Colorful NNLO – Completely local subtractions for fullydifferential predictions at NNLO
Gabor Somogyi
MTA-DE Particle Physics Research Group,Debrecen
with V. Del Duca, C. Duhr, A. Kardos,
F. Tramontano, Z. Trocsanyi
Radcor-Loopfest 2015, UCLA
June 18, 2015
Gabor Somogyi | Colorful NNLO | page 1
Message
Message: can compute NNLO cross sections like you always thought you would
1. Compute relevant IR factorization formulae
2. Use them to construct general, explicit, local subtractions
3. Integrate subtractions once and for all, verify pole cancellation
4. Apply the generic scheme to specific process
Gabor Somogyi | Colorful NNLO | page 2
Outline
1. The problem
2. The recipe
3. Integrating the subtractions
4. Cancellation of poles
5. Application: H → bb
6. Conclusions and outlook
Gabor Somogyi | Colorful NNLO | page 3
The problem
Gabor Somogyi | Colorful NNLO | page 4
The problem
Consider the NNLO correction to a generic m-jet observable
σNNLO = σRRm+2 + σRV
m+1 + σVVm =
∫
m+2dσRR
m+2Jm+2 +
∫
m+1dσRV
m+1Jm+1 +
∫
m
dσVVm Jm .
◮ matrix elements for σRRm+2 (tree) and σRV
m+1 (1-loop) known for many processes
◮ σVVm (2-loop) know for 4 parton, V+3 parton processes, higher multiplicities are on
the horizon
◮ the three contributions are separately infrared divergent in d = 4 dimensions
Double real
◮ kin. singularities asone or two partonsunresolved: up toO(ǫ−4) poles from PSintegration
◮ no explicit ǫ poles
Real-virtual
◮ kin. singularities asone parton unresolved:up to O(ǫ−2) polesfrom PS integration
◮ explicit ǫ poles up toO(ǫ−2)
Double virtual
◮ kin. singularitiesscreened by jetfunction: PSintegration finite
◮ explicit ǫ poles up toO(ǫ−4)
Gabor Somogyi | Colorful NNLO | page 5
The problem
Consider the NNLO correction to a generic m-jet observable
σNNLO = σRRm+2 + σRV
m+1 + σVVm =
∫
m+2dσRR
m+2Jm+2 +
∫
m+1dσRV
m+1Jm+1 +
∫
m
dσVVm Jm .
KLN theorem
Infrared singularities cancel between real and virtual quantum corrections at the sameorder in perturbation theory, for sufficiently inclusive (i.e. IR safe) observables.
However
How to make this cancellation explicit, so that the various contributions can becomputed numerically? Need a method to deal with implicit poles.
Gabor Somogyi | Colorful NNLO | page 5
Several approaches – why this one?
Colorful NNLO: Completely Local subtRactions for Fully differential predictions at NNLO
◮ general and explicit expressions, including color and flavor(automation, color space notation is used)
◮ fully local counterterms, taking account of all color and spin correlations(mathematical rigor, efficiency)
◮ analytic cancellation of explicit ǫ poles in loop amplitudes(mathematical rigor)
◮ option to constrain subtractions to near singular regions(efficiency, important check)
◮ very algorithmic construction(valid at any order in perturbation theory)
Gabor Somogyi | Colorful NNLO | page 6
The recipe
Gabor Somogyi | Colorful NNLO | page 7
Structure of the NNLO correction
Rewrite the NNLO correction as a sum of three terms
σNNLO = σRRm+2 + σRV
m+1 + σVVm = σNNLO
m+2 + σNNLOm+1 + σNNLO
m
each integrable in four dimensions
σNNLOm+2 =
∫
m+2
{
dσRRm+2Jm+2 − dσ
RR,A2m+2 Jm −
[
dσRR,A1m+2 Jm+1 − dσ
RR,A12m+2 Jm
]}
σNNLOm+1 =
∫
m+1
{[
dσRVm+1 +
∫
1dσ
RR,A1m+2
]
