Top Production Cross Section -Matching of Relativistic and Non-Relativistic Regimes

Angelika Widl

in collaboration with

Andre Hoang, Bahman Dehnadi, Maximilian Stahlhofen, Vicent Mateu

Outline

Motivation

Relativistic Regime

Non-Relativistic Regime

Matching

Outlook

1 / 30

Motivation

top quark properties: mpolet ≈ 170 GeV, Γt ≈ 2 GeV

precise measurement of top quark properties for:

– stability of the SM vacuum

– electroweak precision tests

– new physics searches

– ...

up down strange charm bottom top

2 / 30

Motivation

Top production cross section in the pole mass scheme:

σ ( e+e- ⟶ t t )

350 360 370 380 390 4000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σQCD up to order αi

340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σNRQCD up to NNLL

3 / 30

Relativistic Regime

σtt =

∫dΠ

∣∣∣∣∣∣∣e−

e+ t

t

γ

e−

e+ t

t

γ

q

+ + + . . .

∣∣∣∣∣∣∣2

= Im

t

t

e+

e− e−

e++ + + . . .

=(4πα)2

q2Q2

t Im

t

t

+ + + . . .

= A(q2) Im

[−i∫

d4x e iqx 〈0|T jµ(x) jµ(0)|0〉], jµ(x) = ψ(x)γµψ(x)

4 / 30

Relativistic Regime

σtt =

∫dΠ

∣∣∣∣∣∣∣e−

e+ t

t

γ

e−

e+ t

t

γ

q

+ + + . . .

∣∣∣∣∣∣∣2

= Im

t

t

e+

e− e−

e++ + + . . .

=(4πα)2

q2Q2

t Im

t

t

+ + + . . .

= A(q2) Im

[−i∫

d4x e iqx 〈0|T jµ(x) jµ(0)|0〉], jµ(x) = ψ(x)γµψ(x)

4 / 30

Relativistic Regime

σtt =

∫dΠ

∣∣∣∣∣∣∣e−

e+ t

t

γ

e−

e+ t

t

γ

q

+ + + . . .

∣∣∣∣∣∣∣2

= Im

t

t

e+

e− e−

e++ + + . . .

=(4πα)2

q2Q2

t Im

t

t

+ + + . . .

= A(q2) Im

[−i∫

d4x e iqx 〈0|T jµ(x) jµ(0)|0〉], jµ(x) = ψ(x)γµψ(x)

4 / 30

Relativistic Regime

σtt =

∫dΠ

∣∣∣∣∣∣∣e−

e+ t

t

γ

e−

e+ t

t

γ

q

+ + + . . .

∣∣∣∣∣∣∣2

= Im

t

t

e+

e− e−

e++ + + . . .

=(4πα)2

q2Q2

t Im

t

t

+ + + . . .

= A(q2) Im

[−i∫

d4x e iqx 〈0|T jµ(x) jµ(0)|0〉], jµ(x) = ψ(x)γµψ(x)

4 / 30

Relativistic Regime

σtt =

∫dΠ

∣∣∣∣∣∣∣e−

e+ t

t

γ

e−

e+ t

t

γ

q

+ + + . . .

∣∣∣∣∣∣∣2

= Im

t

t

e+

e− e−

e++ + + . . .

=(4πα)2

q2Q2

t Im

t

t

+ + + . . .

= A(q2) Im

[−i∫

d4x e iqx 〈0|T jµ(x) jµ(0)|0〉], jµ(x) = ψ(x)γµψ(x)

4 / 30

Relativistic Regime

X Vacuum Polarization known up to α3 [Hoang, Mateu, Zebarjad ’08]

σQCD up to O[αi]

350 360 370 380 390 400

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

O[α3]

O[α2]

O[α1]

O[α0]

5 / 30

Relativistic Regime

X Vacuum Polarization known up to α3 [Hoang, Mateu, Zebarjad ’08]

σQCD up to O[αi]

350 360 370 380 390 400

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

O[α3]

O[α2]

O[α1]

O[α0]

5 / 30

Non-Relativistic Regime

– scales: m, |~p| ∼ mv , E ∼ mv 2 (mt ∼ 170 GeV , v ∼ α ∼ 0.1)

– large disparity of scales m� mv � mv 2 � ΛQCD

– problems:

1. large logarithms log(

E2

m2

), log

(~p 2

m2

)2. Coulomb singularity contributions of order α2

v3 , α3

v4 , α4

v5 , . . .

