Top-Antitop Production at Hadron Colliders

ggi.fi.infn.it

Top-Antitop Production at Hadron Colliders

Top-Antitop Production at Hadron Colliders

Roberto BONCIANI

Laboratoire de Physique Subatomique et de Cosmologie,

Université Joseph Fourier/CNRS-IN2P3/INPG,

F-38026 Grenoble, France

HP2.3rd GGI-Florence, September 15, 2010 – p.1/25


Plan of the Talk

General Introduction

Top Quark at the Tevatron

LHC Perspectives

Status of the Theoretical calculations

The General Framework

Total Cross Section at NLO

Analytic Two-Loop QCD Corrections

Conclusions

HP2.3rd GGI-Florence, September 15, 2010 – p.2/25


Top Quark

HP2.3rd GGI-Florence, September 15, 2010 – p.3/25


Top Quark

With a mass of m t = 173.1 ± 1.3 GeV, the TOP quark (the up-type quark of the third

generation) is the heaviest elementary particle produced so far at colliders.

Because of its mass, top quark is going to play a unique role in understanding the EW

symmetry breaking ⇒ Heavy-Quark physics crucial at the LHC.

Two production mechanisms

q

t

g

t

Pair

Production

pp(¯p) → t¯t

· · · · · ·

¯q

¯t

g

¯t

Single Top

pp(¯p) → t¯b, tq ′ (¯q ′ ), tW −

q

W +

t

b

W +

t

g

t

· · · · · ·

¯q ′

¯b

q ′ (¯q ′ ) q ′ (¯q ′ )

g

W −

Top quark does not hadronize, since it decays in about 5 ·10 −25 s (one order of magnitude

smaller than the hadronization time) =⇒ opportunity to study the quark as single particle

Spin properties

Interaction vertices

Top quark mass

Decay products: almost exclusively t → W + b (|V tb | ≫ |V td |, |V ts |)

t

V tb

W +

b

HP2.3rd GGI-Florence, September 15, 2010 – p.3/25


Top Quark

With a mass of m t = 173.1 ± 1.3 GeV, the TOP quark (the up-type quark of the third

generation) is the heaviest elementary particle produced so far at colliders.

Because of its mass, top quark is going to play a unique role in understanding the EW

symmetry breaking ⇒ Heavy-Quark physics crucial at the LHC.

Two production mechanisms

q

t

g

t

Pair

Production

pp(¯p) → t¯t

· · · · · ·

¯q

¯t

g

¯t

Single Top

pp(¯p) → t¯b, tq ′ (¯q ′ ), tW −

q

W +

t

b

W +

t

g

t

· · · · · ·

¯q ′

¯b

q ′ (¯q ′ ) q ′ (¯q ′ )

g

W −

Top quark does not hadronize, since it decays in about 5 ·10 −25 s (one order of magnitude

smaller than the hadronization time) =⇒ opportunity to study the quark as single particle

Spin properties

Interaction vertices

Top quark mass

Decay products: almost exclusively t → W + b (|V tb | ≫ |V td |, |V ts |)

t

V tb

W +

b

HP2.3rd GGI-Florence, September 15, 2010 – p.3/25


Top Quark

With a mass of m t = 173.1 ± 1.3 GeV, the TOP quark (the up-type quark of the third

generation) is the heaviest elementary particle produced so far at colliders.

Because of its mass, top quark is going to play a unique role in understanding the EW

symmetry breaking ⇒ Heavy-Quark physics crucial at the LHC.

Two production mechanisms

q

t

g

t

Pair

Production

pp(¯p) → t¯t

· · · · · ·

¯q

¯t

g

¯t

Single Top

pp(¯p) → t¯b, tq ′ (¯q ′ ), tW −

q

W +

t

b

W +

t

g

t

· · · · · ·

¯q ′

¯b

q ′ (¯q ′ ) q ′ (¯q ′ )

g

W −

Top quark does not hadronize, since it decays in about 5 ·10 −25 s (one order of magnitude

smaller than the hadronization time) =⇒ opportunity to study the quark as single particle

Spin properties

Interaction vertices

Top quark mass

Decay products: almost exclusively t → W + b (|V tb | ≫ |V td |, |V ts |)

t

V tb

W +

b

HP2.3rd GGI-Florence, September 15, 2010 – p.3/25


Top Quark

Tevatron

To date the Top quark could be

produced and studied only at the

Tevatron (discovery 1995)

p¯p collisions at √ s = 1.96 TeV

L ∼ 6.5fb −1 reached in 2009

O(10 3 ) t¯t pairs produced so far

Only recently confirmation of single-t

LHC

Running since end 2009

pp collisions at √ s = 7 (14) TeV

LHC will be a factory for heavy quarks

(L ∼ 10 33 −10 34 cm −2 s −1 , t¯t at ∼1Hz!)

