[PPT] - Perturbation theory of computing QCD jet cross sections beyond NLO PowerPoint Presentation

SLIDE 1

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Perturbation theory of computing QCD jet cross sections beyond NLO accuracy

Zoltán Trócsányi

University of Debrecen and Institute of Nuclear Research

in collaboration with

G. Somogyi, V. Del Duca, Z. Nagy

October 4, 2007

SLIDE 2

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Outline

1

Introduction Standard motivation Less standard motivation

2

pQCD computation of jet cross sections Perturbative expansion NLO correction

3

Extension to NNLO Structure of NNLO subtraction Naive generalization of NLO subtraction fails

4

NLO subtraction revisited Separation of collinear and purely-soft subtractions NLO subtraction with new phase-space mappings NLO subtraction with fixed helicities

5

Summary

6

Extra slides

SLIDE 3

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Standard motivation Precision QCD sometimes requires computations beyond NLO to reduce the dependence on unphysical renormalization and factorization scales if the NLO corrections are large (can be more than 100 %), such as Higgs production in hadron collisions the main source of uncertainty in experimental results is due to theory, such as αs measurements the NLO computation is effectively LO, such as energy distribution inside jet cones reliable error estimate is needed, such as, precise measurement of parton luminosity, but rather always . . .

SLIDE 4

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Less standard motivation Deeper understanding of real-radiation requires thinking beyond NLO: fast development in computations of loop amplitudes raises the hope of accessing NLO corrections for multileg processes

SLIDE 5

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Less standard motivation Deeper understanding of real-radiation requires thinking beyond NLO: fast development in computations of loop amplitudes raises the hope of accessing NLO corrections for multileg processes . . . can we compute real radiation fast enough?

SLIDE 6

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Less standard motivation Deeper understanding of real-radiation requires thinking beyond NLO: fast development in computations of loop amplitudes raises the hope of accessing NLO corrections for multileg processes . . . can we compute real radiation fast enough? current approaches to fixed-order and parton shower computations have mutually exclusive elements, which may hamper their combination

SLIDE 7

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Less standard motivation Deeper understanding of real-radiation requires thinking beyond NLO: fast development in computations of loop amplitudes raises the hope of accessing NLO corrections for multileg processes . . . can we compute real radiation fast enough? current approaches to fixed-order and parton shower computations have mutually exclusive elements, which may hamper their combination . . . understanding NNLO helps further development of parton showers

SLIDE 8

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The perturbative expansion at NNLO accuracy σ = σLO + σNLO + σNNLO + . . .

SLIDE 9

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The perturbative expansion at NNLO accuracy σ = σLO + σNLO + σNNLO + . . . Consider e+e− → m jet production

LO

b b b

m m

σLO = R

m dσB m =

R dφm|M(0)

m |2Jm

SLIDE 10

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The perturbative expansion at NNLO accuracy σ = σLO + σNLO + σNNLO + . . . Consider e+e− → m jet production

LO

b b b

m m

σLO = R

m dσB m =

R dφm|M(0)

m |2Jm

NLO

b b b b b b

r m+1 m+1

b b b

m m

σNLO = R

m+1 dσR m+1 +

R

m dσV m

= R dφm+1|M(0)

m+1|2Jm+1 +

R dφm2ReM(1)

m |M(0) m Jm

SLIDE 11

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The perturbative expansion at NNLO accuracy σ = σLO + σNLO + σNNLO + . . . Consider e+e− → m jet production

LO

b b b

m m

σLO = R

m dσB m =

R dφm|M(0)

m |2Jm

NLO

b b b b b b

r m+1 m+1

b b b

m m

σNLO = R

m+1 dσR m+1 +

R

m dσV m

= R dφm+1|M(0)

m+1|2Jm+1 +

R dφm2ReM(1)

m |M(0) m Jm

NNLO

b b b b b b

m+2 m+2 r s

b b b b b b

r m+1 m+1 m+1

b b b

m m

b b b

m m

σNNLO = R

m+2 dσRR m+2 +

R

m+1 dσRV m+1

+ R

m dσVV m

=

= R dφm+2|M(0)

m+2|2Jm+2 +

R dφm+12ReM(1)

m+1|M(0) m+1Jm+1 +

+ R dφm

h

|M(1)

m

|2 + 2ReM(2)

m

|M(0)

m

i

Jm

SLIDE 12

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The perturbative expansion at NNLO accuracy σ = σLO + σNLO + σNNLO + . . . Consider e+e− → m jet production

LO

b b b

m m

σLO = R

m dσB m =

R dφm|M(0)

m |2Jm

NLO

b b b b b b

r m+1 m+1

b b b

m m

σNLO = R

m+1 dσR m+1 +

R

m dσV m

= R dφm+1|M(0)

m+1|2Jm+1 +

R dφm2ReM(1)

m |M(0) m Jm

NNLO

b b b b b b

m+2 m+2 r s

b b b b b b

r m+1 m+1 m+1

b b b

m m

b b b

m m

σNNLO = R dσRR

m+2 +

R dσRV

m+1

+ R dσVV

m

=

= R dφm+2|M(0)

m+2|2Jm+2 +

R dφm+12ReM(1)

m+1|M(0) m+1Jm+1 +

+ R dφm

h

|M(1)

m

|2 + 2ReM(2)

m

|M(0)

m

i

Jm

SLIDE 13

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

b b b b b b

r m+1 m+1 m+1

b b b

m m m

1

dφ(k)

ddk
divergent in d = 4!

