· prologue : state of the art in early 2014 dy xsection : nnlo qcd hamberg, matsuura, van neerven...
TRANSCRIPT
![Page 1: · Prologue : State of the Art in Early 2014 DY Xsection : NNLO QCD Hamberg, Matsuura, Van Neerven (’91) N3LO pSV Moch, Vogt (2005); Ravindran (’06) Rapidity: NNLO QCD Anastasi](https://reader034.vdocuments.net/reader034/viewer/2022051915/600714df1155f11f747e70d1/html5/thumbnails/1.jpg)
Threshold Corrections To DY and Higgsat N3LO QCD
Taushif AhmedInstitute of Mathematical Sciences, IndiaApril 12, 2016
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 1
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Prologue : Processes of Interest
• Drell-Yan (DY) pp→ γ∗/Z : Xsection and rapiditydistributions.
q
q
γ∗/Z
• Higgs boson production through gg : Rapidity distributions.
g
gH
• Higgs boson production through bb : Xsection and rapiditydistributions.
b
b
H
Such processes are very important for precision studies in SM.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 2
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Prologue : State of the Art in Early 2014DY • Xsection : NNLO QCD Hamberg, Matsuura, Van Neerven (’91)
N3LOpSV Moch, Vogt (2005); Ravindran (’06)
• Rapidity: NNLO QCD Anastasiou, Dixon, Melnikov, Petriello (’03);Melnikov, Petriello (’06)
N3LOpSV Ravindran, Smith, van Neerven (’07)
Higgs in gg
• Xsection : NNLO QCD Harlander, Kilgore (’02); Anastasiou,
Melnikov (’02); Ravindran, Smith, van Neerven (’03)
N3LOpSV Moch, Vogt (’05); Ravindran (’06)
• Rapidity : NNLO QCD Anastasiou, Melnikov, Petriello (’04, ’05)
N3LOpSV Ravindran, Smith, van Neerven (’07)
Higgs in bb
• Xsection : NNLO QCD Harlander, Kilgore (’03)
N3LOpSV Ravindran (’06)
• Rapidity : NNLO QCD Buehler, Herzog, Lazopoulos, Mueller (’12)
N3LOpSV Ravindran, Smith, van Neerven (’07)
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 3
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Prologue : What next?
Can we push the theoretical boundary little more?
• Testing SM with more accuracy to answer many unansweredquestions
• Reducing the unphysical scale uncertainties
First crucial step towards this direction
• Higgs production through gg fusion at threshold N3LO byAnastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger (’14)
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 4
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Prologue : What next?
Can we push the theoretical boundary little more?
• Testing SM with more accuracy to answer many unansweredquestions
• Reducing the unphysical scale uncertainties
First crucial step towards this direction
• Higgs production through gg fusion at threshold N3LO byAnastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger (’14)
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 4
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Prologue : Spin-offs from Higgs Productionat Threshold N3LO
Recent results on threshold N3LO QCD correctionsTA, Mahakhud, Mandal, Rana, Ravindran
• dσ/dQ2 of DYPhys. Rev. Lett. 113 (2014) 112002
• dσ/dY of Higgs in gg fusion and d2σ/dQ2/dY of DYPhys. Rev. Lett. 113 (2014) 212003
• σtot and dσ/dY of Higgs in bb annihilationJHEP 1410 (2014) 139
JHEP 1502 (2015) 131
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 5
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Prologue : Spin-offs from Higgs Productionat Threshold N3LO
Recent results on threshold N3LO QCD correctionsTA, Mahakhud, Mandal, Rana, Ravindran
• dσ/dQ2 of DYPhys. Rev. Lett. 113 (2014) 112002 ⇐ Goal
• dσ/dY of Higgs in gg fusion and d2σ/dQ2/dY of DYPhys. Rev. Lett. 113 (2014) 212003
• σtot and dσ/dY of Higgs in bb annihilationJHEP 1410 (2014) 139
JHEP 1502 (2015) 131
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 6
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Plan of the Talk
� What is threshold corrections?
� Prescription of Computation
• Factorization of Soft Gluons
• Overall Operator Renormalization
• Form Factor
• Soft-Collinear Distribution
• Mass Factorization
� Master Formula
� Result of DY SV Xsection at N3LO
� Conclusions
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 7
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Why threshold corrections?
Basic calculational framework for inclusive production : QCDfactorization theorem
σX(τ, q2
)=
1
S
∑ab
∫ 1
τ
dx
xΦab (x)∆ab
(τx, q2)
τ ≡ q2
S , q2 ≡ m2
l+l− for DY and m2H for Higgs
• Perturbatively calculable partonic cross section (CS)
∆ab ≡ s σab
• Non-perturbative partonic flux
Φab (x) =
∫ 1
x
du
ufa (u) fb
(xu
)Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 8
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Why threshold corrections?
Basic calculational framework for inclusive production : QCDfactorization theorem
σX(τ, q2
)=
1
S
∑ab
∫ 1
τ
dx
xΦab (x)∆ab
(τx, q2)
τ ≡ q2
S , q2 ≡ m2
l+l− for DY and m2H for Higgs
• Perturbatively calculable partonic cross section (CS)
∆ab ≡ s σab
• Non-perturbative partonic flux
Φab (x) =
∫ 1
x
du
ufa (u) fb
(xu
)Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 8
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Why Threshold . . .
• Φab (x) becomes large when x→ xmin = τ ⇒ enhances σX
• x→ τ ⇒ s→ q2 ⇒ partonic COM energy is just enough toproduce final state particle ⇒ all the radiations are soft soft limit.
• In this region, Dominantcontributions arise fromVirtual and Soft gluonemission processes (SV).
• For Higgs and DYproductions, substantialcontribution come fromthis region. e.g. for DY91% - 95% at NNLO
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 9
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Threshold Expan : Soft-Plus-Virtual (SV)
Splitting the partonic CS into singular & regular parts around
z ≡ τx = q2
s = 1
∆ (z) = ∆sing(z) + ∆hard(z)
∆sing(z) ≡ ∆SV(z) = ∆SVδ δ(1−z)+
∞∑j=0
∆SVj Dj Dj ≡
(lnj(1− z)
1− z
)+
• These two are the most singular terms when z → 1, sodominant contribution comes from these.
• ∆hard(z) : subleading and polynomial in ln(1− z).
Plus distribution + is defined by its action on test function f(z)∫ 1
0dzDj(z)f(z) ≡
∫ 1
0dz
lnj(1− z)1− z
[f(z)− f(1)]
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 10
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Goal
Goal : Computation of ∆SV(z)
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 11
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Various Methods to Compute SV
I Matrix elements square and phase space integrals in the softlimit.
Catani et al; Harlander and Kilgore
I Form factors and DGLAP kernels :
• Factorization theorem
• Renormalization group invariance
• Sudakov resummation of form factors
Moch and Vogt; Ravindran, Smith and Van Neerven
I Soft Collinear effective theoryBecher and Neubert
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 12
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Diagrams at O(a3s) for DY
Triple virtual : 244 Double virtual : 13 × 1
Double virtual real : 229 × 2 Real virtual squared : 13 × 13
Double real virtual : 134 × 8 Triple real squared: 50 × 50
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 13
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Various Methods to Compute SV
I Matrix elements square and phase space integrals in the softlimit.
Catani et al; Harlander and Kilgore
I Form factors and DGLAP kernels :
• Factorization theorem
• Renormalization group invariance
• Sudakov resummation of form factors
Moch and Vogt; Ravindran, Smith and Van Neerven
I Soft Collinear effective theoryBecher and Neubert
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 14
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Diagrams at O(a3s) for DY
Triple virtual : 244 Double virtual : 13 × 1
We wi
llno
t comp
ute th
ese!!
⇓
Symm
etry
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 15
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Diagrams at O(a3s) for DY
Triple virtual : 244 Double virtual : 13 × 1
We wi
llno
t comp
ute th
ese!!
⇓
Symm
etry
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 15
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Various Methods to Compute SV
I Matrix elements square and phase space integrals in the softlimit.
Catani et al; Harlander and Kilgore
I Form factors and DGLAP kernels :
• Factorization theorem
• Renormalization group invariance
• Sudakov resummation of form factors
Moch and Vogt; Ravindran, Smith and Van Neerven
I Soft Collinear effective theoryBecher and Neubert
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 16
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Various Methods to Compute SV
I Matrix elements square and phase space integrals in the softlimit.
