √ √ 2 1 0 0 2 2 = = ≡ , H H (3) ≡ v√ sin β , v2 √ 0 0 2 2 √ where v = ( 2GF )−1/2 = 246 GeV, and the angle β is defined by tan β ≡ v2 /v1 , the ratio of the vacuum expectation values of the two Higgs doublets. µ is the Higgs mixing parameter.
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
The soft SUSY breaking Lagrangian is 1 M3 g˜ a T C g˜ a + M2 W˜ iT C W˜ i + M1 B˜ T C B˜ + h.c. 2 I J I J J 2 1† 1 2 2† 2 E˜ R H − MH H − L˜ I † ML2 L˜ J − E˜ RI † ME2 − MH 1H 2H IJ ˜ J 2 IJ ˜ J 2 IJ ˜ J ˜ I † MQ −Q Q − D˜ RI † MD DR − U˜ RI † MU2 UR IJ 1 I J + AE H L˜ E˜ R + AIDJ H 1 Q˜ I D˜ RJ + AIUJ H 2 Q˜ I U˜ RJ + BµH 1 H 2 + h.c. . (4)
Lsoft = −
2 , M2 Here MQ U,D and AU,D are the soft broken SU(2) doublet squark mass squared matrix, the SU(2) singlet squark mass squared matrix and the trilinear coupling matrix, respectively. E,U,D , the lepton and quark mass eigenstates are given Using 3 × 3 unitary matrices VL,R by
ν = VLE ΨL1 , l = VLE ΨL2 + VRE C Ψ¯ ET ,
u = VLU ΨQ1 + VRU C Ψ¯ UT , d = VLD ΨQ2 + VRD C Ψ¯ DT .
(5)
Their corresponding diagonal 3 × 3 mass matrices are v cos β ml = − √ VRE YlT VLE† , 2 v cos β D T D† md = − √ VR Yd VL . 2
v sin β mu = √ VRU YuT VLU † , 2
As in the SM, the Cabibbo–Kobayashi–Maskawa (CKM) matrix is K = VLU VLD† . It is convenient to specify the squark mass matrices in the super-CKM basis, in which the mass matrices of the quark fields are diagonalized by rotating the superfields. The super-CKM basis U˜ 0 is defined as U VL U˜ L U˜ 0 = (6) . VRU U˜ R∗ And in this basis, the up squark mass matrix is 6 × 6 matrix, which has the form: 2 2 M ˜ LL + Fu LL + Du LL M ˜ LR + Fu LR U 2 † 2 U M2U˜ ≡ . M ˜ LR + (Fu LR )† M ˜ RR + Fu RR + Du RR U
(7)
U
The F terms and D terms are diagonal 3 × 3 submatrices, which are given by Fu LR = −µ(mu cot β)13 , Du LL = Du RR = cos 2β m2Z
Fu LL = Fu RR = m2u 13 , 1 2 2 − sin θW 13 , 2 3
(8) (9)
where θW is the Weinberg angle, 13 stands for the 3 ×3 unit matrix. And (M 2˜ )LL , (M 2˜ )RR
and (M 2˜ )LR contain the flavor-changing entries
U
MU2˜
LL
2 U† = VLU MQ VL ,
MU2˜
RR
T = VRU MU2 VRU † ,
U
U
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
MU2˜
LR
7
v sin β = − √ VLU A∗U VRU † , 2
(10)
which are directly related to the mechanism of SUSY breaking, and are in general not diagonal in the super-CKM basis. Furthermore, (M 2˜ )LR , arising from the trilinear terms U
in the soft potential, is not hermitian. The matrix M2˜ can further be diagonalized by an U additional 6 × 6 unitary matrix ZU to give the up squark mass eigenvalues
M2U˜
diag
† = ZU M2U˜ ZU .
(11)
As for the down squark mass matrix, we also can define M2˜ as the similar form D
of Eq. (7) with the replacement of (M 2˜ )I J (I, J = L, R) by (M 2˜ )I J . Note that since SU(2)L gauge invariance implies that
U (M 2˜ )LL U
D
= K(M 2˜ )LL K † , the matrices (M 2˜ )LL and D
U
(M 2˜ )LL are correlated and cannot be specified independently. D Thus, in the super-CKM basis, there are new potential sources of flavor-changing neutral current: neutralino–quark–squark coupling and gluino–quark–squark coupling, which arise from the off-diagonal elements of (M 2˜ )LL , (M 2˜ )LR and (M 2˜ )RR . As the former coupling U U U is in general much weaker than the latter one, in this paper, we only consider the SUSYQCD FCNC effects induced by the gluino–quark–squark coupling, which can be written as √
i 2gs T a −(ZU )I i PL + (ZU )(I +3)i PR , I, i = 1, 2, 3. (12) Here PL,R ≡ (1 ∓ γ5)/2 and T a is the SU(3) color matrix. Thus the flavor changing effects 2 , M 2 and A on the observables can be obtained through the of soft broken terms MQ U U matrix ZU . As mentioned in the introduction, most of previous literatures studied the FCNC processes in the MSSM by so-called mass insertion approximation [18]. However, when the off-diagonal elements in the up squark mass matrix become large, the mass insertion approximation is no longer valid [19,20]. Therefore, we use the general mass eigenstate formalism as described above, which has been adopted in the literatures recently [10,19, 20]. So any effects of SUSY FCNC couplings in loops on various observables will provide some information of the SUSY breaking mechanism. In practical calculations, because the squark mass matrix M2˜ is a 6 × 6 matrix, it is U very formidable to calculate the effects arising from the matrix on the observables of FCNC processes. Thus, as shown in Refs. [10,19,20], it is reasonable to calculate the observables by only switching on one off-diagonal element in (M 2˜ )LL , (M 2˜ )LR and (M 2˜ )RR at a U U U time. For the aim of this paper, the following strategy in the calculations of the process pp → t c( ¯ u) ¯ will be used: first we deal with the LL, LR, RL and RR block of the matrix M2˜ separately and in each block we only consider the effects of individual element on the U production cross sections, and then we investigate both the interference effects between different entries within one block and the interference effects between different blocks. To simplify the calculation we further assume that all diagonal entries in (M 2˜ )LL , (M 2˜ )LR , U
U
2 , and then we nor(M 2˜ )RL and (M 2˜ )RR are set to be equal to the common value MSUSY U
U
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
2 malize the off-diagonal elements to MSUSY [19,22],
ij δU LL
ij δU LR
= =
ij U , 2 MSUSY ij (M 2˜ )LR U , 2 MSUSY
(M 2˜ )LL
ij
(M 2˜ )RR ij U δU RR = , 2 MSUSY
ij δU RL
=
ij U 2 MSUSY
(M 2˜ )RL
Thus (M 2˜ )LL can be written as follows: U 12 1 δU LL 2 21 2 MU˜ LL = MSUSY δU LL 1 31 32 δU LL δU LL and analogously for all the other blocks.
(i = j, i, j = 1, 2, 3).