Process independent methods (phase space slicing, residuum, dipole or antennae subtraction) use regularized integrals in d = 4 − 2ǫ dimensions universal soft- and collinear factorization of QCD (squared) matrix elements

Cir is a symbolic operator that takes the collinear limit Cir|M(0)

m+1(pi, pr, . . .)|2 ∝ 1

sir M(0)

m (pir, . . .)|ˆ

P(0)

ir |M(0) m (pir, . . .)

Sr is a symbolic operator that takes the soft limit Sr|M(0)

m+1(pr, . . .)|2 ∝

X

i,k i=k

sik sirskr M(0)

m (. . .)|TiTk|M(0) m (. . .)

SLIDE 14

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

b b b b b b

r m+1 m+1 m+1

b b b

m m m

1

dφ(k)

ddk
divergent in d = 4!

Process independent methods (phase space slicing, residuum, dipole or antennae subtraction) use regularized integrals in d = 4 − 2ǫ dimensions universal soft- and collinear factorization of QCD (squared) matrix elements to construct approximate cross section to regularize real emissions: σNLO =

m+1
dσR

ε=0 −

dσA

ε=0

+
m
dσV+
1

dσA

ε=0

≡

m+1

dσNLO

m+1 +

m

dσNLO

m

SLIDE 15

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Construction of the subtraction terms at NLO The collinear and soft regions overlap:

Cir Sr

SLIDE 16

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Construction of the subtraction terms at NLO The collinear and soft regions overlap:

Cir Sr

The candidate subtraction term. . . A1|M(0)

m+1|2 ?

=

r

i=r

1 2Cir

|M(0)

m+1|2

SLIDE 17

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Construction of the subtraction terms at NLO The collinear and soft regions overlap:

Cir Sr −CirSr

r

−SrCir?

The candidate subtraction term. . . A1|M(0)

m+1|2 ?

=

r

i=r

1 2Cir + Sr

|M(0)

m+1|2

. . . has the correct singularity structure but performs double subtraction in the regions of phase space where the limits overlap

SLIDE 18

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Construction of the subtraction terms at NLO The collinear and soft regions overlap:

Cir Sr −CirSr

The candidate subtraction term. . . A1|M(0)

m+1|2 =

r

i=r

1 2Cir+

Sr−
i=r

CirSr

|M(0)

m+1|2

. . . is now free of double subtractions . . . but only defined in the strict collinear and/or soft limits

SLIDE 19

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Construction of the subtraction terms at NLO Extension over the full phase space requires momentum mappings, {p}m+1 → {˜ p}m leading to phase space factorization, such that

1

dσA = dσB

m ⊗ I(ǫ)

SLIDE 20

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Construction of the subtraction terms at NLO Extension over the full phase space requires momentum mappings, {p}m+1 → {˜ p}m leading to phase space factorization, such that

1

dσA = dσB

m ⊗ I(ǫ)

The pole part of the insertion operator is universal I(ǫ) = αs 2π

i
1

ǫ γi − 1 ǫ2

k=i

Ti · Tk 4πµ2 sik ǫ + O(ε0)

SLIDE 21

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Construction of the subtraction terms at NLO Extension over the full phase space requires momentum mappings, {p}m+1 → {˜ p}m leading to phase space factorization, such that

1

dσA = dσB

m ⊗ I(ǫ)

The pole part of the insertion operator is universal I(ǫ) = αs 2π

i
1

ǫ γi − 1 ǫ2

k=i

Ti · Tk 4πµ2 sik ǫ + O(ε0) and equals that of the virtual correction (up to a sign) dσNLO

m

=

dσV

m +

1

dσA

ε=0

= O(ε0)

SLIDE 22

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Three terms contribute at NNLO accuracy σNNLO = σRR

m+2 + σRV m+1 + σVV m

= =

m+2 dσRR

m+2Jm+2 +

m+1 dσRV

m+1Jm+1 +

m dσVV

m Jm

b b b b b b

m+2 m+2 r s

b b b b b b

r m+1 m+1 m+1

b b b

m m

b b b

m m

2

dφ(k1, k2)

ddk1
1

dφ(k2)

ddk1ddk2
separately divergent in d = 4!