Even in the first low-luminosity phase (2

years ∼ 1fb −1 @7 TeV) ∼ O(10 4 ) registered

t¯t pairs

HP2.3rd GGI-Florence, September 15, 2010 – p.3/25


Top Quark @ Tevatron

HP2.3rd GGI-Florence, September 15, 2010 – p.4/25


Top Quark @ Tevatron

Events measured at Tevatron

[

p¯p → t¯t → W + bW −¯b → lνlνb¯b

σ t¯t ∼ 7pb p¯p → t¯t → W + bW −¯b → lνq¯q ′ b¯b

p¯p → t¯t → W + bW −¯b → q¯q ′ q¯q ′ b¯b

Dilepton ∼ 10%

Lep+jets ∼ 44%

All jets ∼ 46%

HP2.3rd GGI-Florence, September 15, 2010 – p.4/25


Top Quark @ Tevatron

Events measured at Tevatron

[

p¯p → t¯t → W + bW −¯b → lνlνb¯b

σ t¯t ∼ 7pb p¯p → t¯t → W + bW −¯b → lνq¯q ′ b¯b

p¯p → t¯t → W + bW −¯b → q¯q ′ q¯q ′ b¯b

Dilepton ∼ 10%

Lep+jets ∼ 44%

All jets ∼ 46%

2 high-p T lept, ≥ 2 jets and ME

HP2.3rd GGI-Florence, September 15, 2010 – p.4/25


Top Quark @ Tevatron

Events measured at Tevatron

[

p¯p → t¯t → W + bW −¯b → lνlνb¯b

σ t¯t ∼ 7pb p¯p → t¯t → W + bW −¯b → lνq¯q ′ b¯b

p¯p → t¯t → W + bW −¯b → q¯q ′ q¯q ′ b¯b

Dilepton ∼ 10%

Lep+jets ∼ 44%

All jets ∼ 46%

2 high-p T lept, ≥ 2 jets and ME

1 isol high-p T lept, ≥ 4 jets and ME

HP2.3rd GGI-Florence, September 15, 2010 – p.4/25


Top Quark @ Tevatron

Events measured at Tevatron

[

p¯p → t¯t → W + bW −¯b → lνlνb¯b

σ t¯t ∼ 7pb p¯p → t¯t → W + bW −¯b → lνq¯q ′ b¯b

p¯p → t¯t → W + bW −¯b → q¯q ′ q¯q ′ b¯b

Dilepton ∼ 10%

Lep+jets ∼ 44%

All jets ∼ 46%

2 high-p T lept, ≥ 2 jets and ME

1 isol high-p T lept, ≥ 4 jets and ME

NO lept, ≥ 6 jets and low ME

HP2.3rd GGI-Florence, September 15, 2010 – p.4/25


Top Quark @ Tevatron

Events measured at Tevatron

[

p¯p → t¯t → W + bW −¯b → lνlνb¯b

σ t¯t ∼ 7pb p¯p → t¯t → W + bW −¯b → lνq¯q ′ b¯b

p¯p → t¯t → W + bW −¯b → q¯q ′ q¯q ′ b¯b

Dilepton ∼ 10%

Lep+jets ∼ 44%

All jets ∼ 46%

2 high-p T lept, ≥ 2 jets and ME

1 isol high-p T lept, ≥ 4 jets and ME

NO lept, ≥ 6 jets and low ME

Background Processes

g

q ′

g

q

q

g

q, l −

q

W +

g

g

Z, γ

g

q

W +

g

¯q

¯q ¯q, l +

¯q

W −

W+jets

QCD

Drell-Yan

Di-boson

HP2.3rd GGI-Florence, September 15, 2010 – p.4/25


Top Quark @ Tevatron

Events measured at Tevatron

[

p¯p → t¯t → W + bW −¯b → lνlνb¯b

σ t¯t ∼ 7pb p¯p → t¯t → W + bW −¯b → lνq¯q ′ b¯b

p¯p → t¯t → W + bW −¯b → q¯q ′ q¯q ′ b¯b

Dilepton ∼ 10%

Lep+jets ∼ 44%

All jets ∼ 46%

2 high-p T lept, ≥ 2 jets and ME

NO lept, ≥ 6 jets and low ME

1 isol high-p T lept, ≥ 4 jets and ME

Background Processes

g

q

q ′

g

W +

g

q q g q, l − q

Reduction of the background: b-tagging crucial

g

g

¯q ¯q ¯q, l + ¯q

W+jets

QCD

Drell-Yan

Di-boson

Z, γ

W +

g

W −

HP2.3rd GGI-Florence, September 15, 2010 – p.4/25


Top Quark @ Tevatron

Total Cross Section

Combination CDF-D0 (m t = 175 GeV)

σ t¯t = N data − N bkgr

ǫ L

σ t¯t = 7.0 ± 0.6 pb (∆σ t¯t/σ t¯t ∼ 9%)

Top-quark Mass

Fundamental parameter of the SM. A precise

measurement useful to constraint Higgs mass from

radiative corrections (∆r)

A possible extraction: σ t¯t =⇒ need of precise theoretical

determination

∆m t

∼ 1 ∆σ t¯t

m t 5 σ t¯t

Combination CDF-D0

m t = 173.1 ± 1.3 GeV (0.75%)

14

0

CDF-I di-l

D0-I di-l

*

CDF-II di-l

*

D0-II di-l

CDF-I l+j

D0-I l+j

*

CDF-II l+j

*

D0-II l+j

CDF-I all-j

*

CDF-II all-j

*

CDF-II trk

Mass of the Top Quark (*Preliminary)

*

Tevatron March’09

hep-ex/0903.2503

CDF March’07

150 160 170 180 190 200

2

(GeV/c )

m top

167.4 ±10.3 ± 4.9

168.4 ±12.3 ± 3.6

171.2 ± 2.7 ± 2.9

174.7 ± 2.9 ± 2.4

176.1 ± 5.1 ± 5.3

180.1 ± 3.9 ± 3.6

172.1 ± 0.9 ± 1.3

173.7 ± 0.8 ± 1.6

186.0 ±10.0 ± 5.7

174.8 ± 1.7 ± 1.9

175.3 ± 6.2 ± 3.0

173.1 ±

0.6 ± 1.1

(stat.) ± (syst.)

χ 2 /dof = 6.3/10.0 (79%)

12.4 ± 1.5 ± 2.2

HP2.3rd GGI-Florence, September 15, 2010 – p.5/25


Top Quark @ Tevatron

W helicity fractions F i = B(t → bW + (λ W = i)) (i = −1,0,1) measured fitting the

distribution in θ ∗ (the angle between l + in the W + rest frame and W + direction in the t rest frame)

1

Γ


d cos θ ∗ = 3 4 F 0 sin 2 θ ∗ + 3 8 F −(1 − cos θ ∗ ) 2 + 3 8 F +(1 + cos θ ∗ ) 2

F 0 + F + + F − = 1

F 0 = 0.66 ± 0.16 ± 0.05 F + = −0.03 ± 0.06 ± 0.03

Spin correlations measured fitting the double distribution

(θ 1 (θ 2 ) is the angle between the dir of flight of l 1 (l 2 ) in the t(¯t) rest frame and the t(¯t) direction in the t¯t rest frame)

1

N

d 2 N

d cos θ 1 dcos θ 2

= 1 4 (1 + κ cos θ 1 cos θ 2 )

−0.455 < κ < 0.865 (68% CL)

Forward-Backward Asymmetry A FB = N(y t > 0) − N(y t < 0)

N(y t > 0) + N(y t < 0)

A FB = (19.3 ± 6.5(sta) ± 2.4(sys))%

HP2.3rd GGI-Florence, September 15, 2010 – p.6/25


Top Quark @ Tevatron

Tevatron searches of physics BSM in top events

New production mechanisms via new spin-1 or spin-2 resonances: q¯q → Z ′ → t¯t in

lepton+jets and all hadronic events (bumps in the invariant-mass distribution)

