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Perturbative QCDat four and �ve loops
Takahiro Ueda
Nikhef Theory Group
Based onJ. Davies, A. Vogt, B. Ruijl, TU, J.A.M. Vermaseren, NPB915 (2017) 335 [arXiv:1610.07477]F. Herzog, B. Ruijl, TU, J.A.M. Vermaseren, A. Vogt, JHEP 1702 (2017) 090 [arXiv:1701.01404]B. Ruijl, TU, J.A.M. Vermaseren, A. Vogt, JHEP 1706 (2017) 040 [arXiv:1703.08532]B. Ruijl, TU, J.A.M. Vermaseren, arXiv:1704.06650F. Herzog, B. Ruijl, TU, J.A.M. Vermaseren, A. Vogt, JHEP 1708 (2017) 113 [arXiv:1707.01044]S. Moch, B. Ruijl, TU, J.A.M. Vermaseren, A. Vogt, to appear in JHEP [arXiv:1707.08315]
4L5L
4L 5L4L5L4L
29 September 2017, Nikhef, HPP meeting1/27
Perturbative QCDat four and �ve loops
Takahiro Ueda
Nikhef Theory Group
Based onJ. Davies, A. Vogt, B. Ruijl, TU, J.A.M. Vermaseren, NPB915 (2017) 335 [arXiv:1610.07477]F. Herzog, B. Ruijl, TU, J.A.M. Vermaseren, A. Vogt, JHEP 1702 (2017) 090 [arXiv:1701.01404]B. Ruijl, TU, J.A.M. Vermaseren, A. Vogt, JHEP 1706 (2017) 040 [arXiv:1703.08532]B. Ruijl, TU, J.A.M. Vermaseren, arXiv:1704.06650F. Herzog, B. Ruijl, TU, J.A.M. Vermaseren, A. Vogt, JHEP 1708 (2017) 113 [arXiv:1707.01044]S. Moch, B. Ruijl, TU, J.A.M. Vermaseren, A. Vogt, to appear in JHEP [arXiv:1707.08315]
4L5L
4L 5L4L5L4L
29 September 2017, Nikhef, HPP meeting1/27
Outline
The basic building block in this talk is computing4-loop massless propagator diagrams
2/27
Outline
(phenomenological) Motivation
Forcer program (4-loops)and applications
5-loop calculations(theoretical/math fun)
3/27
Precision physics at the LHC
No signals beyond the SMBreakthrough would come from precision physics(?)
NNLO QCD corrections calculated for many processes
Even N3LO, e.g., inclusive gg→ HAnastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Lazopoulos, Mistlberger ’16
N3LO inclusive DIS: Moch, Vermaseren, Vogt ’05; for F3 ’08N3LO inclusive VBF Higgs: Dreyer, Karlberg ’16
4/27
Precision physics at the LHC
No signals beyond the SMBreakthrough would come from precision physics(?)
NNLO QCD corrections calculated for many processes
Even N3LO, e.g., inclusive gg→ HAnastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Lazopoulos, Mistlberger ’16
N3LO inclusive DIS: Moch, Vermaseren, Vogt ’05; for F3 ’08N3LO inclusive VBF Higgs: Dreyer, Karlberg ’16
4/27
Precision physics at the LHC
No signals beyond the SMBreakthrough would come from precision physics(?)
NNLO QCD corrections calculated for many processes
Even N3LO, e.g., inclusive gg→ HAnastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Lazopoulos, Mistlberger ’16
N3LO inclusive DIS: Moch, Vermaseren, Vogt ’05; for F3 ’08N3LO inclusive VBF Higgs: Dreyer, Karlberg ’16
4/27
Missing N3LO PDFsN3LO gg→ H computed with NNLO PDFs
δ(scale) δ(trunc) δ(PDF-TH) δ(EW) δ(t,b, c) δ(1/mt)
+0.10 pb−1.15 pb ±0.18 pb ±0.56 pb ±0.49 pb ±0.40 pb ±0.49 pb+0.21%−2.37% ±0.37% ±1.16% ±1% ±0.83% ±1%
Anastasiou et al. ’16
missingN3LO PDFs
Ideally N3LO analyses must be performedwith N3LO PDFs
4-loop splitting functions?
5/27
Missing N3LO PDFsN3LO gg→ H computed with NNLO PDFs
δ(scale) δ(trunc) δ(PDF-TH) δ(EW) δ(t,b, c) δ(1/mt)
+0.10 pb−1.15 pb ±0.18 pb ±0.56 pb ±0.49 pb ±0.40 pb ±0.49 pb+0.21%−2.37% ±0.37% ±1.16% ±1% ±0.83% ±1%
Anastasiou et al. ’16
missingN3LO PDFs
Ideally N3LO analyses must be performedwith N3LO PDFs
4-loop splitting functions?
5/27
Missing N3LO PDFsN3LO gg→ H computed with NNLO PDFs
δ(scale) δ(trunc) δ(PDF-TH) δ(EW) δ(t,b, c) δ(1/mt)
+0.10 pb−1.15 pb ±0.18 pb ±0.56 pb ±0.49 pb ±0.40 pb ±0.49 pb+0.21%−2.37% ±0.37% ±1.16% ±1% ±0.83% ±1%
Anastasiou et al. ’16
missingN3LO PDFs
Ideally N3LO analyses must be performedwith N3LO PDFs
4-loop splitting functions?
5/27
Computing splitting functionsN-th Mellin moment of splitting function Pab(x)
γab(N) = −∫ 1
0dx xN−1Pab(x)
(i) Matrix elements of (leading-twist) DIS operatorsψ̄γ{µ1Dµ2 ...DµN}ψ
Q Q(ii) Partonic forward scattering
Gorishnii, Larin, Tkachev ’83;Gorishnii, Larin ’87
Q{µ1 ...QµN}
N!
∂N
∂Pµ1 ... ∂PµN
P P
Q Q
∣∣∣∣∣P=0
In the both cases, compute (poles of) masslesspropagator-type diagrams with N-dependence ∑ Q N
3-loop full-N (hence full-x): Moch, Vermaseren, Vogt ’04; for polarized ’146/27
Computing splitting functionsN-th Mellin moment of splitting function Pab(x)
γab(N) = −∫ 1
0dx xN−1Pab(x)
(i) Matrix elements of (leading-twist) DIS operatorsψ̄γ{µ1Dµ2 ...DµN}ψ
Q Q(ii) Partonic forward scattering
Gorishnii, Larin, Tkachev ’83;Gorishnii, Larin ’87
Q{µ1 ...QµN}
N!
∂N
∂Pµ1 ... ∂PµN
P P
Q Q
∣∣∣∣∣P=0
In the both cases, compute (poles of) masslesspropagator-type diagrams with N-dependence ∑ Q N
3-loop full-N (hence full-x): Moch, Vermaseren, Vogt ’04; for polarized ’146/27
Computing splitting functionsN-th Mellin moment of splitting function Pab(x)
γab(N) = −∫ 1
0dx xN−1Pab(x)
(i) Matrix elements of (leading-twist) DIS operatorsψ̄γ{µ1Dµ2 ...DµN}ψ
Q Q(ii) Partonic forward scattering
Gorishnii, Larin, Tkachev ’83;Gorishnii, Larin ’87
Q{µ1 ...QµN}
N!
∂N
∂Pµ1 ... ∂PµN
P P
Q Q
∣∣∣∣∣P=0
In the both cases, compute (poles of) masslesspropagator-type diagrams with N-dependence ∑ Q N
3-loop full-N (hence full-x): Moch, Vermaseren, Vogt ’04; for polarized ’146/27
Computing splitting functionsN-th Mellin moment of splitting function Pab(x)
γab(N) = −∫ 1
0dx xN−1Pab(x)
(i) Matrix elements of (leading-twist) DIS operatorsψ̄γ{µ1Dµ2 ...DµN}ψ
Q Q(ii) Partonic forward scattering
Gorishnii, Larin, Tkachev ’83;Gorishnii, Larin ’87
Q{µ1 ...QµN}
N!
