Cross section and differential distributions for top quarks …...Cross section and differential distributions for top quarks near the production threshold Thomas Rauh AEC, University

Cross section and differential distributions for topquarks near the production threshold

Thomas Rauh

AEC, University of Bern

M. Beneke, Y. Kiyo, P. Marquard, J. Piclum, A. Penin, M. Steinhauser, arXiv:1506.06864M. Beneke, A. Maier, TR, P. Ruiz-Femenía, arXiv:1711.10429

WHIZARD, A. Hoang, M. Stahlhofen, T. Teubner, arXiv:1712.02220F. Simon, arXiv:1902.07246

https://arxiv.org/abs/1506.06864




Top threshold scan 2

Motivation

Consider the top threshold region of the e+e−→W+W−bbX cross section:

• Allows extremely precisedetermination of the topquark mass, goal:

‹mt(mt)≤ 50MeV

• Sensitive to Γt ; ¸s ; yt

Requires very precise theorypredictions:• Inclusive result known at

NNNLO QCD + NNLO SM+ LL ISR + NNNLO Yukawa

• Differential result known at(N)LL + NLO QCD

330 340 350 360 370 380√s [GeV]

0

100

200

300

400

500

600

700

800

900

1000

σ[f

b]

matched, no switch-offNLLmatched, combined, symmetrizedNLO

Matched inclusive W+bW−b cross section, no QED ISR

[Bach, Chokoufé Nejad, Hoang, Kilian, Reuter, Stahlhofen, Teubner, Weiss 2017]

mtΓt

¸syt


Motivation



‹mt(mt)≤ 50MeV





330 340 350 360 370 380√s [GeV]

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σ[f

b]




mt

Γt

¸syt


Motivation



‹mt(mt)≤ 50MeV





330 340 350 360 370 380√s [GeV]

0

100

200

300

400

500

600

700

800

900

1000

σ[f

b]




mt

Γt

¸syt


Motivation



‹mt(mt)≤ 50MeV





330 340 350 360 370 380√s [GeV]

0

100

200

300

400

500

600

700

800

900

1000

σ[f

b]




mtΓt

¸s

yt


Motivation



‹mt(mt)≤ 50MeV





330 340 350 360 370 380√s [GeV]

0

100

200

300

400

500

600

700

800

900

1000

σ[f

b]




mtΓt

¸s

yt


Motivation



‹mt(mt)≤ 50MeV





330 340 350 360 370 380√s [GeV]

0

100

200

300

400

500

600

700

800

900

1000

σ[f

b]




mtΓt

¸syt

Top quarks near threshold 3

Relevant scales and Coulomb effects

Near threshold tops are non-relativistic with velocity v ∼ ¸s• Multiple scales are relevant:

hard mt top masssoft mtv momentumultrasoft mtv

2 energy

• Coulomb singularities (¸s=v)n from n exchanges of potential gluons

t

t

. . .

e−

γ, Z

e+

k0 ∼mtv2; k∼mtv

• Conventional perturbation theory in ¸s fails• Coulomb singularities must be resummed to all orders• Done with potential non-relativistic QCD (PNRQCD)

[Pineda, Soto 1998; Beneke, Signer, Smirnov 1999; Brambilla, Pineda, Soto, Vairo 2000; Beneke, Kiyo, Schuller 2013 ]

QCD cross section 4

Born approximation

Total inclusive cross section from the optical theorem:

fftt(s) = 12ıe2t f (s) Im

hΠ

(v)(s)i∼ ¸2

EWvˆ1 +O(v2)

˜

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1.2

s [GeV]

σtt_[pb]

Born

vector current correlator

QCD cross section 5

Resummed cross section at LO

Coulomb resummation yields narrow toponium resonances

fftt(s)∼ ¸2EWv

∞Xk=0

„¸sv

«k

Σ . . .

≡

Γt = 0

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0.0

0.2

0.4

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1.0

1.2

s [GeV]

σtt_[pb]

Born

LO, stable

QCD cross section 5

Resummed cross section at LO

Coulomb resummation yields narrow toponium resonances which aresmeared out by top decays

fftt(s)∼ ¸2EWv

∞Xk=0

„¸sv

«k

Σ . . .