Top charge measurements (recently excluded Q t = −4/3)

Anomalous couplings

L = − √ g j

ff

γ

2¯b

µ (V L P L + V R P R ) + iσµν (p t − p b ) ν

(g L P L + g R P R )

M W

From helicity fractions

tW − µ

From asymmetries in the final state (for instance A FB = 3/4 (F + − F − ))

Forward-backward asymmetry

Non SM Top decays. Search for charged Higgs: t → H + b → q¯q ′ b(τνb)

Search for heavy t ′ → W + b in lepton+jets

HP2.3rd GGI-Florence, September 15, 2010 – p.7/25


Top Quark @ Tevatron

Tevatron searches of physics BSM in top events

New production mechanisms via new spin-1 or spin-2 resonances: q¯q → Z ′ → t¯t in

lepton+jets and all hadronic events (bumps in the invariant-mass distribution)

Top charge measurements (recently excluded Q t = −4/3)

Anomalous couplings

L = − √ g j

ff

γ

2¯b

µ (V L P L + V R P R ) + iσµν (p t − p b ) ν

(g L P L + g R P R )

M W

of New Physics so far

tWµ


From helicity fractions

From asymmetries in the final state (for instance A FB = 3/4 (F + − F − ))

Forward-backward asymmetry

No Evidence

Non SM Top decays. Search for charged Higgs: t → H + b → q¯q ′ b(τνb)

Search for heavy t ′ → W + b in lepton+jets

HP2.3rd GGI-Florence, September 15, 2010 – p.7/25


Top Quark @ Tevatron

Value Lum fb −1 SM value SM-like?

m t 173.1 ± 0.6 ± 1.1 GeV up to 4.8 / /

σ t¯t 7.0 ± 0.3 ± 0.4 ± 0.4 pb (m t = 175 GeV) 2.8 6.7 pb YES

W -

helicity

F 0 = 0.66 ± 0.16 ± 0.05

F + = −0.03 ± 0.06 ± 0.03

1.9 F 0 = 0.7

F + = 0

YES

Spin

Correlat.

−0.455 < κ < 0.865(68% CL) 2.8 κ = 0.8 YES

A FB 0.19 ± 0.07 ± 0.02 3.2 0.05 @NLO YES

Γ t < 13.1 GeV (95% CL) 1.0 1.5 GeV YES

τ t cτ t < 52.5 µm (95% CL) 0.3 ∼ 10 −16 m YES

BR (t → Wb)/(t → Wq) > 0.61 (95% CL) 0.2 ∼ 100% YES

Charge Exclude Q t = −4/3 (87% CL) 1.5 2/3 YES

HP2.3rd GGI-Florence, September 15, 2010 – p.8/25


LHC Perspectives

Cross Section

Top Mass

With 100 pb −1 of accumulated data an error of ∆σ t¯t/σ t¯t ∼ 15% is expected

(dominated by statistics!)

After 5 years of data taking an error of ∆σ t¯t/σ t¯t ∼ 5% is expected

With 1 fb −1 Mass accuracy: ∆m t ∼ 1 − 3 GeV

Top Properties

W helicity fractions and spin correlations with 10 fb −1 =⇒ 1-5%

Top-quark charge. With 1 fb −1 we could be able to determine Q t = 2/3 with an

accuracy of ∼ 15%

Sensitivity to new physics

all the above mentioned points

Narrow resonances: with 1 fb −1 possible discovery of a Z ′ of M Z ′ ∼ 700 GeV with

σ pp→Z ′ →t¯t ∼ 11 pb

HP2.3rd GGI-Florence, September 15, 2010 – p.9/25


LHC Perspectives

Cross Section

Top Mass

With 100 pb −1 of accumulated data an error of ∆σ t¯t/σ t¯t ∼ 15% is expected

(dominated by statistics!)

After 5 years of data taking an error of ∆σ t¯t/σ t¯t ∼ 5% is expected

With 1 fb −1 Mass accuracy: ∆m t ∼ 1 − 3 GeV

Top Properties

W helicity fractions and spin correlations with 10 fb −1 =⇒ 1-5%

Top-quark charge. With 1 fb −1 we could be able to determine Q t = 2/3 with an

accuracy of ∼ 15%

Sensitivity to new physics

all the above mentioned points

Narrow resonances: with 1 fb −1 possible discovery of a Z ′ of M Z ′ ∼ 700 GeV with

σ pp→Z ′ →t¯t ∼ 11 pb

HP2.3rd GGI-Florence, September 15, 2010 – p.9/25


Top-Anti Top Pair Production

HP2.3rd GGI-Florence, September 15, 2010 – p.10/25


Top-Anti Top Pair Production

According to the factorization theorem, the process h 1 + h 2 → t¯t + X can be sketched as in

the figure:

h 1 {p}

f(x 1 )

t

q, g

¯q, g

H.S.

X

h 2 {p, ¯p}

f(x 2 )

¯t

σ t¯t

h 1 ,h 2

= X i,j

Z 1

0

Z 1

dx 1 dx 2 f h1 ,i(x 1 , µ F )f h2 ,j(x 2 , µ F ) ˆσ ij (ŝ, m t , α s (µ R ), µ F , µ R )

0

s = `p h1 + p h2´2 , ŝ = x1 x 2 s

HP2.3rd GGI-Florence, September 15, 2010 – p.10/25


Top-Anti Top Pair Production

According to the factorization theorem, the process h 1 + h 2 → t¯t + X can be sketched as in

the figure:

h 1 {p}

f(x 1 )

t

h 2 {p, ¯p}

f(x 2 )

q, g

¯q, g

H.S.

¯t

X

PDFs: Universal Part

Evolution with the factorization scale

predicted by the theory

σ t¯t

h 1 ,h 2

= X i,j

Z 1

0

Z 1

dx 1 dx 2 f h1 ,i(x 1 , µ F )f h2 ,j(x 2 , µ F ) ˆσ ij (ŝ, m t , α s (µ R ), µ F , µ R )

0

s = `p h1 + p h2´2 , ŝ = x1 x 2 s

HP2.3rd GGI-Florence, September 15, 2010 – p.10/25


Top-Anti Top Pair Production

According to the factorization theorem, the process h 1 + h 2 → t¯t + X can be sketched as in

the figure:

h 1 {p}

f(x 1 )

t

h 2 {p, ¯p}

f(x 2 )

q, g

¯q, g

H.S.