6 / 30

Non-Relativistic Regime

– scales: m, |~p| ∼ mv , E ∼ mv 2 (mt ∼ 170 GeV , v ∼ α ∼ 0.1)

– large disparity of scales m� mv � mv 2 � ΛQCD

– problems:

1. large logarithms log(

E2

m2

), log

(~p 2

m2

)2. Coulomb singularity contributions of order α2

v3 , α3

v4 , α4

v5 , . . .

6 / 30

Non-Relativistic Regime

– scales: m, |~p| ∼ mv , E ∼ mv 2 (mt ∼ 170 GeV , v ∼ α ∼ 0.1)

– large disparity of scales m� mv � mv 2 � ΛQCD

– problems:

1. large logarithms log(

E2

m2

), log

(~p 2

m2

)2. Coulomb singularity contributions of order α2

v3 , α3

v4 , α4

v5 , . . .

6 / 30

Non-Relativistic Regime

– scales: m, |~p| ∼ mv , E ∼ mv 2 (mt ∼ 170 GeV , v ∼ α ∼ 0.1)

– large disparity of scales m� mv � mv 2 � ΛQCD

– problems:

1. large logarithms log(

E2

m2

), log

(~p 2

m2

)

2. Coulomb singularity contributions of order α2

v3 , α3

v4 , α4

v5 , . . .

6 / 30

Non-Relativistic Regime

– scales: m, |~p| ∼ mv , E ∼ mv 2 (mt ∼ 170 GeV , v ∼ α ∼ 0.1)

– large disparity of scales m� mv � mv 2 � ΛQCD

– problems:

1. large logarithms log(

E2

m2

), log

(~p 2

m2

)2. Coulomb singularity contributions of order α2

v3 , α3

v4 , α4

v5 , . . .

6 / 30

Non-Relativistic Regime

Fields in vNRQCD and pNRQCD:

field (E ,m)

potential quark (mv 2,mv)

soft gluon (mv ,mv)

ultrasoft gluon (mv 2,mv 2)

Example:

p

−p

p′

−p′

(mv2, mv)

7 / 30

∼ Vc(~p−~p ′)2

Non-Relativistic Regime

Fields in vNRQCD and pNRQCD:

field (E ,m)

potential quark (mv 2,mv)

soft gluon (mv ,mv)

ultrasoft gluon (mv 2,mv 2)

Example:

p

−p

p′

−p′

(mv2, mv)

7 / 30

∼ Vc(~p−~p ′)2

Coulomb Singularity: LO

+ + + + . . .

= + + + + . . .

v αs α2s/v α3

s/v2

∼∫

d3p

(2π)3

( −1

E − ~p 2/m + iε

)

+

∫d3p1

(2π)3

d3p2

(2π)3

( −1

E − ~p 21/m + iε

)(4παs · CF

(~p 1 − ~p 2)2

)( −1

E − ~p 22/m + iε

)+ . . .

8 / 30

αs ∼ v !

Coulomb Singularity: LO

+ + + + . . .

= + + + + . . .

v αs α2s/v α3

s/v2

∼∫

d3p

(2π)3

( −1

E − ~p 2/m + iε

)

+

∫d3p1

(2π)3

d3p2

(2π)3

( −1

E − ~p 21/m + iε

)(4παs · CF

(~p 1 − ~p 2)2

)( −1

E − ~p 22/m + iε

)+ . . .

8 / 30

αs ∼ v !

Coulomb Singularity: LO

+ + + + . . .

