Page 1

arXiv:1110.6684v1 [hep-ph] 31 Oct 2011

Next-to-leading order QCD effect of W′on top quark

Forward-Backward Asymmetry

Kai Yan,1Jian Wang,1Ding Yu Shao,1and Chong Sheng Li1,2, ∗

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 calculations of the complete next-to-leading order (NLO) QCD corrections to

the total cross section, invariant mass distribution and the forward-backward asymmetry (AFB) of

top quark pair production mediated by W′boson. Our results show that in the best fit point in

the parameter space allowed by data at the Tevatron, the NLO corrections change the new physics

contributions to the total cross section slightly, but increase the AFBin the large invariant mass

region by about 9%. Moreover, we evaluate the total cross section and charge asymmetry (AC)

of top pair production at the LHC, and find that both total cross section and ACcan be used to

distinguish NP from SM with the integrated luminosity increasing.

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

∗Electronic address: csli@pku.edu.cn

1

Page 2

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 pro-

duction is one of the interesting observables in the top quark sector. Within the SM, AFBis

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

order (NLO) with the prediction AFB∼ 6% in the t¯t rest frame [1–6]. In the last few years,

DØ and CDF Collaborations have measured AFBat the Tevatron [7–10]. Recently, the CDF

Collaborations annouced that, for the invariant mass of the top quark pair mt¯t≥ 450 GeV,

the measured asymmetry in the t¯t rest frame is AFB= 0.475 ± 0.114[9], which differs by

3.4σ from the SM predictions AFB= 0.088±0.013. This deviation has stimulated a number

of theoretical papers on NP models, such as new gauge bosons, axigluons[11–81].

Recent studies are concerned with the problem of top asymmetry by a flavor-changing

interaction mediated by a charged vector boson, W′[13, 82], which can be described by the

following effective Lagrangian [13]:

LNP= −g′W′−

µ¯dγµ(fLPL+ fRPR)t + h.c.,(1)

where PR,L= (1±γ5)/2 are the chirality projection operators, fL,Rare the chiral couplings

of the W′boson with fermions, satisfying f2

L+ f2

R= 1, and g′is the coupling constant.

The study of this model at the leading order (LO) has been explored in Refs. [32, 83]. It is

shown that, for suitable parameters, this model can explain the AFBobserved at the Tevatron

within 1-1.5σ of the data. It is well known that the LO cross sections for process at hardron

colliders suffer form large uncertainties due to the arbitrary choice of the renormalization

and factorization scales, thus it is necessary to include higher order corrections to make

a reliable theoretical prediction. 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 will also contribute to AFB.

Therefore it is necessary to perform complete calculations of NLO contributions in the W′

model.

There is a similar work in the Z′model [84], where the NLO QCD corrections up to

O(α2

sg′2) are taken into account. In this work, we calculate both O(α2

sg′2) and O(αsg′4)

2

Page 3

NP contributions, and the latter term is not definitely smaller than the former so that it

should not be neglected. Based on the above calculation, we fit the data at the Tevatron,

including total cross section, the invariant mass distribution and the AFB, and find the

allowed parameter space. Moreover, we study the top quark pair production at the Large

Hadron Collider (LHC) induced by a W′boson at the NLO QCD level. Since the gluon

fusion channel dominates in the t¯t production process at the LHC, it is difficult to probe NP

effects on AFBfrom early LHC results. However, LHC will be able to detect the potential

NP effect on the Charge Asymmetry(AC) when the integrated luminosity increases in future.

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

top quark pair production. In section III, we present the details of the NLO calculations,

including the virtual and real corrections. In section IV we show the numerical results.

Conclusion is given in section V.

II. LEADING ORDER RESULTS

Up to NLO, the t¯t production amplitudes, including NP contributions, can be written as

Mt¯t= αsfLO

SM+ g′2fLO

NP+ α2

sfNLO

SM + αsg′2fNLO

NP. (2)

And the t¯t amplitude squared is

|Mt¯t|2= α2

sfLO

SMfLO∗

SM+ 2αsg′2Re?fLO

SM

SMfLO∗

NP

?+ g′4fLO

NPfLO∗

NP

+ 2α3

sRe?fLO

SMfNLO∗

?+ 2α2

NP

?.

sg′2?Re?fLO

SMfNLO∗

NP

?+ Re?fLO

NPfNLO∗

SM

??

+ 2αsg′4Re?fLO

NPfNLO∗

(3)

In the first line is the LO result of SM and NP. The first term in the second line is the SM

NLO result and the second term is the interference contribution. The NP NLO result is

given in the third line.

