Top Production Cross Section Matching of Relativistic and Non-Relativistic Regimes
Angelika Widl
in collaboration with
André Hoang, Bahman Dehnadi, Maximilian Stahlhofen, Vicent Mateu
Outline
Motivation
Relativistic Regime
Non-Relativistic Regime
Matching
Outlook
1 / 30
Motivation
top quark properties: mtpole ≈ 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 )
σ[pb]
σ QCD up to order α i
σ NRQCD up to NNLL
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
350
360
370
s [GeV]
380
390
400
340
350
360
370
380
390
s [GeV]
3 / 30
Relativistic Regime
4 / 30
Relativistic Regime
Z
σt t̄ =
dΠ e−
t
γ
+
q
e+
t̄
+
2
+ ... 4 / 30
Relativistic Regime
Z
σt t̄ =
dΠ 
= Im 
e−
e−
t
γ
+
+
q
e+
t̄
t

e−
e+
e+
2
+ ... +
+
+ ...
t̄
4 / 30
Relativistic Regime
Z
σt t̄ =
dΠ 
= Im 
e−
e−
t
γ
+
+
q
e+
t̄
t

e−
e+
e+
2
+ ... +
+
+ ...
t̄

(4πα)2 2
=
Qt Im 
q2

t
+
+
+ ...
t̄
4 / 30
Relativistic Regime
Z
σt t̄ =
dΠ 
= Im 
e−
e−
2
+ ... t
γ
+
+
q
e+
t̄
t
e+
e+

e−
+
+
+ ...
t̄

(4πα)2 2
=
Qt Im 
q2

t
+
+
+ ...
t̄
Z
= A(q 2 ) Im −i d 4 x e iqx h0|T jµ (x) j µ (0)|0i ,
j µ (x) = ψ̄(x)γ µ ψ(x)
4 / 30
Relativistic Regime
X
Vacuum Polarization known up to α3
[Hoang, Mateu, Zebarjad ’08]
5 / 30
Relativistic Regime
Vacuum Polarization known up to α3
[Hoang, Mateu, Zebarjad ’08]
σQCD up to O[αi]
1.4
1.2
1.0
0.8
O[α3 ]
σ[pb]
X
O[α2 ]
0.6
O[α1 ]
O[α0 ]
0.4
0.2
0.0
350
360
370
380
390
400
s [GeV]
5 / 30
Non-Relativistic Regime
6 / 30
Non-Relativistic Regime
– scales: m, |~p | ∼ mv , E ∼ mv 2
(mt ∼ 170 GeV , v ∼ α ∼ 0.1)
6 / 30
Non-Relativistic Regime
– scales: m, |~p | ∼ mv , E ∼ mv 2
– large disparity of scales
(mt ∼ 170 GeV , v ∼ α ∼ 0.1)
m mv mv 2 ΛQCD
6 / 30
Non-Relativistic Regime
– scales: m, |~p | ∼ mv , E ∼ mv 2
– large disparity of scales
(mt ∼ 170 GeV , v ∼ α ∼ 0.1)
m mv mv 2 ΛQCD
– problems:
1. large logarithms
log
E2
m2
, log
~2
p
m2
6 / 30
Non-Relativistic Regime
– scales: m, |~p | ∼ mv , E ∼ mv 2
– large disparity of scales
(mt ∼ 170 GeV , v ∼ α ∼ 0.1)
m mv mv 2 ΛQCD
– problems:
1. large logarithms
2. Coulomb singularity
log
E2
m2
, log
~2
p
m2
contributions of order
α2 α3 α4
, v4 , v5 ,
v3
...
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 )
7 / 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′
(mv 2, mv)
−p
−p′
∼
Vc
(~
p −~
p 0 )2
7 / 30
Coulomb Singularity: LO
+
+
+
+...
8 / 30
Coulomb Singularity: LO
=
+
+
+
+...
+
+
+
+...
8 / 30
Coulomb Singularity: LO
=
v
+
+
+
+
αs
+
+...
+
αs2 /v
+...
αs3 /v 2
αs ∼ v !
8 / 30
Coulomb Singularity: LO
=
+
+
+
+
v
Z
∼
d 3p
(2π)3
+
+
−1
+...
αs2 /v
αs
+...
αs3 /v 2
αs ∼ v !
