Fully differential decay rate of a SM
Higgs boson into a b-pair at NNLO
Zoltán Trócsányi
!
University of Debrecen and MTA-DE Particle Physics Research Group
in collaboration with
V. Del Duca, C. Duhr, G. Somogyi, F. Tramontano
!
!
!
!
!
!
QCD@LHC 2015, London
September 3, 2015
Higgs boson has been discovered
mH [GeV] = 125.09±0.21stat±0.11syst (see Zghiche’s
talk, CMS + ATLAS Run 1: γγ + 4 lepton)
ΓH [MeV] = 1.7+7.7-1.8 (CMS), < 23 (95%, ATLAS)
σ/σSM = 1.00 ± 0.13 (ATLAS)
All measured properties are consistent with SM
expectations within experimental uncertainties
spin zero
parity +
couples to masses of W and Z (with cv=1 within
experimental uncertainty)
Yet it still could be the first element of an
extended Higgs sector (e.g. SUSY neutral Higgs)
Distinction requires high-precision prediction for
both production and decay
2
−
Example: pp → H + X → bb + X in PT
ΓH [MeV]=4.07±0.16theo
can use the narrow width approximation
d
=
dObb̄
"
2 (n)
1
X
dd pp!H+X
n=0
dp?,H d⌘H
known up to NNLO
#
⇥
" P1
(n)
n=0 d H!bb̄ /dObb̄
P1
(n)
n=0 H!bb̄
this talk: up to NNLO
3
#
⇥ Br(H ! bb̄)
known with
1% accuracy
−
pp → H + X → b b + X in PT
Including up to NNLO corrections for production and
decay:
d
=
dObb̄
"
2 (0)
d pp!H+X
(0)
(1)
(2)
d H!bb̄ /dObb̄ + d H!bb̄ /dObb̄ + d H!bb̄ /dObb̄
(0)
(1)
(2)
dp?,H d⌘H
+ H!bb̄ + H!bb̄
H!bb̄
(0)
(1)
2 (1)
d pp!H+X d H!bb̄ /dObb̄ + d H!bb̄ /dObb̄
+
(0)
(1)
dp?,H d⌘H
+ H!bb̄
H!bb̄
#
(2)
(0)
d2 pp!H+X d H!bb̄ /dObb̄
+
⇥ Br(H ! bb̄)
(0)
dp?,H d⌘H
H!bb̄
4
Method
CoLoRFulNNLO is a subtraction scheme with
6
CoLoRFulNNLO is a subtraction scheme with
✓ explicit expressions including flavor and color
(color space notation is used)
6
CoLoRFulNNLO is a subtraction scheme with
✓ explicit expressions including flavor and color
(color space notation is used)
✓ completely general construction
(valid in any order of perturbation theory)
6
CoLoRFulNNLO is a subtraction scheme with
✓ explicit expressions including flavor and color
(color space notation is used)
✓ completely general construction
(valid in any order of perturbation theory)
✓ fully local counter-terms
(efficiency and mathematical rigor)
6
CoLoRFulNNLO is a subtraction scheme with
✓ explicit expressions including flavor and color
(color space notation is used)
✓ completely general construction
(valid in any order of perturbation theory)
✓ fully local counter-terms
(efficiency and mathematical rigor)
✓ fully differential predictions
(with jet functions defined in d = 4)
6
CoLoRFulNNLO is a subtraction scheme with
✓ explicit expressions including flavor and color
(color space notation is used)
✓ completely general construction
(valid in any order of perturbation theory)
✓ fully local counter-terms
(efficiency and mathematical rigor)
✓ fully differential predictions
(with jet functions defined in d = 4)
✓ option to constrain subtraction near singular
regions (important check)
Completely Local SubtRactions for Fully Differential
Predictions@NNLO
6
CoLoRFulNNLO uses known ingredients
•
Universal IR structure of QCD (squared) matrix elements
-
ε-poles of one- and two-loop amplitudes
soft and collinear factorization of QCD matrix