= + + + + . . .

v αs α2s/v α3

s/v2

∼∫

d3p

(2π)3

( −1

E − ~p 2/m + iε

)

+

∫d3p1

(2π)3

d3p2

(2π)3

( −1

E − ~p 21/m + iε

)(4παs · CF

(~p 1 − ~p 2)2

)( −1

E − ~p 22/m + iε

)+ . . .

8 / 30

αs ∼ v !

Coulomb Singularity: LO

+ + + + . . .

= + + + + . . .

v αs α2s/v α3

s/v2

∼∫

d3p

(2π)3

( −1

E − ~p 2/m + iε

)

+

∫d3p1

(2π)3

d3p2

(2π)3

( −1

E − ~p 21/m + iε

)(4παs · CF

(~p 1 − ~p 2)2

)( −1

E − ~p 22/m + iε

)+ . . .

8 / 30

αs ∼ v !

Coulomb Singularity: LO

Green function of the non-relativistic Schrodinger equation:

Free Hamiltonian: H0 = ~p 2/m

(H0 − E)G0(~p, ~p ′) = (2π)3δ(3)(~p − ~p ′)

G0(~p, ~p ′) = (2π)3δ(3)(~p − ~p ′)−1

E − ~p 2/m

G0(~x , ~x ′,E) =

∫d3p

(2π)3

−1

E − ~p 2/me i~p (~x−~x ′)

G0(0, 0,E) ∼

9 / 30

Coulomb Singularity: LO

Green function of the non-relativistic Schrodinger equation:

Free Hamiltonian: H0 = ~p 2/m

(H0 − E)G0(~p, ~p ′) = (2π)3δ(3)(~p − ~p ′)

G0(~p, ~p ′) = (2π)3δ(3)(~p − ~p ′)−1

E − ~p 2/m

G0(~x , ~x ′,E) =

∫d3p

(2π)3

−1

E − ~p 2/me i~p (~x−~x ′)

G0(0, 0,E) ∼

9 / 30

Coulomb Singularity: LO

Green function of the non-relativistic Schrodinger equation:

Free Hamiltonian: H0 = ~p 2/m

(H0 − E)G0(~p, ~p ′) = (2π)3δ(3)(~p − ~p ′)

G0(~p, ~p ′) = (2π)3δ(3)(~p − ~p ′)−1

E − ~p 2/m

G0(~x , ~x ′,E) =

∫d3p

(2π)3

−1

E − ~p 2/me i~p (~x−~x ′)

G0(0, 0,E) ∼

9 / 30

Coulomb Singularity: LO

Green function of the non-relativistic Schrodinger equation:

Free Hamiltonian: H0 = ~p 2/m

(H0 − E)G0(~p, ~p ′) = (2π)3δ(3)(~p − ~p ′)

G0(~p, ~p ′) = (2π)3δ(3)(~p − ~p ′)−1

E − ~p 2/m

G0(~x , ~x ′,E) =

∫d3p

(2π)3

−1

E − ~p 2/me i~p (~x−~x ′)

G0(0, 0,E) ∼

9 / 30

Coulomb Singularity: LO

Full Hamiltonian: H = H0 + V

(H0 − E)G(~p, ~p ′) +

∫d3k

(2π)3V (~p − ~k)G(~k, ~p) = (2π)3δ(3)(~p − ~p ′)

G(~p, ~p ′) = G0(~p, ~p ′) +

∫d3p1

(2π)3

d3p 2

(2π)3G0(~p, ~p 1)V (~p 1−~p 2)G0(~p 2, ~p

′) + . . .

G (0, 0,E) ∼ + + + . . .

G(0, 0,E) = limr→0

m2

4π

[iv +

1

mr− αsCF

(log(−2i mvr)− 1 + 2 γE + ψ

(1− i

CFαs

2v

))]

10 / 30

Coulomb Singularity: LO

Full Hamiltonian: H = H0 + V

(H0 − E)G(~p, ~p ′) +

∫d3k

(2π)3V (~p − ~k)G(~k, ~p) = (2π)3δ(3)(~p − ~p ′)

G(~p, ~p ′) = G0(~p, ~p ′) +

∫d3p1

(2π)3

d3p 2

(2π)3G0(~p, ~p 1)V (~p 1−~p 2)G0(~p 2, ~p

′) + . . .