≡

Γt ∼mt¸EW ∼mt¸2s ∼mtv2

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0.0

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s [GeV]

σtt_[pb]

Born

LO, stable

LO

QCD cross section 6

Resummed cross section at NNNLO

fftt(s)∼ ¸2EWv

∞Xk=0

„¸sv

«k×

8>>><>>>:1 LO¸s ; v NLO¸2s ;¸sv;v

2 NNLO¸3s ;¸

2sv;¸sv

2; v3 NNNLO

δc(1,2,3)v δc(1,2,3)v δc(1,2)v δc(1,2)v

δc(1)v δc(1)v

δV (1,2,3)

δV (1,2) δV (1,2) δV (1) δV (1) δV (1)

ultrasoft

QCD cross section 6


fftt(s)∼ ¸2EWv

∞Xk=0

„¸sv

«k×

8>>><>>>:1 LO¸s ; v NLO¸2s ;¸sv;v

2 NNLO¸3s ;¸

2sv;¸sv

2; v3 NNNLO

δc(1,2,3)v δc(1,2,3)v δc(1,2)v δc(1,2)v

δc(1)v δc(1)v

δV (1,2,3)

δV (1,2) δV (1,2) δV (1) δV (1) δV (1)

ultrasoft

QCD cross section 6


fftt(s)∼ ¸2EWv

∞Xk=0

„¸sv

«k×

8>>><>>>:1 LO¸s ; v NLO¸2s ;¸sv;v

2 NNLO¸3s ;¸

2sv;¸sv

2; v3 NNNLO

δc(1,2,3)v δc(1,2,3)v δc(1,2)v δc(1,2)v

δc(1)v δc(1)v

δV (1,2,3)

δV (1,2) δV (1,2) δV (1) δV (1) δV (1)

ultrasoft

QCD cross section 6


fftt(s)∼ ¸2EWv

∞Xk=0

„¸sv

«k×

8>>><>>>:1 LO¸s ; v NLO¸2s ;¸sv;v

2 NNLO¸3s ;¸

2sv;¸sv

2; v3 NNNLO

δc(1,2,3)v δc(1,2,3)v δc(1,2)v δc(1,2)v

δc(1)v δc(1)v

δV (1,2,3)

δV (1,2) δV (1,2) δV (1) δV (1) δV (1)

ultrasoft

QCD cross section 7


NLO

NNLO

NNNLO

340 342 344 346 348

0.0

0.2

0.4

0.6

0.8

1.0

s [GeV]

σQ

CD[p

b] • NNNLO S-wave

[Beneke, Kiyo, Marquard, Penin, Piclum,Steinhauser 2015]

• NLO P-wave [Beneke, Piclum, TR 2013]

• QQbar_Threshold code[Beneke, Kiyo, Maier, Piclum 2016;Beneke, Maier, TR, Ruiz-Femenía 2017]

• Stabilization of perturbative expansion at NNNLO• 3% uncertainty due to scale variation from 50 to 350 GeV• Similar conclusions at NNLL (5% uncertainty) [Hoang, Stahlhofen 2013]

https://qqbarthreshold.hepforge.org/

Work beyond NNLO QCD 8

2015

2010

2005

2000

Beneke, Kiyo, Schuller 2013

Beneke, Kiyo, Marquard,Penin, Piclum, Steinhauser 2015

Beneke, Piclum, Rauh 2013

Beneke, Kiyo, 2008

Marquard, Piclum, Seidel,Steinhauser 2014

Anzai, Kiyo, Sumino 2009Smirnov, Smirnov, Steinhauser 2009

Kniehl, Penin, Steinhauser, Smirnov 2001Hoang, Manohar, Stewart, Teubner 2001

Hoang, Stahlhofen 2013

Effective field theory QCD cross section Non-QCD effects

Hoang et al., 2000

Beneke, Piclum, Maier, Rauh 2015

Beneke, Jantzen, Ruiz-Femenıa 2010Hoang, Reißer, Ruiz-Femenıa 2010

Jantzen, Ruiz-Femenıa 2013

Penin, Piclum 2011

Ruiz-Femenıa 2014

Beneke, Chapovsky, Signer, Zanderighi 2003Beneke, Chapovsky, Signer, Zanderighi 2004

