Eur. Phys. J. C 9, 463–477 (1999) DOI 10.1007/s100529900027

THE EUROPEAN PHYSICAL JOURNAL C c Springer-Verlag 1999

Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD? Yu Zeng-Hui1,3 , Herbert Pietschmann1 , Ma Wen-Gan2,3 , Han Liang3 , Jiang Yi3 1 2 3

Institut f¨ ur Theoretische Physik, Universit¨ at Wien, A-1090 Vienna, Austria CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, P.R. China Department of Modern Physics, University of Science and Technology of China (USTC), Hefei, Anhui 230027, P.R. China Received: 30 September 1998 / Revised version: 21 January 1999 / Published online: 28 May 1999 Abstract. The QCD corrections to the top-quark pair production via both polarized and unpolarized gluon fusion in pp collisions are calculated in the minimal supersymmetric model (MSSM). We find that the MSSM QCD corrections can reach 4% and may be observable in future precise experiments. Furthermore, we studied CP violation in the MSSM. Our results show that the CP-violating parameter is sensitive to the masses of SUSY particles (it becomes zero when the c.m. energy is less than twice the masses of both the gluino and the stop quarks) and may reach 10−3 .

1 Introduction The minimal supersymmetric model (MSSM) [1] is one of the most interesting extensions of the standard model (SM). Therefore, testing the MSSM has attracted much interest. As is well known, the MSSM predicts supersymmetric (SUSY) partners to all particles expected by the SM, and searching for their existence is very important. Since the top quark was already found experimentally by the CDF and D0 Collaborations at Fermilab [2], we believe that more experimental events including the top quark will be collected in future experiments. That gives us a good chance to study the physics in top-quark pair production from pp or p¯ p collisions with more precise experimental results. Because of the heavy mass of the top quark, this process provides a test of the SM and possible signals of new physics at high energy. The dominant subprocesses of top-quark pair production in pp or p¯ p colliders are quark–antiquark annihilation and gluon–gluon fusion. The lowest order of those two subprocesses has been studied in [3]. There it was found that the former subprocess (q q¯ annihilation) is√more dominant in p¯ p collisions when the c.m. energy ( s) is near the threshold value 2mt , whereas the subprocess via gg fusion will become increasingly important with increasing c.m. energy, and can become the most dominant process when the c.m. energy is much larger than 2mt . In [4], the QCD corrections to top-quark pair production in p¯ p collisions have been studied in the frame of the SM. It may seem natural that the QCD corrections of ?

Supported in part by a Committee of the National Natural Science Foundation of China and Project IV.B.12 of a scientific and technological cooperation agreement between China and Austria.

those processes in the frame of the MSSM are important for distinguishing those two models. Recently, the SUSY QCD corrections to top-pair production via q q¯ annihilation were presented [5]. The SUSY QCD corrections via unpolarized gluon–gluon fusion were given by Li et al. [6]. It is obvious that the correction from the SUSY QCD is related to the masses of the top quark and of SUSY particles. Assuming the SUSY breaking scale to be at about 1 TeV, the masses of SUSY particles would be smaller than 1 TeV. Therefore we can hope that corrections from SUSY particles are significant, since the heavy mass of the top quark (mt = 175.6 ± 5.5 GeV (world average)) may be comparable to some of the light SUSY particle masses. Therefore the SUSY QCD correction would indirectly give us some significant information about the existence of SUSY particles. Recently, the spin structure of the nucleon has been intensively studied by polarized deep-inelastic-scattering experiments at CERN and SLAC. Knowledge about this allows us to find a clear signal beyond the SM, if we collect enough events in the process of top-quark pair production from polarized pp or p¯ p collisions. In the SM QCD there is no CP-violation mechanism, whereas in the SUSY QCD the situation may be different. If we introduce a phase angle of quark SUSY partners, we can get CP violation in the MSSM QCD [7]. Once we get enough statistics of top-quark pairs from pp or p¯ p colliders at higher energy, it will be possible to test CP violation. On the other hand, the spin-dependent parton distributions can be obtained from their polarized structure function data given in [8, 10, 11]. There it is found that the shape of polarized gluon and quark distributions in the nucleon depends on its polarization. Therefore the CP-violation effects through the

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Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD

process of top-quark pair production via gg fusion may be observed in polarized pp or p¯ p collisions. In this work we concentrate on the SUSY QCD corrections to the process pp → gg → tt¯X both in polarized and unpolarized colliding beams. In Sect. 2 we give the treelevel contribution to the subprocess gg → tt¯. In Sect. 3 we give the analytical expressions of the SUSY QCD corrections to gg → tt¯. In Sect. 4 the numerical results of the subprocess gg → tt¯ and the process pp → gg → tt¯X are presented. The conclusion is given in Sect. 5, and some details of the expressions are listed in the Appendix.

3 SUSY QCD corrections (non-SM) to the subprocess gg → tt¯

2 The tree-level subprocess

where

The graphical representation of the process g(λ1 , k1 ) g(λ2 , k2 ) → t(p1 )t¯(p2 ) is shown in Fig. 1a. The Mandelstam variables are defined as usual: sˆ = (p1 + p2 )2 = (k1 + k2 )2 ,

(2.1)

tˆ = (p1 − k1 )2 = (k2 − p2 )2 ,

(2.2)

u ˆ = (p1 − k2 )2 = (k1 − p2 )2 ,

(2.3)

so sˆ + tˆ + u ˆ = 2m2t . The amplitude of tree-level diagrams with polarized gluons can be written as in [3] (a, b are color indices of external gluons, i, j are colors of external top quarks, and T a = λa /2 are the Gell-Mann matrices): (l)

ui (p1 )Γ (l) vj (p2 ), M0 = gs2 µ,a (λ1 , k1 )ν,b (λ2 , k2 )¯ (2.4) (l = s, t, u), with Γ (s) =

Tijc fabc [(/ k1 − / k2 )gµν + (2k2 + k1 )µ γν s −(2k1 + k2 )ν γµ ], (2.5)

a b −iTim Tmj γµ (/ k2 − / p2 + mt )γν , t − m2t b a −iTim Tmj = γν (/ k1 − / p2 + mt )γµ . u − m2t

Γ (t) = Γ

(u)

(2.6) (2.7)

We chose a form in which only physical polarizations of gluons remained:  δλ1 ,λ2 nµ kiν + nν kiµ −g µν +  (λ1 , ki ) (λ2 , ki ) = 2 n · ki  2 µ ν kiσ nρ n ki ki − + iλ1 σµρν , (2.8) 2 (n · ki ) n · ki µ∗

ν

where n = k1 + k2 and λ1,2 = ±1. From that we can get the cross section at the tree level with both polarized and unpolarized gluons.

3.1 Relevant Lagrangian in the MSSM The difference between the MSSM QCD and the SM QCD corrections stems from the interactions of SUSY particles. Thus we can divide SUSY QCD corrections into a standard and a non-standard part. The Lagrangian density of the non-SM part of the SUSY QCD interaction is written as: (3.1.1) L = L1 + L2 + L3 + L4 ,

a (˜ qLj ∂µ q˜Lk − q˜Lk ∂µ q˜Lj ) + (L → R), (3.1.2) L1 = −igs Aµa Tjk

√ a ¯ (g˜a PL q k q˜Lj∗ + q˜j PR g˜a q˜Lk L2 = − 2gˆs Tjk j∗ k −g¯˜a PR q k q˜R − q¯j PL g˜a q˜R ), L3 = 2i gs fabc g¯˜a γµ g˜b Acµ , ∗ qL∗ q˜L + q˜R q˜R ) L4 = 16 gs2 Aaµ Aµa (˜ 1 2 a µb i∗ c j i∗ c j qL Tij q˜L + q˜R Tij q˜R ). + 2 gs dabc Aµ A (˜