elements
tree-level 3-parton splitting, double soft current:
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, soft gluon current:
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
7
CoLoRFulNNLO uses known ingredients
•
Universal IR structure of QCD (squared) matrix elements
-
ε-poles of one- and two-loop amplitudes
soft and collinear factorization of QCD matrix
elements
tree-level 3-parton splitting, double soft current:
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, soft gluon current:
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
•
Simple and general procedure for separating overlapping
singularities (using a physical gauge)
Z. Nagy, G. Somogyi, ZT, 2007
7
CoLoRFulNNLO uses known ingredients
•
Universal IR structure of QCD (squared) matrix elements
-
ε-poles of one- and two-loop amplitudes
soft and collinear factorization of QCD matrix
elements
tree-level 3-parton splitting, double soft current:
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, soft gluon current:
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
•
•
Simple and general procedure for separating overlapping
singularities (using a physical gauge)
Z. Nagy, G. Somogyi, ZT, 2007
Extension over whole phase space using momentum mappings
(not unique):
{p}n+s
{p̃}n
7
Fully local:
kinematic singularities cancel in RR
10
4
Singe collinear limit e+ e ! q q̄ggg
5
y45 = 10
-8
y45 = 10
-10
y45 = 10
5
5
10
10
5
5
2
2
1
0.99
1
0.99
0.995
1.0
R
R
R = subtraction/RR
8
-6
-8
5
10
1.01
-4
2
2
2
1.005
y4Q y5Q = 10
5
2
1.0
y4Q y5Q = 10
10
2
2
0.995
y4Q y5Q = 10
2
3
# events
# events
10
4
5
-6
2
3
10
10
Double soft limit e+ e ! q q̄ggg
1.005
1.01
Fully local:
kinematic singularities cancel in RV
Single collinear limit e+ e ! q q̄ g g
104
5
RV + RR, A1 : Cgi gr
1
1
1
104
4
5
6
103
5
5
2
102
5
10
5
5
2
2
1.01
0.99
ratio
0.995
1.0
ratio
R = subtraction/(RV+∫1RR,A1)
9
5
5
10
1.005
4
102
2
1.0
3
2
2
0.995
yrQ = 10
yrQ = 10
yrQ = 10
2
103
0.99
RV + RR, A1 : Sgr
8
# of events
# of events
2
cos ✓ir = 10
cos ✓ir = 10
cos ✓ir = 10
Single soft limit e+ e ! q q̄ g g
1.005
1.01
Cancellation of poles
‣
we checked the cancellation of the leading and
first subleading poles (defined in our subtraction
scheme) for arbitrary number of m jets
‣
for m=2,
‣
‣
the insertion operators are independent of the
kinematics (momenta are back-to-back, so
MI’s are needed at the endpoints only)
‣
color algebra is trivial: T 1 T 2 =
T 21 =
so can demonstrate the cancellation of poles
10
T 22 =
CF
_
Poles cancel: H→bb at µ = mH
NNLO
m
d
=
Z n
VV
m
d
m
VV
H!bb̄
✓
+
Z h
d
2
◆2
2
RR,A2
m+2
d
⇢
RR,A12
m+2
2CF2
+ 4 +
✏
i
+
✓
Z h
1
d
RV,A1
m+1
+
⇣Z
d
1
⌘
RR,A1 A1
m+2
◆
io
Jm
↵s (µ )
11CA CF
C F nf 1
B
2
=
d H!bb̄
+ 6CF
2⇡
4
2
✏3
✓
◆
✓
◆
2
8 ⇡
17
2CF nf 1
2
2
+
+
CA CF +
2⇡ CF
9 12
2
9
✏2
✓
◆
✓
◆
961 13⇣3
109
65CF nf 1
2
2
+
CA CF +
2⇡
14⇣3 CF +
+
216
2
8
108
✏
!
C. Anastasiou, F. Herzog, A. Lazopoulos, arXiv:0111.2368
XZ
d
A
✓
2
◆2
⇢
2CF2
✏4
✓
◆
↵s (µ )
11CA CF
C F nf 1
B
2
=
d H!bb̄
+ 6CF +
2⇡
4
2
✏3
✓
◆
✓
◆
2
8 ⇡
17
2CF nf 1
2
2
+
CA CF +
2⇡ CF
9 12
2
9
✏2
✓
◆
✓
◆
961 13⇣3
109
65CF nf 1
2
2
+
CA CF +
2⇡
14⇣3 CF +
!
