techniques for higher order qcd...

Techniques forHigher Order QCD Calculations

Thomas Gehrmann

Universitat Zurich

TURIC

EN

SIS

UN

IVE

RSI

TAS

XXXIIIMDCCC

HEPTOOLS School Torino 2008

Techniques for Higher Order QCD Calculations – p.1

Topics

Precision Physics at Colliders

Structure of Higher Order QCD Contributions

Computing Virtual Corrections

Computing Real Corrections

Numerical Implementation

Results


QCD

QCD: SU(3) Yang–Mills theory of quark andgluon interactions

LQCD = −1

4F µνFµν +

∑

q

ψq(iD/−mq)ψq

QCD is experimentally well established

QCD is becoming precision physics

LEP precision physics:Electroweak processes

Tevatron/LHC precision physics:QCD processes


QCD

Precision physics with QCD

precise determination ofstrong coupling constantquark masseselectroweak parametersparton distributionsLHC collider luminosity

precise predictions fornew physics effectsand their backgrounds


Precision Physics

0

1

2

3

4

5

6

10030 300

mH [GeV]

∆χ2

Excluded Preliminary

∆αhad =∆α(5)

0.02758±0.00035

0.02749±0.00012

incl. low Q2 data

Theory uncertainty

mLimit = 144 GeV

LEP EW Working Group

160 165 170 175 180 185mt [GeV]

0.2305

0.2310

0.2315

0.2320

0.2325

0.2330

0.2335

sin2 θ ef

f

SMMSSM

MH = 400 GeV

MH = 114 GeV

heavy scalars

light scalars

m t2~ ,b2

~ / m t1~ ,b1

~ > 2.5SM

MSSMboth models

Heinemeyer, Hollik,

Weber, Weiglein ’07experimental errors 68% CL:

LEP2/Tevatron (today)

Tevatron/LHC

ILC/GigaZ

80.2 80.3 80.4 80.5 80.6MW [GeV]

0.2295

0.2300

0.2305

0.2310

0.2315

0.2320

0.2325

0.2330

sin2 θ ef

f

SM (MH = 114...400 GeV)

mt = 165 ... 175 GeV

MSSM mt

2~

,b2

~ / mt

1~

,b1

~ > 2.5

SMMSSM

both models

Heinemeyer, Hollik,

Weber, Weiglein ’07

experimental errors 68% CL:

LEP2/Tevatron (today)

Tevatron/LHC

ILC/GigaZ

S. Heinemeyer, W. Hollik, A. Weber, G. Weiglein

Indirect determination of new particle masses

Testing self-consistency of Standard Model and its extensions


Precision Physics

Computing new physics signals and backgrounds

typcially multi-particle final states

n-particle QCD process is proportional to αns

uncertainty on normalisationfrom error on αs: nδαs

Higher order corrections

can influence the shape of kinematical distributions

and therefore modify the effects of kinematical cuts

may seriously affect reconstruction of intermediate particle masses


Jet Observables

Observing ”free” quarks and gluons at colliders

QCD describes quarks and gluons;experiments observe hadrons

describe parton −→ hadron transition (fragmentation)

define appropriate final states, independent of particle type in final state (jets)

Jetsexperimentally: hadrons with common momentum direction

theoretically: partons with common momentum direction


Jet Observables

e+e− → 3 jetsevent at LEP

e+

e-

q

q

Z0,γ g


Jet Observables

Formal requirements on jet observablesG. Sterman, S. WeinbergJet observable defined using n-particle final state: On(p1, . . . , pn)

collinear limit: On(p1, p2, . . . , pn)p1‖p2−→ On−1(p1 + p2, . . . , pn)

soft limit: On(p1, p2, . . . , pn)E1→0−→ On−1(p2, . . . , pn)

Jet observables which fulfil these criteria are infrared-safe

Jet algorithmsmeasurement and recombination procedure to combine nearby particle momenta intojets, e.g. JADE-algorithm

recombine pair (ij) with lowest sij = (pi + pj)2 < scut

other jet algorithms: Durham (kT ), Cambridge, Cone


Jet Observables

Event shape variablesassign a number x to a set of final state momenta: pi → x

e.g. Thrust in e+e−

T = max~n

Pni=1 |~pi · ~n|Pn

i=1 |~pi|

limiting values:

back-to-back (two-jet) limit: T = 1

spherical limit: T = 1/2

Ecm=91.2 GeV

Ecm=133 GeV

Ecm=161 GeV

Ecm=172 GeV

Ecm=183 GeV

Ecm=189 GeV

Ecm=200 GeV

Ecm=206 GeV

T

ALEPH

O( s2) + NLLA

1/ d

/dT

10-2

10-1

1

10

10 2

10 3

10 4

10 5

10 6

10 7

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

can be used as precision measurement of αs:

αs(MZ) = 0.1202 ± 0.0003(stat) ± 0.0009(sys) ± 0.0009(had)±0.0047(scale)


Jet Observables

Jets at hadron collidersSingle jet inclusive cross section at Tevatron

test of QCD over wide energy range

D0 Collaboration

precision determination of αs

CDF Collaboration

0.10

0.12

0.14

0.16

αs(MZ) as function of ET for µ=ET

Uncertainties due to the µ scale

(a)

0.10

0.12

0.14

0.16

0 50 100 150 200 250 300 350 400 450

αs(MZ) as function of ET for CTEQ4M

Uncertainties due to the PDF choice

(b)

Transverse Energy (GeV)

Str

ong

Cou

plin

g C

onst

ant α

s(M

Z)


Jet Observables

Jet algorithmstwo types of jet algorithms for hadron colliders

Cone algorithms

attribute all hadronic energy inside a cone in (η − φ) to a given jet

define procedures for finding cone axis and for splitting/merging nearby jets

+ intuitive

+ appropriate for reconstructing heavy particles, e.g. top quark

− theoretical description of splitting/merging not always possible

− sometimes large hadronization corrections

kT algorithms

recombine particles into jets using an iterative procedure on particle pairs

+ theoretically unproblematic

+ certain aspects of jet cross sections can be computed analytically

+ small hadronization corrections

− application time-consuming

− not appropriate for reconstruction of resonances


Jet Observables

Jet algorithms

kT -algorithm

CDF Collaboration

[GeV/c]JETTp

0 100 200 300 400 500 600 700

[n

b/(

GeV

/c)]

JET

T d

pJE

T /

dy

σ2d

-810

-610

-410

-210

1

210|<0.7

JET D=0.7 0.1<|yTK

Data

Systematic errors

NLO: JETRAD CTEQ6.1Mcorrected to hadron level

0µ / 2 = JETT = max pFµ = Rµ

-1 L = 385 pb∫

Cone algorithm

CDF Collaboration

(GeV/c)Tp0 100 200 300 400 500 600 700

[n

b/(

GeV

/c)]

T

/dyd

pσ2

d

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

1

10/2)Jet

T=pµNLO pQCD EKS CTEQ 6.1M, (

=1.3 Sep

=0.75, Rmerge=0.7, fconeMidpoint R0.1<|y|<0.7

-1 L = 385 pb∫

CDF Run II

Total systematic uncertainty

Data corrected to parton level

NLO pQCD


Jet Observables

Hadron collider cross sectionsCross section for producing final state X (e.g. vector boson, Higgs boson, multi-jet, . . .)