G (0, 0,E) ∼ + + + . . .

G(0, 0,E) = limr→0

m2

4π

[iv +

1

mr− αsCF

(log(−2i mvr)− 1 + 2 γE + ψ

(1− i

CFαs

2v

))]

10 / 30

Coulomb Singularity: LO

Full Hamiltonian: H = H0 + V

(H0 − E)G(~p, ~p ′) +

∫d3k

(2π)3V (~p − ~k)G(~k, ~p) = (2π)3δ(3)(~p − ~p ′)

G(~p, ~p ′) = G0(~p, ~p ′) +

∫d3p1

(2π)3

d3p 2

(2π)3G0(~p, ~p 1)V (~p 1−~p 2)G0(~p 2, ~p

′) + . . .

G (0, 0,E) ∼ + + + . . .

G(0, 0,E) = limr→0

m2

4π

[iv +

1

mr− αsCF

(log(−2i mvr)− 1 + 2 γE + ψ

(1− i

CFαs

2v

))]

10 / 30

Coulomb Singularity: LO

Full Hamiltonian: H = H0 + V

(H0 − E)G(~p, ~p ′) +

∫d3k

(2π)3V (~p − ~k)G(~k, ~p) = (2π)3δ(3)(~p − ~p ′)

G(~p, ~p ′) = G0(~p, ~p ′) +

∫d3p1

(2π)3

d3p 2

(2π)3G0(~p, ~p 1)V (~p 1−~p 2)G0(~p 2, ~p

′) + . . .

G (0, 0,E) ∼ + + + . . .

G(0, 0,E) = limr→0

m2

4π

[iv +

1

mr− αsCF

(log(−2i mvr)− 1 + 2 γE + ψ

(1− i

CFαs

2v

))]

10 / 30

Coulomb Singularity: LO

Full Hamiltonian: H = H0 + V

(H0 − E)G(~p, ~p ′) +

∫d3k

(2π)3V (~p − ~k)G(~k, ~p) = (2π)3δ(3)(~p − ~p ′)

G(~p, ~p ′) = G0(~p, ~p ′) +

∫d3p1

(2π)3

d3p 2

(2π)3G0(~p, ~p 1)V (~p 1−~p 2)G0(~p 2, ~p

′) + . . .

G (0, 0,E) ∼ + + + . . .

G(0, 0,E) = limr→0

m2

4π

[iv +

1

mr− αsCF

(log(−2i mvr)− 1 + 2 γE + ψ

(1− i

CFαs

2v

))]

10 / 30

Coulomb Singularity

(+ + + + . . .

)c1(ν)

v αs (1, L) α2s/v α3

s/v2 LO

αsv α2s (1, L, L2) α3

s/v (1, L) NLO

v3 αsv2 (1, L) α2

s v (1, L) α3s (1, L, L2, L3) NNLO

L = log (v)

−→ NNLO Schrodinger equation solved numerically [Hoang, Teubner ’99]

11 / 30

Coulomb Singularity

(+ + + + . . .

)c1(ν)

v αs (1, L) α2s/v α3

s/v2 LO

αsv α2s (1, L, L2) α3

s/v (1, L) NLO

v3 αsv2 (1, L) α2

s v (1, L) α3s (1, L, L2, L3) NNLO

L = log (v)

−→ NNLO Schrodinger equation solved numerically [Hoang, Teubner ’99]

