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arXiv:1107.4012v1 [hep-ph] 20 Jul 2011

Model independent analysis of top quark forward-backward

asymmetry at the Tevatron up to O(α2

s/Λ2)

Ding Yu Shao,1Chong Sheng Li,1,2, ∗Jian Wang,1

Jun Gao,1Hao Zhang,1and Hua Xing Zhu1

1Department of Physics and State Key Laboratory of Nuclear Physics and Technology,

Peking University, Beijing 100871, China

2Center for High Energy Physics, Peking University, Beijing 100871, China

Abstract

We present the complete calculations of the forward-backward asymmetry (AFB) and the total

cross section of top quark pair production induced by dimension-six four quark operators at the

Tevatron up to O(α2

can change AFB and the total cross section by about 10%. Moreover, NLO QCD corrections

s/Λ2). Our results show that next-to-leading order (NLO) QCD corrections

reduce the dependence of AFB and total cross section on the renormalization and factorization

scales significantly. We also evaluate the total cross section and the charge asymmetry (AC)

induced by these operators at the Large Hadron Collider (LHC) up to O(α2

space allowed by the Tevatron data. We find that the value of ACinduced by these operators is

s/Λ2), for the parameter

much larger than SM prediction, and LHC has potential to discover these NP effects when the

measurement precision increases.

PACS numbers: 14.65.Ha, 12.38.Bx, 12.60.-i

∗Electronic address: csli@pku.edu.cn

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I. INTRODUCTION

The top quark is the heaviest particle discovered so far, with a mass close to the elec-

troweak symmetry breaking scale. Thus it is a wonderful probe for the electroweak breaking

mechanism and new physics (NP) beyond the standard model (SM) through its productions

and decays at colliders. The forward-backward asymmetry (AFB) of the top quark pair

production is one of the interesting observables at hadron colliders. Within the SM, AFB

is absent at the tree level in QCD due to charge symmetry, and occurs at next-to-leading

order (NLO) O(α3

the last few years, DØ and CDF Collaborations measured AFBat the Tevatron [7–10]. Re-

s) in QCD with the prediction AFB∼ 6% in the t¯t rest frame [1–6]. In

cently, the CDF Collaborations annouced that, for the invariant mass of the top quark pair

mt¯t≥ 450 GeV, the measured asymmetry, AFB= 0.475 ± 0.114[9], differs by 3.4σ from the

SM predictions AFB= 0.088±0.013, which has aroused many discussions of explaining this

deviation in NP model, including new gauge bosons, axigluons and so on[11–57].

Since we do not know which type of NP will be responsible for this deviation, it is interest-

ing to study the AFBin a model independent way, using an effective Lagrangian. In general,

NP scale relevant to AFBis large enough so that the heavy fields have been integrated out

at the low energy scale. At the Tevatron, the subprocess q¯ q → t¯t dominates over top quark

pair production, so only contributions from dimension-six four quark operators to the t¯t

production are considered. Similar approach had been adopted for the dijet production to

constrain the composite scale of light quarks [58–62]. The relevant effective Lagrangian can

be written as

LNP=

1

Λ2

?

A,B

?C1

AB(¯ qAγµqA)(¯tBγµtB) + C8

AB(¯ qATaγµqA)(¯tBTaγµtB)?, (1)

where {A,B} = {L,R} with q = (u,d)T,(c,s)T. The contributions to AFBat leading order

(LO) from such operators has been explored in Refs. [16, 24, 63]. It is shown that the AFB

observed at the Tevatron can be explained by above operators for suitable parameters. As

we know, the LO cross section at hardron colliders suffers from large uncertainties due to

the arbitrary choice of the renormalization scale and factorization scale, thus it is important

to include NLO corrections to improve theoretical predictions. Besides, at the NLO level,

virtual corrections, real gluon emission and massless (anti)quark emission can lead to a

sizeable difference between the differential top and anti-top production process [1, 2], which

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will contribute to AFB.