(3.1.3) (3.1.4) (3.1.5)

q stands for quark, q˜ for the corresponding squark, g˜ for gluino, and PL and PR for left and right helicity projections, respectively. The mixing between the left- and righthanded stop quarks t˜L and t˜R can be very large due to the large mass of the top quark, and the lightest scalar topquark mass eigenstate t˜1 can be much lighter than the top quark and all the scalar partners of the light quarks. Therefore the left–right mixing for the SUSY partners of the top quark plays an important role. Here we only considered the SUSY QCD effect from the stop quark, because we assume that other scalar SUSY quarks are much heavier than the stop quark and hence decoupled. Furthermore, we introduce the phase angle φA in the stop mixing matrix. Defining θ as mixing angle of stop quark, we have −iφA (3.1.6) t˜L = e 2 (t˜1 cos θ + t˜2 sin θ), t˜R = e

iφA 2

(−t˜1 sin θ + t˜2 cos θ),

(3.1.7)

where we suppose mt˜1 ≤ mt˜2 . 3.2 Analytical results of the MSSM QCD corrections The one-loop SUSY QCD correction diagrams are shown in Fig. 1b. In the following we only present the amplitude expressions of the s-channel and the t-channel. The amplitude of the u-channel can be obtained from the tchannel expressions by the following variable exchanges: t ↔ u, k1 ↔ k2 , aµ (k1 ) ↔ bν (k2 ) and T a ↔ T b . The oneloop diagrams can be divided into three groups: the selfenergy diagrams of the gluon and the top quark shown in Fig. 1b.1; gtt¯ and ggg vertex correction diagrams shown in

Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD

εaµ(k1) εbν(k2)

t(p1) t(p2)

g (a)

465

t t

g t g

t

g g

g

(b-1)

g

t

g

t

(b-3) Fig. 1b.2; and box diagrams shown in Fig. 1b.3. The ultraviolet divergence is controlled by dimensional regularization (n = 4 − ). The strong coupling-constants are renormalized by using the modified minimal subtraction (MS) scheme at charge-renormalization scale µR . This scheme violates SUSY explicitly, and the q q˜g˜ Yukawa coupling gˆs , which should be the same with the qqg gauge coupling gs in supersymmetry, takes a finite shift at one-loop or-

(b-2)

Fig. 1. Feynman diagrams at the tree level and one-loop level in the SUSY QCD for the gg → tt¯ subprocess. Figure 1(a) Tree-level diagrams. Figure 1(b.1) Self-energy diagrams (for top quark and gluon). Figure 1(b.2) Vertex diagrams (including tri-gluon and gluon–top–top interactions). Figure 1(b.3) Box diagrams (only t-channel). Dashed lines represent t˜1 , t˜2 in Fig. 1(b)

der. Therefore we take this shift to be between gˆs and gs as shown in (3.2.1) into account in our calculation, so the physical amplitudes are independent of the renormalization scheme, and we subtract the contribution of the false, non-supersymmetric degrees of freedom (also called  scalars) [12]:   αs 2 1 (3.2.1) ( CA − CF ) , gˆs = gs 1 + 4π 3 2

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Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD

where CA = 3 and CF = 4/3 are the Casimir invariants of the SU(3) gauge group. The heavy particles (top quarks, gluino, stop quarks, etc.) are removed from the µR evolution of αs (µ2R ); then they are decoupled smoothly when momenta are smaller than their masses [13]. We define masses of heavy particles as pole masses. The renormalized amplitude corresponding to all SUSY QCD one-loop corrections (as shown in Fig. 1) can be split into the following components:

Here we define ˆkl (p) = CF (HL /pPL + HR /pPR − H S PL − H S PR )δkl , Σ L R (3.3.7) with g ˆ2 HL = 8πs2 x1 x3 B1 [p, mg˜ , mt˜1 ] (3.3.8) +(mt˜1 → mt˜2 , xi → yi ) + 12 (δZL + δZL† ),

HR =

(3.2.2)

δM = δMs + δMv + δMbox + δMd ,

µ2

µ2

+ 16 log( mR2 ) + t

1 2

˜1 t

1 24

µ2

˜2 t

log( mR2 )]. g ˜

The amplitude of self-energy diagrams δMs (Fig. 1b.1) can be decomposed into δMsg (gluon self-energy) and δMsq (top-quark self-energy), i.e. δMs = δMsg + δMsq g(s) g(t) g(u) q(t) q(u) = δMs + δMs +δMs + δMs + δMs . (3.3.1) g(s) g(t) g(u) are for the s-, The amplitudes δMs , δMs and δMs t- and u-channel, respectively. They can be expressed as: =

(s) 1 2 2 M0 [Π(k1 )

+ Π(k22 ) + 2Π(s)],

g(t)

= 12 M0 [Π(k12 ) + Π(k22 )],

g(u)

= 12 M0 [Π(k12 ) + Π(k22 )],

δMs δMs

(3.3.2)

(t)

(3.3.3)

(u)

(3.3.4)

where M0 is the tree-level amplitude defined in (2.4). s ¯ ¯ ¯ Π(k 2 ) = − α 4π (TF (B0 + 4B1 + 4B21 )[k, mt˜1 , mt˜1 ] ¯ ¯ ¯ +TF (B0 + 4B1 + 4B21 )[k, mt˜2 , mt˜2 ] ¯1 + B ¯21 )[k, mg˜ , mg˜ ] − 1 CA ). −4CA (B 3 (3.3.5) where CF = 4/3, TF = 1/2 and CA = 3 are invariants in the SU(3) color group, and Bi and Bij are Passarino– Veltman two-point functions [14,15]. The definitions of ¯1 and B ¯21 are listed in Appendix A. The amplitude ¯0 , B B q(t) δMs is written as:

q(t)

δMs

=

HRS =

(3.2.3)

3.3 Self-energy corrections to the amplitude

g(s) δMs

HLS =

µ2

log( mR2 )

a b −igs2 Tik Tlj (t−m2t )2

µ,a (k1 )ν,b (k2 )¯ ui (p1 )γµ h i ˆkl (k2 − p2 ) (/ k2 − / p2 + mt ) Σ p2 + mt )γν vj (p2 ). ×(/ k2 − /

g ˆs2 ˜ B0 [p, mg ˜ , mt˜1 ] 8π 2 x2 x3 mg

(3.3.10)

+(mt˜1 → mt˜2 , xi → yi ) † ) + δmt , + 12 mt (δZL + δZR g ˆs2 ˜ B0 [p, mg ˜ , mt˜1 ] 8π 2 x1 x4 mg

(3.3.11)

+(mt˜1 → mt˜2 , xi → yi ) + 12 mt (δZR + δZL† ) + δmt ,

where we abbreviate φ = φA , x1 = cos θe−iφ , x2 = sin θeiφ , x3 = cos θeiφ , x4 = sin θe−iφ , y1 = sin θe−iφ , y2 = − cos θeiφ , y3 = sin θeiφ , y4 = − cos θe−iφ , and θ is the mixing angle of stop quarks (see (3.1.6) and (3.1.7)). The explicit expressions of the top-quark wave-function renormalization constants have the following forms: g ˆ2

m

δZL = − 8πs2 (x1 x3 Re[B1 ] − mg˜t (x1 x4 − x2 x3 )Re[B0 ] 0 + m2t (x1 x3 + x2 x4 )Re[B1 ] 0 − mt mg˜ (x2 x3 + x1 x4 )Re[B0 ])[p, mg˜ , mt˜1 ]|p2 =m2t , (3.3.12) g ˆ2

0

δZR = − 8πs2 (x2 x4 Re[B1 ] + m2t (x1 x3 + x2 x4 )Re[B1 ] 0 − mt mg˜ (x2 x3 + x1 x4 )Re[B0 ])[p, mg˜ , mt˜1 ]|p2 =m2t , (3.3.13) g ˆ2

δmt = 16πs 2 ((x1 x3 + x2 x4 )mt Re[B1 ] − (x2 x3 + x1 x4 )mg˜ Re[B0 ])[p, mg˜ , mt˜1 ]|p2 =m2t . (3.3.14) 0 We use the following abbreviations: Bi,ij [p, m1 , m2 ] = ∂Bi,ij [p, m1 , m2 ]/∂p2 . 3.4 Vertex-corrections to the amplitude The amplitudes for vertex diagrams can be expressed as: (l)

ui (p1 )Λ(l) vj (p2 ), δMv = gs µ,a (k1 )ν,b (k2 )¯ where

h

i

c Tij (3g) Λµνρ (k1 , k2 ) s − fabc µν s [(k1 − k2 )ρ gh

Λ(s) = − (3.3.6)

(3.3.9)