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

NP and the SM QCD interactions are shown in Fig. 1, and the LO partonic cross section

can be written as

ˆ σLO= ˆ σLO

SM+ ˆ σLO

INT+ ˆ σLO

NPS

(4)

where subscripts SM, INT and NPS denote the SM channel contributions, the interference

between SM and NP channels, and NP channel contributions, respectively. The LO partonic

3

Page 4

d

d

t

t

g

d

d

t

t

W'

FIG. 1: LO Feynman diagrams for d¯d → t¯t.

differential cross sections are given by

dˆ σLO

dcosθ

SM

=

2πβ

9ˆ s

α2

ˆ s2

s

?6m4

t− 4m2

t(t + u) + t2+ u2?, (5)

dˆ σLO

dcosθ

INT

=

2β

9ˆ s

+m2

αsg′2

m2

W′s(m2

?t2− 2m2

g′4

W′(m2

?4m4

+2f2

L

?m8

W′ − t)

W′(t + 3u)?+ 2m2

?f2

R+ f2

L

??3m6

W′u2?,

t+ m4

t

?6m2

W′ − 3t − u?

t

(6)

dˆ σLO

dcosθ

NPS

=

β

8πˆ s

+m4

m4

W′ − t)2

W′ + 4m2

?(f4

W′s + t2?− 8m2

t

R+ f4

L)?m8

t− 2m6

tm4

tt

t

W′u + 4m4

W′u2?

Rf2

t− 2m6

tt + m4

?4m2

W′s + t2?− 8m2

W′m4

W′s + 4m4

W′s2??,(7)

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

s = (p1+ p2)2,t = (p1− p3)2,u = (p1− p4)2.(8)

The relations between them are

m2

t− t = s(1 − β cosθ)/2,m2

t− u = s(1 + β cosθ)/2, (9)

where β =

?1 − 4m2

t/s, and θ is the polar angle of the outgoing top quark in the t¯t

rest frame. The colors and spins of the incoming (outgoing) particles have been averaged

(summed) over. Integrating over cosθ, we obtain the LO result of d¯d → t¯t partonic cross

section.

4

Page 5

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

cross section with the parton distribution functions (PDF) Gi/A(B)for the initial hadrons A

and B:

σLO=

?

a,b

?1

τ

dxa

?1

τ/xa

dxbGa/A(xa,µf)Gb/B(xb,µf)ˆ σLO, (10)

where τ = 4m2

t/s.

III. NEXT-TO-LEADING ORDER QCD CORRECTIONS

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

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

of a real gluon or a massless (anti)quark. For the real corrections, we used the two cutoff

phase space slicing method to subtract the infrared (IR) divergences[85].

A.Virtual corrections

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

triangle diagrams, and self-energy diagrams induced by SM QCD and NP interactions as

shown in Fig. 2 and Fig. 3, respectively.The renormalized virtual amplitudes are given as

follows

Mren

SM=

αs

4πCǫCF

αs

4πCǫCF

?

?

2

ǫUV

2

ǫUV

?

?

MLO

SM+?δZq

MLO

2+ δZt

2+ 2δgs

?MLO

NP+ Mfin

SM+ Mfin

SM, (11)

Mren

NP=

NP+?δZq

NPare ultraviolet(UV) finite terms for SM and NP

2+ δZt

2

?MLO

NP,(12)

where Cǫ= (4π)ǫ

processes. All the UV divergences in the loop diagrams are canceled by counterterms δZq

1

Γ(1−ε). Mfin

SMand Mfin

2for

the wave functions of the external fields in on-shell scheme, and δgsfor the strong coupling

constant in the MS scheme modified to decouple the top quark[86],

δZq

2= −αs

3πCǫ

?

?

1

ǫUV

1

ǫUV

−11

−

1

ǫIR

2

ǫIR

?

?

+ 4 + 3lnµ2

,(13)

δZt

2= −αs

3πCǫ

αs

4πCǫ

+

r

m2

1

ǫUV

t

?

+ lnµ2

,(14)

δgs =

?nf

32

+

αs

12πCǫ

?

r

m2

t

?