11 / 30

Large Logarithms

−→ resum logs with vNRQCD

m� |~p | � E � ΛQCD

correlated scales E = ~p 2

m

renormalization scales µus , µs

logs log(

E2

µ2us

), log

(~p 2

µ2s

)subtraction velocity ν µus = mν2, µs = mν

12 / 30

Large Logarithms

−→ resum logs with vNRQCD

m� |~p | � E � ΛQCD

correlated scales E = ~p 2

m

renormalization scales µus , µs

logs log(

E2

µ2us

), log

(~p 2

µ2s

)subtraction velocity ν µus = mν2, µs = mν

12 / 30

Large Logarithms

−→ resum logs with vNRQCD

m� |~p | � E � ΛQCD

correlated scales E = ~p 2

m

renormalization scales µus , µs

logs log(

E2

µ2us

), log

(~p 2

µ2s

)subtraction velocity ν µus = mν2, µs = mν

12 / 30

Large Logarithms

−→ resum logs with vNRQCD

m� |~p | � E � ΛQCD

correlated scales E = ~p 2

m

renormalization scales µus , µs

logs log(

E2

µ2us

), log

(~p 2

µ2s

)

subtraction velocity ν µus = mν2, µs = mν

12 / 30

Large Logarithms

−→ resum logs with vNRQCD

m� |~p | � E � ΛQCD

correlated scales E = ~p 2

m

renormalization scales µus , µs

logs log(

E2

µ2us

), log

(~p 2

µ2s

)subtraction velocity ν µus = mν2, µs = mν

12 / 30

Large Logarithms

Expansion of currents:

j i = ψ(x) γ i ψ(x) =∑~p

(c1(ν)Oi

~p,1 + c2(ν)Oi~p,2

),

Cross section:

σtt = A(q2) Im

[−i∫

d4x e iqx 〈0|T jµ(x) jµ(0)|0〉]

= A(q2) Im [c1(ν) A1(ν, v ,m) + 2 c1(ν)c2(ν) A2(ν,m, v)]

13 / 30

Oi~p,1 = ψ

†~p σ

i (iσ2) χ∗−~p

Oi~p,2 =

1

m2ψ†~p~p 2σi (iσ2) χ∗−~p

A1 = i∑~p,~p ′

∫d4x e iqx 〈0|TO~p,1(x) O†

~p ′,1(0)|0〉

A2 =i

2

∑~p,~p ′

∫d4x e iqx 〈0|TO~p,1 O†~p ′,2

+ O~p,2 O†~p ′,1|0〉

Large Logarithms

Expansion of currents:

j i = ψ(x) γ i ψ(x) =∑~p

(c1(ν)Oi

~p,1 + c2(ν)Oi~p,2

),

Cross section:

σtt = A(q2) Im

[−i∫

d4x e iqx 〈0|T jµ(x) jµ(0)|0〉]

= A(q2) Im [c1(ν) A1(ν, v ,m) + 2 c1(ν)c2(ν) A2(ν,m, v)]

13 / 30

Oi~p,1 = ψ

†~p σ

i (iσ2) χ∗−~p

Oi~p,2 =

1

m2ψ†~p~p 2σi (iσ2) χ∗−~p

A1 = i∑~p,~p ′

∫d4x e iqx 〈0|TO~p,1(x) O†

~p ′,1(0)|0〉

A2 =i

2

∑~p,~p ′

∫d4x e iqx 〈0|TO~p,1 O†~p ′,2

+ O~p,2 O†~p ′,1|0〉

Large Logarithms

c1(ν) A1(ν, v ,m)

QCD vNRQCD

c1

log(

E2

m2

), log

(~p 2

m2

)log(

E2

µ2us

), log

(~p 2

µ2s

), 1εn

– ν = 1 µus = m, µs = m → determine c1

14 / 30

Large Logarithms

c1(ν) A1(ν, v ,m)

QCD vNRQCD

c1

log(

E2

m2

), log

(~p 2

m2

)log(

E2

µ2us

), log

(~p 2

µ2s

), 1εn

– ν = 1 µus = m, µs = m → determine c1

14 / 30

Large Logarithms

c1(ν) A1(ν, v ,m)

– logs in A1: log(

E2

µ2us

)∼ log

(v4

ν4

),

log(~p 2

µ2s

)∼ log

(v2

ν2

)