In this paper, we present the complete NLO QCD calculations of AFBand the total cross

section of top quark pair production at the Tevatron induced by above operators, and we

also study the top quark pair production at the Large Hadron Collider (LHC) induced by

these operators at the NLO QCD level. Last year, LHC reported their first observation of

top quark pair production, and will soon become a major top quark factory. At the LHC, the

top quark pairs can be produced through quark antiquark annihilation q¯ q → t¯t and gluon

fusion gg → t¯t. Since gluon fusion channel dominates at the LHC, it is difficult to probe

these four quark effective operators from early LHC results. However, it is still possible

to detect these effects from above effective operators on the Charge Asymmetry(AC) at the

LHC, in the parameter space allowed by the Tevatron data, when the measurement precision

increases.

The arrangement of this paper is as follows. In Sec. II we show the LO results. In Sec. III,

we present the details of the NLO calculations, including the virtual and real corrections to

the top quark pair production. Section IV contains the numerical results, and Section V is

a brief summary.

II. LO RESULTS

Throughout our calculation, we adopt the same conventions as in Ref. [64] (see Sec. III

A), and present the helicity amplitudes for q¯ q → t¯t in the Four-Dimensional Helicity (FDH)

regularization scheme [65]. The t¯t production amplitudes, including NP contributions, can

be written as

Mt¯t= αsfSM

LO+

1

Λ2fNP

LO+ α2

sfSM

NLO+αs

Λ2fNP

NLO+ ··· ,(2)

and thus the partonic cross section, up to O(α2

s/Λ2), can be written as

ˆ σt¯t = α2

sfSM

LOfSM∗

LO + 2αs

Λ2Re?fSM

LOfSM∗

NLO

LOfNP∗

LO

?

+2α3

sRe?fSM

?+ 2α2

s

Λ2

?Re?fNP

LOfSM∗

NLO

?+ Re?fSM

LOfNP∗

NLO

??.(3)

The LO Feynman diagrams for the subprocess q(p1)¯ q(p2) → t(p3)¯t(p4) induced by the SM

QCD and the NP interactions are shown in Fig. 1, and their (+ − ++) helicity amplitudes

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FIG. 1: LO Feynman diagrams for q¯ q → t¯t induced by SM QCD and NP interactions.

are

MSM

MNP

LO(+ − ++) =

i8παsmt

s

i2mt

Λ2

(M1+ M2)C8,

??M1C1

(4)

LO(+ − ++) =

RR+ M2C1

RL

?C1+?M1C8

RR+ M2C8

RL

?C8

?, (5)

where the SM QCD and NP contributions are denoted by superscipts SM and NP, and we

define the following abbreviations for the color structures and matrix elements,

M1=?η41??η3|3|2]

M3=?η42??η3|3|1]

C1= δi2i1δi3i4,

?3♭η3??η44♭?,

?3♭η3??η44♭?,

M2=?η31??η4|4|2]

M4=?η32??η4|4|1]

C8= Ta

?3♭η3??η44♭?,

?3♭η3??η44♭?,

i2i1Ta

i3i4, (6)

where i1...4are the color indices of the external quarks and the boldface momenta denotes

massive vectors. We use the modified spinor helicity method suited for massive particles [66]

in our calculations, and a recent application of this method can be found in the Ref [67].

The (− + ++) amplitudes are given by

MSM

MNP

LO(− + ++) =

i8παsmt

s

i2mt

Λ2

(M3+ M4)C8,

??M3C1

(7)

LO(− + ++) =

LR+ M4C1

LL

?C1+?M3C8

LR+ M4C8

LL

?C8

?. (8)

At the LO, there is only vector current coupling¯ψγµψ at the massive quark vertex. At the

NLO, however, magnetic-momentum coupling¯ψ(iσµν(p3+p4)µ)ψ/(2mt) is induced from loop

diagrams. For completeness we list the matrix elements for magnetic-moment interaction,

M(m)

1

=m2

t?η31??η41?[21]

?3♭η3??η44♭?

