1
Recent Results on Finite Harmonic Sums
in Quantum Field Theory
J. Blümlein, DESY
Structural Relations of Harmonic Sums up to w = 6
• From Moments to Functions in HO QCD
•
Based on :
arXiv:0901.3106, arXiv:0901.0837;
arxiv:0902.4091 [hep-ph], CPC in print with M. Kauers (RISC), S. Klein (DESY), and C. Schneider (RISC)
J. Blümlein
IHES Bures sur Yvette
June 2009
2
1. Introduction
Why are zero- and single scale quantities in Quantum Field Theory related to
ζ-values, nested harmonic sums and related objects ?
• The former quantities can be obtained from the latter putting the Mellin
variable N either to fixed values or N → ∞.
Perturbation Theory [fixed order]
Scalar Propagators:
i
p2 + iǫ
Combine Momenta using the Feynman Trick
1
=
A·B
Z
1
0
dx
[xA + (1 − x)B]2
Feynman parameter integrals substitute all non-trivial angular integrals
in D = 4 + ε–dimensions.
3
Introduction
Momentum Integrals yield rational functions of Γ–functions
Γ(n1 + α1 ε)...Γ(nk + αk ε) Γ(m1 + β1 ε)...Γ(ml + βk ε) mi ,ni ǫZ,αi ,βi ǫQ
• The scale-ratio in the diagrams factors form the Feynman parameter integrals
for single scale processes
The Feynman parameter integrals can be transformed into Mellin-Barnes
Integrals
Z
1
1
=
(A + B)q
2πi
γ+i∞
dσAσ B −q−σ
γ−i∞
Γ(−σ)Γ(q + σ)
Γ(q)
Then all Feynman parameter integrals can again be integrated =⇒ rational
function of Γ–functions
One or more Feynman parameters contain N as power xN
i , ... in the
numerators. =⇒ the Γ–functions contain N as argument.
The Mellin-Barnes Integrals can be carried out using the Residue Theorem.
=⇒ several infinite sums over the rational functions of Γ–functions.
4
Introduction
• Seek compact representations for these in terms of
(Generalizations of) generalized hypergeometric functions.
• The Mellin variable N is a discrete quantity in the first place for
physical reasons
=⇒ Light-cone expansion; cut vertex method + dispersion relations
• Perform the ε–expansion.
• The respective coefficients obey Difference Equations of finite order.
• The ε–expansion of Pochhammer-Symbols & Γ–functions leads to products of
single finite harmonic sums and MZV’s.
• The infinite sums over the Mellin–Barnes parameters lead to the respective
Nesting.
• Observation: Most of the sums occurring are Nested Finite Harmonic Sums.
• However, other related sums are possible too for individual Feynman
diagrams. [Vermaseren et al. (2005)]
• General solution formalisms like, Sigma, will reveal this uniquely.
cf. C. Schneider.
5
Introduction
• Single scale processes in massless Quantum Field Theories or being considered
in the limit m2 /Q2 → 0 exhibit significant simplifications when calculated in
Mellin space.
• This is, to some extent, due to structure of Feynman parameter intergrals
which posess a Mellin symmetry.
• Harmonic sums form the appropriate language to derive compact expressions in
the respective calculations.
• We will line out the relations of the harmonic sums, resp. their continuations to
N ǫ Q, R, C.
x-space results :
Nielsen-type integrals, resp. harmonic polylogarithms (E. Remiddi and J. Vermaseren (1999))
Z
(−1)n+p+q−1 1 dz (n−1)
Sn,p,q (x) =
ln
(z) lnp (1 − zx) lnq (1 + zx)
Γ(n)p!q!
0 z
6
2 Loop Wilson Coefficients
W.L. van Neerven et al.: (1992) 79 functions
80 objects would be maximal.
• The high complexity is partly caused applying the the IBP–Method.
• x–space usually is not the best space to work in.
3 Loop Anomalous Dimensions & Wilson Coefficients
•
•
•
•
=⇒ Harmonic Sums in linear representation.
Still high complexity of terms.
Compactification possible applying algebraic and structural relations.