13 δ LL U 23 , δU LL 1
(13)
(14)
3. The process pp → t c( ¯ u) ¯ induced by SUSY FCNC couplings The process pp → t c( ¯ u) ¯ at the LHC can be induced through the SUSY-QCD FCNC ¯ s s¯ , cc, couplings with the initial partonic states uu, ¯ d d, ¯ uc, ¯ cu¯ and gg, as shown in Fig. 1. ¯ b s¯ and s b¯ initial states Our practical calculations show that the contributions from the bb, are negligibly small, so we do not discuss them below. As the t u¯ process is completely similar to the t c¯ one, here we only discuss the situations of t c¯ in detail. Neglecting the charm quark mass, the amplitude of the process gg → t c¯ can be written as gg
gg
gg
Mgg = Mself + Mvertex + Mbox, gg Mself ,
gg Mvertex
(15)
gg Mbox
and are the amplitudes of self-energy diagrams Fig. 1(a)–(e), where vertex diagrams Fig. 1(f), (g) and (h) and box diagrams Fig. 1(i)–(o), respectively. They can be further expressed as gg Mself
=
14
gg
fsi Mi ,
(16)
i=1 gg
Mvertex =
30
gg
fvi Mi ,
(17)
gg
(18)
i=1 gg
Mbox =
40
fbi Mi ,
i=1
respectively. Here fsi , fvi and fbi are form factors corresponding to the self-energy diagrams, vertex diagrams and box diagrams, respectively, and their expressions are given gg explicitly in Appendix B. Mi are the standard matrix elements, which are defined by gg
M1,2 = u(p ¯ t )/ε(k1 )k1 · ε(k2 )PL,R v(pc ), gg
M3,4 = u(p ¯ t )/ε(k2 )k2 · ε(k1 )PL,R v(pc ),
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
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Fig. 1. Gluon initial state subprocess Feynman diagrams for the top quark and anti-charm quark associated production. gg
M5,6 = u(p ¯ t )/ε(k2 )pt · ε(k1 )PL,R v(pc ), gg
M7,8 = u(p ¯ t )/k1 ε(k1 ) · ε(k2 )PL,R v(pc ), gg
¯ t )ε(k1 ) · ε(k2 )PL,R v(pc ), M9,10 = u(p gg
M11,12 = u(p ¯ t )/ε(k1 )/k1 ε/(k2 )PL,R v(pc ),
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32 gg
M13,14 = u(p ¯ t )/ε(k1 )/ε(k2 )PL,R v(pc ), gg
¯ t )/ε(k1 )/k1 k1 · ε(k2 )PL,R v(pc ), M15,16 = u(p gg
M17,18 = u(p ¯ t )/ε(k1 )pt · ε(k2 )PL,R v(pc ), gg
¯ t )/ε(k1 )/k1 pt · ε(k2 )PL,R v(pc ), M19,20 = u(p gg
M21,22 = u(p ¯ t )/ε(k2 )/k1 k2 · ε(k1 )PL,R v(pc ), gg
¯ t )/ε(k2 )/k1 pt · ε(k1 )PL,R v(pc ), M23,24 = u(p gg
M25,26 = u(p ¯ t )k1 · ε(k2 )pt · ε(k1 )PL,R v(pc ), gg
¯ t )pt · ε(k2 )k2 · ε(k1 )PL,R v(pc ), M27,28 = u(p gg
M29,30 = u(p ¯ t )pt · ε(k2 )pt · ε(k1 )PL,R v(pc ), gg
M31,32 = u(p ¯ t )/k1 k1 · ε(k2 )k2 · ε(k1 )PL,R v(pc ), gg
¯ t )/k1 k1 · ε(k2 )pt · ε(k1 )PL,R v(pc ), M33,34 = u(p gg
M35,36 = u(p ¯ t )k1 · ε(k2 )k2 · ε(k1 )PL,R v(pc ), gg
¯ t )/k1 pt · ε(k2 )k2 · ε(k1 )PL,R v(pc ), M37,38 = u(p gg
M39,40 = u(p ¯ t )/k1 pt · ε(k2 )pt · ε(k1 )PL,R v(pc ),
(19)
where k1,2 denote the momenta of incoming partons, while pt and pc are used for the outgoing top and anti-charm-quarks, respectively. For the quarks initiated subprocesses, we also define the standard matrix elements as q q¯
¯ 2 )Pα γµ u(k1 )u(p ¯ t )Pβ γ µ v(pc ), M1αβ = v(k q q¯
M2αβ = u(p ¯ t )Pα v(pc )v(k ¯ 2 )Pβp / t u(k1 ), q q¯
M3αβ = v(k ¯ 2 )Pα u(k1 )u(p ¯ t )Pβ v(pc ), q q¯
¯ 2 )Pα u(k1 )u(p ¯ t )Pβ k/1 v(pc ), M4αβ = v(k q q¯
M5αβ = v(k ¯ 2 )Pα v(pc )u(p ¯ t )Pβ u(k1 ), q q¯
M6αβ = v(k ¯ 2 )Pα v(pc )u(p ¯ t )Pβ k/2 u(k1 ), q q¯
M7αβ = u(p ¯ t )Pα u(k1 )v(k ¯ 2 )Pβ k/1 v(pc ), q q¯
¯ 2 )Pα k/1 v(pc )u(p ¯ t )Pβ k/2 u(k1 ), M8αβ = v(k q q¯
M9αβ = v(k ¯ 2 )Pαp / t u(k1 )u(p ¯ t )Pβ k/1 v(pc ), q q¯
M10αβ = v(k ¯ 2 )Pα γµ v(pc )u(p ¯ t )Pβ γ µ u(k1 ), q q¯
M11αβ = u(p ¯ t )Pα u(k2 )v(p ¯ c )Pβ v(k1 ), q q¯
M12αβ = u(p ¯ t )Pα u(k2 )v(p ¯ c )Pβ k/2 v(k1 ), q q¯
¯ t )Pβ k/1 u(k2 )v(p ¯ c )Pα v(k1 ), M13αβ = u(p
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
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Fig. 2. Quark initial state subprocess Feynman diagrams for the top quark and anti-charm quark associated production.
q q¯
M14αβ = u(p ¯ t )Pα k/1 u(k2 )v(p ¯ c )Pβ k/2 v(k1 ), q q¯
M15αβ = u(p ¯ t )Pα γµ u(k2 )v(p ¯ c )Pβ γ µ v(k1 ),
(20)
where (α, β) = (L, L), (L, R), (R, L) and (R, R), respectively. And the amplitude for the quarks initiated subprocesses can be expressed as
Mq q¯ =
15
q q¯
g iαβ Miαβ ,
(21)
i=1 α,β=L,R
where the g iαβ are form factors which can be fixed by the straightforward calculations of the relevant diagrams. However, the Feynman diagrams contributing to the subprocesses with the different initial states can be different. We first describe the amplitude of the subprocess cc¯ → t c, ¯ which has the largest set of Feynman diagrams, i.e., all diagrams in Fig. 2, and the explicit expressions of the non-zero g iαβ of each diagram can be found in Appendix B. In Table 1 we list all other channels that can contribute to t c( ¯ u) ¯ production and their related Feynman diagrams, and their corresponding form factors can be obtained from ones of cc¯ → t c¯ by modifying some couplings as shown in the table.