dσpp→X (spp, µ2f ) =

X

f1,2=q,q,g

Zdx1 dx2

hf1(x1, µ2

f ) f2(x2, µ2f )i

×dσf1f2→X (x1, x2, spp, µ2f )

e+

e-

p

p q

Z0,γqp

gp

f1(x1, µ2f ): parton distribution in hadron 1

f2(x1, µ2f ): parton distribution in hadron 2

dσf1f2→X : parton level cross section to produce final state X

kinematics of final state particles: (pT , φ)ppi = (pT , φ)f1f2

i , ηppi = ηf1f2

i + 12

log x1x2

partonic centre-of-mass energy: sf1f2= x1x2spp


Jet Observables

e+e−

strong coupling constant from e+e− → 3j

study of non-perturbative power corrections from e+e− → 3j

QCD gauge structure from e+e− → 4j

ep

strong coupling constant from ep → (2 + 1)j

αZEUSs (MZ) = 0.1190 ± 0.0017(stat)+0.0049

−0.0023(sys)±0.0026(th)

αH1s (MZ) = 0.1186 ± 0.0030(exp)+0.0039

−0.0045(scale) ± 0.0023(pdf)

gluon distribution in proton from ep → (2 + 1)j

pp

strong coupling constant from pp → 1j + X

αCDFs (MZ) = 0.1178 ± 0.0001(stat)+0.0081

−0.0095(sys) +0.0071−0.0047(scale) ± 0.0059(pdf)

gluon distribution in proton from pp → 2j

multijet-signatures often background to new physics searches


Jets in Perturbation Theory

TheoreticallyPartons are combined into jets using the same jet algorithm as in experiment

LO

each

parton

forms 1 jet

on its own

NLO

2 partons in

1 jet, 1 parton

experimentally

unresolved

NNLO

3 partons in

1 jet, 2 partons

experimentally

unresolved



Reasons to compute higher orders

reduction of theoretical error

better matching of parton level jet algorithm with experimental hadron level jetalgorithm

better description of transverse momentum of final states at hadron colliders dueto double radiation in the initial state

modified power corrections as higher perturbative powers 1/ ln(Q2/Λ2) can mimicgenuine power corrections Q/Λ

allow full NNLO global fits to parton distributions −→ lower error on benchmarkprocesses at LHC and Tevatron



General structure:m jets, n–th order in perturbation theory

mpartons, n loop...

m + n − 1 partons, 1 loop

m + n partons, tree

-

-

-*

@@R

Jet algorithmto selectmjetfinal state

Jet cross sectionEvent shapes

Jet algorithm acts differently on different partonic final states

Divergencies from soft and collinear real and virtual contributions must beextracted before application of jet algorithm


End of Lecture 1


Perturbation Theory

Dimensional regularisationuse dimensionality of space-time to regularise infrared and ultraviolet divergences

work in d = 4 − 2ǫ dimensions, perform Laurent expansion around limit ǫ → 0

loop integrals: Zd4k → ddk

phase-space integrals: Zd3p

2E→

dd−1p

2E

in practise: generalise to d-dimensional polar coordinates:

dd−1p = |p|d−2d|p|dΩd−2 withZ

dΩd =2πd/2

Γ(d/2)

metric tensor: gµµ = d (gauge boson polarisations: d − 2)

Dirac matrices: tr1 = 4 (fermion helicities: 2)


Perturbation Theory

Example: e+e− → hadrons at NLOtree-level:

= T(2)

qq (q2) = 〈M(0)|M(0)〉qq = 4N(1 − ǫ)q2

virtual one-loop correction to e+e− → qq

=

„N −

1

N

«T

(2)qq (q2)

24−

1

ǫ2−

3

2ǫ− 4 +

7π2

12

35

real radiation correction: e+e− → qqg

= 8π2

ZdΦD〈M(0)|M(0)〉qqg

=

„N −

1

N

«T

(2)qq (q2)

24 1

ǫ2+

3

2ǫ+

19

4−

7π2

12

35


Perturbation Theory

Example: e+e− → hadrons at NLOtotal correction is finite:

Poles(1×0)qq + Poles

(0×0)qqg = 0

sum of contributions yields hadronic R-ratio

R = 1 +“αs

2π

”0@Finite

(1×0)qq + Finite

(0×0)qqg

4Nq2

1A = 1 +

“αs

2π

” „N2 − 1

2N

«3

2

Pole structure is universal (S. Catani)

Poles(1×0)qq = 2ℜ〈M(0)|I

(1)qq (ǫ)|M(0)〉

I(1)qq (ǫ) = −

eǫγ

2Γ(1 − ǫ)

24N2 − 1

2N

„2

ǫ2+

3

ǫ

«„−

µ2

q2

«ǫ35


Perturbation Theory

Example: e+e− → hadrons at NLOFinal state multiplicity:

always two jets for γ∗ → qq

two or three jets for γ∗ → qqg

Matrix element for γ∗ → qqg

|M|2 = 32π2(e2qααs)CF

„sqg

sqg+

sqg

sqg+

2sqqsqqg

sqgsqg

«

Singularities: gluon becomes unresolved

quark-gluon collinear: sqg → 0

antiquark-gluon collinear: sqg → 0

gluon soft: sqg,qg → 0

always correspond to two-jet configurations

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

@@

sqg

q ‖ g

sqg

q ‖ g

g → 0


Perturbation Theory

Example: e+e− → hadrons at NLOThree-jet cross section is finite at O(αs), take JADE algorithm with sij > q2 y

dσ

dy= σ0

αs

2πCF

»23y2 − 3y + 2

y(1 − y)ln

1 − 2y

y− 3

(1 − 3y)(1 + y)

y

–

is leading order in αs for this observable

diverges for y → 0

infrared cancellations take place only in two-jet region:

soft singularity: 1/ǫ2

collinear singularities: 1/ǫ

want to extract these singularities independent of jet algorithm


Perturbation Theory

Structure of e+e− → 3 jets at NLO

Subprocess partonicfinal state

partonsin jets

γ∗ → 3 partons, 1 loope.g.

3 partons (1) (1) (1)

γ∗ → 4 partons, treee.g. 4 partons

(3+1) partons(2) (1) (1)(1) (1) (1)

Partons in red are soft or collinear: theoretically unresolved.

explicit infrared singularities in three-parton, one-loop matrix elements(ǫ−2, ǫ−1 in dimensional regularisation)

infrared singularities in four-parton, tree-level matrix elements appear only afterintegration over four-parton phase space


Perturbation Theory

Structure of e+e− → 3 jets at NNLO

Subprocess partonicfinal state

partonsin jets

γ∗ → 3 partons, 2 loope.g.

3 partons (1) (1) (1)

γ∗ → 4 partons, 1 loope.g. 4 partons

(3+1) partons(2) (1) (1)(1) (1) (1)

γ∗ → 5 partons, treee.g.