~ 2 /m + iε
E −p
Z
+
d 3 p1 d 3 p2
(2π)3 (2π)3
−1
~ 21 /m + iε
E −p
4παs · CF
~ 2 )2
(~
p1 −p
−1
~ 22 /m + iε
E −p
+ ...
8 / 30
Coulomb Singularity: LO
Green function of the non-relativistic Schrödinger equation:
Free Hamiltonian: H0 = p~ 2 /m
9 / 30
Coulomb Singularity: LO
Green function of the non-relativistic Schrödinger equation:
Free Hamiltonian: H0 = p~ 2 /m
~ 0 ) = (2π)3 δ (3) (~
~ 0)
(H0 − E ) G0 (~
p, p
p−p
9 / 30
Coulomb Singularity: LO
Green function of the non-relativistic Schrödinger equation:
Free Hamiltonian: H0 = p~ 2 /m
~ 0 ) = (2π)3 δ (3) (~
~ 0)
(H0 − E ) G0 (~
p, p
p−p
~ 0 ) = (2π)3 δ (3) (~
~ 0)
G0 (~
p, p
p−p
G0 (~
x , x~ ,0 E ) =
Z
−1
~ 2 /m
E −p
0
d 3p
−1
e i~p (~x −~x )
(2π)3 E − p
~ 2 /m
9 / 30
Coulomb Singularity: LO
Green function of the non-relativistic Schrödinger equation:
Free Hamiltonian: H0 = p~ 2 /m
~ 0 ) = (2π)3 δ (3) (~
~ 0)
(H0 − E ) G0 (~
p, p
p−p
~ 0 ) = (2π)3 δ (3) (~
~ 0)
G0 (~
p, p
p−p
G0 (~
x , x~ ,0 E ) =
Z
−1
~ 2 /m
E −p
0
d 3p
−1
e i~p (~x −~x )
(2π)3 E − p
~ 2 /m
G0 (0, 0, E ) ∼
9 / 30
Coulomb Singularity: LO
Full Hamiltonian: H = H0 + V
10 / 30
Coulomb Singularity: LO
Full Hamiltonian: H = H0 + V
~ 0) +
(H0 − E ) G (~
p, p
Z
d 3k
~) = (2π)3 δ (3) (~
~ 0)
V (~
p−~
k) G (~
k, p
p−p
(2π)3
10 / 30
Coulomb Singularity: LO
Full Hamiltonian: H = H0 + V
~ 0) +
(H0 − E ) G (~
p, p
~ 0 ) = G0 (~
~ 0) +
G (~
p, p
p, p
Z
Z
d 3k
~) = (2π)3 δ (3) (~
~ 0)
V (~
p−~
k) G (~
k, p
p−p
(2π)3
d 3 p1 d 3 p 2
~ 1 )V (~
~ 0) + . . .
G0 (~
p, p
p 1 −~
p 2 )G0 (~
p 2, p
(2π)3 (2π)3
10 / 30
Coulomb Singularity: LO
Full Hamiltonian: H = H0 + V
~ 0) +
(H0 − E ) G (~
p, p
~ 0 ) = G0 (~
~ 0) +
G (~
p, p
p, p
G (0, 0, E ) ∼
Z
Z
d 3k
~) = (2π)3 δ (3) (~
~ 0)
V (~
p−~
k) G (~
k, p
p−p
(2π)3
d 3 p1 d 3 p 2
~ 1 )V (~
~ 0) + . . .
G0 (~
p, p
p 1 −~
p 2 )G0 (~
p 2, p
(2π)3 (2π)3
+
+
+...
10 / 30
Coulomb Singularity: LO
Full Hamiltonian: H = H0 + V
~ 0) +
(H0 − E ) G (~
p, p
~ 0 ) = G0 (~
~ 0) +
G (~
p, p
p, p
G (0, 0, E ) ∼
G (0, 0, E ) = lim
r →0
Z
Z
d 3k
~) = (2π)3 δ (3) (~
~ 0)
V (~
p−~
k) G (~
k, p
p−p
(2π)3
d 3 p1 d 3 p 2
~ 1 )V (~
~ 0) + . . .
G0 (~
p, p
p 1 −~
p 2 )G0 (~
p 2, p
(2π)3 (2π)3
+
+
+...
