Progress on
NNLO subtraction
Vittorio Del Duca
INFN Torino
Les Houches 7 May 2005
NLO features
Jet structure: final-state collinear radiation
PDF evolution: initial-state collinear radiation
Opening of new channels
Reduced sensitivity to fictitious input scales: µR , µF
predictive normalisation of observables
first step toward precision measurements
accurate estimate of signal and background
for Higgs and new physics
Matching with parton-shower MC’s: MC@NLO
Jet structure
the jet non-trivial structure shows up first to NLO
leading order
NLO
NNLO
NNLO corrections may be relevant if
the main source of uncertainty in extracting info from
data is due to NLO theory: αS measurements
NLO corrections are large:
Higgs production from gluon fusion in hadron collisions
NLO uncertainty bands are too large to test
theory vs. data: b production in hadron collisions
NLO is effectively leading order:
energy distributions in jet cones
in short, NNLO is relevant where NLO fails to do its job
Summary of αS (MZ )
S. Bethke hep-ex/0407021
world average of αS (MZ )
using MS and NNLO results only
αS (MZ ) = 0.1182 ± 0.0027
(cf. 2002 αS (MZ ) = 0.1183 ± 0.0027
outcome almost identical
because new entries wrt 2002
- LEP jet shape observables and
4-jet rate, and HERA jet rates
and shape variables - are NLO )
filled symbols are NNLO results
Is NLO enough to describe data ?
b cross section in pp̄ collisions at 1.96 TeV
dσ(pp̄ → Hb X, Hb → J/ψ X)/dpT (J/ψ)
NLO + NLL
good agreement
with data (with use
of updated FF’s by
Cacciari & Nason)
The CDF value in the
inset was preliminary.
The published value is
(CDF hep-ex/0412071)
19.4 ± 0.3(stat)+2.1
−1.9 (syst) nb
Cacciari, Frixione, Mangano, Nason, Ridolfi 2003
Is NLO enough to describe data ?
Inclusive jet pT cross section at Tevatron
good agreement between
NLO and data over
several orders of
magnitude
constrains the gluon
distribution at high x
Is NLO enough to describe data ?
di-lepton rapidity distribution for (Z, γ ∗ ) production vs. Tevatron Run I data
LO and NLO curves are
for the MRST PDF set
no spin correlations
C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003
Is NLO enough to describe data ?
Drell-Yan W cross section at LHC with leptonic decay of the W
|MC@NLO − NLO| = O(2%)
S. Frixione M.L. Mangano 2004
NNLO useless without spin correlations
Precisely evaluated Drell-Yan W, Z cross sections could be used
as ``standard candles’’ to measure the parton luminosity at LHC
Is NLO enough to describe data ?
Total cross section for inclusive Higgs production at LHC
contour bands are
lower
µR = 2MH
µF = MH /2
upper
µR = MH /2
µF = 2MH
scale uncertainty
is about 10%
NNLO prediction stabilises the perturbative series
NNLO Drell-Yan Z production at LHC
Rapidity distribution for
an on-shell Z boson
30%(15%) NLO increase wrt to LO at central Y’s (at large Y’s)
NNLO decreases NLO by 1 − 2%
scale variation: ≈ 30% at LO; ≈ 6% at NLO; less than 1% at NNLO
C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003
Scale variations in Drell-Yan Z production
solid: vary µR and µF together
dashed: vary µF only
dotted: vary µR only
C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003
Drell-Yan W production at LHC
Rapidity distribution
for an on-shell
W − boson (left)
W + boson (right)
distributions are symmetric in Y
NNLO scale variations are 1%(3%) at central (large) Y
C. Anastasiou L. Dixon K. Melnikov F. Petriello 2003
Higgs production at LHC
a fully differential cross section:
bin-integrated rapidity distribution, with a jet veto
C. Anastasiou K. Melnikov F. Petriello 2004
jet veto: require
R = 0.4
|pjT | < pveto
= 40 GeV
T
for 2 partons
2
R12
= (η1 − η2 )2 + (φ1 − φ2 )2
if R12 > R
|p1T |, |p2T | < pveto
T
if R12 < R
|p1T + p2T | < pveto
T
MH = 150 GeV (jet veto relevant in the H → W + W − decay channel)
K factor is much smaller for the vetoed x-sect than for the inclusive one:
j
average |pT | increases from NLO to NNLO: less x-sect passes the veto
NLO assembly kit
e+ e− → 3 jets
leading order
2
|Mtree
|
n
IR
NLO real
⊗
2
|Mtree
n+1 |
2
|Mtree
n |
→
+
=
!
dP S|Psplit |2
!
"
A
B
− 2+
!
!
NLO virtual
d = 4 − 2!
!
∗
tree
dd l 2(Mloop
n ) Mn
=
"
A
B
+
!2
!
#
2
|Mtree
n | + f in.
NLO production rates
Process-independent procedure devised in the 90’s
slicing
subtraction
dipole
antenna
Giele Glover & Kosower
Frixione Kunszt & Signer; Nagy & Trocsanyi
Catani & Seymour
Kosower; Campbell Cullen & Glover
!