Catani et al; Harlander and Kilgore
I Form factors and DGLAP kernels :
• Factorization theorem
• Renormalization group invariance
• Sudakov resummation of form factors
Moch and Vogt; Ravindran, Smith and Van Neerven
I Soft Collinear effective theoryBecher and Neubert
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 17
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The Prescription
⇓Master Formula
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 18
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Structure of Partonic XSection
General structure of ∆ upon taking radiative corrections :
∆(z, q2) = δ(1− z)
+ as(q2) [c
(1)1 δ (1− z) + c
(1)2
(ln (1− z)(1− z)
)+
+R1(z, q2)]
+ a2s
(q2) [· · ·+ · · ·+R2(z, q2)
]+ · · ·
z = q2
s , Ri(z, q2) are remaining terms.
• Virtual δ(1− z)
• Real emission δ(1− z),Dj , Rk
• Soft approximation z → 1
Defn.: as ≡ αs/4π
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Structure of Partonic XSection
General structure of ∆ upon taking radiative corrections :
∆(z, q2) = δ(1− z)
+ as(q2) [c
(1)1 δ (1− z) + c
(1)2
(ln (1− z)(1− z)
)+
+R1(z, q2)]
+ a2s
(q2) [· · ·+ · · ·+R2(z, q2)
]+ · · ·
z = q2
s , Ri(z, q2) are remaining terms.
• Virtual δ(1− z)
• Real emission δ(1− z),Dj , Rk
• Soft approximation z → 1
Defn.: as ≡ αs/4π
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Structure of Partonic XSection
General structure of ∆ upon taking radiative corrections :
∆(z, q2) = δ(1− z)
+ as(q2) [c
(1)1 δ (1− z) + c
(1)2
(ln (1− z)(1− z)
)+
+R1(z, q2)]
+ a2s
(q2) [· · ·+ · · ·+R2(z, q2)
]+ · · ·
z = q2
s , Ri(z, q2) are remaining terms.
• Virtual δ(1− z)
• Real emission δ(1− z),Dj , Rk
• Soft approximation z → 1
Defn.: as ≡ αs/4π
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 19
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Structure of Partonic XSection
General structure of ∆ upon taking radiative corrections :
∆(z, q2) = δ(1− z)
+ as(q2) [c
(1)1 δ (1− z) + c
(1)2
(ln (1− z)(1− z)
)+
+R1(z, q2)]
+ a2s
(q2) [· · ·+ · · ·+R2(z, q2)
]+ · · ·
z = q2
s , Ri(z, q2) are remaining terms.
• Virtual δ(1− z)
• Real emission δ(1− z),Dj , Rk
• Soft approximation z → 1
Defn.: as ≡ αs/4π
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 19
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Factorization of Soft Gluons
Soft distribution factorizes as exponentiation
∆(z, q2
)= S
(z, q2, µ2
R
)⊗[δ (1− z) + asR1(z, q2) + a2
sR2(z, q2)]
and
S(z, q2, µ2
R
)= C exp
(Φ(z, q2, µ2
R)
), Φ is a finite distribution
Defn:
Cef(z) = δ(1− z) +1
1!f(z) +
1
2!f(z)⊗ f(z) + · · ·
⊗ is Mellin convolution.
Drop all regular functions.
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Factorization of Soft Gluons
Soft distribution factorizes as exponentiation
∆(z, q2
)= S
(z, q2, µ2
R
)⊗[δ (1− z) + asR1(z, q2) + a2
sR2(z, q2)]
and
S(z, q2, µ2
R
)= C exp
(Φ(z, q2, µ2
R)
), Φ is a finite distribution
Defn:
Cef(z) = δ(1− z) +1
1!f(z) +
1
2!f(z)⊗ f(z) + · · ·
⊗ is Mellin convolution.
Drop all regular functions.
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SV Cross Section : Conjecture
We propose UV renormalized
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2|F I (as, Q2, µ2
)|2δ(1− z)
I = q, g, b for DY, Higgs through gg and Higgs in bb.
• Soft contribution
• Form factor factorizes due to Born kinematics.
• Overall operator ren. const. required for
1. effective Lagrangian of Higgs in gg
2. Yukawa coupling for Higgs in bb
and ZI = 1 for DY.
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SV Cross Section : Conjecture
We propose UV renormalized
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2|F I (as, Q2, µ2
)|2δ(1− z)
I = q, g, b for DY, Higgs through gg and Higgs in bb.
• Soft contribution
• Form factor factorizes due to Born kinematics.
• Overall operator ren. const. required for
1. effective Lagrangian of Higgs in gg
2. Yukawa coupling for Higgs in bb
and ZI = 1 for DY.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 21
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SV Cross Section : Conjecture
We propose UV renormalized
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2|F I (as, Q2, µ2
)|2δ(1− z)
I = q, g, b for DY, Higgs through gg and Higgs in bb.
• Soft contribution
• Form factor factorizes due to Born kinematics.
• Overall operator ren. const. required for
1. effective Lagrangian of Higgs in gg
2. Yukawa coupling for Higgs in bb
and ZI = 1 for DY.
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SV Cross Section : Conjecture
We propose UV renormalized
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2|F I (as, Q2, µ2
)|2δ(1− z)
I = q, g, b for DY, Higgs through gg and Higgs in bb.
• Soft contribution
• Form factor factorizes due to Born kinematics.
• Overall operator ren. const. required for
1. effective Lagrangian of Higgs in gg
2. Yukawa coupling for Higgs in bb
and ZI = 1 for DY.
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Cancellation of Divergences
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2 |F I (as, Q2, µ2
)|2δ(1− z)
• as renormalization and ZI makes partonic CS UV finite.
• Soft divergences from real radiation and virtual correctioncancel upon summing over all the final states (KLN).
• Final state collinear singularities also cancel upon summingover all the final states (KLN)
• Still XSection is NOT fully finite due to initial state collinearsingularities
• This can arise from virtual as well as real corrections
• Mass factorization to make collinear finite.
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Cancellation of Divergences
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2 |F I (as, Q2, µ2
)|2δ(1− z)
• as renormalization and ZI makes partonic CS UV finite.
• Soft divergences from real radiation and virtual correctioncancel upon summing over all the final states (KLN).
• Final state collinear singularities also cancel upon summingover all the final states (KLN)
• Still XSection is NOT fully finite due to initial state collinearsingularities
• This can arise from virtual as well as real corrections
• Mass factorization to make collinear finite.
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Cancellation of Divergences
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2 |F I (as, Q2, µ2
)|2δ(1− z)
• as renormalization and ZI makes partonic CS UV finite.
• Soft divergences from real radiation and virtual correctioncancel upon summing over all the final states (KLN).
• Final state collinear singularities also cancel upon summingover all the final states (KLN)
• Still XSection is NOT fully finite due to initial state collinearsingularities
• This can arise from virtual as well as real corrections
• Mass factorization to make collinear finite.
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Cancellation of Divergences
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2 |F I (as, Q2, µ2
)|2δ(1− z)
• as renormalization and ZI makes partonic CS UV finite.
• Soft divergences from real radiation and virtual correctioncancel upon summing over all the final states (KLN).
• Final state collinear singularities also cancel upon summingover all the final states (KLN)
• Still XSection is NOT fully finite due to initial state collinearsingularities
• This can arise from virtual as well as real corrections
• Mass factorization to make collinear finite.
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Cancellation of Divergences
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2 |F I (as, Q2, µ2
)|2δ(1− z)
• as renormalization and ZI makes partonic CS UV finite.
• Soft divergences from real radiation and virtual correctioncancel upon summing over all the final states (KLN).
• Final state collinear singularities also cancel upon summingover all the final states (KLN)
• Still XSection is NOT fully finite due to initial state collinearsingularities
• This can arise from virtual as well as real corrections
• Mass factorization to make collinear finite.
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Cancellation of Divergences
∆SV,I(z, q2, µ2
R
)= Ce2ΦI(as,q2,µ2,z)
⊗(ZI(as, µ
2R, µ
2))2 |F I (as, Q2, µ2
)|2δ(1− z)
• as renormalization and ZI makes partonic CS UV finite.
• Soft divergences from real radiation and virtual correctioncancel upon summing over all the final states (KLN).
• Final state collinear singularities also cancel upon summingover all the final states (KLN)
• Still XSection is NOT fully finite due to initial state collinearsingularities
• This can arise from virtual as well as real corrections
• Mass factorization to make collinear finite.
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Factorization of Collinear Singularities
∆SV,I(z, q2, µ2
R
)= ΓT(z, µ2
F )⊗ ∆SV,I(z, q2, µ2
R, µ2F
)⊗ Γ(z, µ2
F )
↗UV, Soft & Collinear finite
• Kernels Γ absorb all the initial state collinear singularities.
• Only diagonal elements contribute to threshold limit.
• Substituting this we arrive at the master formula to computeSV CS.
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Factorization of Collinear Singularities
∆SV,I(z, q2, µ2
R
)= ΓT(z, µ2
F )⊗ ∆SV,I(z, q2, µ2
R, µ2F
)⊗ Γ(z, µ2
F )
↗UV, Soft & Collinear finite
• Kernels Γ absorb all the initial state collinear singularities.