Attempt to apply the same strategy as at NLO: regularize in d = 4 − 2ǫ dimensions and use universal IR structure to subtract divergences

SLIDE 23

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Thanks to many people, the universal IR structure at NNLO is well-known: Tree level 3-parton splitting functions and double soft gg and q¯ q currents

J. M. Campbell, E. W. N. Glover 1997, S. Catani, M. Grazzini 1998
V. Del Duca, A. Frizzo, F. Maltoni, 1999, D. Kosower, 2002

One-loop 2-parton splitting fucntions and soft gluon current

Z. Bern, L. J. Dixon, D. C. Dunbar, D. A. Kosower 1994
Z. Bern, V. Del Duca, W. B. Kilgore, C. R. Schmidt 1998-9
D. A. Kosower, P

. Uwer 1999, S. Catani, M. Grazzini 2000

D. A. Kosower 2003

SLIDE 24

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Thanks to many people, the universal IR structure at NNLO is well-known: Tree level 3-parton splitting functions and double soft gg and q¯ q currents

J. M. Campbell, E. W. N. Glover 1997, S. Catani, M. Grazzini 1998
V. Del Duca, A. Frizzo, F. Maltoni, 1999, D. Kosower, 2002

One-loop 2-parton splitting fucntions and soft gluon current

Z. Bern, L. J. Dixon, D. C. Dunbar, D. A. Kosower 1994
Z. Bern, V. Del Duca, W. B. Kilgore, C. R. Schmidt 1998-9
D. A. Kosower, P

. Uwer 1999, S. Catani, M. Grazzini 2000

D. A. Kosower 2003

A r e t h e s e u s e f u l i n N N L O c

m

p u t a t i

n

s ?

SLIDE 25

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Successful computations use different strategy Antennae subtraction uses complete squared matrix elements instead of IR structure

See talk by T. Gehrmann

For processes involving massive particles and/or simple kinematics, direct numerical evaluation of the coefficients in the Laurent expansion of the three contributions (based on sector decomposition) has been more successful

C. Anastasiou, K. Melnikov, F. Petriello 2004–2007

SLIDE 26

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Successful computations use different strategy Antennae subtraction uses complete squared matrix elements instead of IR structure

See talk by T. Gehrmann

For processes involving massive particles and/or simple kinematics, direct numerical evaluation of the coefficients in the Laurent expansion of the three contributions (based on sector decomposition) has been more successful

C. Anastasiou, K. Melnikov, F. Petriello 2004–2007

Existing NLO subtraction schemes cannot naively be extended to NNLO

G. Somogyi, V. Del Duca, Z.T. 2006–2007

SLIDE 27

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Structure of subtraction is governed by the jet function

σNNLO = σNNLO

m+2 + σNNLO m+1 + σNNLO m

= =

m+2
dσRR

m+2Jm+2 − dσRR,A2 m+2 Jm −

dσRR,A1

m+2 Jm+1 − dσRR,A12 m+2

Jm

+
m+1
dσRV

m+1 +

1 dσRR,A1

m+2

Jm+1 −
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm
+
m
dσVV

m +

2
dσRR,A2

m+2 −dσRR,A12 m+2

+
1
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm

SLIDE 28

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Structure of subtraction is governed by the jet function

σNNLO = σNNLO

m+2 + σNNLO m+1 + σNNLO m

= =

m+2
dσRR

m+2Jm+2 − dσRR,A2 m+2 Jm −

dσRR,A1

m+2 Jm+1 − dσRR,A12 m+2

Jm

+
m+1
dσRV

m+1 +

1 dσRR,A1

m+2

Jm+1 −
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm
+
m
dσVV

m +

2
dσRR,A2

m+2 −dσRR,A12 m+2

+
1
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm

The approximate cross section dσRR,A2

m+2

regularizes the doubly-unresolved limits of dσRR

m+2

SLIDE 29

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Structure of subtraction is governed by the jet function

σNNLO = σNNLO

m+2 + σNNLO m+1 + σNNLO m

= =

m+2
dσRR

m+2Jm+2 − dσRR,A2 m+2 Jm −

dσRR,A1

m+2 Jm+1 − dσRR,A12 m+2

Jm

+
m+1
dσRV

m+1 +

1 dσRR,A1

m+2

Jm+1 −
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm
+
m
dσVV

m +

2
dσRR,A2

m+2 −dσRR,A12 m+2

+
1
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm

The approximate cross section dσRR,A2

m+2

regularizes the doubly-unresolved limits of dσRR

m+2

The approximate cross section dσRR,A1

m+2

regularizes the singly-unresolved limits of dσRR

m+2

SLIDE 30

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Structure of subtraction is governed by the jet function