– ν = 1→ ν = v use RGE running for c1(ν)

– ν = v evaluate c1(ν) A1(ν, v ,m)

– matching at h ·m −→ log(h)

evaluate at ν = v · f −→ log(f )

15 / 30

Large Logarithms

c1(ν) A1(ν, v ,m)

– logs in A1: log(

E2

µ2us

)∼ log

(v4

ν4

),

log(~p 2

µ2s

)∼ log

(v2

ν2

)– ν = 1→ ν = v use RGE running for c1(ν)

– ν = v evaluate c1(ν) A1(ν, v ,m)

– matching at h ·m −→ log(h)

evaluate at ν = v · f −→ log(f )

15 / 30

Large Logarithms

c1(ν) A1(ν, v ,m)

– logs in A1: log(

E2

µ2us

)∼ log

(v4

ν4

),

log(~p 2

µ2s

)∼ log

(v2

ν2

)– ν = 1→ ν = v use RGE running for c1(ν)

– ν = v evaluate c1(ν) A1(ν, v ,m)

– matching at h ·m −→ log(h)

evaluate at ν = v · f −→ log(f )

15 / 30

Large Logarithms

(+ + + + . . .

)c1(ν)

v αs (1, L) α2s/v α3

s/v2 LO

αsv α2s (1, L, L2) α3

s/v (1, L) NLO

v3 αsv2 (1, L) α2

s v (1, L) α3s (1, L, L2, L3) NNLO

L = log (v)

16 / 30

Large Logarithms

(+ + + + . . .

)c1(ν)

v αs (1, L) α2s/v α3

s/v2 LL

αsv α2s α3

s/v NLL

v 3 αsv2 α2

sv α3s NNLL

L = log (v)

17 / 30

Non-Relativistic Regime

µm = hm, µs = hm(ν f ), µus = hm(ν f )2

Renormalization Group Improved (RGI) :[Hoang, Manohar, Stewart, Teubner ’01]

[Hoang, Stahlhofen ’14]

– ν ∼ v

– f and h variation

Fixed Order (FO) :[Hoang, Manohar, Stewart, Teubner ’01]

[Hoang, Stahlhofen ’14]

– ν ∼ 1, h ∼√v

– h variation

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

RGI

LL NLL NNLL

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

FO

LO NLO NNLO

18 / 30

Non-Relativistic Regime

µm = hm, µs = hm(ν f ), µus = hm(ν f )2

Renormalization Group Improved (RGI) :[Hoang, Manohar, Stewart, Teubner ’01]

[Hoang, Stahlhofen ’14]

– ν ∼ v

– f and h variation

Fixed Order (FO) :[Hoang, Manohar, Stewart, Teubner ’01]

[Hoang, Stahlhofen ’14]

– ν ∼ 1, h ∼√v

– h variation

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

RGI

LL NLL NNLL

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

FO

LO NLO NNLO

18 / 30

Non-Relativistic Regime

µm = hm, µs = hm(ν f ), µus = hm(ν f )2

Renormalization Group Improved (RGI) :[Hoang, Manohar, Stewart, Teubner ’01]

[Hoang, Stahlhofen ’14]

– ν ∼ v

– f and h variation

Fixed Order (FO) :[Hoang, Manohar, Stewart, Teubner ’01]

[Hoang, Stahlhofen ’14]

– ν ∼ 1, h ∼√v

– h variation

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

RGI

LL NLL NNLL

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

FO

LO NLO NNLO

18 / 30

Matching

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σNRQCD (RGI, pole mass)

LL NLL NNLL

19 / 30

Matching

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σNRQCD (RGI, pole mass)

LL NLL NNLL

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σQCD

⇒ include leading order finite width effects:

v =√

q−2mm→√

q−2m+i Γtm

19 / 30

Matching

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σNRQCD (RGI, pole mass)

LL NLL NNLL

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σQCD (pole mass scheme)

⇒ include leading order finite width effects:

v =√

q−2mm→√

q−2m+i Γtm

19 / 30

Matching

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σNRQCD (RGI, pole mass)

LL NLL NNLL

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σQCD (pole mass scheme)

q2 > 400 GeV σmatched = σQCD

q2 ∼ (2m)2 σmatched = σQCD + (σNRQCD − σexpanded )

19 / 30

Matching

QCD α0 α1 α2 α3 α4

LO v α α2

vα3

v2α4

v3...