B
σ = σ LO + σ NLO =
dσm
Jm + σ NLO
m
!
!
R
V
σ NLO =
dσm+1
Jm+1 +
dσm
Jm
m+1
m
the 2 terms on the rhs are divergent in d=4
σ NLO
use universal IR structure to subtract divergences
%
!
!
"
# ! $
R,A
R,A
R
V
=
dσm+1
Jm+1 − dσm+1
Jm +
dσm
+ dσm+1
Jm
m+1
m
the 2 terms on the rhs are finite in d=4
1
Observable (jet) functions
Jm vanishes when one parton becomes soft or collinear to another one
Jm (p1 , ..., pm ) → 0 ,
if
pi · pj → 0
B
dσm
is integrable over 1-parton IR phase space
Jm+1 vanishes when two partons become simultaneously soft and/or collinear
Jm+1 (p1 , ..., pm+1 ) → 0 ,
if
pi · pj and pk · pl → 0 (i "= k)
R and V are integrable over 2-parton IR phase space
observables are IR safe
Jn+1 (p1 , .., pj = λq, .., pn+1 ) → Jn (p1 , ..., pn+1 )
Jn+1 (p1 , .., pi , .., pj , .., pn+1 ) → Jn (p1 , .., p, .., pn+1 )
for all n ≥ m
if
if
λ→0
pi → zp , pj → (1 − z)p
NLO IR limits
collinear operator
(0)
Cir |Mm+2 (pi , pr , . . .)|2 ∝
1
(0)
"Mm+1 (0)(pir , . . .)|P̂fi fr |Mm+1 (0)(pir , . . .)#
sir
soft operator
(0)
Sr |Mm+2 (pr , . . .)|2 ∝
counterterm
sik
"Mm+1 (0)(. . .)|Ti · Tk |Mm+1 (0)(. . .)#
sir srk


! !1
(0)

Cir + Sr  |Mm+2 (pi , pr , . . .)|2
2
r
i!=r
performs double subtraction in overlapping regions
NLO overlapping divergences
Cir Sr can be used to cancel double subtraction
(0)
Cir (Sr − Cir Sr ) |Mm+2 |2 = 0
(0)
Sr (Cir − Cir Sr ) |Mm+2 |2 = 0
the NLO counterterm



! !1
!
(0) 2
(0)

A1 |Mm+2 | =
Cir + Sr −
Cir Sr  |Mm+2 (pi , pr , . . .)|2
2
r
i!=r
i!=r
has the same singular behaviour as SME, and is free of double subtractions
(0)
Cir (1 − A1 ) |Mm+1 |2 = 0
(0)
Sr (1 − A1 ) |Mm+1 |2 = 0
contains spurious singularities when parton s != r
becomes unresolved, but they are screened by Jm
σ NNLO =
!
NNLO subtraction
RR
dσm+2
Jm+2 +
m+2
!
RV
dσm+1
Jm+1 +
m+1
!
VV
dσm
Jm
m
the 3 terms on the rhs are divergent in d=4
use universal IR structure to subtract divergences
!
"
#
RR,A2
RR
σ NNLO =
dσm+2
Jm+2 − dσm+2
Jm
m+2
takes care of doubly-unresolved regions,
but still divergent in singly-unresolved ones
!
"
#
RV,A1
RV
+
dσm+1
Jm+1 − dσm+1
Jm
m+1
still contains 1/! poles in regions away from 1-parton IR regions
#
! "
!
!
RR,A2
RV,A1
VV
+
dσm
+ dσm+2
+ dσm+1
Jm
m
2
1
! Use IR factorization formulae valid in the doublyunresolved regions Catani-Grazzini, Campbell-Glover
NNLO counterterm
!
To account
for double subtraction
in overlapping
construct
the 2-unresolved-parton
counterterm
using the IR currents
regions: subtract"the collinear
limit of the soft one
#
!! ! 1
! 1
1
(0) 2
A2 |Mm+2 | =
Cirs +
Cir;js + Srs
6
8
2
r s!=r
i!=r,s
j!=i,r,s
$
%&
!
1
+
CSir;s − Cirs CSir;s −
Cir;js CSir;s
2
j!=i,r,s
$
%
!#
1
−
CSir;s Srs + Cirs
Srs − CSir;s Srs
2
i!=r,s
$
% &'
!