5 partons

(4+1) partons(3+2) partons

(3) (1) (1)(2) (2) (1)(2) (1) (1)(1) (1) (1)

explicit infrared singularities in three-parton, two-loop matrix elements(ǫ−4, . . ., ǫ−1 in dimensional regularisation)


Perturbation Theory

Computing Collider Observablesmany observables require higher order corrections

precision observables: NNLO to match experimental precision

multi-parton production: NLO to obtain reliable prediction

Ingredients to higher order calculationsvirtual corrections

(multi-)loop corrections to multi-leg scattering amplitudes(focus here on multi-loop)

systematic understanding of their infrared pole structure

ultraviolet renormalisation

real corrections

observable-independent method to extract soft and collinear divergentcontributions from real radiation matrix elements (infrared subtraction)

efficient organisation of subtraction terms for multi-leg processes

integration of subtraction terms over relevant phase space

interplay of real and virtual correctionsTechniques for Higher Order QCD Calculations – p.27

Virtual Corrections

Generic structure of scalar two-loop integrals:

It,r,s(p1, . . . , pn) =

Zddk

(2π)d

ddl

(2π)d

1

Dm11 . . . Dmt

t

Sn11 . . . S

nqq

Di : massless scalar propagators

Si : scalar products involving

loop momenta

t : number of different propagators

r =P

i mi : dimension of denominator

s =P

i ni : dimension of numerator

Topology of Feynman graph defined by specifying the set of different propagators

D1, . . . , Dt

Typcially 1000’s of different integrals in a single calculation


Virtual Corrections: Reduction

Reduction of Two-Loop Four-Point FunctionsIdentities:

Integration-by-parts (IBP)K. Chetyrkin, F. Tkachov

Zddk

(2π)d

ddl

(2π)d

∂

∂aµ[bµf(k, l, pi)] = 0

with: aµ = kµ, lµ and bµ = kµ, lµ, pµi

Lorentz Invariance (LI)E. Remiddi, TG

Zddk

(2π)d

ddl

(2π)dδεµ

ν

X

i

pνi

∂

∂pµi

!f(k, l, pi) = 0

For each two-loop four-point integral, one has 10 IBP and 3 LI identities.



Reduction of Two-Loop Four-Point FunctionsIBP and LI identities for It,r,s relate:

It,r,s : the integral itself

It−1,r,s : simpler topology

It,r+1,s, It,r+1,s+1 : same topology, more complicated than It,r,s

It,r−1,s, It,r−1,s−1 : same topology, simpler than It,r,st = 7dierent It;r;s a umulated equationsunknownsHHHHHr s 0 1 2 3 47 1 2 3 4 58 7 14 21 28 359 28 56 84 112 14010 84 168 252 336 420HHHHHr s 0 1 2 313 39 78 1307 22 45 76 115104 312 624 10408 106 213 354 535468 1404 2808 46809 358 717 1196 1795

1# equations grows faster than # unknowns

−→ Reduction possible (using MAPLE and FORM) Techniques for Higher Order QCD Calculations – p.30


Reduction of Two-Loop Four-Point Functions

Using IBP and LI identities, any two-loop four-point integral can be expressed as linearcombination of a small number of two-loop four-point master integrals:

on-shell (e.g. gg → gg): 6 master integrals

off-shell (e.g. γ∗ → qqg): 14 master integrals

plus simpler two- and three-point master integrals.



--- p123p1 p2p3 =2(2d 9)(d 4)(d 6) 2s12 + s13 + s23s13s23 ---up123p1 p2p33(d 4)d 6 (s13 + s23)2s213s223 ---p123p1 p2p36(d 3)(3d 14)(d 4)(d 6) (s12 + s23)2s213s223 --- p123p2 p1p36(d 3)(3d 14)(d 4)(d 6) (s12 + s13)2s213s223 --- p123p1 p2p3+3(d 3)(3d 10)(3d 14)2(d 4)2(d 5)(d 6) (2d 10)s12 + (3d 14)s23s213s223 - --p13 p1p3+3(d 3)(3d 10)(3d 14)2(d 4)2(d 5)(d 6) (2d 10)s12 + (3d 14)s13s213s223 - --p23 p2p33(d 3)(3d 10)(d 4)2(d 6) (3d 14)s12 + (4d 18)s13 (d 4)s23s213s223 - --p123 p13p23(d 3)(3d 10)(d 4)2(d 6) (3d 14)s12 (d 4)s13 + (4d 18)s23s213s223 - --p123 p23p13(d 3)(3d 8)(3d 10)(d 4)3(d 5)(d 6) (d 5)(3d 14)(s12 + s13) + (d 4)2s23s313s223 -p133(d 3)(3d 8)(3d 10)(d 4)3(d 5)(d 6) (d 5)(3d 14)(s12 + s23) + (d 4)2s13s213s323 -p23Example of a reducible topology inγ∗ → qqg

express t = 7 integral by masterintegrals with at most t = 5

reduction is exact in d



Reduction of Two-Loop Four-Point Functions

Two-loop four-point master integrals in γ∗ → qqg

Planar topologies-

-

-

@@

@

-

-

-

Non-planar topologies-

AAA

AAA

-

AAA

-

AAA


End of Lecture 2


Virtual Corrections: Integrals

Techniques to compute master integralsSchwinger parameters

DF,n(kn) =i

k2n − m2

n + i0=

Z ∞

0dαnei(k2

n−m2n)αn

Feynman parameters

1Q

Aλll

=Γ(P

λl)QΓ(λl)

Z 1

0dξ1 . . .

Z 1

0dξL

Y

L

ξλl−1l

δ(P

ξl − 1)

(P

Alξl)P

λl

Mellin-Barnes integration (V. Smirnov, J.B. Tausk)

Nested sums (S. Moch, P. Uwer, S. Weinzierl)

Differential equations (E. Remiddi, TG)

Sector decomposition (K. Hepp; A. Denner, M. Roth; T. Binoth, G. Heinrich)



Mellin-Barnes integrationStarting point:

1

(X + Y )λ=

1

Γ(λ)

1

2πi

Z +i∞

−i∞dz Γ(λ + z) Γ(−z)

Y z

Xλ+z

allows to disentangle complicated denominators

applied typcially on Feynman parameter representation

Feynman parameter integral then becomes trivial

Z 1

0dxxα(1 − x)β =

Γ(1 + α)Γ(1 + β)

Γ(2 + α + β)

inversion is contour integral

contour must be chosen such that all Γ(. . . + z) are left, all Γ(. . . − z) are right

computer-algebra tools to derive Mellin-Barnes representation availableAMBRE: J. Gluza, K. Kajda and T. Riemann



Mellin-Barnes integrationInversion of Mellin-Barnes integral is usually well-defined only in a band around ǫ = ǫ0,which often does not include ǫ → 0

ε=0ε=ε0

Im(w)

Re(w) Re(w)

Im(w)

poles move to wrong side of contour in ǫ → 0 limit

perform step-by-step analytic continuation from ǫ0 on the Mellin-Barnesrepresentation

R(ǫ = ǫ0) = R(ǫ = ǫ1) − Res(R)

˛˛ǫ0→ǫ1

where the residues are enclosed by moving ǫ0 → ǫ1

automated computer algebra tools availableC. Anastasiou, A. DaleoMB: M. Czakon Techniques for Higher Order QCD Calculations – p.37


Mellin-Barnes integrationInversion of Mellin-Barnes integrals

Barnes first lemma

1

2πi

Z +i∞

−i∞dzΓ(λ1+z)Γ(λ2+z)Γ(λ3−z)Γ(λ4−z)=

Γ(λ1+λ3)Γ(λ1+λ4)Γ(λ2+λ3)Γ(λ2+λ3)

Γ(λ1+λ2+λ3+λ4)

plus corollaries and generalisations to six Γ-functions (second lemma)

Hypergeometric functions:

2F1(a, b; c; x) =Γ(c)

Γ(a)Γ(b)

1

2πi

Z +i∞

−i∞dz

Γ(a + z)Γ(b + z)Γ(−z)

Γ(c + z)(−x)z

Many integrals tabulated in the literatureBateman Manuscipts Vol. 1,4,5Brychkov, Prudnikov: Integrals and Series