Partonic Cross Section

Process dependent part

Calculation in Perturbation Theory

¯t

X

σ t¯t

h 1 ,h 2

= X i,j

Z 1

0

Z 1

dx 1 dx 2 f h1 ,i(x 1 , µ F )f h2 ,j(x 2 , µ F ) ˆσ ij (ŝ, m t , α s (µ R ), µ F , µ R )

0

s = `p h1 + p h2´2 , ŝ = x1 x 2 s

HP2.3rd GGI-Florence, September 15, 2010 – p.10/25


The Cross Section: LO

HP2.3rd GGI-Florence, September 15, 2010 – p.11/25


The Cross Section: LO

q(p 1 ) + ¯q(p 2 ) −→ t(p 3 ) + ¯t(p 4 )

q

¯t

Dominant at Tevatron

∼ 85%

¯q

t

g(p 1 ) + g(p 2 ) −→ t(p 3 ) + ¯t(p 4 )

g

¯t

Dominant at LHC

∼ 90%

g

t

σ LO

t¯t

(LHC, m t = 171 GeV) = 583 pb ± 30%

σ LO

t¯t

(Tev, m t = 171 GeV) = 5.92 pb ± 44%

HP2.3rd GGI-Florence, September 15, 2010 – p.11/25


The Cross Section: NLO

HP2.3rd GGI-Florence, September 15, 2010 – p.12/25


The Cross Section: NLO

Fixed Order

The NLO QCD corrections are quite sizable: + 25% at Tevatron and +50% at LHC.

Scale variation ±15%.

Nason, Dawson, Ellis ’88-’90; Beenakker, Kuijf, van Neerven, Smith ’89-’91;

Mangano, Nason, Ridolfi ’92; Frixione et al. ’95; Czakon and Mitov ’08

Melnikov and Schulze ’09; Bernreuther and Si ’10

Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC.

Beenakker et al. ’94 Bernreuther, Fuecker, and Si ’05-’08

Kühn, Scharf, and Uwer ’05-’06; Moretti, Nolten, and Ross ’06.

HP2.3rd GGI-Florence, September 15, 2010 – p.12/25


The Cross Section: NLO

Fixed Order

The NLO QCD corrections are quite sizable: + 25% at Tevatron and +50% at LHC.

Scale variation ±15%.

Nason, Dawson, Ellis ’88-’90; Beenakker, Kuijf, van Neerven, Smith ’89-’91;

Mangano, Nason, Ridolfi ’92; Frixione et al. ’95; Czakon and Mitov ’08

Melnikov and Schulze ’09; Bernreuther and Si ’10

Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC.

Beenakker et al. ’94 Bernreuther, Fuecker, and Si ’05-’08

Kühn, Scharf, and Uwer ’05-’06; Moretti, Nolten, and Ross ’06.

The QCD corrections to processes involving at least two large energy scales

(ŝ, m 2 t ≫ Λ2 QCD

) are characterized by a logarithmic behavior in the vicinity of the

boundary of the phase space

σ ∼ X n,m

C n,m α n S lnm (1 − ρ) m ≤ 2n

HP2.3rd GGI-Florence, September 15, 2010 – p.12/25


The Cross Section: NLO

Fixed Order

The NLO QCD corrections are quite sizable: + 25% at Tevatron and +50% at LHC.

Scale variation ±15%.

Nason, Dawson, Ellis ’88-’90; Beenakker, Kuijf, van Neerven, Smith ’89-’91;

Mangano, Nason, Ridolfi ’92; Frixione et al. ’95; Czakon and Mitov ’08

Melnikov and Schulze ’09; Bernreuther and Si ’10

Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC.

Inelasticity parameter

Beenakker et al. ’94 Bernreuther, Fuecker, and Si ’05-’08

Kühn, Scharf, and Uwer ’05-’06; Moretti, Nolten, and Ross ’06.

ρ = 4m2 t

→ 1

ŝ

The QCD corrections to processes involving at least two large energy scales

(ŝ, m 2 t ≫ Λ2 QCD

) are characterized by a logarithmic behavior in the vicinity of the

boundary of the phase space

σ ∼ X n,m

C n,m α n S lnm (1 − ρ) m ≤ 2n

HP2.3rd GGI-Florence, September 15, 2010 – p.12/25


The Cross Section: NLO

Fixed Order

The NLO QCD corrections are quite sizable: + 25% at Tevatron and +50% at LHC.

Scale variation ±15%.

Nason, Dawson, Ellis ’88-’90; Beenakker, Kuijf, van Neerven, Smith ’89-’91;

Mangano, Nason, Ridolfi ’92; Frixione et al. ’95; Czakon and Mitov ’08

Melnikov and Schulze ’09; Bernreuther and Si ’10

Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC.

Inelasticity parameter

Beenakker et al. ’94 Bernreuther, Fuecker, and Si ’05-’08

Kühn, Scharf, and Uwer ’05-’06; Moretti, Nolten, and Ross ’06.

ρ = 4m2 t

→ 1

ŝ

The QCD corrections to processes involving at least two large energy scales

(ŝ, m 2 t ≫ Λ2 QCD

) are characterized by a logarithmic behavior in the vicinity of the

boundary of the phase space

σ ∼ X n,m

C n,m α n S lnm (1 − ρ) m ≤ 2n

Even if α S ≪ 1 (perturbative region) we can have at all orders

Resummation =⇒ improved perturbation theory

α n S lnm (1 − ρ) ∼ O(1)

HP2.3rd GGI-Florence, September 15, 2010 – p.12/25


The Cross Section: NLO

Fixed Order

The NLO QCD corrections are quite sizable: + 25% at Tevatron and +50% at LHC.

Scale variation ±15%.

Nason, Dawson, Ellis ’88-’90; Beenakker, Kuijf, van Neerven, Smith ’89-’91;

Mangano, Nason, Ridolfi ’92; Frixione et al. ’95; Czakon and Mitov ’08

Melnikov and Schulze ’09; Bernreuther and Si ’10

Mixed NLO QCD-EW corrections are small: - 1% at Tevatron and -0.5% at LHC.

