Lectures on multiloop calculations
Andrey Grozin
Institut fur Theoretische Teilchenphysik, Universitat Karlsruhe
CALC-2003, Dubna, June 1321 p. 1/128
Plan Massless propagator diagrams
HQET propagator diagrams
Massive on-shell propagator diagrams
Expansion of hypergeometric functions in
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Plan Massless propagator diagrams
1 loop 2 loops 3 loops master integral G with non-integer index master integral non-planar
HQET propagator diagrams
Massive on-shell propagator diagrams
Expansion of hypergeometric functions in
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Plan Massless propagator diagrams
HQET propagator diagrams crash course of HQET 1 loop 2 loops 3 loops master integral (my) master integral (Beneke-Braun)
Massive on-shell propagator diagrams
Expansion of hypergeometric functions in
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Plan Massless propagator diagrams
HQET propagator diagrams
Massive on-shell propagator diagrams 1 loop 2 loops 2 loops, 2 non-zero masses 3 loops 3-loop HQET master integral(Czarnecki-Melnikov)
Expansion of hypergeometric functions in
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Plan Massless propagator diagrams
HQET propagator diagrams
Massive on-shell propagator diagrams
Expansion of hypergeometric functions in multiple values expansion example expansion algorithm
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Scalar integrals
Tensor integrals can be expanded in tensor structuresCoefficients scalar integrals, projectors
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Scalar integrals
Tensor integrals can be expanded in tensor structuresCoefficients scalar integrals, projectors
Expressing scalar products in the numeratorvia the denominatorsSometimes, there are irreducible scalar products
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Scalar integrals
Tensor integrals can be expanded in tensor structuresCoefficients scalar integrals, projectors
Expressing scalar products in the numeratorvia the denominatorsSometimes, there are irreducible scalar products
=n1 n2
n1 + n2
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1 loop
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Diagram
k + p
k
n1
n2
ddk
Dn11 Dn22
= id/2(p2)d/2n1n2G(n1, n2)
D1 = (k + p)2 i0 D2 = k
2 i0
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Integer n1 0
= 0
Numerator D|n1|1
ddk
(k2 i0)n= 0
Dimensionality d 2n, no dimensional parameter
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Fourier transform
eipx
(p2 i0)nddp
(2)d=
i22nd/2(d/2 n)
(n)
1
(x2 + i0)d/2n
eipx
(x2 + i0)nddx =
i2d2nd/2(d/2 n)
(n)
1
(p2 i0)d/2n
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x space
22(n1+n2)d(d/2 n1)(d/2 n2)
(n1)(n2)
1
(x2)dn1n2
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x space
22(n1+n2)d(d/2 n1)(d/2 n2)
(n1)(n2)
1
(x2)dn1n2
G(n1, n2) =
(d/2 + n1 + n2)(d/2 n1)(d/2 n2)
(n1)(n2)(d n1 n2)
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Inversion
Euclidean momentum k0 = ikE0, k2 = k2E
Dimensionless K = kE/p2
ddK
[(K + n)2]n1 [K2]n2= d/2G(n1, n2)
n Euclidean vector, n2 = 1
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Inversion
Euclidean momentum k0 = ikE0, k2 = k2E
Dimensionless K = kE/p2
ddK
[(K + n)2]n1 [K2]n2= d/2G(n1, n2)
n Euclidean vector, n2 = 1Inversion K = K /K 2
ddK = ddK /(K 2)d
K2 =1
K 2(K + n)2 =
(K + n)2
K 2
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Inversion
=
n1
n2
n1
d n1 n2
G(n1, n2) = G(n1, d n1 n2)
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2 loops
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Diagramk1 k2
k1 + p k2 + p
k1 k2n1 n2
n3 n4n5
ddk1 d
dk2Dn11 D
n22 D
n33 D
n44 D
n55
=
d(p2)d
niG(n1, n2, n3, n4, n5)
D1 = (k1 + p)2 D2 = (k2 + p)
2
D3 = k21 D4 = k
22 D5 = (k1 k2)
2
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Trivial case n5 = 0
n1 n2
n3 n4
G(n1, n2, n3, n4, 0) = G(n1, n3)G(n2, n4)
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Trivial case n1 = 0
=
n2
n3 n4
n5
n5
n3
n2
n4 + n3 + n5 d/2
G(0, n2, n3, n4, n5) = G(n3, n5)G(n2, n4+n3+n5d/2)
Inner loopG(n3, n5)
(k22)n3+n5d/2
Symmetric: n2 = 0, n3 = 0, n4 = 0
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Integration by parts
When applied to the integrand, derivative
k2
n2D2
2(k2 + p) +n4D4
2k2 +n5D5
2(k2 k1)
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Integration by parts
When applied to the integrand, derivative
k2
n2D2
2(k2 + p) +n4D4
2k2 +n5D5
2(k2 k1)
Applying (/k2) k2 to the integrand and using
k22 = D4 2(k2 + p) k2 = (p2)D2 D4
2(k2 k1) k2 = D3 D4 D5
we obtain
d n2 n5 2n4 +n2D2
((p2)D4) +n5D5
(D3D4)
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Integration by parts
Notation
1G(n1, n2, n3, n4, n5) = G(n1 1, n2, n3, n4, n5)
Triangle relation[d n2 n5 2n4
+ n22+(1 4) + n55
+(3 4)]G = 0
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Integration by parts
Notation
1G(n1, n2, n3, n4, n5) = G(n1 1, n2, n3, n4, n5)
Triangle relation[d n2 n5 2n4
+ n22+(1 4) + n55
+(3 4)]G = 0
Applying (/k2) (k2 k1)[d n2 n4 2n5
+ n22+(1 5) + n44
+(3 5)]G = 0
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Homogeneity
Applying p (/p) we get 2(d
ni):[2(d n3 n4 n5) n1 n2
+ n11+(1 3) + n22
+(1 4)]I = 0
This is the sum of the (/k2) k2 relationand the symmetric (/k1) k1 one
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Larin relationInsert (k1 + p)
to the integrand:
k1 + p(k1 + p) p
p2p =
(1 +
D1 D3p2
)p
2
Taking /p(32d
ni
)(1 +
D1 D3p2
)
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Larin relationExplicit differentiation
d+n1D1
2(k1 + p)2 +
n2D2
2(k2 + p) (k1 + p)
2(k2 + p) (k1 + p) = D5 D1 D2
Therefore[12d+ n1 n3 n4 n5 +
(32d
ni
)(1 3)
+ n22+(1 5)
]G = 0
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Integration by parts
G =n22
+(5 1) + n44+(5 3)
d n2 n4 2n5G
n1 + n3 + n5 reduces by 1
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Integration by parts
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Integration by parts
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Integration by parts
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Topology
1 generic topology (for all integer ni)
Basis (all ni = 1)
= G21 = G2
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Sunset
Gn =1(
n+ 1 nd2)n
((n+ 1)d2 2n 1
)n
(1 + n)n+1(1 )
(1 (n+ 1))
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Inversion
=
n1 n2
n3 n4
n5
n1 n2
d n1 n3 n5
d n2 n4 n5
n5
K 1 = K1/K21 , K
2 = K2/K
22
(K1 K2)2 =
(K 1 K2)
2
K 21 K22
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Inversion
=
n1 n2
n3 n4
n5
n1 n2
d n1 n3 n5
d n2 n4 n5
n5
K 1 = K1/K21 , K
2 = K2/K
22
(K1 K2)2 =
(K 1 K2)
2
K 21 K22
Z2 S6 symmetry group 1440 elements
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3 loops
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Topologies
3 generic topologies (for all integer ni, numerators)
Chetyrkin, Tkachov 1981; Mincer
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Basis
= G31 = G1G2
= G3 G3G21G2
= G1G(1, 1, 1, 1, )
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Inversion
=
=
n1 n2
n3 n4
n5
n6
n7 n8
n1 n2
d n1 n3 n5 n7
d n2 n4 n5 n8
n5
d n6 n7 n8
n7 n8
n1 n2
n3
n4 n5n6 n7
n8
n1 n2
n3
d n1 n3 n6
d n2 n4 n7
n6 n7
d n3 n6 n7 n8
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InversionAll ni = 1, d = 4
=
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G with non-integer indices
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G(1, 1, 1, 1, n)
G(1, 1, 1, 1, n) = 2(d2 1
)(d2 n 1
)(n d+ 3)[
2(d2 1
)(d 2n 4)(n+ 1)
(32d n 4
) 3F2
(1, d 2, n d2 + 2
n+ 1, n d2 + 3
1) cot (d n)(d 2)]
Kotikov 1996Kazakov 1985Broadhurst 199397
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G(1, 1, 1, 1, n)
Shifting n5 by 1[(d 2n5 4)5
+ + 2(d n5 3)]G(1, 1, 1, 1, n5)
= 21+(3 25+)G(1, 1, 1, 1, n5)
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G(1, 1, 1, 1, n + )
G(1, 1, 1, 1, 2 + ) =4(1 + 3)(1 )
1 + 2[(1 2)(1 )
2(2 + )(1 + )2(1 4)(1 + )
3F2
(1, 2 2, 2 + 2
3 + , 3 + 2
1)+ cot 34(1 2)]
3F2
(1, 2 2, 2 + 2
3 + , 3 + 2
1) = 4(2 1)+6(43 + 32 1)+ 2(414 543 + 222 9)
2
+3(1245 + 2423 + 1234 883 + 302 8)3 +
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Non-planar diagram
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Gluing
=205 +O()
(p2)2+3
= 205 1
4+O(1)
=205 +O()
(p2)2+3
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HQET
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Lagrangian
Single heavy quark |~p | . E, |p0 m| . E
Light quarks and gluons |~ki| . E, |k0i| . EHQET = QCD expanded to some order in E/mResidual energy p0 = p0 m(vacuum has energym)
Dispersion law p0 =m2 + ~p 2 m 0
Two-component spinor Q(or 4-component 0Q = Q)
L = Q+iD0Q+
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Feynman rules
Propagator
S(p) =1
p0 + i0S(x) = i(x0)(~x )
Vertex ig0 ta
Loops vanish
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Covariant notations
Lv = Qviv DQv +
p = mv + p |p | m /vQv = Qv
S(p) =1 + /v
2
1
p v + i0
Vertex igvta
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Covariant notations
Lv = Qviv DQv +
p = mv + p |p | m /vQv = Qv
S(p) =1 + /v
2
1
p v + i0
Vertex igvta
QCD propagator
S(p) =/p+m
p2 m2=m(1 + /v) + /p
2mp v + p 2=
1 + /v
2
1
p v+
Vertex igta igvta between (1 + /v)/2
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1 loop
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Diagram
k0 +
k
n1
n2
ddk
Dn11 Dn22
= id/2(2)d2n2I(n1, n2)
D1 =k0 +
D2 = k
2
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Fourier transform
+
eit
( i0)nd
2=
in
(n)tn1(t)
0
eittn dt =(i)n+1(n+ 1)
( i0)n+1
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Fourier transform
+
eit
( i0)nd
2=
in
(n)tn1(t)
0
eittn dt =(i)n+1(n+ 1)
( i0)n+1
x-space x2 = (it)2 (t = itE)
22n2d/2(d/2 n2)
(n1)(n2)(it)n1+2n2d1(t)
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II(n1, n2) =(d+ n1 + 2n2)(d/2 n2)
(n1)(n2)
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2 loops
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Diagram I
k1 k2
k10 + k20 +
k1 k2n1 n2
n3 n4n5
ddk1 d
dk2Dn11 D
n22 D
n33 D
n44 D
n55
=
d(2)2(dn3n4n5)I(n1, n2, n3, n4, n5)
D1 =k10 +
D2 =
k20 +
D3 = k
21 D4 = k
22 D5 = (k1 k2)
2
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Trivial case n5 = 0
n1 n2
n3 n4
I(n1, n2, n3, n4, 0) = I(n1, n3)I(n2, n4)
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Trivial cases n1 = 0, n3 = 0
=
n2
n3 n4
n5
n5
n3
n2
n4 + n3 + n5 d/2
I(0, n2, n3, n4, n5) = G(n3, n5)I(n2, n4+n3+n5d/2)
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Trivial cases n1 = 0, n3 = 0
=
n2
n3 n4
n5
n5
n3
n2
n4 + n3 + n5 d/2
I(0, n2, n3, n4, n5) = G(n3, n5)I(n2, n4+n3+n5d/2)
=
n1 n2
n4
n5n1
n5
n2 + n1 + 2n5 d
n4
I(n1, n2, 0, n4, n5) = I(n1, n5)I(n2 + n1 +2n5 d, n4)
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Integration by parts