∂N
∂Pµ1 ... ∂PµN
P P
Q Q
∣∣∣∣∣P=0
In the both cases, compute (poles of) masslesspropagator-type diagrams with N-dependence ∑ Q N
3-loop full-N (hence full-x): Moch, Vermaseren, Vogt ’04; for polarized ’146/27
Approximation from �xed-N results
images/pns2pf0-eps-converted-to.pdf
3-loop: exact vs. approx.(non-singlet, nf -independent part)from Moch, Vermaseren, Vogt, NPB 688 (2004) 101
[arXiv:hep-ph/0403192]
Full N-dependence in 4-loops:out of reach
Approximation from�xed N = 2, 4, 6, ... results:promissingThe more N the more accurate
Very time-comsuming for high NRequires e�ciency
7/27
Approximation from �xed-N results
600
800
1000
1200
0 0.2 0.4 0.6 0.8 1
x
(1−x) P (2)
(x)+,0
exact
N = 2...12
x
(1−x) P (2)
(x)+,0
exact
Lx
NLx
N2Lx
N3Lx
-10000
0
10000
20000
10-5
10-4
10-3
10-2
10-1
1
3-loop: exact vs. approx.(non-singlet, nf -independent part)from Moch, Vermaseren, Vogt, NPB 688 (2004) 101
[arXiv:hep-ph/0403192]
Full N-dependence in 4-loops:out of reach
Approximation from�xed N = 2, 4, 6, ... results:promissingThe more N the more accurate
Very time-comsuming for high NRequires e�ciency
7/27
Approximation from �xed-N results
600
800
1000
1200
0 0.2 0.4 0.6 0.8 1
x
(1−x) P (2)
(x)+,0
exact
N = 2...12
x
(1−x) P (2)
(x)+,0
exact
Lx
NLx
N2Lx
N3Lx
-10000
0
10000
20000
10-5
10-4
10-3
10-2
10-1
1
3-loop: exact vs. approx.(non-singlet, nf -independent part)from Moch, Vermaseren, Vogt, NPB 688 (2004) 101
[arXiv:hep-ph/0403192]
Full N-dependence in 4-loops:out of reach
Approximation from�xed N = 2, 4, 6, ... results:promissingThe more N the more accurate
Very time-comsuming for high NRequires e�ciency
7/27
Outline
(phenomenological) Motivation: 4-loop splitting functions
Forcer program (4-loops)and applications to splitting functions
5-loop calculations(theoretical/math fun)
8/27
Feynman integral calculus
(1) Generate Feynman diagrams
(2) Apply Feynman rules
(3) Multiply projection operator to scalarize integrals
At this point, a bunch of (scalar) Feynman integralsThe standard way to proceed is:
(4) Reduce many integrals toa small set of master integrals
(5) Evaluate the master integrals
4-loop massless propagator master integrals are known:Baikov, Chetyrkin ’10; Lee, Smirnov, Smirnov ’11
9/27
Feynman integral calculus
(1) Generate Feynman diagrams
(2) Apply Feynman rules
(3) Multiply projection operator to scalarize integrals
At this point, a bunch of (scalar) Feynman integralsThe standard way to proceed is:
(4) Reduce many integrals toa small set of master integrals
(5) Evaluate the master integrals
4-loop massless propagator master integrals are known:Baikov, Chetyrkin ’10; Lee, Smirnov, Smirnov ’11
9/27
Feynman integral calculus
(1) Generate Feynman diagrams
(2) Apply Feynman rules
(3) Multiply projection operator to scalarize integrals
At this point, a bunch of (scalar) Feynman integralsThe standard way to proceed is:
(4) Reduce many integrals toa small set of master integrals
(5) Evaluate the master integrals
4-loop massless propagator master integrals are known:Baikov, Chetyrkin ’10; Lee, Smirnov, Smirnov ’11
Problem!!!
9/27
Integration-by-parts identities (IBPs)The divergence theorem in the D-dimensionalspace-time Chetyrkin, Tkachov ’81∫
dDk ∂
∂kµ fµ = 0
(We work in D = 4− 2ε space-time)
Give linear relations among Feynman integralsExample:
F(n1,n2) =
∫dDp 1
(p2)n1[(Q− p)2]n2 QQ
n1
n2Q2 = 1
(D− 2n1 − n2)F(n1,n2)− n2F(n1 − 1,n2 + 1) + n2F(n1,n2 + 1) = 0
(D− n1 − 2n2)F(n1,n2)− n1F(n1 + 1,n2 − 1) + n1F(n1 + 1,n2) = 0
10/27
Integration-by-parts identities (IBPs)The divergence theorem in the D-dimensionalspace-time Chetyrkin, Tkachov ’81∫
dDk ∂
∂kµ fµ = 0
(We work in D = 4− 2ε space-time)
Give linear relations among Feynman integralsExample:
F(n1,n2) =
∫dDp 1
(p2)n1[(Q− p)2]n2 QQ
n1
n2Q2 = 1
(D− 2n1 − n2)F(n1,n2)− n2F(n1 − 1,n2 + 1) + n2F(n1,n2 + 1) = 0
(D− n1 − 2n2)F(n1,n2)− n1F(n1 + 1,n2 − 1) + n1F(n1 + 1,n2) = 0
10/27
Parametric reduction rules
One can “solve” the IBPs (by hand) as
F(n1,n2) = −(D− n1 − n2)(D− 2n1 − 2n2 + 2)
(n1 − 1)(D− 2n1)F(n1 − 1,n2)
F(n1,n2) = −(D− n1 − n2)(D− 2n1 − 2n2 + 2)
(n2 − 1)(D− 2n2)F(n1,n2 − 1)
Recursively apply these rules
F(3, 3) =14(D− 3)(D− 5)(D− 8)(D− 10)F(1, 1)
11/27
Parametric reduction rules
One can “solve” the IBPs (by hand) as
F(n1,n2) = −(D− n1 − n2)(D− 2n1 − 2n2 + 2)
(n1 − 1)(D− 2n1)F(n1 − 1,n2)
F(n1,n2) = −(D− n1 − n2)(D− 2n1 − 2n2 + 2)
(n2 − 1)(D− 2n2)F(n1,n2 − 1)
Recursively apply these rules
F(3, 3) =14(D− 3)(D− 5)(D− 8)(D− 10)F(1, 1)
11/27
The Laporta’s algorithmAutomatic IBP reduction by Gaussian eliminationwith ordering of integrals Laporta ’00; many public programs in the market
(1) Generate enough equations by setting n1, n2(n1 , n2) = (3, 2): −2F(2, 3) + 2F(3, 3) + (D− 8)F(3, 2) = 0 −3F(4, 1) + 3F(4, 2) + (D− 7)F(3, 2) = 0
(n1 , n2) = (3, 1): −F(2, 2) + F(3, 2) + (D− 7)F(3, 1) = 0 −3F(4, 0) + 3F(4, 1) + (D− 5)F(3, 1) = 0
(n1 , n2) = (2, 2): −2F(1, 3) + 2F(2, 3) + (D− 6)F(2, 2) = 0 −2F(3, 1) + 2F(3, 2) + (D− 6)F(2, 2) = 0
(n1 , n2) = (2, 1): −F(1, 2) + F(2, 2) + (D− 5)F(2, 1) = 0 −2F(3, 0) + 2F(3, 1) + (D− 4)F(2, 1) = 0
(n1 , n2) = (1, 2): −2F(0, 3) + 2F(1, 3) + (D− 4)F(1, 2) = 0 −F(2, 1) + F(2, 2) + (D− 5)F(1, 2) = 0
(n1 , n2) = (1, 1): −F(0, 2) + F(1, 2) + (D− 3)F(1, 1) = 0 −F(2, 0) + F(2, 1) + (D− 3)F(1, 1) = 0
(n1 , n2) = (0, 3): −3F(−1, 4) + 3F(0, 4) + (D− 3)F(0, 3) = 0 (D− 6)F(0, 3) = 0
(n1 , n2) = (0, 2): −2F(−1, 3) + 2F(0, 3) + (D− 2)F(0, 2) = 0 (D− 4)F(0, 2) = 0
(2) Solve for/eliminate “di�cult” integrals
F(3, 3) =14(D− 3)(D− 5)(D− 8)(D− 10)F(1, 1)
12/27
The Laporta’s algorithmAutomatic IBP reduction by Gaussian eliminationwith ordering of integrals Laporta ’00; many public programs in the market
(1) Generate enough equations by setting n1, n2(n1 , n2) = (3, 2): −2F(2, 3) + 2F(3, 3) + (D− 8)F(3, 2) = 0 −3F(4, 1) + 3F(4, 2) + (D− 7)F(3, 2) = 0
(n1 , n2) = (3, 1): −F(2, 2) + F(3, 2) + (D− 7)F(3, 1) = 0 −3F(4, 0) + 3F(4, 1) + (D− 5)F(3, 1) = 0
(n1 , n2) = (2, 2): −2F(1, 3) + 2F(2, 3) + (D− 6)F(2, 2) = 0 −2F(3, 1) + 2F(3, 2) + (D− 6)F(2, 2) = 0
(n1 , n2) = (2, 1): −F(1, 2) + F(2, 2) + (D− 5)F(2, 1) = 0 −2F(3, 0) + 2F(3, 1) + (D− 4)F(2, 1) = 0
(n1 , n2) = (1, 2): −2F(0, 3) + 2F(1, 3) + (D− 4)F(1, 2) = 0 −F(2, 1) + F(2, 2) + (D− 5)F(1, 2) = 0
(n1 , n2) = (1, 1): −F(0, 2) + F(1, 2) + (D− 3)F(1, 1) = 0 −F(2, 0) + F(2, 1) + (D− 3)F(1, 1) = 0
(n1 , n2) = (0, 3): −3F(−1, 4) + 3F(0, 4) + (D− 3)F(0, 3) = 0 (D− 6)F(0, 3) = 0
(n1 , n2) = (0, 2): −2F(−1, 3) + 2F(0, 3) + (D− 2)F(0, 2) = 0 (D− 4)F(0, 2) = 0