, (15)

5

Page 6

d

d

t

t

g

g

t

t

d

d

t

t

g

t

g

g

d

d

t

t

g

g

d

d

d

d

t

t

g

d

g

g

d

d

t

t

d

g

g

t

d

d

t

t

d

g

g

t

d

d

t

t

g

g

g

d

d

t

t

g

g

t

t

d

d

t

t

g

g

d

d

d

d

t

t

g

g

ug

ug

d

d

t

t

g

g

g

g

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

d

d

t

t

g

W'

d

d

d

d

t

t

W'

g

dt

d

d

t

t

W'

g

dt

d

d

t

t

g

W'

t

t

d

d

t

t

W'

t

t

g

d

d

t

t

g

d

d

W'

d

d

t

t

t

W'

g

d

d

d

t

t

d

gW'

t

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

where nf= 5 and µris the renormalization scale. The renormalized amplitudes Mren

Mren

SMand

NPare UV finite, but still contains IR divergences. The virtual corrections for subprocess

6

Page 7

q¯ q → t¯t can be expressed as:

dˆ σvirt= dˆ σvirt

SM+ dˆ σvirt

NPS+ dˆ σvirt

INT

=

1

2sdΓ2

?2Re?Mren

SMMLO∗

NP+ Mren

SM

?+ 2Re?Mren

SM

??.

NPMLO∗

NP

?

+ 2Re?Mren

SMMLO∗

NPMLO∗

(16)

We have calculated the SM contribution, and find the result agrees with that in the Ref. [87].

The one-loop correction for the cross section induced by NP interactions, with IR singular-

ities separated from finite terms, is given by

dˆ σvirt=

αs

2πCǫ

αs

2πCǫ

?(Av

?(Av

2)INT

ǫ2

IR

2)NPS

ǫ2

IR

+(Av

1)INT

ǫIR

1)NPS

ǫIR

?

?

dˆ σLO

INT

(17)

+

+(Av

dˆ σLO

NPS+ dˆ σvirt,fin

(18)

where

(Av

2)INT = −2CF,

(Av

(19)

1)INT =

CF

4

?

16ln−t1

µ2

r

+ 2ln−u1

µ2

r

+ 9lnµ2

r

m2

t

+ lnµ2

r

s

−

1 + β2

2β

lnβ + 1

1 − β− 20

?

,(20)

and

(Av

2)NPS = −2CF,

(Av

(21)

1)NPS = 2CF

?

2ln−t1

µ2

r

+ lnµ2

r

m2

t

−5

2

?

,(22)

with t1= t−m2

crosss section ˆ σLO

t, u1= u−m2

INTand ˆ σLO

t. The IR divergent terms are proportional to the LO partonic

NPS. σvirt,finis the finite terms of the virtual cross section.

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 as shown in Fig.4 and Fig.5.

The phase space integration for the real gluon emission will produce soft and collinear

singularities, which can be isolated by slicing the phase space into different regions using

suitable cutoffs. In this paper, we use the two cutoff phase space slicing method [85], which

7

Page 8

d

d

t

t

g

g

g

d

d

t

t

g

g

t

d

d

t

t

g

g

t

d

d

t

t

g

d

g

d

d

t

t

g

d

g

FIG. 4: Feynman diagrams for a gluon emission induced by SM QCD interactions. The diagrams

for a (anti)quark emission can be obtained by crossing the initial-state (anti)quark with the final-

state gluon.

d

d

t

t

g

W'

d

d

d

t

t

g

W'

t

d

d

t

t

g

d

W'

d

d

t

t

g

W'

t

FIG. 5: Feynman diagrams for a gluon emission induced by NP interactions. The diagrams for a

(anti)quark emission can be obtained by crossing the initial-state (anti)quark with the final-state

gluon.

introduces two arbitrary small cutoff parameters, i.e. soft cutoff parameters δsand collinear

parameters δc, to decompose the three-body phase space into three regions.

First, the phase space is separated into two regions by the soft cutoff parameters δs,

according to whether the energy of the emitted gluon is soft, i.e. E5? δs√s12/2, or hard,

i.e. E5 > δs√s12/2. Then the collinear cutoff parameters δc is introduced to divide the

hard gluon phase space into two regions, according to whether the Mandelstam variables

ti5≡ (pi− p5)2(i=1,2) satisfy the collinear condition |ti5| < δcs12or not. Thus we have

dˆ σReal= dˆ σS+ dˆ σHC+ dˆ σHC. (23)

The hard non-collinear term dˆ σH¯Ccan be written as,

dˆ σHC=

1

2s12

?

|M3|2dΓ3|HC

(24)

which can be evaluated numerically using standard Monte-Carlo techniques [88]. In the

following sections, we discuss the parts containing the soft and hard collinear sigularities.