Pineda, Soto 1998

Beneke, Signer, Smirnov 1999Brambilla, Pineda, Soto, Vairo 1999

Eiras, Steinhauser 2006Hoang, Reißer 2006

WHIZARD, Hoang, Stahlhofen, Teubner 2017

Beneke, Kiyo, Maier, Piclum 2016

Hoang, Reißer 2004

Beneke, Maier, Rauh, Ruiz-Femenıa 2017

Kiyo, Seidel, Steinhauser 2008

Fadin, Khoze 1987Grzadkowski, Kuhn, Krawczyk, Stuart 1987

Guth, Kuhn 1992

Lee, Smirnov, Smirnov, Steinhauser 2016

Pineda, Signer 2006Hoang, Stahlhofen 2006


Wuster 2003

Hoang 2003

Beneke et al., 2007

Actis, Beneke, Falgari, Schwinn, Signer 2008

Beneke, Kiyo, Penin 2007

Luke, Manohar, Rothstein 2000Manohar, Stewart 2000

Hoang, Stewart 2002


Beneke, Kiyo, Schuller, in preparation

PNRQCD

vNRQCD

UnstableParticle ET

PNRQCD

NNLO QCD

NNLL QCDNNNLO QCD

(N)LL+NLO diff. NNLO SM+LL ISR+NNNLO Yukawa

Work beyond NNLO QCD 8

2015

2010

2005

2000


Beneke, Kiyo, Marquard,Penin, Piclum, Steinhauser 2015

Beneke, Piclum, Rauh 2013

Beneke, Kiyo, 2008

Marquard, Piclum, Seidel,Steinhauser 2014

Anzai, Kiyo, Sumino 2009Smirnov, Smirnov, Steinhauser 2009

Kniehl, Penin, Steinhauser, Smirnov 2001Hoang, Manohar, Stewart, Teubner 2001


Effective field theory QCD cross section Non-QCD effects

Hoang et al., 2000

Beneke, Piclum, Maier, Rauh 2015

Beneke, Jantzen, Ruiz-Femenıa 2010Hoang, Reißer, Ruiz-Femenıa 2010

Jantzen, Ruiz-Femenıa 2013

Penin, Piclum 2011

Ruiz-Femenıa 2014

Beneke, Chapovsky, Signer, Zanderighi 2003Beneke, Chapovsky, Signer, Zanderighi 2004

Pineda, Soto 1998

Beneke, Signer, Smirnov 1999Brambilla, Pineda, Soto, Vairo 1999

Eiras, Steinhauser 2006Hoang, Reißer 2006

WHIZARD, Hoang, Stahlhofen, Teubner 2017

Beneke, Kiyo, Maier, Piclum 2016

Hoang, Reißer 2004

Beneke, Maier, Rauh, Ruiz-Femenıa 2017

Kiyo, Seidel, Steinhauser 2008

Fadin, Khoze 1987Grzadkowski, Kuhn, Krawczyk, Stuart 1987

Guth, Kuhn 1992

Lee, Smirnov, Smirnov, Steinhauser 2016

Pineda, Signer 2006Hoang, Stahlhofen 2006


Wuster 2003

Hoang 2003

Beneke et al., 2007

Actis, Beneke, Falgari, Schwinn, Signer 2008

Beneke, Kiyo, Penin 2007

Luke, Manohar, Rothstein 2000Manohar, Stewart 2000

Hoang, Stewart 2002


Beneke, Kiyo, Schuller, in preparation

PNRQCD

vNRQCD

UnstableParticle ET

PNRQCD

NNLO QCD

NNLL QCDNNNLO QCD

(N)LL+NLO diff. NNLO SM+LL ISR+NNNLO Yukawa

Non-QCD effects 9

Non-resonant contributions

The physical final state is W+W−bbX

• Γt ∼mt¸∼mt¸2s is not suppressed with respect to the ultrasoft scale

• Narrow width approximation is unphysical!• Top decay modifies cross section in non-perturbative way (smearing of

toponium resonances)

Top instability implies existence of contributions to the cross section fromhard subgraphs that connect to the initial and final state

γ, Z γ, Z

t

t

b

W

e

e

e

e

νγ, Z

W

W

t

b

e

e

e

eb

Non-QCD effects 10

Effective theory setup

Contributions can be organized systematically within Unstable ParticleEffective Theory [Beneke, Chapovsky, Signer, Zanderighi 2003-4]

Resonant contributioninvolving non-rel. tops.Width resummed intopropagators E→ E+ iΓt

O(l) O(k)† O(l) O(k)†+ + . . .

ff(s)∼ Im

(Xk;l

C(k)C(l)

Zd4x 〈e−e+|T[iO(k)†(0) iO(l)(x)]|e−e+〉EFT

+Xk

C(k)4e 〈e−e+|iO(k)

4e (0)|e−e+〉EFT

)

Non-resonant contributionfrom W+W−bb productionin hard process

O(k)4e

+ . . .

Both parts containspuriousdivergences! Onlythe sum is finite.Calculations mustbe done in thesame regularizationscheme.