When applied to the integrand
k2
n2D2
v
+
n4D4
2k2 +n5D5
2(k2 k1)
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Integration by parts
When applied to the integrand
k2
n2D2
v
+
n4D4
2k2 +n5D5
2(k2 k1)
Applying (/k2) k2, (/k2) (k2 k1) and usingk2v/ = D2 1, 2(k2 k1) k2 = D3 D4 D5:
d n2 n5 2n4 +n2D2
+n5D5
(D3 D4)
d n2 n4 2n5 +n2D2
D1 +n4D4
(D3 D5)
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Integration by parts
[d n2 n5 2n4 + n22
+ + n55+(3 4)
]I = 0[
d n2 n4 2n5 + n22+1 + n44
+(3 5)]I = 0
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Integration by parts
[d n2 n5 2n4 + n22
+ + n55+(3 4)
]I = 0[
d n2 n4 2n5 + n22+1 + n44
+(3 5)]I = 0
Applying (/k2) v[2n22
+ + n44+(2 1) + n55
+(2 1)]I = 0
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Homogeneity
Applying (d/d) to n1n2I:[2(d n3 n4 n5) n1 n2 + n11
+ + n22+]I = 0
This is the sum of the (/k2) k2 relationand the symmetric (/k1) k1 one
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Homogeneity
Applying (d/d) to n1n2I:[2(d n3 n4 n5) n1 n2 + n11
+ + n22+]I = 0
This is the sum of the (/k2) k2 relationand the symmetric (/k1) k1 oneThe (/k2) (k2 k1) relation minus 1
times thehomogeneity relation:[
d n1 n2 n4 2n5 + 1
(2(d n3 n4 n5) n1 n2 + 1
)1
+ n44+(3 5)
]I = 0
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Integration by parts
I =
(2(d n3 n4 n5) n1 n2 + 1)1 + n44
+(5 3)
d n1 n2 n4 2n5 + 1I
n1 + n3 + n5 reduces by 1
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Diagram J
k2
k1
k10 + k10 + k20 +
k20 + n1 n3
n2
n4
n5ddk1 d
dk2Dn11 D
n22 D
n33 D
n44 D
n55
=
d(2)2(dn4n5)J(n1, n2, n3, n4, n5)
D1 =k10 +
D2 =
k20 +
D3 =(k1 + k2)0 +
D4 = k
21 D5 = k
22
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Partial fractioning
Trivial cases: n3 = 0, n1,2 = 0Denominators are linearly dependent:
D1 +D2 D3 = 1
J = (1 + 2 3)J
n1 + n2 + n3 reduces by 1Numerator (k1 k2)
n not a problem
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Topologies
2 generic topologies (for all integer ni)
Broadhurst, Grozin 1991
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Topologies
2 generic topologies (for all integer ni)
Broadhurst, Grozin 1991
Basis (all ni = 1)
= I21 = I2
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Sunset
In =(1 + 2n)n(1 )
(1 n(d 2))2n
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3 loops
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Topologies
Grozin 2000Grinder
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Basis
= I31 = I1I2
= I3
I3I21I2
I3G21G2
= G1I(1, 1, 1, 1, )
= I1J(1, 1,1 + 2, 1, 1)
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J with non-integer indices
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J(1, 1, n, 1, 1)
0
t1t2
tn
0
J(1, 1, n, 1, 1)
0
t1t2
tn
0
J(1, 1, n, 1, 1)
0
t1t2
1n
J(1, 1, n, 1, 1) =(n 2d+ 6)2(d/2 1)
(n)J
J =
0
J(1, 1, n, 1, 1)
0
t1t2
1n
J(1, 1, n, 1, 1) =(n 2d+ 6)2(d/2 1)
(n)J
J =
0
J(1, 1, n, 1, 1)
(1 xt)2d =k=0
(d 2)kk!
(xt)k
(x)k =k1i=0
(x+ i) =(x+ k)
(x)
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J(1, 1, n, 1, 1)
(1 xt)2d =k=0
(d 2)kk!
(xt)k
(x)k =k1i=0
(x+ i) =(x+ k)
(x)
J = (n)k=0
(d 2)k(n d+ k + 3)(n+ k + 1)
=1
n(n d+ 3)
k=0
(d 2)k(n d+ 3)k(n+ 1)k(n d+ 4)k
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J(1, 1, n, 1, 1)
3F2
(a1, a2, a3b1, b2
x) = k=0
(a1)k(a2)k(a3)k(b1)k(b2)k
xk
k!
(1)k = k!
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J(1, 1, n, 1, 1)
3F2
(a1, a2, a3b1, b2
x) = k=0
(a1)k(a2)k(a3)k(b1)k(b2)k
xk
k!
(1)k = k!
J =1
n(n d+ 3)3F2
(1, d 2, n d+ 3
n+ 1, n d+ 4
1)J(1, 1, n, 1, 1) =
(n 2d+ 6)2(d/2 1)
(n d+ 3)(n+ 1)
3F2
(1, d 2, n d+ 3
n+ 1, n d+ 4
1)CALC-2003, Dubna, June 1321 p. 63/128
J(n1, n2, n3, n4, n5)
J(n1, n2, n3, n4, n5) =
(n1 + n2 + n3 + 2(n4 + n5 d))
(n1 + n3 + 2n4 d)
(n1 + n2 + n3 + 2n4 d)
(d/2 n4)(d/2 n5)
(n4)(n5)(n1 + n3)
3F2
(n1, d 2n5, n1 + n3 + 2n4 d
n1 + n3, n1 + n2 + n3 + 2n4 d
1)Grozin 2000
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J(1, 1, n + 2, 1, 1)
Shifting n3 by 1: (1 1 2 + 3)J = 0
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J(1, 1, n + 2, 1, 1)
Shifting n3 by 1: (1 1 2 + 3)J = 0
J(1, 1, 2 + 2, 1, 1) =1
3(d 4)(d 5)(d 6)(2d 9)
(1 + 6)2(1 )
(1 + 2)3F2
(1, 2 2, 1 + 4
3 + 2, 2 + 4
1)3F2
(1, 2 2, 1 + 4
3 + 2, 2 + 4
1) = 2 + 6(22 + 3)+12(103 112 + 6)
2 + 24(284 + 553 272 + 9)3
+48(945 1623 1544 + 1353 452 + 12)4 +
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I with non-integer indices
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I(1, 1, 1, 1, n)
0 vt1v(t1 t)
x
Euclidean x-space ( = d/2 1)
t0
dt1
ddx
1
(x2)d/2n [(x vt1)2] [(x+ v(t t1))2]
=2d/2
(d/2)It2(nd)+5
CALC-2003, Dubna, June 1321 p. 67/128
I(1, 1, 1, 1, n)
0 vt1v(t1 t)
x
Euclidean x-space ( = d/2 1)
t0
dt1
ddx
1
(x2)d/2n [(x vt1)2] [(x+ v(t t1))2]
=2d/2
(d/2)It2(nd)+5
I(1, 1, 1, 1, n) =2
(2(n d+ 3))(d/2 n)2(d/2 1)
(d/2)(n)I
ddx =2d/2
(d/2)xd1dx dx
dx = 1
CALC-2003, Dubna, June 1321 p. 67/128
Gegenbauer polynomials
0 vtv(t 1)
x
I =
10
dt
0
dx dxx2n1
[(x vt)2] [(x+ v(1 t))2]
CALC-2003, Dubna, June 1321 p. 68/128
Gegenbauer polynomials
0 vtv(t 1)
x
I =
10
dt
0
dx dxx2n1
[(x vt)2] [(x+ v(1 t))2]
1
[(x y)2]=
1
max2(x, y)
k=0
T k(x, y)Ck (x y)
T (x, y) = min
(x
y,y
x
)CALC-2003, Dubna, June 1321 p. 68/128
Gegenbauer polynomials
dx Ck1(a x)C
k2(b x) = k1k2
+ k1Ck1(a b)
Ck (1) = (1)kCk (1) = (1)
k(2+ k)
k! (2)
CALC-2003, Dubna, June 1321 p. 69/128
I(1, 1, 1, 1, n)
I =d 2
(d 2)
10
dt
k=0
(1)k
k!