=m2

?3♭η3??η44♭?

,M(m)

2

=?12??η3|3|2]?η4|4|2]

?3♭η3??η44♭?

=?12??η3|3|1]?η4|4|1]

?3♭η3??η44♭?

,

M(m)

3

t?η32??η42?[21]

,M(m)

4

.(9)

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Mandelstam variables s, t and u are defined as follows:

s = (p1+ p2)2,t = (p1− p3)2,u = (p1− p4)2. (10)

After phase space integration, the O(αs/Λ2) partonic differential cross section is

dˆ σNP

LO

dcosθ 18Λ2

?1 − ρ2, and θ is the polar angle between the incoming quark and the

outgoing top quark in the t¯t rest frame. The color and spin indices are averaged(summed)

=

βαs

?1

4(1 + ρ + β2cos2θ)(C8

LR+ C8

RR) +1

2β cosθ(C8

RR− C8

LR)

?

, (11)

where ρ = 4m2

t/s, β =

over initial(final) states. In Eq. (11) the term linear in cosθ could generate AFBproportional

to (C8

RR− C8

RR+ C8

LR) and the rest terms contribute to the total cross section proportional to

(C8

LR). These relations will be changed at the NLO level.

The LO total cross section at the hadron collider is obtained by convoluting the partonic

level cross section with the Parton Distribution Function (PDF) fi/Afor the initial hadron

A:

σLO=

?

a,b

?1

τ

dxa

?1

τ/xa

dxbfa/A(xa,µf)fb/B(xb,µf)ˆ σLO, (12)

where τ = 4m2

t/s. The sum is over all possible initial partons.

III. NLO QCD CORRECTIONS

The NLO corrections to the top pair production consist of the virtual corrections, gen-

erated by loop diagrams of colored particles, and the real corrections with the radiation of

a real gluon or a massless (anti)quark. We carried out all the calculations in the ’t Hooft-

Feynman gauge and used the FDH scheme to regularize all the divergences. Moveover, for

the real corrections, we used the dipole substraction method with massive partons [68] to

separate the infrared (IR) divergences, which is convenient for the case of massive Feynman

diagrams and provides better numerical accuracy.

A. Virtual corrections

The virtual corrections for the top quark pair production include the box diagrams,

triangle diagrams, and self-energy diagrams in SM QCD and NP as shown in Fig. 2 and

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Fig. 3. We have calculated the one-loop helicity amplitudes for the SM process, and find

complete agreement with those in the Ref. [64]. Here we only list the NP contributions.

FIG. 2: One-loop virtual Feynman diagrams for q¯ q → t¯t induced by SM QCD interactions.

FIG. 3: One-loop virtual Feynman diagrams for q¯ q → t¯t induced by NP interactions.

All the ultraviolet (UV) divergences in the loop diagrams are canceled by counterterms

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for the wave functions of the external fields (δZq,δZt), and the Wilsion coefficients δZCi

AB.

For the external fields, we fix all the renormalization constants using on-shell subtraction,

and, therefore, they also have IR singularities

δZq = −αs

δZt = −αs

4πCεCF

?

?

1

εUV

1

εUV

−

1

εIR

2

εIR

?

+ 5 − 3lnm2

,(13)

4πCεCF

+

t

µ2

r

?