(2) Solve for/eliminate “di�cult” integrals
F(3, 3) =14(D− 3)(D− 5)(D− 8)(D− 10)F(1, 1)
12/27
The Laporta’s algorithmAutomatic IBP reduction by Gaussian eliminationwith ordering of integrals Laporta ’00; many public programs in the market
(1) Generate enough equations by setting n1, n2(n1 , n2) = (3, 2): −2F(2, 3) + 2F(3, 3) + (D− 8)F(3, 2) = 0 −3F(4, 1) + 3F(4, 2) + (D− 7)F(3, 2) = 0
(n1 , n2) = (3, 1): −F(2, 2) + F(3, 2) + (D− 7)F(3, 1) = 0 −3F(4, 0) + 3F(4, 1) + (D− 5)F(3, 1) = 0
(n1 , n2) = (2, 2): −2F(1, 3) + 2F(2, 3) + (D− 6)F(2, 2) = 0 −2F(3, 1) + 2F(3, 2) + (D− 6)F(2, 2) = 0
(n1 , n2) = (2, 1): −F(1, 2) + F(2, 2) + (D− 5)F(2, 1) = 0 −2F(3, 0) + 2F(3, 1) + (D− 4)F(2, 1) = 0
(n1 , n2) = (1, 2): −2F(0, 3) + 2F(1, 3) + (D− 4)F(1, 2) = 0 −F(2, 1) + F(2, 2) + (D− 5)F(1, 2) = 0
(n1 , n2) = (1, 1): −F(0, 2) + F(1, 2) + (D− 3)F(1, 1) = 0 −F(2, 0) + F(2, 1) + (D− 3)F(1, 1) = 0
(n1 , n2) = (0, 3): −3F(−1, 4) + 3F(0, 4) + (D− 3)F(0, 3) = 0 (D− 6)F(0, 3) = 0
(n1 , n2) = (0, 2): −2F(−1, 3) + 2F(0, 3) + (D− 2)F(0, 2) = 0 (D− 4)F(0, 2) = 0
(2) Solve for/eliminate “di�cult” integrals
F(3, 3) =14(D− 3)(D− 5)(D− 8)(D− 10)F(1, 1)
12/27
4-loop IBP reduction?(D − n1 − n4 − 2n8 − n9 − n11)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (−n1)F(n1 + 1, n2 , n3 , n4 , n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 , n12 , n13 , n14) + (−n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 , n12 , n13 , n14)
+ (−n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 − 1, n10 , n11 , n12 , n13 , n14) + (−n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 − 1, n12 , n13 , n14)
+ (−n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 − 1, n14) + (−n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 , n12 , n13 , n14)
+ (−n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 , n9 , n10 , n11 − 1, n12 , n13 , n14) + (−n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 − 1, n14)
+ (−n5)F(n1 , n2 , n3 , n4 − 1, n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14) + (−n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 − 1, n12 , n13 , n14)
+ (−n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 − 1, n14) + (−n9)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 − 1, n9 + 1, n10 , n11 , n12 , n13 , n14)
+ (−n11)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 + 1, n12 , n13 , n14) + (−n11)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 + 1, n12 , n13 − 1, n14)
+ (−n12)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 − 1, n12 + 1, n13 , n14) + (−n12)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 + 1, n13 − 1, n14)
+ (−n14)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 , n12 , n13 , n14 + 1) + (−n14)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14 + 1)+ (n1)F(n1 + 1, n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14) + (n3)F(n1 − 1, n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n3)F(n1 , n2 − 1, n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14) + (n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 − 1, n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n4)F(n1 − 1, n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14) + (n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 − 1, n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 − 1, n13 , n14) + (n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 − 1, n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 , n12 − 1, n13 , n14) + (n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n9)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 − 1, n8 , n9 + 1, n10 , n11 , n12 , n13 , n14) + (n11)F(n1 , n2 − 1, n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 + 1, n12 , n13 , n14)
+ (n11)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 + 1, n12 , n13 , n14) + (n12)F(n1 − 1, n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 + 1, n13 , n14)
+ (n12)F(n1 , n2 − 1, n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 + 1, n13 , n14) + (n14)F(n1 − 1, n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14 + 1) = 0
+ other 19 IBPs
Parametric reduction rules by hand: no wayLaporta or other generic algorithms: very slow
13/27
4-loop IBP reduction?(D − n1 − n4 − 2n8 − n9 − n11)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (−n1)F(n1 + 1, n2 , n3 , n4 , n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 , n12 , n13 , n14) + (−n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 , n12 , n13 , n14)
+ (−n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 − 1, n10 , n11 , n12 , n13 , n14) + (−n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 − 1, n12 , n13 , n14)
+ (−n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 − 1, n14) + (−n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 , n12 , n13 , n14)
+ (−n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 , n9 , n10 , n11 − 1, n12 , n13 , n14) + (−n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 − 1, n14)
+ (−n5)F(n1 , n2 , n3 , n4 − 1, n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14) + (−n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 − 1, n12 , n13 , n14)
+ (−n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 − 1, n14) + (−n9)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 − 1, n9 + 1, n10 , n11 , n12 , n13 , n14)
+ (−n11)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 + 1, n12 , n13 , n14) + (−n11)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 + 1, n12 , n13 − 1, n14)
+ (−n12)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 − 1, n12 + 1, n13 , n14) + (−n12)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 + 1, n13 − 1, n14)
+ (−n14)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 − 1, n9 , n10 , n11 , n12 , n13 , n14 + 1) + (−n14)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14 + 1)+ (n1)F(n1 + 1, n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14) + (n3)F(n1 − 1, n2 , n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n3)F(n1 , n2 − 1, n3 + 1, n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14) + (n3)F(n1 , n2 , n3 + 1, n4 , n5 , n6 , n7 − 1, n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n4)F(n1 − 1, n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14) + (n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 − 1, n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n4)F(n1 , n2 , n3 , n4 + 1, n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 − 1, n13 , n14) + (n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 − 1, n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 , n12 − 1, n13 , n14) + (n5)F(n1 , n2 , n3 , n4 , n5 + 1, n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14)
