Lectures on Multiloop Calculations (Grozin)

Lectures on multiloop calculations

Andrey Grozin

[email protected]

Institut fur Theoretische Teilchenphysik, Universitat Karlsruhe

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Plan Massless propagator diagrams

HQET propagator diagrams

Massive on-shell propagator diagrams

Expansion of hypergeometric functions in

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1 loop 2 loops 3 loops master integral G with non-integer index master integral non-planar




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HQET propagator diagrams crash course of HQET 1 loop 2 loops 3 loops master integral (my) master integral (Beneke-Braun)



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Massive on-shell propagator diagrams 1 loop 2 loops 2 loops, 2 non-zero masses 3 loops 3-loop HQET master integral(Czarnecki-Melnikov)


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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


Expressing scalar products in the numeratorvia the denominatorsSometimes, there are irreducible scalar products

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Scalar integrals


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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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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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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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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G =n22

+(5 1) + n44+(5 3)

d n2 n4 2n5G

n1 + n3 + n5 reduces by 1

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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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When applied to the integrand

k2

n2D2

v

+

n4D4

2k2 +n5D5

2(k2 k1)

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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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[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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[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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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


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

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

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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

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

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Gegenbauer polynomials

0 vtv(t 1)

x

I =

10

dt

0

dx dxx2n1

[(x vt)2] [(x+ v(1 t))2]

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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


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?


Magnetic moments, charge radii

CALC-2003, Dubna, June 1321 p. 77/128

Why?


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


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


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


Broadhurst 199092 RecursorFleischer, Tarasov 1992 SHELL2

CALC-2003, Dubna, June 1321 p. 84/128

Topologies


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

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


10

dx1x1

x10

dx2x2

x20

dx3x3

xn3 1

n

CALC-2003, Dubna, June 1321 p. 107/128


10

dx1x1

x10

dx2x2

xn2 1

n2

CALC-2003, Dubna, June 1321 p. 107/128


10

dx1x1

xn1 1

n3

CALC-2003, Dubna, June 1321 p. 107/128


1

n4

CALC-2003, Dubna, June 1321 p. 107/128


1

ns=

1>x1>>xs>0

dx1x1

dxsxs

xns

CALC-2003, Dubna, June 1321 p. 107/128


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


Short notation:

0 =dx

x1 =

dx

1 x

CALC-2003, Dubna, June 1321 p. 108/128


Short notation:

0 =dx

x1 =

dx

1 x

s =

s10 1

CALC-2003, Dubna, June 1321 p. 108/128


Short notation:

0 =dx

x1 =

dx

1 x

s =

s10 1

s1s2 =

s110 1

s210 1

CALC-2003, Dubna, June 1321 p. 108/128


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


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


3 =

001 =

011 = 21

CALC-2003, Dubna, June 1321 p. 111/128

Duality


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)

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Step 3

Pn() =(1 +

n

)Pn1()

Expanding and partial-fractioning

CALC-2003, Dubna, June 1321 p. 119/128

Step 3

Pn() =(1 +

n

)Pn1()


F =n=2

1

n2Pn1(2)

Pn1() 3

n=2

1

n3+O(2)

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Step 3

Pn() =(1 +

n

)Pn1()


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)

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Step 5

Pn1() =

n>n>0

(1 +

n

)= 1+ z1(n)+ z11(n)

2 +

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Step 5

Pn1() =

n>n>0

(1 +

n

)= 1+ z1(n)+ z11(n)

2 +

zs(n) =

n>n1>0

1

ns

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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

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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)+

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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)+

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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

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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)

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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)

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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)

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Step 2

Expand

R(n, ) = R0(n) +R1(n)+

and partial-fraction each Rj(n)

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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)

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Step 3

Pn+m(l) = Pn1(l)

(1 +

l

n

)

(1 +

l

n+m

)

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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)

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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 $n_1=0$, $n_3=0$



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

Lectures on Multiloop Calculations (Grozin)

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Transcript of Lectures on Multiloop Calculations (Grozin)