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
Table 1 The form factors of other channels obtained from ones of cc¯ → t c. ¯ Here, ViL(R) are defined in the Appendix A. The subscript ρ represents the corresponding diagram Channel
Related diagrams in Fig. 2
Form factors
uu¯ → t u¯
b1 , b2 , b3 , b4 , v1 , v2 , v3 , v4 , s1 , s2 , s3 , s4
gρ (V5L(R) ↔ V4L(R) )
iαβ iαβ
gρ (V5L(R) ↔ V4L(R) )
uu¯ → t c¯ cc¯ → t u¯
b1 , b2 , b3 , b4 , v1 , v2 , s1 , s2
iαβ
gρ (V5L(R) ↔ V4L(R) ) iαβ
gρ (V5L(R) ↔ V4L(R) )
uc¯ → t c¯ cu¯ → t u¯
b1 , b2 , b3 , b4 , v3 , v4 , s3 , s4
iαβ
gρ (V5L(R) ↔ V4L(R) ) iαβ
gρ (V5L(R) ↔ V4L(R) )
cu¯ → t c¯ uc¯ → t u¯
iαβ
gρ (V5L(R) ↔ V4L(R) )
b1 , b2 , b3 , b4
iαβ
gρ (V6L(R) ↔ V5L(R) )
s s¯ → t c( ¯ u) ¯ d d¯ → t c( ¯ u) ¯
iαβ
gρ (V6L(R) ↔ V4L(R) )
b2 , b3 , v1 , v2 , s1 , s2
The partonic level cross section for κλ → t c¯ is tˆ+ σˆ
κλ
= tˆ−
1 κλ 2 M d tˆ 16π sˆ 2
(22)
with m2 + m2t − sˆ 1 tˆ± = c (23) ± (ˆs − (mc + mt )2 )(ˆs − (mc − mt )2 ), 2 2 √ ¯ states. The total hadronic cross section where sˆ is the c.m. energy of the κλ (gg or q q) for pp → κλ → t c¯ can be written in the form σ (s) =
κ,λ
Here as
1
dz √ (mc +mt )/ s
dL κλ σˆ dz
(κλ → t c¯ at sˆ = z2 s).
(24)
√ s is the c.m. energy of the pp states, and dL/dz is the parton luminosity, defined dL = 2z dz
1
dx fκ/p (x, µ)fλ/p z2 /x, µ , x
(25)
z2
where fκ/p (x, µ) and fλ/p (z2 /x, µ) are the κ and λ parton distribution functions, respectively.
4. Numerical calculation and discussion In the following we present some numerical results for the total cross section of the single top quark production induced by SUSY-QCD FCNC couplings at the LHC.
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
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Fig. 3. The total cross sections for the pp → t u¯ as a function of the mass of gluino.
In our numerical calculations the SM parameters were taken to be mt = 174.3 GeV, MW = 80.423 GeV, MZ = 91.1876 GeV, mc = 1.2 GeV, sin2 (θW ) = 0.23113 and αs (MZ ) = 0.1172 [2]. And we used the CTEQ6L PDF [23] and took the factorization scale and the renormalization scale as µf = µr = mt /2. The relevant SUSY parameters are µ, tan β, MSUSY and mg˜ , which are unrelated to flavor changing mechanism, and may be fixed from flavor conserving observables at the future colliders. And they are chosen as follows: MSUSY = 400, 1000 GeV, tan β = 3, 30, mg˜ = 200, 300 GeV and µ = 200 GeV. ij As for the range of the flavor mixing parameters, (δU )LL are constrained by correspondij 12 ) ¯ mixing [26], and ing (δD )LL [19,22,24,25], in which (δU LL also is constrained by K–K 31 12 12 12 D0 –D¯ 0 mixing makes constraints on (δU )LL , (δU )LR and (δU )RL [27]. And (δU )LL , 32 31 32 ¯ (δU )LL , (δU )RL and (δU )RL are constrained by the chargino contributions to Bd –Bd mixing [24]. Finally, there also are constraints on the up squark mass matrix from the chargino contributions to b → sγ [19,28]. Taking into account above constraints, in our numerical calculations, we use the following limits: 12 ) , (δ 12 ) 12 (i) (δU LL U LR and (δU )RL are less than 0.08MSUSY /(1 TeV); 13 12 (ii) (δU )RR and (δU )LL are limited below 0.2MSUSY /(1 TeV); 23 23 23 23 13 13 13 (iii) (δU )LL , (δU )LR , (δU )RL , (δU )RR , (δU )LR , (δU )RL and (δU )RR vary from 0 to 1.
Our results are shown in Figs. 3–10. There are two common features of these curves: one is that the cross sections increase rapidly with the mixing parameters increasing, and the other is that the cross sections depend strongly on the gluino mass mg˜ , but weakly depend on tan β (so we just discuss the results for tan β = 30 below). Comparing with above cases, the dependence on MSUSY is medium.
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
Fig. 4. The total cross sections for the pp → t c( ¯ u) ¯ with LL off-diagonal elements. Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
13 ) Fig. 5. The total cross sections for the pp → t c( ¯ u) ¯ with RR off-diagonal elements (δU RR . Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
15
23 ) Fig. 6. The total cross sections for the pp → t c( ¯ u) ¯ with RR off-diagonal elements (δU RR . Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
13 ) Fig. 7. The total cross sections for the pp → t c( ¯ u) ¯ with LR off-diagonal elements (δU LR . Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
23 ) Fig. 8. The total cross sections for the pp → t c( ¯ u) ¯ with LR off-diagonal elements (δU LR . Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
Fig. 3 shows that in general the production rates of gluon fusion processes are several times larger than ones of the q q¯ annihilation, and these rates all depend strongly on the 13 ) gluino mass mg˜ . For example, assuming (δU LR = 0.7 and tan β = 30, the production rates of gg → t u¯ decrease from 700 fb at mg˜ = 200 GeV to 70 fb at mg˜ = 500 GeV. 23 ) in the LL block. For Fig. 4 shows the dependence of the cross sections on the (δU 23 ) tan β = 30 the cross section of t c¯ production can reach around 12 fb when (δU LL = 0.7, MSUSY = 400 GeV, and mg˜ = 200 GeV, as shown in Fig. 4(b), but for t u¯ production it 13 ) is one order of magnitude smaller. The production cross sections arising from (δU LL are 13 less than 1 fb as (δU )LL is limited below 0.2MSUSY/(1 TeV), so we do not show the corresponding curves here. 13 ) and (δ 23 ) in the RR block, Figs. 5 and 6 give the cross sections as the functions of (δU U respectively. We can see that the cross sections of t u¯ production are larger than ones of 13 ) t c¯ production for the case where (δU RR = 0 and all the other mixing parameters are set to be 0, and in contrast with this case, the cross sections of t c¯ production are larger than 23 ones of t u¯ production for the case of (δU )RR = 0. For example, assuming tan β = 30, MSUSY = 400 GeV, and mg˜ = 200 GeV, the cross sections of t c¯ and t u¯ productions are 23 ) 13 12 fb and 1 fb for (δU RR = 0.7, respectively, but 2 fb and 13 fb for (δU )RR = 0.7, respectively. 13 ) and (δ 23 ) in the LR Figs. 7 and 8 show the dependence of the cross sections on (δU U block, respectively. We can see that the cross sections are much larger than ones arising from the mixing in the LL and RR blocks, which indicates that the SUSY FCNC effects
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
17
Fig. 9. Typical interference effects between different matrix elements within one block in (a) and (b), and between different blocks in (c) and (d). Here, solid line: mg˜ = 200 GeV, MSUSY = 400 GeV; dashed line: mg˜ = 300 GeV, MSUSY = 400 GeV; dotted line: mg˜ = 200 GeV, MSUSY = 1000 GeV; dash-dotted line: mg˜ = 300 GeV, MSUSY = 1000 GeV.