(d+ k 2)
d+ 2k 2Ik(t)
CALC-2003, Dubna, June 1321 p. 70/128
I(1, 1, 1, 1, n)
I =d 2
(d 2)
10
dt
k=0
(1)k
k!
(d+ k 2)
d+ 2k 2Ik(t)
Ik(t) =
0
dxx2n1 [T (x, t)T (x, 1 t)]k
[max(x, t)max(x, 1 t)]d2
CALC-2003, Dubna, June 1321 p. 70/128
I(1, 1, 1, 1, n)
I =d 2
(d 2)
10
dt
k=0
(1)k
k!
(d+ k 2)
d+ 2k 2Ik(t)
Ik(t) =
0
dxx2n1 [T (x, t)T (x, 1 t)]k
[max(x, t)max(x, 1 t)]d2
t < 12 : Ik(t) =
t0
dxx2n1
[xt
x1t
]k[t(1 t)]d2
+
1tt
dxx2n1
[tx
x1t
]k[x(1 t)]d2
+
1t
dxx2n1
[tx1tx
]k[x2]d2
CALC-2003, Dubna, June 1321 p. 70/128
I(1, 1, 1, 1, n)
I =d 2
(d 2n 2)(d 2)
1/20
dt
k=0
(1)k
k!(d+ k 2)
[td+2n+k+2(1 t)dk+2
n+ ktk(1 t)2d+2nk+4
d n+ k 2
]
CALC-2003, Dubna, June 1321 p. 71/128
I(1, 1, 1, 1, n)
I(1, 1, 1, 1, n) = 2
(d
2 1
)
(d
2 n 1
)[
(2n 2d+ 6)
(2n d+ 3)(n+ 1)3F2
(1, d 2, 2n d+ 3
n+ 1, 2n d+ 4
1)
(d n 2)2(n d+ 3)
(d 2)
]
CALC-2003, Dubna, June 1321 p. 72/128
I(1, 1, 1, 1, n)
I(1, 1, 1, 1, n) =(d2 1
)(d2 n 1
)(d 2)[
2(2n d+ 3)(2n 2d+ 6)
(n d+ 3)(3n 2d+ 6)
3F2
(n d+ 3, n d+ 3, 2n 2d+ 6
n d+ 4, 3n 2d+ 6
1) (d n 2)2(n d+ 3)
]Beneke, Braun 1994
CALC-2003, Dubna, June 1321 p. 73/128
I(1, 1, 1, 1, n)
Shifting n5 by 1:[(d 2n5 4)5
+ 2(d n5 3)]I(1, 1, 1, 1, n5) =[
(2d 2n5 7)15+ 34+5 + 13+
]I(1, 1, 1, 1, n5)
CALC-2003, Dubna, June 1321 p. 74/128
I(1, 1, 1, 1, n + )
I(1, 1, 1, 1, 1 + ) =4(1 )
9(d 3)(d 4)2[(1 + 4)(1 + 6)
(1 + 7)3F2
(3, 3, 6
1 + 3, 1 + 7
1)
2(1 + 3)(1 3)
]
3F2
(3, 3, 6
1 + 3, 1 + 7
1) = 1 + 5433 51344+ 54(255 + 2823)
5 +
CALC-2003, Dubna, June 1321 p. 75/128
On-shell diagrams
CALC-2003, Dubna, June 1321 p. 76/128
Why?
On-shell mass and wave-function renormalization(needed for all scattering amplitudes)
CALC-2003, Dubna, June 1321 p. 77/128
Why?
On-shell mass and wave-function renormalization(needed for all scattering amplitudes)
Magnetic moments, charge radii
CALC-2003, Dubna, June 1321 p. 77/128
Why?
On-shell mass and wave-function renormalization(needed for all scattering amplitudes)
Magnetic moments, charge radii
QCD/HQET matching
CALC-2003, Dubna, June 1321 p. 77/128
1 loop
k +mv
k
n1
n2
ddk
Dn11 Dn22
= id/2md2(n1+n2)M(n1, n2)
D1 = m2 (k +mv)2 D2 = k
2
CALC-2003, Dubna, June 1321 p. 78/128
InversionDimensionless Euclidean momentum K = kE/m
ddK
(K2 2iK0)n1(K2)n2= d/2M(n1, n2)
CALC-2003, Dubna, June 1321 p. 79/128
InversionDimensionless Euclidean momentum K = kE/m
ddK
(K2 2iK0)n1(K2)n2= d/2M(n1, n2)
HQET K = kE/(2)ddK
(1 2iK0)n1(K2)n2= d/2I(n1, n2)
CALC-2003, Dubna, June 1321 p. 79/128
InversionDimensionless Euclidean momentum K = kE/m
ddK
(K2 2iK0)n1(K2)n2= d/2M(n1, n2)
HQET K = kE/(2)ddK
(1 2iK0)n1(K2)n2= d/2I(n1, n2)
Inversion K = K /K 2
K2 2iK0 =1 2iK 0
K 2
CALC-2003, Dubna, June 1321 p. 79/128
Inversion
=
n1
n2
n1
d n1 n2
M(n1, n2) = I(n1, d n1 n2) =
(d n1 2n2)(d/2 + n1 + n2)
(n1)(d n1 n2)
CALC-2003, Dubna, June 1321 p. 80/128
2 loops
CALC-2003, Dubna, June 1321 p. 81/128
DiagramM
k1 k2
k1 +mv k2 +mv
k1 k2n1 n2
n3 n4n5
ddk1 d
dk2Dn11 D
n22 D
n33 D
n44 D
n55
=
dm2(d
ni)M(n1, n2, n3, n4, n5)
D1 = m2 (k1 +mv)
2 D2 = m2 (k2 +mv)
2
D3 = k21 D4 = k
22 D5 = (k1 k2)
2
CALC-2003, Dubna, June 1321 p. 82/128
Diagram N
k2
k1
k1 +mv k1 + k2 +mv
k2 +mvn1 n3
n2
n4
n5ddk1 d
dk2Dn11 D
n22 D
n33 D
n44 D
n55
=
dm2(d
ni)N(n1, n2, n3, n4, n5)
D1 = m2 (k1 +mv)
2 D2 = m2 (k2 +mv)
2
D3 = m2 (k1 + k2 +mv)
2
D4 = k21 D5 = k
22
CALC-2003, Dubna, June 1321 p. 83/128
Topologies
2 generic topologies (for all integer ni)
Broadhurst 199092 RecursorFleischer, Tarasov 1992 SHELL2
CALC-2003, Dubna, June 1321 p. 84/128
Topologies
2 generic topologies (for all integer ni)
Broadhurst 199092 RecursorFleischer, Tarasov 1992 SHELL2
Basis (all ni = 1)
CALC-2003, Dubna, June 1321 p. 84/128
Sunset
Mn =(nd 4n+ 1)2(n1)(
n+ 1 nd2)n
((n+ 1)d2 2n 1
)n
1(nd2 2n+ 1
)n1
(n (n 1)d2
)n1
(1 + (n 1))(1 + n)(1 2n)n(1 )
(1 n)(1 (n+ 1))
CALC-2003, Dubna, June 1321 p. 85/128
Sunset
Mn =(nd 4n+ 1)2(n1)(
n+ 1 nd2)n
((n+ 1)d2 2n 1
)n
1(nd2 2n+ 1
)n1
(n (n 1)d2
)n1
(1 + (n 1))(1 + n)(1 2n)n(1 )