C F αs
m2
1
iv +
− αs CF log(−2i mvr ) − 1 + 2 γE + ψ 1 − i
4π
mr
2v
10 / 30
Coulomb Singularity
+
v
+
αs (1, L)
+ . . . c1 (ν)
+
α2s /v
α3s /v 2
LO
L = log (v )
11 / 30
Coulomb Singularity
+
v
v3
+
+ . . . c1 (ν)
+
αs (1, L)
α2s /v
α3s /v 2
LO
αs v
α2s (1, L, L2 )
α3s /v (1, L)
NLO
αs v 2 (1, L)
α2s v (1, L)
α3s (1, L, L2 , L3 )
NNLO
L = log (v )
−→ NNLO Schrödinger equation solved numerically [Hoang, Teubner ’99]
11 / 30
Large Logarithms
−→ resum logs with vNRQCD
m |~
p | E ΛQCD
12 / 30
Large Logarithms
−→ resum logs with vNRQCD
m |~
p | E ΛQCD
correlated scales
E =
~2
p
m
12 / 30
Large Logarithms
−→ resum logs with vNRQCD
m |~
p | E ΛQCD
~2
p
m
correlated scales
E =
renormalization scales
µus , µs
12 / 30
Large Logarithms
−→ resum logs with vNRQCD
m |~
p | E ΛQCD
~2
p
m
correlated scales
E =
renormalization scales
µus , µs
2
2
log µE2 , log p~µ2
logs
us
s
12 / 30
Large Logarithms
−→ resum logs with vNRQCD
m |~
p | E ΛQCD
~2
p
m
correlated scales
E =
renormalization scales
µus , µs
2
2
log µE2 , log p~µ2
logs
subtraction velocity ν
us
s
2
µus = mν , µs = mν
12 / 30
Large Logarithms
Expansion of currents:
i
j i = ψ̄(x) γ i ψ(x) =
X
~
p
c1 (ν) O~pi ,1 + c2 (ν) O~pi ,2 ,
†
i
∗
O~p,1 = ψ~p σ (iσ2 ) χ−~p
i
O~p,2 =
1 † 2 i
~ σ (iσ2 ) χ∗
ψ p
−~
p
m2 ~p
13 / 30
Large Logarithms
Expansion of currents:
i
j i = ψ̄(x) γ i ψ(x) =
X
c1 (ν) O~pi ,1 + c2 (ν) O~pi ,2 ,
i
O~p,2 =
~
p
†
i
∗
O~p,1 = ψ~p σ (iσ2 ) χ−~p
1 † 2 i
~ σ (iσ2 ) χ∗
ψ p
−~
p
m2 ~p
Cross section:
Z
σt t̄ = A(q 2 ) Im −i d 4 x e iqx h0|T jµ (x) j µ (0)|0i
= A(q 2 ) Im [c1 (ν) A1 (ν, v , m) + 2 c1 (ν)c2 (ν) A2 (ν, m, v )]
A1 = i
XZ
4
d x e
iqx
†
h0|T O~p,1 (x) O~
(0)|0i
p 0 ,1
~
p ,~
p0
A2 =
Z
i X
4
iqx
†
†
d x e h0|T O~p,1 O~
+ O~p,2 O~
|0i
p 0 ,2
p 0 ,1
2
0
~
p ,~
p
13 / 30
Large Logarithms
c1 (ν) A1 (ν, v , m)
QCD
vNRQCD
c1
log
E2
m2
, log
~2
p
m2
log
E2
µ2
us
, log
~2
p
µ2
s
,
1
εn
14 / 30
Large Logarithms
c1 (ν) A1 (ν, v , m)
QCD
vNRQCD
c1
log
– ν=1
E2
m2
, log
~2
p
m2
log
E2
µ2
us
, log
~2
p
µ2
s
,
1
εn
µus = m, µs = m → determine c1
14 / 30
Large Logarithms
c1 (ν) A1 (ν, v , m)
– logs in A1 : log
log
E2
µ2
us
∼ log
v4
ν4
~2
p
µ2
s
∼ log
v2
ν2
,
15 / 30
Large Logarithms
c1 (ν) A1 (ν, v , m)
– logs in A1 : log
log
E2
µ2
us
∼ log
v4
ν4
~2
p
µ2
s
∼ log
v2
ν2
,
– ν = 1 → ν = v use RGE running for c1 (ν)