8

Page 9

In the limit that the energy of the emitted gluon becomes small, i.e. E5? δs√s12/2, the

three-body cross section dˆ σScan be factorized as

dˆ σS=

?αs

2πC′

ǫ

?

?ǫ. The color charge factors Cijare

= CINT

= CINT

= CINT

4

?

i,j=1

?

dˆ σLO

INT

CINT

CINT

ij

0

+ dˆ σLO

NPS

CNPS

CNPS

ij

0

??

dS

−pi· pj

(pi· p5)(pj· p5), (25)

where C′

ǫ=

Γ(1−ǫ)

Γ(1−2ǫ)

?

CINT

CINT

CINT

4πµ2

s12

r

1234

= CF/2,CINT

CINT

33

= CINT

= CINT

44

= CAC2

F,

13 24

= −CAC2

= 0,

F,

1423

= −CF/2,

1122

(26)

and

CNPS

CNPS

13

= CNPS

= CNPS

24

= 3CACF,CNPS

= CNPS

33

= CNPS

= CNPS

44

= −3CACF,

= CNPS

11

22

= 0,CNPS

12

34

14

23

= 0.(27)

Here, CINT

The integration over the soft phase space is given by [85]:

0

= CACF and CNPS

0

= 9 are the color factors of LO diagrams in Fig. 1.

?

dS =1

π

?4

s12

?−ǫ?δs√s12/2

0

dE5E1−2ǫ

5

sin1−2ǫθ1dθ1× sin−2ǫθ2dθ2. (28)

We define

Iij=

?

dS

1

(pi· p5)(pj· p5).(29)

Then we have

I11 = I22= 0,

I33 = I44= −1

2

s

2

s

I13 = I24=1

m2

1

βlnβ + 1

1

ǫIR(−2lnδs)

?

?

t

1

ǫIR

+ Ifin

33,

I34 =

?

?1

−1

ǫIR

1 − β

?

+ Ifin

?

?

?

34,

I12 =

ǫ2

IR

++ Ifin

2ln−t1

12,

t1

1

u1

−1

−1

ǫ2

IR

+

1

ǫIR

1

ǫIR

s

+ ln

s

m2

t

s

+ 2lnδs

??

??

+ Ifin

13,

I14 = I23=

ǫ2

IR

+2ln−u1

s

+ ln

m2

t

+ 2lnδs

+ Ifin

14,(30)

9

Page 10

where all the IR sigularities in Iijhave been extracted out and for briefness, the finite terms

Ifin

αs

2πCǫ

ǫ2

IR

+αs

2πCǫ

ǫ2

IR

in which

ij

are not shown here. Now, the Eq.(25) can be rewritten as

dˆ σS=

?(AS

?(AS

2)INT

+(AS

1)int

ǫIR

+ (AS

0)INT

?

dˆ σLO

INT,

2)NPS

+(AS

1)NPS

ǫIR

+ (AS

0)NPS

?

dˆ σLO

NPS, (31)

(AS

2)INT = 2CF,

1)INT = −1

(AS

CA

?

16ln−t1

µ2

r

+ 2ln−u1

µ2

r

+ 9lnµ2

r

m2

?

t

+ lnµ2

r

s

+16lnδs−(1 + β2)

2β

ln1 + β

1 − β− 8,(32)

and

(AS

2)NPS = 2CF,

(AS

1)NPS = −2CF

?

2ln−t1

µ2

r

+ lnµ2

r

m2

t

+ 2lnδs− 1

?

. (33)

In the hard collinear region, E5> δs√s12/2 and |ti5| < δcs12, the emitted hard gluon is

collinear to one of the incoming partons and the three-body cross section is factorized as

dσHC= dσLO?αs

2πC′

ǫ

??

−1

ǫ

?

δ−ǫ

c

?Pdd(z,ǫ)Gd/p(x1/z)G¯d/p(x2)

+P¯d¯d(z,ǫ)G¯d/p(x1/z)Gd/p(x2) + (x1↔ x2)?dz

where Pijare the unregulated splitting functions in n = 4−2ǫ dimension for 0 < z < 1, which

is related to the usual Altarelli-Parisi splitting kernels [89] as Pij(z,ǫ) = Pij(z) + ǫP′

z

?1 − z

z

?−ǫ

dx1dx2,(34)

ij(z).