+ (n9)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 − 1, n8 , n9 + 1, n10 , n11 , n12 , n13 , n14) + (n11)F(n1 , n2 − 1, n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 + 1, n12 , n13 , n14)
+ (n11)F(n1 , n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 + 1, n12 , n13 , n14) + (n12)F(n1 − 1, n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 + 1, n13 , n14)
+ (n12)F(n1 , n2 − 1, n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 + 1, n13 , n14) + (n14)F(n1 − 1, n2 , n3 , n4 , n5 , n6 , n7 , n8 , n9 , n10 , n11 , n12 , n13 , n14 + 1) = 0
+ other 19 IBPs
Parametric reduction rules by hand: no wayLaporta or other generic algorithms: very slow
13/27
The Forcer programFORM program for 4-loop massless propagator-type(scalar) Feynman integrals
Ruijl, TU, Vermaseren ’17; https://github.com/benruijl/forcerWorks with FORM 4.2
Extends the Mincer approach to 4-loopsChetyrkin, Tkachov ’81;Schoonschip version: Gorishny, Larin, Surguladze, Tkachov ’89;FORM version: Larin, Tkachov, Vermaseren ’91
When possible (determined by the topology),one-loop integration/triangle rule
:Chetyrkin, Kataev, Tkachov ’80;Chetyrkin, Tkachov ’81
: and/or
Chetyrkin, Tkachov ’81;Diamond extension: Ruijl, TU, Vermaseren ’15 14/27
The Forcer programFORM program for 4-loop massless propagator-type(scalar) Feynman integrals
Ruijl, TU, Vermaseren ’17; https://github.com/benruijl/forcerWorks with FORM 4.2
Extends the Mincer approach to 4-loopsChetyrkin, Tkachov ’81;Schoonschip version: Gorishny, Larin, Surguladze, Tkachov ’89;FORM version: Larin, Tkachov, Vermaseren ’91
When possible (determined by the topology),one-loop integration/triangle rule
:Chetyrkin, Kataev, Tkachov ’80;Chetyrkin, Tkachov ’81
: and/or
Chetyrkin, Tkachov ’81;Diamond extension: Ruijl, TU, Vermaseren ’15 14/27
4-loop reduction �owchartAll possible topoloties automatically classi�ed
no2
d387d394 d420d424
no1
no6 d391 d399d406d416
haha la4
d401 d425 d426 d427
j1
d392 d396d417d421d422 d423
j6
d386d418d419
j2
d388d397 d411d412d413d414 d415
j5
d402d403 d404d405 d407d408d409 d410
j11
d398 d400
no3
d393
j4
d389 d390
bubud306 d378d308 lala
d207 d220d234 d235d222
nono
bebed217 d218cross
d181d159 d160 d184d176d167 d185
d212 d206d209 d213 lastar5
fastar2 d114d115 d119
nostar0
d118 d137
nostar5
bustar5 d135d128
nostar6
fastar3d136 d127d144
d83 d89 d103d104d58 d57 d98 d110d93 d76 d73d75 d69
t1star05
d15
t1star24
d18 d25
t1star34
d16
t1star45
d19
t1star55
d20
d5d8
d1
d2
d10
d4
d6d11
d7
d51 d36d35 d42d52
d22 d24
d3 d9
d27
d34d38 d45d49
d12 d23
d41 d39d30
d17d14 d13
d86 d70d72 d64 d74
d47
d21d29
d94 d71d66
d43
d26
d77 d81d60 d87 d92 d79
d33d50 d32d37
d28
d91d90 d88d111d113
d112
d170d166 d171d163 d153
d102 d101 d95d100d96 d97
d53
d99
d155d156 d172 d180d178d187 d186d173 d189
d281 d280 d215 d221d208d270 d271
d165 d157 d168d140d175
d65d59d63
d336 d381 d339d327d334 d332 d333d328d324 d326d305 d325d310 d385d321 d345d353 d349d354 d356d311 d344d316 d379
d285d284 d257 d211d197 d236d263d205d239d229
d121
d216
d162 d161d158
d62d68
d164 d146
d80d78
d124
d108
d190 d191
d219 d266d256 d232 d244
d174 d179
d260d250d259d273 d243
d142d130
d254 d262d275 d264d252 d246
d141 d143
d196
d338d337 d383d384 d314d315d320 d323 d347 d352
d272d282 d278 d267d279 d276
d133
d194d287
d145 d134d147d154 d169
d61 d109
d283 d198
d357 d300 d303d342 d370 d366 d296d299 d302 d295 d293d377d375
d231 d248d258d253 d255d233 d241d247d225
d116
d193 d289
d150d122 d125 d117 d152d183 d182 d151d188
d56d105d84 d82 d85
d223 d249d261 d251d224 d242d245d201
d123 d120
d358d362 d367d372d341 d376 d359d363 d368d360d364 d369d373d343 d361d365 d371
d199
d126d131
d240d265 d230d226
d177
d67
d238d228 d237d227
d340
d192
d346d350 d355d322 d348d351d317d318 d312
d277d274
d148 d149
d55d106d107 d54
d268d269
d309
d195
d331 d335d329d330
d286
d138
d307d313d319 d292d298d301 d304 d294d297
d290d288 d291
416 out of 437 topologies have good struturesCan be simpli�ed by 1-loop integration/triangle rule etc.
Python program with 2794 lines automatically generatesFORM code with 39406 lines
The rest (21 topologies) require special rulesConstructed manually from IBPs, but considerablycomputer-assisted, heuristics/brute-force search,optimized by trial and error
15/27
4-loop reduction �owchartAll possible topoloties automatically classi�ed
no2
d387d394 d420d424
no1
no6 d391 d399d406d416
haha la4
d401 d425 d426 d427
j1
d392 d396d417d421d422 d423
j6
d386d418d419
j2
d388d397 d411d412d413d414 d415
j5
d402d403 d404d405 d407d408d409 d410
j11
d398 d400
no3
d393
j4
d389 d390
bubud306 d378d308 lala
d207 d220d234 d235d222
nono
bebed217 d218cross
d181d159 d160 d184d176d167 d185
d212 d206d209 d213 lastar5
fastar2 d114d115 d119
nostar0
d118 d137
nostar5
bustar5 d135d128
nostar6
fastar3d136 d127d144
d83 d89 d103d104d58 d57 d98 d110d93 d76 d73d75 d69
t1star05
d15
t1star24
d18 d25
t1star34
d16
t1star45
d19
t1star55
d20
d5d8
d1
d2
d10
d4
d6d11
d7
d51 d36d35 d42d52
d22 d24
d3 d9
d27
d34d38 d45d49
d12 d23
d41 d39d30
d17d14 d13
d86 d70d72 d64 d74
d47
d21d29
d94 d71d66
d43
d26
d77 d81d60 d87 d92 d79
d33d50 d32d37
d28
d91d90 d88d111d113
d112
d170d166 d171d163 d153
d102 d101 d95d100d96 d97
d53
d99
d155d156 d172 d180d178d187 d186d173 d189
d281 d280 d215 d221d208d270 d271
d165 d157 d168d140d175
d65d59d63
d336 d381 d339d327d334 d332 d333d328d324 d326d305 d325d310 d385d321 d345d353 d349d354 d356d311 d344d316 d379
d285d284 d257 d211d197 d236d263d205d239d229
d121
d216
d162 d161d158
d62d68
d164 d146
d80d78
d124
d108
d190 d191
d219 d266d256 d232 d244
d174 d179
d260d250d259d273 d243
d142d130
d254 d262d275 d264d252 d246
d141 d143
d196
d338d337 d383d384 d314d315d320 d323 d347 d352
d272d282 d278 d267d279 d276
d133
d194d287
d145 d134d147d154 d169
d61 d109
d283 d198
d357 d300 d303d342 d370 d366 d296d299 d302 d295 d293d377d375
d231 d248d258d253 d255d233 d241d247d225
d116
d193 d289
d150d122 d125 d117 d152d183 d182 d151d188
d56d105d84 d82 d85
d223 d249d261 d251d224 d242d245d201
d123 d120
d358d362 d367d372d341 d376 d359d363 d368d360d364 d369d373d343 d361d365 d371
d199
d126d131
d240d265 d230d226
d177
d67
d238d228 d237d227
d340
d192
d346d350 d355d322 d348d351d317d318 d312
d277d274
d148 d149
d55d106d107 d54
d268d269
d309
d195
d331 d335d329d330
d286
d138
d307d313d319 d292d298d301 d304 d294d297
d290d288 d291
416 out of 437 topologies have good struturesCan be simpli�ed by 1-loop integration/triangle rule etc.