• Only diagonal elements contribute to threshold limit.
• Substituting this we arrive at the master formula to computeSV CS.
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Master Formula for SV CS
∆SV,I(z, q2, µ2
R, µ2F
)= C exp
(ΨI(z, q2, µ2
R, µ2F , ε))|ε=0
where
ΨI =
(ln[ZI(as, µ
2R, µ
2, ε)]2
+ ln∣∣∣F I(as, Q2, µ2, ε)
∣∣∣2) δ(1− z)+ 2ΦI
(as, q
2, µ2, z, ε)− 2C ln ΓII
(as, µ
2, µ2F , z, ε
)Ravindran
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Ingredients for SV CS
1. Logarithm of the overall operator renormalization lnZI
2. Logarithm of the form factor ln F I
3. Soft distribution ΦI
4. Logarithm of the mass factorization kernel ln ΓII
Goal boils down to compute these quantities.
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Ingredients for SV CS
1. Logarithm of the overall operator renormalization lnZI
2. Logarithm of the form factor ln F I
3. Soft distribution ΦI
4. Logarithm of the mass factorization kernel ln ΓII
Goal boils down to compute these quantities.
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The Prescription
⇓
Master Formula
⇓How do we compute these?
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Ingredient 1 : Overall OperatorRenormalization lnZI
It satisfies the RGE
µ2R
d
dµ2R
lnZI(as, µ2R, µ
2, ε) =
∞∑i=1
ais(µ2R)γIi−1
• γIi ’s are anomalous dimensions.
• γI are available up to O(a3s) for I = b, g and γq = 0 (q 6= b).
• Solution of this RGE provides lnZI .
• OOR is not required for DY ZI = 1 .
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Ingredient 2 : Form factor ln F I
I ln |F I |2 can be computed from available results up to 3-loop.
I Instead, we have followed an alternative way. The reasonwould be clear soon.
I Gauge and RG invariance implies KG eqn
Q2 d
dQ2ln F I =
1
2
[KI
(as,
µ2R
µ2, ε
)+GI
(as,
Q2
µ2R
,µ2R
µ2, ε
)]Ashoke Sen, Mueller, Collins, Magnea
• KI contains all the poles in ε where, ST dimensions d = 4 + ε.
• GI collects finite terms at ε→ 0.
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Ingredient 2 . . .
RG invariance of F implies
µ2R
d
dµ2R
KI
(as,
µ2R
µ2, ε
)= −µ2
R
d
dµ2R
GI(as,
Q2
µ2R
,µ2R
µ2, ε
)= −AI
(as(µ
2R
)• AI ’s are finite, called cusp anomalous dimensions.
• AI ’s are maximally non-Abelian
Ag =CACF
Aq
CA = N, CF = (N2 − 1)/2N
• Solve by expanding KI , GI and AI in powers of as(µ2R
).
• Substitute these KI & GI in KG eqn to solve for ln F I toorder by order.
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Ingredient 2 . . .
RG invariance of F implies
µ2R
d
dµ2R
KI
(as,
µ2R
µ2, ε
)= −µ2
R
d
dµ2R
GI(as,
Q2
µ2R
,µ2R
µ2, ε
)= −AI
(as(µ
2R
)• AI ’s are finite, called cusp anomalous dimensions.
• AI ’s are maximally non-Abelian
Ag =CACF
Aq
CA = N, CF = (N2 − 1)/2N
• Solve by expanding KI , GI and AI in powers of as(µ2R
).
• Substitute these KI & GI in KG eqn to solve for ln F I toorder by order.
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Ingredient 2 . . . Solution of ln F I
The solution up to 3 loops are:
ln FI(as, Q
2, µ
2, ε) =
∞∑i=1
ais
(Q2
µ2
)i ε2Siε L
I,(i)F
(ε)
Moch, Vogt, Vermaseren; Ravindran; Magnea
with
LI,(1)F
(ε) =1
ε2
{− 2A
I1
}+
1
ε
{GI1(ε)
},
LI,(2)F
(ε) =1
ε3
{β0A
I1
}+
1
ε2
{−
1
2AI2 − β0G
I1(ε)
}+
1
ε
{1
2GI2(ε)
},
LI,(3)F
(ε) =1
ε4
{−
8
9β
20A
I1
}+
1
ε3
{2
9β1A
I1 +
8
9β0A
I2 +
4
3β
20G
I1(ε)
}
+1
ε2
{−
2
9AI3 −
1
3β1G
I1(ε)−
4
3β0G
I2(ε)
}+
1
ε
{1
3GI3(ε)
}
At every order, all the poles except 1ε can be predicted from
previous order.
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Ingredient 2 . . . Solution of ln F I
The solution up to 3 loops are:
ln FI(as, Q
2, µ
2, ε) =
∞∑i=1
ais
(Q2
µ2
)i ε2Siε L
I,(i)F
(ε)
Moch, Vogt, Vermaseren; Ravindran; Magnea
with
LI,(1)F
(ε) =1
ε2
{− 2A
I1
}+
1
ε
{GI1(ε)
},
LI,(2)F
(ε) =1
ε3
{β0A
I1
}+
1
ε2
{−
1
2AI2 − β0G
I1(ε)
}+
1
ε
{1
2GI2(ε)
},
LI,(3)F
(ε) =1
ε4
{−
8
9β
20A
I1
}+
1
ε3
{2
9β1A
I1 +
8
9β0A
I2 +
4
3β
20G
I1(ε)
}
+1
ε2
{−
2
9AI3 −
1
3β1G
I1(ε)−
4
3β0G
I2(ε)
}+
1
ε
{1
3GI3(ε)
}
At every order, all the poles except 1ε can be predicted from
previous order.
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Ingredient 2 . . . Single Pole Also Predictable
Ravindran, Smith, van Neerven
2-loop results for F q & F g in SU(N) dictates
GI1(ε) = 2(BI
1 − γI1)
+ f I1 +
∞∑k=1
εkg I,k1
GI2(ε) = 2(BI
2 − γI2)
+ f I2 − 2β0gI,1
1 +
∞∑k=1
εkg I,k2
• Origin : BIi collinear, γIi UV and f Ii soft region.
• BIi are collinear anomalous dimensions.
• Single pole can also be predicted!
• f Ii are soft anomalous dimensions. These satisfy
fgi =CACF
f qi i = 1, 2
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Ingredient 2 . . . Single Pole Also Predictable
Ravindran, Smith, van Neerven
2-loop results for F q & F g in SU(N) dictates
GI1(ε) = 2(BI
1 − γI1)
+ f I1 +
∞∑k=1
εkg I,k1
GI2(ε) = 2(BI
2 − γI2)
+ f I2 − 2β0gI,1
1 +
∞∑k=1
εkg I,k2
• Origin : BIi collinear, γIi UV and f Ii soft region.
• BIi are collinear anomalous dimensions.
• Single pole can also be predicted!
• f Ii are soft anomalous dimensions. These satisfy
fgi =CACF
f qi i = 1, 2
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Ingredient 2 . . . Single Pole Also Predictable
Ravindran, Smith, van Neerven
2-loop results for F q & F g in SU(N) dictates
GI1(ε) = 2(BI
1 − γI1)
+ f I1 +
∞∑k=1
εkg I,k1
GI2(ε) = 2(BI
2 − γI2)
+ f I2 − 2β0gI,1
1 +
∞∑k=1
εkg I,k2
• Origin : BIi collinear, γIi UV and f Ii soft region.
• BIi are collinear anomalous dimensions.
• Single pole can also be predicted!
• f Ii are soft anomalous dimensions. These satisfy
fgi =CACF
f qi i = 1, 2
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Ingredient 2 . . . Single Pole Also Predictable
Ravindran, Smith, van Neerven
2-loop results for F q & F g in SU(N) dictates
GI1(ε) = 2(BI
1 − γI1)
+ f I1 +
∞∑k=1
εkg I,k1
GI2(ε) = 2(BI
2 − γI2)
+ f I2 − 2β0gI,1
1 +
∞∑k=1
εkg I,k2
• Origin : BIi collinear, γIi UV and f Ii soft region.
• BIi are collinear anomalous dimensions.
• Single pole can also be predicted!
• f Ii are soft anomalous dimensions. These satisfy
fgi =CACF
f qi i = 1, 2
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 31
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Ingredient 2 . . . Single Pole Also Predictable
Ravindran, Smith, van Neerven
2-loop results for F q & F g in SU(N) dictates
GI1(ε) = 2(BI
1 − γI1)
+ f I1 +
∞∑k=1
εkg I,k1
GI2(ε) = 2(BI
2 − γI2)
+ f I2 − 2β0gI,1
1 +
∞∑k=1
εkg I,k2
• Origin : BIi collinear, γIi UV and f Ii soft region.