Smirnov: Feynman Integral Calculus

often case-by-case solutions, no computer algebra package available



Differential equationsMulti-scale master integrals fulfil inhomogeneous differential equations in externalinvariants

express derivatives by external momenta

example: for massless four-point integrals

s12∂

∂s12=

1

2

„+pµ

1

∂

∂pµ1

+ pµ2

∂

∂pµ2

− pµ3

∂

∂pµ3

«

s13∂

∂s13=

1

2

„+pµ

1

∂

∂pµ1

− pµ2

∂

∂pµ2

+ pµ3

∂

∂pµ3

«

s23∂

∂s23=

1

2

„−pµ

1

∂

∂pµ1

+ pµ2

∂

∂pµ2

+ pµ3

∂

∂pµ3

«

apply IBP and LI reduction identities after momentum differentiation

yield: master integral plus simpler subtopologies



Differential equationsExample: one-loop box with one off-shell leg

s12∂

∂s12 -

--

p1 p3

p2q= −

d − 4

2 -

--

p1 p3

p2q

+2(d − 3)

s12 + s13

[

1

s123

-

p123−

1

s23

-

p23]

+2(d − 3)

s12 + s23

[

1

s123

-

p123−

1

s13

-

p13]

s13∂

∂s13 -

--

p1 p3

p2q=

d − 6

2 -

--

p1 p3

p2q

−2(d − 3)

s12 + s13

[

1

s123

-

p123−

1

s23

-

p23]

s23∂

∂s23 -

--

p1 p3

p2q=

d − 6

2 -

--

p1 p3

p2q

−2(d − 3)

s12 + s23

[

1

s123

-

p123−

1

s13

-

p13]



Differential equationsBoundary conditions from above equation in sij → 0, contain only:

-

p=

"(4π)

4−d2

16π2

Γ(3 − d/2)Γ2(d/2 − 1)

Γ(d − 3)

#−2i

(d − 4)(d − 3)

`−p2

´ d−42

Integration yields:

-

--

p1 p3

p2q= −

4(d − 3)

d − 4A2,LO

1

s13s23

24„

s13s23

s13 − s123

« d2−2

2F1

„d/2 − 2, d/2 − 2; d/2 − 1;

s123 − s13 − s23

s123 − s13

«

+

„s13s23

s23 − s123

« d2−2

2F1

„d/2 − 2, d/2 − 2; d/2 − 1;

s123 − s13 − s23

s123 − s23

«

−

„−s123s13s23

(s13 − s123)(s23 − s123)

« d2−2

2F1

„d/2 − 2, d/2 − 2; d/2 − 1;

s123(s123 − s13 − s23)

(s123 − s13)(s123 − s23)

«35



Differential equationsExample at two loops: off-shell vertex function

s123∂

∂s123

-

-

-

p123p12

p3

= +d − 4

2

2s123 − s12

s123 − s12

-

-

-

p123p12

p3

−3d − 8

2

1

s123 − s12

-

p12

s12∂

∂s12

-

-

-

p123p12

p3

= −d − 4

2

s12

s123 − s12

-

-

-

p123p12

p3

+3d − 8

2

1

s123 − s12

-

p12

is a hypergeometric differential equation

boundary conditions are again two-point functions

Laurent-series obtained by expansion of hypergeometric functions in theirparametersHypExp: T. Huber, D. Maitre; XSummer: S. Moch, P. Uwer Techniques for Higher Order QCD Calculations – p.42


Harmonic polylogarithms (HPL)generalisation of the Nielsen polylogarithms Lii(x) and Si,j(x)

E. Remiddi, J. Vermaseren

Definition of the HPL at weight w = 1:

H(1; x) ≡ − ln(1 − x) ,

H(0; x) ≡ ln x ,

H(−1; x) ≡ ln(1 + x)

and the rational fractions in x

f(1; x) ≡1

1 − x,

f(0; x) ≡1

x,

f(−1; x) ≡1

1 + x.



Harmonic polylogarithms (HPL)For w > 1:

H(0, . . . , 0; x) ≡1

w!lnw x ,

H(a,~b; x) ≡

Z x

0dxf(a; x)H(~b; x) ,

which results in∂

∂xH(a,~b; x) = f(a; x)H(~b; x) .

Properties:

HPL are linear independent

HPL fulfill a product algebra:

H(~a; x)H(~b; x) =P

H(~a ⊕~b; x)

HPL form a closed set under the class of integrations

Z x

0dx

„1

x,

1

1 − x,

1

1 + x

«H(~b; x)



Harmonic polylogarithms (HPL)Implementations

FORTRAN (E. Remiddi, TG)

Mathematica (D. Maitre)

Extensions

Two-dimensional Harmonic Polylogarithms (2dHPL) H(~m(z); y) by construction;closed set under

Z y

0dy

„1

y,

1

1 − y,

1

1 − y − z,

1

y + z

«H(~b(z); y)

2dHPL are basis functions for two-loop four-point functions with one off-shell leg

further extensions can be tailored to problem under considerationU. Agletti, R. Bonciani



Sector decompositionstarting point: Feynman (or other) parametrisation

decoposose integration region into sectors, which contain each only a singlesingularity on the edges of the sector

expand integrand in distributions:

x−1+ǫ =1

ǫδ(x) +

∞X

n=0

ǫn

n!

»lnn(x)

x

–

+

Z 1

0dx

»lnn(x)

x

–

+

f(x) =

Z 1

0dx lnn(x)

»f(x) − f(0)

x

–

integrate sector integrals numerically

was used at least as cross-check for all two-loop and three-loop multi-legcalculationsT. Binoth, G. Heinrich


Virtual Corrections: Amplitudes

Computer algebra implementationQGRAF

QCD modelin/out states

?

diagrams

?

FORM Feynman rules

?

Sum of all loop diagrams in terms ofloop integrals:

|M(2)〉 =

α1

∫

ddk

(2π)d

ddl

(2π)d

Sn11 . . . S

nqq

Dm11 . . . Dmt

t

+ α2

∫

. . .

?

Sum of all loop diagrams in terms ofmaster integrals:

|M(2)〉 =∑

i

βiMIi

?

|M(2)〉 =f4

ǫ4+

f3

ǫ3+

f2

ǫ2+

f1

ǫ+ f0

FORMIBP/LIidentities

?

linear system ofloop integrals

?

MAPLEsolve formaster integrals

?

∫

ddk

(2π)d

ddl

(2π)d

Sn11 . . . S

nqq

Dm11 . . .Dmt

t

=∑

i

aiMIi

Database of integrals

-

FORMderivatives ofmaster integrals

?

Derivatives of master integrals interms of loop integrals:

∂

∂sjk

MI =

γ1

∫

ddk

(2π)d

ddl

(2π)d

Sn11 . . . S

nqq

Dm11 . . .Dmt

t

+ γ2

∫

. . .

?

Derivatives of master integrals interms of master integrals:

∂

∂sjk

MI = γMI +∑

i

γiMIi

Ansatz

?

?