Python program with 2794 lines automatically generatesFORM code with 39406 lines
The rest (21 topologies) require special rulesConstructed manually from IBPs, but considerablycomputer-assisted, heuristics/brute-force search,optimized by trial and error
no2
d387d394 d420d424
no1
no6 d391 d399d406d416
haha la4
d401 d425 d426 d427
j1
d392 d396d417d421d422 d423
j6
d386d418d419
j2
d388d397 d411d412d413d414 d415
j5
d402d403 d404d405 d407d408d409 d410
j11
d398 d400
no3
d393
j4
d389 d390
bubud306 d378d308 lala
d207 d220d234 d235d222
nono
bebed217 d218cross
d181d159 d160 d184d176d167 d185
d212 d206d209 d213 lastar5
fastar2 d114d115 d119
nostar0
d118 d137
nostar5
bustar5 d135d128
nostar6
fastar3d136 d127d144
d83 d89 d103d104d58 d57 d98 d110d93 d76 d73d75 d69
t1star05
d15
t1star24
d18 d25
t1star34
d16
t1star45
d19
t1star55
d20
d5d8
d1
d2
d10
d4
d6d11
d7
d51 d36d35 d42d52
d22 d24
d3 d9
d27
d34d38 d45d49
d12 d23
d41 d39d30
d17d14 d13
d86 d70d72 d64 d74
d47
d21d29
d94 d71d66
d43
d26
d77 d81d60 d87 d92 d79
d33d50 d32d37
d28
d91d90 d88d111d113
d112
d170d166 d171d163 d153
d102 d101 d95d100d96 d97
d53
d99
d155d156 d172 d180d178d187 d186d173 d189
d281 d280 d215 d221d208d270 d271
d165 d157 d168d140d175
d65d59d63
d336 d381 d339d327d334 d332 d333d328d324 d326d305 d325d310 d385d321 d345d353 d349d354 d356d311 d344d316 d379
d285d284 d257 d211d197 d236d263d205d239d229
d121
d216
d162 d161d158
d62d68
d164 d146
d80d78
d124
d108
d190 d191
d219 d266d256 d232 d244
d174 d179
d260d250d259d273 d243
d142d130
d254 d262d275 d264d252 d246
d141 d143
d196
d338d337 d383d384 d314d315d320 d323 d347 d352
d272d282 d278 d267d279 d276
d133
d194d287
d145 d134d147d154 d169
d61 d109
d283 d198
d357 d300 d303d342 d370 d366 d296d299 d302 d295 d293d377d375
d231 d248d258d253 d255d233 d241d247d225
d116
d193 d289
d150d122 d125 d117 d152d183 d182 d151d188
d56d105d84 d82 d85
d223 d249d261 d251d224 d242d245d201
d123 d120
d358d362 d367d372d341 d376 d359d363 d368d360d364 d369d373d343 d361d365 d371
d199
d126d131
d240d265 d230d226
d177
d67
d238d228 d237d227
d340
d192
d346d350 d355d322 d348d351d317d318 d312
d277d274
d148 d149
d55d106d107 d54
d268d269
d309
d195
d331 d335d329d330
d286
d138
d307d313d319 d292d298d301 d304 d294d297
d290d288 d291
15/27
4-loop reduction �owchartAll possible topoloties automatically classi�ed
no2
d387d394 d420d424
no1
no6 d391 d399d406d416
haha la4
d401 d425 d426 d427
j1
d392 d396d417d421d422 d423
j6
d386d418d419
j2
d388d397 d411d412d413d414 d415
j5
d402d403 d404d405 d407d408d409 d410
j11
d398 d400
no3
d393
j4
d389 d390
bubud306 d378d308 lala
d207 d220d234 d235d222
nono
bebed217 d218cross
d181d159 d160 d184d176d167 d185
d212 d206d209 d213 lastar5
fastar2 d114d115 d119
nostar0
d118 d137
nostar5
bustar5 d135d128
nostar6
fastar3d136 d127d144
d83 d89 d103d104d58 d57 d98 d110d93 d76 d73d75 d69
t1star05
d15
t1star24
d18 d25
t1star34
d16
t1star45
d19
t1star55
d20
d5d8
d1
d2
d10
d4
d6d11
d7
d51 d36d35 d42d52
d22 d24
d3 d9
d27
d34d38 d45d49
d12 d23
d41 d39d30
d17d14 d13
d86 d70d72 d64 d74
d47
d21d29
d94 d71d66
d43
d26
d77 d81d60 d87 d92 d79
d33d50 d32d37
d28
d91d90 d88d111d113
d112
d170d166 d171d163 d153
d102 d101 d95d100d96 d97
d53
d99
d155d156 d172 d180d178d187 d186d173 d189
d281 d280 d215 d221d208d270 d271
d165 d157 d168d140d175
d65d59d63
d336 d381 d339d327d334 d332 d333d328d324 d326d305 d325d310 d385d321 d345d353 d349d354 d356d311 d344d316 d379
d285d284 d257 d211d197 d236d263d205d239d229
d121
d216
d162 d161d158
d62d68
d164 d146
d80d78
d124
d108
d190 d191
d219 d266d256 d232 d244
d174 d179
d260d250d259d273 d243
d142d130
d254 d262d275 d264d252 d246
d141 d143
d196
d338d337 d383d384 d314d315d320 d323 d347 d352
d272d282 d278 d267d279 d276
d133
d194d287
d145 d134d147d154 d169
d61 d109
d283 d198
d357 d300 d303d342 d370 d366 d296d299 d302 d295 d293d377d375
d231 d248d258d253 d255d233 d241d247d225
d116
d193 d289
d150d122 d125 d117 d152d183 d182 d151d188
d56d105d84 d82 d85
d223 d249d261 d251d224 d242d245d201
d123 d120
d358d362 d367d372d341 d376 d359d363 d368d360d364 d369d373d343 d361d365 d371
d199
d126d131
d240d265 d230d226
d177
d67
d238d228 d237d227
d340
d192
d346d350 d355d322 d348d351d317d318 d312
d277d274
d148 d149
d55d106d107 d54
d268d269
d309
d195
d331 d335d329d330
d286
d138
d307d313d319 d292d298d301 d304 d294d297
d290d288 d291
416 out of 437 topologies have good struturesCan be simpli�ed by 1-loop integration/triangle rule etc.
Python program with 2794 lines automatically generatesFORM code with 39406 lines
The rest (21 topologies) require special rulesConstructed manually from IBPs, but considerablycomputer-assisted, heuristics/brute-force search,optimized by trial and error
15/27
4-loop reduction �owchartAll possible topoloties automatically classi�ed
no2
d387d394 d420d424
no1
no6 d391 d399d406d416
haha la4
d401 d425 d426 d427
j1
d392 d396d417d421d422 d423
j6
d386d418d419
j2
d388d397 d411d412d413d414 d415
j5
d402d403 d404d405 d407d408d409 d410
j11
d398 d400
no3
d393
j4
d389 d390
bubud306 d378d308 lala
d207 d220d234 d235d222
nono
bebed217 d218cross
d181d159 d160 d184d176d167 d185
d212 d206d209 d213 lastar5
fastar2 d114d115 d119
nostar0
d118 d137
nostar5
bustar5 d135d128
nostar6
fastar3d136 d127d144
d83 d89 d103d104d58 d57 d98 d110d93 d76 d73d75 d69
t1star05
d15
t1star24
d18 d25
t1star34
d16
t1star45
d19
t1star55
d20
d5d8
d1
d2
d10
d4
d6d11
d7
d51 d36d35 d42d52
d22 d24
d3 d9
d27
d34d38 d45d49
d12 d23
d41 d39d30
d17d14 d13
d86 d70d72 d64 d74
d47
d21d29
d94 d71d66
d43
d26
d77 d81d60 d87 d92 d79
d33d50 d32d37
d28
d91d90 d88d111d113
d112
d170d166 d171d163 d153
d102 d101 d95d100d96 d97
d53
d99
d155d156 d172 d180d178d187 d186d173 d189
d281 d280 d215 d221d208d270 d271
d165 d157 d168d140d175
d65d59d63
d336 d381 d339d327d334 d332 d333d328d324 d326d305 d325d310 d385d321 d345d353 d349d354 d356d311 d344d316 d379
d285d284 d257 d211d197 d236d263d205d239d229
d121
d216
d162 d161d158
d62d68
d164 d146
d80d78
d124
d108
d190 d191
d219 d266d256 d232 d244
d174 d179
d260d250d259d273 d243
d142d130
d254 d262d275 d264d252 d246
d141 d143
d196
d338d337 d383d384 d314d315d320 d323 d347 d352
d272d282 d278 d267d279 d276
d133
d194d287
d145 d134d147d154 d169
d61 d109
d283 d198
d357 d300 d303d342 d370 d366 d296d299 d302 d295 d293d377d375
d231 d248d258d253 d255d233 d241d247d225
d116
d193 d289
d150d122 d125 d117 d152d183 d182 d151d188
d56d105d84 d82 d85
d223 d249d261 d251d224 d242d245d201
d123 d120
d358d362 d367d372d341 d376 d359d363 d368d360d364 d369d373d343 d361d365 d371
d199
d126d131
d240d265 d230d226
d177
d67
d238d228 d237d227
d340
d192
d346d350 d355d322 d348d351d317d318 d312
d277d274
d148 d149
d55d106d107 d54
d268d269
d309
d195
d331 d335d329d330
d286
d138
d307d313d319 d292d298d301 d304 d294d297
d290d288 d291
416 out of 437 topologies have good struturesCan be simpli�ed by 1-loop integration/triangle rule etc.