(1 n)(1 (n+ 1))
= 1
2
d 2
d 3
CALC-2003, Dubna, June 1321 p. 85/128
N(1, 1, 1, 1, 1)
N(1, 1, 1, 1, 1) =
42(1 + )
3(1 42)
[2 3F2
(1, 12 ,
12
32 + ,
32
1)+
1
1 + 23F2
(1, 12 ,
12
32 + ,
32 +
1)+
3(1 + 2)
162
((1 4)(1 + 2)2(1 )
(1 3)(1 2)(1 + ) 1
)]
CALC-2003, Dubna, June 1321 p. 86/128
N(1, 1, 1, 1, 1)
N(1, 1, 1, 1, 1) = 2 log 23
23 +O()
Broadhurst, 1992
CALC-2003, Dubna, June 1321 p. 87/128
Inversion
=
n1 n2
n3 n4n5
n1 n2
d n1 n3 n5
d n2 n4 n5
n5
CALC-2003, Dubna, June 1321 p. 88/128
2 masses
CALC-2003, Dubna, June 1321 p. 89/128
Topology
Davydychev, Grozin 1999
CALC-2003, Dubna, June 1321 p. 90/128
Topology
Davydychev, Grozin 1999
CALC-2003, Dubna, June 1321 p. 90/128
Topology
Davydychev, Grozin 1999
Basis (all ni = 1)
I0 I1We could also choose a sunset diagram with index 2instead of I1
CALC-2003, Dubna, June 1321 p. 90/128
I0
I02(1 + )
= (m2)12
2(1 ){
1
1 23F2
(1, 12 ,1 + 2
2 , 12 +
m2m2)
+
(m2
m2
)13F2
(1, , 32
3 2, 32
m2m2)}
CALC-2003, Dubna, June 1321 p. 91/128
I1
I12(1 + )
= (m2)2
2{
1
2(1 )(1 + 2)3F2
(1, 12 , 2
2 , 32 +
m2m2)
(m2
m2
)11
1 23F2
(1, , 12
2 2, 32
m2m2)}
CALC-2003, Dubna, June 1321 p. 92/128
I0
I02(1 + )
=
m24[
1
22+
5
4+ 2(1 r2)2(L+ + L) 2 log
2 r +11
8
]m24
[1
2+
3
2 log r + 6
]+O()
r = m/m
CALC-2003, Dubna, June 1321 p. 93/128
I1
I12(1 + )
= m4
[1
22+
5
2
+ 2(1 + r)2L+ + 2(1 r)2L 2 log
2 r +19
2
]+O()
CALC-2003, Dubna, June 1321 p. 94/128
L
L+ = Li2(r) +12 log
2 r log r log(1 + r) 162
= Li2(r1) + log r1 log(1 + r1)
L = Li2(1 r) +12 log
2 r + 162
= Li2(1 r1) + 16
2
= Li2(r) +12 log
2 r log r log(1 r) + 132
= Li2(r1) + log r1 log(1 r1)
L+ + L =12 Li2(1 r
2) + log2 r + 1122
= 12 Li2(1 r2) + 112
2
CALC-2003, Dubna, June 1321 p. 95/128
3 loops
CALC-2003, Dubna, June 1321 p. 96/128
Topologies
CALC-2003, Dubna, June 1321 p. 97/128
Basis
Melnikov, van Ritbergen 2000 SHELL3
CALC-2003, Dubna, June 1321 p. 98/128
Inversion
n1 n2 n1 n2
n3 n4d n1 n3 n5 n7
d n2 n4 n5 n8
n5 n5n7 n8 n7 n8
n6 d n6 n7 n8
=
n1 n3 n2 n1 n3 n2
n4 n5d n1 n4
n6d n2 n5 n7n6n7 n6n7
n8 d n3 n6 n7 n8
=
CALC-2003, Dubna, June 1321 p. 99/128
Inversiond = 4
= = 55 + 1223
Czarnecki, Melnikov 2002
CALC-2003, Dubna, June 1321 p. 100/128
Multiple values
CALC-2003, Dubna, June 1321 p. 101/128
Definition
s =n>0
1
ns
CALC-2003, Dubna, June 1321 p. 102/128
Definition
s =n>0
1
ns
s1s2 =
n1>n2>0
1
ns11 ns22
CALC-2003, Dubna, June 1321 p. 102/128
Definition
s =n>0
1
ns
s1s2 =
n1>n2>0
1
ns11 ns22
s1s2s3 =
n1>n2>n3>0
1
ns11 ns22 n
s33
CALC-2003, Dubna, June 1321 p. 102/128
Definition
s =n>0
1
ns
s1s2 =
n1>n2>0
1
ns11 ns22
s1s2s3 =
n1>n2>n3>0
1
ns11 ns22 n
s33
Converges at s1 > 1
CALC-2003, Dubna, June 1321 p. 102/128
Definition
s =n>0
1
ns
s1s2 =
n1>n2>0
1
ns11 ns22
s1s2s3 =
n1>n2>n3>0
1
ns11 ns22 n
s33
Converges at s1 > 1Depth k Weight s = s1 + + sk
CALC-2003, Dubna, June 1321 p. 102/128
Examples
Weight 2
2
CALC-2003, Dubna, June 1321 p. 103/128
Examples
Weight 2
2
Weight 3
3 21
CALC-2003, Dubna, June 1321 p. 103/128
Examples
Weight 2
2
Weight 3
3 21
Weight 4
4 31 22 211
CALC-2003, Dubna, June 1321 p. 103/128
Examples
Weight 2
2
Weight 3
3 21
Weight 4
4 31 22 211
Weight 5
5 41 32 23 311 221 212 2111
CALC-2003, Dubna, June 1321 p. 103/128
Stuffling
ss1s2 =n > 0
n1 > n2 > 0
1
nsns11 ns22
CALC-2003, Dubna, June 1321 p. 104/128
Stuffling
ss1s2 =n > 0
n1 > n2 > 0
1
nsns11 ns22
n
n1 n2
n>n1>n2>0
1
nsns11 ns22
= ss1s2
CALC-2003, Dubna, June 1321 p. 104/128
Stuffling
ss1s2 =n > 0
n1 > n2 > 0
1
nsns11 ns22
n
n1 n2
n=n1>n2>0
1
nsns11 ns22
= s+s1,s2
CALC-2003, Dubna, June 1321 p. 104/128
Stuffling
ss1s2 =n > 0
n1 > n2 > 0
1
nsns11 ns22
n
n1 n2
n1>n>n2>0
1
nsns11 ns22
= s1ss2
CALC-2003, Dubna, June 1321 p. 104/128
Stuffling
ss1s2 =n > 0
n1 > n2 > 0
1
nsns11 ns22
n
n1 n2
n1>n=n2>0
1
nsns11 ns22
= s1,s+s2
CALC-2003, Dubna, June 1321 p. 104/128
Stuffling
ss1s2 =n > 0
n1 > n2 > 0
1
nsns11 ns22
n
n1 n2
n1>n2>n>0
1
nsns11 ns22
= s1s2s
CALC-2003, Dubna, June 1321 p. 104/128
Stuffling
ss1s2 =n > 0