ǫ-expansion

MI = R(sjk)(g4

ǫ4+

g3

ǫ3+

g2

ǫ2+

g1

ǫ+ g0

)

gi ∋ Polylogarithms



Virtual two-loop matrix elements have been computed for:

Bhabha-Scattering: e+e− → e+e−

Z. Bern, L. Dixon, A. Ghinculov

Hadron-Hadron 2-Jet production: qq′ → qq′, qq → qq, qq → gg, gg → gg

C. Anastasiou, N. Glover, C. Oleari, M. Yeomans-TejedaZ. Bern, A. De Freitas, L. Dixon [SUSY-YM]

Photon pair production at LHC: gg → γγ, qq → γγ

Z. Bern, A. De Freitas, L. DixonC. Anastasiou, N. Glover, M. Yeomans-Tejeda

Three-jet production: e+e− → γ∗ → qqg

L. Garland, N. Glover, A.Koukoutsakis, E. Remiddi, TGS. Moch, P. Uwer, S. Weinzierl

DIS (2+1) jet production: γ∗g → qq, Hadronic (V+1) jet production: qg → V q

E. Remiddi, TG

Matrix elements with internal masses: γ∗ → QQ, qq → QQ, gg → QQ

W.Bernreuther, R.Bonciani, R.Heinesch, T.Leineweber, P.Mastrolia, E.Remiddi, TGM. Czakon, A. Mitov, S. Moch



Ultraviolet renormalisationFor massless QCD amplitudes: MS renormalisation amounts to replacement of barecoupling α0 by renormalised coupling αs(µ2)

α0µ2ǫ0 Sǫ = αsµ2ǫ

»1 −

β0

ǫ

“αs

2π

”+

„β20

ǫ2−

β1

2ǫ

«“αs

2π

”2+ O(α3

s)

–

with

Sǫ = (4π)ǫe−ǫγ

Massive amplitudes: matrix elements always (implicitly) calculated with on-shell masses

either convert OS to MS amplitudes

or perform mixed renormalisation: MS coupling, but OS mass counter terminsertions

mixed wave function and vertex renormalisation factors, determined bySlavnov-Taylor identities



Infrared pole structure at two loopsS. Catani

Poles(2×0)qq = 2ℜ

24−1

2〈M(0)|I(1)(ǫ)I(1)(ǫ)|M(0)〉 −

β0

ǫ〈M(0)|I(1)(ǫ)|M(0)〉

+ 〈M(0)|I(1)(ǫ)|M(1)〉

+e−ǫγ Γ(1 − 2ǫ)

Γ(1 − ǫ)

„β0

ǫ+ K

«〈M(0)|I(1)(2ǫ)|M(0)〉

+ 〈M(0)|H(2)(ǫ)|M(0)〉

35

Poles(1×1)qq = ℜ

242〈M(1)|I(1)(ǫ)|M(0)〉 − 〈M(0)|I(1)†(ǫ)I(1)(ǫ)|M(0)〉

35

with H(2)(ǫ) ∼ 1/ǫ



Infrared pole structure at two loopsInfrared singularity operators depend only on nature of external partons(colour-ordered form, massless)

I(1)qq (ǫ, sqq) = −

eǫγ

2Γ(1 − ǫ)

»1

ǫ2+

3

2ǫ

–ℜ(−sqq)−ǫ

I(1)qg (ǫ, sqg) = −

eǫγ

2Γ(1 − ǫ)

»1

ǫ2+

5

3ǫ−

NF

N

1

6ǫ

–ℜ(−sqg)−ǫ

I(1)gg (ǫ, sgg) = −

eǫγ

2Γ(1 − ǫ)

»1

ǫ2+

11

6ǫ−

NF

N

1

3ǫ

–ℜ(−sgg)−ǫ

Higher multiplicities:

I(1)qqg(ǫ, sij) = N

“I(1)qg (ǫ, sqg) + I

(1)qg (ǫ, sqg)

”−

1

NI(1)qq (ǫ, sqq)

H(2)(ǫ) = nqH

(2)q + ngH

(2)g

Quark masses act as collinear regulator:can infer dominant mass-effect from massless amplitudes and massive form factorsA. Mitov, S. Moch Techniques for Higher Order QCD Calculations – p.51

End of Lecture 3


Real Corrections: Outline

Single real radiation

dσ(n+1) = |Mn+1|2dΦn+1F

(n+1)n (p1, . . . , pn+1) ∼

1

ǫ2

with F(n+1)n jet definition for combining n + 1 partons into n jets

Singular configurations:

collinear i ‖ j

|Mn+1|2 →

1

sijPij(z)|Mn|

2

Altarelli-Parisi splitting functions

Pqg→Q(z) = CF

„1 + (1 − z)2 − ǫz2

z

«

Pqq→G(z) = TF

„z2 + (1 − z)2 − ǫ

1 − ǫ

«

Pgg→G(z) = 2CA

„z

1 − z+

1 − z

z+ z(1 − z)

«

soft j → 0

|Mn+1|2 →

X

ik

Sijk|Mn|2

Soft eikonal factor

Sabc ≡2sac

sabsbc



Double real radiation

dσ(n+2) = |Mn+2|2dΦn+2F

(n+2)n (p1, . . . , pn+2) ∼

1

ǫ4

with F(n+2)n jet definition for combining n + 2 partons into n jets

Singular configurations (J. Campbell, E.W.N. Glover; S. Catani, M. Grazzini)

triple collinear i ‖ j ‖ k

|Mn+2|2 → Pijk(x, y, z, sij , sik, sjk)|Mn|

2

double single collinear i ‖ j, k ‖ l

|Mn+2|2 →

1

sijPij(x)

1

sklPkl(y) |Mn|

2

soft/collinear i ‖ j, k → 0

|Mn+2|2 → Sijk(x, sij , sik, sjk, sijl, skl)|Mn|

2

double soft j, k → 0

|Mn+2|2 → Si,jk,l(sij , sik, sjk, sil, sjl, skl)|Mn|

2



Double real radiationExample: triple collinear splitting function ggg → G (colour-ordered)

Pabc→G(w, x, y, sab, sbc, sabc) =

2

8<:+

(1 − ǫ)

s2abs

2abc

(xsabc − (1 − y)sbc)2

(1 − y)2+

2(1 − ǫ)sbc

sabs2abc

+3(1 − ǫ)

2s2abc

+1

sabsabc

„(1 − y(1 − y))2

yw(1 − w)− 2

x2 + xy + y2

1 − y+

xw − x2y − 2

y(1 − y)+ 2ǫ

x

(1 − y)

«

+1

2sabsbc

„3x2 −

2(2 − w + w2)(x2 + w(1 − w))

y(1 − y)+

1

yw+

1

(1 − y)(1 − w)

«9=;

+ (sab ↔ sbc, w ↔ y) ,

momentum conservation: w = 1 − x − y

symmetric under a ↔ c

no poles in sac



Colour orderingQCD matrix elements are tensors in colour space

all-gluon amplitudes

|Mg1...gn〉 ∼X

Pn

tr(Taσ(1) . . . Taσ(n) )|Mgσ(1)...gσ(n)〉

amplitudes with a qq pair

|Mqiqjg1...gn 〉 ∼X

Pn

j(Taσ(1) . . . Taσ(n) )i|Mqiqjgσ(1)...gσ(n)

〉

amplitudes with several qq pairs−→ several independent colour lines

Colour-ordered subamplitudes: |M〉 contain only soft and collinear singularities between

colour-adjacent particles



Colour orderingSquared matrix elements computed by contraction of colour matrices

fabc = −2i tr“Ta [T b, T c]

”

TaijTa

kl =1

2

„δilδkj −

1

Nδijδkl

«

leading colour: no interference between different colour orderings

〈Mg1...gn ||Mg1...gn 〉 ∼X

Pn

〈Mgσ(1)...gσ(n)||Mgσ(1)...gσ(n)

〉

subleading colour: combine amplitudes with different permutations

0 = |M123...n〉 + |M213...n〉 + . . . + |M23...1n〉 + |M23...n1〉

|Mqqγ123...n〉 = |Mqq123...n〉 + |Mqq213...n〉 + . . . + |Mqq23...1n〉 + |Mqq23...n1〉

gluon permuted over all possible positions behaves like photonTechniques for Higher Order QCD Calculations – p.57