σNNLO = σNNLO

m+2 + σNNLO m+1 + σNNLO m

= =

m+2
dσRR

m+2Jm+2 − dσRR,A2 m+2 Jm −

dσRR,A1

m+2 Jm+1 − dσRR,A12 m+2

Jm

+
m+1
dσRV

m+1 +

1 dσRR,A1

m+2

Jm+1 −
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm
+
m
dσVV

m +

2
dσRR,A2

m+2 −dσRR,A12 m+2

+
1
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm

Structure of subtraction is governed by the jet function

σNNLO = σNNLO

m+2 + σNNLO m+1 + σNNLO m

= =

m+2
dσRR

m+2Jm+2 − dσRR,A2 m+2 Jm −

dσRR,A1

m+2 Jm+1 − dσRR,A12 m+2

Jm

+
m+1
dσRV

m+1 +

1 dσRR,A1

m+2

Jm+1 −
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm
+
m
dσVV

m +

2
dσRR,A2

m+2 −dσRR,A12 m+2

+
1
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm

and

1 dσRR,A1 m+2

A1

regularize the singly-unresolved limits of dσRV

m+1 and

1 dσRR,A1

m+2

SLIDE 32

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Structure of subtraction is governed by the jet function

σNNLO = σNNLO

m+2 + σNNLO m+1 + σNNLO m

= =

m+2
dσRR

m+2Jm+2 − dσRR,A2 m+2 Jm −

dσRR,A1

m+2 Jm+1 − dσRR,A12 m+2

Jm

+
m+1
dσRV

m+1 +

1 dσRR,A1

m+2

Jm+1 −
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm
+
m
dσVV

m +

2
dσRR,A2

m+2 −dσRR,A12 m+2

+
1
dσRV,A1

m+1 + 1 dσRR,A1 m+2

A1
Jm

and

1 dσRR,A1 m+2

A1

regularize the singly-unresolved limits of dσRV

m+1 and

1 dσRR,A1

m+2

SLIDE 33

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The m + 1-parton contribution:

m+1
dσRV

m+1 +

1

dσRR,A1

m+2

Jm+1 −
dσRV,A1

m+1

+

1

dσRR,A1

m+2

A1

Jm

Construction of approximate cross section dσRV,A1

m+1

that regularizes the kinematical singularities of dσRV

m+1 in the

singly-unresolved regions is straightforward

SLIDE 34

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The m + 1-parton contribution:

m+1
dσRV

m+1 +

1

dσRR,A1

m+2

Jm+1 −
dσRV,A1

m+1

+

1

dσRR,A1

m+2

A1

Jm

Construction of approximate cross section dσRV,A1

m+1

that regularizes the kinematical singularities of dσRV

m+1 in the

singly-unresolved regions is straightforward . . . but

dσRV

m+1 − dσRV,A1 m+1

ǫ=0 = infinite

SLIDE 35

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The m + 1-parton contribution:

m+1
dσRV

m+1 +

1

dσRR,A1

m+2

Jm+1 −
dσRV,A1

m+1

+

1

dσRR,A1

m+2

A1

Jm

Construction of approximate cross section dσRV,A1

m+1

that regularizes the kinematical singularities of dσRV

m+1 in the

singly-unresolved regions is straightforward . . . but need to subtract the universal pole part too

m+1
dσRV

m+1 − dσRV,A1 m+1 +

dσB

m+1 ⊗ I(ǫ) − ?

ǫ=0 = finite

and recall I(ǫ) ∝ αs 2π

i
1

ǫ γi − 1 ǫ2

k=i

Ti · Tk 4πµ2 sik ǫ + O(ε0)

SLIDE 36

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The standard integrated approximate cross sections do not obey universal IR collinear factorization Due to coherent soft-gluon emission from unresolved partons only the sum M(0)

m+1|(Tj·Tk + Tr·Tk)|M(0) m+1

factorizes in the collinear limit (Tjr = Tj + Tr) CjrM(0)

m+1|(Tj·Tk + Tr·Tk)|M(0) m+1 ∝ 1

sjr M(0)

m |Tjr·Tk ˆ

P(0)

jr |M(0) m

SLIDE 37

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The standard integrated approximate cross sections do not obey universal IR collinear factorization Due to coherent soft-gluon emission from unresolved partons only the sum M(0)

m+1|(Tj·Tk + Tr·Tk)|M(0) m+1

factorizes in the collinear limit (Tjr = Tj + Tr) CjrM(0)

m+1|(Tj·Tk + Tr·Tk)|M(0) m+1 ∝ 1

sjr M(0)

m |Tjr·Tk ˆ

P(0)

jr |M(0) m

This factorization is violated by the factors s−ǫ

ik /ǫ2

Cjr 1 ǫ2M(0)

m+1|(Tj·Tk s−ǫ jk + Tr·Tk s−ǫ rk )|M(0) m+1 ∝

× 1 sjr

M(0)

m |Tjr·Tk ˆ

P(0)

jr

1

ǫ2 − 1 ǫ ln s(jr)k

|M(0)

m

−1 ǫ M(0)

m |Tj·Tk ln zj + Tr·Tk ln zr |M(0) m

SLIDE 38

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The standard integrated approximate cross sections do not obey universal IR collinear factorization Due to coherent soft-gluon emission from unresolved partons only the sum M(0)

m+1|(Tj·Tk + Tr·Tk)|M(0) m+1

factorizes in the collinear limit (Tjr = Tj + Tr) CjrM(0)

m+1|(Tj·Tk + Tr·Tk)|M(0) m+1 ∝ 1

sjr M(0)

m |Tjr·Tk ˆ

P(0)

jr |M(0) m

This factorization is violated by the factors s−ǫ

ik /ǫ2

⇒ need to use either colour stripped amplitudes (as in antennae subtraction)

r properly defined new approximate cross

sections → remark about dσRR,A1

m+2

SLIDE 39

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Pure soft limit of the squared matrix element Using the soft insertion rules one obtains

Sr|M(0)

m+1(pr, . . . )|2 ∝ m

i=1

m

k=1
hel.