Jm+1 −[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1
]
Jm
}
σNNLOm =
∫
m
{
dσVVm +
∫
2
[
dσRR,A2m+2 − dσ
RR,A12m+2
]
+
∫
1
[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1]}
Jm
Gabor Somogyi | Colorful NNLO | page 8
Structure of the NNLO correction
Rewrite the NNLO correction as a sum of three terms
σNNLO = σRRm+2 + σRV
m+1 + σVVm = σNNLO
m+2 + σNNLOm+1 + σNNLO
m
each integrable in four dimensions
σNNLOm+2 =
∫
m+2
{
dσRRm+2Jm+2 − dσ
RR,A2m+2 Jm −
[
dσRR,A1m+2 Jm+1 − dσ
RR,A12m+2 Jm
]}
σNNLOm+1 =
∫
m+1
{[
dσRVm+1 +
∫
1dσ
RR,A1m+2
]
Jm+1 −[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1
]
Jm
}
σNNLOm =
∫
m
{
dσVVm +
∫
2
[
dσRR,A2m+2 − dσ
RR,A12m+2
]
+
∫
1
[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1]}
Jm
1. dσRR,A2m+2 regularizes the doubly-unresolved limits of dσRR
m+2
Gabor Somogyi | Colorful NNLO | page 8
Structure of the NNLO correction
Rewrite the NNLO correction as a sum of three terms
σNNLO = σRRm+2 + σRV
m+1 + σVVm = σNNLO
m+2 + σNNLOm+1 + σNNLO
m
each integrable in four dimensions
σNNLOm+2 =
∫
m+2
{
dσRRm+2Jm+2 − dσ
RR,A2m+2 Jm −
[
dσRR,A1m+2 Jm+1 − dσ
RR,A12m+2 Jm
]}
σNNLOm+1 =
∫
m+1
{[
dσRVm+1 +
∫
1dσ
RR,A1m+2
]
Jm+1 −[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1
]
Jm
}
σNNLOm =
∫
m
{
dσVVm +
∫
2
[
dσRR,A2m+2 − dσ
RR,A12m+2
]
+
∫
1
[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1]}
Jm
1. dσRR,A2m+2 regularizes the doubly-unresolved limits of dσRR
m+2
2. dσRR,A1m+2 regularizes the singly-unresolved limits of dσRR
m+2
Gabor Somogyi | Colorful NNLO | page 8
Structure of the NNLO correction
Rewrite the NNLO correction as a sum of three terms
σNNLO = σRRm+2 + σRV
m+1 + σVVm = σNNLO
m+2 + σNNLOm+1 + σNNLO
m
each integrable in four dimensions
σNNLOm+2 =
∫
m+2
{
dσRRm+2Jm+2 − dσ
RR,A2m+2 Jm −
[
dσRR,A1m+2 Jm+1 − dσ
RR,A12m+2 Jm
]}
σNNLOm+1 =
∫
m+1
{[
dσRVm+1 +
∫
1dσ
RR,A1m+2
]
Jm+1 −[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1
]
Jm
}
σNNLOm =
∫
m
{
dσVVm +
∫
2
[
dσRR,A2m+2 − dσ
RR,A12m+2
]
+
∫
1
[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1]}
Jm
1. dσRR,A2m+2 regularizes the doubly-unresolved limits of dσRR
m+2
2. dσRR,A1m+2 regularizes the singly-unresolved limits of dσRR
m+2
3. dσRR,A12m+2 accounts for the overlap of dσRR,A1
m+2 and dσRR,A2m+2
Gabor Somogyi | Colorful NNLO | page 8
Structure of the NNLO correction
Rewrite the NNLO correction as a sum of three terms
σNNLO = σRRm+2 + σRV
m+1 + σVVm = σNNLO
m+2 + σNNLOm+1 + σNNLO
m
each integrable in four dimensions
σNNLOm+2 =
∫
m+2
{
dσRRm+2Jm+2 − dσ
RR,A2m+2 Jm −
[
dσRR,A1m+2 Jm+1 − dσ
RR,A12m+2 Jm
]}
σNNLOm+1 =
∫
m+1
{[
dσRVm+1 +
∫
1dσ
RR,A1m+2
]
Jm+1 −[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1
]
Jm
}
σNNLOm =
∫
m
{
dσVVm +
∫
2
[
dσRR,A2m+2 − dσ
RR,A12m+2
]
+
∫
1
[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1]}
Jm
1. dσRR,A2m+2 regularizes the doubly-unresolved limits of dσRR
m+2
2. dσRR,A1m+2 regularizes the singly-unresolved limits of dσRR
m+2
3. dσRR,A12m+2 accounts for the overlap of dσRR,A1
m+2 and dσRR,A2m+2
4. dσRV,A1m+1 regularizes the singly-unresolved limits of dσRV
m+1
Gabor Somogyi | Colorful NNLO | page 8
Structure of the NNLO correction
Rewrite the NNLO correction as a sum of three terms
σNNLO = σRRm+2 + σRV
m+1 + σVVm = σNNLO