n1 > n2 > 0
1
nsns11 ns22
= ss1s2 + s+s1,s2 + s1ss2 + s1,s+s2 + s1s2s
CALC-2003, Dubna, June 1321 p. 104/128
Stuffling
CALC-2003, Dubna, June 1321 p. 105/128
Examples
22 = 22 + 4 + 22
CALC-2003, Dubna, June 1321 p. 106/128
Examples
22 = 22 + 4 + 22
23 = 23 + 5 + 32
CALC-2003, Dubna, June 1321 p. 106/128
Examples
22 = 22 + 4 + 22
23 = 23 + 5 + 32
221 = 221 + 41 + 221 + 23 + 212
CALC-2003, Dubna, June 1321 p. 106/128
Integral representation
10
dx1x1
x10
dx2x2
x20
dx3x3
x30
dx4x4
xn4
CALC-2003, Dubna, June 1321 p. 107/128
Integral representation
10
dx1x1
x10
dx2x2
x20
dx3x3
xn3 1
n
CALC-2003, Dubna, June 1321 p. 107/128
Integral representation
10
dx1x1
x10
dx2x2
xn2 1
n2
CALC-2003, Dubna, June 1321 p. 107/128
Integral representation
10
dx1x1
xn1 1
n3
CALC-2003, Dubna, June 1321 p. 107/128
Integral representation
1
n4
CALC-2003, Dubna, June 1321 p. 107/128
Integral representation
1
ns=
1>x1>>xs>0
dx1x1
dxsxs
xns
CALC-2003, Dubna, June 1321 p. 107/128
Integral representation
1
ns=
1>x1>>xs>0
dx1x1
dxsxs
xns
s =
1>x1>>xs>0
dx1x1
dxs1xs1
dxs1 xs
CALC-2003, Dubna, June 1321 p. 107/128
Integral representation
Short notation:
0 =dx
x1 =
dx
1 x
CALC-2003, Dubna, June 1321 p. 108/128
Integral representation
Short notation:
0 =dx
x1 =
dx
1 x
s =
s10 1
CALC-2003, Dubna, June 1321 p. 108/128
Integral representation
Short notation:
0 =dx
x1 =
dx
1 x
s =
s10 1
s1s2 =
s110 1
s210 1
CALC-2003, Dubna, June 1321 p. 108/128
Integral representation
Short notation:
0 =dx
x1 =
dx
1 x
s =
s10 1
s1s2 =
s110 1
s210 1
s1s2s3 =
s110 1
s210 1
s310 1
CALC-2003, Dubna, June 1321 p. 108/128
Integral representation
Short notation:
0 =dx
x1 =
dx
1 x
s =
s10 1
s1s2 =
s110 1
s210 1
s1s2s3 =
s110 1
s210 1
s310 1
Always 1 > x1 > > xs > 0
CALC-2003, Dubna, June 1321 p. 108/128
Shuffling
22 =
1>x1>x2>0
01
1>x
1>x
2>0
01
CALC-2003, Dubna, June 1321 p. 109/128
Shuffling
22 =
1>x1>x2>0
01
1>x
1>x
2>0
01
0 1
0 1= 22
CALC-2003, Dubna, June 1321 p. 109/128
Shuffling
22 =
1>x1>x2>0
01
1>x
1>x
2>0
01
0
01
1= 31
CALC-2003, Dubna, June 1321 p. 109/128
Shuffling
22 =
1>x1>x2>0
01
1>x
1>x
2>0
01
0
0 11 = 31
CALC-2003, Dubna, June 1321 p. 109/128
Shuffling
22 =
1>x1>x2>0
01
1>x
1>x
2>0
01
0
0 11
= 31
CALC-2003, Dubna, June 1321 p. 109/128
Shuffling
22 =
1>x1>x2>0
01
1>x
1>x
2>0
01
0
01
1 = 31
CALC-2003, Dubna, June 1321 p. 109/128
Shuffling
22 =
1>x1>x2>0
01
1>x
1>x
2>0
01
0 1
0 1 = 22
CALC-2003, Dubna, June 1321 p. 109/128
Shuffling
22 =
1>x1>x2>0
01
1>x
1>x
2>0
01
= 431 + 222
CALC-2003, Dubna, June 1321 p. 109/128
Shuffling
22 =
1>x1>x2>0
01
1>x
1>x
2>0
01
= 431 + 222
23 = 6311 + 3221 + 212
CALC-2003, Dubna, June 1321 p. 109/128
Shuffling
CALC-2003, Dubna, June 1321 p. 110/128
Duality
xi 1 xi: 0 11 > x1 > xs > 0: opposite order
CALC-2003, Dubna, June 1321 p. 111/128
Duality
xi 1 xi: 0 11 > x1 > xs > 0: opposite order
3 =
001 =
011 = 21
CALC-2003, Dubna, June 1321 p. 111/128
Duality
xi 1 xi: 0 11 > x1 > xs > 0: opposite order
3 =
001 =
011 = 21
4 =
0001 =
0111 = 211
CALC-2003, Dubna, June 1321 p. 111/128
Duality
5 =
00001 =
01111 = 2111
CALC-2003, Dubna, June 1321 p. 112/128
Duality
5 =
00001 =
01111 = 2111
41 =
00011 =
00111 = 311
CALC-2003, Dubna, June 1321 p. 112/128
Duality
5 =
00001 =
01111 = 2111
41 =
00011 =
00111 = 311
32 =
00101 =
01011 = 221
CALC-2003, Dubna, June 1321 p. 112/128
Duality
5 =
00001 =
01111 = 2111
41 =
00011 =
00111 = 311
32 =
00101 =
01011 = 221
23 =
01001 =
01101 = 212
CALC-2003, Dubna, June 1321 p. 112/128
Weight 4
4 = 211 31 22
CALC-2003, Dubna, June 1321 p. 113/128
Weight 4
4 = 211 31 22
22 = 4 + 222 22 =344
22 = 431 + 222 31 =144
2 =2
64 =
4
90
CALC-2003, Dubna, June 1321 p. 113/128
Weight 5
5 = 2111 41 = 311 32 = 221 23 = 212
CALC-2003, Dubna, June 1321 p. 114/128
Weight 5
5 = 2111 41 = 311 32 = 221 23 = 21223 = 32 + 23 + 523 = 641 + 332 + 2323 = 221 = 41 + 232 + 223
CALC-2003, Dubna, June 1321 p. 114/128
Weight 5