• BIi are collinear anomalous dimensions.
• Single pole can also be predicted!
• f Ii are soft anomalous dimensions. These satisfy
fgi =CACF
f qi i = 1, 2
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Ingredient 2 . . . Single Pole Also Predictable
Ravindran, Smith, van Neerven
2-loop results for F q & F g in SU(N) dictates
GI1(ε) = 2(BI
1 − γI1)
+ f I1 +
∞∑k=1
εkg I,k1
GI2(ε) = 2(BI
2 − γI2)
+ f I2 − 2β0gI,1
1 +
∞∑k=1
εkg I,k2
• Origin : BIi collinear, γIi UV and f Ii soft region.
• BIi are collinear anomalous dimensions.
• Single pole can also be predicted!
• f Ii are soft anomalous dimensions. These satisfy
fgi =CACF
f qi i = 1, 2
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Ingredient 2 . . . Single Pole Also Predictable
3-loop results for the F I by Moch et al, Gehrmann et al. confirmthis conclusion
GI3(ε) = 2(BI3 − γI3) + f I3 − 2β1g
I,11 − 2β0
(g I,1
2 + 2β0gI,2
1
)+
∞∑k=1
εkg I,k3
with
fg3 =CACF
f q3
• f I ’s are universal.
• Explicit computation of FF gives ε dependent gI,ki .
• This understanding is crucial to get soft part.
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Ingredient 2 . . . Single Pole Also Predictable
3-loop results for the F I by Moch et al, Gehrmann et al. confirmthis conclusion
GI3(ε) = 2(BI3 − γI3) + f I3 − 2β1g
I,11 − 2β0
(g I,1
2 + 2β0gI,2
1
)+
∞∑k=1
εkg I,k3
with
fg3 =CACF
f q3
• f I ’s are universal.
• Explicit computation of FF gives ε dependent gI,ki .
• This understanding is crucial to get soft part.
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Ingredient 3 : Mass Factorization ln Γ
DGLAP kernel Γ satisfies the RGE
µ2F
d
dµ2F
Γ(z, µ2F , ε) =
1
2P(z, µ2
F
)⊗ Γ
(z, µ2
F , ε)
• P ’s are Altarelli Parisi splitting functions (matrix valued).
• Diagonal parts of the kernel ONLY contribute to SV limit.
• By solving we get Γ ln Γ
• Γ is available up to O(a4s).
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Where are we?
X ln[ZI ]2
X ln |F I |2
X ln ΓII
?? ΦI
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Ingredient 4 : Strategy to Calculate Soft ΦI
Revisiting cancellation of poles
ln F I
lnZI ln ΓII
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Ingredient 4 : Strategy to Calculate Soft ΦI
Revisiting cancellation of poles
ln F I
lnZI ln ΓII
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Ingredient 4 : Strategy to Calculate Soft ΦI
Revisiting cancellation of poles
ln F I
lnZI ln ΓII
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Ingredient 4 : Strategy to Calculate Soft ΦI
Revisiting cancellation of poles
ln F I
lnZI ln ΓII
ΦI
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Ingredient 4 : Strategy to Calculate Soft ΦI
Revisiting cancellation of poles
ln F I
lnZI ln ΓII
ΦI
• Demanding finiteness of partonic CS ⇒ ΦI must have soft aswell as collinear contributions.
• Now onward we will call ΦI as Soft-Collinear distribution.
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Ingredient 4 : Strategy to Calculate Soft ΦI
Revisiting cancellation of poles
ln F I
lnZI ln ΓII
ΦI
• Demanding finiteness of partonic CS ⇒ ΦI must have soft aswell as collinear contributions.
• Now onward we will call ΦI as Soft-Collinear distribution.
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Ingredient 4 . . . Ansatz for Soft-Collinear ΦI
Demand of
1. finiteness of ∆SV,I(z) at ε→ 0 and
2. the RGE
µ2R
d
dµ2R
ΦI(as, q
2, µ2, z, ε)
= 0
can be achieved if we make an ansatz that ΦI satisfies KG typeDE
q2 d
dq2ΦI =
1
2
[KI(as,
µ2R
µ2, z, ε
)+G
I(as,
q2
µ2R
,µ2R
µ2, z, ε
)]
• KIcontains all the poles in ε. G
Icontains finite terms in
ε→ 0.
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Ingredient 4 . . . Ansatz for Soft-Collinear ΦI
Demand of
1. finiteness of ∆SV,I(z) at ε→ 0 and
2. the RGE
µ2R
d
dµ2R
ΦI(as, q
2, µ2, z, ε)
= 0
can be achieved if we make an ansatz that ΦI satisfies KG typeDE
q2 d
dq2ΦI =
1
2
[KI(as,
µ2R
µ2, z, ε
)+G
I(as,
q2
µ2R
,µ2R
µ2, z, ε
)]
• KIcontains all the poles in ε. G
Icontains finite terms in
ε→ 0.
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Ingredient 4 . . . Ansatz for Soft-Collinear ΦI
Demand of
1. finiteness of ∆SV(z) at ε→ 0 and
2. the RGE
µ2R
d
dµ2R
ΦI(as, q
2, µ2, z, ε)
= 0
can be achieved if we make an ansatz that Φ satisfies KG type DE
q2 d
dq2ΦI =
1
2
[KI(as,
µ2R
µ2, z, ε
)+G
I(as,
q2
µ2R
,µ2R
µ2, z, ε
)]
KI
contains all the poles in ε. GI
contains finite terms inε→ 0.
• The solutions of KG consistent with our demands
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Ingredient 4 . . . Solution of ΦI
ΦI(as, q2, µ2, z, ε) =
∞∑i=1
ais
(q2(1− z)2
µ2
)i ε2
Siε
(iε
1− z
)φI,(i)(ε)
where
φI,(i)(ε) = LI,(i)F (ε)
(AI → −AI , GI(ε)→ GI(ε)
)This implies
GI1(ε) = −fI1 +∞∑k=1
εkGI,k1
GI2(ε) = −fI2 − 2β0GI,11 +
∞∑k=1
εkGI,k2
GI3(ε) = −fI3 − 2β1GI,11 − 2β0
(GI,12 + 2β0G
I,21
)+∞∑k=1
εkGI,k3
Explicit computation is required to get ε dependent cof GI,ki .
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Ingredient 4 . . . Solution of ΦI
ΦI(as, q2, µ2, z, ε) =
∞∑i=1
ais
(q2(1− z)2
µ2
)i ε2
Siε
(iε
1− z
)φI,(i)(ε)
where
φI,(i)(ε) = LI,(i)F (ε)
(AI → −AI , GI(ε)→ GI(ε)
)This implies
GI1(ε) = −fI1 +∞∑k=1
εkGI,k1
GI2(ε) = −fI2 − 2β0GI,11 +
∞∑k=1
εkGI,k2
GI3(ε) = −fI3 − 2β1GI,11 − 2β0
(GI,12 + 2β0G
I,21
)+∞∑k=1
εkGI,k3
Explicit computation is required to get ε dependent cof GI,ki .
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Ingredient 4 . . . Crucial Property of ΦI
Explicit computation shows
Φq =CFCA
Φg and Gq,ki =CFCAGg,ki
up to O(a2s).
soft-collinear distributions are universal
ΦI = CIΦ
with
CI =
{CA for Higgs in gg
CF for DY and for Higgs in bb
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Ingredient 4 . . . Crucial Property of ΦI
Explicit computation shows
Φq =CFCA
Φg and Gq,ki =CFCAGg,ki
up to O(a2s).
soft-collinear distributions are universal
ΦI = CIΦ
with
CI =
{CA for Higgs in gg
CF for DY and for Higgs in bb
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Where are we?
X ln[ZI ]2
X ln |F I |2
X ln ΓII
X ΦI
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The Prescription
⇓
Master Formula
⇓
How do we compute these?
⇓What about DY at N3LO?
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Results : SV XSection for DY at N3LO
∆SV,(3) = ∆SV,(3)|δδ(1− z) +∑5
j=0 ∆SV,(3)|DjDj
I All the Dj , j = 0, 1, . . . , 5 and partial δ(1− z) contributionswere known for a decade.
Moch and Vogt
I We completed the full computation of δ(1− z) part.TA, Mahakhud, Rana, Ravindran
• All the required quantities were available except Gq,13 fromΦq,(3).
X ln[Zq]2 up to O(a3s)
X ln |F q|2 up to O(a3s)
X ln Γqq up to O(a3s)
X Φq up to O(a2s) BUT ? O(a3
s)
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Results : SV XSection for DY at N3LO
∆SV,(3) = ∆SV,(3)|δδ(1− z) +∑5
j=0 ∆SV,(3)|DjDj
I All the Dj , j = 0, 1, . . . , 5 and partial δ(1− z) contributionswere known for a decade.