m+2 + σNNLOm+1 + σNNLO
m
each integrable in four dimensions
σNNLOm+2 =
∫
m+2
{
dσRRm+2Jm+2 − dσ
RR,A2m+2 Jm −
[
dσRR,A1m+2 Jm+1 − dσ
RR,A12m+2 Jm
]}
σNNLOm+1 =
∫
m+1
{[
dσRVm+1 +
∫
1dσ
RR,A1m+2
]
Jm+1 −[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1
]
Jm
}
σNNLOm =
∫
m
{
dσVVm +
∫
2
[
dσRR,A2m+2 − dσ
RR,A12m+2
]
+
∫
1
[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1]}
Jm
1. dσRR,A2m+2 regularizes the doubly-unresolved limits of dσRR
m+2
2. dσRR,A1m+2 regularizes the singly-unresolved limits of dσRR
m+2
3. dσRR,A12m+2 accounts for the overlap of dσRR,A1
m+2 and dσRR,A2m+2
4. dσRV,A1m+1 regularizes the singly-unresolved limits of dσRV
m+1
5. (∫
1 dσRR,A1m+2 )
A1 regularizes the singly-unresolved limit of∫
1 dσRR,A1m+2
Gabor Somogyi | Colorful NNLO | page 8
Use known ingredients
Collinear and soft factorization of QCD matrix elements at NNLO known
◮ Tree level 3-parton splitting functions and double soft gg and qq currents
(Campbell, Glover 1997; Catani, Grazzini 1998;Del Duca, Frizzo, Maltoni 1999; Kosower 2002)
◮ One-loop 2-parton splitting functions and soft gluon current
(Bern, Dixon, Dunbar, Kosower 1994; Bern, Del Duca, Kilgore,Schmidt 1998-9; Kosower, Uwer 1999; Catani, Grazzini 2000;
Kosower 2003)
Gabor Somogyi | Colorful NNLO | page 9
Use known ingredients
Collinear and soft factorization of QCD matrix elements at NNLO known
◮ Tree level 3-parton splitting functions and double soft gg and qq currents
(Campbell, Glover 1997; Catani, Grazzini 1998;Del Duca, Frizzo, Maltoni 1999; Kosower 2002)
◮ One-loop 2-parton splitting functions and soft gluon current
(Bern, Dixon, Dunbar, Kosower 1994; Bern, Del Duca, Kilgore,Schmidt 1998-9; Kosower, Uwer 1999; Catani, Grazzini 2000;
Kosower 2003)
But note
◮ unresolved regions in phase space overlap
◮ quantities appearing in factorization formulae are only well-defined in the strict limit
Gabor Somogyi | Colorful NNLO | page 9
Colorful NNLO - general features
Construction based on universal IR limit formulae
◮ Altarelli-Parisi splitting functions, soft currents (tree and one-loop, triple APfunctions)
◮ simple and general procedure for matching of limits using physical gauge
◮ extension based on momentum mappings that can be generalized to any number ofunresolved partons
Fully local in color ⊗ spin space
◮ no need to consider the color decomposition of real emission ME’s
◮ azimuthal correlations correctly taken into account in gluon splitting
◮ can check explicitly that the ratio of the sum of counterterms to the real emissioncross section tends to unity in any IR limit
Straightforward to constrain subtractions to near singular regions
◮ gain in efficiency
◮ independence of physical results on phase space cut is strong check
Given completely explicitly for any process with non colored initial stateGabor Somogyi | Colorful NNLO | page 10
Kinematic singularities cancel in RR
Can check the ratio of the double real emission matrix element and the sum of allsubtractions for all IR limits tends to one.
ratio = subtractions/RR
Gabor Somogyi | Colorful NNLO | page 11
Kinematic singularities cancel in RV
Can check the ratio of the real-virtual matrix element and the sum of all subtractions forall IR limits tends to one.