5 = 2111 41 = 311 32 = 221 23 = 21223 = 32 + 23 + 523 = 641 + 332 + 2323 = 221 = 41 + 232 + 223
41 = 25 23
32 = 11
25 + 323
23 =9
25 223
CALC-2003, Dubna, June 1321 p. 114/128
Expanding
hypergeometric functions in
Example
CALC-2003, Dubna, June 1321 p. 115/128
Step 1
Example
F =n=0
(2 2)n(2 + 2)n(3 + )n(3 + 2)n
CALC-2003, Dubna, June 1321 p. 116/128
Step 1
Example
F =n=0
(2 2)n(2 + 2)n(3 + )n(3 + 2)n
(2 + 2)n(3 + 2)n
=2 + 2
n+ 2 + 2
CALC-2003, Dubna, June 1321 p. 116/128
Step 1
Example
F =n=0
(2 2)n(2 + 2)n(3 + )n(3 + 2)n
(2 + 2)n(3 + 2)n
=2 + 2
n+ 2 + 2
(2 2)n =(1 2)n+1
1 2
(3 + )n =(1 + )n+2
(1 + )(2 + )
CALC-2003, Dubna, June 1321 p. 116/128
Step 1
F =2(2 + )(1 + )2
1 2F = 2(2 + 9+ )F
F =n=0
1
n+ 2 + 2
(1 2)n+1(1 + )n+2
CALC-2003, Dubna, June 1321 p. 117/128
Step 1
F =2(2 + )(1 + )2
1 2F = 2(2 + 9+ )F
F =n=0
1
n+ 2 + 2
(1 2)n+1(1 + )n+2
(1 + )n = n!Pn() Pn() =n
n=1
(1 +
n
)
CALC-2003, Dubna, June 1321 p. 117/128
Step 1
F =2(2 + )(1 + )2
1 2F = 2(2 + 9+ )F
F =n=0
1
n+ 2 + 2
(1 2)n+1(1 + )n+2
(1 + )n = n!Pn() Pn() =n
n=1
(1 +
n
)
F =n=0
1
(n+ 2)(n+ 2 + 2)
Pn+1(2)
Pn+2()
CALC-2003, Dubna, June 1321 p. 117/128
Step 2
Expanding in and partial-fractioning
1
(n+ 2)(n + 2 + 2)=
1
(n+ 2)2
2
(n+ 2)3+
CALC-2003, Dubna, June 1321 p. 118/128
Step 2
Expanding in and partial-fractioning
1
(n+ 2)(n + 2 + 2)=
1
(n+ 2)2
2
(n+ 2)3+
F =n=0
1
(n+ 2)2Pn+1(2)
Pn+2() 2
n=0
1
(n+ 2)3+O(2)
=n=2
1
n2Pn1(2)
Pn() 2
n=2
1
n3+O(2)
CALC-2003, Dubna, June 1321 p. 118/128
Step 3
Pn() =(1 +
n
)Pn1()
Expanding and partial-fractioning
CALC-2003, Dubna, June 1321 p. 119/128
Step 3
Pn() =(1 +
n
)Pn1()
Expanding and partial-fractioning
F =n=2
1
n2Pn1(2)
Pn1() 3
n=2
1
n3+O(2)
CALC-2003, Dubna, June 1321 p. 119/128
Step 3
Pn() =(1 +
n
)Pn1()
Expanding and partial-fractioning
F =n=2
1
n2Pn1(2)
Pn1() 3
n=2
1
n3+O(2)
Step 4
F =n=1
1
n2Pn1(2)
Pn1() 1 3(3 1)+O(
2)
CALC-2003, Dubna, June 1321 p. 119/128
Step 5
Pn1() =
n>n>0
(1 +
n
)= 1+ z1(n)+ z11(n)
2 +
CALC-2003, Dubna, June 1321 p. 120/128
Step 5
Pn1() =
n>n>0
(1 +
n
)= 1+ z1(n)+ z11(n)
2 +
zs(n) =
n>n1>0
1
ns
CALC-2003, Dubna, June 1321 p. 120/128
Step 5
Pn1() =
n>n>0
(1 +
n
)= 1+ z1(n)+ z11(n)
2 +
zs(n) =
n>n1>0
1
ns
zs1s2(n) =
n>n1>n2>0
1
ns11 ns22
CALC-2003, Dubna, June 1321 p. 120/128
Step 5
Pn1() =
n>n>0
(1 +
n
)= 1+ z1(n)+ z11(n)
2 +
zs(n) =
n>n1>0
1
ns
zs1s2(n) =
n>n1>n2>0
1
ns11 ns22
Stuffling
zs(n)zs1s2(n) = zss1s2(n) + zs+s1,s2(n)
+ zs1ss2(n) + zs1,s+s2(n) + zs1s2s(n)CALC-2003, Dubna, June 1321 p. 120/128
Step 5
Pn1(2)
Pn1()=
1 2z1(n)+
1 + z1(n)+ = 1 3z1(n)+
CALC-2003, Dubna, June 1321 p. 121/128
Step 5
Pn1(2)
Pn1()=
1 2z1(n)+
1 + z1(n)+ = 1 3z1(n)+
F =n>0
1
n2 3
n>n1>0
1
n2n1 1 3(3 1)+
= 2 1 3(21 + 3 1)+
CALC-2003, Dubna, June 1321 p. 121/128
Step 5
Pn1(2)
Pn1()=
1 2z1(n)+
1 + z1(n)+ = 1 3z1(n)+
F =n>0
1
n2 3
n>n1>0
1
n2n1 1 3(3 1)+
= 2 1 3(21 + 3 1)+
Result 21 = 3
F = 2(2 + 9+ ) [2 1 3(23 1)+ ]
= 4(2 1) + 6(43 + 32 1)+
CALC-2003, Dubna, June 1321 p. 121/128
Expanding
hypergeometric functions in
Algorithm
CALC-2003, Dubna, June 1321 p. 122/128
Step 1
F =n=0
i(mi + li)ni(m
i + l
i)n
CALC-2003, Dubna, June 1321 p. 123/128
Step 1
F =n=0
i(mi + li)ni(m
i + l
i)n
(m+ l)n =(1 + l)n+m1
(1 + l) (m 1 + l)
CALC-2003, Dubna, June 1321 p. 123/128
Step 1
F =n=0
i(mi + li)ni(m
i + l
i)n
(m+ l)n =(1 + l)n+m1
(1 + l) (m 1 + l)
(1 + l)n+m1 = (n+m 1)!Pn+m1(l)
CALC-2003, Dubna, June 1321 p. 123/128
Step 1
F =n=0
i(mi + li)ni(m
i + l
i)n
(m+ l)n =(1 + l)n+m1
(1 + l) (m 1 + l)
(1 + l)n+m1 = (n+m 1)!Pn+m1(l)
F =n=0
R(n, )
i Pn+mi1(li)i Pn+mi1(l
i)
CALC-2003, Dubna, June 1321 p. 123/128
Step 2
Expand
R(n, ) = R0(n) +R1(n)+
and partial-fraction each Rj(n)
CALC-2003, Dubna, June 1321 p. 124/128
Step 2
Expand
R(n, ) = R0(n) +R1(n)+
and partial-fraction each Rj(n)Convergent sum combination of logarithmicallydivergent ones
CALC-2003, Dubna, June 1321 p. 124/128
Step 2
Expand
R(n, ) = R0(n) +R1(n)+
and partial-fraction each Rj(n)Convergent sum combination of logarithmicallydivergent onesShift indices. F = sum of terms