Python program with 2794 lines automatically generatesFORM code with 39406 lines
The rest (21 topologies) require special rulesConstructed manually from IBPs, but considerablycomputer-assisted, heuristics/brute-force search,optimized by trial and error
15/27
4-loop reduction �owchartAll possible topoloties automatically classi�ed
no2
d387d394 d420d424
no1
no6 d391 d399d406d416
haha la4
d401 d425 d426 d427
j1
d392 d396d417d421d422 d423
j6
d386d418d419
j2
d388d397 d411d412d413d414 d415
j5
d402d403 d404d405 d407d408d409 d410
j11
d398 d400
no3
d393
j4
d389 d390
bubud306 d378d308 lala
d207 d220d234 d235d222
nono
bebed217 d218cross
d181d159 d160 d184d176d167 d185
d212 d206d209 d213 lastar5
fastar2 d114d115 d119
nostar0
d118 d137
nostar5
bustar5 d135d128
nostar6
fastar3d136 d127d144
d83 d89 d103d104d58 d57 d98 d110d93 d76 d73d75 d69
t1star05
d15
t1star24
d18 d25
t1star34
d16
t1star45
d19
t1star55
d20
d5d8
d1
d2
d10
d4
d6d11
d7
d51 d36d35 d42d52
d22 d24
d3 d9
d27
d34d38 d45d49
d12 d23
d41 d39d30
d17d14 d13
d86 d70d72 d64 d74
d47
d21d29
d94 d71d66
d43
d26
d77 d81d60 d87 d92 d79
d33d50 d32d37
d28
d91d90 d88d111d113
d112
d170d166 d171d163 d153
d102 d101 d95d100d96 d97
d53
d99
d155d156 d172 d180d178d187 d186d173 d189
d281 d280 d215 d221d208d270 d271
d165 d157 d168d140d175
d65d59d63
d336 d381 d339d327d334 d332 d333d328d324 d326d305 d325d310 d385d321 d345d353 d349d354 d356d311 d344d316 d379
d285d284 d257 d211d197 d236d263d205d239d229
d121
d216
d162 d161d158
d62d68
d164 d146
d80d78
d124
d108
d190 d191
d219 d266d256 d232 d244
d174 d179
d260d250d259d273 d243
d142d130
d254 d262d275 d264d252 d246
d141 d143
d196
d338d337 d383d384 d314d315d320 d323 d347 d352
d272d282 d278 d267d279 d276
d133
d194d287
d145 d134d147d154 d169
d61 d109
d283 d198
d357 d300 d303d342 d370 d366 d296d299 d302 d295 d293d377d375
d231 d248d258d253 d255d233 d241d247d225
d116
d193 d289
d150d122 d125 d117 d152d183 d182 d151d188
d56d105d84 d82 d85
d223 d249d261 d251d224 d242d245d201
d123 d120
d358d362 d367d372d341 d376 d359d363 d368d360d364 d369d373d343 d361d365 d371
d199
d126d131
d240d265 d230d226
d177
d67
d238d228 d237d227
d340
d192
d346d350 d355d322 d348d351d317d318 d312
d277d274
d148 d149
d55d106d107 d54
d268d269
d309
d195
d331 d335d329d330
d286
d138
d307d313d319 d292d298d301 d304 d294d297
d290d288 d291
416 out of 437 topologies have good struturesCan be simpli�ed by 1-loop integration/triangle rule etc.
Python program with 2794 lines automatically generatesFORM code with 39406 lines
The rest (21 topologies) require special rulesConstructed manually from IBPs, but considerablycomputer-assisted, heuristics/brute-force search,optimized by trial and error
15/27
Parton evolutionsDGLAP equation
dd lnµ2f
fa(x,µ2f ) =∑b
[Pab(αs(µ
2f ))⊗ fb(µ2f )
](x)
2 nf (anti-)quark distributions decomposed as• q±ns,ik = (qi ± q̄i)− (qk ± q̄k),�avor non-singlet, 2(nf − 1) components,evolving with P±ns
• qvns =∑nf
i=1(qi − q̄i): �avor non-singlet (“valence”),evolving with Pvns = P−ns +O(α3s)
• qs =∑nf
i=1(qi + q̄i): �avor singlet,mixing with gluons. Pqq = P+
ns +O(α2s)
dd lnµ2f
(qsg
)=
(Pqq PqgPgq Pgg
)⊗(qsg
)16/27
Parton evolutionsDGLAP equation
dd lnµ2f
fa(x,µ2f ) =∑b
[Pab(αs(µ
2f ))⊗ fb(µ2f )
](x)
2 nf (anti-)quark distributions decomposed as• q±ns,ik = (qi ± q̄i)− (qk ± q̄k),�avor non-singlet, 2(nf − 1) components,evolving with P±ns
• qvns =∑nf
i=1(qi − q̄i): �avor non-singlet (“valence”),evolving with Pvns = P−ns +O(α3s)
• qs =∑nf
i=1(qi + q̄i): �avor singlet,mixing with gluons. Pqq = P+
ns +O(α2s)
dd lnµ2f
(qsg
)=
(Pqq PqgPgq Pgg
)⊗(qsg
)16/27
4-loop splitting functions by Forcer (I)
Computation via Q{µ1 ...QµN}
N!
∂N
∂Pµ1 ... ∂PµN
P P
Q Q
∣∣∣∣∣P=0
Davies, Vogt, Ruijl, TU, Vermaseren ’16
Con�rmed known low-N momentsup to N = 4 for NS Baikov, Chetyrkin, Kühn, Rittinger; Velizhanin ’11; ’14
New results: up to N = 6 for NS, up to N = 4 for S
P±ns, Pvns(Pqq PqgPgq Pgg
)Up to N > 40 for high-nf parts:enough to reconstruct full-N results
17/27
4-loop splitting functions by Forcer (I)
Computation via Q{µ1 ...QµN}
N!
∂N
∂Pµ1 ... ∂PµN
P P
Q Q
∣∣∣∣∣P=0
Davies, Vogt, Ruijl, TU, Vermaseren ’16
Con�rmed known low-N momentsup to N = 4 for NS Baikov, Chetyrkin, Kühn, Rittinger; Velizhanin ’11; ’14
New results: up to N = 6 for NS, up to N = 4 for S
P±ns, Pvns(Pqq PqgPgq Pgg
)Up to N > 40 for high-nf parts:enough to reconstruct full-N results
17/27
4-loop splitting functions by Forcer (I)
Computation via Q{µ1 ...QµN}
N!
∂N
∂Pµ1 ... ∂PµN
P P
Q Q
∣∣∣∣∣P=0
Davies, Vogt, Ruijl, TU, Vermaseren ’16
Con�rmed known low-N momentsup to N = 4 for NS Baikov, Chetyrkin, Kühn, Rittinger; Velizhanin ’11; ’14
New results: up to N = 6 for NS, up to N = 4 for S
P±ns, Pvns(Pqq PqgPgq Pgg
)Up to N > 40 for high-nf parts:enough to reconstruct full-N results
17/27
4-loop splitting functions by Forcer (II)
Computation viaONns = ψ̄λαγ{µ1Dµ2 ...DµN}ψ
Q QMoch, Ruijl, TU, Vermaseren, Vogt ’17
Easier: done up to N = 16 for NS(for S conceptually more involved; in progress)
Up to N = 20 for the large-nc limitFull-N (large-nc) result reconstructed
18/27
4-loop splitting functions by Forcer (II)
Computation viaONns = ψ̄λαγ{µ1Dµ2 ...DµN}ψ
Q QMoch, Ruijl, TU, Vermaseren, Vogt ’17
Easier: done up to N = 16 for NS(for S conceptually more involved; in progress)
Up to N = 20 for the large-nc limitFull-N (large-nc) result reconstructed
18/27
4-loop splitting functions by Forcer (II)
Computation viaONns = ψ̄λαγ{µ1Dµ2 ...DµN}ψ
Q QMoch, Ruijl, TU, Vermaseren, Vogt ’17
Easier: done up to N = 16 for NS(for S conceptually more involved; in progress)
Up to N = 20 for the large-nc limitFull-N (large-nc) result reconstructed
18/27
Moments for NS splitting functions
γ±ns(N) = αsγ(0)±ns (N) + α2sγ
(1)±ns (N) + α3sγ
(2)±ns (N) + α4sγ
(3)±ns (N) + ...