induced by the mixing in the LR block can enhance the cross sections significantly. For example, for tan β = 30, MSUSY = 400 GeV, and mg˜ = 200 GeV, the cross section can reach 23 ) 13 643 fb for (δU LR = 0.7, and 75 fb for (δU )LR = 0.7, as shown in Figs. 7(b) and 8(b), respectively. Here we see again that the production cross sections of t c¯ and t u¯ depend 23 ) 13 strongly on (δU LR and (δU )LR , respectively. The cases of the RL block are similar to ones of the LR block, so we do not show them here. From Figs. 7 and 8 we can find that the production rates may be as large as a few pb for favorable parameter values. In the above discussion for Figs. 3–8, we only show the results for mg˜ = 200 GeV and MSUSY = 400 GeV, and the corresponding results for other cases (mg˜ = 200 GeV and MSUSY = 1000 GeV, mg˜ = 300 GeV and MSUSY = 400 or 1000 GeV) decrease significantly, but they are still remarkable and vary from a few fb to hundreds of fb. In Fig. 9, we give the results to uncover the interference effects between different entries within one block and between different blocks for t c¯ production. Fig. 9(a) and (b) show 13 ) 23 the typical interference effects between (δU LR and (δU )LR within the LR block and 13 23 between (δU )RR and (δU )RR within the RR block, respectively. Fig. 9(c) and (d) show the interference effects between the LL and LR block and between the LL and RR block for the typical parameters, respectively. In general, those interference effects enhance the cross sections. And the interference effects for t u¯ production are very similar to the case of t c¯
18
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
Fig. 10. Similar as Fig. 9, but for t u. ¯ Table 2 Cross sections in fb for the t c¯ and t u¯ productions. Here tan β = 30, MSUSY = 400 GeV and mg˜ = 200 GeV, the off-diagonal element in the table equals to 0.7 and others are set to zero Subprocess
23 ) (δU LL
13 ) (δU RR
23 ) (δU RR
13 ) (δU LR
23 ) (δU LR
gg → t u¯ q q¯ → t u¯ gg → t c¯ q q¯ → t c¯
0 0.2 11.0 1.6
14.4 4.7 0 1.8
0 0.2 10.5 1.5
650.4 124.1 0 74.5
0 12.6 633.0 10.0
Total
12.8
20.9
12.2
849.0
655.6
production and are shown in Fig. 10. Although the interference effects do not change the cross sections of single top quark production at the LHC significantly, the consideration of them is of importance to reveal the details of flavor changing mechanism. Finally, for convenience, we list some typical results in Table 2.
5. Conclusions We have evaluated the t c¯ and t u¯ productions at the LHC within the general unconstrained MSSM framework. These single top quark productions are induced by SUSY-
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
19
QCD FCNC couplings and have remarkable cross sections for favorable parameter values allowed by current low energy data, which can be as large as a few pb. It has been pointed out [13] that the top quark FCNC production signals are more accessible than the top quark FCNC decay signals. And according to the model independent analysis in Ref. [14], it is possible to observe FCNC effects through single top quark productions at the LHC. Thus, our above results show that once large rates of the t c¯ and t u¯ productions are detected at the LHC, they may be induced by SUSY FCNC couplings, and such flavor changing must come from the LR or RL block, which is related to the soft trilinear couplings AU . Therefore, we believe that the precise measurement of single top quark production cross sections at the LHC is a powerful probe for the details of the SUSY FCNC couplings.
Acknowledgements This work was supported in part by the National Natural Science Foundation of China and Specialized Research Fund for the Doctoral Program of Higher Education.
Appendix A. Couplings Here we list the relevant couplings in the amplitudes. i, j stand for generation indices and r, s stand for color indices. V1 = −igs T a , V2 = −gs fabc , V3 = −igs fabc , √ √ q a q V4R = i 2gs Trsa Z6i , V4L = −i 2gs Trs Z3i , √ √ q q V5R = i 2gs Trsa Z5i , V5L = −i 2gs Trsa Z2i , √ √ q q V6R = i 2gs Trsa Z4i , V6L = −i 2gs Trsa Z1i , i 1 V7 = gs2 δab + dabc T c . 2 3
(A.1) (A.2) (A.3) (A.4) (A.5)
Appendix B. Form factors This appendix lists all the form factors in the amplitude of various subprocess of pp → t c, ¯ in terms of 2-, 3- and 4-points one-loop integrals [29]. For convenience, we define the abbreviations of one-loop integrals as following: B a = B s, m2g˜ , m2g˜ , B d = B t, m2g˜ , m2g˜ , B t = B 0, m2g˜ , m2q˜ , B q = B m2t , m2g˜ , m2q˜ , B g = B t, m2g˜ , m2q˜ , B m = B u, m2g˜ , m2q˜ , C = C m2t , s, 0, m2q˜ , m2g˜ , m2g˜ , C d = C m2t , s, 0, m2g˜ , m2q˜ , m2q˜ , C t = C m2t , t, 0, m2q˜ , m2g˜ , m2g˜ , C q = C m2t , t, 0, m2g˜ , m2q˜ , m2q˜ , C g = C 0, m2t , s, m2g˜ , m2q˜ , m2g˜ , C f = C m2t , 0, s, m2g˜ , m2q˜ , m2g˜ ,
20
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C n = C 0, 0, t, m2g˜ , m2g˜ , m2q˜ , C m = C 0, 0, t, m2q˜ , m2q˜ , m2g˜ , C l = C 0, t, 0, m2g˜ , m2g˜ , m2q˜ , C o = C 0, t, 0, m2g˜ , mq2˜ , m2g˜ , C r = C 0, u, 0, m2q˜ , m2g˜ , m2q˜ , C p = C 0, u, 0, m2g˜ , m2q˜ , m2g˜ , C x = C 0, u, m2t , m2g˜ , m2g˜ , m2q˜ , C y = C 0, u, m2t , m2q˜ , m2q˜ , m2g˜ , C w = C m2t , t, 0, m2q˜ , m2g˜ , mq2˜ , C z = C m2t , t, 0, m2g˜ , m2q˜ , m2g˜ , D = D 0, 0, m2t , 0, t, s, m2g˜ , m2q˜ , m2g˜ , m2q˜ , D v = D 0, 0, m2t , 0, t, s, m2q˜ , m2g˜ , m2q˜ , m2g˜ , D t = D 0, m2t , 0, 0, u, s, m2q˜ , m2g˜ , m2q˜ , m2g˜ , D q = D m2t , 0, 0, 0, u, t, m2q˜ , m2g˜ , m2q˜ , m2q˜ , D g = D 0, 0, m2t , 0, t, s, m2g˜ , m2g˜ , m2q˜ , m2g˜ , D f = D 0, 0, m2t , 0, t, s, m2q˜ , mq2˜ , m2g˜ , m2q˜ , D s = D 0, m2t , 0, 0, u, s, m2g˜ , m2g˜ , mq2˜ , m2g˜ , D m = D 0, m2t , 0, 0, u, s, m2q˜ , m2q˜ , m2g˜ , mq2˜ , D n = D m2t , 0, 0, 0, u, t, m2g˜ , m2q˜ , m2q˜ , m2g˜ , D l = D m2t , 0, 0, 0, u, s, m2q˜ , m2g˜ , m2g˜ , m2q˜ . Below we display the non-zero form factors for the gluon fusion subprocess (we only display the odd number form factors, and the even number ones can be obtained by the replacement of V4L ↔ V4R and V5L ↔ V5R , correspondingly). For self-energy diagram (see Fig. 1(a)–(e)), fsi are:
