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On the four-loop cusp anomalous dimension
On the four-loop cusp anomalous
dimension
Johannes M. Henn
Mainz University
PRISMA Cluster of Excellence
Nordita, Aspects of Amplitudes, June 30, 2016
On the four-loop cusp anomalous dimension
Based on collaboration with Alexander Smirnov, Vladimir
Smirnov and Matthias Steinhauser
J. Henn, A. Smirnov, V. Smirnov and M. Steinhauser,arXiv:1604.03126, JHEP 1605 (2016) 066
On the four-loop cusp anomalous dimension
J. Henn, A. Smirnov, V. Smirnov and M. Steinhauser,arXiv:1604.03126, JHEP 1605 (2016) 066The first next-to-next-to-next-to-next-to-leading order (N4LO)contribution to a three-point function within QCD.
On the four-loop cusp anomalous dimension
J. Henn, A. Smirnov, V. Smirnov and M. Steinhauser,arXiv:1604.03126, JHEP 1605 (2016) 066The first next-to-next-to-next-to-next-to-leading order (N4LO)contribution to a three-point function within QCD.
Evaluating four-loop QCD form factors
On the four-loop cusp anomalous dimension
J. Henn, A. Smirnov, V. Smirnov and M. Steinhauser,arXiv:1604.03126, JHEP 1605 (2016) 066The first next-to-next-to-next-to-next-to-leading order (N4LO)contribution to a three-point function within QCD.
Evaluating four-loop QCD form factors
Massless planar four-loop vertex integrals
On the four-loop cusp anomalous dimension
J. Henn, A. Smirnov, V. Smirnov and M. Steinhauser,arXiv:1604.03126, JHEP 1605 (2016) 066The first next-to-next-to-next-to-next-to-leading order (N4LO)contribution to a three-point function within QCD.
Evaluating four-loop QCD form factors
Massless planar four-loop vertex integrals
Perspectives
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
The photon-quark form factor, which is a building block forN4LO cross sections.
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
The photon-quark form factor, which is a building block forN4LO cross sections.It is a gauge-invariant part of virtual forth-order corrections forthe process e+e− → 2 jets, or for Drell-Yan production athadron colliders.
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
The photon-quark form factor, which is a building block forN4LO cross sections.It is a gauge-invariant part of virtual forth-order corrections forthe process e+e− → 2 jets, or for Drell-Yan production athadron colliders.Let Γµq be the photon-quark vertex function.The scalar form factor is
Fq(q2) = −
1
4(1 − ǫ)q2Tr
(
p2/ Γµqp1/ γµ)
,
where D = 4 − 2ǫ, q = p1 + p2 and p1 (p2) is the incoming(anti-)quark momentum.
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
The photon-quark form factor, which is a building block forN4LO cross sections.It is a gauge-invariant part of virtual forth-order corrections forthe process e+e− → 2 jets, or for Drell-Yan production athadron colliders.Let Γµq be the photon-quark vertex function.The scalar form factor is
Fq(q2) = −
1
4(1 − ǫ)q2Tr
(
p2/ Γµqp1/ γµ)
,
where D = 4 − 2ǫ, q = p1 + p2 and p1 (p2) is the incoming(anti-)quark momentum.The large-Nc asymptotics of Fq(q
2) → planar Feynmandiagrams.
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
Three-loop results[P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnovand M. Steinhauser’09,T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, andC. Studerus’10]
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
Three-loop results[P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnovand M. Steinhauser’09,T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, andC. Studerus’10]Analytic results for the three missing coefficients[R. N. Lee, A. V. Smirnov and V. A. Smirnov’10]
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
Three-loop results[P. A. Baikov, K. G. Chetyrkin, A. V. Smirnov, V. A. Smirnovand M. Steinhauser’09,T. Gehrmann, E. W. N. Glover, T. Huber, N. Ikizlerli, andC. Studerus’10]Analytic results for the three missing coefficients[R. N. Lee, A. V. Smirnov and V. A. Smirnov’10]Analytic results for the three-loop master integrals up toweight 8[R. N. Lee and V. A. Smirnov’10]motivated by a future four-loop calculation.
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
The fermionic corrections (∼ nf ) to Fq in the large-Nc limit,to the four-loop order.