1
(0)
+
Cir;js
Srs − CSir;s Srs
|Mm+2 |2
2
j!=i,r,s
Zoltán Trócsányi: Matching of Singly- and Doubly-Unresolved Limits of Tree-level QCD Squared Matrix Elements
G.ETH
Somogyi
Z. Trocsanyi
VDD
2005
Zürich, March
8, 2005
– p.16/22
performing double and triple subtractions in overlapping regions
(0)
Cirs (1 − A2 ) |Mm+2 |2 = 0
(0)
Cir;js (1 − A2 ) |Mm+2 |2 = 0
(0)
Srs (1 − A2 ) |Mm+2 |2 = 0
(0)
CSir;s (1 − A2 ) |Mm+2 |2 = 0
needs a NLO-type
subtraction
The subtraction
scheme
at NNLO
between the m+2- and the m+1-parton contributions
The subtraction
scheme at NNLO
σ NNLO = σ NNLO + σ NNLO + σ NNLO
{m+2}
{m+1}
{m}
! ="σ NNLO + σ NNLO + σ NNLO
σ NNLO
{m+2}
{m+1}
{m}
RR,A2
NNLO
RR
σ{m+2} =
dσm+2 Jm+2 − dσm+2 Jm
! m+2"
RR,A2
#
NNLO
RR
σ
=
dσ
J
−
dσ
J
m+2
m
m+2 RR,A
m+2RR,A
must be finite{m+2}
in
1
12
m+2
−dσ
J
+
dσ
Jm
m+1
m+2
m+2
the doubly-unresolved regions
# d=4
G.
Somogyi
Z. Trocsanyi VDD
"
RR,A
RR,A2005
!
1
12
−dσ
J
+
dσ
Jm
m+1
m+2
m+2
RV,A
NNLO
RV
A1 takes
care of the
and A12 of 1the over-subtracting
σ{m+1}
= singly-unresolved
dσm+1 Jregions
d=4
m+1 − dσm+1 Jm
! m+1"
#
RV,A1
! $J
NNLO
RV
%
σ{m+1}
=
dσm+1
−
dσ
J
m+1
m RR,A
RR,A1 m+1
+
dσm+2 Jm+1 − dσm+2 12 Jm
m+1
! 1$
% # d=4
! " +
!dσ RR,A1 J !− dσ RR,A12# J
m
m+2
m+2 1
RR,A2m+1
RV,A
NNLO
VV
σ{m}
=
dσm 1+ dσm+2 + dσm+1
Jm d=4
"
#
!m
!2
!1
d=4
RR,A
RV,A
NNLO
VV
σ{m}
=
dσm
+ dσm+2 2 + dσm+1 1
Jm
Zoltán Trócsányi: Matching of Singly- and Doubly-Unresolved
m Limits of Tree-level QCD
2 Squared Matrix Elements
1
ETH Zürich, Marchd=4
8, 2005
– p.13/22
RR,A12
Constructing
dσ
A
need to construct 12 such that all m+2
overlapping regions in
1-parton and 2-parton IR phase space regions
RR,A12 are counted only once
Need to construct
A
such
that
12
Constructing
dσ
m+2
(0) 2
A12 )|Mm+2 |
(0) 2
Cir |Mm+2 |
! Cir (A1 + A2 −
=
Need to construct A12 such that
(0) 2
(0) 2
! Sr (A1 + A2 − A12 )|M
(0)m+2
(0)
2 | = Sr |M
2|
m+2
! Cir (A
+
A
−
A
)|M
|
=
C
|M
|
1
2
12
ir
m+2
m+2
(0)
(0) 2
! Cirs (A1 + A2 − A12(0)
)|M2m+2 |2 = C(0)
|M
irs 2 m+2 |
! Sr (A1 + A2 − A12 )|Mm+2 | = Sr |Mm+2
|
(0) 2
(0) 2
! Cir;js (A1 + A2 − A12 )|M
|
=
C
|M
(0) m+2
(0)
ir;js
2 m+2 |
! Cirs (A1 + A2 − A12 )|Mm+2 |2 = Cirs |M
|
m+2
(0) 2
(0) 2
! CSir;s (A1 + A2 − A12 )|M
|
=
CS
|M
(0) m+2
(0)
ir;s
2 |
m+2
! Cir;js (A1 + A2 − A12 )|Mm+2 |2 = Cir;js |M
|
m+2
(0) 2
(0) 2
! Srs (A1 + A2 − A12 )|M(0)
|
=
S
|M
(0)| 2
rs
m+2|2 = CS
m+2
! CSir;s (A1 + A2 − A12 )|Mm+2
|M
ir;s
m+2 |
(0)
(0)
2
2
!the
Srsdefinition
(A1 + Aof
)|M
|
=
S
|M
|
AA
2−
12
rs
m+2simple
m+2
12 is rather
(0)
(0)
A12 |Mm+2 |2 ≡ A1 A2 |Mm+2 |2 does the job
(0)
−→
(0)
2
2
need
to
compute
S
A
|M
|
,
C
A
|M
|
and
but
showing
that
it
has
the
right
properties
is
non
trivial,
and
requires
t
2
kt
2
Zoltán Trócsányi: Matching of Singly- and Doubly-Unresolved Limits of Tree-level QCD
Squared Matrix Elements
ETH Zürich,
2005
m+2
m+2March 8,considering
– p.17/22
iterated singly-unresolved
(0) 2 limits and strongly-ordered doubly-unresolved limits
Ckt St A2 |Mm+2 |
Conclusions
in the last few years, a lot of progress on the computation of
NNLO cross sections
sector decomposition is already up and running
subtraction is making progress (stay tuned)