+(mt˜1 → mt˜2 , xi → yi ) † ), + 12 (δZR + δZR

where δMs , δMv , δMbox and δMd are the one-loop amplitudes corresponding to the self-energy, vertex, and box correction diagrams and the decoupling part, respectively. The δMd stems from the decoupling of the heavy flavors from the running strong coupling, and is given explicately by (see also [12, 13]): 1 log( mR2 ) + δMd = M0 ( αsπ(µ) )[ 24

g ˆs2 ˜ , mt˜1 ] 8π 2 x2 x4 B1 [p, mg

(l = s, t, u), (3.4.1)

γρ + (2k2 + k1i)µ gνρ

−(2k1 + k2 )ν gµρ ] Λcρ,(ij) (p1 , p2 )

(3.4.2)

Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD

and

i n h −i b k2 − / p2 + mt )γν Tmj Λaµ,(im) (p1 , k1 − p1 ) (/ Λ(t) = t−m 2 t h io a +Tim γµ (/ k2 − / p2 + mt ) Λbν,(mj) (k2 − p2 , p2 ) . (3.4.3) (3g) a The functions Λµνρ and Λµ,(ij) are listed in Appendix B. 3.5 Box corrections to the amplitude The box-diagram corrections in the t-channel (Fig. 1b.3) are given as follows: (t)

(t1)

ui (p1 )((T c T a T b T c )ij Fµν δMbox = 2gs2 µ,a (k1 )ν,b (k2 )¯ (t2) −ifbcd (T c T a T d )ij Fµν (t3) −facm fbmd (T c T d )ij Fµν (t4) −[T c (T a T b + T b T a )T c ]ij Fµν )vj (p2 ), (3.5.1) where fabc is defined as [T a , T b ] = ifabc T c . The form fac(ti) tors Fµν (i = 1–4) correspond to the kernel of the four Feynman diagrams in Fig. 1b.3 respectively, and are given explicitly in Appendix C. 3.6 Total cross section Collecting all terms in (3.2.2), we can get the total cross section: σ(λ1 , λ2 ) = σ0 (λ1 , λ2 )(1 + δσ(λ1 , λ2 )) R t+ P 1 dt spins [|M0 |2 + 2Re(M0† δM )], = 16πs 2 t− (3.6.1) p where t± = (m2t − (1/2)s) ± (1/2)sβt , βt = 1 − 4m2t /s, and the spin sum is performed only over the final topquark pair when we considered polarized gluons.

4 Numerical results We write σ ˆ0 for the Born cross section and σ ˆ for the cross section including one-loop SUSY QCD corrections of subprocess gg → tt¯, and define its relative correction σ0 . For polarized gluon fusions, σ ˆ++ , as δˆ = (ˆ σ−σ ˆ0 )/ˆ ˆ+− are the cross sections with positive, negσ ˆ−− and σ ative and mixed polarization of the gluons, respectively. In order to inspect the CP-violating effects we introduce the CP-violation parameter for the subprocess defined by σ++ − σ ˆ−− )/(ˆ σ++ + σ ˆ−− ). The possible SUSY ξˆCP = (ˆ QCD effects in gg → tt¯ should be observed in pp colliders. By analogy we can also define the relative correction and the CP-violating parameter for the process pp → gg → tt¯x as δ = (σ − σ0 )/σ0 and ξCP = (σ++ − σ−− )/(σ++ + σ−− ), respectively. The SUSY QCD contribution to the process p(P1 , x)p(P2 , y) → gg → tt¯X (x and y are polarizations of protons) can be obtained by convoluting the subprocess with gluon distribution functions; R (4.1) σ (ˆ s, αs (µ)), σ(s) = dx1 dx2 G(x1 , Q)G(x2 , Q)ˆ

467

with k1 = x1 P1 , k2 = x2 P2 and τ = x1 x2 = sˆ/s. G(xi , Q) (i = 1, 2) are gluon distribution functions of protons. We take Q = µR = 2mt . In order to get results of top-quark pair production from polarized pp collisions, we need to consider the polarized gluon distributions in protons. The cross sections of polarized pp → gg → tt¯X can be written as R σ(x, y) = Σλ1 ,λ2 =± dx1 dx2 Gxλ1 (x1 , Q) (4.2) σλ1 ,λ2 (ˆ s, αs (µ)), ×Gyλ2 (x2 , Q)ˆ where x and y are the polarizations of incoming protons, and λ1 and λ2 are the polarizations of gluons inside protons. Gxλ1 (x, Q), Gyλ2 (x, Q) = G± (x, Q) for equal (+) and opposite (−) polarizations, where G+ (x, Q) and G− (x, Q) are polarized gluon distribution functions in the proton. We used the unpolarized proton structure functions of Gl¨ uck et al. [9] in our numerical calculations. For the polarized proton structure functions, we use the evolution equations of Gl¨ uck et al. [10] with input parameters from the paper of Stratmann et al. [11] (next-to-leading order). Since the structure functions belong to the least certain inputs of our calculation, we checked the result against another set, i.e. the polarized structure functions G± (x, Q) of Brodsky et al. [8] (using leading order only). This tests the stability of our results against the particular form of the input structure functions. The two different sets of input are compared in Fig. 2, which gives the relative √ SUSY QCD correction (δ) and ξCP versus c.m. energy s for the process pp → gg → tt¯X. Though the SUSY QCD corrections from the two sets of structure functions are not very different for δ, there is some noticeable change for ξCP . Because ξCP depends strongly on the c.m. energy of the subprocess gg → tt¯ (shown in Fig. 3b), a small modification of the structure functions may lead to a large change of ξCP . Thus we can infer that the NLO-QCD calculation is required and the precise numerical prediction does depend on the reliability of the structure functions. The SUSY QCD relative corrections are about 2–4% and decrease with increasing c.m. energy (see Fig. 2). These correction effects are within reach of future precision experiments and provide a possible discrimination of the SM and the MSSM effects. From Fig. 2c we can see that the CP-violation parameter ξCP can be 10−3 . Therefore, CP violation in this process stemming from the SUSY QCD can in principle be tested in future precision experiments. That would help us to learn more about the sources of CP violation. In order to explore the effects of the SUSY QCD correction for future arrangements of optimal experimental conditions, we also investigate the subprocess gg → tt¯. The relative SUSY QCD correction and CP-violating √ parameter versus c.m. energy ( sˆ) for different polarization gluons are plotted in Fig. 3a–c with mg˜ = 200 GeV, mt˜1 = 250 GeV, mt˜2 = 450 GeV, and θ = φ = 45◦ . In Fig. 3a, δˆ++ and δˆ−− are shown by a solid line and a dashed line, respectively. ξˆCP as a function of c.m. energy √ is depicted in Fig. 3b, and δˆ+− as a function of sˆ is

468

Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD

Fig. 2. a Relative corrections to polarized and unpolarized ¯ cross sections √ of the tt production process in pp colliders as a function of s with input structure functions of Brodsky et al. [8] (LO). b Relative corrections to polarized and unpolar¯ ized cross sections √of the tt production process in pp colliders as a function of s with input structure functions of Gl¨ uck et al. [9–11] (NLO). In both a and b, the solid line is for the MSSM QCD correction with unpolarized protons, the dashed line is for the MSSM QCD correction with proton(+) proton(+) polarization, the dotted line is for the MSSM QCD correction with proton(−) proton(−) polarization, and the dot-dashed line is for the MSSM QCD correction with proton(+) proton(−) polarization. c The CP-violating parame√ ter ξCP as a function of s. The solid line is for input structure functions of Gl¨ uck et al. (NLO), the dashed line is for input structure functions of Brodsky et al. (LO) mg˜ = 200 GeV, mt˜1 = 250 GeV, mt˜2 = 450 GeV and θ = φ = 45◦

plotted in Fig. 3c. Each curve in Fig. 3a has an obvious peak near the position of the threshold of top pair production. That large enhancement is the combined effect of the √ threshold, when sˆ is just√larger than 2mt = 350 GeV, and the resonance when sˆ ∼ 2m √g˜ = 400 GeV. The sˆ = 900 GeV, where small spikes around the position of √ sˆ ∼ 2mt˜2 = 900 GeV, also shows the resonance effect. Although Fig. 3a shows that δˆ++ and δˆ−− approach equal values when the c.m. energy is far beyond its threshold value 2mt , the quantitative difference between δˆ+− and δˆ++ still exists in the whole energy range plotted in these figures. Figure 3b also shows that ξˆCP will be zero if the c.m. energy is√below the threshold of SUSY particles in the loop (i.e. sˆ ≤ 2mg˜ = 400 GeV in Fig. 3b. This is reasonable because only beyond this point we can have