ratio = subtractions/(RV + RR,A1)
Gabor Somogyi | Colorful NNLO | page 12
RR and RV contributions finite
Regularized RR and RV contributions finite, can be computed by standard MCtechniques. Implementation for general m in progress ⇒ see Adam Kardos’ talk
NNLO∗ = B+ R+ V+ RR+ RV
Gabor Somogyi | Colorful NNLO | page 13
Integrating the subtractions
Gabor Somogyi | Colorful NNLO | page 14
Integrating the subtractions
Momentum mappings used to define the counterterms
{p}n+pR
−→ {p}n ⇒ dφn+p({p};Q) = dφn({p}(R)n ;Q)[dp
(R)p,n ]
◮ implement exact momentum conservation, recoil distributed democratically (can begeneralized to any p)
◮ different collinear and soft mappings (R labels precise limit)
◮ exact factorization of phase space
Counterterms are products (in color and spin space) of
◮ factorized ME’s independent of variables in [dp(R)p,n ]
◮ singular factors (AP functions, soft currents), to be integrated over [dp(R)p,n ]
XR ({p}n+p) =(
8παsµ2ǫ)p
SingR (p(R)p )⊗ |M
(0)n ({p}
(R)n )|2
Can compute once and for all the integral over unresolved partons
∫
p
XR ({p}n+p) =(
8παsµ2ǫ)p
[ ∫
p
SingR (p(R)p )
]
⊗ |M(0)n ({p}
(R)n )|2
Gabor Somogyi | Colorful NNLO | page 15
List of basic integrals
Int status
I(k)1C ,0
✔
I(k)1C ,1
✔
I(k)1C ,2
✔
I(k)1C ,3
✔
I(k)1C ,4
✔
I(k,l)1C ,5
✔
I(k,l)1C ,6
✔
I(k)1C ,7
✔
I1C ,8 ✔
Int status
I(k)12S ,1
✔
I(k)12S ,2
✔
I(k)12S ,3
✔
I(k)12S ,4
✔
I(k)12S ,5
✔
I12S ,6 ✔
I12S ,7 ✔
I12S ,8 ✔
I12S ,9 ✔
I12S ,10 ✔
I12S ,11 ✔
I12S ,12 ✔
I12S ,13 ✔
Int status
I1S ,0 ✔
I1S ,1 ✔
I1S ,2 ✔
I(k)1S ,3
✔
I1S ,4 ✔
I1S ,5 ✔
I1S ,6 ✔
I1S ,7 ✔
Int status
I(k)12CS ,1
✔
I12CS ,2 ✔
I12CS ,3 ✔
Int status
I1CS ,0 ✔
I1CS ,1 ✔
I(k)1CS ,2
✔
I1CS ,3 ✔
I1CS ,4 ✔
Int status
I(j,k,l,m)2C ,1
✔
I(j,k,l,m)2C ,2
✔
I(j,k,l,m)2C ,3
✔
I(j,k,l,m)2C ,4
✔
I(−1,−1,−1,−1)2C ,5
✔
I(k,l)2C ,6
✔
Int status
I(k,l)12C ,1
✔
I(k,l)12C ,2
✔
I(k)12C ,3
✔
I(k,l)12C ,4
✔
I(k)12C ,5
✔
I(k)12C ,6
✔
I(k)12C ,7
✔
I(k)12C ,8
✔
I(k)12C ,9
✔
I(k)12C ,10
✔
Int status
I(k)2CS ,1
✔
I(k)2CS ,2
✔
I(2)2CS ,2
✔
I(k)2CS ,3
✔
I(k)2CS ,4
✔
I(k)2CS ,5
✔
Int status
I2S ,1 ✔
I2S ,2 ✔
I2S ,3 ✔
I2S ,4 ✔
I2S ,5 ✔
I2S ,6 ✔
I2S ,7 ✔
I2S ,8 ✔
I2S ,9 ✔
I2S ,10 ✔
I2S ,11 ✔
I2S ,12 ✔
I2S ,13 ✔
I2S ,14 ✔
I2S ,15 ✔
I2S ,16 ✔
I2S ,17 ✔
I2S ,18 ✔
I2S ,19 ✔
I2S ,20 ✔
I2S ,21 ✔
I2S ,22 ✔
I2S ,23 ✔
✔: pole coefficients known analytically, finite numerically
Gabor Somogyi | Colorful NNLO | page 16
An example
The double soft subtraction term leads to the following integral, among others:
I2S ,2(Yik,Q ; ǫ, y0, d′0) = −
4Γ4(1− ǫ)
πΓ2(1 − ǫ)
By0 (−2ǫ, d ′0)
ǫYik,Q