Beenakker et al. ’94 Bernreuther, Fuecker, and Si ’05-’08

Kühn, Scharf, and Uwer ’05-’06; Moretti, Nolten, and Ross ’06.

All-order Soft-Gluon Resummation

Leading-Logs (LL)

Next-to-Leading-Logs (NLL)

Laenen et al. ’92-’95; Berger and Contopanagos ’95-’96; Catani et al. ’96.

Kidonakis and Sterman ’97; R. B., Catani, Mangano, and Nason ’98-’03.

Next-to-Next-to-Leading-Logs (NNLL)

Moch and Uwer ’08; Beneke et al. ’09-’10; Czakon et al. ’09; Kidonakis ’09;

Ahrens et al. ’10

HP2.3rd GGI-Florence, September 15, 2010 – p.12/25


NLO+NLL Theoretical Prediction

TEVATRON

σ NLO+NLL (Tev, m

t¯t

t = 171 GeV,CTEQ6.5) = 7.61 +0.30(3.9%)

−0.53(6.9%)

(scales)

+0.53(7%)

−0.36(4.8%)

(PDFs) pb

LHC

σ NLO+NLL (LHC, m

t¯t

t = 171 GeV,CTEQ6.5) = 908 +82(9.0%)

−85(9.3%)

(scales)

+30(3.3%)

−29(3.2%)

(PDFs) pb

M. Cacciari, S. Frixione, M. Mangano, P. Nason, and G. Ridolfi, JHEP 0809:127,2008

12

10

σ pp

– → tt

- [pb] at Tevatron

1400

1200

σ pp → tt

- [pb] at LHC

8

6

4

1000

800

600

400

2

NLL res (CTEQ65)

200

NLL res (CTEQ65)

0

165 170 175 180

m t

[GeV]

0

165 170 175 180

m t

[GeV]

S. Moch and P. Uwer, Phys. Rev. D 78 (2008) 034003

HP2.3rd GGI-Florence, September 15, 2010 – p.13/25


Measurement Requirements for σ t¯t

HP2.3rd GGI-Florence, September 15, 2010 – p.14/25


Measurement Requirements for σ t¯t

Experimental requirements for σ t¯t:

Tevatron ∆σ/σ ∼ 12% =⇒ ∼ ok!

LHC (14 TeV, high luminosity) ∆σ/σ ∼ 5% ≪ current theoretical prediction!!

HP2.3rd GGI-Florence, September 15, 2010 – p.14/25


Measurement Requirements for σ t¯t

Experimental requirements for σ t¯t:

Tevatron ∆σ/σ ∼ 12% =⇒ ∼ ok!

LHC (14 TeV, high luminosity) ∆σ/σ ∼ 5% ≪ current theoretical prediction!!

Different groups presented approximated higher-order results for σ t¯t

Including scale dep at NNLO, NNLL soft-gluon contributions, Coulomb corrections

σ NNLOappr (Tev, m

t¯t

t = 173 GeV,MSTW2008) = 7.04 +0.24

−0.36

σ NNLOappr (LHC, m

t¯t

t = 173 GeV,MSTW2008) = 887 +9

−33

Integration of the Invariant mass distribution at NLO+NNLL

(scales)

+0.14

−0.14

(scales)

+15

−15

(PDFs) pb

(PDFs) pb

Kidonakis and Vogt ’08; Moch and Uwer ’08; Langenfeld, Moch, and Uwer ’09

σ NLO+NNLL (Tev, m

t¯t

t = 173.1 GeV,MSTW2008) = 6.48 +0.17

−0.21

σ NLO+NNLL (LHC, m

t¯t

t = 173.1 GeV,MSTW2008) = 813 +50

−36

(scales)

+0.32

−0.25

(scales)

+30

−35

(PDFs) pb

(PDFs) pb

V. Ahrens, A. Ferroglia, M. Neubert, B. D. Pecjak, L. L. Yang, arXiv:1006.4682

HP2.3rd GGI-Florence, September 15, 2010 – p.14/25


Next-to-Next-to-Leading Order

HP2.3rd GGI-Florence, September 15, 2010 – p.15/25


Next-to-Next-to-Leading Order

The NNLO calculation of the top-quark pair hadro-production requires several ingredients:

HP2.3rd GGI-Florence, September 15, 2010 – p.15/25


Next-to-Next-to-Leading Order

The NNLO calculation of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop matrix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Aybat ’08

HP2.3rd GGI-Florence, September 15, 2010 – p.15/25


Next-to-Next-to-Leading Order

The NNLO calculation of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop matrix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Aybat ’08

Real Corrections

one-loop matrix elements for the hadronic production of t¯t + 1 parton

tree-level matrix elements for the hadronic production of t¯t + 2 partons

Dittmaier, Uwer and Weinzierl ’07-’08

HP2.3rd GGI-Florence, September 15, 2010 – p.15/25


Next-to-Next-to-Leading Order

p¯p → t¯t + 1 jet

Important for a deeper understanding of the t¯t prod

(possible structure of the top-quark)

Important for the charge asymmetry at Tevatron

Technically complex involving multi-leg NLO diagrams

σ t¯t+j (LHC) = 376.2 +17

−48 pb (with p T,jet,cut = 50 GeV)

confirmed by G. Bevilacqua, M. Czakon, C.G. Papadopoulos, M. Worek, Phys. Rev. Lett. 104 (2010) 162002

( ) [ ] dσ fb

dp T,jet GeV

( ) [ ] dσ fb

dp T,jet GeV

10

p¯p → t¯t + jet + X

√ s = 1.96TeV

1000

pp → t¯t + jet + X

√ s = 14TeV

t¯t + 2j

1

100

0.1

NLO

LO

10

NLO

LO

0.01

0

2.0

1.5

1.0

0.5

0

50

50

100 150 200

K = NLO/LO

100

150

200

p T,jet [GeV]

250

250

300

300

0

2.0

1.5

1.0

0.5

0

100

100

200

200

300 400 500

K = NLO/LO

300

400

500

p T,jet [GeV]

600

600

700

700

S. Dittmaier, P. Uwer and S. Weinzierl,

Eur. Phys. J. C 59 (2009) 625

HP2.3rd GGI-Florence, September 15, 2010 – p.15/25


Next-to-Next-to-Leading Order

The NNLO calculation of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop matrix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Aybat ’08

Real Corrections

one-loop matrix elements for the hadronic production of t¯t + 1 parton

tree-level matrix elements for the hadronic production of t¯t + 2 partons

Dittmaier, Uwer and Weinzierl ’07-’08

Subtraction Terms

Both matrix elements known for t¯t + j calculation, BUT subtraction up to 1

unresolved parton, while in a complete NNLO computation of σ t¯t we need

subtraction terms with up to 2 unresolved partons.