εµ(pr)ε∗

ν(pr)2pµ i pν k

sirskr M(0)

m (. . . )|Ti·Tk|M(0) m (. . . )

dµν(pr, n) =

hel.

εµ(pr)ε∗

ν(pr) = −gµν + pµ r nν + pν r nµ

pr·n .

SLIDE 40

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Pure soft limit of the squared matrix element Using the soft insertion rules one obtains

Sr|M(0)

m+1(pr, . . . )|2 ∝ m

i=1

m

k=1
hel.

εµ(pr)ε∗

ν(pr)2pµ i pν k

sirskr M(0)

m (. . . )|Ti·Tk|M(0) m (. . . )

dµν(pr, n) =

hel.

εµ(pr)ε∗

ν(pr) = −gµν + pµ r nν + pν r nµ

pr·n .

Soft-collinear contributions are given by the colour-diagonal terms,

Sr|M(0)

m+1(pr, . . . )|2 ∝ m

i=1
1

2

m

k=i
sik

sirsrk − 2sin srnsir − 2skn srnskr

M(0)

m (. . . )|Ti·Tk|M(0) m (. . . )

− T2

i

2 sir sin srn |M(0)

m (. . . )|2

sin = 2pi·n

SLIDE 41

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Pure soft limit of the squared matrix element Using the soft insertion rules one obtains

Sr|M(0)

m+1(pr, . . . )|2 ∝ m

i=1

m

k=1
hel.

εµ(pr)ε∗

ν(pr)2pµ i pν k

sirskr M(0)

m (. . . )|Ti·Tk|M(0) m (. . . )

dµν(pr, n) =

hel.

εµ(pr)ε∗

ν(pr) = −gµν + pµ r nν + pν r nµ

pr·n .

choose Coulomb gauge and keep the pure soft only

Sr|M(0)

m+1(pr, . . . )|2 −

→

m

i=1
1

2

m

k=i
sik

sirsrk − 2siQ srQsir − 2skQ srQskr

M(0)

m (. . . )|Ti·Tk|M(0) m (. . . )

with nµ = Qµ − pµ

r Q2/srQ and colour − conservation

SLIDE 42

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Construction of the subtraction terms at NLO Collinear and soft limits are automatically disjunct

Cir Sr

SLIDE 43

Cir Sr

Extension over the full phase space requires momentum mappings,{p}m+1 → {˜ p}m that implement exact momentum conservation lead to exact phase-space factorization can be generalized to any number of unresolved partons

SLIDE 49

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Construction of the subtraction terms at NLO Collinear and soft limits are automatically disjunct

Cir Sr

Extension over the full phase space requires momentum mappings,{p}m+1 → {˜ p}m that implement exact momentum conservation lead to exact phase-space factorization can be generalized to any number of unresolved partons We use separate phase space mappings for the collinear and soft limits

SLIDE 50

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Collinear mapping ˜ pµ

ir =

1 1 − αir (pµ

i +pµ r −αirQµ) ,

˜ pµ

n =

1 1 − αir pµ

n ,

n = i, r αir = 1 2

y(ir)Q −
y2

(ir)Q − 4yir

Q

1

b b b

i r

b b b

m + 1 m + 1 Cir Q ˜ 1

b b b

ir

b b b

m + 1

m ⊗ 2 (ir) i r

SLIDE 51

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Collinear mapping ˜ pµ

ir =

1 1 − αir (pµ

i +pµ r −αirQµ) ,

˜ pµ

n =

1 1 − αir pµ

n ,

n = i, r αir = 1 2

y(ir)Q −
y2

(ir)Q − 4yir

Q

1

b b b

i r

b b b

m + 1 m + 1 Cir Q ˜ 1

b b b

ir

b b b

m + 1

m ⊗ 2 (ir) i r

momentum is conserved ˜ pµ

ir + n ˜

pµ

n = pµ i + pµ r + n pµ n

phase-space factorization is exact dφm+1(p1, . . . ; Q) = dφm( p1, . . . ; Q) ⊗ dφ2(pi, pr; p(ir))