0
0.2
0.4
0.6
0.8
0 5 10 15 20 25
large nc
N
γ ns
γ (3)±
(N)
nf = 3
nf = 4
points: ± at even/odd N
N
γ ns
γ (3)±
(N)
expansion in αS
nf = 5
nf = 6
-0.3
-0.2
-0.1
0
0.1
0 5 10 15 20 25
even for +, odd for −19/27
Approximated NS splitting functions
P±ns(x) = αsP(0)±ns (x) + α2sP
(1)±ns (x) + α3sP
(2)±ns (x) + α4sP
(3)±ns (x) + ...
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 0.2 0.4 0.6 0.8 1x
(1-x) P
ns(1-x)
P
(3)± (x)
appr. +
appr. −
large nc
x
(1-x) P
ns(1-x)
P
(3)± (x)
exp. in αS
, nf = 4
-6
-4
-2
0
2
4
6
8
10
10-4
10-3
10-2
10-1
1
20/27
Outline
(phenomenological) Motivation: 4-loop splitting functions
Forcer program (4-loops)and applications to splitting functions
: Approximation for NS
5-loop calculations(theoretical/math fun)
21/27
Infrared rearrangementSuper�cial (or overall) UV divergence: coming fromthe region where all loop momenta go to∞All mass scales (internal massess and externalmomenta) can be ignored
sup.UV div.
[ ]= sup.UV div.
[ ]
= sup.UV div.
[ ]
= sup.UV div.
[M
]Log-divergences are mass-independentin dimensional regularization. Rearrange diagrams
22/27
Computing 5-loop pole parts
Div.
[ ]= sup.UV div.
[ ]+(UV/IR subdivergences
)L-loops L-loops lower loops
= sup.UV div.
[ ]
= sup.UV div.
[ ] integrate1-loop
R∗
(local) R∗-operation:generaization of the BPHZ R-operation,diagrammatically identifying/subtracting both UV and IR
Chetyrkin, Tkachov ’82; Chetyrkin, Smirnov ’83, ’84;Extension for arvitrary numerator structure: Herzog, Ruijl ’17
Poles of 5-loop diagrams can be computedas 4-loop massless propagator diagrams
23/27
Computing 5-loop pole parts
Div.
[ ]= sup.UV div.
[ ]+(UV/IR subdivergences
)L-loops e�ectively (L− 1)-loops lower loops
= sup.UV div.
[ ]
= sup.UV div.
[ ] integrate1-loop
R∗
(local) R∗-operation:generaization of the BPHZ R-operation,diagrammatically identifying/subtracting both UV and IR
Chetyrkin, Tkachov ’82; Chetyrkin, Smirnov ’83, ’84;Extension for arvitrary numerator structure: Herzog, Ruijl ’17
Poles of 5-loop diagrams can be computedas 4-loop massless propagator diagrams
23/27
Computing 5-loop pole parts
Div.
[ ]= sup.UV div.
[ ]+(UV/IR subdivergences
)L-loops e�ectively (L− 1)-loops lower loops
= sup.UV div.
[ ]
= sup.UV div.
[ ] integrate1-loop
R∗
(local) R∗-operation:generaization of the BPHZ R-operation,diagrammatically identifying/subtracting both UV and IR
Chetyrkin, Tkachov ’82; Chetyrkin, Smirnov ’83, ’84;Extension for arvitrary numerator structure: Herzog, Ruijl ’17
Poles of 5-loop diagrams can be computedas 4-loop massless propagator diagrams
23/27
Computing 5-loop pole parts
Div.
[ ]= sup.UV div.
[ ]+(UV/IR subdivergences
)L-loops e�ectively (L− 1)-loops lower loops
= sup.UV div.
[ ]
= sup.UV div.
[ ] integrate1-loop
R∗
(local) R∗-operation:generaization of the BPHZ R-operation,diagrammatically identifying/subtracting both UV and IR
Chetyrkin, Tkachov ’82; Chetyrkin, Smirnov ’83, ’84;Extension for arvitrary numerator structure: Herzog, Ruijl ’17
Poles of 5-loop diagrams can be computedas 4-loop massless propagator diagrams
23/27
5-loop QCD beta functionBackground �eld method + local R∗ + Forcer
Generic color groupHerzog, Ruijl, TU, Vermaseren, Vogt Jan-’17
3 day computations with more than 500 cores (total CPU time ∼ 1.2 years)Later improved: the current record is 6 days on a (fast) 32 core machine
βMS4 = C 5A(82962353888 − 1630
81 ζ3 +1216 ζ4 −
10459 ζ5
)+d abcdA d abcdA
NACA(−5143 +
187163 ζ3 − 968ζ4 −
154003 ζ5
)+ C 4A TFnf
(−5048959972 +
1050581 ζ3 −
5833 ζ4 + 1230ζ5
)+ C 3A CFTFnf
(81419951944 + 146ζ3 +
9023 ζ4 −
87203 ζ5
)+ C 2A C 2F TFnf
(−54873281 − 50581
27 ζ3 −4843 ζ4 +
128203 ζ5
)+ CAC 3F TFnf
(3717+
56963 ζ3 −
74803 ζ5
)− C 4F TFnf
(41576 + 128ζ3
)+d abcdA d abcdA
NATFnf
(9049 −
207529 ζ3 + 352ζ4 +
40009 ζ5
)+d abcdF d abcdA
NACAnf
(113129 − 127736
9 ζ3 + 2288ζ4 +675209 ζ5
)+d abcdF d abcdA
NACFnf
(−320+
12803 ζ3 +
64003 ζ5
)+ C 3A T 2F n 2f
(843067486 +
1844627 ζ3 −
1043 ζ4 −
22003 ζ5
)+ C 2A CFT 2F n 2f
(5701162 +
2645227 ζ3 −
9443 ζ4 +
16003 ζ5
)+ C 2F CAT 2F n 2f
(3158318 − 28628
27 ζ3 +11443 ζ4 −
44003 ζ5
)+ C 3F T 2F n 2f
(−50189 − 2144
3 ζ3 +46403 ζ5
)+d abcdF d abcdA
NATFn 2f
(−36809 +
401609 ζ3 − 832ζ4 −
12809 ζ5
)+d abcdF d abcdF
NACAn 2f
(−71843 +
403369 ζ3 − 704ζ4 +
22409 ζ5
)+d abcdF d abcdF
NACFn 2f
(41603 +
51203 ζ3 −
128003 ζ5
)+ C 2A T 3F n
3f
(−207727 −
973681 ζ3 +
1123 ζ4 +
3209 ζ5
)+ CACFT 3F n
3f
(−73681 −
568027 ζ3 +
2243 ζ4
)+ C 2F T 3F n
3f
(−992281 +
761627 ζ3 −
3523 ζ4
)+d abcdF d abcdF
NATFn 3f
(35209 − 2624
3 ζ3 + 256ζ4 +12803 ζ5
)+ CAT 4F n 4f
(916243 −
64081 ζ3
)− CFT 4F n 4f
(856243 +
12827 ζ3
)24/27
5-loop QCD beta functionBackground �eld method + local R∗ + Forcer
Generic color groupHerzog, Ruijl, TU, Vermaseren, Vogt Jan-’17
3 day computations with more than 500 cores (total CPU time ∼ 1.2 years)Later improved: the current record is 6 days on a (fast) 32 core machine
βMS4 = C 5A(82962353888 − 1630
81 ζ3 +1216 ζ4 −
10459 ζ5
)+d abcdA d abcdA
NACA(−5143 +
187163 ζ3 − 968ζ4 −
154003 ζ5
)+ C 4A TFnf
(−5048959972 +
1050581 ζ3 −
5833 ζ4 + 1230ζ5
)+ C 3A CFTFnf
(81419951944 + 146ζ3 +
9023 ζ4 −
87203 ζ5
)+ C 2A C 2F TFnf
(−54873281 − 50581
27 ζ3 −4843 ζ4 +
128203 ζ5
)+ CAC 3F TFnf
(3717+
56963 ζ3 −
74803 ζ5
)− C 4F TFnf
(41576 + 128ζ3
)+d abcdA d abcdA
NATFnf
(9049 −
207529 ζ3 + 352ζ4 +
40009 ζ5
)+d abcdF d abcdA
NACAnf
(113129 − 127736
9 ζ3 + 2288ζ4 +675209 ζ5
)+d abcdF d abcdA
NACFnf
(−320+
12803 ζ3 +
64003 ζ5
)+ C 3A T 2F n 2f
(843067486 +
1844627 ζ3 −
1043 ζ4 −
22003 ζ5
)+ C 2A CFT 2F n 2f
(5701162 +
2645227 ζ3 −
9443 ζ4 +
16003 ζ5