HP2.3rd GGI-Florence, September 15, 2010 – p.15/25


Next-to-Next-to-Leading Order

The NNLO calculation of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop matrix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Aybat ’08

Real Corrections

one-loop matrix elements for the hadronic production of t¯t + 1 parton

tree-level matrix elements for the hadronic production of t¯t + 2 partons

Dittmaier, Uwer and Weinzierl ’07-’08

Subtraction Terms

Both matrix elements known for t¯t + j calculation, BUT subtraction up to 1

unresolved parton, while in a complete NNLO computation of σ t¯t we need

subtraction terms with up to 2 unresolved partons.

=⇒ Need an extension of the subtraction methods at the NNLO.

Gehrmann-De Ridder, Ritzmann ’09, Daleo et al. ’09,

Boughezal et al. ’10, Glover, Pires ’10, Czakon ’10

HP2.3rd GGI-Florence, September 15, 2010 – p.15/25


Next-to-Next-to-Leading Order

The NNLO calculation of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop matrix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Aybat ’08

Real Corrections

one-loop matrix elements for the hadronic production of t¯t + 1 parton

tree-level matrix elements for the hadronic production of t¯t + 2 partons

Dittmaier, Uwer and Weinzierl ’07-’08

Subtraction Terms

Both matrix elements known for t¯t + j calculation, BUT subtraction up to 1

unresolved parton, while in a complete NNLO computation of σ t¯t we need

subtraction terms with up to 2 unresolved partons.

=⇒ Need an extension of the subtraction methods at the NNLO.

Gehrmann-De Ridder, Ritzmann ’09, Daleo et al. ’09,

Boughezal et al. ’10, Glover, Pires ’10, Czakon ’10

HP2.3rd GGI-Florence, September 15, 2010 – p.15/25


Two-Loop Corrections to q¯q → t¯t

HP2.3rd GGI-Florence, September 15, 2010 – p.16/25


Two-Loop Corrections to q¯q → t¯t

|M| 2 (s, t, m, ε) = 4π2 α 2 » “

s αs

” “ αs

” 2

A 0 + A 1 + A2 + O `αs´–

3

N c π π

A 2 = A (2×0)

2

+ A (1×1)

2

A (2×0)

2

= N c C F

h

N 2 c A + B + C N 2 c

+ N l


N c D l + E l

N c

«


+N h N c D h + E «

h

+ Nl 2 N F l + N l N h F lh + Nh 2 F h

c

i

218 two-loop diagrams contribute to the 10 different color coefficients

HP2.3rd GGI-Florence, September 15, 2010 – p.16/25


Two-Loop Corrections to q¯q → t¯t

|M| 2 (s, t, m, ε) = 4π2 α 2 » “

s αs

” “ αs

” 2

A 0 + A 1 + A2 + O `αs´–

3

N c π π

A 2 = A (2×0)

2

+ A (1×1)

2

A (2×0)

2

= N c C F

h

N 2 c A + B + C N 2 c

+ N l


N c D l + E l

N c

«


+N h N c D h + E «

h

+ Nl 2 N F l + N l N h F lh + Nh 2 F h

c

i

218 two-loop diagrams contribute to the 10 different color coefficients

The whole A (2×0)

2

is known numerically

Czakon ’08.

HP2.3rd GGI-Florence, September 15, 2010 – p.16/25


Two-Loop Corrections to q¯q → t¯t

|M| 2 (s, t, m, ε) = 4π2 α 2 » “

s αs

” “ αs

” 2

A 0 + A 1 + A2 + O `αs´–

3

N c π π

A 2 = A (2×0)

2

+ A (1×1)

2

A (2×0)

2

= N c C F

h

N 2 c A + B + C N 2 c

+ N l


N c D l + E l

N c

«


+N h N c D h + E «

h

+ Nl 2 N F l + N l N h F lh + Nh 2 F h

c

i

218 two-loop diagrams contribute to the 10 different color coefficients

The whole A (2×0)

2

is known numerically

Czakon ’08.

The coefficients D i , E i , F i , and A are known analytically (agreement with known res)

R. B., Ferroglia, Gehrmann, Maitre, and Studerus ’08-’09

HP2.3rd GGI-Florence, September 15, 2010 – p.16/25


Two-Loop Corrections to q¯q → t¯t

|M| 2 (s, t, m, ε) = 4π2 α 2 » “

s αs

” “ αs

” 2

A 0 + A 1 + A2 + O `αs´–

3

N c π π

A 2 = A (2×0)

2

+ A (1×1)

2

A (2×0)

2

= N c C F

h

N 2 c A + B + C N 2 c

+ N l


N c D l + E l

N c

«


+N h N c D h + E «

h

+ Nl 2 N F l + N l N h F lh + Nh 2 F h

c

i

218 two-loop diagrams contribute to the 10 different color coefficients

The whole A (2×0)

2

is known numerically

Czakon ’08.

The coefficients D i , E i , F i , and A are known analytically (agreement with known res)

R. B., Ferroglia, Gehrmann, Maitre, and Studerus ’08-’09

The poles of A (2×0)

2

(and therefore of B and C) are known analytically

Ferroglia, Neubert, Pecjak, and Li Yang ’09

HP2.3rd GGI-Florence, September 15, 2010 – p.16/25


Two-Loop Corrections to q¯q → t¯t

D i , E i , F i

come from the corrections involving a closed (light or heavy) fermionic loop:

A the leading-color coefficient, comes from the planar diagrams:

The calculation is carried out analytically using:

Laporta Algorithm for the reduction of the dimensionally-regularized scalar integrals

(in terms of which we express the |M| 2 ) to the Master Integrals (MIs)

Differential Equations Method for the analytic solution of the MIs

HP2.3rd GGI-Florence, September 15, 2010 – p.17/25


Laporta Algorithm and Diff. Equations

Decomposition of the Amplitude

in terms of Scalar Integrals

(DIM. REGULARIZATION)

Identity relations among Scalar Integrals:

Generation of IBPs, LI and symmetry relations

(codes written in FORM)

Output: Algebraic Linear System of equations

on the unknown integrals

Solution of the algebraic system with a C program

Output: Relations that link Scalar Integrals to the MIs

Generation (in FORM) of the

System of DIFF. EQs. on the ext. kin.

invariants (calculation of the MIs)

IBPs, LI, Symm. rel.