– ν = v evaluate c1 (ν) A1 (ν, v , m)
15 / 30
Large Logarithms
c1 (ν) A1 (ν, v , m)
– logs in A1 : log
log
E2
µ2
us
∼ log
v4
ν4
~2
p
µ2
s
∼ 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
+
v
v3
+
+ . . . c1 (ν)
+
αs (1, L)
α2s /v
α3s /v 2
LO
αs v
α2s (1, L, L2 )
α3s /v (1, L)
NLO
αs v 2 (1, L)
α2s v (1, L)
α3s (1, L, L2 , L3 )
NNLO
L = log (v )
16 / 30
Large Logarithms
+
v
v3
+
+ . . . c1 (ν)
+
αs (1, L)
αs2 /v
αs3 /v 2
LL
αs v
αs2
αs3 /v
NLL
αs v 2
αs2 v
αs3
NNLL
L = log (v )
17 / 30
Non-Relativistic Regime
µm = h m, µs = h m(ν f ), µus = h m(ν f )2
18 / 30
Non-Relativistic Regime
µm = h m, µs = h m(ν f ), µus = h m(ν f )2
Renormalization Group Improved (RGI) :
[Hoang, Manohar, Stewart, Teubner ’01]
[Hoang, Stahlhofen ’14]
– ν∼v
– f and h variation
RGI
1.4
1.2
σ[pb]
1.0
0.8
0.6
0.4
0.2
LL
NLL
NNLL
0.0
330
340
350
360
370
380
390
s [GeV]
18 / 30
Non-Relativistic Regime
µm = h m, µs = h m(ν f ), µus = h m(ν f )2
Renormalization Group Improved (RGI) :
Fixed Order (FO) :
[Hoang, Manohar, Stewart, Teubner ’01]
[Hoang, Manohar, Stewart, Teubner ’01]
[Hoang, Stahlhofen ’14]
[Hoang, Stahlhofen ’14]
– ν∼v
– ν ∼ 1, h ∼
– f and h variation
– h variation
FO
1.4
1.4
1.2
1.2
1.0
1.0
0.8
0.8
σ[pb]
σ[pb]
RGI
√
v
0.6
0.4
0.6
0.4
0.2
0.2
LL
NLL
NNLL
0.0
LO
NLO
NNLO
0.0
330
340
350
360
s [GeV]
370
380
390
330
340
350
360
370
380
390
s [GeV]
18 / 30
Matching
19 / 30
Matching
σNRQCD (RGI, pole mass)
σQCD
1.4
1.4
1.2
1.2
1.0
σ[pb]
σ[pb]
1.0
0.8
0.6
0.8
0.6
0.4
0.4
0.2
LL
NLL
NNLL
0.0
330
340
350
360
s [GeV]
370
380
390
0.2
0.0
330
340
350
360
370
380
390
s [GeV]
⇒ include leading order finite width effects:
q
q
q−2m+i Γt
v = q−2m
→
m
m
19 / 30
Matching
σNRQCD (RGI, pole mass)
σQCD (pole mass scheme)
1.4
1.4
1.2
1.2
1.0
σ[pb]
σ[pb]
1.0
0.8
0.6
0.8
0.6
0.4
0.4
0.2
LL
NLL
NNLL
0.2
390
0.0
0.0
330
340
350
360
s [GeV]
370
380
330
340
350
360
370
380
390
s [GeV]
⇒ include leading order finite width effects:
q
q
Γt
v = q−2m
→ q−2m+i
m
m
19 / 30
Matching
σNRQCD (RGI, pole mass)
σQCD (pole mass scheme)
1.4
1.4
1.2
1.2
1.0
σ[pb]
σ[pb]
1.0
0.8
0.6
0.8
0.6
0.4
0.4
0.2
LL
NLL
NNLL
0.2
390
0.0
0.0
330
340
350
360
370
s [GeV]
380
330
340
350
360
370
380
390
s [GeV]
q 2 > 400 GeV
2
2
q ∼ (2m)
σmatched = σQCD
σmatched = σQCD + (σNRQCD − σexpanded )
19 / 30
Matching
NNLO
3
v
4
...