SLIDE 52

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Collinear mapping ˜ pµ

ir =

1 1 − αir (pµ

i +pµ r −αirQµ) ,

˜ pµ

n =

1 1 − αir pµ

n ,

n = i, r αir = 1 2

y(ir)Q −
y2

(ir)Q − 4yir

Q

1

b b b

i r

b b b

m + 1 m + 1 Cir Q ˜ 1

b b b

ir

b b b

m + 1

m ⊗ 2 (ir) i r

momentum is conserved ˜ pµ

ir + n ˜

pµ

n = pµ i + pµ r + n pµ n

phase-space factorization is exact dφm+1(p1, . . . ; Q) = dφm( p1, . . . ; Q) ⊗ dφ2(pi, pr; p(ir)) integral over convolution can be constrained to improve numerical efficiency

SLIDE 53

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Soft mapping ˜ pµ

n = Λµ ν[Q, (Q−pr)/λr](pν n/λr) ,

n = r , λr =

1 − yrQ ,

Λµ

ν[K,

K] = gµ

ν − 2(K +

K)µ(K + K)ν (K + K)2 + 2Kµ Kν K2

Q 1

b b b

r

b b b

m + 1 m + 1 Sr Q r ⊗ Q 2 ˜ 1

b b b

r

b b b b b b

m + 1

m

SLIDE 54

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Soft mapping ˜ pµ

n = Λµ ν[Q, (Q−pr)/λr](pν n/λr) ,

n = r , λr =

1 − yrQ ,

Λµ

ν[K,

K] = gµ

ν − 2(K +

K)µ(K + K)ν (K + K)2 + 2Kµ Kν K2

Q 1

b b b

r

b b b

m + 1 m + 1 Sr Q r ⊗ Q 2 ˜ 1

b b b

r

b b b b b b

m + 1

m

momentum is conserved

n ˜

pµ

n = pµ r + n pµ n

phase-space factorization is exact dφm+1(p1, . . . ; Q) = dφm( p1, . . . ; Q) ⊗ dφ2(pr, p; Q) integral over convolution can be constrained to improve numerical efficiency

SLIDE 55

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The approximate cross section at NLO The collinear and soft momentum mappings define extensions of the limit formulae over the full phase space Cir|M(0)

m+2|2

− → C(0,0)

ir

Sr|M(0)

m+2|2

− → S(0,0)

r

The true subtraction term is A1|M(0)

m+2|2 =

r

i=r

1 2C(0,0)

ir

+ S(0,0)

r

C(0,0)

ir

, S(0,0)

r

are functions of the original momenta that inherit the notation of the operators, but have nothing to do with taking limits the soft subtraction S(0,0)

r

is universal

SLIDE 56

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

The approximate cross section at NLO The collinear and soft momentum mappings define extensions of the limit formulae over the full phase space Cir|M(0)

m+2|2

− → C(0,0)

ir

Sr|M(0)

m+2|2

− → S(0,0)

r

The true subtraction term is A1|M(0)

m+2|2 =

r

i=r

1 2C(0,0)

ir

+ S(0,0)

r

C(0,0)

ir

, S(0,0)

r

are functions of the original momenta that inherit the notation of the operators, but have nothing to do with taking limits the soft subtraction S(0,0)

r

is universal

The approximate cross section is written formally as dσRR,A1

m+2

= dφm+1[dp1]A1|M(0)

m+2|2

SLIDE 57

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Similarly and completely systematically one can define the fully differencial (m + 2)-parton contriubtion

dσNNLO

m+2

= dσRR

m+2Jm+2 − dσRR,A2 m+2 Jm −

dσRR,A1

m+2 Jm+1 − dσRR,A12 m+2

Jm

dσ

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Similarly and completely systematically one can define the fully differencial (m + 2)-parton contriubtion

dσNNLO

m+2

= dσRR

m+2Jm+2 − dσRR,A2 m+2 Jm −

dσRR,A1

m+2 Jm+1 − dσRR,A12 m+2

Jm

dσ

dσNNLO

m+1

=

dσRV

m+2+

1

dσRR,A1

m+2

Jm+1−
dσRV,A1

m+1 + 1

dσRR,A1

m+2

A1

Jm

R

1 dσRR,A1 m+2

= dφm+1|M(0)

m+1|2 ⊗ I(m, ε)

dσRV,A1

m+1

= dφm[dp1]A12ReM(0)

m+1|M(1) m+1

“ R

1 dσRR,A1 m+2

”

A1 = dφm[dp1]A1

“ |M(0)

m+1|2 ⊗ I(m, ε)

”

both are integrable in d = 4 dimensions

SLIDE 59

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Monte Carlo summation over helicity in NLO computations proved to be useful to gain speed in multileg computations at Born level

P . Draggiotis, R. H. P . Kleiss, C. G. Papadopoulos 1998, 2002

Gaining speed is even more important in computing real radiation

SLIDE 60

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Monte Carlo summation over helicity in NLO computations proved to be useful to gain speed in multileg computations at Born level

P . Draggiotis, R. H. P . Kleiss, C. G. Papadopoulos 1998, 2002

Gaining speed is even more important in computing real radiation The pure-soft factorization is independent of the soft-gluon helicity