1 V mg˜ m2t tB0t V1 V5L sV1 + t − m2t V3 fs1 = 3 4L 2 2 mt π s(mt − t)t g + mt V1 −B0 mg˜ m3t sV1 V5L
q q g , + m2t − t mg˜ mt B0 V5R − B1 m2t V5L (sV2 + tV3 ) + m2t stV1 V5R B1 fs3
1
=
(B.1)
q V4L −mt m2t − u V1 mg˜ mt B0 V5R
m3t π 2 s(m2t − t)t
q − B1 m2t V5L (uV3 − sV1 ) − m2t V1 mg˜ B0t uV5L u − m2t V3 − sV2 + mt sV1 B0m mg˜ mt V5L − uV5R B1m , mg˜ B0t V1 V5L mg˜ B0t V2 V5L 1 fs5 = V V + 1 4L 8π 2 mt t − m3t mt u − m3t q
+
q
V1 (B1 m2t V5L − mg˜ mt B0 V5R ) m2t u
q
+
q
V2 (B1 m2t V5L − mg˜ mt B0 V5R ) m2t t
(B.2)
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32 g
g
V1 (tV5R B1 − B0 mg˜ mt V5L )
V1 (uV5R B1m − B0m mg˜ mt V5L )
21
+ , t (t − m2t ) u(u − m2t ) 1 q −mt m2t − u V1 mg˜ mt B0 V5L V fs7 = 3 4R 2 2 mt π s(mt − t)t
q − B1 m2t V5R (uV3 − sV1 ) − m2t V1 mg˜ B0t uV5R u − m2t V3 − sV2 + mt sV1 B0m mg˜ mt V5R − uV5L B1m , 1 fs9 = V1 V4R 2mg˜ m2t s B0m V1 − B0t V2 V5R 2 2 16mt π 2 s(mt − u) q q − m2t − u B1 V5R m3t − mg˜ m2t B0 V5L + mg˜ m2t B0t V5R V3 − 2m3t sV1 V5L B1m ,
(B.4)
1 fs11 = fs5 , 2
(B.6)
+
fs13 =
1
16m2t π 2 t (t − m2t )(m2t − u) g + m2t V1 mg˜ B0 m2t − u (s
V4R
(B.3)
(B.5)
q q m2t − u V1 V2 B1 m3t V5R − mg˜ m2t B0 V5L
+ u)V1 + m2t − t t B0m V1 − B0t V2 V5R . + mt V1 V5L t t − m2t B1m (B.7)
For vertex diagram Fig. 1(f), (g) and (h), the non-zero form factors fti are: l 1 fv1 = V 4L stV1 C0 mg˜ mt V2 V5L 2 2 8π st (t − mt )
g l l o o − B0 V2 − 2C00 V2 + 2C00 V1 − (s + u) C12 V2 − C12 V1 V5R
d + m2t − t V5L t −C0 m2g˜ + C0 m2q + B0a − 2C00 + C1 m2t V2 − 2C00 V1 V3 g
z w + sV1 V2 B0 − 2C00 + m2t C1z − 2V1 C00 − mg˜ mt V2 V5R C0 tV3 + sV1 C0z (B.8) , 1 2 V m fv3 = − u mg˜ mt V2 V5R C0 uV3 − sV1 C0x 4L t 8π 2 su(u − m2t ) d
V1 − −C0 m2g˜ + C0 m2q + B0a − 2C00 + C1 m2t V2 V3 + V5L u 2C00 y x + sV1 V2 B0m − 2C00 + m2t C2x − 2V1 C00 p − suV1 mg˜ mt C1 V2 + C1r V1 V5L
p p r r + B0m − 2C00 + C12 u V2 + 2C00 V1 − C12 uV1 V5R (B.9) , o g l V ) 2C00 V1 V5R V2 (C0l mg˜ mt V5L − B0 V5R + 2C00 1 5R V + V fv5 = 1 4L 2 2 8π 2 t − mt mt − t r V r V1 (−mg˜ mt V5LC1r − 2C00 5R + C12 V5R ) + m2t − u
22
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32 p
− −
p
p
V2 (C1 mg˜ mt V5L + (B0m − 2C00 + C00 u)V5R ) 1 t
m2t − t
w w w w V1 mg˜ mt V5R C1w + V5L 2C00 − tC12 + m2t C1w + C11 + C12 y
x + m2 C x ) − m m V C x ) 2V1 V5L C00 V2 (V5L (B0m − 2C00 g˜ t 5R 0 t 2 − + u u 1 g z z z z + V2 mg˜ mt V5R C1z + V5L B0 − 2C00 + tC12 − m2t C11 + C12 , t (B.10) 1 p V4L suV1 mg˜ mt C0 V2 V5L fv7 = 8π 2 su(u − m2t )
p r − B0m V2 − 2C00 + 2C00 V1 V5R + m2t − u mg˜ mt V2 V5R C0 uV3 − sV1 C0x d
V1 − −C0 m2g˜ + C0 m2q + B0a − 2C00 + C1 m2t V2 V3 + V5L u 2C00 y x + sV1 V2 B0m − 2C00 + m2t C2x − 2V1 C00 (B.11) , 1 1 p 2sV1 mt −B0m V2 + 2C00 V2 V fv9 = 4R 2 2 16π s(mt − u)
p r − 2C00 V1 V5L + mg˜ C0 uV2 V5R + m2t − u mg˜ C0 m2t V2 + C2 (t − u)V2 + (t − u) C1 V2 − C0d + C1d + C2d V1 V5L + mt −B0a V2 + 2C00 + C0 (mg˜ − mq )(mg˜ + mq ) + (C11 + C12 )(t − u)
d d d (t − u)V1 V5R V3 V1 + C1d + C11 + C12 − C1 (s + 2u) V2 + 2C00 2V1 V2 (mt uV5R C2x − mg˜ uV5LC0x ) , + (B.12) u 2 1 1 11 o r V1 2 − C00 fv = mt + C00 V 4L 2 2 2 16π (mt − u)(mt − t) p
r o + C00 t + C00 u V1 V5R + V2 mg˜ mt C0 m2t − t + C0l m2t − u V5L 2
p g l mt − u + B0 u − m2t V5R + − B0m − 2C00 m2t − t + 2C00
g
1 z x V1 V2 V5L t V0m − 2C00 + m2t uC1z + m2t C2x + u B0 − 2C00 tu w y − mg˜ mt V5R tC0x + uC0z − 2V1 V5L uC00 (B.13) + tC00 , +
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
23
r V V 2mt C00 C0l mg˜ (s + u)V2 V5R 1 1 5L x + m = V V C V − V 1 4R 2 5L g ˜ 0 16π 2 m2t − u m2t − t
1 m p p + 2 V2 mt B0 − 2C00 V5L − mg˜ C0 uV5R mt − u w 2mt V1 V5R C00 + − mt V2 V5R C2x t g 1 z + V2 mg˜ (s + u)V5L C0z − mt V5R B0 − 2C00 + (s + u)C1z , (B.14) t l 1 o V1 V4R C12 mt V2 V5L − mt C12 V1 V5L fv15 = − 2 2 8π (mt − t)
+ mg˜ C0l + C2l V2 + C2o V1 V5R , (B.15)
fv13
1 l o V2 − C12 V1 V4L (s + u)V1 C12 2 8π (m2t − t) y x + m2t − t V1 V5L V2 C12 + V1 C12 ,
fv17 = −
fv19 =
fv21 =
fv23 =
fv25 =
V5R
l 1 o V 4R uV1 C12 mt V2 V5L − mt C12 V1 V5L 2 2 8π (mt − t)u
+ mg˜ C0l + C2l V2 + C2o V1 V5R
y + m2t − t V1 mg˜ V5L V1 C2 − V2 C0x + C2x x y y y x + mt V5R V2 C12 + C2x + C22 + V1 C12 + C2 + C22 , p 1 r V1 V4R mt C12 V2 − C12 V1 V5L 2 2 8π (mt − u) p
p + mg˜ C0 + C1 V2 + C1r V1 V5R ,
1 V m2t − u V1 mg˜ V5L V2 C0z + C1z − V1 C1w 4R 2 2 8π t (mt − u) z z w w − mt V5R V1 C1w + C11 + V2 C1z + C11 + C12 + C12 p r − tV1 mt C12 V2 − C12 V1 V5L p
p , + mg˜ C0 + C1 V2 + C1r V1 V5R l 1 o V4R stV1 C12 mt V2 V5L − mt C12 V1 V5L 2 4π 2 s(mt − t)t
+ mg˜ C0l + C2l V2 + C2o V1 V5R
+ m2t − t t mg˜ (C1 + C2 )V2 − C0d + C1d + C2d V1 V5L
d d V1 V5R V3 + mt (C1 + C11 + C12 )V2 + C1d + C11 + C12
+ sV1 mg˜ V5L V1 C1w − V2 C0z + C1z z z w w + C12 + C12 + mt V5R V1 C1w + C11 + V2 C1z + C11 ,
(B.16)
(B.17)
(B.18)
(B.19)
(B.20)
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
1 y sV1 mg˜ V5L V1 C2 − V2 C0x + C2x V 4R 2 4π su y y y x x + V1 C12 + C2 + C22 + C2x + C22 + mt V5R V2 C12
− u mg˜ (C1 + C2 )V2 − C0d + C1d + C2d V1 V5L
d d V1 V5R V3 , + mt (C1 + C11 + C12 )V2 + C1d + C11 + C12 (B.21) 1 y
V4R m2t − t V1 mg˜ V5L V2 C0x + C2x − V1 C2 fv29 = 2 2 4π (mt − t)u x y y y x + V1 C12 + C2 + C22 + C2x + C22 − mt V5R V2 C12 l
o mt V2 V5L − mt C12 V1 V5L + mg˜ C0l + C2l V2 + C2o V1 V5R − uV1 C12 .