absorptive terms which give contributions to ξˆCP . ξˆCP √ has an obvious resonance √ effect in the regions around sˆ ∼ 2mg˜ = 400 GeV and sˆ ∼ 2mt˜i (i = 1, 2) = 500 GeV, 900 GeV. We also find that the two stop quarks give opposite contributions to ξˆCP , and when their masses √ are degenerate ξˆCP will vanish. When the c.m. energy sˆ is larger than 1 TeV, ξˆCP will be near zero, because the contributions from the two stop quarks will cancel each other. Therefore a quantitative strong change of ξˆCP as a function of c.m. energy can be an indication for the signals of stop quarks and gluino. σ ˆ (±, ±) and ξˆCP as functions of mg˜ are shown √ in Fig. 4a and b, respectively. In Fig. 4 we take sˆ = 500 GeV, mt˜1 = 100 GeV, mt˜2 = 450 GeV, and θ = φ = 45◦ . We can see from Fig. 4b that ξˆCP changes its sign when mg˜ is near mt = 175 GeV. The curves in Fig. 4a, b

Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD

469

Fig. 3. a Relative corrections to the cross section √ of the tt¯ production subprocess, δˆ±± as a function of sˆ. The solid line is for the MSSM QCD correction with gluon(+) gluon(+) polarization and the dashed line is for the MSSM QCD correction with gluon(−) gluon(−) polarization. b The CP-violating parameter ξˆCP of the subprocess as a function of √ sˆ. c Relative corrections √to the cross section of the subprocess δˆ+− as a function of sˆ. mg˜ = 200 GeV, mt˜1 = 250 GeV, mt˜2 = 450 GeV and θ = φ = 45◦

√ again show the resonance effect when sˆ ∼ 2mg˜ = 500 GeV. Note that for each line there is a steep change of the value of σ ˆ (±, ±) or ξˆCP around the position of mg˜ = 250 GeV. Dependences of the relative correction δˆ±± and ξˆCP for the subprocess gg → tt¯ on mt˜1 are plotted in Fig. 5a, b. δˆ±± and ξˆCP as functions of mt˜2 are shown in Fig. 6a and b, respectively. In all parts of √ Figs. 5 and 6, we take the common parameter set with sˆ = 500 GeV, mg˜ = 200 GeV and θ = φ = 45◦ . In Fig. 5 we set mt˜2 = 450 GeV, whereas mt˜1 = 100 GeV in Fig. 6. We find that ξˆCP in fact increases with mass splitting of stop quarks (i.e. mt˜2 − mt˜1 ), and when mt˜1 = mt˜2 , ξˆCP is equal to zero. The res√ onance effect of stop quarks, when sˆ ∼ 2mt˜i (i = 1, 2), is superimposed on the curves in Fig. 5a, b and Fig. 6a, b

around the positions of mt˜1 = 250 GeV in Fig. 5a, b and mt˜2 = 250 GeV in Fig. 6a, b. Around those points the relatively sharp changes of the values of ξˆCP and the relative corrections are shown in these figures. Finally, the dependence of δˆ±± and ξˆCP √ on the phase φ is shown in Fig. 7a, b. In Fig. 7, we take sˆ = 500 GeV, mg˜ = 200 GeV, θ = 45◦ and mt˜1 = 150 GeV. We find that ξˆCP is directly proportional to sin (2φ) and reaches its maximum value when φ = π/4.

5 Conclusion In this work we have studied the one-loop supersymmetric QCD corrections to the subprocess gg → tt¯ and the process pp → gg → tt¯X. The calculations show that the

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Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD

Fig. 4. a Cross section of the tt¯ production subprocess via gg fusion, σ ˆ±± as a function of mg˜ . The solid line is for the MSSM QCD correction with gluon(+) gluon(+) polarization and the dashed line is for the MSSM QCD correction with gluon(−) gluon(−) polarization. b The CP-violating parameter ξˆCP of the subprocess as a function of mg˜ . mt˜1 = 100 GeV, √ mt˜2 = 450 GeV, sˆ = 500 GeV and θ = φ = 45◦

SUSY QCD effects are significant. The absolute values of the corrections are about 2–4%, so they may be observable in future precision experiments. Furthermore, we find ξCP depends strongly on the masses of SUSY particles and can reach 10−3 when we take plausible SUSY parameters. The results show that there is an obvious difference between the corrections for the protons polarized with parallel spin and those with anti-parallel spin. Hence there is a possibility to study spin dependence in the frame of the MSSM QCD.

Fig. 5. a Relative corrections to the cross section of the tt¯ production subprocess via gg fusion, δˆ±± as a function of mt˜1 . The solid line is for the MSSM QCD correction with gluon(+) gluon(+) polarization and the dashed line is for the MSSM QCD correction with gluon(−) gluon(−) polarization. b The CP-violating parameter ξˆCP of the subprocess as a function √ of mt˜1 . mg˜ = 200 GeV, mt˜2 = 450 GeV, sˆ = 500 GeV and θ = φ = 45◦

We also presented and discussed the results of the subprocess gg → tt¯. We find that, when the c.m. energy passes through the value 2mg˜ or 2mt˜i (i = 1, 2), the value of the CP-violating parameter ξˆCP changes considerably. If the c.m. energy is less than both 2mg˜ and 2mt˜i (i = 1, 2), ξˆCP will be zero. If in future experiments a sharp change in √ ξˆCP is found with sˆ running from low c.m. energy to high c.m. energy, it would be interpreted as a signal of

Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD

471

Fig. 6. a Relative corrections to the cross section of the tt¯ production subprocess via gg fusion, δˆ±± as a function of mt˜2 . The solid line is for the MSSM QCD correction with gluon(+) gluon(+) polarization and the dashed line is for the MSSM QCD correction with gluon(−) gluon(−) polarization. b The CP-violating parameter ξˆCP of the subprocess √ as a function of mt˜2 . mg˜ = 200 GeV, mt˜1 = 100 GeV and sˆ = 500 GeV and θ = φ = 45◦

Fig. 7. a Relative corrections to the cross section of the tt¯ production subprocess via gg fusion, δˆ±± as a function of φ. The solid line is for the MSSM QCD correction with gluon(+) gluon(+) polarization and the dashed line is for the MSSM QCD correction with gluon(−) gluon(−) polarization. b The CP-violating parameter ξˆCP of the subprocess as a function √ of φ. mg˜ = 200 GeV,mt˜1 = 150 GeV, mt˜2 = 450 GeV, sˆ = ◦ 500 GeV and θ = 45

SUSY particles. Furthermore, because the CP-violating parameter ξˆCP is sensitive to the mass of the gluino (as shown in Fig. 4b) and the mass splitting of stop quarks mt˜2 − mt˜1 (as shown in Figs. 5 and 6), we can also get information on SUSY particles from precise measurements of ξˆCP .

Acknowledgements. The authors would like to thank Prof. A. Bartl for useful discussions and comments. One of the authors, Yu Zeng-Hui, would like to thank Prof. H. Stremnitzer for his help.

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Appendix

where

{pk}µν = pµ kν + kµ pν ,

A. Loop integrals

{pkl}µνρ = pµ kν lρ + lµ pν kρ + kµ lν pρ ,

We adopt the definitions of two-, three- and four-point one-loop Passarino–Veltman integral functions of [14, 15]. 1. The two-point integrals are: Z (2πµ)4−n dn q {B0 ; Bµ ; Bµν }(p, m1 , m2 ) = iπ 2 {1; qµ ; qµ qν } , (A.1) × 2 [q − m21 ][(q + p)2 − m22 ]

The numerical calculation of the vector and tensor loop integral functions can be traced back to the four scalar loop integrals A0 , B0 , C0 and D0 in [14, 15] and the references therein.

The function Bµ is proportional to pµ :

B. Vertex corrections

Bµ (p, m1 , m2 ) = pµ B1 (p, m1 , m2 ).