Sr|M(0)

m+1(pλ r , . . . )|2 ∝

1 2

m

i=1
1

2

m

k=i
sik

sirsrk − 2siQ srQsir − 2skQ srQskr

M(0)

m (. . . )|Ti·Tk|M(0) m (. . . )

−T2

i

2 sir sin srn |M(0)

m (. . . )|2

SLIDE 61

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Monte Carlo summation over helicity in NLO computations proved to be useful to gain speed in multileg computations at Born level

P . Draggiotis, R. H. P . Kleiss, C. G. Papadopoulos 1998, 2002

Gaining speed is even more important in computing real radiation The pure-soft factorization is independent of the soft-gluon helicity Can define collinear and pure-soft subtractions for fixed helicities

Sr|M(0)

m+1(pλ r , . . . )|2 −

→ 1 2

m

i=1
1

2

m

k=i
sik

sirsrk − 2siQ srQsir − 2skQ srQskr

M(0)

m (. . . )|Ti·Tk|M(0) m (. . . )

SLIDE 62

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Summary Existing subtraction methods cannot straightforwardly be generalized to NNLO

SLIDE 63

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Summary Existing subtraction methods cannot straightforwardly be generalized to NNLO Proposed new NLO scheme, that can be generalized to any order in PT, based on

SLIDE 64

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Summary Existing subtraction methods cannot straightforwardly be generalized to NNLO Proposed new NLO scheme, that can be generalized to any order in PT, based on

simple separation of collinear and soft singularities new phase-space mappings

SLIDE 65

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Summary Existing subtraction methods cannot straightforwardly be generalized to NNLO Proposed new NLO scheme, that can be generalized to any order in PT, based on

simple separation of collinear and soft singularities new phase-space mappings

The new scheme can be formulated such that MC summation over helicities in multileg NLO computations is possible

SLIDE 66

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Summary Existing subtraction methods cannot straightforwardly be generalized to NNLO Proposed new NLO scheme, that can be generalized to any order in PT, based on

simple separation of collinear and soft singularities new phase-space mappings

The new scheme can be formulated such that MC summation over helicities in multileg NLO computations is possible Complete scheme for NNLO computations is worked

ut (details not shown here)

SLIDE 67

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Summary Existing subtraction methods cannot straightforwardly be generalized to NNLO Proposed new NLO scheme, that can be generalized to any order in PT, based on

simple separation of collinear and soft singularities new phase-space mappings

The new scheme can be formulated such that MC summation over helicities in multileg NLO computations is possible Complete scheme for NNLO computations is worked

ut (details not shown here)

Only partially completed (nothing shown here): integrations over the unresolved phase spaces

SLIDE 68

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Summary Existing subtraction methods cannot straightforwardly be generalized to NNLO Proposed new NLO scheme, that can be generalized to any order in PT, based on

simple separation of collinear and soft singularities new phase-space mappings

The new scheme can be formulated such that MC summation over helicities in multileg NLO computations is possible Complete scheme for NNLO computations is worked

ut (details not shown here)

Only partially completed (nothing shown here): integrations over the unresolved phase spaces Thanks for your attention!

SLIDE 69

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Towards event shapes Constructed dσNNLO

5

(RR) and dσNNLO

4

(RV) for e+e− → 3 jets (e+e− → q¯ qggg and e+e− → q¯ qgg subprocesses respectively)

SLIDE 70

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Towards event shapes Constructed dσNNLO

5

(RR) and dσNNLO

4

(RV) for e+e− → 3 jets (e+e− → q¯ qggg and e+e− → q¯ qgg subprocesses respectively) Checked numerically that (J = C or 1 − T)

In all singly- and doubly-unresolved limits dσ

RR,A2 5

J3 + dσ

RR,A1 5

J4 − dσ

RR,A12 5

J3 dσRR

5

→ 1 In all singly-unresolved limits dσ

RV,A1 4

J3 − R

1 dσ RR,A1 5

J4 − “ R

1 dσ RR,A1 5

”

A1J3

dσRV

4

→ 1

⇒ The counterterms are fully local, azimuthal and color correlations fully included

SLIDE 71

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Towards event shapes Computed the RR and RV contributions to first three moments of 3-jet event shape variables thurst (T) and C-parameter

On ≡

On dσ

σ0 = αs(Q) 2π

A(n)

O +

αs(Q) 2π 2 B(n)

O +

αs(Q) 2π 3 C(n)

O

where the NNLO contribution C(n)

O is a sum of the RR,

RV and VV pieces C(n)

O = C(n) O;5 + C(n) O;4 + C(n) O;3

The quantities C(n)

O;5 and C(n) O;4 are found to be finite

(O = C or O = τ ≡ 1 − T; n = 1, 2, 3) Up to NLO accuracy perfect agreement with known results for e+e− → 3 jets

SLIDE 72

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Towards event shapes – RR contribution Prediction for moments of event shapes – RR contribution n C(n)

τ;5

C(n)