, (14)

where Cε= (4π)ε

1

Γ(1−ε). For contourterms of the Wilsion coefficients δZCi

AB, we adopted

the MS scheme

δZCi

AB=αs

4πCεCF

1

εUV

0

9

8

000000

1

4

0

nf−5

16

0

0

1

16

−9

nf−20

16

0

0

nf

16

0

00

0

8

000

0

1

16

0

−1

0

4

000

nf

16

000−9

nf−20

16

0

8

0

0

nf

16

0

00−1

0

4

0

1

16

0000

9

8

000

nf

16

0

1

16

1

4

nf−5

16

, (15)

where nf= 5 and the order of the Wilsion coefficients is

?C1

LL,C8

LL,C1

LR,C8

LR,C1

RL,C8

RL,C1

RR,C8

RR

?. (16)

We have considered mixing effects of different color and chiral operators, and the evolution

equations of the Wilson coefficients are given in the Appendix B. The renormalized virtual

amplitudes can be written as

MV= Munren+ Mcon. (17)

Here Munrencontains the self-energy and vertex corrections, and Mconare the correspond-

ing counterterms.The renormalized amplitude MVis UV finite, but still contains IR

divergences, which are given by

MIR

MIR

SM=

αs

2πCεCF

αs

2πCεCF

?fIR

osC1MSM

?fIR

8

+ fIR

ooC8MSM

+ fIR

8

?,

+ fIR

(18)

NP=

ssC1MNP

1soC8MNP

1 osC1MNP

8

+ fIR

ooC8MNP

8

?, (19)

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where

fIR

ss = −1

ε2

IR

+

1

εIR

?

?

1

εIR

?1

1

εIR,

1

εIR,

?

?

2

?

β +1

β

?

ln

?β + 1

β − 1

?

+ ln

?−s

µ2

r

?

−5

2

?

, (20)

fIR

so =

3

2ln

1

3ln

?t1

?t1

u1

(21)

fIR

os =

u1

(22)

fIR

oo= −1

ε2

IR

+−1

16

?

?u1

β +1

β

?

−5

ln

?β + 1

?

β − 1

?

−9

8ln

?m2

t

µ2

r

?

−1

8ln

?−s

µ2

r

?

+7

4ln

?t1

µ2

r

+1

2lnµ2

r

?

2

,(23)

where t1= m2

t− t, u1= m2

t− u and MNP

1, MNP

8

and MSM

8

are defined as follows,

MSM

MNP

LO= MSM

LO= MNP

8C8,

1C1+ MNP

(24)

8C8. (25)

Since we only consider high order corrections up to O(α2

virtual corrections can be written as

s/Λ2), the IR divergences of the

2Re?MIR

πCεCF

SMMLO∗

??9fIR

NPare given in the Appendix A.

NP

?+ 2Re?MIR

so

NPMLO∗

SM

?

8

=

αs

os+ 2fIR

?Re?MNP∗

1

MSM

?+ 4fIR

ooRe?MNP∗

8

MSM

8

??. (26)

The finite terms in MV

B.Real corrections

At the NLO level the real corrections consist of the radiations of an additional gluon or

massless (anti)quark in the final states, including the subprocess

q¯ q → t¯tg, gq(¯ q) → t¯tq(¯ q)(27)

as shown in Fig.4 and Fig.5.

Before performing the numerical calculations, we need to extract the IR divergences in

the real corrections. In the dipole formalism this is done by subtracting some dipole terms

from the real corrections to cancel the singularities and large logarithms exactly, and then

the real corrections become integrable in four dimensions. On the other hand, these dipole

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FIG. 4: Feynman diagrams for the real gluon emission contributions induced by SM QCD and NP

interactions.

FIG. 5: Feynman diagrams for the massless quark emission contributions induced by SM QCD

and NP interactions.

subtraction terms are analytically integrable in n dimensions over one-parton subspaces,

which give ε poles that represent the soft and collinear divergences. Then we can add them

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to the virtual corrections to cancel the ε poles, and ensure the virtual corrections are also

integrable in four dimensions. This whole procedure can be illustrated by the formula [68]:

?

where m is the number of final state particles at the LO, and dˆ σAis a sum of the dipole

ˆ σNLO=

m+1

??dˆ σR?

ε=0−?dˆ σA?

ε=0

?+

?

m

?

dˆ σV+

?

1

dˆ σA

?