∫ y0
0dy y−1−2ǫ(1− y)d
′0−1+ǫ
×
∫ 1
−1d(cos ϑ) (sin ϑ)−2ǫ
∫ 1
−1d(cosϕ) (sinϕ)−1−2ǫ
[
f (ϑ, ϕ; 0)]−1[
f (ϑ,ϕ;Yik,Q)]−1
×[
Y (y , ϑ, ϕ;Yik,Q)]−ǫ
2F1
(
− ǫ,−ǫ, 1− ǫ,1− Y (y , ϑ, ϕ;Yik,Q))
where
f (ϑ, ϕ;Yik,Q) = 1− 2√
Yik,Q(1 − Yik,Q) sinϑ cosϕ− (1 − 2Yik,Q)χ cos ϑ
Y (y , ϑ, ϕ;χ) =4(1 − y)Yik,Q
[2(1 − y) + y f (ϑ, ϕ; 0)][2(1− y) + y f (ϑ, ϕ;Yik,Q)]
Gabor Somogyi | Colorful NNLO | page 17
An example
This integral is equal to (y0 = 1, d ′0 = 3− 3ǫ)
I2S ,2(Y ; ǫ, 1, 3− 3ǫ) =
=1
2ǫ4−
1
ǫ3
[
ln(Y ) − 3
]
+1
ǫ2
[
2Li2(1− Y ) + ln2(Y )− π2 −
(
2
1− Y
−1
2(1− Y )2+
9
2
)
ln(Y ) +1
2(1− Y )+ 16
]
+1
ǫ
[
5
3
(
18Li3(1 − Y )
5+
6Li3(Y )
5
−6Li2(1− Y ) ln(Y )
5−
2
5ln3(Y ) +
3
5ln(1− Y ) ln2(Y ) + π2 ln(Y )−
78ζ3
5
)
+
(
3
1− Y−
3
4(1 − Y )2+
15
4
)
(
2Li2(1− Y ) + ln2(Y ))
− 6π2 −
(
27
2(1 − Y )
−13
4(1− Y )2+
91
4
)
ln(Y ) +19
4(1 − Y )+
163
2
]
+O(ǫ0)
◮ Note the Y → 1 limit is finite
limY→1
I2S ,2(Y ; ǫ, 1, 3− 3ǫ) =1
2ǫ4+
3
ǫ3+
1
ǫ2
(
71
4− π2
)
+1
ǫ
(
393
4− 6π2 − 24ζ3
)
+O(ǫ0)
◮ Finite term is computed numerically
Gabor Somogyi | Colorful NNLO | page 17
Solving the integrals
Strategy for computing the master integrals
1. write phase space in terms ofangles and energies
2. angular integrals in terms ofMellin-Barnes representations
3. resolve the ǫ poles by analyticcontinuation
4. MB integrals to Euler-typeintegrals, pole coefficients are finiteparametric integrals
1. choose explicit parametrization ofphase space
2. write the parametric integralrepresentation in chosen variables
3. resolve the ǫ poles by sectordecomposition
4. pole coefficients are finiteparametric integrals
5. evaluate the parametric integrals in terms of multiple polylogs
6. simplify result (optional)
Gabor Somogyi | Colorful NNLO | page 18
Cancellation of poles
Gabor Somogyi | Colorful NNLO | page 19
Integrated approximate cross sections
Recall the NNLO correction is a sum of three terms
σNNLO = σRRm+2 + σRV
m+1 + σVVm = σNNLO
m+2 + σNNLOm+1 + σNNLO
m
each integrable in four dimensions
σNNLOm+2 =
∫
m+2
{
dσRRm+2Jm+2 − dσ
RR,A2m+2 Jm −
[
dσRR,A1m+2 Jm+1 − dσ
RR,A12m+2 Jm
]}
σNNLOm+1 =
∫
m+1
{[
dσRVm+1 +
∫
1dσ
RR,A1m+2
]
Jm+1 −[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1]
Jm
}
σNNLOm =
∫
m
{
dσVVm +
∫
2
[
dσRR,A2m+2 − dσ
RR,A12m+2
]
+
∫
1
[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1]}
Jm
Integrated approximate cross sections
◮ After summing over unobserved flavors, all integrated approximate cross sections canbe written as products (in color space) of various insertion operators with lowerpoint cross sections.