(A.2)

Similarly we define: (A.3) Bµν = pµ pν B21 + gµν B22 . ¯ ¯ ¯ We define B0 = B0 − ∆, B1 = B1 + (1/2)∆ and B21 = B21 − (1/3)∆, with ∆ = 2/ − γ + log(4π),  = 4 − n. µ is the scale parameter. 2. The three-point integrals are: (2πµ)4−n {C0 ; Cµ ; Cµν ; Cµνρ }(p, k, m1 , m2 , m3 ) = − iπ 2 Z {1; qµ ; qµ qν ; qµ qν qρ } . × dn q 2 [q − m21 ][(q + p)2 − m22 ][(q + p + k)2 − m23 ] (A.4) We define form factors as follows: Cµ = pµ C11 + kµ C12 , Cµν = pµ pν C21 + kµ kν C22 + (pµ kν + kµ pµ )C23 +gµν C24 , Cµνρ = pµ pν pρ C31 + kµ kν kρ C32 +(kµ pν pρ + pµ kν pρ + pµ pν kρ )C33 +(kµ kν pρ + pµ kν kρ + kµ pν kρ )C34 × + (pµ gνρ + pν gµρ + pρ gµν )C35 +(kµ gνρ + kν gµρ + kρ gµν )C36 . (A.5)

{pg}µνρ = pµ gνρ + pν gµρ + pρ gµν .

The 3-gluon-vertex can be written as (a, b and c are the color indices of the external gluons): n h i (3g) (1) ig 3 Λµνρ (k1 , k2 ) = 16πs2 Tr(T b T c T a ) Λµνρ (k1 , k2 ) io h (2) +if cmn f anl f blm Λµνρ (k1 , k2 ) ; (B.1) (1) (2) the vertex functions Λµνρ , Λµνρ are expressed as follows: (a)

(a)

(a)

Λµνρ (k1 , k2 ) = f1 gµρ k1ν + f2 gµν k1ρ (a) (a) +f3 gνρ k2µ + f4 gµν k2ρ (a) (a) +f5 k1ν k1ρ k2µ + f6 k1ν k2ρ k2µ +(mt˜1 → mt˜2 , xi → yi ), (1)

(B.2)

(2)

where a = 1, 2, and the fi , fi are given in terms of the Passarino–Veltman functions with internal stop lines (1) Cij (= Cij [−k1 , −k2 , mt˜1 , mt˜1 , mt˜1 ]) and internal gluino (2)

lines Cij (= Cij [−k1 , −k2 , mg˜ , mg˜ , mg˜ ]). For simplicity, we abbreviate the definite part of C integral functions (us(a) (a) ing the definitions of [14, 15]) as follows: C¯24 = C24 − 14 ∆, (a) (a) (a) (a) 1 ∆ (a = 1, 2): C¯35 = C35 + 16 ∆, C¯36 = C35 + 12 (1)

= −8C¯24 − 8C¯35 ,

(1)

(1) (1) = −4C¯24 − 8C¯35 ,

f1

3. The four-point integrals are:

f2

{D0 ; Dµ ; Dµν ; Dµνρ ; Dµνρα }(p, k, l, m1 , m2 , m3 , m4 ) = Z (2πµ)4−n dn q{1; qµ ; qµ qν ; qµ qν qρ ; qµ qν qρ qα } iπ 2  × [q 2 − m21 ][(q + p)2 − m22 ] × [(q + p + k)2 − m23 ][(q + p + k + l)2 − m24 ] .(A.6)

(1)

(1)

f3 (1)

f4 (1)

f5

Again we define form factors of D functions: Dµ = pµ D11 + kµ D12 + lµ D13 , Dµν = pµ pν D21 + kµ kν D22 + lµ lν D23 + {pk}µν D24 +{pl}µν D25 + {kl}µν D26 + gµν D27 , Dµνρ = pµ pnu pρ D31 + kµ knu kρ D32 + lµ lnu lρ D33 +{kpp}µνρ D34 + {lpp}µνρ D35 + {pkk}µνρ D36 +{pll}µνρ D37 + {lkk}µνρ D38 + {kll}µνρ D39 +{pkl}µνρ D310 + {pg}µνρ D311 + {kg}µνρ D312 +{lg}µνρ D313 , (A.7)

(A.8)

(1)

f6

(1)

= −8C¯36 , (1)

(B.3)

(1) (1) = −4C¯24 − 8C¯36 , (1)

(1)

(1)

= 4C12 + 12C23 + 8C33 , (1)

(1)

(1)

(1)

= 4C12 + 8C22 + 4C23 + 8C34 ,

and (2)

f1

(2) (2) (2) = −8m2g˜ C0 − 4m2g˜ C11 − 16C¯24 (2) (2) (2) +12C24 − 8C¯35 + 6C35 (2) (2) (2) +8k1 · k2 C12 + 16k1 · k2 C23 + 8k1 · k2 C33 , (2)

f2

(2) (2) (2) = −4m2g˜ C11 − 8C¯35 + 6C35 (2) (2) +8C23 k1 · k2 + 8C33 k1 · k2 ,

Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD (2)

f3

(2)

f4

(2) (2) (2) = 4m2g˜ C0 − 4m2g˜ C12 + 8C¯24 (2) (2) (2) −6C24 − 8C¯36 + 6C36 (2) (2) +8k1 · k2 C22 + 8k1 · k2 C34 ,

and (B.4)

(2L)

h1

(2) (2) (2) = −4m2g˜ C0 − 4m2g˜ C12 − 8C¯24 (2) (2) (2) +6C24 − 8C¯36 + 6C36 (2) (2) (2) +8k1 · k2 C12 + 8k1 · k2 C22 + 8k1 · k2 C23 (2) +8k1 · k2 C34 , (2)

f5 (2)

f6

(2)

(2)

(4)

(2)

(2)

(1L) gs g ˆs2 a − 16π (p1 , p2 )PL 2 Tij {(2CF − CA )(Λµ (1R) +Λµ (p1 , p2 )PR ) (2L) +CA (Λµ (p1 , p2 )PL (2R) +Λµ (p1 , p2 )PR )} (CT) +(mt˜1 → mt˜2 , xi → yi ) + Λµ .

(n)

(n)

(n)

(n)

Λµ (p1 , p2 ) = h1 γµ + h2 p1µ + h3 p2µ + h4 /p1 p1µ (n) (n) +h5 / p1 p2µ + h6 / p2 p1µ (n) (n) +h7 / p2 p2µ + h8 γµ / p1 (n) (n) +h9 γµ / p2 + +h10 γµ / p1 / p2 . (B.6) We define (3)

(3)

C0 , Cij = C0 , Cij [−p1 , −p2 , mt˜1 , mg˜ , mt˜1 ] and (4) (4) C0 , Cij

(n)

as follows (i = 1, 2, · · · , 10):

(1L)

h1

= x2 x3 mg˜ (C0 + 2C11 ),

(1L)

= x2 x3 mg˜ (C0 + 2C12 ),

h3 (1L)

h4 (1L)

h5

(3)

= −2x2 x4 C24 ,

(1L)

h2

(3)

(3)

(3)

(3)

(3)

(3)

(3)

= x2 x4 (C0 + 3C11 + 2C21 ), (3)

(3)

(3)

(3)

= x2 x4 (C0 + C11 + 2C12 + 2C23 ), (1L)

= x2 x4 (C12 + 2C23 ),

(1L)

= x2 x4 (C12 + 2C22 ),

h6 h7

(1L)

h8

(1L)

= h9

(3)

(3)

(3)

(3)

(1L)

= h10

=0

(2L)

= −2x2 x4 (C12 + C23 ),

(2L)

= −2x2 x4 (C11 + C23 ),

(2L)

= −2x2 x4 (C12 + C22 ),

(2L)

= h9

h8

(2L)

h10

(B.7)

(2L)

(4)

(4)

(4)

(4)

(4)

(4)

(4)

(4)

(B.8)

(4)

= x2 x3 m2g˜ C0 , (4)

(4)

= x2 x4 (C11 − C12 ).

(2R)

and hi can be obtained by exchanging x1 ↔ x2 hi (1L) (2L) and x3 ↔ x4 in hi and hi (i = 1, 2, · · · , 10). The counterterms are given by: h (CT) = −CF g2s Tija γµ (δZL + δZL† )PL Λµ i (B.9) † +(δZR + δZR )PR . The wave-function renormalization constants can be obtained from (3.3.12) and (3.3.13). C. Box corrections ti as given in (3.5.1) Finally, we list the four form factors Fµν in terms of Passarino–Veltman functions. First we define FktiL and FktiR by:

Fµν =

= C0 , Cij [−p1 , −p2 , mg˜ , mt˜1 , mg˜ ].