Real Corrections: NLO

Structure of NLO m-jet cross section (subtraction formalism):Z. Kunszt, D. Soper

dσNLO =

Z

dΦm+1

“dσR

NLO − dσSNLO

”+

"Z

dΦm+1

dσSNLO +

Z

dΦm

dσVNLO

#

dσSNLO : local counter term for dσR

NLO , conincide in all soft and collinear limits

dσRNLO − dσS

NLO : free of divergences, can be integrated numerically

dσSNLO must be suffieciently simple to be integrated analytically

infrared poles from dσSNLO cancel poles in dσV

NLO

General methods at NLO

Dipole subtraction (S. Catani, M. Seymour)

E-prescription (S. Frixione, Z. Kunszt, A. Signer)

Antenna subtraction (D. Kosower; J. Campbell, M. Cullen, E.W.N. Glover)



Construction of subtraction termgeneral structure

dσSNLO = N dΦm+1(p1, ..., pm+1, Q)

×X

conf,ijk

Cijk|Mm(p1, . . . , pm)|2F(m)m (p1, . . . , pm)

sum runs over singular configurations

Cijk contains singular behaviour of configuration, depends on three momenta

only Cijk is integrated analytically

all (or at least some) momenta in the process are redefined (mapped) such thatthe correct unresolved behaviour is recovered in each configuration

pa → pa in soft limit, pi → pi + pj in collinear i ‖ j limit

methods differ in: choice of configurations, momentum mapping



Dipole subtractionS. Catani, M. Seymour

Configurations: pairs of partons ij correlated with a single extra parton k

m+1

m+1

1

1

m

k

iV

jij,kΣ m

contains full collinear i ‖ j limit

contains half of soft limits Sijk and Sjik

form of subtraction term

dσSNLO = N dΦm+1(p1, ..., pm+1, Q)

X

pairs i,j

X

k 6=i,j

〈Mm(p1, ..pij , pk, ..., pm+1)|Dij,k|Mm(p1, ..pij , pk, ..., pm+1)〉

J(m)m (p1, ..pij , pk, ..., pm+1)



Dipole subtractionPhase space mapping: yij,k = sij/sijk

pµk =

1

1 − yij,kpµ

k , pµij = pµ

i + pµj −

yij,k

1 − yij,kpµ

k

ensures correct collinear and soft limits

new momenta are on-shell

Dipole function

Dij,k =1

sij

Tk · T ij

T2ij

V ij,k

tensor in colour space

dipole factors constructed using zi = sik/(sik + sjk), e.g. quark-gluon dipole

V qigj ,k = 8παsCF

»2

1 − zi(1 − yij,k)− (1 + zi) − ǫ(1 − zi)

–

yields collinear splitting function, and part of soft eikonal factorTechniques for Higher Order QCD Calculations – p.61


Dipole subtractionPhase space factorisation

dΦm+1(p1, . . . , pm+1; q) = dΦm(p1, . . . , pij , pk, . . . , pm+1; q) · dΦD(pi, pj , pk; pij + pk)

for m = 2: dΦ2 = 1/8π constant −→ dΦD = 8πdΦ3

three-particle phase space expressed in invariantsZ

dΦ3(p1, p2, p3, Q) =1

2d+1

Z(s12s13s23)

d−42 ds12 ds13ds23dΩd−1 dΩd−2

δ(s12 + s13 + s23 − Q2) (Q2)2−d2 .

Integrated dipole factors

for example quark-gluon dipole

ZdΦDV qigj ,k = αsCF

»1

ǫ2+

3

2ǫ+ 5 −

π2

2

–

pole terms given by I(1)qq



Antenna subtraction D. Kosower; J. Campbell, M. Cullen, E.W.N. Glover

Configurations: colour-ordered pairs of partons ik radiating parton j in between them

1 1

i

j

k

I

i

j

k

I

m+1 m+1

K

K

contains both collinear j ‖ i and j ‖ k limits

contains soft limit j → 0: Sijk

relation to dipole subtraction: two dipoles ≃ one antenna

form of subtraction term

dσSNLO = N

X

m+1

dΦm+1(p1, . . . , pm+1; q)1

Sm+1

×X

j

X0ijk |Mm(p1, . . . , pI , pK , . . . , pm+1)|

2 J(m)m (p1, . . . , pI , pK , . . . , pm+1)



Antenna subtractionSystematic construction of antenna functions from matrix elements

X0ijk = Sijk,IK

|M0ijk|

2

|M0IK |2

dΦXijk=

dΦ3

P2

Phase space factorisation

dΦm+1(p1, . . . , pm+1; q) = dΦm(p1, . . . , pI , pK , . . . , pm+1; q) · dΦXijk(pi, pj , pk; pI + pK)

Integrated subtraction term (analytically)

|Mm|2 J(m)m dΦm

ZdΦXijk

X0ijk ∼ |Mm|2 J

(m)m dΦm

ZdΦ3|M

0ijk|

2 ∼ |Mm|2 J(m)m X 0

ijk

can be combined with dσVNLO



Partons in the initial stateSubtraction for radation off initial state partons

must include all configurations with initial/final state partons,e.g. in antenna subtraction



Partons in the initial state

subtraction functions are the same

phase space factorisation and mapping is different, e.g. initial-final antenna

dΦm+1(k1, . . . , km+1; p, r) = dΦm(k1, . . . , KK , . . . , km+1; xp, r)

×Q2

2πdΦ2(kj , kk; p, q)

dx

x.

integrated subtraction terms depend on parton momentum fraction,e.g. initial-final antenna

dσS,(if)(p, r) =X

m+1

X

j

Sm

Sm+1

Zdξ1

ξ1

Zdξ2

ξ2

Z 1

ξ1

dx

xfi/1

„ξ1

x

«fb/2 (ξ2)

×C(ǫ)Xi,jk(x) dσB(ξ1H1, ξ2H2) .


End of Lecture 4


Real Corrections: NNLO

Structure of NNLO m-jet cross section:

dσNNLO =

Z

dΦm+2

“dσR

NNLO − dσSNNLO

”

+

Z

dΦm+1

“dσV,1

NNLO − dσV S,1NNLO

”

+

Z

dΦm+2

dσSNNLO +

Z

dΦm+1

dσV S,1NNLO +

Z

dΦm

dσV,2NNLO ,

dσSNNLO : real radiation subtraction term for dσR

NNLO

dσV S,1NNLO : one-loop virtual subtraction term for dσV,1

NNLO

dσV,2NNLO : two-loop virtual corrections

each line is finite and can be implemented numerically



Tree-level real radiation contribution to m jets at NNLO

dσRNNLO = N

X

m+2

dΦm+2(p1, . . . , pm+2; q)1

Sm+2

× |Mm+2(p1, . . . , pm+2)|2 J

(m+2)m (p1, . . . , pm+2)

dΦm+2: full m + 2-parton phase space

J(m+2)m : ensures m + 2 partons → m jets

−→ two partons must be experimentally unresolved

Up to two partons can be theoretically unresolved (soft and/or collinear)

Building blocks of subtraction terms:

products of two three-parton antenna functions

single four-parton antenna function



Distinct Configurations for m + 2 partons → m jets

one unresolved parton (a)

three parton antenna function X0ijk can be used (as at NLO)

this will not yield a finite contribution in all single unresolved limits

two colour-connected unresolved partons (b)i j k Il L

four-parton antenna function X0ijkl

two almost colour-unconnected unresolved partons (common radiator) (c)i j k l Im MK

strongly ordered product of non-independent three-parton antenna functions

two colour-unconnected unresolved partons (d)j k I Ki n o p N P

product of independent three-parton antenna functions



Two colour-connected unresolved partonsdσS,b

NNLO = NX

m+2

dΦm+2(p1, . . . , pm+2; q)1

Sm+2

×

24X

jk

“X0

ijkl − X0ijkX0

IKl − X0jklX

0iJL

”