Moch and Vogt
I We completed the full computation of δ(1− z) part.TA, Mahakhud, Rana, Ravindran
• All the required quantities were available except Gq,13 fromΦq,(3).
X ln[Zq]2 up to O(a3s)
X ln |F q|2 up to O(a3s)
X ln Γqq up to O(a3s)
X Φq up to O(a2s) BUT ? O(a3
s)
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Results : SV XSection for DY at N3LO
∆SV,(3) = ∆SV,(3)|δδ(1− z) +∑5
j=0 ∆SV,(3)|DjDj
I All the Dj , j = 0, 1, . . . , 5 and partial δ(1− z) contributionswere known for a decade.
Moch and Vogt
I We completed the full computation of δ(1− z) part.TA, Mahakhud, Rana, Ravindran
• All the required quantities were available except Gq,13 fromΦq,(3).
X ln[Zq]2 up to O(a3s)
X ln |F q|2 up to O(a3s)
X ln Γqq up to O(a3s)
X Φq up to O(a2s) BUT ? O(a3
s)
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Results : SV XSection for DY at N3LO . . .
• Recently, complete SV XSec at N3LO for the Higgs productionin gluon fusion was computed by Anastasiou et. al.
• From which we extracted Φg at O(a3s).
• Recall
Φq =CFCA
Φg
was established up to O(a2s) by explicit computation.
• We conjectured the relation to be hold true even at O(a3s).
• Using this we got Φq Gq,13 .
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Results : SV XSection for DY at N3LO . . .
• Recently, complete SV XSec at N3LO for the Higgs productionin gluon fusion was computed by Anastasiou et. al.
• From which we extracted Φg at O(a3s).
• Recall
Φq =CFCA
Φg
was established up to O(a2s) by explicit computation.
• We conjectured the relation to be hold true even at O(a3s).
• Using this we got Φq Gq,13 .
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Results : SV XSection for DY at N3LO . . .
• Recently, complete SV XSec at N3LO for the Higgs productionin gluon fusion was computed by Anastasiou et. al.
• From which we extracted Φg at O(a3s).
• Recall
Φq =CFCA
Φg
was established up to O(a2s) by explicit computation.
• We conjectured the relation to be hold true even at O(a3s).
• Using this we got Φq Gq,13 .
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Results : SV XSection for DY at N3LO . . .
• This gives the ONLY missing part to achieve δ(1− z) part.
• In addition, now D7, . . .D1 are available at N4LO exactly.TA, Mahakhud, Rana, Ravindran; de Florian, Mazzitelli, Moch, Vogt
• Later the result was reconfirmed by two groups : Catani, Cieri, de
Florian, Ferrera, Frazzini; Li, von Manteuffel, Schabinger, Zhu
• This in turn establishes our conjecture at O(a3s).
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Results : Numerical Implications
At 14 TeV LHC with µF = µR = Q = ml+l−
•[Q2 dσ
dQ2
]δ≈ −
∑j
[Q2 dσ
dQ2
]Dj⇒ Most of the contributions
from Dj ’s are cancelled against δ(1− z) making N3LOSV
more subleading.
Q (GeV) 50 90 200 400 600 1000
δ (nb) 2.561 10−3 140.114 10−3 4.567 10−5 3.153 10−6 6.473 10−7 7.755 10−8
D (nb) -2.053 10−3 -124.493 10−3 -4.421 10−5 -3.368 10−6 -7.455 10−7 -9.959 10−8
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Results : Numerical Implications . . .
• Correction at N3LOSV is very very small Good newz!
Q (GeV) 50 90 200 400 600 1000
NNLO (nb) 0.158 11.296 5.233 10−3 4.694 10−4 1.116 10−4 1.607 10−5
N3LOSV (nb) 0.158 11.311 5.234 10−3 4.692 10−4 1.116 10−4 1.605 10−5
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Results : Numerical Implications . . .
• Scale dependence reduces.
/QR
µ0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(i)
R
0.98
1
1.02
1.04
1.06
Q = 200 GeV
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(i)
R
0.9
0.95
1
1.05
1.1
1.15NLO
NNLO
SVLO3NQ = 20 GeV
R(i) ≡[Q2 d
dQ2σ(i)(µ2
R)]/[
Q2 ddQ2σ
(i)(Q2)]
and µF = Q
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Summary
• A systematic way of computing threshold corrections toinclusive Higgs and DY productions in pQCD is prescribed.
• Underlying principles : factorization of soft and collineardivergences, RG invariance, gauge invariance and Sudakovresummation.
• δ(1− z) part of N3LOSV DY Xsection is computed for thefirst time.
• Importantly, the calculation is done even without performingthe explicit computation of the real emission diagrams for DYat O(a3
s).
• Necessary information arising from real emission processes isextracted from the computation of Higgs boson in gg atN3LO threshold.
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Summary
• A systematic way of computing threshold corrections toinclusive Higgs and DY productions in pQCD is prescribed.
• Underlying principles : factorization of soft and collineardivergences, RG invariance, gauge invariance and Sudakovresummation.
• δ(1− z) part of N3LOSV DY Xsection is computed for thefirst time.
• Importantly, the calculation is done even without performingthe explicit computation of the real emission diagrams for DYat O(a3
s).
• Necessary information arising from real emission processes isextracted from the computation of Higgs boson in gg atN3LO threshold.
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Summary . . .
• We find that the impact of the δ(1− z) contribution is quitelarge to the pure N3LOSV correction.
• This method has been later employed to compute some otherinclusive and exclusive observables at threshold by us in
Phys. Rev. Lett. 113 (2014) 212003
JHEP 1410 (2014) 139
JHEP 1502 (2015) 131
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Other Areas of Interest
RG Improved Higgs Boson Production toN3LO in QCD
TA, Das, Kumar, Rana, Ravindran
arXiv:1505.07422
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 54
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Large Scale Uncertainties
State of the art : The production cross section of Higgs bosonthrough gg at N3LO by Anastasiou et. al.
Consequence
(a) Spectacular accuracy
(b) Significant reduction in unphysical renormalization &factorization scales.
Points of concern
(a) Significant increase in scale uncertainties upon increasing therange of scale variation : µR < mH/4
(b) Total cross section may become negative in this region.
Reason : The presence of large logarithms of the scales at everyorder ⇒ ans (µ2
R) lnk(µ2R/m
2H).
Remedy : (a) Go beyond N3LO, (b) Perform all orderresummations
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 55
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Large Scale Uncertainties
State of the art : The production cross section of Higgs bosonthrough gg at N3LO by Anastasiou et. al.
Consequence
(a) Spectacular accuracy
(b) Significant reduction in unphysical renormalization &factorization scales.
Points of concern
(a) Significant increase in scale uncertainties upon increasing therange of scale variation : µR < mH/4
(b) Total cross section may become negative in this region.
Reason : The presence of large logarithms of the scales at everyorder ⇒ ans (µ2
R) lnk(µ2R/m
2H).
Remedy : (a) Go beyond N3LO, (b) Perform all orderresummations
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 55
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Large Scale Uncertainties
State of the art : The production cross section of Higgs bosonthrough gg at N3LO by Anastasiou et. al.
Consequence
(a) Spectacular accuracy
(b) Significant reduction in unphysical renormalization &factorization scales.
Points of concern
(a) Significant increase in scale uncertainties upon increasing therange of scale variation : µR < mH/4
(b) Total cross section may become negative in this region.
Reason : The presence of large logarithms of the scales at everyorder ⇒ ans (µ2
R) lnk(µ2R/m
2H).
Remedy : (a) Go beyond N3LO, (b) Perform all orderresummations
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 55
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Large Scale Uncertainties
Figure: Higgs boson production in gg fusion
Duhr, Talk at RADCOR ’15
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 56
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Large Scale Uncertainties
State of the art : The production cross section of Higgs bosonthrough gg at N3LO by Anastasiou et. al.
Consequence
(a) Spectacular accuracy
(b) Significant reduction in unphysical renormalization &factorization scales.
Points of concern
(a) Significant increase in scale uncertainties upon increasing therange of scale variation : µR < mH/4
(b) Total cross section may become negative in this region.
Reason : The presence of large logarithms of the scales at everyorder ⇒ ans (µ2
R) lnk(µ2R/m
2H).
Remedy : (a) Go beyond N3LO, (b) Perform all orderresummations
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 57
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Large Scale Uncertainties
State of the art : The production cross section of Higgs bosonthrough gg at N3LO by Anastasiou et. al.
Consequence
(a) Spectacular accuracy
(b) Significant reduction in unphysical renormalization &factorization scales.
Points of concern
(a) Significant increase in scale uncertainties upon increasing therange of scale variation : µR < mH/4
(b) Total cross section may become negative in this region.