(4)

= −2x2 x4 (C11 + C21 ),

h7

(1R)

(4)

(2L)

h6

(ti)

Then we arrive at hi

= 2x2 x3 m2g˜ C12 ,

h5

(B.5) (n) The expressions for Λµ , n = 1L, 1R, 2L, 2R are given as: (n)

(2L)

h4

Similarly, the gtt vertex functions are composed of lefthanded and right-handed contributions plus a counterterm (we define a as the color index of the external gluon and i, j as colors of external top quarks): =

= 2x2 x3 m2g˜ C11 ,

h3

= −8C12 − 16C22 − 8C23 − 16C34 .

Λaµ,(ij) (p1 , p2 )

(4)

(2L)

h2

(2)

(2)

(4)

= x2 x4 (−m2g˜ C0 − 2C24 + C24 ) (4) (4) (4) (4) +x2 x4 p21 (C11 + C21 ) + 2x2 x4 p1 p˙2 (C12 + C23 ) (4) (4) +x2 x4 p22 (C12 + C22 ),

= −8C12 − 24C23 − 16C33 , (2)

473

(tiR) (tiR) iˆ gs2 + γν γµ F2 16π 2 PR [γµ γν F1 (tiR) (tiR) +p1ν γµ F3 + p2ν γµ F4 (tiR) (tiR) +p1µ γν F5 + p2µ γν F6 (tiR) (tiR) +γµ γν /k1 F7 + γν γµ /k1 F8 (tiR) (tiR) +/ k1 p1µ p2ν F9 + /k1 p2µ p1ν F10 (tiR) (tiR) +γµ /k1 p1ν F11 + γµ /k1 p2ν F12 (tiR) (tiR) +γν /k1 p1µ F13 + γν /k1 p2µ F14 (tiR) (tiR) +p1µ p2ν F15 + p1µ p1ν F16 (tiR) (tiR) +p2µ p1ν F17 + /k1 p1µ p1ν F18 (tiR) (tiR) +/ k1 p2µ p2ν F19 + p2µ p2ν F20 ] (tiR) (tiL) → Fk ), +(PR → PL , Fk

(C.1)

(k = 1–20, i = 1–4).

In the following we only give the expressions for FktiR (k = 1, 2, · · · , 20 and i = 1–4). The expressions for FktiL can be obtained from FktiR by exchanging x1 ↔ x2 and

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Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD

x3 ↔ x4 . Furthermore, the form factors in the u-channel are given by (ui)

(ti)

(C.2)

Fµν (k1 , k2 , p1 , p2 ) = Fνµ (k2 , k1 , p1 , p2 ). (t1R)

The expressions of Fk (t1R)

F1

(t1R)

(t1R) F4 (t1R) F5

=

(1)

(t1R)

=

(1) 4x1 x3 (D27

(t1R)

(1) D311

−

(t2R)

F5

(1)

(t1R)

(1)

= F8

(1)

= 2x1 x3 (D313 − D312 ), (1)

(1)

(1)

(1)

(t1R)

(t1R)

F15

(t1R)

(1)

(1)

(1)

(t1R)

= F12

(t1R)

= F13

(t1R)

= F14

(1)

(1)

(1)

(1)

= 4x1 x3 mt (D13 − D23 + D25 (1) (1) (1) +D26 − D39 + D310 ) (1) (1) (1) −4x1 x4 mg˜ (D0 + D11 + D12 (1) (1) (1) −D13 + D24 − D26 ) (1) (1) (1) +4x2 x4 mt (D11 − D13 + D21 (1) (1) (1) +D23 + D24 − 2D25 (1) (1) (1) (1) −D26 + D34 + D39 − 2D310 ), (1)

(1)

=

(t1R)

F18

(1)

(1)

(1)

(1)

(t1R)

(t1R)

(1)

(1)

(1)

(1)

= 4x1 x3 (−D23 + D26 − D39 + D38 ), (1)

(1)

= 4x1 x3 mt (−D23 − D39 ) (1) (1) +4x1 x4 mg˜ (D13 + D26 ) (1) (1) (1) (1) +4x2 x4 mt (D23 − D25 + D39 − D310 ),

(2)

(2)

(2)

(2)

(t2R)

F9

(2)

(2)

= 2x1 x3 D312 , (2)

(2)

(2)

(2)

= 4x1 x3 (D12 − D13 + D22 + D24 − D25 (2) (2) (2) −D26 + D36 − D310 ), (t2R)

F10

(2)

(2)

= 4x1 x3 (D38 − D310 ), (t2R)

F11 (t2R)

(2)

= 2x1 x3 (D27 + D312 ), (t2R)

F13

(2)

(2)

F8

(1) D26 )

(1)

(2)

= 2x1 x3 m2g˜ D13 (2) (2) (2) (2) −2x1 x3 m2t (D23 + D25 + D33 + D35 ) (2) +2(x2 x3 + x1 x4 )mg˜ mt D13 (2) (2) +2x2 x4 m2t (D23 − D25 ) (2) (2) +4x1 x3 k1 · p1 (D26 + D310 ) (2) (2) +4x1 x3 k1 · p2 (D23 + D39 ) (2) (2) −4x1 x3 p1 · p2 (D23 + D37 ) (2) +8x1 x3 D313 , (t2R)

= 4x1 x3 (−D22 + D24 − D25 + D26 + D34 (1) (1) (1) (1) (1) −D35 − D36 + D37 + D38 − D39 ),

F19

(2)

F7

(1)

(1) (1) (1) 4x1 x3 mt (D37 − D39 ) − 4x1 x4 mg˜ (D25 − (1) (1) (1) (1) +4x2 x4 mt (D35 − D37 + D39 − D310 ),

(2)

= 4x1 x3 (D312 − D313 ),

(2)

(t2R)

= 4x1 x3 mt (−D25 + D26 − D35 (1) (1) (1) +D37 − D39 + D310 ) (1) (1) (1) +4x1 x4 mg˜ (D11 − D12 + D21 (1) (1) (1) −D24 − D25 + D26 ) (1) (1) (1) (1) (1) −4x2 x4 mt (D21 − D24 − D25 + D26 + D31 (1) (1) (1) (1) (1) −D34 − 2D35 + D37 − D39 + 2D310 ),

(2)

= 8x1 x3 (D27 + D311 ) + 2x1 x3 m2g˜ (D0 + D11 ) (2) (2) (2) (2) +2x1 x3 m2t (−D11 − D13 − 2D21 − D23 (2) (2) (2) −D25 − D31 − D37 ) (2) (2) +2(x2 x3 + x1 x4 )mg˜ mt (D0 + D11 ) (2) (2) (2) (2) −2x2 x4 m2t (D11 − D13 + D21 − D25 ) (2) (2) (2) +4x1 x3 k1 · p1 (D12 + 2D24 + D34 ) (2) (2) (2) (2) +4x1 x3 k1 · p2 (D13 + D25 + D26 + D310 ) (2) (2) (2) −4x1 x3 p1 · p2 (D13 + 2D25 + D35 ),

F6

= 0,

(C.4)

(2)

(2)

(t2R)

(1) D313 ),

(2)

= 4x1 x3 (−D27 − D311 + D312 ),

F4

= 4x1 x3 (D37 − D39 + D38 − D310 ),

F11

F20

+

+

(2)

(2)

(t2R)

= −4x1 x3 D313 ,

(1)

F10

(2)

F3

(1) D312 ),

(1)

= 2x1 x3 mt D313 + 2x1 x4 mg˜ D27 − 2x2 x4 mt D311 ,

= 4x1 x3 (D13 − D12 − D22 − D23 − D24 (1) (1) (1) (1) (1) (1) +D25 + 2D26 − D36 + D38 − D39 + D310 ),

F9

(t1R) F17

(t2R)

F2

(1)

(1) −4x1 x3 (D27

(t1R)

F7

(C.3)

(1)

= 2x1 x3 mt (D27 + D313 ) + 2x1 x4 mg˜ D27 (2) (2) −2x2 x4 mt (D27 + D311 ),

= 4x1 x3 (D311 − D312 ),

F6

F16

(t2R)

F1

(t1R)