Non-QCD effects 10




O(l) O(k)† O(l) O(k)†+ + . . .

ff(s)∼ Im

(Xk;l

C(k)C(l)


+Xk


4e (0)|e−e+〉EFT

)


O(k)4e

+ . . .


Non-QCD effects 10




O(l) O(k)† O(l) O(k)†+ + . . .

ff(s)∼ Im

(Xk;l

C(k)C(l)


+Xk


4e (0)|e−e+〉EFT

)


O(k)4e

+ . . .


Non-QCD effects 10




O(l) O(k)† O(l) O(k)†+ + . . .

ff(s)∼ Im

(Xk;l

C(k)C(l)


+Xk


4e (0)|e−e+〉EFT

)


O(k)4e

+ . . .


Public code 11

Implementation of NNNLO QCD + NNLO SM + LL ISR + NNNLO Yukawaresults is available on HEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForgeHEPForge


Non-QCD effects 12

NNLO SM and NNNLO Yukawa contributions

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s [GeV]

σ[p

b]

full QCD

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0.9

1.0

1.1

1.2

1.3

s [GeV]

σX/σ

full(μ=

80

Ge

V)

full

QCD

• Uncertainty due to renormalization scale variation between 50 GeV and350 GeV

• Effects significantly larger than QCD uncertainty• Shape changes particularly in the important region at and below threshold

Non-QCD effects 13

Initial state radiation

340 342 344 346 3480.0

0.2

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1.0

s [GeV]

σX(μ=

80

Ge

V)[

pb]

without

ISR

with ISR

with ISR0

• ISR reduces cross section by 30-45 %• Band is envelope of different LL accurate implementations• NLL precision is a must for a lepton collider (not just for ttbar)

Determination of SM parameters 14

Results of a full simulation assuming ILC luminosity spectrum [Simon 2019]

171.3 171.4 171.5 171.6 171.7 [GeV]tfitted m

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55 [G

eV]

tΓfit

ted

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2 0

.005

GeV

×pr

obab

ility

[%] /

0.0

05

-1ILC, 8 point scan, 200.0 fb = 171.5 GeVPS

tm = 1.37 GeVtΓ

2D template fit

efficiencies and signal yieldsfrom EPJ C73, 2530 (2013)

February 2019

MPV contourσ1 contourσ2

171.3 171.4 171.5 171.6 171.7 [GeV]tfitted m

0.4

0.6

0.8

1

1.2

1.4

1.6tfit

ted

y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6 0.0

1×

prob

abili

ty [%

] / 0

.005

GeV

-1ILC, 8 point scan, 200.0 fb = 171.5 GeVPS

tm = 1.37 GeVtΓ

2D template fit

efficiencies and signal yieldsfrom EPJ C73, 2530 (2013)

February 2019

MPV contourσ1 contourσ2

parameter 8 point scan 10 point scan1D fitmt (±10:3(stat) ± 44(theo)) MeV (12.2(stat) ± 40(theo)) MeV2D fit mt and Γt

mt (+20:7−24:3 (stat) ± 45(theo)) MeV (+29:7

−25:3 (stat) ± 43(theo)) MeVΓt (+50

−55(stat) ± 32(theo)) MeV (+80−55(stat) ± 39(theo)) MeV

2D fit mt and ytmt (±35(stat) ± 45(theo)) MeV (+34

−31(stat) ± 42(theo)) MeVyt

+0:12−0:14(stat) ± 0.09(theo)

+0:128−0:112(stat) ± 0.132(theo)

Differential distributions 15

Implementation in WHIZARD [Bach, Chokoufé Nejad, Hoang, Kilian, Reuter, Stahlhofen, Teubner, Weiss 2017]

Matched cross section:

ffmatched = ffNLO(¸H) +ffresum(fs¸H; fs¸S; fs¸US)−ffexpandresum (fs¸H)

Fixed order W+W−bbcross section at NLO inQCD from WHIZARD

Resummed cross section at (N)LLwith form factors FNLL = FNLL−1in the resonant contribution