)+ C 2F CAT 2F n 2f
(3158318 − 28628
27 ζ3 +11443 ζ4 −
44003 ζ5
)+ C 3F T 2F n 2f
(−50189 − 2144
3 ζ3 +46403 ζ5
)+d abcdF d abcdA
NATFn 2f
(−36809 +
401609 ζ3 − 832ζ4 −
12809 ζ5
)+d abcdF d abcdF
NACAn 2f
(−71843 +
403369 ζ3 − 704ζ4 +
22409 ζ5
)+d abcdF d abcdF
NACFn 2f
(41603 +
51203 ζ3 −
128003 ζ5
)+ C 2A T 3F n
3f
(−207727 −
973681 ζ3 +
1123 ζ4 +
3209 ζ5
)+ CACFT 3F n
3f
(−73681 −
568027 ζ3 +
2243 ζ4
)+ C 2F T 3F n
3f
(−992281 +
761627 ζ3 −
3523 ζ4
)+d abcdF d abcdF
NATFn 3f
(35209 − 2624
3 ζ3 + 256ζ4 +12803 ζ5
)+ CAT 4F n 4f
(916243 −
64081 ζ3
)− CFT 4F n 4f
(856243 +
12827 ζ3
)24/27
5-loop QCD beta functionBackground �eld method + local R∗ + Forcer
Generic color groupHerzog, Ruijl, TU, Vermaseren, Vogt Jan-’17
3 day computations with more than 500 cores (total CPU time ∼ 1.2 years)Later improved: the current record is 6 days on a (fast) 32 core machine
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
0 0.1 0.2 0.3 0.4 0.5α
s
βN LO
/ βNLO
n
n = 2
n = 3
n = 4
nf = 4
µ2
αs,N LO
/ αs,NLO
n
n = 2
n = 3
n = 4
nf = 4, fixed value at 40 GeV
2
1
1.02
1.04
1.06
1 10 102
103
104
24/27
5-loop QCD beta functionBackground �eld method + local R∗ + Forcer
Generic color groupHerzog, Ruijl, TU, Vermaseren, Vogt Jan-’17
3 day computations with more than 500 cores (total CPU time ∼ 1.2 years)Later improved: the current record is 6 days on a (fast) 32 core machine
Con�rmed and extended SU(3) resultsBaikov, Chetyrkin, Kühn Jun-’16Leading/subleading nf -terms (generic group): Gracey ’06;
Luthe, Maier, Marquard, Schröder Jun-’16
Now by another team. EstablishedLuthe, Maier, Marquard, Schröder Sep-’17
Renormalization of QCD at 5-loop �nishedLuthe, Maier, Marquard, Schröder Dec-’16; Jan-’17;Baikov, Chetyrkin, Kühn Feb-’17;Chetyrkin, Falcioni, Herzog, Vermaseren Sep-’17;β4 in MiniMOM also known: Ruijl, TU, Vermaseren, Vogt Mar-’17
24/27
5-loop QCD beta functionBackground �eld method + local R∗ + Forcer
Generic color groupHerzog, Ruijl, TU, Vermaseren, Vogt Jan-’17
3 day computations with more than 500 cores (total CPU time ∼ 1.2 years)Later improved: the current record is 6 days on a (fast) 32 core machine
Con�rmed and extended SU(3) resultsBaikov, Chetyrkin, Kühn Jun-’16Leading/subleading nf -terms (generic group): Gracey ’06;
Luthe, Maier, Marquard, Schröder Jun-’16
Now by another team. EstablishedLuthe, Maier, Marquard, Schröder Sep-’17
Renormalization of QCD at 5-loop �nishedLuthe, Maier, Marquard, Schröder Dec-’16; Jan-’17;Baikov, Chetyrkin, Kühn Feb-’17;Chetyrkin, Falcioni, Herzog, Vermaseren Sep-’17;β4 in MiniMOM also known: Ruijl, TU, Vermaseren, Vogt Mar-’17
24/27
5-loop QCD beta functionBackground �eld method + local R∗ + Forcer
Generic color groupHerzog, Ruijl, TU, Vermaseren, Vogt Jan-’17
3 day computations with more than 500 cores (total CPU time ∼ 1.2 years)Later improved: the current record is 6 days on a (fast) 32 core machine
Con�rmed and extended SU(3) resultsBaikov, Chetyrkin, Kühn Jun-’16Leading/subleading nf -terms (generic group): Gracey ’06;
Luthe, Maier, Marquard, Schröder Jun-’16
Now by another team. EstablishedLuthe, Maier, Marquard, Schröder Sep-’17
Renormalization of QCD at 5-loop �nishedLuthe, Maier, Marquard, Schröder Dec-’16; Jan-’17;Baikov, Chetyrkin, Kühn Feb-’17;Chetyrkin, Falcioni, Herzog, Vermaseren Sep-’17;β4 in MiniMOM also known: Ruijl, TU, Vermaseren, Vogt Mar-’17
24/27
N4LO Higgs decay to gluons
E�ective Hgg ... -coupling in Mt →∞N4LO Wilson coe�cient: Baikov, Chetyrkin, Kühn ’16
The optical theorem leads to
ΓH→gg ∼ ImΠGG(−M2H − iδ) = Im
{eiπεLΠGG(M2
H)}
= Lπε [1+O(ε2)] ΠGG(M2H)
H H
Only the pole terms contribute to the �nite result
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N4LO Higgs decay to gluonsComputed by local R∗ + Forcer Herzog, Ruijl, TU, Vermaseren, Vogt ’17
1.2
1.4
1.6
1.8
2
100 200 300µ
/ GeV
G∼
(M
H ) / G∼
0G∼
(M
2
N LO2
NLO
N LO4
N LO3
µ / GeV
Γ (H
→
gg)
/ Γ
0
SI
N LO2
NLO
N LO4
N LO3
1.2
1.4
1.6
1.8
2
100 200 300
Renormalization-scale dependence of K-factor(αs(M2Z) =0.118, MH =125GeV, µt =164GeV)
Computationally challenging:Took 2 monthscf. ΓH→bb̄ and R-ratio (known):much easier, few hours
N4LO correction < 1%(-0.6% at µ = MH)Slightly smaller than 1/mtop e�ectsat NNLO
Uncertainty due to the truncationof the perturbation series (±0.6%)is much smaller than that due touncertainty of αs(MZ)
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Summary
Forcer: program for analytically computing masslesspropagator-type Feynman integrals up to 4-loops
Moments of 4-loop splitting functions computedUp to N = 16 for non-singlet. Approximated x-resultSinglet in progress
5-loop pole parts by Forcer + local R∗
• QCD beta function• Higgs decay
27/27
Summary
Forcer: program for analytically computing masslesspropagator-type Feynman integrals up to 4-loops
Moments of 4-loop splitting functions computedUp to N = 16 for non-singlet. Approximated x-resultSinglet in progress
5-loop pole parts by Forcer + local R∗
• QCD beta function• Higgs decay
27/27
Summary
Forcer: program for analytically computing masslesspropagator-type Feynman integrals up to 4-loops
Moments of 4-loop splitting functions computedUp to N = 16 for non-singlet. Approximated x-resultSinglet in progress
5-loop pole parts by Forcer + local R∗
• QCD beta function• Higgs decay
27/27
Backup
1/2
Self-energy benchmark @ 4-loops
0 1 2 3 4 5 6 7 8Maximum number of powers of
1
10
100
1000El
apse
d tim
e (m
)
ghost
quark
gluon
background
Our setup with Forcer on a 32-core machine with SSD(8 × “tform -w4” jobs): 4-loop beta in 3 minutes by BFM
2/2