= F2 (1) (1) = −2x1 x4 mg˜ D27 + 2x2 x4 mt D311 (1) +2(x1 x3 − x2 x4 )mt D313 , F3

(t1R)

(k = 1–20) are as given below:

(1)

where we denote Di , Dij , Dijk = Di , Dij , Dijk [−p1 , k1 , k2 , mg˜ , mt˜1 , mt˜1 , mt˜1 ]. (t2R) (k = 1–20) are as follows: The expressions for Fk

(t2R)

= F12 (2)

= 0, (2)

(2)

(2)

= 2x1 x3 mt (−D12 + D13 − D24 + D25 ) (2) (2) +2x1 x4 mg˜ (D0 + D11 ) (2) (2) (2) (2) −2x2 x4 mt (D11 − D12 + D21 − D24 ), (t2R)

F14

(2)

(2)

= 2x1 x3 mt (D23 − D26 ) (2) +2x1 x4 mg˜ D13 (2) (2) −2x2 x4 mt (D25 − D26 ),

Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD (t2R)

(2)

F15

(2)

(2)

(2)

(2)

= 4x1 x3 mt (D12 − D13 − D23 + D24 − D25 (2) (2) (2) +D26 − D37 + D310 ) (2) (2) (2) (2) +4x1 x4 mg˜ (D12 − D13 + D24 − D25 ) (2) (2) (2) −4x2 x4 mt (D12 − D13 + 2D24 (2) (2) (2) −2D25 + D34 − D35 ),

(t2R)

F16

(t2R)

F17

(2)

(2)

(2)

(t3R)

The expressions for Fk (t3R)

F1

(2)

= 4x1 x3 mt (D12 − D13 + D24 − 2D25 (2) (2) (2) +D26 − D35 + D310 ) (2) (2) +4x1 x4 mg˜ (−D0 − 2D11 (2) (2) (2) +D12 − D21 + D24 ) (2) (2) (2) +4x2 x4 mt (D11 − D12 + 2D21 (2) (2) (2) −2D24 + D31 − D34 ),

(2)

(2)

(2)

(2)

= 4x1 x3 mt (−D23 + D26 − D37 + D39 ) (2) (2) (2) +4x1 x4 mg˜ (−D13 − D25 + D26 ) (2) (2) (2) (2) +4x2 x4 mt (D25 − D26 + D35 − D310 ),

(t2R)

F18

(2)

(2)

(2)

(2)

= 4x1 x3 (D22 − D24 − D34 + D36 ),

(3)

(t3R)

(t2R)

(2)

(2)

(2)

(2)

(2)

(2)

(2)

= 4x1 x3 mt (−D23 + D26 − D33 + D39 ) (2) (2) +4x1 x4 mg˜ (−D23 + D26 ) (2) (2) (2) (2) +4x2 x4 mt (D23 − D26 + D37 − D310 ),

(2)

(3)

(3)

= −2x1 x4 mg˜ D27 − 2x1 x3 mt D313 (3) (3) (3) −2x2 x4 mt (D27 + D311 − D313 ),

(2)

(t3R)

(t2R)

(3)

= 4x1 x3 (−D23 + D26 + D38 − D39 ), F3

F20

(k = 1–20) are written as:

= 2x1 x3 mt (D27 + 2D313 ) (3) (3) +x1 x3 mt m2g˜ (D0 + D13 ) (3) (3) (3) (3) −x1 x3 m3t (D0 + 2D11 − D13 + D21 (3) (3) (3) +2D33 + 2D35 − 2D37 ) (3) (3) +2x1 x4 mg˜ D27 + x1 x4 m3g˜ D0 (3) (3) (3) −x1 x4 m2t mg˜ (D0 + 2D11 − 2D13 (3) (3) (3) +D21 + 2D23 − 2D25 ) (3) (3) (3) +2x2 x4 mt (D27 + 2D311 − 2D313 ) (3) (3) +x2 x4 mt m2g˜ (D11 − D13 ) (3) (3) (3) (3) (3) −x2 x4 m3t (D11 − D13 + 2D21 + 2D23 − 4D25 (3) (3) (3) (3) +D31 − 2D33 − 3D35 + 4D37 ) (3) (3) (3) (3) +2x1 x3 mt k1 · p1 (D12 − D13 − D23 + D24 (3) (3) (3) (3) +D33 − D37 − D39 + D310 ) (3) (3) (3) +2x1 x4 mg˜ k1 · p1 (D11 + D12 − 2D13 (3) (3) (3) (3) +D23 + D24 − D25 − D26 ) (3) (3) (3) (3) +2x2 x4 mt k1 · p1 (D11 − D13 + D21 + 2D23 (3) (3) (3) (3) (3) (3) +D24 − 3D25 − D26 − D33 + D34 − D35 (3) (3) (3) +2D37 + D39 − 2D310 ) (3) (3) (3) (3) +2x1 x3 mt k1 · p2 (−D13 − D26 + D33 − D39 ) (3) (3) (3) +2x1 x4 mg˜ k1 · p2 (−D13 + D23 − D26 ) (3) (3) (3) (3) +2x2 x4 mt k1 · p2 (D23 − D25 − D33 + D37 (3) (3) +D39 − D310 ) (3) (3) (3) (3) +2x1 x3 mt p1 · p2 (D13 + D25 − D33 + D37 ) (3) (3) (3) +2x1 x4 mg˜ p1 · p2 (D13 − D23 + D25 ) (3) (3) (3) +2x2 x4 mt p1 · p2 (−D23 + D25 + D33 (3) (3) +D35 − D37 ), (C.5) F2

F19

475

(2)

(2)

where Di , Dij , Dijk = Di , Dij , Dijk [−p1 , k1 , −p2 , mg˜ , mt˜1 , mt˜1 , mg˜ ].

(3)

(3)

(3)

(3)

= 4x1 x3 (D27 + 2D311 + D312 − 3D313 ) (3) (3) (3) +2x1 x3 m2g˜ (−D0 + D11 − D13 ) (3) (3) (3) (3) +2x1 x3 m2t (D13 − D21 + 2D25 − D31 (3) (3) (3) +2D33 + 3D35 − 4D37 ) (3) (3) −2x2 x3 mt mg˜ D0 − 2x1 x4 mt mg˜ D0 (3) (3) −2x2 x4 m2t (D0 + D11 ) (3) (3) (3) (3) +4x1 x3 k1 · p1 (D23 + D24 − D25 − D26 (3) (3) (3) (3) (3) (3) −D33 + D34 − D35 + 2D37 + D39 − 2D310 ) (3) (3) (3) +4x1 x3 k1 · p2 (−D25 + D26 − D33 (3) (3) (3) +D37 + D39 − D310 ) (3) (3) (3) +4x1 x3 p1 · p2 (D33 + D35 − 2D37 ),

476

Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD (t3R)

(3)

(t3R)

F5

(3)

(3)

(t3R)

(3)

(t3R)

(t3R)

F13

(3)

(3)

(3)

(3)

= 2x1 x3 (D27 + D312 − D313 ),

(t3R)

= 4x1 x3 (D22 + D23 − D25 − D26 (3) (3) (3) (3) +D36 − D38 + D39 − D310 ),

(3)

(t3R)

F10 (t3R)

F11

(3)

(3)

(3)

(3)

(3)

(3)

(t3R)

F14

= −2x1 x3 mt (D13 + D26 ) (3) −2x1 x4 mg˜ D13 (3) (3) +2x2 x4 mt (−D25 + D26 ),

(t3R)

= −4x1 x3 mt (D12 − D23 + D24 (3) (3) (3) +D25 − D39 + D310 ) (3) (3) +4x1 x4 mg˜ (−D11 − D12 (3) (3) (3) +D13 − D24 + D26 ) (3) (3) +4x2 x4 mt (−D11 + D13 (3) (3) −D21 − D23 (3) (3) (3) −D24 + 2D25 + D26 (3) (3) (3) −D34 − D39 + 2D310 ),

F15

(3)

(3)

(3)

(t3R)

F17

(3)

(3)

(3)

(3)

(3)

(3)

(3)

(3)