× |Mm(p1, . . . , pI , pL, . . . , pm+2)|2 J

(m)m (p1, . . . , pI , pL, . . . , pm+2)

35

X0ijkl: four-parton tree-level antenna function contains all double unresolved

pj , pk limits of |Mm+2|2, but is also singular in single unresolved limits of pj or pk

X0ijkX0

IKl: cancels single unresolved limit in pj of X0ijkl

X0jklX

0iKL: cancels single unresolved limit in pk of X0

ijkl

Triple-collinear, soft-collinear, double soft limits: X0ijkX0

IKl, X0jklX

0iKL → 0

Double single collinear limit: X0ijkX0

IKl, X0jklX

0iKL 6= 0

cancels with double single collinear limit of dσS,aNNLO


Double Real Subtraction

Two colour-connected unresolved partons

1 1

i I

i

I

m+2 m+2

Ll

l

L

j

j

k

k X0ijkl = Sijkl,IL

|M0ijkl|

2

|M0IL|2

dΦXijkl=

dΦ4

P2

Phase space factorisation

dΦm+2(p1, . . . , pm+2; q) = dΦm(p1, . . . , pI , pL, . . . , pm+2; q)·dΦXijkl(pi, pj , pk, pl; pI + pL)

Integrated subtraction term (analytically)

|Mm|2 J(m)m dΦm

ZdΦXijkl

X0ijkl ∼ |Mm|2 J

(m)m dΦm

ZdΦ4|M

0ijkl|

2



Four-particle phase space integrals∫

dφ4|M|24use (R. Cutkosky; M. Veltman)

dd−1p

2E= ddpδ+(p2) =

1

2πiddp

„1

p2 + iǫ−

1

p2 − iǫ

«

to convert to cuts of three-loop propagator integrals

Z ˛˛˛

˛˛˛

2

dφ4 =

ZIm dp1,2,3

use IBP to reduce to master integrals

evaluate only master integralsTechniques for Higher Order QCD Calculations – p.73


Single unresolved limit of one-loop amplitudes

Loopm+1j unresolved

−→ Splittree × Loopm + Splitloop × Treem

Z. Bern, L.D. Dixon, D. Dunbar, D. Kosower; S. Catani, M. Grazzini; D. Kosower, P. UwerZ. Bern, V. Del Duca, W.B. Kilgore, C.R. SchmidtZ. Bern, L.D. Dixon, D. Kosower; S. Badger, E.W.N. Glover

Accordingly: Splittree → X0ijk , Splitloop → X1

ijk

1

i

j

k

m+1

1

I

i

j

k

I

m+1

K

K

1

I

i

j

k

I

m+1

K

K

X1ijk = Sijk,IK

|M1ijk|

2

|M0IK |2

− X0ijk

|M1IK |2

|M0IK |2



tree level one loop

quark-antiquark

qgq A03(q, g, q) A1

3(q, g, q), A13(q, g, q), A1

3(q, g, q)

qggq A04(q, g, g, q), A0

4(q, g, g, q)

qq′q′q B04(q, q′, q′, q)

qqqq C04 (q, q, q, q)

quark-gluon

qgg D03(q, g, g) D1

3(q, g, g), D13(q, g, g)

qggg D04(q, g, g, g)

qq′q′ E03(q, q′, q′) E1

3(q, q′, q′), E13(q, q′, q′), E1

3(q, q′, q′)

qq′q′g E04(q, q′, q′, g), E0

4 (q, q′, q′, g)

gluon-gluon

ggg F 03 (g, g, g) F 1

3 (g, g, g), F 13 (g, g, g)

gggg F 04 (g, g, g, g)

gqq G03(g, q, q) G1

3(g, q, q), G13(g, q, q), G1

3(g, q, q)

gqqg G04(g, q, q, g), G0

4(g, q, q, g)

qqq′q′ H04 (q, q, q′, q′)


Example: Real NNLO corrections

Partonic channels in e+e− → 3j, 1/N2

γ∗ → qqggg and γ∗ → qqqqg at tree-levelK. Hagiwara, D. Zeppenfeld; F.A. Berends, W.T. Giele, H. Kuijf;N. Falck, D. Graudenz, G. Kramer

γ∗ → qqgg and γ∗ → qqqq at one loopZ. Bern, L. Dixon, D. Kosower, S. Weinzierl;J. Campbell, D.J. Miller, E.W.N. Glover; Z. Nagy, Z. Trocsanyi

γ∗ → qqg at two loopsL. Garland, E.W.N. Glover, A. Koukoutsakis, E. Remiddi, TG

O(1/N2) receives contributions from all partonic processes (also identical qqqq)

Discuss here: subtraction terms for γ∗ → qqggg, γ∗ → qqgg in O(1/N2)

Gluons are photon-like (no self-coupling)



Five-parton contributionsMatrix element for γ∗ → qqggg

˛M0

qq3g

˛2= N5

1

N2A0

5(1q, 3g, 4g , 5g, 2q)

with

Nn = 4παX

q

e2q

`g2´(n−2) `

N2 − 1´ ˛

M0qq

˛2

˛M0

qq

˛2= 4(1 − ǫ)q2

NNLO real radiation contribution to three jet final state

dσRNNLO,A

=N5

N2dΦ5(p1, . . . , p5; q)

1

3!A0

5(1q, 3g, 4g , 5g, 2q)J(5)3 (p1, . . . , p5)

Only one possible hard radiator pair: (q − q) for this colour factor→ No singularities associated to (almost) colour unconnected configurations

dσS,cNNLO = dσS,d

NNLO = 0Techniques for Higher Order QCD Calculations – p.77


Five-parton contributionsSingle unresolved parton subtraction

dσS,a

NNLO,A=

N5

N2dΦ5(p1, . . . , p5; q)

1

3!

×X

i,j,k∈PC (3,4,5)

A03(1q , ig, 2q) A0

4(g(1i)q, jg, kg ,g(2i)q) J

(4)3 (fp1i, pj , pk, fp2i)

Colour connected double unresolved subtraction

dσS,b

NNLO,A=

N5

N2dΦ5(p1, . . . , p5; q)

1

3!

X

i,j,k∈PC (3,4,5)

A0

4(1q , ig, jg, 2q)

−A03(1q, ig, 2q) A0

3(g(1i)q , jg,g(2i)q) − A03(1q, jg, 2q) A0

3(g(1j)q , ig, g(2j)q)

!

×A03((1ij)q, kg, (2ij)q) J

(3)3 (gp1ij , pk, gp2ij)

dσRNNLO,A

− dσS,a

NNLO,A− dσS,b

NNLO,Ais finite and can be integrated numerically over dΦ5



Four-parton contributionsOne-loop O(1/N2) contribution to γ∗ → qqgg

dσV,1

NNLO,A=

N4

N2

“αs

2π

”dΦ4(p1, . . . , p4; q)

1

2!A1,b

4 (1q, 3g, 4g, 2q) J(4)3 (p1, . . . , p4)

Three types of subtraction terms (explicit poles, implicit unresolved poles, compensation)

dσV S,1,a

NNLO,A= −dσS,a

NNLO,A

= −N4

N2

“αs

2π

”dΦ4(p1, . . . , p4; q)

1

2!

×A03(s12)A0

4(1q, 3g , 4g, 2q)J(4)3 (p1, p2, p3, p4)

with

A03 =

RΦD

A03: integrated three-parton antenna function

A04: four-parton tree-level matrix element



Four-parton contributionsSubtraction for single unresolved one-loop contributions

dσV S,1,b

NNLO,A=

N4

N2

“αs

2π

”dΦ4(p1, . . . , p4; q)

1

2!