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
Numerical four-loop calculations[R. H. Boels, B. A. Kniehl & G. Yang’16]
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
Numerical four-loop calculations[R. H. Boels, B. A. Kniehl & G. Yang’16]
Partial results for some individual integrals[A. von Manteuffel, E. Panzer & R. M. Schabinger’15]
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We apply
qgraf for the generation of Feynman amplitudes;
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We apply
qgraf for the generation of Feynman amplitudes;
q2e and exp for writing down form factors in terms ofFeynman integrals
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We apply
qgraf for the generation of Feynman amplitudes;
q2e and exp for writing down form factors in terms ofFeynman integrals
FIRE and LiteRed for the IBP reduction to masterintegrals.
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We apply
qgraf for the generation of Feynman amplitudes;
q2e and exp for writing down form factors in terms ofFeynman integrals
FIRE and LiteRed for the IBP reduction to masterintegrals.
Calculations in generic ξ-gauge for checks.
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We apply
qgraf for the generation of Feynman amplitudes;
q2e and exp for writing down form factors in terms ofFeynman integrals
FIRE and LiteRed for the IBP reduction to masterintegrals.
Calculations in generic ξ-gauge for checks.
Fq = 1 +∑
n≥1
(
α0s
4π
)n (µ2
−q2
)(nǫ)
F (n)q .
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We apply
qgraf for the generation of Feynman amplitudes;
q2e and exp for writing down form factors in terms ofFeynman integrals
FIRE and LiteRed for the IBP reduction to masterintegrals.
Calculations in generic ξ-gauge for checks.
Fq = 1 +∑
n≥1
(
α0s
4π
)n (µ2
−q2
)(nǫ)
F (n)q .
Our result is the fermionic contribution to F(4)q in the large-Nc
limit.
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
F(4)q |large-Nc
=
1
ǫ7
[
1
12N3
c nf
]
+1
ǫ6
[
41
648N2
c n2f −
37
648N3
c nf
]
+1
ǫ5
[
1
54Ncn
3f +
277
972N2
c n2f
+
(
41π2
648−
6431
3888
)
N3c nf
]
+1
ǫ4
[
(
215ζ3
108−
72953
7776−
227π2
972
)
N3c nf
+11
54Ncn
3f +
(
5
24+
127π2
1944
)
N2c n
2f
]
+1
ǫ3
[
(
229ζ3
486−
630593
69984+
293π2
2916
)
N2c n
2f
+
(
2411ζ3
243−
1074359
69984−
2125π2
1296+
413π4
3888
)
N3c nf +
(
127
81+
5π2
162
)
Ncn3f
]
+O
(
1
ǫ2
)
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
Exponentiation of infrared (soft and collinear) divergences
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
Exponentiation of infrared (soft and collinear) divergences
log(Fq)|pole part|(αs4π)4 =
{
1
ǫ5
[
25
96β3
0CFγ0cusp
]
+1
ǫ4
[
CF
(
−13
96β2
0γ1cusp −
5
12β1β0γ
0cusp
)
−1
4β3
0γ0q
]
+1
ǫ3
[
CF
(
5
32β2γ
0cusp +
3
32β1γ
1cusp +
7
96β0γ
2cusp
)
+1
4β2
0γ1q +
1
2β1β0γ
0q
]
+1
ǫ2
[
−1
4β2γ
0q −
1
4β1γ
1q −
1
4β0γ
2q −
1
32CFγ
3cusp
]
+1
ǫ
[
γ3q
4
]}
,
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
The coefficients of the cusp and collinear anomalous dimensions
γx =∑
n≥0
(
αs(µ2)
4π
)n
γnx ,
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
The coefficients of the cusp and collinear anomalous dimensions
γx =∑
n≥0
(
αs(µ2)
4π
)n
γnx ,
with x ∈ {cusp, q}.
γ0cusp
= 4 ,
γ1cusp
=
(
−4π2
3+
268
9
)
Nc −40nf
9,
γ2cusp
=
(
44π4
45+
88ζ3
3−
536π2
27+
490
3
)
N2c
+
(
−64ζ3
3+
80π2
27−
1331
27
)
Ncnf −16n2
f
27,
γ3cusp
=
(
−32π4
135+
1280ζ3
27−
304π2
243+
3463
81
)
Ncn2f +
(
128π2ζ3
9+ 224ζ5
−44π4
27−
16252ζ3
27+
13346π2
243−
60391
81
)
N2c nf +
(
64ζ3
27−
32
81
)
n3f + . . .