= −4x1 x3 mt (D12 − D13 + D24 (3) (3) (3) (3) (3) −D25 − D35 + D37 − D39 + D310 ) (3) (3) (3) +4x1 x4 mg˜ (−D12 + D13 + D21 (3) (3) (3) −D24 − D25 + D26 ) (3) (3) (3) (3) +4x2 x4 mt (D21 − D24 − D25 + D26 (3) (3) (3) +D31 − D34 − 2D35 (3) (3) (3) +D37 − D39 + 2D310 ), (3)

(3)

(3)

(3)

= 4x1 x3 mt (D13 + D26 − D37 + D39 ) (3) (3) (3) +4x1 x4 mg˜ (D13 − D25 + D26 ) (3) (3) (3) (3) +4x2 x4 mt (−D35 + D37 − D39 + D310 ),

(t3R)

F18

(3)

(3)

(3)

(3)

(3)

(3)

(3)

= 4x1 x3 (D22 − D24 + D25 − D26 (3) (3) (3) −D34 + D35 + D36 (3) (3) (3) −D37 − D38 + D39 ), (t3R)

F19

= 4x1 x3 (D23 − D26 (3) (3) −D38 + D39 ),

(3)

= 4x1 x3 (D25 − D26 − D37 (3) (3) (3) −D38 + D39 + D310 ), (3)

(t3R)

F16

(3)

F8 F9

(3)

= 2x1 x3 mt (D12 − D13 + D24 − D26 ) (3) (3) +2x1 x4 mg˜ (D11 − D13 ) (3) (3) +2x2 x4 mt (D11 − D12 (3) (3) (3) (3) +D21 − D24 − D25 + D26 ),

(3)

= −4x1 x3 (D312 − D313 ) (3) (3) (3) +x1 x3 m2g˜ (D0 − D12 + D13 ) (3) (3) (3) (3) +x1 x3 m2t (−D11 + D12 − D13 − D21 (3) (3) (3) (3) (3) +2D24 − 2D26 − 2D33 + D34 − D35 (3) (3) (3) +2D37 + 2D39 − 2D310 ) (3) +(x2 x3 + x1 x4 )mt mg˜ D0 (3) (3) +x2 x4 m2t (D0 + D11 ) (3) (3) +2x1 x3 k1 · p1 (−D22 − D23 (3) (3) (3) +2D26 + D33 − D36 (3) (3) (3) (3) −D37 + D38 − 2D39 + 2D310 ) (3) (3) (3) +2x1 x3 k1 · p2 (D33 + D38 − 2D39 ) (3) (3) +2x1 x3 p1 · p2 (D25 − D26 (3) (3) (3) (3) −D33 + D37 + D39 − D310 ), (t3R)

(3)

= 2x1 x3 mt (D13 + D26 ) + 2x1 x4 mg˜ D12 (3) (3) (3) (3) +2x2 x4 mt (D12 − D13 + D24 − D26 ),

(3)

= −8x1 x3 D313 − 2x1 x3 m2g˜ D13 (3) (3) (3) (3) +2x1 x3 m2t (D25 + 2D33 + D35 − 2D37 ) (3) −2x2 x3 mt mg˜ D13 (3) (3) (3) −2x1 x4 mt mg˜ D13 − 2x2 x4 m2t (D13 + D25 ) (3) (3) (3) +4x1 x3 k1 · p1 (D23 − D25 − D33 (3) (3) (3) +D37 + D39 − D310 ) (3) (3) +4x1 x3 k1 · p2 (−D33 + D39 ) (3) (3) +4x1 x3 p1 · p2 (D33 − D37 ),

F7

(3)

F12

= 4x1 x3 (−D311 + D313 ) + 2x1 x3 m2g˜ D0 (3) (3) −2x1 x3 m2t (D13 + D25 ) (3) (3) (3) +2x2 x3 mt mg˜ (D0 + D11 − D13 ) (3) (3) (3) +2x1 x4 mt mg˜ (D0 + D11 − D13 ) (3) (3) (3) (3) (3) +2x2 x4 m2t (D0 + 2D11 − D13 + D21 − D25 ) (3) (3) +4x1 x3 k1 · p2 (D25 − D26 ),

(t3R)

F6

(3)

= 4x1 x3 D312 − 2x1 x3 m2g˜ D0 (3) (3) (3) (3) +2x1 x3 m2t (D11 + D21 + 2D23 − 2D25 ) (3) (3) −2x2 x3 mt mg˜ D0 − 2x1 x4 mt mg˜ D0 (3) (3) −2x2 x4 m2t (D0 + D11 ) (3) (3) +4x1 x3 k1 · p1 (−D23 + D25 ) (3) (3) +4x1 x3 k1 · p2 (−D23 + D26 ) (3) (3) +4x1 x3 p1 · p2 (D23 − D25 ),

F4

= 2x1 x3 mt (−D25 + D26 ) (3) (3) +2x1 x4 mg˜ (−D11 + D12 ) (3) (3) (3) +2x2 x4 mt (−D11 + D12 − D21 (3) (3) (3) +D24 + D25 − D26 ),

(t3R)

F20

(3)

(3)

(3)

(3)

= 4x1 x3 mt (D13 + D23 + D26 + D39 ) (3) (3) +4x1 x4 mg˜ (D13 + D26 ) (3) (3) (3) (3) +4x2 x4 mt (−D23 + D25 − D39 + D310 ),

(3)

(3)

(3)

where Di , Dij , Dijk = Di , Dij , Dijk [−p1 , k1 , k2 , mt˜1 , mg˜ , mg˜ , mg˜ ].

Y. Zeng-Hui et al.: Top-quark pair production via polarized and unpolarized protons in the supersymmetric QCD (t4R)

The Fk

(k = 1–20) are written explicitly as:

(t4R)

F1

(t4R)

Fi

(t4R)

= F2 = 12 ((x1 x3 mt (C11 − C12 ) +x2 x4 mt C12 − x2 x3 mg˜ C0 ) [−p1 , p1 + p2 , mg˜ , mt˜1 , mt˜1 ]), = 0,

(C.6)

(i = 3, 4, · · · 20).

References 1. H.E. Haber, G.L. Kane, Phys. Rep. 117, 75 (1985); J.F. Gunion, H.E. Haber, Nucl. Phys. B 272, 1 (1986) 2. P.C. Bhat, for the D0 collaboration, talk presented at the Wine and Cheese Seminar at Fermilab, February 1997 3. M. Gl¨ uck, J.F. Owens, E. Reya, Phys. Rev. D 17, 2324 (1978); B.L. Combridge, Nucl. Phys. B 151, 429 (1979); H. Georgi et al., Ann. Phys. (N.Y.) 114, 273 (1978) 4. P. Nason, S. Dawson, R.K. Ellis, Nucl. Phys. B 303, 607 (1988); G. Altarelli, M. Diemoz, G. Martinelli, P. Nason, Nucl. Phys. B 308, 724 (1988); W. Beenakker, H. Kujif, W.L. van Neerven, J. Smith, Phys. Rev. D 40, 54 (1989)

477

5. C. Li, B. Hu, J. Yang, C. Hu, Phys. Rev. D 52, 5014 (1995); Z. Sullivan, hep-ph/9611302 6. H.Y. Zhou, C.S. Li, Phys. Rev. D 55, 4421 (1997) 7. A. Bartl, E. Christova, W. Majerotto, Nucl. Phys. B 460, 235 (1996) 8. T. Gehrmann, W.J. Stirling, Z. Phys. C 65, 461 (1995); S.J. Brodsky, M. Burkardt, I. Schmidt, Nucl. Phys. B 441, 197 (1995) and the references therein 9. M. Gl¨ uck, E. Reya, A. Vogt, Z. Phys. C 48, 471 (1990); M. Gl¨ uck, E. Reya, A. Vogt, Z. Phys. C 67, 433 (1995) 10. M. Gl¨ uck, E. Reya, M. Stratmann, Phys. Rev. D 53, 4775 (1996) 11. M. Stratmann, hep-ph/9710379 12. S.P. Martin, M.T. Vaughn, Phys. Lett. B 318, 331 (1993); W. Beenakker, R. H¨ opker, P.M. Zerwas, Phys. Lett. B 378, 159 (1996) 13. W. Beenaker, R. H¨ opker, T. Plehn, P.M. Zerwas, DESY.96-178 (1996) 14. B.A. Kniehl, Phys. Rep. 240, 211 (1994) 15. G. Passarino, M. Veltman, Nucl. Phys. B 160, 151 (1979)