fv27 =
(B.22) And for box diagrams Fig. 1(i)–(o), all form factors
fbi
are non-zero:
f s 1 g g g g s s V4L C2 + C0 − 2D00 + 2D002 + 2D003 − 2D003 u + D13 + D23 2 8π g
g g g g g + D12 + D13 + D22 (s + u) + D23 (s + u) + D2 m2t V5L − D0 mg˜ mt V5R V22 l n n + 2D003 + C2m − 2D003 + D23 u V1 V5L V2 f 2 f m + 2 D002 + D003 − D003 (B.23) V1 V5L , 1 f g g g g s C0 − C0 − C1 − C2 − 2D001 − 2D00 fb3 = − 2 V4L 8π s s s s s s s s u + −D12 u + 2 D001 + D002 − D13 + D23 + D13 − D22 + D23 g g g − D13 (s + u) + D12 (t − s) V5L + mg˜ mt −D1 + D1s + D2s V5R V22
l l n l + V1 −2D002 + D2l m2q − m2g˜ + D12 m2t + C1n + 2D002 + D22 u V5L 2 f m m V1 V5L , − D2l mg˜ mt V5R V2 + 2 −D001 + D001 + D002 (B.24) f 1 f g g g g s fb5 = − 2 V4L C1 + C2 + C1 + C2 + 2D00 − 2D002 + 2D00 8π g
g g s s s − 2D002 + D23 u + D22 u + D23 t + D12 t V5L − mg˜ mt D2 + D2s V5R V22 l l 2 l l + V1 2D001 + 2D002 + D1l + D2l m2q − m2g˜ − D11 + D12 mt
n n l l − D12 u V5L + C0n + C2n − 2 D001 + D002 + D22 l 2 f l m + D1 + D2 mg˜ mt V5R V2 − 2 D002 + D002 V1 V5L , (B.25) f 1 g g g V mg˜ mt D0s V5R − C0 + 2D001 + 2D002 + 2D003 fb7 = 4L 8π 2 s
s s s + D2s u V5L V22 − 2 2D00 + D001 + D002 + D003 l l n n n + 2 −D002 − D003 + D00 + D002 + D003 V1 V5L V2 f 2
f f f m m m m − 2 D00 + D001 + D002 + D003 − D00 − D001 − D002 − D003 V1 V5L , fb1 = −
(B.26)
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
fb9 =
25
f g
1 s V mg˜ V5L 2 C0 − 2 D00 + D00 − D0s u V22 4R 2 16π 2 l f n m V1 V2 + 4 D00 + D00 V1 + C0d V7 − 4 D00 + D00 f f g g s + mt V5R −2 − D2s + D3s m2t + C1 + C2 − 2D001 2D002 + 2D00 l s l n V1 V2 + D001 + m2t D2s + 2D3s − D2s u − D3s m2t V22 − 4 D00 − D001 − D001 2
f f f m d , + 4 D00 + D001 + D002 − D001 V3 − C1 V7 (B.27)
fb11 =
fb13 =
fb15 =
fb17 =
fb19 =
fb21 =
fb23 =
f 1 g g g s s V + C − 4D − 4D + D u + D t V5L V V C 2 4L 2 00 2 0 0 00 2 16π 2 g
l n − mg˜ mt D0 + D0s V5R − 2 D00 + D00 (B.28) V3 V5L , f 1 f g g s V2 V4R mt C1 + C2 + C2 + 2D00 + 2D00 − D2s u V2 16π 2
l + 2D00 V1 V5R f
g g g g − mg˜ C0 − C0 − D2 s − D0 s − D0 + D0s u V2 V5L , (B.29) 1 n V2 V4R −mg˜ D3n V1 V5L + mt D13 V1 V5R 2 8π g g g + V2 mg˜ D0 + D2 + D3 − D3s V5L g
g g g g s V5R , − mt D12 + D13 + D2 + D22 + D23 + D13 (B.30) s 1 f g g g s s s C + C2 + 2D00 + 2D002 + 2 D00 + mt2 D23 + D002 − D23 (s + t) 8π 2 2 s g g g s s − D12 + D2s + D22 + D23 u + D12 + D2 + D22 (s + u) l l m n n n − 2D001 − 2D002 − 2D001 + 2D002 − C2 − 2D00
2 n f n m u V1 V2 + 2 D002 + D002 V3 V4L V5L , + D12 + D2n + D22 (B.31) g
1 g s n n V2 + D1n + D11 V1 V5R + D12 V2 V4R mt D12 + D2G + D22 − D12 2 8π g
g − mg˜ D0 + D2 + D0s + D2s V2 + D0n + D1n + D2n V1 V5L , (B.32) g
1 V2 V4R mg˜ D1 V2 − D0s + D1s + D2s V2 + D2l V1 V5L 2 8π g s s s s V2 + mt D13 + D12 + D13 + D2s + D22 + D23 l
l l l + D12 + D2 + D22 + D23 V1 V5R , (B.33) g
1 g V2 V4R mg˜ D0 + D2 + D0s + D2s V2 − D1n + D2l V1 V5L 2 8π g s s V2 − mt −D23 + D2s + D22 + D23 l
l l l l l + D1 + D11 + 2D12 + D13 + D2l + D22 + D23 (B.34) V1 V5R ,
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
g 2 1 g g s V4R mg˜ D0 − D22 − D23 + D23 V2 2 4π l f 2
f l n n m + D13 + D23 + D23 + D13 V1 V2 + D22 + D23 − D23 V1 V5L g 2 g g g g g m + mt D122 + D123 + D22 + D222 + D223 + D23 − D23 V2 l l n n V1 V2 + D133 + D123 − D133 − D123 2
f f f f f f m V3 V5R , + D122 + D123 + D22 + D222 + D223 + D23 + D123 (B.35) 1 g g s s V5L + D2s + D22 V4R mg˜ −D1 − D12 + D1s + D12 fb27 = 2 4π g
g g s s + mt D112 + D122 + D12 + D112 + D122 V5R V22 l l n n + V1 mg˜ D12 + D22 + D12 + D2n + D22 V5L l
l l n n n V5R V2 + mt D112 + D12 + D122 − D112 − D12 − D122 f m m mg˜ V5L + V12 D12 − D12 − D22
f f f m m mt V5R , + D112 + D12 + D122 + D112 + D122 (B.36) 1 g g s fb29 = − 2 V4R mg˜ (D2 + D22 + D2s + D22 )V5L 4π g
g g s V5R V22 − mt D122 + D22 + D222 − D122 l l l n n n V5L + V1 mg˜ D11 + 2D12 + D22 + D1n + D11 + 2D12 + D2n + D22 l l l l l n n n + mt D11 + D111 + 2D112 + D12 + D122 − D11 − D111 − 2D112