System of 1st-order linear DIFF. EQs.

Solution in Laurent series of (D-4). Coeff expressedin terms of HPLs

PUBLIC

PROGRAMS

AIR – Maple package

(C. Anastasiou and A. Lazopoulos, JHEP 0407 (2004) 046)

FIRE – Mathematica package (A. V. Smirnov, JHEP 0810 (2008) 107)

REDUZE – C++/GiNaC package

(C. Studerus, Comput. Phys. Commun. 181 (2010) 1293)

HP2.3rd GGI-Florence, September 15, 2010 – p.18/25


Master Integrals for N l and N h

1 MI 1 MI 1 MI 2 MIs 1 MI 1 MI

1 MI 1 MI 2 MIs 1 MI 2 MIs 3 MIs

2 MIs 1 MI 1 MI 1 MI 2 MIs 2 MIs

18 irreducible two-loop topologies (26 MIs)

R. B., A. Ferroglia, T. Gehrmann, D. Maitre, and C. Studerus, JHEP 0807 (2008) 129.

HP2.3rd GGI-Florence, September 15, 2010 – p.19/25


Master Integrals for the Leading Color Coeff

2 MIs 2 MIs 2 MIs 2 MIs 2 MIs

2 MIs 2 MIs 2 MIs 3 MIs

For the leading color coefficient there are 9 additional irreducible topologies (19 MIs)

R. B., A. Ferroglia, T. Gehrmann, and C. Studerus, JHEP 0908 (2009) 067.

HP2.3rd GGI-Florence, September 15, 2010 – p.20/25


Example

= 1 X−1

m 6

i=−4

A i ǫ i + O(ǫ 0 )

A −4 =

A −3 =

A −2 =

A −1 =

x 2

24(1 − x) 4 (1 + y) ,

x 2

96(1 − x) 4 (1 + y)

h

i

−10G(−1; y) + 3G(0; x) − 6G(1; x) ,

x 2 h

i

48(1 − x) 4 −5ζ(2) − 6G(−1; y)G(0; x)+12G(−1; y)G(1; x)+8G(−1,−1; y) ,

(1 + y)

x 2

48(1 − x) 4 (1 + y)

h

−13ζ(3) + 38ζ(2)G(−1; y) + 9ζ(2)G(0; x) + 6ζ(2)G(1; x) − 24ζ(2)G (−1/y; x)

+24G(0; x)G(−1, −1; y) − 24G(1; x)G(−1, −1; y) − 12G (−1/y; x) G(−1, −1; y)

−12G(−y; x)G(−1, −1; y) − 6G(0; x)G(0, −1; y) + 6G (−1/y; x) G(0, −1; y) + 6G(−y; x)G(0, −1; y)

+12G(−1; y)G(1, 0; x) − 24G(−1; y)G(1, 1; x) − 6G(−1; y)G (−1/y, 0; x) + 12G(−1; y)G (−1/y, 1; x)

−6G(−1; y)G(−y, 0; x) + 12G(−1; y)G(−y, 1; x) + 16G(−1, −1, −1; y) − 12G(−1, 0, −1; y)

−12G(0, −1, −1; y) + 6G(0, 0, −1; y) + 6G(1, 0, 0; x) − 12G(1, 0, 1; x) − 12G(1, 1, 0; x) + 24G(1, 1, 1; x)

−6G (−1/y, 0, 0; x) + 12G (−1/y, 0, 1; x) + 6G (−1/y, 1, 0; x) − 12G (−1/y, 1, 1; x) + 6G(−y, 1, 0; x)

i

−12G(−y, 1, 1; x)

HP2.3rd GGI-Florence, September 15, 2010 – p.21/25


Example

= 1 X−1

m 6

i=−4

A i ǫ i + O(ǫ 0 )

A −4 =

A −3 =

A −2 =

A −1 =

x 2

24(1 − x) 4 (1 + y) ,

x 2

96(1 − x) 4 (1 + y)

h

i

−10G(−1; y) + 3G(0; x) − 6G(1; x) ,

1- and 2-dim GHPLs

x 2 h

i

48(1 − x) 4 −5ζ(2) − 6G(−1; y)G(0; x)+12G(−1; y)G(1; x)+8G(−1,−1; y) ,

(1 + y)

x 2 h

48(1 − x) 4 −13ζ(3) + 38ζ(2)G(−1; y) + 9ζ(2)G(0; x) + 6ζ(2)G(1; x) − 24ζ(2)G (−1/y; x)

(1 + y)

ρ = 4m2 t → 1

+24G(0; x)G(−1, −1; y) − 24G(1; x)G(−1, −1; y) − 12G (−1/y; x) G(−1, −1; y)

ŝ

−12G(−y; x)G(−1, −1; y) − 6G(0; x)G(0, −1; y) + 6G (−1/y; x) G(0, −1; y) + 6G(−y; x)G(0, −1; y)

+12G(−1; y)G(1, 0; x) − 24G(−1; y)G(1, 1; x) − 6G(−1; y)G (−1/y, 0; x) + 12G(−1; y)G (−1/y, 1; x)

−6G(−1; y)G(−y, 0; x) + 12G(−1; y)G(−y, 1; x) + 16G(−1, −1, −1; y) − 12G(−1, 0, −1; y)

−12G(0, −1, −1; y) + 6G(0, 0, −1; y) + 6G(1, 0, 0; x) − 12G(1, 0, 1; x) − 12G(1, 1, 0; x) + 24G(1, 1, 1; x)

−6G (−1/y, 0, 0; x) + 12G (−1/y, 0, 1; x) + 6G (−1/y, 1, 0; x) − 12G (−1/y, 1, 1; x) + 6G(−y, 1, 0; x)

i

−12G(−y, 1, 1; x)

HP2.3rd GGI-Florence, September 15, 2010 – p.21/25


GHPLs

One- and two-dimensional Generalized Harmonic Polylogarithms (GHPLs) are defined

as repeated integrations over set of basic functions. In the case at hand

(

)