X

i,j∈P (3,4)

J(3)3 (fp1i, pj , fp2i)

×

A1

3(1q, ig, 2q)A03(g(1i)q, jg,g(2i)q)

+A03(1q, ig, 2q)

hA1

3(g(1i)q , jg,g(2i)q) + A12(s1234)A

03(g(1i)q , jg,g(2i)q)

i!

Structure: X13 |M3,tree|2 + X0

3 |M3,1l|2 subtracts simple unresolved limits

introduces spurious poles outside the limits, compensated by

dσV S,1,c

NNLO,A=

N4

N2

“αs

2π

”dΦ4(p1, . . . , p4; q)

1

2!

×X

i,j∈P (3,4)

A03(s12)A0

3(1q , ig, 2q)A03(g(1i)q, jg ,g(2i)q)J

(3)3 (fp1i, pj , fp2i)

dσV,1

NNLO,A− dσV S,1

NNLO,Ais free of 1/ǫ poles, and can be integrated numerically over dΦ4



Three-parton contributionsTwo-loop three-parton matrix element plus integrated five-parton and four-partonsubtraction terms

dσV,2NNLO + dσS

NNLO + dσV S,1NNLO

with

dσSNNLO,A

− dσTNNLO,A

=1

N2

1

2A0

4(s12)A03(1q, 3g, 2q) dσ3 ,

dσV S,1

NNLO,A+ dσT

NNLO,A=

1

N2

A1

3(s12)A03(1q, 3g , 2q)

+A03(s12)

hA1

3(1q, 3g, 2q) + A12(s123)A

03(1q, 3g, 2q)

i!dσ3

Together with the contributions from γ∗ → qqqq(g) find at O(1/N2):

Poles(dσSNNLO + dσV S,1

NNLO + dσVNNLO) = 0

dσV,2NNLO + dσV S,1

NNLO + dσSNNLO is free of 1/ǫ and can be integrated numerically over dΦ3

Relation also holds for all other colour factorsTechniques for Higher Order QCD Calculations – p.81

e+e− → 3 jets at NNLO

Structure of e+e− → 3 jets program:EERAD3: A. Gehrmann-De Ridder, E.W.N. Glover, G. Heinrich, TG

5 parton

channel

4 parton

channel

3 parton

channel

dΦqqggg

dΦqqgg

dΦqqg

Monte Carlo

Phase Space

dσRNNLO − dσS

NNLO

dσV,2NNLO

+

∫

dσV S,1NNLO dΦX3

+

∫

dσSNNLO dΦX4

dσV,1NNLO − dσ

V S,1NNLO

-pi5

-pi4

-pi3

Cross section

-pi5, w

-pi4, w

-pi3, w

Definition of Observables

5 parton

→ 3 jet

4 parton

→ 3 jet

3 parton

→ 3 jet

w, C, S, T

w, C, S, T

w, C, S, T -

-

-

-

⊕

Histograms

σ3j

dσ/dT

dσ/dS

dσ/dC



Event shape distributionsPerturbative expression for event shapes to NNLO

1

σhad

dσ

dy(s, µ2, y) =

„αs(µ)

2π

«dA

dy+

„αs(µ)

2π

«2 „dB

dy+

dA

dyβ0 log

µ2

s

«

+

„αs(µ)

2π

«3

dC

dy+ 2

dB

dyβ0 log

µ2

s+

dA

dy

„β20 log2 µ2

s+ β1 log

µ2

s

«!

+O(α4s)

Computation of perturbative coefficients A, B, C: parton-level event generator program.

Higher order corrections stabilize scale-dependence

observe: magnitude of NNLO corrections: (5-20)%, substantial differencesbetween different event shapes

Attempt: new fit of αs from event shapesTechniques for Higher Order QCD Calculations – p.83


ALEPH dataEcm=91.2 GeV

fit range

NNLO, αs=0.1261 ±0.0003, χ2/ndof=1.2

NLO, αs=0.1354 ±0.0003, χ2/ndof=4.3

NLO + NLLA, αs=0.1198 ±0.0002, χ2/ndof=7.6

1/σ

dσ/d

MH

MH

(dat

a-fi

t)/d

ata stat. ⊕ exp. uncertainty

statistical uncertainty

10-4

10-3

10-2

10-1

1

10

-0.4

-0.2

0

0.2

0.4

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

clear improvement of NNLO over NLO

good fit quality

extended range of good description in3-jet region

matched NLO+NNLA still yields a betterprediction in 2-jet region

value of αs lower than at NLO,but still rather high



Uncertainty from renormalisation scale

NNLO

NLO

NLO+NLLA

using T at LEPI

xµ

α s(M

z)

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.155

0.16

0 0.5 1 1.5 2 2.5

NNLO

NLO

NLO+NLLA

using MH at LEPI

xµ

α s(M

z)

0.11

0.115

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0 0.5 1 1.5 2 2.5



αs

NNLO

αs

NLO

αs

NLO+NLLA

T

MH

C

BW

BT

y3

0.11

0.12

0.13

0.14

0.15

0.11

0.12

0.13

0.14

0.15

0.11

0.12

0.13

0.14

0.15

scale uncertainty reduced byfactor 2 compared to NLO;factor 1.3 compared toNLLA+NLO

scatter among values fromdifferent observables reducedvery substantially at NNLO−→ genuine NNLO effect

Result for all ALEPH event shapes of LEP1/LEP2αs(MZ) = 0.1240 ± 0.0008(stat) ± 0.0010(exp) ± 0.0011(had) ± 0.0029(theo)


Vector Boson Production at NNLO

Inclusive cross sectioncan be measured precisely

are theoretically well understood

NNLO corrections known

relevant partons well constrained

benchmark reaction for LHC(luminosity monitor)M. Dittmar, F. Pauss, D. Zürcher

14

15

16

17

18

19

20

21

22

23

24

NLONNLO

LO

LHC Z(x10)W

σ . B

l (

nb)

A. Martin, J. Stirling, R. Roberts, R. Thorne

Rapidity distributionallows to account forlimited detector coverage

known to NNLOC. Anastasiou, L. Dixon, K. Melnikov,F. Petriello

NNLO also for lepton distributionK. Melnikov, F. Petriello


Higgs Boson Production at NNLO

NNLO calculations, including Higgs decays and full final state information−→ allow implementation of experimental cuts

sector decomposition for real radiation (C. Anastasiou, K. Melnikov, F. Petriello)

NNLO subtraction for real radiation (S. Catani, M. Grazzini)

modified behaviour of transverse momentum distribution

reduced uncertainty on normalisation of experimentally measureable cross section

implies better extraction of Higgs branching ratios−→ first step to precision Higgs physics at LHC


Summary

Precision physics at collidersindirect constraints on new physics

improved predictions for background and signal processes

at hadron colliders: require higher order QCD

Structure of higher order correctionsobservables involve jet definitions and experimental cuts

must therefore consider all partonic channels separately

infrared singularities from virtual and real corrections cancel only in sum of allcontributions

Computing virtual correctionsrequire (multi-)loop integrals with many legs

reduction to master integrals using IBP and LI identities

techniques: Mellin-Barnes, differential equations, sector decomposition, . . .


Summary

Computing real correctionscomplex definition of final state: require separation of universal divergentcontributions → subtraction

subtraction methods: dipole, antenna

involve: subtraction functions, phase space mapping, analytic integration ofsubtraction terms

Evaluating observablesimplement all contributions in parton-level event generator

apply cuts at parton-level

Resultsreduced scale uncertainty

better consistency data versus theory

improved predictions for LHC benchmark reactions and searches


End of Lecture 5


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