Explicitely, in our case,

Pdd(z) = P¯d¯d(z) = CF1 + z2

1 − z,

(35)

P′

dd(z) = P′¯d¯d(z) = −CF(1 − z). (36)

For massless d(¯d) emission, we decompose the phase space into two regions, collinear and

non-collinear, and give the expression for gd → t¯td cross section,

dσ(qg → t¯tq) =

α=d,¯d

+ dσLO?αs

+ P¯dg(z,ǫ)Gg/p(x1/z)Gd/p(x2) + (x1↔ x2)]dz

?

ˆ σC(αg → t¯tα)[Gα/p(x1)Gg/p(x2) + (x1↔ x2)]dx1dx2

2πC′

ǫ

??

−1

ǫ

?

δ−ǫ

c[Pdg(z,ǫ)Gg/p(x1/z)G¯d/p(x2)

z

?1 − z

z

?−ǫ

dx1dx2, (37)

10

Page 11

where

Pdg(z) = P¯dg(z) =1

2[z2+ (1 − z)2],P′

dg(z) = P′¯dg(z) = −z(1 − z), (38)

and

ˆ σC(αg → t¯tα) =

1

2s12

?

|M3|2(αg → t¯tα)dΓ3|HC.(39)

In order to factorize the collinear singularity into the PDF, we introduce a scale dependent

PDF in the MS convention [85],

Gα/p(x,µf) = Gα/p(x) +

?

β

?

−1

ǫ

??

αs

2π

?4πµ2

f

µ2

r

?ǫ??1

x

dz

zPαβ(z)Gβ/p(x/z).(40)

As in Ref. [85], the O(αs) collinear contribution is

dσcoll= dˆ σLO?αs

+

?

+(x1↔ x2)?dx1dx2,

where

2πC′

?Asc

ǫ

??˜Gd/p(x1,µf)G¯d/p(x2,µf) + Gd/p(x1,µf)˜G¯d/p(x2,µf)

ǫ

α=d,¯d

1(α → αg)

+ Asc

0(α → αg)

?

Gd/p(x1,µf)G¯d/p(x2,µf)

(41)

Asc

1(d → dg) = Asc

Asc

1(¯d →¯dg) = CF(2δs+ 3/2),

0 = Asc

µ2

f

1lns12

, (42)

and

Gα/p(x1,µf) =

?

β

?1−δsδαβ

x

dy

yGβ/p(x1,µf)˜Pαβ(y),(43)

with

˜Pαβ(y) = Pαβ(y)ln

?

δc1 − y

y

s12

µ2

f

?

− P′

αβ(y). (44)

Finally the NLO correction of d¯d → t¯t process can be written as

??

+

?

σNLO= dx1dx2

?Gd/p(x1,µf)G¯d/p(x2,µf) + (x1↔ x2)?(σvirt+ σS+ σH¯C) + σcoll?

dx1dx2

?Gg/p(x1,µf)Gα/p(x2,µf) + (x1↔ x2)?σC(gα → t¯tα).

α=d,¯d

?

(45)

11

Page 12

Note that all the IR divergences in the NLO total cross section are proportional to the

LO cross sections. and we find the following relations

(Av

2)INT+ (AS

2)INT= 0,(Av

1)INT+ (AS

1)INT+

?

?

α=d,¯d

Asc

1(α → αg) = 0,

(Av

2)NPS+ (AS

2)NPS= 0,(Av

1)NPS+ (AS

1)NPS+

α=d,¯d

Asc

1(α → αg) = 0. (46)

Now all the IR divergences are canceled exactly.

IV. NUMERICAL RESULTS

In the numerical calculations, we set mW′ = 400 GeV, because such a W′is readily

observed at Tevatron with an integrated luminosity of 10 fb−1, and at the LHC with an

integrated luminosity of 100 pb−1[32]. There are two independent parameters in the NP

Lagrangian. For the convenience of calculations we define the a parameter set (CV,CA),

where CV = g′(fR+ fL)/2 and CA = g′(fR− fL)/2. The mass of top quark is chosen to

be mt = 172.5 GeV. The CTEQ6L and CTEQ6M PDF sets [90] and the associated αs

functions are used for LO and NLO calculation, respectively. Both the renormalization and

factorization scales are fixed to the top quark mass unless specified otherwise.

A.scale dependence

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

From Fig. 6, we can see that the NLO corrections significantly reduce the scale dependence

for all three cases, making the theoretical predictions more reliable.

B. Tevatron constraints

AFBof top quark pair productions is defined as

AFB =

σF− σB

σF+ σB

= ANP

FB× R + ASM

FB× (1 − R)

12

Page 13

t

/m

µ

0.60.81 1.2 1.41.6 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

LO

NLO

=1

R

=0, f

L

f

µ

=

µ

,

µ

,

f

µ

µ

µ

=

r

µ

µ

µ

t

=m

=m

f

=

=

r

t

r

f

FIG. 6: Scale dependences of the total cross sections at the Tevetron. The black and the red lines

represent the LO and NLO results, respectively.