Reason : The presence of large logarithms of the scales at everyorder ⇒ ans (µ2
R) lnk(µ2R/m
2H).
Remedy : (a) Go beyond N3LO, (b) Perform all orderresummations
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 57
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Our Proposal to Improve µR Dependence
Our proposal : Perform resummations by including RG accessibleµR dependent logarithms of all orders.
Consequence : Quite remarkable!
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
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What are we resumming? . . .
Collecting the highest logs to all orders
σ(0)Σ (S,m2
H, µ2R) ≡
∞∑n=0
σ(n)n (asLR)
n
Collecting the next-to-highest logs to all orders
σ(1)Σ (S,m2
H, µ2R) ≡
∞∑n=1
σ(n)n−1 (asLR)
n−1
and so on. We re-write the Xsection in terms of these as
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
= a2s(µ2
R)
∞∑i=0
ais(µ2R)σ
(i)Σ (S,m2
H, µ2R)
Goal : Compute σ(i)Σ for i = 0, 1, 2, 3 for gg → H.
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Our Proposal to Improve µR Dependence
Our proposal : Perform resummations by including RG accessibleµR dependent logarithms of all orders.
Consequence : Quite remarkable!
Hm/R
µ0.1 1 10
[pb
]σ
10
20
30
40
50
60
70LO NLO
NNLO LO3N
H = mF
µ
MSTW2008
13 TeV LHC
FO RESUM
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Our Proposal to Improve µR Dependence . . .
[ TeV ]s
7 8 9 10 11 12 13 14
[ p
b ]
σ
10
20
30
40
50
60FO NNLO RESUM NNLO
LO3FO N LO3RESUM N
H = mF
µ
]H
/10,10mH
= [mR
µ
MSTW2008
Figure: Dependence of µR in both the fixed order and resummedXsections on
√S
Demerit : Method fails to improve the central value of Xsection,unlike other resummation.
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Other Areas of Interest . . .
Two-loop QCD corrections to Higgs → b+ b+ gamplitude
TA, Mahakhud, Mathews, Rana, Ravindran
JHEP 1408 (2014) 075
• Theory : SM.
• Bottom quark mass VFS scheme.
• No of diagrams : 251
• Mostly in-house codes written in FORM and Mathematica.
• For IBP, LI identities LiteRed by Lee.
• Agrees with the universal IR-pole structures of QCD.
• Result will be used for Higgs + jet production through bb atNNLO QCD in hadron colliders.
• Analytically computed.
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Other Areas of Interest . . .
Two-Loop QCD Correction to massive spin-2 resonance→ 3 gluons
TA, Mahakhud, Mathews, Rana, Ravindran
JHEP 1405 (2014) 107
• Spin-2 couples to SM minimally through SMenergy-momentum tensor.
• No of diagrams : 2362
• Mostly in-house codes written in FORM and Mathematica.
• For IBP, LI identities LiteRed by Lee.
• Explicit verification of the universal IR-pole structure of QCDwhen spin-2 presents.
• Result will be used for spin-2 + jet production at NNLO QCDin hadron colliders.
• Analytically computed.
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Extra Slides
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Extra Slide
Extra 1 : RG Resum in Details
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Understanding The Source Of The Problem
The inclusive cross section of Higgs production
σH(S,m2H) = σ0a2
s(µ2R)∑a,b
∫dx1dx2fa(x1, µ
2F )fb(x2, µ
2F )
× C2H
(as(µ
2R))
∆Hab
(τ
x1x2,m2
H, µ2R, µ
2F
)≡ a2
s(µ2R) σ(S,m2
H, µ2R)
• RG invariance wrt µR of the observable µ2R
ddµ2RσH = 0
• The Solution
σ(µ2R) = σ(µ2
0) exp
[−∫ µ2
R
µ20
dµ2
µ2
2 β(as(µ2))
as(µ2)
]
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 68
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Understanding The Source Of The Problem
The inclusive cross section of Higgs production
σH(S,m2H) = σ0a2
s(µ2R)∑a,b
∫dx1dx2fa(x1, µ
2F )fb(x2, µ
2F )
× C2H
(as(µ
2R))
∆Hab
(τ
x1x2,m2
H, µ2R, µ
2F
)≡ a2
s(µ2R) σ(S,m2
H, µ2R)
• RG invariance wrt µR of the observable µ2R
ddµ2RσH = 0
• The Solution
σ(µ2R) = σ(µ2
0) exp
[−∫ µ2
R
µ20
dµ2
µ2
2 β(as(µ2))
as(µ2)
]
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 68
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Understanding The Source Of The Problem
The inclusive cross section of Higgs production
σH(S,m2H) = σ0a2
s(µ2R)∑a,b
∫dx1dx2fa(x1, µ
2F )fb(x2, µ
2F )
× C2H
(as(µ
2R))
∆Hab
(τ
x1x2,m2
H, µ2R, µ
2F
)≡ a2
s(µ2R) σ(S,m2
H, µ2R)
• RG invariance wrt µR of the observable µ2R
ddµ2RσH = 0
• The Solution
σ(µ2R) = σ(µ2
0) exp
[−∫ µ2
R
µ20
dµ2
µ2
2 β(as(µ2))
as(µ2)
]
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 68
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Understanding The Source Of The Problem
• Considering µ0 as central scale mH and using naive evolutionof as, the solution
σ(µ2R) =
∞∑n=0
n∑k=0
ans (µ2R) Rn,k LkR
where, LR ≡ ln(µ2R
m2H
).
• The RG invariance dictates
Rn,n−m =1
(n−m)
m∑i=0
(n− i+ 1)βiRn−i−1,n−m−1
i.e. the coefficients of the logarithms Rn,k(0 < k ≤ n) can beexpressed in terms of the lower order ones Rn−1,0.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 69
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Understanding The Source Of The Problem
• Considering µ0 as central scale mH and using naive evolutionof as, the solution
σ(µ2R) =
∞∑n=0
n∑k=0
ans (µ2R) Rn,k LkR
where, LR ≡ ln(µ2R
m2H
).
• The RG invariance dictates
Rn,n−m =1
(n−m)
m∑i=0
(n− i+ 1)βiRn−i−1,n−m−1
i.e. the coefficients of the logarithms Rn,k(0 < k ≤ n) can beexpressed in terms of the lower order ones Rn−1,0.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 69
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Understanding The Source Of The Problem
• Coefficients of the highest logarithms at nth order in as growsas (n+ 1)ansβ
n0R0,0
can be potentially large contributions!
fixed order predictions become unreliable
• RG invariance can be used to resum these large logarithms toall orders.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 70
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Understanding The Source Of The Problem
• Coefficients of the highest logarithms at nth order in as growsas (n+ 1)ansβ
n0R0,0
can be potentially large contributions!
fixed order predictions become unreliable
• RG invariance can be used to resum these large logarithms toall orders.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 70
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The Remedy : Our Prescription
We extend the approach by Ahmady et. al. to resum these largecontributions.
• We rewrite the solution σ(µ2R) as
σ(µ2R) =
∞∑m=0
ams (µ2R)
∞∑n=m
Rn,n−m(asLR)n−m
≡∞∑m=0
ams (µ2R) σ
(m)Σ
(as(µ
2R)LR
),
σ(m)Σ resums as(µ
2R)LR to all orders.
• Our Goal : Determine σ(m)Σ ’s.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 71
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The Remedy : Our Prescription
We extend the approach by Ahmady et. al. to resum these largecontributions.
• We rewrite the solution σ(µ2R) as
σ(µ2R) =
∞∑m=0
ams (µ2R)
∞∑n=m
Rn,n−m(asLR)n−m
≡∞∑m=0
ams (µ2R) σ
(m)Σ
(as(µ
2R)LR
),
σ(m)Σ resums as(µ
2R)LR to all orders.
• Our Goal : Determine σ(m)Σ ’s.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 71
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The Remedy : Our Prescription
Using the recursion relation we show σ(m)Σ satisfies
[ωd
dω+ (m+ 2)
]σ
(m)Σ
= Θm−1
m∑i=1
ηi
[(1− ω)
d
dω− (m− i+ 2)
]σ
(m−i)Σ
ηi ≡ βi/β0 and ω ≡ 1− β0as(µ2R)LR
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 72
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The Remedy : The Solutions
σ(0)Σ =
1
ω2
{R0,0
}, σ
(1)Σ =
1
ω3
{R1,0 − 2η1R0,0 ln(ω)
},
σ(2)Σ =
1
ω3
{2R0,0
(η1
2 − η2
)}+
1
ω4
{R2,0 + 2R0,0
(η2 − η1
2)
+ ln(ω)(− 2η1
2R0,0 − 3η1R1,0
)+ 3η1
2R0,0 ln2(ω)},
σ(3)Σ =
1
ω3
{R0,0
(−η1
3+ 2η1η2 − η3
)}+
1
ω4
{R0,0
(2η1
3 − 2η1η2
)+R1,0
(3η1
2 − 3η2
)+R0,0
(6η1η2 − 6η1
3)
ln(ω)}
+1
ω5
{R3,0 +R0,0
(η3 − η1
3)
+R1,0
(3η2 − 3η1
2)
+ ln(ω)(R0,0
(6η1
3 − 8η1η2
)− 3η1
2R1,0 − 4η1R2,0
)+ ln
2(ω)(7η1
3R0,0 + 6η12R1,0
)− 4η1
3R0,0 ln3(ω)},
σ(4)Σ = Big expression
• We have resummed only µR dependent logarithms.