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
γ0q = −
3Nc
2, γ1
q =
(
π2
6+
65
54
)
Ncnf +
(
7ζ3 −5π2
12−
2003
216
)
N2c ,
γ2q =
(
−π4
135−
290ζ3
27+
2243π2
972+
45095
5832
)
N2c nf +
(
−4ζ3
27−
5π2
27+
2417
1458
)
N
+ N3c
(
−68ζ5 −22π2ζ3
9−
11π4
54+
2107ζ3
18−
3985π2
1944−
204955
5832
)
,
γ3q = N3
c
[(
−680ζ2
3
9−
1567π6
20412+
83π2ζ3
9+
557ζ5
9+
3557π4
19440−
94807ζ3
972
+354343π2
17496+
145651
1728
)
nf
]
+
(
−8π4
1215−
356ζ3
243−
2π2
81+
18691
13122
)
Ncn3f
+
(
−2
3π2ζ3 +
166ζ5
9+
331π4
2430−
2131ζ3
243−
68201π2
17496−
82181
69984
)
N2c n
2f + . . .
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We reproduce results up to three loops[A. Vogt’01; C.F. Berger’02; S. Moch, J.A.M. Vermaseren &A. Vogt’04,05; P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov,V.A. Smirnov & M. Steinhauser’09; T. Becher &M. Neubert’09; T. Gehrmann, E.W.N. Glover, T. Huber,N. Ikizlerli & C. Studerus’10]
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We reproduce results up to three loops[A. Vogt’01; C.F. Berger’02; S. Moch, J.A.M. Vermaseren &A. Vogt’04,05; P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov,V.A. Smirnov & M. Steinhauser’09; T. Becher &M. Neubert’09; T. Gehrmann, E.W.N. Glover, T. Huber,N. Ikizlerli & C. Studerus’10]
The N3c n
3f term of γ3
cusp agrees with[M. Beneke & V.M. Braun’94]
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We reproduce results up to three loops[A. Vogt’01; C.F. Berger’02; S. Moch, J.A.M. Vermaseren &A. Vogt’04,05; P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov,V.A. Smirnov & M. Steinhauser’09; T. Becher &M. Neubert’09; T. Gehrmann, E.W.N. Glover, T. Huber,N. Ikizlerli & C. Studerus’10]
The N3c n
3f term of γ3
cusp agrees with[M. Beneke & V.M. Braun’94]
Agreement of the n2f term with
[Davies, B. Ruijl, T.Ueda, J. Vermaseren & A. Vogt]talk at Loops and Legs 2016 by A. Vogt
On the four-loop cusp anomalous dimension
Evaluating four-loop QCD form factors
We reproduce results up to three loops[A. Vogt’01; C.F. Berger’02; S. Moch, J.A.M. Vermaseren &A. Vogt’04,05; P.A. Baikov, K.G. Chetyrkin, A.V. Smirnov,V.A. Smirnov & M. Steinhauser’09; T. Becher &M. Neubert’09; T. Gehrmann, E.W.N. Glover, T. Huber,N. Ikizlerli & C. Studerus’10]
The N3c n
3f term of γ3
cusp agrees with[M. Beneke & V.M. Braun’94]
Agreement of the n2f term with
[Davies, B. Ruijl, T.Ueda, J. Vermaseren & A. Vogt]talk at Loops and Legs 2016 by A. Vogt
All the other four-loop terms in γ3cusp and γ3
q , and for the finitepart, are new.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
All planar four-loop on-shell form-factor integrals withp2
1 = p22 = 0, with q2 ≡ p2
3 = (p1 + p2)2
Fa1, . . . , a18 =
∫
. . .
∫
dDk1 . . . d
Dk4
(−(k1 + p1)2)a1(−(k2 + p1)2)a2(−(k3 + p1)2)a3
×1
(−(k4 + p1)2)a4(−(k1 − p2)2)a5(−(k2 − p2)2)a6(−(k3 − p2)2)a7
×1
(−(k4 − p2)2)a8(−k21 )
a9(−k22 )
a10(−k23 )
a11(−k24 )
a12
×1
(−(k1 − k2)2)−a13(−(k1 − k3)2)−a14(−(k1 − k4)2)−a15
×1
(−(k2 − k3)2)−a16(−(k2 − k4)2)−a17(−(k3 − k4)2)−a18
.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
All planar four-loop on-shell form-factor integrals withp2
1 = p22 = 0, with q2 ≡ p2
3 = (p1 + p2)2
Fa1, . . . , a18 =
∫
. . .