N LO
v
3
q 2 > 400 GeV
2
2
q ∼ (2m)
α4
α
α
v
3
α
v2
α4
v3
...
α v
α2
α3
v
α4
v2
...
3
4
α v
2
2
α
v
α
α v
3
2
2
3
α
v
α
α
v
v
4
α
...
...
...
NLO
α3
2
...
v
α2
...
LO
α1
...
α0
QCD
σmatched = σQCD
σmatched = σQCD + (σNRQCD − σexpanded )
20 / 30
Matching
NNLO
v
3
3
4
v
...
N LO
q 2 > 400 GeV
2
2
q ∼ (2m)
α4
3
α
v2
α4
v3
2
3
4
α
v2
3
4
α
v
α
α
v
α v
α
2
2
α
v
α
3
2
2
3
α v
α v
α
v
α
α
v
v
4
α
...
...
...
...
...
NLO
α3
2
...
v
α2
...
LO
α1
...
α0
NRQCD
σmatched = σQCD
σmatched = σQCD + (σNRQCD − σexpanded )
20 / 30
Matching
v
v
4
...
NNLO
3
q 2 > 400 GeV
2
2
q ∼ (2m)
α3
v2
α4
v3
...
2
α4
v2
...
...
α
v
α
α v
α
α3
v
α v
2
2
α
v
3
α
α4
v
α v
3
2
2
3
4
α
v
α
v
α
...
...
NLO
α3
2
...
v
α2
...
LO
α1
...
α0
NRQCD
σmatched = σQCD
σmatched = σQCD + (σNRQCD − σexpanded )
20 / 30
Matching
Switch-Off Function fs
1.0
0.8
0.6
0.4
0.2
0.0
330
340
350
360
370
380
390
q 2 > 400 GeV
σmatched = σQCD
q 2 ∼ (2m)2
σmatched = σQCD + (σNRQCD − σexpanded )
2
2
(2m) < q < 400 GeV
σmatched = σQCD + (σNRQCD − σexpanded ) · fs
20 / 30
Matching
Switch-Off Function fs
1.0
0.8
0.6
0.4
0.2
0.0
330
340
350
360
370
380
390
q 2 > 400 GeV
σmatched = σQCD
q 2 ∼ (2m)2
σmatched = σQCD + (σNRQCD − σexpanded )
2
2
(2m) < q < 400 GeV
σmatched = σQCD + (σNRQCD − σexpanded ) · fs
20 / 30
Matching
σNRQCD
σQCD
←→
←→
←→
LL
NLL
NNLL
matching:
α1
α2
α3
results in the pole mass scheme:
σmatched (RGI, pole mass)
1.4
1.2
σ[pb]
1.0
LL, O[α1 ]
0.8
NLL, O[α2 ]
0.6
NNLL, O[α3 ]
0.4
0.2
0.0
330
340
350
360
370
380
390
s [GeV]
21 / 30
Matching
σNRQCD
LL
NLL
NNLL
matching:
σQCD
←→
←→
←→
α1
α2
α3
results in the pole mass scheme:
σmatched (RGI, pole mass)
1.4
1.2
σ[pb]
1.0
LL, O[α1 ]
0.8
NLL, O[α2 ]
0.6
NNLL, O[α3 ]
0.4
0.2
0.0
340
342
344
346
s [GeV]
21 / 30
Matching
1S mass
X renormalon free
[Hoang, Ligeti, Manohar ’98]
m1S = mpole + CF αs (µ)mpole
∞ n−1
X
X
n=1 k=0
= mpole
cn,k αs (µ)n log
µ
CF αs (µ)mpole
2
− α2s mpole + . . .
9
22 / 30
Matching
1S mass
X renormalon free
[Hoang, Ligeti, Manohar ’98]
m1S = mpole + CF αs (µ)mpole
∞ n−1
X
X
cn,k αs (µ)n log
n=1 k=0
= mpole
µ
CF αs (µ)mpole
2
− α2s mpole + . . .
9
σmatched (RGI, 1S mass)
1.4
1.2
σ[pb]
1.0
LL, O[α1 ]
0.8
NLL, O[α2 ]
0.6
NNLL, O[α3 ]
0.4
0.2
0.0
340
342
344
346
s [GeV]
22 / 30
Matching
m1S = mpole + CF αs (µ)mpole
∞ n−1
X
X
cn,k αs (µ)n log
n=1 k=0
= mpole −
µ
CF αs (µ)mpole
2 2
α mpole + . . .