f n n m V5R V2 − V12 D22 + D22 mg˜ V5L − D12 − D122
f f f f m mt V5R , + D122 + D22 + D222 + D122 − D122 (B.37)
fb25 = −
1 g g g g g g g D112 + D113 + D12 + D122 + 2D123 + D13 + D133 4π 2 2 s s s s s s s − D113 − 2D123 − D13 − D133 − D23 − D223 − D233 V2 l l l n n n + D223 + D23 + D233 − D23 − D223 − D233 V1 V2 f f f f f f f m m + D112 + D113 + D12 + D122 + 2D123 + D13 + D133 − D113 − 2D123 2
m m m m m V1 V4L V5L , − D13 − D133 − D223 − D23 − D233 (B.38) 1 g g g g g g g g fb33 = D12 + D122 + D123 + D13 + D2 + 2D22 + D222 + 2D223 4π 2 2 g g s s s s s + 2D23 + D233 + D123 + D13 + D223 + D23 + D233 V2 l l l l l l n n n − D123 + D13 + D133 + D223 + D23 + D233 − D123 − D223 − D133 f f f f f f n n V1 V2 + D122 + D123 + D22 + D222 + 2D223 + D23 − D23 − D233 2
f m m m m V1 V4L V5L , + D233 + D123 + D223 + D23 + D233 (B.39)
fb31 =
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
fb35 =
27
g 1 g g g s s V mg˜ D12 + D13 + D13 V5L − mt D112 + D113 + D23 4R 2 4π
g g g g s s V5R V22 + D12 + D122 + D123 + D13 − D113 − D123 l l
n n + V1 mg˜ D23 + D23 − D123 V5L + mt D123 V5R V2 f f m m mg˜ V5L − V12 D12 + D13 + D13 + D23
f f f f f f m m mt V5R , − D123 + D112 + D113 + D12 + D122 + D123 + D13 − D113
(B.40) 1 g g g g g s = − 2 D112 + D12 + D122 + D123 + D13 + D112 4π 2 s s s s s s s s V2 + 2D12 + 2D122 + D123 + D13 + D2s + 2D22 + D222 + D223 + D23 l l l l l n n + −D122 − D123 − D22 − D222 − D223 + D12 + D122 n n n n n V1 V2 + D123 + D2n + 2D22 + D222 + D223 + D23 f f f f m m + D112 + D123 + D12 + D122 + D112 + D12
m m m m m V12 V4L V5L , + 2D122 + D123 + D22 + D222 + D223 (B.41) 1 g g g g g g fb39 = −D12 − D122 − D2 − 2D22 − D222 − D223 4π 2 2 g s s s s s s V2 − D23 + D12 + D122 + D2s + 2D22 + D222 + D223 + D23 l l l l l l l l + −D112 − D113 − D12 − 2D122 − 2D123 − D22 − D222 − D223
fb37
n n n n n n n + D112 + D113 + 2D12 + 2D122 + 2D123 + D13 + D2n + 2D22 f f f f n n n + D222 + D223 + D23 V1 V2 + −D122 − D22 − D222 − D223 2
m m m m + D122 + D22 + D222 + D223 V1 V4L V5L .
(B.42)
For the quarks initiated subprocesses, we list all form factors of cc¯ initial state: = gs14LR = gs6LR 4 4
mg˜ B0t V12 V4R V5R , 8mt π 2 t
gs6RL = gs14LL = gs6LR (V4L ↔ V4R , V5L ↔ V5R ), 4 4 4 gs6LR 3 gs6RL 3 gs5LR 2 gs5RL 2
q V12 V4R (B1 m2t V5R − mg˜ mt B0 V5L ) , = gs14LR = 3 8m2t π 2 t = gs14LL = gs6LR (V4L ↔ V4R , V5L ↔ V5R ), 3 3 q q 2 V1 V4R (mg˜ mt B0 V5L − B1 m2t V5R , = gs14RR = 2 8m2t π 2 s = gs14RL = gs5LR (V4L ↔ V4R , V5L ↔ V5R ), 2 2
= gs14RR =− gs5LR 1 1
mg˜ B0t V12 V4R V5R , 8mt π 2 s
gs5RL = gs14RL = gs5LR (V4L ↔ V4R , V5L ↔ V5R ), 1 1 1
(B.43) (B.44) (B.45) (B.46) (B.47) (B.48) (B.49) (B.50)
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
gv2LL = gv2LR = 4 4
q 1 q q V1 V1 V4R mt C1 + C11 + C12 V5R 2 8π t
q q q − mg˜ (C0 + C1 + C2 )V5L ,
= gv2LL (V4L ↔ V4R , V5L 4 q C V1 V1 V4R V5R , = gv14LR = 00 gv6LR 4 4 4π 2 t
gv2RL 4
= gv2RR 4
(B.51)
↔ V5R ),
(B.52) (B.53)
gv6RL = gv14LL = gv6LR (V4L ↔ V4R , V5L ↔ V5R ), 4 4 4 1 = gv2LR = − 2 V1 V2 V4R mg˜ C1t + C2t V5L gv2LL 3 3 8π t
t t V5R , + mt C1t + C11 + C12
(B.55)
↔ V4R , V5L ↔ V5R ), 1 = gv14LR = − 2 V1 V2 V4R C0t V5R m2g˜ + mt C0t V5Lmg˜ gv6LR 3 3 8π t
t − C0t m2q + B0d − 2C00 + m2t C1t V5R ,
gv2RL 3
= gv2RR 3
(B.54)
= gv2LL (V4L 3
= gv6LR (V4L ↔ V4R , V5L 3 d C V1 V1 V4R V5R , gv5LR = gv14RR = 00 2 2 4π 2 s
gv6RL 3
= gv14LL 3
↔ V5R ),
= gv8RR 2
(B.59) (B.60)
(B.61)
= gv8LL (V4L 2
↔ V4R , V5L ↔ V5R ), 1 V1 V2 V4R C0 mg˜ mt V5L = gv14RR = gv5LR 1 1 8π 2 s
− −C0 m2g˜ + C0 m2q + B0a − 2C00 + C1 mt2 V5R ,
(B.62)
(B.63)
(B.64) ↔ V4R , V5L ↔ V5R ), V1 V2 V4R ((C1 + C2 )mg˜ V5L + (C1 + C11 + C12 )mt V5R ) , (B.65) = gv8LR = gv8LL 1 1 8π 2 s gv8RL (B.66) = gv8RR = gv8LL (V4L ↔ V4R , V5L ↔ V5R ), 1 1 1
gv5RL 1
= gv14RL 1
(B.57) (B.58)
gv5RL = gv14RL = gv5LR (V4L ↔ V4R , V5L ↔ V5R ), 2 2 2 1 d d gv8LL mt V5R = gv8LR = + C12 V1 V1 V4R C1d + C11 2 2 2 8π s
d − C0 + C1d + C2d mg˜ V5L , gv8RL 2