1

f w (x) =

x − w , with w ∈ 0,1, −1, −y, − 1 y , 1 2 ± i√ 3

2

j

1

f w (y) =

y − w , with w ∈ 0, 1, −1, −x, − 1 x ,1 − 1 ff

x − x

The weight-one GHPLs are defined as

G(0; x) = ln x , G(w; x) =

Z x

0

dtf w (t)

Higher weight GHPLs are defined by iterated integrations

G(0,0, · · · , 0; x) = 1 | {z } n! lnn x , G(w, · · · ; x) =

n

Z x

0

dtf w (t)G(· · · ; t)

Shuffle algebra. Integration by parts identities

Remiddi and Vermaseren ’99, Gehrmann and Remiddi ’01-’02, Aglietti and R. B. ’03,

Vollinga and Weinzierl ’04, R. B., A. Ferroglia, T. Gehrmann, and C. Studerus ’09

HP2.3rd GGI-Florence, September 15, 2010 – p.22/25


Coefficient A

Finite part of A

Threshold expansion versus exact result

18.0

17.5

100

A

17.0

50

16.5

0

0.8

η =

0.6

Φ

0.4

s

4m 2 − 1 ,

0.2

0

5

Η

− m2

φ = −t

s

10

16.0

15.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6

β =

s

Β

1 − 4m2

s

partonic c.m. scattering angle = π 2

Numerical evaluation of the GHPLs with GiNaC C++ routines.

Vollinga and Weinzierl ’04

HP2.3rd GGI-Florence, September 15, 2010 – p.23/25


Two-Loop Corrections to gg → t¯t

HP2.3rd GGI-Florence, September 15, 2010 – p.24/25


Two-Loop Corrections to gg → t¯t

|M| 2 (s, t, m, ε) = 4π2 α 2 » “

s αs

” “ αs

” 2

A 0 + A 1 + A2 + O `αs´–

3

N c π π

A 2 = A (2×0)

2

+ A (1×1)

2

A (2×0)

2

= (N 2 c − 1) „

N 3 c A + N cB + 1 N c

C + 1

N 3 c

D + N 2 c N lE l + N 2 c N hE h

+N l F l + N h F h + N l

N 2 c

G l + N h

N 2 c

G h + N c N 2 l H l + N c N 2 h H h

+N c N l N h H lh + N2 l

N c

I l + N2 h

N c

I h + N lN h

N c

I lh

«

789 two-loop diagrams contribute to 16 different color coefficients

No numeric result for A (2×0)

2

yet

The poles of A (2×0)

2

are known analytically

Ferroglia, Neubert, Pecjak, and Li Yang ’09

The coefficients A, E l –I l can be evaluated analytically as for the q¯q channel

R. B., Ferroglia, Gehrmann, von Manteuffel and Studerus, in preparation

HP2.3rd GGI-Florence, September 15, 2010 – p.24/25


Two-Loop Corrections to gg → t¯t

|M| 2 (s, t, m, ε) = 4π2 α 2 » “

s αs

” “ αs

” 2

A 0 + A 1 + A2 + O `αs´–

3

N c π π

A 2 = A (2×0)

2

+ A (1×1)

2

A (2×0)

2

= (N 2 c − 1) „

N 3 c A + N cB + 1 N c

C + 1

N 3 c

D + N 2 c N lE l + N 2 c N hE h

+N l F l + N h F h + N l

N 2 c

G l + N h

G h + N c Nl 2 H l + N c Nh 2 H h

N 2 c

+N c N l N h H lh + N2 l

I l + N2 h

I h + N «

lN h

I lh

N c N c N c

789 two-loop diagrams contribute to 16 different color coefficients

No numeric result for A (2×0)

2

yet

The poles of A (2×0)

2

are known analytically

- We finished the evaluation

of the leading-color coefficient

NO additional MI

Ferroglia, Neubert, Pecjak, and Li Yang ’09

The coefficients A, E l –I l can be evaluated analytically as for the q¯q channel

R. B., Ferroglia, Gehrmann, von Manteuffel and Studerus, in preparation

HP2.3rd GGI-Florence, September 15, 2010 – p.24/25


Two-Loop Corrections to gg → t¯t

|M| 2 (s, t, m, ε) = 4π2 α 2 » “

s αs

” “ αs

” 2

A 0 + A 1 + A2 + O `αs´–

3

N c π π

A 2 = A (2×0)

2

+ A (1×1)

2

A (2×0)

2

= (N 2 c − 1) „

N 3 c A + N cB + 1 N c

C + 1

N 3 c

D + N 2 c N lE l + N 2 c N hE h

+N l F l + N h F h + N l

N 2 c

G l + N h

N 2 c

G h + N c N 2 l H l + N c N 2 h H h

+N c N l N h H lh + N2 l

N c

I l + N2 h

N c

I h + N lN h

N c

I lh

«

789 two-loop diagrams contribute to 16 different color coefficients

- For the light-fermion contrib

one missing topology to reduce

- up to now

No numeric result for A (2×0)

2

yet

7 additional MIs

The poles of A (2×0)

2

are known analytically

Ferroglia, Neubert, Pecjak, and Li Yang ’09

The coefficients A, E l –I l can be evaluated analytically as for the q¯q channel

R. B., Ferroglia, Gehrmann, von Manteuffel and Studerus, in preparation

HP2.3rd GGI-Florence, September 15, 2010 – p.24/25


Conclusions

In the last 15 years, Tevatron explored top-quark properties reaching a remarkable

experimental accuracy. The top mass could be measured with ∆m t /m t = 0.75% and the

production cross section with ∆σ t¯t/σ t¯t = 9%. Other observables could be measured

only with bigger errors.

At the LHC the situation will further improve. The production cross section of t¯t pairs is

expected to reach the accuracy of ∆σ t¯t/σ t¯t = 5%!!

This experimental precision requires a complete NNLO theoretical analysis.

In this talk I briefly reviewed the analytic evaluation of the two-loop matrix elements.

The corrections involving a fermionic loop (light or heavy) in the q¯q channel are

completed, together with the leading color coefficient.

Analogous corrections in the gg channel can be calculated with the same technique

and are at the moment under study.

The calculation of the crossed diagrams and of the diagrams with a heavy loop have still

to be afforded.

HP2.3rd GGI-Florence, September 15, 2010 – p.25/25

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