NLO α v α2 α3

vα4

v2...

NNLO v3 α v2 α2 v α3 α4

v...

N3LO v4 α v3 α2 v2 α3 v α4 ...

...

...

...

...

...

q2 > 400 GeV σmatched = σQCD

q2 ∼ (2m)2 σmatched = σQCD + (σNRQCD − σexpanded )

20 / 30

Matching

NRQCD α0 α1 α2 α3 α4

LO v α α2

vα3

v2α4

v3...

NLO α v α2 α3

vα4

v2...

NNLO v3 α v2 α2 v α3 α4

v...

N3LO v4 α v3 α2 v2 α3 v α4 ...

...

...

...

...

...

q2 > 400 GeV σmatched = σQCD

q2 ∼ (2m)2 σmatched = σQCD + (σNRQCD − σexpanded )

20 / 30

Matching

NRQCD α0 α1 α2 α3

LO v α α2

vα3

v2α4

v3...

NLO α v α2 α3

vα4

v2...

NNLO v3 α v2 α2 v α3 α4

v...

v4 α v3 α2 v2 α3 v α4 ...

...

...

...

...

...

q2 > 400 GeV σmatched = σQCD

q2 ∼ (2m)2 σmatched = σQCD + (σNRQCD − σexpanded )

20 / 30

Matching

330 340 350 360 370 380 390

0.0

0.2

0.4

0.6

0.8

1.0

Switch-Off Function fs

q2 > 400 GeV σmatched = σQCD

q2 ∼ (2m)2 σmatched = σQCD + (σNRQCD − σexpanded )

(2m)2 < q2 < 400 GeV σmatched = σQCD + (σNRQCD − σexpanded ) · fs

20 / 30

Matching

330 340 350 360 370 380 390

0.0

0.2

0.4

0.6

0.8

1.0

Switch-Off Function fs

q2 > 400 GeV σmatched = σQCD

q2 ∼ (2m)2 σmatched = σQCD + (σNRQCD − σexpanded )

(2m)2 < q2 < 400 GeV σmatched = σQCD + (σNRQCD − σexpanded ) · fs

20 / 30

Matching

matching:

σNRQCD σQCD

LL ←→ α1

NLL ←→ α2

NNLL ←→ α3

results in the pole mass scheme:

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σmatched (RGI, pole mass)

LL, O[α1]

NLL, O[α2]

NNLL, O[α3]

21 / 30

Matching

matching:

σNRQCD σQCD

LL ←→ α1

NLL ←→ α2

NNLL ←→ α3

results in the pole mass scheme:

340 342 344 3460.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σmatched (RGI, pole mass)

LL, O[α1]

NLL, O[α2]

NNLL, O[α3]

21 / 30

Matching

1S mass X renormalon free[Hoang, Ligeti, Manohar ’98]

m1S = mpole +(CF αs(µ)mpole

) ∞∑n=1

n−1∑k=0

cn,kαs(µ)nlog

(µ

CF αs(µ)mpole

)= mpole −

2

9α2smpole + . . .

340 342 344 3460.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σmatched (RGI, 1S mass)

LL, O[α1]

NLL, O[α2]

NNLL, O[α3]

22 / 30

Matching

1S mass X renormalon free[Hoang, Ligeti, Manohar ’98]

m1S = mpole +(CF αs(µ)mpole

) ∞∑n=1

n−1∑k=0

cn,kαs(µ)nlog

(µ

CF αs(µ)mpole

)= mpole −

2

9α2smpole + . . .