216
2
8
108
✏
V. Del Duca, C. Duhr, G. Somogyi, F. Tramontano, Z. Trócsányi, arXiv:1501.07226
11
+ -
2
Example: e e → m(=3) jets at µ = s
NNLO
m
=
d
Poles
Z n
VV
m
d
m
VV
3
+
Z h
2
= Poles
(2⇥0)
A3
+
d
RR,A2
m+2
(2⇥0)
A3
(1⇥1)
A3
+
d
RR,A12
m+2
(1⇥1)
A3
+ Poles
XZ
i
+
Z h
1
d
RV,A1
m+1
+ Finite
d
A
+
⇣Z
(2⇥0)
A3
d
1
+
(1⇥1)
A3
io
Jm
= 200k Mathematica lines
= zero numerically in any phase space point:
0.
2
0. nf!
0. + --- + 0. Nc + ----- + 0. Nc nf!
2
Nc!
Nc!
Out[1]= ------------------------------------ + !
2!
e!
!
0.
2 0. nf!
0. + --- + 0. Nc + ----- + 0. Nc nf!
2
Nc!
Nc
0!
----------------------------------------------- + O[e]!
e
!
12
⌘
RR,A1 A1
m+2
+ -
2
Example: e e → m(=3) jets at µ = s
NNLO
m
=
d
Poles
Z n
VV
m
d
m
VV
3
+
Z h
2
= Poles
(2⇥0)
A3
+
d
RR,A2
m+2
(2⇥0)
A3
(1⇥1)
A3
+
d
RR,A12
m+2
(1⇥1)
A3
+ Poles
XZ
i
+
Z h
d
1
RV,A1
m+1
+ Finite
d
A
+
⇣Z
(2⇥0)
A3
d
1
+
(1⇥1)
A3
Message:
Z n
XZ
=
d 3VV !+
d
3
A
o
✏=0
J3
indeed finite in d=4 dimensions
13
io
Jm
= 200k Mathematica lines
= zero analytically according to C. Duhr
NNLO
3
⌘
RR,A1 A1
m+2
Application
−
Example: H→bb
NNLO
H!bb̄ (µ
LO
H!bb̄ (µ
= mH ) =

= mH ) 1
⇣↵ ⌘
s
⇡
5.666667
⇣ ↵ ⌘2
s
⇡
29.149 + O(↵s3 )
(µ)/
LO
(mH )
2.0
1.8
Colorful NNLO
1.6
analytic predictions
NNLO
NLO
LO
1.4
1.2
NNLO
NLO
1.0
0.8
1.1
1.0
0.9
0.0
0.5
1.0
1.5
2.0
µ/mH
−
!
Scale dependence of the inclusive decay rate Γ(H -> bb)
analytic: K.G. Chetyrkin hep-ph/9608318
15
Constrained subtractions
Can constrain subtractions
Constrained subtractions
We can constrain subtractions to near singular regions: α0 ∈ (0, 1]
We can
subtractions
to near singular regions:
α0singular
∈ (0, 1]
Weconstrain
can constrain
near
! poles subtractions
cancel numerically (α0 = 0.1)
cancel
numerically
! Poles
"0
poles cancel
numerically
(α0 = 0.1!
) (α
= 0.1)
−8
5.4
×
10
VV
A
dσ
+
dσ
=−5
+
"
−8
−3
H→b
b̄
! VV
4
!
5.4 × 10
3.9 × 10
3.3
ϵ × 10
A
dσH→bb̄ +
dσ =
+
#4 ! "+
$3
2
−5
ϵ
ϵ
ϵ
3.1
×
10
!
A
#!"
$
+
Err
dσ
=−4 8.1
−3
−5
5.0
×
10
×
10
3.1
×
10
4
+
+ ϵ
Err
dσA =
!
ϵ4
ϵ3
ϵ2
! results unchanged
!