(B.56)
= gv5LR (V4L 1
q
3 V mg˜ D3 V4R 5L , 16π 2 gb1RR = gb1LL (V4L ↔ V4R , V5L ↔ V5R ), 4 4
= gb1LL 4
= gb1LR 4
(B.67) (B.68)
q 2 V4L V5R mg˜ D3 V4R , 16π 2
(B.69)
gb1RL = gb1LR (V4L ↔ V4R , V5L ↔ V5R ), 4 4 = gb4LL 4
2 (m D q V mg˜ V4L V4R g˜ 0 5L
q + mt (D0 16π 2
(B.70) q + D1
q + D2 )V5R )
,
(B.71)
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
29
gb4RR = gb4LL (V4L ↔ V4R , V5L ↔ V5R ), 4 4 = gb4LR 4
3 (m D q V mg˜ V4L g˜ 0 5L
q + mt (D0 16π 2
q + D1
(B.72) q + D2 )V5R )
,
(B.73)
= gb4LR (V4L ↔ V4R , V5L ↔ V5R ), gb4RL 4 4 gb9LL = gb9LR = 4 4
(B.74)
q 2 V D13 V4L V4R 5L , 16π 2
(B.75)
= gb9RR = gb9LL (V4L ↔ V4R , V5L ↔ V5R ), gb9RL 4 4 4 gb11LL = gb11LR = 4 4
2 V (m D q V V4L 4R g˜ 1 5L
q + mt (D1 16π 2
(B.76)
q + D11
= gb11RR = gb11LL (V4L ↔ V4R , V5L ↔ V5R ), gb11RL 4 4 4 gb13LL = gb13RL = 4 4
q + D12 )V5R )
,
(B.77) (B.78)
q 2 V D00 V4L V4R 5L , 8π 2
(B.79)
= gb13RR = gb13LL (V4L ↔ V4R , V5L ↔ V5R ), gb13LR 4 4 4
(B.80)
3 V mg˜ D3t V4R 5L gb1LL = − , 3 16π 2
(B.81)
= gb1LL (V4L ↔ V4R , V5L ↔ V5R ), gb1RR 3 3
(B.82)
2 V V mg˜ D3t V4R 4L 5R , 16π 2 = gb1LR (V4L ↔ V4R , V5L ↔ V5R ), gb1RL 3 3
gb1LR =− 3
= gb4LL 3
2 (m D t V mg˜ V4L V4R g˜ 0 5L 16π 2
− mt D2t V5R )
(B.83) (B.84) ,
(B.85)
= gb4LL (V4L ↔ V4R , V5L ↔ V5R ), gb4RR 3 3 gb4LR = 3
(B.86)
3 (m D t V t mg˜ V4L g˜ 0 5L − mt D2 V5R ) , 16π 2
(B.87)
= gb4LR (V4L ↔ V4R , V5L ↔ V5R ), gb4RL 3 3
(B.88)
t 2 V D13 V4L V4R 5L , 16π 2 = gb9RR = gb9LL (V4L ↔ V4R , V5L ↔ V5R ), gb9RL 3 3 3
gb9LL = gb9LR = 3 3
= gb11LR = gb11LL 3 3
2 V (m D t V V4L 4R t 12 5R 16π 2
− mg˜ D1t V5L )
(B.89) (B.90) ,
(B.91)
= gb11RR = gb11LL (V4L ↔ V4R , V5L ↔ V5R ), gb11RL 3 3 3
(B.92)
t 2 V D00 V4LV4R 5L 13RL = g = − , gb13LL b3 3 2 8π
(B.93)
= gb13RR = gb13LL (V4L ↔ V4R , V5L ↔ V5R ), gb13LR 3 3 3
(B.94)
gb2LL = 2
2 V (m V D v − m V D v ) V4L 4R t 5R 23 g˜ 5L 3 , 16π 2
(B.95)
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J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
gb2RR = gb2LL (V4L ↔ V4R , V5L ↔ V5R ), 2 2 = gb2LR 2
(B.96)
v ) − mt V5R D23 , 16π 2
3 (m V D v V4R g˜ 5L 3
(B.97)
= gb2LR (V4L ↔ V4R , V5L ↔ V5R ), gb2RL 2 2 gb5RR = gb5RL = 2 2
2 V (m V D v mg˜ V4L 4R g˜ 5R 0 16π 2
− mt V5L D2v )
(B.98) ,
= gb5LR = gb5RR (V4L ↔ V4R , V5L ↔ V5R ), gb5LL 2 2 2 2 V V Dv V4L 4R 5R 00 , 8π 2 = gb6LR (V4L ↔ V4R , V5L ↔ V5R ), gb6RL 2 2
gb6LR = 2
(B.99) (B.100) (B.101) (B.102)
3 V Dv V4L 5L 13 gb10LL = − , 2 16π 2
(B.103)
= gb10LL (V4L ↔ V4R , V5L ↔ V5R ), gb10RR 2 2
(B.104)
2 V V Dv V4L 4R 5L 13 = − , gb10LR 2 16π 2
(B.105)
= gb10LR (V4L ↔ V4R , V5L ↔ V5R ), gb10RL 2 2
(B.106)
2 V V Dv mg˜ V4L 4R 5L 1 , 16π 2 = gb12RR = gb12LL (V4L ↔ V4R , V5L ↔ V5R ), gb12RL 2 2 2
= gb12LR =− gb12LL 2 2
= gb14LL 2
3 V Dv V4L 5L 00 , 8π 2
= gb14LL (V4L ↔ V4R , V5L ↔ V5R ), gb14LR 2 2 2 V V (D0 + D1 + D2 + D3 )mg˜ V4L 4R 5L , 2 16π = gb3RR = gb3LL (V4L ↔ V4R , V5L ↔ V5R ), gb3LR 1 1 1
gb3LL = gb3RL = 1 1
2 V V D00 V4L 4R 5R , 8π 2 = gb5LR (V4L ↔ V4R , V5L ↔ V5R ), gb5RL 1 1
gb5LR =− 1
+ (D0 + D1 + D2 )mt V5R ) , 16π 2 = gb6RR = gb6LL (V4L ↔ V4R , V5L ↔ V5R ), gb6LR 1 1 1 = gb6RL =− gb6LL 1 1
2 (D m V mg˜ V4L V4R 0 g˜ 5L
3 V (D12 + D2 + D22 + D23 )V4L 5L , 2 16π gb7RR = gb7LL (V4L ↔ V4R , V5L ↔ V5R ), 1 1
= gb7LL 1
= gb7LR 1
2 V V (D12 + D2 + D22 + D23 )V4L 4R 5R , 16π 2
(B.107) (B.108) (B.109) (B.110) (B.111) (B.112) (B.113) (B.114) (B.115) (B.116) (B.117) (B.118) (B.119)
J.J. Liu et al. / Nuclear Physics B 705 (2005) 3–32
gb7RL = gb7LR (V4L ↔ V4R , V5L ↔ V5R ), 1 1 + (D12 + D2 + D22 )mt V5R ) , 16π 2 = gb8LL (V4L ↔ V4R , V5L ↔ V5R ), gb8RR 1 1 =− gb8LL 1
2 V (D m V V4L 4R 2 g˜ 5L
+ (D12 + D2 + D22 )mt V5R ) , 16π 2 gb8RL = gb8LR (V4L ↔ V4R , V5L ↔ V5R ), 1 1 gb8LR =− 1
3 (D m V V4R 2 g˜ 5L
3 V5L D00 V4L , 8π 2 gb14RR = gb14RL (V4L ↔ V4R , V5L ↔ V5R ). 1 1
=− gb14RL 1
31
(B.120) (B.121) (B.122) (B.123) (B.124) (B.125) (B.126)
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