340 342 344 3460.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σmatched (RGI, 1S mass)

LL, O[α1]

NLL, O[α2]

NNLL, O[α3]

22 / 30

Matching

m1S = mpole +(CF αs(µ)mpole

) ∞∑n=1

n−1∑k=0

cn,kαs(µ)nlog

(µ

CF αs(µ)mpole

)= mpole −

2

9α2smpole + . . .

Pole mass:

NRQCD α0 α1 α2 α3

LO v ...

NLO ...

NNLO ...

...

...

...

...

1S mass:

NRQCD α0 α1 α2 α3

LO v α2

v...

NLO α3

v...

NNLO α2 v ...

...

...

...

...

23 / 30

Matching

m1S = mpole +(CF αs(µ)mpole

) ∞∑n=1

n−1∑k=0

cn,kαs(µ)nlog

(µ

CF αs(µ)mpole

)= mpole −

2

9α2smpole + . . .

Pole mass:

NRQCD α0 α1 α2 α3

LO v ...

NLO ...

NNLO ...

...

...

...

...

1S mass:

NRQCD α0 α1 α2 α3

LO v α2

v...

NLO α3

v...

NNLO α2 v ...

...

...

...

...

23 / 30

Matching

MS mass :

mpole = m + m∞∑n=1

an(nl , nh)αs(m)n

= m + m αs a1 + . . . ( m = m(nl+1)(m(nl+1)) )

320 330 340 350 3600.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σQCD up to O[α3]

pole mass (mpole = 172.6 GeV)

MS mass (mMS_ = 163 GeV)

24 / 30

Matching

MS mass :

mpole = m + m∞∑n=1

an(nl , nh)αs(m)n

= m + m αs a1 + . . . ( m = m(nl+1)(m(nl+1)) )

320 330 340 350 3600.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σQCD up to O[α3]

pole mass (mpole = 172.6 GeV)

MS mass (mMS_ = 163 GeV)

24 / 30

Matching

Pole mass:

NRQCD α0 α1 α2 α3

LO v ...

NLO ...

NNLO ...

...

...

...

...

MS mass:

NRQCD α0 α1 α2 α3

α3

v5...

α2

v3...

α

vα3

v3...

LO v α2

v...

NLO α v ...

NNLO ...

...

...

...

...

25 / 30

Matching

MSR mass :[Hoang, Jain, Scimemi, Stewart ’08]

mpole = m + m∞∑n=1

an(nl , nh)αs(m)n = m + m αs a1 + . . .

mpole = mMSRn(R) + R∞∑n=1

an(nl , 0) αs(R)n =mMSRn(R) + αs R a1 + . . .

⇒ choose R ∼ mv

26 / 30

Matching

MSR mass with R ∼ mv :

320 340 360 380 4000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σQCD up to O[α3]

pole mass

MSR mass

27 / 30

Matching

Pole mass:

NRQCD α0 α1 α2 α3

LO v ...

NLO ...

NNLO ...

...

...

...

...

MSR mass:

NRQCD α0 α1 α2 α3

LO v α

vRm

α2

v3R2

m2α3

v5R3

m3...

NLO α2

vRm

α3

v3R2

m2...

NNLO α v Rm

α2

vR2

m2α3

v3R3

m3, α

3

vRm

...

...

...

...

...

R ∼ mv ∼ mα

28 / 30

Matching

Results in the MSR mass scheme:

σmatched (RGI, MSR mass)

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

σ[pb]

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

σmatched (FO, MSR mass)

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

σ[pb]

330 340 350 360 370 380 3900.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

s [GeV]

σ[pb]

29 / 30

Conclusion

Summary

– matched σNRQCD and σQCD

– implemented of the MSR mass scheme

Outlook

– N3LO corrections for the fixed order cross section

– higher order electroweak effects

Thank you for your attention!

30 / 30

Conclusion

Summary

– matched σNRQCD and σQCD

– implemented of the MSR mass scheme

Outlook

– N3LO corrections for the fixed order cross section

– higher order electroweak effects

Thank you for your attention!

30 / 30