Subtraction toremove doublecounting

ffNLO+NLL = ffNLO +

0@`FNLL −F expNLL

é−

e+

b

W−

W+

b 8>>>>>;e−

e+

b

W−

W+

b1A

+

˛˛FNLL

e−

e+

b

W−

W+

b˛˛

2

+

˛˛FNLL

e−

e+

b

W−

W+

b

g˛˛

2

+

˛˛FNLL

e−

e+

b

W−

g

W+

b˛˛

2

+

0@FNLL

0@e−

e+

b

W−

W+

b

αs

+

e−

e+

b

W−

W+

b

αs

1A8>>>>>;e−

e+

b

W−

W+

b

FNLL

1A

https://whizard.hepforge.org/



Implementation in WHIZARD



fs = switch-off function to turn offresummation in relativistic regime




ffNLO+NLL = ffNLO +

0@`FNLL −F expNLL

é−

e+

b

W−

W+

b 8>>>>>;e−

e+

b

W−

W+

b1A

+

˛˛FNLL

e−

e+

b

W−

W+

b˛˛

2

+

˛˛FNLL

e−

e+

b

W−

W+

b

g˛˛

2

+

˛˛FNLL

e−

e+

b

W−

g

W+

b˛˛

2

+

0@FNLL

0@e−

e+

b

W−

W+

b

αs

+

e−

e+

b

W−

W+

b

αs

1A8>>>>>;e−

e+

b

W−

W+

b

FNLL

1A




Implementation in WHIZARD






(N)LL+NLO accuracy depending on observable:

ff ∼ ¸2EWv

Xk;i

“¸sv

”k(¸s lnv)i ×

(1 LL¸s ; v NLL

Ultrasoft gluon exchanges involving the decayproducts are missing, but cancel in sufficientlyinclusive quantities.

FLL

e−

e+

W−

b

b

W+




Examples at the peak√s = 2m1S

t

top inv. mass distribution

NLO

matched

10−1

1

10 1

10 2

e+e− → W+bW−b, Njets ≥ 2,√s = 344GeV

dσ

dm[fb/GeV

]

160 165 170 175 180

2468

101214161820

mW+jb [GeV]

σ/σNLO

b-jet energy distribution

NLO

matched

10−4

10−3

10−2

10−1

1

10 1


dσ

dE[fb/GeV

]

0 20 40 60 80 100 120 140

2468

101214161820

Eb[GeV]

σ/σNLO

E∗b ≈m2

t−m2W

2mt≈ 68GeV

(RIVET event analysis; FASTJET generalized kT algorithm, R = 0:4, p = −1; Ejet > 1GeV)[Bach, Chokoufé Nejad, Hoang, Kilian, Reuter, Stahlhofen, Teubner, Weiss 2017]


NLO

matched

10−4

10−3

10−2

10−1

1

10 1


dσ

dE[fb/GeV

]

50 100 150 200 250

2468

101214161820

Ejbjb [GeV]

σ/σNLO

NLO

matched

10 1

10 2

10 3


dσ

dΔφ[fb/GeV

]

0 0.5 1 1.5 2 2.5 3

2468

101214161820

ΔφW+jb

σ/σNLO

NLO

matched

10−2

10−1

1

10 1

10 2


dσ

dE[fb/G

eV]

80 90 100 110 120 130 140 150 160

2468

101214161820

EW+

[GeV]

σ/σ

NLO

NLO

matched

10−3

10−2

10−1

1

10 1


dσ

dpT

[fb/GeV

]

0 20 40 60 80 100 120 140

2468

101214161820

pW+

T [GeV]

σ/σNLO

NLO

matched

10−3

10−2

10−1

1

10 1


dσ

dpT

[fb/GeV

]

0 10 20 30 40 50 60 70 80

2468

101214161820

pj2T [GeV]

σ/σNLO

NLO

matched

10 1

10 2

10 3

10 4


dσ

dO[fb/[a.u.]]

0 0.05 0.1 0.15 0.2 0.25 0.3

2468

101214161820

σ/σ

NLO

NLO

matched

10 2

10 3


dσ

dO[fb/[a.u.]]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

2468

101214161820

O

σ/σ

NLO

NLO

matched

10 1

10 2

10 3


dσ

dO[fb/[a.u.]]

0.1 0.2 0.3 0.4 0.5 0.6 0.7

2468

101214161820

Tmajor

σ/σ

NLO

NLO

matched

10−3

10−2

10−1

1

10 1

10 2

10 3


dσ

dO[fb/[a.u.]]

0 0.2 0.4 0.6 0.8 1

2468

101214161820

σ/σNLO

NLO

matched

10−1

1

10 1

10 2

10 3


dσ

dO[fb/[a.u.]]

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

2468

101214161820

1− T

σ/σNLO

NLO

matched

10 1

10 2

10 3

10 4


dσ

dO[fb/[a.u.]]