• µF has been chosen to some specific value mH µFdependence remains unchanged.
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The Remedy : The Solutions
σ(0)Σ =
1
ω2
{R0,0
}, σ
(1)Σ =
1
ω3
{R1,0 − 2η1R0,0 ln(ω)
},
σ(2)Σ =
1
ω3
{2R0,0
(η1
2 − η2
)}+
1
ω4
{R2,0 + 2R0,0
(η2 − η1
2)
+ ln(ω)(− 2η1
2R0,0 − 3η1R1,0
)+ 3η1
2R0,0 ln2(ω)},
σ(3)Σ =
1
ω3
{R0,0
(−η1
3+ 2η1η2 − η3
)}+
1
ω4
{R0,0
(2η1
3 − 2η1η2
)+R1,0
(3η1
2 − 3η2
)+R0,0
(6η1η2 − 6η1
3)
ln(ω)}
+1
ω5
{R3,0 +R0,0
(η3 − η1
3)
+R1,0
(3η2 − 3η1
2)
+ ln(ω)(R0,0
(6η1
3 − 8η1η2
)− 3η1
2R1,0 − 4η1R2,0
)+ ln
2(ω)(7η1
3R0,0 + 6η12R1,0
)− 4η1
3R0,0 ln3(ω)},
σ(4)Σ = Big expression
• We have resummed only µR dependent logarithms.
• µF has been chosen to some specific value mH µFdependence remains unchanged.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 73
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The Remedy : Numerical Implications
(a) Result is almost µR independent for a wide range of µR ∈[0.1mH, 10mH]
LO NLO NNLO N3LO
FO (%) 167.26 143.40 54.99 27.01
RESUM (%) 6.11 5.47 3.39 1.23
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 74
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The Remedy : Numerical Implications
Hm/R
µ0.1 1 10
[pb
]σ
10
20
30
40
50
60
70LO NLO
NNLO LO3N
H = mF
µ
MSTW2008
13 TeV LHC
FO RESUM
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The Remedy : Numerical Implications
[ TeV ]s
7 8 9 10 11 12 13 14
[ p
b ]
σ
10
20
30
40
50
60FO NNLO RESUM NNLO
LO3FO N LO3RESUM N
H = mF
µ
]H
/10,10mH
= [mR
µ
MSTW2008
(b) Cross section is always positive and reliable.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 76
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The Remedy : Numerical Implications
[ TeV ]s
7 8 9 10 11 12 13 14
[ p
b ]
σ
10
20
30
40
50
60FO NNLO RESUM NNLO
LO3FO N LO3RESUM N
H = mF
µ
]H
/10,10mH
= [mR
µ
MSTW2008
(b) Cross section is always positive and reliable.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 76
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Extra Slide
Extra 2 : What are we resumming?
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 77
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 78
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
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What are we resumming?
The inclusive cross section of Higgs production
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
Structure of perturbation theory :
σ = σ(0)0
+ as(µ2R
){σ
(1)0 + σ
(1)1 LR
}+ a2
s
(µ2R
){σ
(2)0 + σ
(2)1 LR + σ
(2)2 L2
R
}+ a3
s
(µ2R
){σ
(3)0 + σ
(3)1 LR + σ
(3)2 L2
R + σ(3)3 L3
R
}+ . . .
where
LR ≡ ln
(µ2R
m2H
)
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What are we resumming? . . .
Collecting the highest logs to all orders
σ(0)Σ (S,m2
H, µ2R) ≡
∞∑n=0
σ(n)n (asLR)
n
Collecting the next-to-highest logs to all orders
σ(1)Σ (S,m2
H, µ2R) ≡
∞∑n=1
σ(n)n−1 (asLR)
n−1
and so on. We re-write the Xsection in terms of these as
σH(S,m2H) = a2
s(µ2R) σ(S,m2
H, µ2R)
= a2s(µ2
R)
∞∑i=0
ais(µ2R)σ
(i)Σ (S,m2
H, µ2R)
Goal : Compute σ(i)Σ for i = 0, 1, 2, 3 for gg → H.
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Extra Slide
Extra 3 : SV Xsection of Higgs in bb
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Result 2 : SV CS for Higgs in bb at N3LO
∆SV,(3) for Higgs production in bb annihilation
• All the Di and partial δ(1− z) contributions were known.
• We completed the full computation of δ(1− z) part.
• All the required quantities were available except Gb,13 from Φb
and gb,13 from 3-loop form factor F b.
• Being flavor independent
Gb,13 = Gq,13
which is available from our DY result.
• gb,13 is extracted from recent result of 3-loop Hbb form factorby Gehrmann et. al.
• This completes the calculation of threshold N3LO correctionsto Higgs production in bb annihilation.
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Result 2 : SV CS for Higgs in bb at N3LO
∆SV,(3) for Higgs production in bb annihilation
• All the Di and partial δ(1− z) contributions were known.
• We completed the full computation of δ(1− z) part.
• All the required quantities were available except Gb,13 from Φb
and gb,13 from 3-loop form factor F b.
• Being flavor independent
Gb,13 = Gq,13
which is available from our DY result.
• gb,13 is extracted from recent result of 3-loop Hbb form factorby Gehrmann et. al.
• This completes the calculation of threshold N3LO correctionsto Higgs production in bb annihilation.
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Result 2 : SV CS for Higgs in bb at N3LO
∆SV,(3) for Higgs production in bb annihilation
• All the Di and partial δ(1− z) contributions were known.
• We completed the full computation of δ(1− z) part.
• All the required quantities were available except Gb,13 from Φb
and gb,13 from 3-loop form factor F b.
• Being flavor independent
Gb,13 = Gq,13
which is available from our DY result.
• gb,13 is extracted from recent result of 3-loop Hbb form factorby Gehrmann et. al.
• This completes the calculation of threshold N3LO correctionsto Higgs production in bb annihilation.
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Result 2 : SV CS for Higgs in bb at N3LO
∆SV,(3) for Higgs production in bb annihilation
• All the Di and partial δ(1− z) contributions were known.
• We completed the full computation of δ(1− z) part.
• All the required quantities were available except Gb,13 from Φb
and gb,13 from 3-loop form factor F b.
• Being flavor independent
Gb,13 = Gq,13
which is available from our DY result.
• gb,13 is extracted from recent result of 3-loop Hbb form factorby Gehrmann et. al.
• This completes the calculation of threshold N3LO correctionsto Higgs production in bb annihilation.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 81
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Result 2 : SV CS for Higgs in bb at N3LO
∆SV,(3) for Higgs production in bb annihilation
• All the Di and partial δ(1− z) contributions were known.
• We completed the full computation of δ(1− z) part.
• All the required quantities were available except Gb,13 from Φb
and gb,13 from 3-loop form factor F b.
• Being flavor independent
Gb,13 = Gq,13
which is available from our DY result.
• gb,13 is extracted from recent result of 3-loop Hbb form factorby Gehrmann et. al.
• This completes the calculation of threshold N3LO correctionsto Higgs production in bb annihilation.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 81
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Result 2 : SV CS for Higgs in bb at N3LO
∆SV,(3) for Higgs production in bb annihilation
• All the Di and partial δ(1− z) contributions were known.
• We completed the full computation of δ(1− z) part.
• All the required quantities were available except Gb,13 from Φb
and gb,13 from 3-loop form factor F b.
• Being flavor independent
Gb,13 = Gq,13
which is available from our DY result.
• gb,13 is extracted from recent result of 3-loop Hbb form factorby Gehrmann et. al.
• This completes the calculation of threshold N3LO correctionsto Higgs production in bb annihilation.