ε=0

, (28)

terms. Besides, at hadron colliders, we have to include the well-known collinear subtraction

counterterms in order to cancel the collinear divergences arising from the splitting processes

of the initial state massless partons. Here we use the MS scheme and the corresponding

NLO PDFs.

For the process with two initial state hadrons, the dipole terms can be classified into four

groups, the final-state emitter and final-state spectator type,

Dij,k(p1,...,pm+1) =

−

(pi+ pj)2− m2

1

ij

m?...,?ij,...,?k,...|Tk· Tij

T2

ij

Vij,k|...,?ij,...,?k,...?m, (29)

the final-state emitter and initial-state spectator type,

Da

ij(p1,...,pm+1;pa,...) =

−

1

(pi+ pj)2− m2

ij

1

xij,a

m,a?...,?ij,...;? a,...|Ta· Tij

T2

ij

Va

ij|...,?ij,...;? a,...?m,a, (30)

the initial-state emitter and final-state spectator type,

Dai

j(p1,...,pm+1;pa,...) =

1

2papi

xij,a

−

1

m,? ai?...,?j,...;? ai,...|Tj· Tai

T2

ai

Vai

j|...,?j,...;? ai,...?m,? ai, (31)

and the initial-state emitter and initial-state spectator type,

Dai,b(p1,...,pm+1;pa,pb) =

−

2papi

xi,ab

1

1

m,? ai?...;?ai,b|Tb· Tai

T2

ai

Vai,b|...;?ai,b?m,? ai,(32)

where a,b and i,j,... are the initial and final state partons, and T and V are the color charge

operators and dipole functions acting on the LO amplitudes, respectively. The explicit

expressions for xi,ab, xij,aand V can be found in Ref. [68]. The integrated dipole functions

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together with the collinear counterterms can be written in the following factorized form

?

+

0

∼dΦ(m)(pa,pb)m,ab?...;pa,pb|Im+a+b(ε)|...;pa,pb?m,ab

?

+(a ↔ b),

a′

?1

dx

?

dΦ(m)(xpa,pb)m,a′b?...;xpa,pb|Pa,a′

m+b(x) + Ka,a′

m+b(x)|...;xpa,pb?m,a′b

(33)

where x is the momentum fraction of the splitting parton, dΦ(m)contains all the factors

apart from the squared amplitudes, I, P, and K are insertion operators defined in [68].

For simplicity, in all the above formulas we do not show the jet functions that define the

observables and are included in our numerical calculations.

The operators P and K provide finite contributions to the NLO corrections, and only

the operator I contains the IR divergences

I|IR = −αs

2π

(4π)ε

Γ(1 − ε)

?

+Ta· Tb

??

?

?ε?

j

?

r

sja

k?=j

?ε

1

ε2

IR

Tj· Tk

??µ2

r

sjk

?ε

V(sjk,mj,mk;εIR) +

1

T2

j

Γj(mj,εIR)

?

+

j

Tj· Ta

??µ2

2

?µ2

V(sja,mj,0;εIR) +

1

T2

j

Γj(mj,εIR) +

1

T2

a

γa

εIR

?

r

sab

+

1

T2

a

γa

εIR

??

+ (a ↔ b)

?

,(34)

with

V(sjk,mj,mk;εIR) =

1

vjk

γj

εIR,

?Q2

jk

sjk

Γj(mj?= 0,εIR) =CF

?ε

×

?

1 −1

2ρ−2ε

j

−1

2ρ−2ε

k

?

1

ε2

IR

,

Γj(0,εIR) =

εIR, (35)

where CF = 4/3, γq= 2, and γg= 11/2 − nf/3. And sjk, Q2

variables defined as follows

jk, vjk, and ρnare kinematic

sjk = 2pjpk,Q2

jk= sjk+ m2

j+ m2

k,vjk=

?

1 −

m2

(pjpk)2,

jm2

k

ρn =

?