∫
dDk1 . . . d
Dk4
(−(k1 + p1)2)a1(−(k2 + p1)2)a2(−(k3 + p1)2)a3
×1
(−(k4 + p1)2)a4(−(k1 − p2)2)a5(−(k2 − p2)2)a6(−(k3 − p2)2)a7
×1
(−(k4 − p2)2)a8(−k21 )
a9(−k22 )
a10(−k23 )
a11(−k24 )
a12
×1
(−(k1 − k2)2)−a13(−(k1 − k3)2)−a14(−(k1 − k4)2)−a15
×1
(−(k2 − k3)2)−a16(−(k2 − k4)2)−a17(−(k3 − k4)2)−a18
.
At most 12 indices can be positive.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
FIRE → 99 master integrals.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
FIRE → 99 master integrals.
[J. Henn, A.&V. Smirnov’13]: introduce an additional scale.p2
2 6= 0
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
FIRE → 99 master integrals.
[J. Henn, A.&V. Smirnov’13]: introduce an additional scale.p2
2 6= 0
504 master integrals
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
FIRE → 99 master integrals.
[J. Henn, A.&V. Smirnov’13]: introduce an additional scale.p2
2 6= 0
504 master integrals
Use differential equations[A.V. Kotikov’91, Bern, Dixon & Kosower’94, E. Remiddi’97,T. Gehrmann & E. Remiddi’00, J. Henn’13,...]
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
FIRE → 99 master integrals.
[J. Henn, A.&V. Smirnov’13]: introduce an additional scale.p2
2 6= 0
504 master integrals
Use differential equations[A.V. Kotikov’91, Bern, Dixon & Kosower’94, E. Remiddi’97,T. Gehrmann & E. Remiddi’00, J. Henn’13,...]
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
FIRE → 99 master integrals.
[J. Henn, A.&V. Smirnov’13]: introduce an additional scale.p2
2 6= 0
504 master integrals
Use differential equations[A.V. Kotikov’91, Bern, Dixon & Kosower’94, E. Remiddi’97,T. Gehrmann & E. Remiddi’00, J. Henn’13,...]
[J. Henn’13]: use basis of integrals with constant leadingsingularities
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
Let f = (f1, . . . , fN) be primary master integrals (MI) for agiven family of dimensionally regularized (with D = 4 − 2ǫ)Feynman integrals.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
Let f = (f1, . . . , fN) be primary master integrals (MI) for agiven family of dimensionally regularized (with D = 4 − 2ǫ)Feynman integrals.
Let x = (x1, . . . , xn) be kinematical variables and/or masses,or some new variables introduced to ‘get rid of square roots’.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
Let f = (f1, . . . , fN) be primary master integrals (MI) for agiven family of dimensionally regularized (with D = 4 − 2ǫ)Feynman integrals.
Let x = (x1, . . . , xn) be kinematical variables and/or masses,or some new variables introduced to ‘get rid of square roots’.
DE:
∂i f (ǫ, x) = Ai(ǫ, x)f (ǫ, x) ,
where ∂i =∂∂xi
, and each Ai is an N × N matrix.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
Let f = (f1, . . . , fN) be primary master integrals (MI) for agiven family of dimensionally regularized (with D = 4 − 2ǫ)Feynman integrals.
Let x = (x1, . . . , xn) be kinematical variables and/or masses,or some new variables introduced to ‘get rid of square roots’.
DE:
∂i f (ǫ, x) = Ai(ǫ, x)f (ǫ, x) ,
where ∂i =∂∂xi
, and each Ai is an N × N matrix.
[J. Henn’13]: turn to a new basis where DE take the form
∂i f (ǫ, x) = ǫAi(x)f (ǫ, x) .
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
In the case of two scales, i.e. with one variable in the DE, i.e.n = 1.
f ′(ǫ, x) = ǫ∑
k
ak
x − x (k)f (ǫ, x) .
where x (k) is the set of singular points of the DE and N × N
matrices ak are independent of x and ǫ.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
In the case of two scales, i.e. with one variable in the DE, i.e.n = 1.
f ′(ǫ, x) = ǫ∑
k
ak
x − x (k)f (ǫ, x) .
where x (k) is the set of singular points of the DE and N × N
matrices ak are independent of x and ǫ.