0 0.1 0.2 0.3 0.4 0.5

2468

101214161820

Tminor

σ/σ

NLO

NLO

matched

0

2

4

6

8

10

12


dσ

dm[fb/G

eV]

20 40 60 80 100 120 140 160

2468

101214161820

mjj [GeV]

σ/σ

NLO

��

��

��

Summary & Outlook 18

• Determination of several SM parameters possible from scan of thetotal e+e−→W+W−bbX cross section near the top threshold

• NNNLO QCD + NNLO SM + LL ISR + NNNLO Yukawa predictionknown and available in QQbar_Threshold

• Theoretical uncertainty of 2-5% (energy-dependent), translates to

parameter 8 point scan 10 point scanmt (±10:3(stat) ± 44(theo)) MeV (12.2(stat) ± 40(theo)) MeV

[Simon 2019]

• Fully differential results at (N)LL+NLO implemented in WHIZARD

• Increase precision: NNNLO+NNLL QCD, NLL ISR, N4LO Yukawa,NLL differential, : : :

• Phenomenology of differential distributions, parameter sensitivity



Power counting 19

¸EW ∼ ¸t ≡–2t

4ı∼ ¸2

s ∼ v2;

ffQCD only ∼ ¸2EWv

∞Xk=0

“¸s

v

”k×

8<:1 LO¸s ; v NLO¸2s ;¸sv;v

2 NNLO¸3s ;¸

2sv;¸sv

2; v3 NNNLO

;

ff ∼¸2EWv

∞Xk=0

“¸s

v

”k×

8>>>>>><>>>>>>:

¸em

vNLO“

¸em

v

”2

;¸em

v×{¸s ; v};¸EW;

√¸EW¸t ;¸t NNLO“

¸em

v

”3

;

“¸em

v

”2

×{¸s ; v};¸em

v×{¸2

s ;¸sv;v2;√¸EW¸t};

¸t ×{¸em

v;¸s ; v}; : : : NNNLO

+ ¸2EW×

(¸EW NLO¸EW¸s NNLO: : : NNNLO

;

Organization of the calculation 20

Split cross section into three separately finite parts (I), (II) and (III):

ffNNLO =hffsq +ffres, rest

i| {z }

(I)

+

»ff

(EP div)int +ff

C(k)Abs,bare

–| {z }

(II)

+hff

(EP fin)int +ffaut

i| {z }

(III)

:

• (I): computational scheme for ’squared contribution’ fixed by existing QCDresults (Dim reg with NDR for ‚5)

• (II): Use freedom of scheme choice to simplify calculation (some partsdone in four dimensions)

• (III): Endpoint finite part of ’interference contribution’ must be computedconsistent with MadGraph

Divergence structure 21

UV finite IR finite EP finite(I) X X X

ffsq X X –

ff(h1a;:::;h1g )

sq X – –ff

(g1;:::;g6)sq X – ?

ffres, rest X X –ffQCD X X –ffP-wave X X –ffH X X X

ff‹VQED X X XffΓ X X –ffC

(k)

EWX X X

ffC

(k)

Abs,Zt

X X –

ffconvIS X X X

(II) X X X

ff(EP div)

int X X –ffC

(k)

Abs,bareX X –

(III) X X X

ff(EP fin)

int – X Xffaut – X X

Non-QCD effects 22

Top-quark decay width

Lbilinear = †

"i@0 +

~@2

2mt+iΓt2

+(~@2 + imtΓt)

2

8m3t

+ : : :

# + anti-quark

• † iΓt2 : same order as kinetic term, shifts E→ E+ iΓt (E =√s−2mt)

Causes divergences at NNLO: ff ⊃ ff0ImÊ›

˜→ ff0Im

Ê+iΓt›

˜

• − † Γ2t

8mt : additional shift→ E+ iΓt −Γt

2=(8mt), but treated perturbatively

• † iΓt ~@2

4m2t : time dilatation, reduces toponium width Γn = 2Γt −

Γt¸2sC

2F

4n2 + : : :

Non-Hermitian Hamiltonian H ⇒ eigenstates do not form a basis

H |n〉= En |n〉 ; H† |m〉= Em |m〉 ; En = E∗n = (En− iΓn=2)∗

exponentially exponentially 〈n |m〉= ‹nmdecaying states growing states

Non-relativistic Green function: G(E) =D~0˛G(E)

˛~0E

=PRn

n(~0) ∗n(~0)

En−E

Non-QCD effects 22


Lbilinear = †

"i@0 +

~@2

2mt+iΓt2

+(~@2 + imtΓt)

2

8m3t

+ : : :