1 − vjk+ 2m2

1 + vjk+ 2m2

n/(Q2

n/(Q2

jk− m2

jk− m2

j− m2

j− m2

k)

k)

(n = j,k). (36)

When inserting Eq. (34) into the LO amplitudes for the q¯ q and qg subprocesses as shown

in Eq. (33), we can see that the IR divergences, including the 1/ǫIRterms, can be written

as combinations of the LO color correlated squared amplitudes and all the IR divergences

from the virtual corrections in Eq. (26) are canceled exactly, as we expected.

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IV. NUMERICAL RESULTS

In the numerical calculations, for simplicity we set C1

LL= C8

LL= C1

RL= C8

RL= 0 in

order to avoid the SM constrains from B physics. Up to O(α2

singlet operators due to mixing effects are much less than contributions from color octet

s/Λ2), contributions from color

operators, so we only consider color octet interactions. As a result, there are only three free

NP parameters in the Lagrangian, i.e. C8

LR, C8

RRand Λ.

Top quark mass is taken to be

mt= 172.5 GeV. (37)

We choose CTEQ6L and CTEQ6M PDF sets [69] and the associated αsfunctions for LO

and NLO calculation, respectively. Both the renormalization and factorization scales are

fixed to the top quark mass unless specified. We have used the modified MadDipole [70]

package for the real corrections.

t

/m

µ

0.6 0.81 1.21.41.61.82

)

=m

µ

(

σ

)/

µ

(

σ

t

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

=0

=

µ

,

µ

,

8

LR

µ

µ

µ

=1, C

µ

µ

µ

8

RR

C

µ

f

=

r

t

=m

=m

f

=

=

r

t

r

f

t

/m

µ

0.6 0.81 1.21.4 1.6 1.82

)

=m

µ

(

σ

)/

µ

(

σ

t

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

=1

=

µ

,

µ

,

8

LR

µ

µ

µ

=0, C

µ

µ

µ

8

RR

C

µ

f

=

r

t

=m

=m

f

=

=

r

t

r

f

FIG. 6: Scale dependence of the total cross section at the Tevetron, the black and the red lines

represent LO and NLO results, respectively.

In Fig. 6 we show the scale dependence of the LO and NLO total cross section at the

Tevatron for three cases: (1) the renormalization scale dependence µr= µ, µf= mt, (2) the

factorization scale dependence µr= mt, µf= µ, and (3) total scale dependence µr= µf= µ.

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It can be seen that the NLO corrections reduce the scale dependence significantly for all

three cases, which makes the theoretical predictions more reliable.

FIG. 7: Values of C8

RRand C8

LRallowed by the Tavetron data at 1σ CL: σt¯t=(7.50 ± 0.48)pb

and AFB(mt¯t> 450 GeV)=0.475±0.114. The green star (5.31,−4.15) and red star (4.92,−3.88)

represent the best-fit points at LO and NLO level, respectively.

AFBof top quark pair productions is defined as

AFB =σF− σB

σF+ σB

= ANP

FB× R + ASM

FB× (1 − R)

where

ANP

FB= (σNP

F − σNP

− σSM

tot/(σSM

B)/(σNP

F + σNP

B),

ASM

FB= (σSM

FB)/(σSM

F

+ σSM

B),

R = σNP

tot+ σNP

tot) (38)

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are the asymmetries induced by NP and SM, and R is the fraction of NP contribution

to the total cross section. σF and σB denote the total cross sections in the forward(F)

and backward(B) rapidity regions, respectively. Up to order O(α2

s/Λ2), total cross sections

induced by NP can be written as

σNP

LO=

?(0.428+0.103

−0.076)C8

RR+ (0.428+0.101

−0.075)C8

LR

??1TeV

??1TeV

Λ

?2

pb, (39)

σNP

NLO=

?(0.442+0.018

−0.032)C8

RR+ (0.435+0.022

−0.022)C8

LR

Λ

?2

pb, (40)

and the difference of the cross section in the forward and backward rapidity regions can be

written as

?σNP

F − σNP

B

?mt¯ t>450 GeV

LO

=

?(0.118+0.031

−0.023)C8

RR− (0.118+0.031

−0.023)C8

LR

??1TeV

??1TeV

Λ

?2

pb, (41)

