Top-Antitop Production at Hadron Colliders

**Top**-**Antitop** **Production** **at** **Hadron** **Colliders**

Roberto BONCIANI

Labor**at**oire de Physique Sub**at**omique 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 Tev**at**ron

LHC Perspectives

St**at**us of the Theoretical calcul**at**ions

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

gener**at**ion) 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 hadroniz**at**ion 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

gener**at**ion) 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 hadroniz**at**ion 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

gener**at**ion) 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 hadroniz**at**ion 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

Tev**at**ron

To d**at**e the **Top** quark could be

produced and studied only **at** the

Tev**at**ron (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 confirm**at**ion 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 @ Tev**at**ron

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

**Top** Quark @ Tev**at**ron

Events measured **at** Tev**at**ron

[

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 @ Tev**at**ron

Events measured **at** Tev**at**ron

[

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 @ Tev**at**ron

Events measured **at** Tev**at**ron

[

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 @ Tev**at**ron

Events measured **at** Tev**at**ron

[

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 @ Tev**at**ron

Events measured **at** Tev**at**ron

[

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 @ Tev**at**ron

Events measured **at** Tev**at**ron

[

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 @ Tev**at**ron

Total Cross Section

Combin**at**ion CDF-D0 (m t = 175 GeV)

σ t¯t = N d**at**a − 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

radi**at**ive corrections (∆r)

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

determin**at**ion

∆m t

∼ 1 ∆σ t¯t

m t 5 σ t¯t

Combin**at**ion 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)

*

Tev**at**ron 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

(st**at**.) ± (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 @ Tev**at**ron

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

dΓ

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 correl**at**ions 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 @ Tev**at**ron

Tev**at**ron 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 st**at**e (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 @ Tev**at**ron

Tev**at**ron 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 st**at**e (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 @ Tev**at**ron

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

Correl**at**.

−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 accumul**at**ed d**at**a an error of ∆σ t¯t/σ t¯t ∼ 15% is expected

(domin**at**ed by st**at**istics!)

After 5 years of d**at**a 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 correl**at**ions 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 accumul**at**ed d**at**a an error of ∆σ t¯t/σ t¯t ∼ 15% is expected

(domin**at**ed by st**at**istics!)

After 5 years of d**at**a 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 correl**at**ions 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 factoriz**at**ion 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 factoriz**at**ion 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 factoriz**at**ion 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 factoriz**at**ion 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

Calcul**at**ion in Perturb**at**ion 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** Tev**at**ron

∼ 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** Tev**at**ron and +50% **at** LHC.

Scale vari**at**ion ±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** Tev**at**ron 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** Tev**at**ron and +50% **at** LHC.

Scale vari**at**ion ±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** Tev**at**ron 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** Tev**at**ron and +50% **at** LHC.

Scale vari**at**ion ±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** Tev**at**ron 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** Tev**at**ron and +50% **at** LHC.

Scale vari**at**ion ±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** Tev**at**ron 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 (perturb**at**ive region) we can have **at** all orders

Resumm**at**ion =⇒ improved perturb**at**ion 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** Tev**at**ron and +50% **at** LHC.

Scale vari**at**ion ±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** Tev**at**ron 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 Resumm**at**ion

Leading-Logs (LL)

Next-to-Leading-Logs (NLL)

Laenen et al. ’92-’95; Berger and Contopanagos ’95-’96; C**at**ani et al. ’96.

Kidonakis and Sterman ’97; R. B., C**at**ani, 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** Tev**at**ron

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:

Tev**at**ron ∆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:

Tev**at**ron ∆σ/σ ∼ 12% =⇒ ∼ ok!

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

Different groups presented approxim**at**ed 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

Integr**at**ion 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 calcul**at**ion 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 calcul**at**ion of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop m**at**rix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Ayb**at** ’08

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

Next-to-Next-to-Leading Order

The NNLO calcul**at**ion of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop m**at**rix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Ayb**at** ’08

Real Corrections

one-loop m**at**rix elements for the hadronic production of t¯t + 1 parton

tree-level m**at**rix 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** Tev**at**ron

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 calcul**at**ion of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop m**at**rix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Ayb**at** ’08

Real Corrections

one-loop m**at**rix elements for the hadronic production of t¯t + 1 parton

tree-level m**at**rix elements for the hadronic production of t¯t + 2 partons

Dittmaier, Uwer and Weinzierl ’07-’08

Subtraction Terms

Both m**at**rix elements known for t¯t + j calcul**at**ion, BUT subtraction up to 1

unresolved parton, while in a complete NNLO comput**at**ion 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 calcul**at**ion of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop m**at**rix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Ayb**at** ’08

Real Corrections

one-loop m**at**rix elements for the hadronic production of t¯t + 1 parton

tree-level m**at**rix elements for the hadronic production of t¯t + 2 partons

Dittmaier, Uwer and Weinzierl ’07-’08

Subtraction Terms

Both m**at**rix elements known for t¯t + j calcul**at**ion, BUT subtraction up to 1

unresolved parton, while in a complete NNLO comput**at**ion 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 calcul**at**ion of the top-quark pair hadro-production requires several ingredients:

Virtual Corrections

two-loop m**at**rix elements for q¯q → t¯t and gg → t¯t

interference of one-loop diagrams

Körner et al. ’05-’08; Anastasiou and Ayb**at** ’08

Real Corrections

one-loop m**at**rix elements for the hadronic production of t¯t + 1 parton

tree-level m**at**rix elements for the hadronic production of t¯t + 2 partons

Dittmaier, Uwer and Weinzierl ’07-’08

Subtraction Terms

Both m**at**rix elements known for t¯t + j calcul**at**ion, BUT subtraction up to 1

unresolved parton, while in a complete NNLO comput**at**ion 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 calcul**at**ion 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 Equ**at**ions Method for the analytic solution of the MIs

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

Laporta Algorithm and Diff. Equ**at**ions

Decomposition of the Amplitude

in terms of Scalar Integrals

(DIM. REGULARIZATION)

Identity rel**at**ions among Scalar Integrals:

Gener**at**ion of IBPs, LI and symmetry rel**at**ions

(codes written in FORM)

Output: Algebraic Linear System of equ**at**ions

on the unknown integrals

Solution of the algebraic system with a C program

Output: Rel**at**ions th**at** link Scalar Integrals to the MIs

Gener**at**ion (in FORM) of the

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

invariants (calcul**at**ion 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 – M**at**hem**at**ica 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 repe**at**ed integr**at**ions 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 iter**at**ed integr**at**ions

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

n

Z x

0

dtf w (t)G(· · · ; t)

Shuffle algebra. Integr**at**ion 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. sc**at**tering angle = π 2

Numerical evalu**at**ion 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 evalu**at**ed analytically as for the q¯q channel

R. B., Ferroglia, Gehrmann, von Manteuffel and Studerus, in prepar**at**ion

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 evalu**at**ion

of the leading-color coefficient

NO additional MI

Ferroglia, Neubert, Pecjak, and Li Yang ’09

The coefficients A, E l –I l can be evalu**at**ed analytically as for the q¯q channel

R. B., Ferroglia, Gehrmann, von Manteuffel and Studerus, in prepar**at**ion

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 evalu**at**ed analytically as for the q¯q channel

R. B., Ferroglia, Gehrmann, von Manteuffel and Studerus, in prepar**at**ion

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

Conclusions

In the last 15 years, Tev**at**ron 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 situ**at**ion 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 evalu**at**ion of the two-loop m**at**rix 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 calcul**at**ed with the same technique

and are **at** the moment under study.

The calcul**at**ion 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