# + anti-quark



˜→ ff0Im

Ê+iΓt›

˜• − † Γ2

t8mt

: additional shift→ E+ iΓt −Γt2=(8mt), but treated perturbatively

• † iΓt ~@2


Γt¸2sC

2F

4n2 + : : :





˛~0E

=PRn

n(~0) ∗n(~0)

En−E

Non-QCD effects 22


Lbilinear = †

"i@0 +

~@2

2mt+iΓt2

+(~@2 + imtΓt)

2

8m3t

+ : : :

# + anti-quark



˜→ ff0Im

Ê+iΓt›

˜• − † Γ2

t8mt

: additional shift→ E+ iΓt −Γt2=(8mt), but treated perturbatively

• † iΓt ~@2


Γt¸2sC

2F

4n2 + : : :





˛~0E

=PRn

n(~0) ∗n(~0)

En−E

NNLO non-resonant contribution 23

Contains endpoint divergences when the hard tops go on-shell [Jantzen, Ruiz-Femenía ’13]

W

t

t

b

h1a

W

t

t

b

h1b

W

t

t

b

h1c

W

t

t

b

h1d

W

t

t

b

−i δmtmt

h1e

W

t

t

b

δb2

h1f

W

t

t

b

δt2

h1g

W

t

t

b

g1

W

t

t

b

g2

W

t

t

b

g3

W

t

t

b

g4

W

t

t

b

g5

W

t

t

b

g6

’Squared contribution’: Gluoncorrections to h1, endpointdivergent but UV & IR finite

t

tW

b

b

h2a

W

W

t

t

b

h3a

t

t

b

W

W

e+

e−

ν

e+

e−

h4a

’Interference contribution’:endpoint & UV divergent

+ O(100) endpoint finite diagrams(not drawn)

’Automated contribution’: endpointfinite but UV divergent, computedwith automated tools (MadGraph)

cancel withresonant part cancel



W

t

t

b

h1a

W

t

t

b

h1b

W

t

t

b

h1c

W

t

t

b

h1d

W

t

t

b

−i δmtmt

h1e

W

t

t

b

δb2

h1f

W

t

t

b

δt2

h1g

W

t

t

b

g1

W

t

t

b

g2

W

t

t

b

g3

W

t

t

b

g4

W

t

t

b

g5

W

t

t

b

g6


t

tW

b

b

h2a

W

W

t

t

b

h3a

t

t

b

W

W

e+

e−

ν

e+

e−

h4a







W

t

t

b

h1a

W

t

t

b

h1b

W

t

t

b

h1c

W

t

t

b

h1d

W

t

t

b

−i δmtmt

h1e

W

t

t

b

δb2

h1f

W

t

t

b

δt2

h1g

W

t

t

b

g1

W

t

t

b

g2

W

t

t

b

g3

W

t

t

b

g4

W

t

t

b

g5

W

t

t

b

g6


t

tW

b

b

h2a

W

W

t

t

b

h3a

t

t

b

W

W

e+

e−

ν

e+

e−

h4a







W

t

t

b

h1a

W

t

t

b

h1b

W

t

t

b

h1c

W

t

t

b

h1d

W

t

t

b

−i δmtmt

h1e

W

t

t

b

δb2

h1f

W

t

t

b

δt2

h1g

W

t

t

b

g1

W

t

t

b

g2

W

t

t

b

g3

W

t

t

b

g4

W

t

t

b

g5

W

t

t

b

g6


t

tW

b

b

h2a

W

W

t

t

b

h3a

t

t

b

W

W

e+

e−

ν

e+

e−

h4a




cancel withresonant part

cancel



W

t

t

b

h1a

W

t

t

b

h1b

W

t

t

b

h1c

W

t

t

b

h1d

W

t

t

b

−i δmtmt

h1e

W

t

t

b

δb2

h1f

W

t

t

b

δt2

h1g

W

t

t

b

g1

W

t

t

b

g2

W

t

t

b

g3

W

t

t

b

g4

W

t

t

b

g5

W

t

t

b

g6


t

tW

b

b

h2a

W

W

t

t

b

h3a

t

t

b

W

W

e+

e−

ν

e+

e−

h4a





Dependence on —w scale 24

Regularizing width-related/endpoint divergences dimensionally splits some ofthe large logarithms by introducing the scale —w

fffull ⊃ lnv = ln—wmt| {z }

⊂ffnon-res

+lnmtv

—w| {z }⊂ffres

:

The dependence on —w cancels exactly at a given order.