?σNP

F − σNP

B

?mt¯ t>450 GeV

NLO

=

?(0.149+0.003

−0.009)C8

RR− (0.103+0.008

−0.025)C8

LR

Λ

?2

pb, (42)

where the errors are obtained by varying the scale between µr = µf = mt/2 and

µr = µf = 2mt. From the expressions Eqs.(39 – 42) we can see that NLO corrections

reduce the dependence of σNP

F

− σNP

B

and σNPon the renormalization and factorization

scales significantly.

In Fig. 7, we show the allowed region in the (C8

RR,C8

LR) plane that is consistent with both

AFB[9] in the large mt¯tinvariant mass region and the t¯t production cross section σt¯t

σEX

t¯t

= (7.50 ± 0.48)pb,

FB= 0.475 ± 0.114,AEX

for mt¯t> 450 GeV (43)

at the Tevatron. We use Monte Carlo programm MCFM [71] to get the SM predictions of

σt¯tat the NLO QCD level

σSM

t¯t

= 7.00+0.36

−0.76pb,(44)

where we have considered scale uncertainty in the calculations. We have used the SM QCD

predicted values of AFB(mt¯t≥ 450GeV) = 0.088 ± 0.013. In Fig. 7, green and red regions

correspond to NP LO and NLO results at 1σ confidence level(CL), where we have consid-

ered theoretical and experimental uncertainty in the total cross section and only consider

14

Page 15

experimental uncertainty in the AFBcalculation. It can be seen that NLO corrections obvi-

ously change the allowed region of C8

RRand C1

RR. The green star (5.31,−4.15) and red star

(4.92,−3.88) represent the best-fit points at LO and NLO level, respectively, from which we

can see higher order corrections reduce the best-fit value by about 7%. The t¯t total cross

sections induced by NP at the NLO best-fit point (4.92,−3.88) are

σNP

t¯t,LO= 0.445 pb,

σNP

t¯t,NLO= 0.497 pb, (45)

where the K factor is about 1.12. AFBcontaining NP contributions at the NLO best-fit point

are shown together in Table I. The NLO QCD corrections to AFBcan reach about 10%,

and the theoretical predictions containing NP NLO effects are consistent with experimental

results at 2σ CL.

In Fig. 8, we show differential cross section dσ/dmt¯twhen we consider NP effects at the

SM NLO QCD + NP LO SM NLO QCD + NP NLO

Ap¯ p

FB

At¯ t

FB

0.1750.189 (∼ 0.7 σ)

0.275 (∼ 1.6 σ)

0.136 (∼ 1.6 σ)

0.475 (∼ 0 σ)

0.161 (∼ 0.9 σ)

0.681 (∼ 0.3 σ)

0.252

At¯ t

FB(mt¯ t< 450 GeV)

At¯ t

FB(mt¯ t> 450 GeV)

At¯ t

FB(|∆y| < 1)

At¯ t

FB(|∆y| > 1)

0.132

0.452

0.170

0.719

TABLE I: AFBwith C8

RR(1TeV/Λ)2= 4.92 and C8

LR(1TeV/Λ)2= −3.88 at the Tevtron. Here we

list the CL when containing NP effects at NLO level.

NLO best-fit point, from which we can see that higher order corrections do not change the

distribution very much.

The process of top quark pair production has been measured at the LHC, and the cross

section [72, 73] is

σATLAS

t¯t

(√S = 7 TeV) = 180 ± 18pb,

σCMS

t¯t

(√S = 7 TeV) = 158 ± 19pb, (46)

which is consistent with the SM predictions. The NP contributions at the NLO best-fit

point (4.92,−3.88) is about 3 pb, which is much smaller than the experimental uncertainty.

15