For example, if xk = 0,−1, 1 then results are expressed interms of HPLs [E. Remiddi & J.A.M. Vermaseren]
H(a1, a2, . . . , an; x) =
∫ x
0
f (a1; t)H(a2, . . . , an; t)dt ,
where f (±1; t) = 1/(1 ∓ t), f (0; t) = 1/t
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
How to turn to a UT basis?
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
How to turn to a UT basis?
[J. Henn’13] Select basis integrals that have constantleading singularities [F. Cachazo’08] which aremultidimensional residues of the integrand. (Replacepropagators by delta functions).Based on experience in super Yang-Mills and a conjectureby [N. Arkani-Hamed et al.’12]
Transformations of the system of differential equationsbased on its singularities[Moser’59],[J. Henn’14],[R.N. Lee’14]
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
How to turn to a UT basis?
[J. Henn’13] Select basis integrals that have constantleading singularities [F. Cachazo’08] which aremultidimensional residues of the integrand. (Replacepropagators by delta functions).Based on experience in super Yang-Mills and a conjectureby [N. Arkani-Hamed et al.’12]
Transformations of the system of differential equationsbased on its singularities[Moser’59],[J. Henn’14],[R.N. Lee’14]
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
How to turn to a UT basis?
[J. Henn’13] Select basis integrals that have constantleading singularities [F. Cachazo’08] which aremultidimensional residues of the integrand. (Replacepropagators by delta functions).Based on experience in super Yang-Mills and a conjectureby [N. Arkani-Hamed et al.’12]
Transformations of the system of differential equationsbased on its singularities[Moser’59],[J. Henn’14],[R.N. Lee’14]
If you have almost reached the ε-form, make a small finalrotation of the current basis.See, e.g., [S. Caron-Huot, J. Henn,’14], [T. Gehrmann,A. von Manteuffel, L. Tancredi and E. Weihs’14]
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
We obtain differential equations with respect to x = p22/p
23
∂x f (x , ǫ) = ǫ
[
a
x+
b
1 − x
]
f (x , ǫ)
where a and b are x- and ǫ-independent 504 × 504 matrices.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
We obtain differential equations with respect to x = p22/p
23
∂x f (x , ǫ) = ǫ
[
a
x+
b
1 − x
]
f (x , ǫ)
where a and b are x- and ǫ-independent 504 × 504 matrices.
Solving these equations in terms of HPL with letters 0 and 1.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
Asymptotic behaviour at the points x = 0 and x = 1
f (x , ǫ)x→0=
[
1 +∑
k≥1
pk(ǫ)xk
]
x ǫaf0(ǫ) ,
f (x , ǫ)x→1=
[
1 +∑
k≥1
qk(ǫ)(1 − x)k
]
(1 − x)−ǫbf1(ǫ) ,
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
Asymptotic behaviour at the points x = 0 and x = 1
f (x , ǫ)x→0=
[
1 +∑
k≥1
pk(ǫ)xk
]
x ǫaf0(ǫ) ,
f (x , ǫ)x→1=
[
1 +∑
k≥1
qk(ǫ)(1 − x)k
]
(1 − x)−ǫbf1(ǫ) ,
Natural boundary conditions at the point x = 1:there is no singularity and it corresponds to propagator-typeintegrals which are known[P. Baikov and K. Chetyrkin’10; R. Lee and V. Smirnov’11]
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
Asymptotic behaviour at the points x = 0 and x = 1
f (x , ǫ)x→0=
[
1 +∑
k≥1
pk(ǫ)xk
]
x ǫaf0(ǫ) ,
f (x , ǫ)x→1=
[
1 +∑
k≥1
qk(ǫ)(1 − x)k
]
(1 − x)−ǫbf1(ǫ) ,
Natural boundary conditions at the point x = 1:there is no singularity and it corresponds to propagator-typeintegrals which are known[P. Baikov and K. Chetyrkin’10; R. Lee and V. Smirnov’11]
To obtain analytical results for the 99 one-scale MI, weperform (with the help of the HPL package [D. Maıtre’06])matching at the point x = 0.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
Transporting boundary conditions at x = 1 to the point x = 0.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
Transporting boundary conditions at x = 1 to the point x = 0.