◮ Can be computed once and for all (though admittedly lots of tedious work).
◮ Poles are computed analytically, finite part numerically.
Gabor Somogyi | Colorful NNLO | page 20
Poles cancel: VV contribution finite
After adding all integrated approximate cross sections the double virtual contributionmust be finite in ǫ.
σNNLOm =
∫
m
{
dσVVm +
∫
2
[
dσRR,A2m+2 − dσ
RR,A12m+2
]
+
∫
1
[
dσRV,A1m+1 +
(
∫
1dσ
RR,A1m+2
)
A1
]}
Jm
◮ Have checked the cancellation of the 1ǫ4
and 1ǫ3
poles analytically for any number ofjets (i.e., with m symbolic).
◮ Have checked m = 2 (e+e− → qq, H → bb) explicitly and we find that all polescancel.
◮ Have checked m = 3 (e+e− → qqg) explicitly and we find that all poles cancel.
Gabor Somogyi | Colorful NNLO | page 21
Example: H → bb
The double virtual contribution has the following pole structure (µ2 = m2H)
dσVV
H→bb=
(
αs(µ2)
2π
)2
dσB
H→bb
{
2C2F
ǫ4+
(
11CACF
4+ 6C2
F −CFnf
2
)
1
ǫ3
+
[(
8
9+
π2
12
)
CACF +
(
17
2− 2π2
)
C2F −
2CFnf
9
]
1
ǫ2
+
[(
−961
216+
13ζ3
2
)
CACF +
(
109
8− 2π2 − 14ζ3
)
C2F+
65CFnf
108
]
1
ǫ
}
(Anastasiou, Herzog, Lazopoulos, arXiv:0111.2368)
Gabor Somogyi | Colorful NNLO | page 22
Example: H → bb
The double virtual contribution has the following pole structure (µ2 = m2H)
dσVV
H→bb=
(
αs(µ2)
2π
)2
dσB
H→bb
{
2C2F
ǫ4+
(
11CACF
4+ 6C2
F −CFnf
2
)
1
ǫ3
+
[(
8
9+
π2
12
)
CACF +
(
17
2− 2π2
)
C2F −
2CFnf
9
]
1
ǫ2
+
[(
−961
216+
13ζ3
2
)
CACF +
(
109
8− 2π2 − 14ζ3
)
C2F+
65CFnf
108
]
1
ǫ
}
(Anastasiou, Herzog, Lazopoulos, arXiv:0111.2368)
The sum of the integrated approximate cross sections gives (µ2 = m2H)
∑
∫
dσA =
(
αs(µ2)
2π
)2
dσB
H→bb
{
−2C2F
ǫ4+
(
−11CACF
4− 6C2
F +CFnf
2
)
1
ǫ3
+
[(
−8
9−
π2
12
)
CACF +
(
−17
2+ 2π2
)
C2F+
2CFnf
9
]
1
ǫ2
+
[(
961
216−
13ζ3
2
)
CACF +
(
−109
8+ 2π2 + 14ζ3
)
C2F −
65CFnf
108
]
1
ǫ
}
(Del Duca, Duhr, GS, Tramontano, Trocsanyi,arXiv:1501.07226)
Gabor Somogyi | Colorful NNLO | page 22
Example: e+e− → 3 jets
The double virtual contribution has the following pole structure (µ2 = s)
dσVV3 = Poles
(
A(2×0)3 + A
(1×1)3
)
+ F inite(
A(2×0)3 + A
(1×1)3
)
where
Poles(
A(2×0)3 + A
(1×1)3
)
= 2
[
−(
I(1)qqg (ǫ)
)2−
β0
ǫI(1)qqg (ǫ)
+ e−ǫγΓ(1 − 2ǫ)
Γ(1 − ǫ)
(
β0
ǫ+ K
)
I(1)qqg (2ǫ) + H
(2)qqg
]
A03(1q , 3g , 2q)
+ 2I(1)qqg (ǫ)A
1×03 (1q , 3g , 2q)
with
H(2)qqg =
eǫγ
4ǫΓ(1 − ǫ)
[(
4ζ3 +589
432−
11π2
72
)
Nc +
(
−1
2ζ3 −
41
54−
π2
48
)
+
(
− 3ζ3 −3
16+
π2
4
)
1
Nc
+
(
−19
18+
π2
36
)
Ncnf +
(
−1
54−
π2
24
)
nf
Nc
+5
27n2f
]
(Gehrmann-De Ridder, Gehrmann, Glover, Heinrich,arXiv:0710.0346)
Gabor Somogyi | Colorful NNLO | page 23
Example: e+e− → 3 jets
The double virtual contribution has the following pole structure (µ2 = s)
dσVV3 = Poles
(
A(2×0)3 + A
(1×1)3
)
+ F inite(
A(2×0)3 + A