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 81
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Extra Slide
Extra 4 : Fixing soft-collinear distribution ΦI
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Fixing Forms of ΦI
∆SV,I
= C exp(ΨI) with
ΨI
=
[ln(ZI)2
+ ln |F I |2]δ(1− z) + 2Φ
I − 2C ln ΓII
Considering only poles at O(as) with µR = µF in d = 4 + ε
ln(ZI,(1)
)2= as(µ
2F )
4γI0
ε
ln |F I,(1)|2 = as(µ2F )
(q2
µ2F
)ε/2 [−
4AI1
ε2+
1
ε
(2fI1 + 4B
I1 − 4γ
I0
)]
2C ln ΓII,(1) = 2as(µ2F )
[2BI1
εδ(1− z) +
2AI1
εD0
]
Collecting the coefficient of as(µ2F )[
neglecting ln(q2/µ2
F
)terms
]
ΨI,(1)
= as(µ2F )
��4γI0
ε−
4AI1
ε2+
1
ε
(2fI1 +ZZ4B
I1 −��4γ
I0
) δ(1− z)−@@4BI1
εδ(1− z) +
4AI1
εD0
+ 2ΦI,(1)
= as(µ
2F )
[{−
4AI1
ε2+
2fI1
ε
}δ(1− z)−
4AI1
εD0
]+ 2Φ
I
Demand 1 : Hence to cancel all the poles, we must have at O(as)
2ΦI,(1)|poles = as(µ
2F )
[{4AI1
ε2−
2fI1
ε
}δ(1− z) +
4AI1
εD0
]
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Fixing Forms of ΦI
∆SV,I
= C exp(ΨI) with
ΨI
=
[ln(ZI)2
+ ln |F I |2]δ(1− z) + 2Φ
I − 2C ln ΓII
Considering only poles at O(as) with µR = µF in d = 4 + ε
ln(ZI,(1)
)2= as(µ
2F )
4γI0
ε
ln |F I,(1)|2 = as(µ2F )
(q2
µ2F
)ε/2 [−
4AI1
ε2+
1
ε
(2fI1 + 4B
I1 − 4γ
I0
)]
2C ln ΓII,(1) = 2as(µ2F )
[2BI1
εδ(1− z) +
2AI1
εD0
]
Collecting the coefficient of as(µ2F )[
neglecting ln(q2/µ2
F
)terms
]
ΨI,(1)
= as(µ2F )
��4γI0
ε−
4AI1
ε2+
1
ε
(2fI1 +ZZ4B
I1 −��4γ
I0
) δ(1− z)−@@4BI1
εδ(1− z) +
4AI1
εD0
+ 2ΦI,(1)
= as(µ
2F )
[{−
4AI1
ε2+
2fI1
ε
}δ(1− z)−
4AI1
εD0
]+ 2Φ
I
Demand 1 : Hence to cancel all the poles, we must have at O(as)
2ΦI,(1)|poles = as(µ
2F )
[{4AI1
ε2−
2fI1
ε
}δ(1− z) +
4AI1
εD0
]
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Fixing Forms of ΦI
∆SV,I
= C exp(ΨI) with
ΨI
=
[ln(ZI)2
+ ln |F I |2]δ(1− z) + 2Φ
I − 2C ln ΓII
Considering only poles at O(as) with µR = µF in d = 4 + ε
ln(ZI,(1)
)2= as(µ
2F )
4γI0
ε
ln |F I,(1)|2 = as(µ2F )
(q2
µ2F
)ε/2 [−
4AI1
ε2+
1
ε
(2fI1 + 4B
I1 − 4γ
I0
)]
2C ln ΓII,(1) = 2as(µ2F )
[2BI1
εδ(1− z) +
2AI1
εD0
]
Collecting the coefficient of as(µ2F )[
neglecting ln(q2/µ2
F
)terms
]
ΨI,(1)
= as(µ2F )
��4γI0
ε−
4AI1
ε2+
1
ε
(2fI1 +ZZ4B
I1 −��4γ
I0
) δ(1− z)−@@4BI1
εδ(1− z) +
4AI1
εD0
+ 2ΦI,(1)
= as(µ
2F )
[{−
4AI1
ε2+
2fI1
ε
}δ(1− z)−
4AI1
εD0
]+ 2Φ
I
Demand 1 : Hence to cancel all the poles, we must have at O(as)
2ΦI,(1)|poles = as(µ
2F )
[{4AI1
ε2−
2fI1
ε
}δ(1− z) +
4AI1
εD0
]
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Fixing Form of ΦI . . .Demand 2 : Also, ΦI has to be RG-invariant i.e. µ2
Rd
dµ2R
ΦI = 0.
We make an ansatz that if ΦI satisfies KG-type DE, then we can accomplish this
q2 d
dq2ΦI(as, q
2, µ
2, z, ε
)=
1
2
[KI
(as,
µ2R
µ2, z, ε
)+G
I
(as,
q2
µ2R
,µ2R
µ2, z, ε
)]
using Demand 2
µ2R
d
dµ2R
KI
= −µ2R
d
dµ2R
GI ≡ −XI
The solution
ΦI
=∞∑i=1
ais
(q2
µ2
)iε/2SiεΦI,(i)
(z, ε)
with
ΦI,(i)
(z, ε) = LI,(i)(AI → X
I, GI → G
I(z, ε)
).
So
2ΦI,(1)
(z, ε) =1
ε2
(−4X
I1
)+
2
εGI1 (z, ε))
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Fixing Form of ΦI . . .Demand 2 : Also, ΦI has to be RG-invariant i.e. µ2
Rd
dµ2R
ΦI = 0.
We make an ansatz that if ΦI satisfies KG-type DE, then we can accomplish this
q2 d
dq2ΦI(as, q
2, µ
2, z, ε
)=
1
2
[KI
(as,
µ2R
µ2, z, ε
)+G
I
(as,
q2
µ2R
,µ2R
µ2, z, ε
)]
using Demand 2
µ2R
d
dµ2R
KI
= −µ2R
d
dµ2R
GI ≡ −XI
The solution
ΦI
=∞∑i=1
ais
(q2
µ2
)iε/2SiεΦI,(i)
(z, ε)
with
ΦI,(i)
(z, ε) = LI,(i)(AI → X
I, GI → G
I(z, ε)
).
So
2ΦI,(1)
(z, ε) =1
ε2
(−4X
I1
)+
2
εGI1 (z, ε))
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Fixing Form of ΦI . . .where,
ΦI
=∞∑i=1
ais
(q2
µ2
)iε/2SiεΦI,(i)
(z, ε)
=∞∑i=1
ais
(µ
2F
)( q2µ2
)iε/2Zias
ΦI,(i)
(z, ε)
≡∞∑i=1
aisΦI,(i)R
(z, ε)
At O(as(µ2F ))⇒ ΦI,(1)(z, ε) = Φ
I,(1)R
(z, ε)
XI
=∞∑i=1
ais(µ
2F )X
Ii
GI
(z, ε)) =∞∑i=1
ais(µ
2F )G
Ii (z, ε))
Hence,
XI1 = −AI1δ(1− z)
GI1 (z, ε)) = −fI1 δ(1− z) + 2A
I1D0 +
∞∑k=1
εkgI,k1 (z)
Explicit computation is required to determine O(ε) terms gI,k1 (z).
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Fixing Form of ΦI . . .Alternative Method : We say our demands can be fulfilled if we assume the solution of KG Eq.
ΦI
=∞∑i=1
ais
(q2
µ2
)iε/2SiεΦI,(i)
(z, ε)
with
ΦI,(i)
(z, ε) ≡{iε
1
1− z
[(1− z)2
]iε/2 }φI,(i)
(ε)
=
{δ(1− z) +
∞∑j=0
(iε)j+1
j!Dj
}φI,(i)
(ε)
RGE invarinace of ΦI
µ2R
d
dµ2R
KI
= −µ2R
d
dµ2R
GI ≡ −Y I
The solution
φI,(i)
(ε) = LI,(i)(AI → Y
I, GI → GI (ε)
).
So
2ΦI,(1)
(z, ε) =
{1
ε2
(−4Y
I1
)+
2
εGI1 (ε))
}δ(1− z) +
{−
4Y I1
ε2+
2
εGI1(ε)
} ∞∑j=0
εj+1
j!Dj
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Fixing Form of ΦI . . .
where
YI
=∞∑i=1
ais(µ
2F )X
Ii
GI (ε) =∞∑i=1
ais(µ
2F )GIi (ε)
Hence,
YI1 = −AI1
GI1 (ε) = −fI1 +∞∑k=1
εkGI,k1
Explicit computation is required to determine the O(ε) terms GI,k1 .
• The methodology holds at every order.
• Both methods are equivalent. We have followed the 2nd one to perform our computations.
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Extra Slide
Extra 5 : Higgs N3LO
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 88
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Band Plot for Higgs Boson
Figure: Higgs boson production in gg fusion
Anastasiou, Duhr, Dulat, Herzog, Mistlberger
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 89
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N3LL Threshold Resum for Higgs Boson
Figure: Higgs boson production in gg fusion
Duhr, Talk at RadCor ’15
Threshold Corrections To DY and Higgs at N3LO QCD Bergische Universitat Wuppertal 90