Top Production Cross Section · Motivation Top production cross section in the pole mass scheme:...
Transcript of Top Production Cross Section · Motivation Top production cross section in the pole mass scheme:...
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
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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)
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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
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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
)+ . . .
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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) ∼
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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