0 1
p1 p2
p3
Re(x)
Im(x)
p1 p2
p3
x = 0
p2
p3
x = 1
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
In mathematics, the object transporting the boundaryinformation is called the Drinfeld associator.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
In mathematics, the object transporting the boundaryinformation is called the Drinfeld associator.We construct the Drinfeld associator perturbatively in ǫ.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
In mathematics, the object transporting the boundaryinformation is called the Drinfeld associator.We construct the Drinfeld associator perturbatively in ǫ.
x ǫa =∑
x jεaj
Results for the 99 one-scale MI can be obtained from the‘naive’ part of the asymptotic expansion at x = 0, i.e. fromthe part corresponding to the zero eigenvalue of the matrix a,i.e. to j = 0.
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
In mathematics, the object transporting the boundaryinformation is called the Drinfeld associator.We construct the Drinfeld associator perturbatively in ǫ.
x ǫa =∑
x jεaj
Results for the 99 one-scale MI can be obtained from the‘naive’ part of the asymptotic expansion at x = 0, i.e. fromthe part corresponding to the zero eigenvalue of the matrix a,i.e. to j = 0.
Two alternative descriptions of the asymptotic expansion:with DE and with with expansion by regions[M. Beneke and V. Smirnov’98]
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
In mathematics, the object transporting the boundaryinformation is called the Drinfeld associator.We construct the Drinfeld associator perturbatively in ǫ.
x ǫa =∑
x jεaj
Results for the 99 one-scale MI can be obtained from the‘naive’ part of the asymptotic expansion at x = 0, i.e. fromthe part corresponding to the zero eigenvalue of the matrix a,i.e. to j = 0.
Two alternative descriptions of the asymptotic expansion:with DE and with with expansion by regions[M. Beneke and V. Smirnov’98]
j = 0 corresponds to the hard-. . . -hard region while terms withj < 0 to other regions (soft, collinear, . . . ).
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
In mathematics, the object transporting the boundaryinformation is called the Drinfeld associator.We construct the Drinfeld associator perturbatively in ǫ.
x ǫa =∑
x jεaj
Results for the 99 one-scale MI can be obtained from the‘naive’ part of the asymptotic expansion at x = 0, i.e. fromthe part corresponding to the zero eigenvalue of the matrix a,i.e. to j = 0.
Two alternative descriptions of the asymptotic expansion:with DE and with with expansion by regions[M. Beneke and V. Smirnov’98]
j = 0 corresponds to the hard-. . . -hard region while terms withj < 0 to other regions (soft, collinear, . . . ). No positive j .
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
An example of our result
replacements
p1 p2k1 k2 k3
k4 + p1
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
An example of our result
replacements
p1 p2k1 k2 k3
k4 + p1
I12 =
∫
. . .
∫ 4∏
j=1
dDkj
(k24 )
2
k21k
22k
23 (k1 − k2)2(k2 − k3)2(k1 − k4)2
×1
(k2 − k4)2(k3 − k4)2(k1 + p1)2(k4 + p1)2(k4 − p2)2(k3 − p2)2
On the four-loop cusp anomalous dimension
Massless planar four-loop vertex integrals
=1
576ε8+
1
216π2 1
ǫ6+
151
864ζ3
1
ǫ5+
173
10368π4 1
ǫ4+
[
505
1296π2ζ3 +
5503
1440ζ5
]
1
ǫ3+
+
[
6317
155520π6 +
9895
2592ζ23
]
1
ǫ2+
[
89593
77760π4ζ3 +
3419
270π2ζ5 −
169789
4032ζ7
]
1
ǫ
+
[
407
15s8a +
41820167
653184000π8 +
41719
972π2ζ2
3 −263897
2160ζ3ζ5
]
+O(ǫ) ,
On the four-loop cusp anomalous dimension
Perspectives
General Nc . Non-planar diagrams
On the four-loop cusp anomalous dimension
Perspectives
General Nc . Non-planar diagrams
Other form-factors. More complicated integrals.
On the four-loop cusp anomalous dimension
Perspectives
General Nc . Non-planar diagrams
Other form-factors. More complicated integrals.
No conceptual problems. More powerful algorithms forIBP reduction. More powerful machines.