C;5

1 −(9.27 ± 0.34) · 101 −(3.44 ± 0.14) · 102 2 −3.07 ± 0.43 −(1.42 ± 0.03) · 102 3 2.01 ± 0.12 6.29 ± 1.87 Technical details

No. of MC points used: n = 40 × 2.5 · 105 (VEGAS)

χ2/d.o.f. as reported by VEGAS: χ2/d.o.f. = 0.79

No. of subtractions: 535 at 139 different PS points for each event

[compare with 12 subtractions at 12 different PS points for e+e− → 4 jets at NLO needed in this scheme (q¯ qggg subprocess)] Speed of code on an AMD Athlon 1.3 GHz machine with 256 MB RAM: 2.5 · 105 pts. ≈ 2.5 h

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Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Towards event shapes – RV contribution Prediction for moments of event shapes – RV contribution n C(n)

τ;4

C(n)

C;4

1 (1.23 ± 0.01) · 103 (4.33 ± 0.05) · 103 2 (2.55 ± 0.02) · 102 (3.25 ± 0.02) · 103 3 (4.79 ± 0.03) · 101 (1.80 ± 0.01) · 103 Technical details

No. of MC points used: n = 20 × 2.5 · 105 (VEGAS)

χ2/d.o.f. as reported by VEGAS: χ2/d.o.f. = 1.24

No. of subtractions: 15 at 7 different PS points for each event

Speed of code on an AMD Athlon 1.3 GHz machine with 256 MB RAM: 2.5 · 105 pts. ≈ 7 h

SLIDE 74

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Collinear limit of color-connected SME Due to color coherence only the sum |M(0)

m+1;(i,l)|2 + |M(0) m+1;(r,l)|2

has a universal collinear limit as pi||pr, not |M(0)

m+1;(i,l)|2

r |M(0)

m+1;(r,l)|2 separately

Generally we have I = Vil|M(0)

m+1;(i,l)({˜

p}(il)

m+1)|2+Vrl|M(0) m+1;(r,l)({˜

p}(rl)

m+1)|2+. . .

CirI exists iff

CirVil = CirVrl ≡ V(ir)l {˜ p}(il)

m+1 i||r

− → {˜ p}[(ir)l]

m

, {˜ p}(rl)

m+1 i||r

− → {˜ p}[(ir)l]

m

Then CirI ∝

1 sir V(ir)lM(0) m ({˜

p}[(ir)l]

m

)|TirTlˆ Pir|M(0)

m ({˜

p}[(ir)l]

m

)

SLIDE 75

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Rate of convergence RR part

5 6 7 8 9

No. of points [10

6] 3.5 4.0 4.5 5.0 5.5 6.0

Rel. error [%]

C ;5

(1) 5 6 7 8 9

No. of points [10

6] 3.5 4.0 4.5 5.0 5.5 6.0

Rel. error [%]

CC;5

(1) 5 6 7 8 9

No. of points [10

6] 12 14 16 18 20 22

Rel. error [%]

C ;5

(2) 5 6 7 8 9

No. of points [10

6] 2.2 2.4 2.6 2.8 3.0 3.2

Rel. error [%]

CC;5

(2) 5 6 7 8 9

No. of points [10

6] 5 6 7 8 9 10 11

Rel. error [%]

C ;5

(3) 5 6 7 8 9

No. of points [10

6] 25 30 35 40 45 50 55 60 65

Rel. error [%]

CC;5

(3)

RV part

2 2.5 3 3.5 4

No. of points [10

6] 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Rel. error [%]

C ;4

(1) 2 2.5 3 3.5 4

No. of points [10

6] 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Rel. error [%]

CC;4

(1) 2 2.5 3 3.5 4

No. of points [10

6] 0.4 0.6 0.8 1.0 1.2 1.4

Rel. error [%]

C ;4

(2) 2 2.5 3 3.5 4

No. of points [10

6] 0.4 0.6 0.8 1.0 1.2

Rel. error [%]

CC;4

(2) 2 2.5 3 3.5 4

No. of points [106]

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

Rel. error [%]

C ;4

(3) 2 2.5 3 3.5 4

No. of points [106]

0.5 0.6 0.7 0.8 0.9 1.0 1.1

Rel. error [%]

CC;4

(3)

SLIDE 76

Introduction pQCD computation of jet cross sections Extension to NNLO NLO subtraction revisited Summary Extra slides

Event shape distributions (still unphysical at NNLO) C-parameter

.2
.1

.1

1/

0 C d /dC .2 .4 .6 .8 1 RR piece RV piece .2 .4 .6 .8 1

C

.1 .2 .3 .4 .5

1/

0 C d /dC

C-parameter distribution

LO result NLO result NLO+RR+RV

Thrust

.2
.1

.1

1/

0 (1-T) d /dT .5 .6 .7 .8 .9 1 RR piece RV piece .5 .6 .7 .8 .9 1

1 - T

.1 .2 .3 .4

1/

0 (1-T) d /dT

Thrust distribution

LO result NLO result NLO+RR+RV