n=n0
1
nk
i Pn+mi(li)i Pn+mi(l
i)
CALC-2003, Dubna, June 1321 p. 124/128
Step 3
Pn+m(l) = Pn1(l)
(1 +
l
n
)
(1 +
l
n+m
)
CALC-2003, Dubna, June 1321 p. 125/128
Step 3
Pn+m(l) = Pn1(l)
(1 +
l
n
)
(1 +
l
n+m
)
n=n0
R(n, )
nk
i Pn1(li)i Pn1(l
i)
CALC-2003, Dubna, June 1321 p. 125/128
Step 3
Pn+m(l) = Pn1(l)
(1 +
l
n
)
(1 +
l
n+m
)
n=n0
R(n, )
nk
i Pn1(li)i Pn1(l
i)
Expand and partial-fraction (only affects higherorders in ). F = sum of terms
n=n0
1
nk
i Pn1(li)i Pn1(l
i)
CALC-2003, Dubna, June 1321 p. 125/128
Step 4
Add and subtract terms with n = 1, . . . n0 1. F =sum of terms
n=1
1
nk
i Pn1(li)i Pn1(l
i)
and rational functions of
CALC-2003, Dubna, June 1321 p. 126/128
Step 5
Pn1(l) = 1 + z1(n)l + z11(l)2 +
Expand, find products of z sums by the stufflingrelations
n=1
1
nkzs1...sj(n) = ks1...sj
F = expansion in coefficients combinations of multiple values
CALC-2003, Dubna, June 1321 p. 127/128
Implementation
Grozin 2000 (unpublished) REDUCE
Moch, Uwer, Weinzierl 2002 Algorithm A
Weinzierl 2002 C++ library nestedsums(based on GiNaC)
CALC-2003, Dubna, June 1321 p. 128/128
PlanPlanPlanPlanPlan
Scalar integralsScalar integralsScalar integrals
1 loopDiagramInteger $n_1le 0$Fourier transform$x$ space$x$ space
InversionInversion
Inversion2 loopsDiagramTrivial case $n_5=0$Trivial case $n_1=0$Integration by partsIntegration by parts
Integration by partsIntegration by parts
HomogeneityLarin relationLarin relationIntegration by partsIntegration by partsIntegration by partsIntegration by parts
TopologySunsetInversionInversion
3 loopsTopologiesBasisInversionInversion$G$ with non-integer indices$G(1,1,1,1,n)$$G(1,1,1,1,n)$$G(1,1,1,1,n+varepsilon )$Non-planar diagramGluingHQETLagrangianFeynman rulesCovariant notationsCovariant notations
1 loopDiagramFourier transformFourier transform
$I$2 loopsDiagram $I$Trivial case $n_5=0$Trivial cases $n_1=0$, $n_3=0$Trivial cases $n_1=0$, $n_3=0$
Integration by partsIntegration by parts
Integration by partsIntegration by parts
HomogeneityHomogeneity
Integration by partsDiagram $J$Partial fractioningTopologiesTopologies
Sunset3 loopsTopologiesBasis$J$ with non-integer indices$J(1,1,n,1,1)$$J(1,1,n,1,1)$
$J(1,1,n,1,1)$$J(1,1,n,1,1)$
$J(1,1,n,1,1)$$J(1,1,n,1,1)$
$J(1,1,n,1,1)$$J(1,1,n,1,1)$
$J(n_1,n_2,n_3,n_4,n_5)$$J(1,1,n+2varepsilon ,1,1)$$J(1,1,n+2varepsilon ,1,1)$
$I$ with non-integer indices$I(1,1,1,1,n)$$I(1,1,1,1,n)$
Gegenbauer polynomialsGegenbauer polynomials
Gegenbauer polynomials$I(1,1,1,1,n)$$I(1,1,1,1,n)$$I(1,1,1,1,n)$
$I(1,1,1,1,n)$$I(1,1,1,1,n)$$I(1,1,1,1,n)$$I(1,1,1,1,n)$$I(1,1,1,1,n+varepsilon )$On-shell diagramsWhy?Why?Why?
1 loopInversionInversionInversion
Inversion2 loopsDiagram $M$Diagram $N$TopologiesTopologies
SunsetSunset
$N(1,1,1,1,1)$$N(1,1,1,1,1)$Inversion2 massesTopologyTopologyTopology
$I_0$$I_1$$I_0$$I_1$$L_pm $3 loopsTopologiesBasisInversionInversionMultiple $zeta $ valuesDefinitionDefinitionDefinitionDefinitionDefinition
ExamplesExamplesExamplesExamples
StufflingStufflingStufflingStufflingStufflingStufflingStuffling
StufflingExamplesExamplesExamples
Integral representationIntegral representationIntegral representationIntegral representationIntegral representationIntegral representationIntegral representation
Integral representationIntegral representationIntegral representationIntegral representationIntegral representation
ShufflingShufflingShufflingShufflingShufflingShufflingShufflingShufflingShuffling
ShufflingDualityDualityDuality
DualityDualityDualityDuality
Weight 4Weight 4
Weight 5Weight 5Weight 5
Expanding\[2mm]hypergeometric functions in $varepsilon $\[2mm]ExampleStep 1Step 1Step 1
Step 1Step 1Step 1
Step 2Step 2
Step 3Step 3Step 3
Step 5Step 5Step 5Step 5
Step 5Step 5Step 5
Expanding\[2mm]hypergeometric functions in $varepsilon $\[2mm]AlgorithmStep 1Step 1Step 1Step 1
Step 2Step 2Step 2
Step 3Step 3Step 3
Step 4Step 5Implementation