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) (47)

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. The LO and NLO total cross sections of the

interference and NP contributions can be written in terms of C2

Vand C2

A,

[σINT

t¯t

]LO =

?−(1.14)+0.22

?2.06+0.35

−0.31(C2

−0.27(C2

V+ C2

A)?

pb,(48)

?σNPS

t¯t

?

LO=

V+ C2

A)2− (2.51)+0.32

−0.40(C2

V· C2

A)?

pb,(49)

13

Page 14

FIG. 7: Values of CV and CA allowed by Tavetron data at 95% CL: σt¯t=(7.50 ± 0.48) pb ,

AFB(mt¯t> 450 GeV)=0.475±0.114, and (dσt¯t/dmt¯t)mt¯ t∈[800,1400] GeV= (0.068 ± 0.036) fb/GeV.

The blue dot (0.78, 0.78) and brown star (0.74, 0.74) represent the BFPs at LO and NLO level,

respectively. The allowed parameter region is symmetric with respect to the CAand CVaxes, so

we only display the contours where CACV> 0.

and

[σINT

t¯t

]NLO =

?−(1.42)+0.06

?2.39+0.06

−0.10)(C2

−0.09(C2

V+ C2

A)?

pb,(50)

?σNPS

t¯t

?

NLO=

V+ C2

A)2− (2.82)+0.04

−0.10(C2

V· C2

A)?

pb,(51)

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

2mt. The differences of the cross sections in the forward and backward rapidity region are

14

Page 15

given by

?σINT

?σNPS

F

− σINT

− σNPS

B

?

?

LO=

?−(0.26)−0.05

?0.68−0.10

+0.08(C2

V+ C2

A)?

pb, (52)

FB

LO=

+0.12(C2

V+ C2

A)2+ 0.054−0.007

+0.009(C2

V· C2

A)?

pb, (53)

and

?σINT

?σNPS

F

− σINT

− σNPS

B

?

?

NLO=

?−(0.40)−0.05

?0.94−0.03

+0.04(C2

V+ C2

A)?

pb, (54)

FB

NLO=

+0.04(C2

V+ C2

A)2− (0.127)−0.000

+0.002(C2

V· C2

A)?

pb. (55)

For t¯t invariant mass spectrum, we restrict our attention in the large invariant mass region,

i.e. mt¯t∈ [800,1400] GeV, where the AFBis the most obvious. The results are presented as

?mt¯ t∈[800,1400]

?dσNPS

and

?dσINT

t¯t

dmt¯t

LO

=

?−(0.014)+0.006

?0.082+0.020

−0.004(C2

V+ C2

A)?

pb

GeV, (56)

t¯t

dmt¯t

?mt¯ t∈[800,1400]

LO

=

−0.018(C2

V+ C2

A)2− (0.064)+0.007

−0.008(C2

V· C2

A)?

pb

GeV, (57)

?dσINT

?dσNPS

t¯t

dmt¯t

?mt¯ t∈[800,1400]

?mt¯ t∈[800,1400]

NLO

=

?−(0.012)+0.004

?0.117+0.014

−0.002(C2

V+ C2

A)?

pb

GeV, (58)

t¯t

dmt¯t

NLO

=

−0.010(C2

V+ C2

A)2− (0.094)+0.003

−0.004(C2

V· C2

A)?

pb

GeV, (59)

From the errors in Eqs.(48 - 59) we can see that NLO corrections reduce the dependence of

the cross sections on the renormalization and factorization scales.

In Fig. 7, we show the allowed region in the (CV,CA) plane that is consistent with the

total cross section σt¯t, AFB[9] and the spectrum of mt¯tin the large mass region [91], which

are given by

σEX

t¯t

= (7.50 ± 0.48)pb,

FB= 0.475 ± 0.114,

(60)

AEX

for mt¯t> 450GeV,(61)

?dσt¯t

dmt¯t

?mt¯ t∈[800,1400]GeV

EX

= (0.068 ± 0.036) fb/GeV.(62)

We use Monte Carlo programm MCFM [92] to get the cross section of the gluon fusion

channel gg → t¯t at the NLO QCD level. As for the process of q¯ q → t¯t, we have checked our

value with the results given by MCFM at QCD NLO level, which are well consistent in the

15