! results unchanged
clustering at y
Predictions remain theDurham
same:
3.9 × 10−5
3.3 × 10−3
6
+
+
−3
3
2
6.7ϵ × 10
ϵ
+
+ O(1)
−4
ϵ
5.0 × 10
8.1 × 10−3
7
+
7.7ϵ3× 10−2+
2
+
+ O(1)ϵ
ϵ
Durham clustering at ycut = 0.05, µ = mH
cut
3.0 µ = mH
= 0.05,
3.0
2.0
Γ(α0 = 1) 2.5
Γ2 (α0 = 1)
Γ3 (α0 = 1)
Γ4 (α0 = 1) 2.0
1.5
1.5
2.5
[MeV]
1.0
0.5
Γ(α0 = 0.1)
Γ3 (α0 = 0.1)
Γ2 (α0 = 0.1)
Γ4 (α0 = 0.1)
1.0
0.5
0.0
0.0
Γ(α0 = 1)
Γ2 (α0 = 1)
Γ3 (α0 = 1)
Γ4 (α0 = 1)
Γ(α0 = 0.1)
Γ3 (α0 = 0.1)
Γ2 (α0 = 0.1)
Γ4 (α0 = 0.1)
dΓ
d|η1 |
dΓ
d|η1 |
[MeV]
rapidity distribution of the leading
jet in the rest frame of the Higgs
boson. jets are clustered using the
Durham algorithm (flavour blind)
with ycut = 0.05
regions (α0<1)
0.5
1.0
1.5
|η1|
2.0
0.0
0.0
2.5
3.0
0.5
1.0
1.5
2.0
2.5
|η1|
16
Gábor Somogyi | Colorful NNLO | page 26
3.0
ained subtractions
Subtractions
may help efficiency
constrain
to near
singular regions:
α0 ∈ (0, 1]
We subtractions
can constrain
subtractions
near singular
regions (α0<1),
to fewer calls of subtractions:
provedleading
efficiency
sub ⟩
!
!
!
!
α0
timing (rel.)
⟨Nsub ⟩
1
1
52
0.1
0.40
14.5
!
is the average number of subtraction terms computed
⟨Nsub⟩ is the average number of subtraction calls
17
IR safe predictions w flavour-k⊥
At NNLO accuracy the Durham algorithm is not infrared safe if
the jet is tagged because soft gluon splitting can spoil the flavor
of jets
in RR:
18
IR safe predictions w flavour-k⊥
At NNLO accuracy the Durham algorithm is not infrared safe if
the jet is tagged because soft gluon splitting can spoil the flavor
of jets
in RR:
in RR,A2:
18
IR safe predictions w flavour-k⊥
At NNLO accuracy the Durham algorithm is not infrared safe if
the jet is tagged because soft gluon splitting can spoil the flavor
of jets
in RR:
in RR,A2:
Possible solutions
treat the b-quarks massive only in the parts of the
Feynman graphs that contain the gluon splitting into a bquark pair, while keeping mb = 0 in the Hbb vertex
Use flavour-k⊥ algorithm
A. Banfi et al hep-ph/0601139
18
IR safe predictions w flavour-k⊥
Flavour k? clustering at ycut = 0.05
3.0
µ 2 [0.5, 2]mH
LO
NLO
NNLO
2.5
10
4
5
1.5
10
1.0
2
5
5
2
0.5
10
0.0
0.0
µ 2 [0.5, 2]mH
LO
NLO
NNLO
2
d
dp?,b,1
d
[MeV]
d|⌘b,1 |
2.0
Flavour k? clustering at ycut = 0.05
5
6
5
0.5
1.0
1.5
2.0
2.5
3.0
0
|⌘b,1 |
10
20
30
40
50
p?,b,1 [GeV]
rapidity distribution
pT spectrum
of the leading b-jet in the rest frame of the Higgs boson.
jets are clustered using the flavour-k⊥ algorithm with ycut = 0.05
19
60
Conclusions
Conclusions
21
Conclusions
✓
Defined a general subtraction scheme for computing
NNLO fully differential jet cross sections (presently only
for processes with no colored particles in the initial state)
21
Conclusions
✓
Defined a general subtraction scheme for computing
NNLO fully differential jet cross sections (presently only
for processes with no colored particles in the initial state)
✓
Subtractions are
✓
fully local
✓
exact and explicit in color (using color state
formalism)
21
Conclusions
✓
Defined a general subtraction scheme for computing
NNLO fully differential jet cross sections (presently only
for processes with no colored particles in the initial state)
✓
Subtractions are
✓
✓
fully local
✓
exact and explicit in color (using color state
formalism)
Demonstrated the cancellation of ε-poles
✓
✓
analytically (numerically for constrained subtractions)
First application: Higgs-boson decay into a b-quark pair
(combining with production at NNLO in progress)
21