(1×1)3
)
Adding the sum of the integrated approximate cross sections gives
Poles(
A(2×0)3 + A
(1×1)3
)
+ Poles∑
∫
dσA = 117k terms
Gabor Somogyi | Colorful NNLO | page 23
Example: e+e− → 3 jets
The double virtual contribution has the following pole structure (µ2 = s)
dσVV3 = Poles
(
A(2×0)3 + A
(1×1)3
)
+ F inite(
A(2×0)3 + A
(1×1)3
)
Adding the sum of the integrated approximate cross sections gives
Poles(
A(2×0)3 + A
(1×1)3
)
+ Poles∑
∫
dσA = 117k terms
◮ zero numerically in any phase space point
◮ zero analytically after simplification using symbol technology (C. Duhr)
Gabor Somogyi | Colorful NNLO | page 23
Application: H → bb
Gabor Somogyi | Colorful NNLO | page 24
Higgs decay to b-quarks
Consider H → bb decay at NNLO:
◮ admittedly the simplest case
◮ but this just amounts to having to sum less terms in general formulae
Inclusive decay rate
Gabor Somogyi | Colorful NNLO | page 25
Higgs decay to b-quarks
Consider H → bb decay at NNLO:
◮ admittedly the simplest case
◮ but this just amounts to having to sum less terms in general formulae
Differential distributions
◮ pseudorapidity of highest energy jet (right) and leading jet energy (left)
Gabor Somogyi | Colorful NNLO | page 25
Constrained subtractions
We can constrain subtractions to near singular regions: α0 ∈ (0, 1]
◮ poles cancel numerically (α0 = 0.1)
dσVV
H→bb+
∑
∫
dσA =5.4× 10−8
ǫ4+
3.9× 10−5
ǫ3+
3.3× 10−3
ǫ2+
6.7× 10−3
ǫ+O(1)
Err
(
∑
∫
dσA
)
=3.1× 10−5
ǫ4+
5.0× 10−4
ǫ3+
8.1× 10−3
ǫ2+
7.7× 10−2
ǫ+O(1)
◮ results unchanged
0.0
0.5
1.0
1.5
2.0
2.5
3.0
dΓ
d|η
1|[M
eV]
0.0 0.5 1.0 1.5 2.0 2.5 3.0
|η1|
Durham clustering at ycut = 0.05, µ = mH
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)
Gabor Somogyi | Colorful NNLO | page 26
Constrained subtractions
We can constrain subtractions to near singular regions: α0 ∈ (0, 1]
◮ improved efficiency
α0 1 0.1
timing (rel.) 1 0.40
〈Nsub〉 52 14.5
〈Nsub〉 is the average number of subtraction terms computed
Gabor Somogyi | Colorful NNLO | page 26
Conclusions and outlook
Gabor Somogyi | Colorful NNLO | page 27
Conclusions and outlook
Colorful NNLO framework
◮ Completely Local subtRactions for Fully differential predictions at NNLO
◮ construction of subtraction terms based on IR limit formulae
◮ analytic integration of subtraction terms is feasible with modern integrationtechniques
◮ demonstrated cancellation of ǫ poles for m = 2 and m = 3
◮ worked out in full detail for processes with no colored particles in the initial state
First application: Higgs boson decay into a b and anti-b quark
Next steps
◮ e+e− → 3 jets is almost finished
◮ extension to hadronic initial states conceptually understood and on the way
Gabor Somogyi | Colorful NNLO | page 28