340 342 344 346 348

0.0

0.2

0.4

0.6

0.8

1.0

s [GeV]

σ[p

b]

full

resonant

+ NLO non-res.

340 342 344 346 348

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

s [GeV]

σX/σ

full(μ

w=

35

0G

eV)

full

resonant

+ NLO non-res.

We choose a central scale of —w = 350GeV to minimize the unknownlogarithms from the NNNLO non-resonant part.

Invariant mass cut 25

Consider "loose" invariant mass cuts

(mt −∆Mt)2 ≤ p2

t;t ≤ (mt + ∆Mt)2;

with ∆Mt � Γt . Since the off-shellness in the resonant part is parametricallyof the order Γt they only affect the non-resonant part:

NLO

NNLO

aNNLO

NLO+NNLO

5 10 20 50

-0.15

-0.10

-0.05

0.00

ΔMt[GeV]

σn

r(μ

w=

35

0G

eV)[

pb]

Non-QCD effects 26

Individual contributions

NNLO Higgs

NNNLO Higgs

340 342 344 346 3481.00

1.02

1.04

1.06

1.08

1.10

s [GeV]

σQ

CD+

H/σ

QC

D(μ=

80

Ge

V)

δVQED

NNLO EW

340 342 344 346 3480.85

0.90

0.95

1.00

1.05

1.10

s [GeV]

σQ

CD+

H+

EW/σ

QC

D+

H(μ=

80

Ge

V)

NLO non-res.

NNLO non-res.

340 342 344 346 3480.75

0.80

0.85

0.90

0.95

1.00

s [GeV]

σfu

ll/σ

QC

D+

H+

EW(μ=

80

Ge

V)

Determination of SM parameters 27

Correlation between top Yukawa and strong coupling

●

●

●

●

●

●

●

●

●

●

●●●●

343.9 344.0 344.1 344.2

0.56

0.58

0.60

0.62

0.64

s [GeV]

σIS

R[p

b]

κt=1.5

κt=1.2

κt=0.8

κt=0.5

αs(mZ )=0.1204

αs(mZ )=0.1194

αs(mZ )=0.1174

αs(mZ )=0.1164

Peak height and width

• Estimate theory uncertainty by determining what parameter shift is neededto obtain curves outside the scale variation band

• Naive expectation: ‹»t ≈+20−25 % and ‹¸s ≈ 0:0015

• Effects from variation of Yukawa coupling and strong coupling very similar• Need full simulation to see how well they can be disentangled

NNLL cross section 28

��

��

��

��

��

��

��

√s (GeV)

σ(p

b)

1/2 ≤ h ≤ 2

mν2∗/2 ≤ µusoft ≤ 2mν2

∗

M1St = 172 GeV, Γt = 1.5 GeV

δσ/σ ≈ 5 %��

��

�� 20<δm<100 MeV

��

��

��

Implementation of threshold resummation 29

�� W+W−bb

��

��

��

F× FLL =�c1,3(ν)G

S,P(0, pt,E+iΓt, ν) − 1�

+ + + + . . .

��

��

��

Double pole approximation 30

Mfact = F×�

ht,ht

i

p2t −m2 − imΓt

i

p2t −m2 − imΓtMht,ht

prod (pt, pt)Mhtdec,t(pt)M

htdec,t(pt)

��

e−

e+

b

W−

W+

b

p2t = p2t = m2� ��

� ��

� ��

√s ≥ 2m

√s < 2m pt, pt

√s = 2m

��

��

��

(N)LL + NLO matched cross section 31

��

330 340 350 360 370 380√s [GeV]

0

100

200

300

400

500

600

700

800

900

1000

σ[f

b]


300 320 340 360 380 400√s [GeV]

0.00

0.20

0.40

0.60

0.80

1.00

Γt = 1.4 GeV

v1 = 0.1

v2 = 0.3

|v(√s)|Re[v(

√s)]

Im[v(√s)]

fs(|v(√s)|)

fs(Re[v(√s)])

fs(Im[v(√s)])

��

σmatched = σNLO[αH] + σresum[fsαH, fsαS, fsαUS]− σexpandresum [fsαH]

� ��

��

0.1 ≤ v1 < v2 ≤ 0.4

��

��

��

Initial state radiation 32

��

330 340 350 360 370 380√s [GeV]

0

250

500

750

1000

σ[f

b]

matched, no switch-offmatched, combined, symmetrizedNLO

��

��

��

��

��

��

��

��

��

Cross section and differential distributions for top quarks …...Cross section and differential distributions for top quarks near the production threshold Thomas Rauh AEC, University

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