9 s
NRQCD
α0
LO
v
α1
α2
α3
...
...
...
...
...
NNLO
...
NLO
...
Pole mass:
23 / 30
Matching
m1S = mpole + CF αs (µ)mpole
∞ n−1
X
X
cn,k αs (µ)n log
n=1 k=0
2 2
α mpole + . . .
9 s
NRQCD
α0
LO
v
α1
α3
...
LO
α0
α1
...
...
...
...
...
NNLO
NRQCD
α2
α3
α2
v
v
...
α3
v
NLO
...
...
α2 v
NNLO
...
1S mass:
α2
NLO
...
Pole mass:
...
...
...
= mpole −
µ
CF αs (µ)mpole
23 / 30
Matching
MS mass :
mpole = m + m
∞
X
an (nl , nh ) αs (m)n
n=1
= m + m αs a1 + . . .
( m = m(nl+1) (m(nl+1) ) )
24 / 30
Matching
MS mass :
mpole = m + m
∞
X
an (nl , nh ) αs (m)n
n=1
( m = m(nl+1) (m(nl+1) ) )
= m + m αs a1 + . . .
σQCD up to O[α3 ]
1.4
1.2
σ[pb]
1.0
pole mass (mpole = 172.6 GeV)
0.8
_ = 163 GeV)
MS mass (mMS
0.6
0.4
0.2
0.0
320
330
340
350
360
s [GeV]
24 / 30
Matching
α0
LO
v
α1
α2
α3
...
NRQCD
α0
α1
...
...
...
...
NNLO
...
NLO
...
Pole mass:
NRQCD
α2
α3
α3
v5
...
2
α
v3
α3
v3
α
v
MS mass:
LO
...
α
v
v
...
α v
NLO
...
2
...
NNLO
...
...
...
...
...
25 / 30
Matching
MSR mass :
[Hoang, Jain, Scimemi, Stewart ’08]
mpole =
m
+m
mpole = mMSRn (R) + R
∞
X
n=1
∞
X
an (nl , nh ) αs (m)n
an (nl , 0) αs (R)n
=
m
+ m αs a1 + . . .
= mMSRn (R) + αs R a1 + . . .
n=1
⇒ choose R ∼ m v
26 / 30
Matching
MSR mass with R ∼ m v :
σQCD up to O[α3 ]
1.4
1.2
σ[pb]
1.0
0.8
pole mass
0.6
MSR mass
0.4
0.2
0.0
320
340
360
380
400
s [GeV]
27 / 30
Matching
v
α2
α3
...
v
α1
α2
α R
v m
2
NLO
...
R
m
...
α v
NNLO
α3
α R
v3 m2
α R3
v5 m3
α2 R
v m
α3 R 2
v3 m2
2
2
2
α R
v m2
...
LO
α0
...
...
...
...
NNLO
...
NLO
NRQCD
MSR mass:
α1
3
3
3
α R
v3 m3
...
...
3
, αv
R
m
...
...
α0
LO
...
Pole mass:
NRQCD
R ∼ mv ∼ mα
28 / 30
Matching
Results in the MSR mass scheme:
σmatched (FO, MSR mass)
1.4
1.2
1.2
1.0
1.0
σ[pb]
σ[pb]
σmatched (RGI, MSR mass)
1.4
0.8
0.6
0.4
0.4
0.2
0.2
0.0
0.0
330
340
350
360
370
380
390
1.4
1.4
1.2
1.2
1.0
1.0
σ[pb]
σ[pb]
0.8
0.6
0.8
340
350
360
370
380
390
330
340
350
360
370
380
390
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
330
0.0
330
340
350
360
s [GeV]
370
380
390
s [GeV]
29 / 30
Conclusion
Summary
– matched σNRQCD and σQCD
– implemented of the MSR mass scheme
Outlook
– N 3 LO corrections for the fixed order cross section
– higher order electroweak effects
30 / 30
Conclusion
Summary
– matched σNRQCD and σQCD
– implemented of the MSR mass scheme
Outlook
– N 3 LO corrections for the fixed order cross section
– higher order electroweak effects
Thank you for your attention!
30 / 30