Observation : In all single scale calculations the same Basic Functions occur
in the resp. weight.
• =⇒ Derive these Universal Functions and their complex analysis.
7
2. Algebraic Relations
cf. J.Blümlein, Comput. Phys. Commun. 159 (2004) 19
Number of harmonic sums up to weight w : 3w−1 .
Harmonic sums form a quasi-shuffle algebra through . (M.E. Hoffman, J. Algebraic Combin. 11 (2000)
49 )
Sa1 ,a2 Sa3 ,a4
= Sa1 ,a2 ,a3 ,a4 + Sa1 ,a3 ,a2 ,a4 + Sa1 ,a2 ,a4 ,a2
+Sa3 ,a4 ,a1 ,a2 + Sa3 ,a1 ,a4 ,a2 + Sa3 ,a1 ,a2 ,a4
Solve all the linear equations possible for the harmonic sums
etc.
=⇒ algebraic basis.
Let {a, a, a, ..., b, b, ..., ..., z, z} a set of n1 a’s, n2 b’s etc. The number of basis elements
corresponding to all words formed by ALL the above letters is:
X
(n/d)!
1X
,
ni = n
µ(d)
ln (n1 , ..., nq ) =
n
(n1 /d)!...(nd /d)!
i
d|ni
(E. Witt, 1937)
=⇒ # Lyndon words
w
1
2
3
4
5
6
#c
2
8
26
80
242
728
#r
2
5
13
31
79
195
Algebraic Relations
Observation in Quantum Field Theory :
At least up to O(αs3 ) the contributing harmonic sums exhibit never any index ak = −1
applying a compact representation.
The number of sums of this type is
√ w √ w i
1 h
1− 2 + 1+ 2
N¬{−1} (w) =
2
basic
N¬{−1}
(w)
2 X w basic
µ
N¬{−1} (d) .
=
w
d
d|w
w
1
2
3
4
5
6
#c
1
4
11
28
69
168
#r
1
3
7
14
30
60
• Here #c is smaller than #r in the general case.
8
9
Algebraic Relations
Remark:
Harmonic, Generalized Harmonic Polylogarithms and Multiple Polylogarithms also form shuffle
algebras. As shuffle algebras are sub-sets of the quasi-shuffle algebra studied above, the
respective algebraic relations can be derived directly.
• Form the index alphabet.
• Solve the shuffle-relations =⇒ Basis
As the relations in J.B., Comput. Phys. Commun. 159 (2004) 19 are of arbitrary weight (general
alphabet) and depth d ≤ 6 the corresponding relations can be read off there.
Algorithms to extend this scenario are available.
10
3. Structural Relations
w = 1:
1
1−x
1
1+x
&
1
1
1
1
+
=
1 − x2
2 1−x 1+x
M
−ψ
N
2
"
1
1−x
− γE
#
+
N
2
=M
"
1
1−x
#
+
1
(N ) + M
(N ) + ln(2)
1+x
= −ψ(N ) − γE + β(N ) + ln(2);
• S−1 (N ) depends on S1 (N ) for N ǫ Q
1
N +1
N
β(N ) =
ψ
−ψ
2
2
2
Structural Relations
N ǫR:
S2 (N ) = −
d
S1 (N ) + ζ2
dN
(etc.)
For N ǫ R : only one independent single sum occurs.
N
X
1
S1 (N ) =
= ψ(N + 1) + γE
k
k=1
Harmonic sums ∪ ζk1 ,...kn are closed under differentiation.
w = 2:
ln(1 + x)
ln(1 − x)
(N ) = −M
(N ) − [ψ(N ) + γE + ln(2)]β(N ) + β ′ (N )
M
1+x
1+x
ln(1 + x)
(N ) → S1,−1 (N )
F1 (N ) := M
1+x
11
12
Structural Relations
The relations for w = 2 were explored by N. Nielsen (1906).
ln(1 − x)
ln(1 + x) − ln(2)
(N );
η(N ) = M
(N )
1+x
1−x
+
ln(1 − x)
ln(1 + x)
(N );
− ξ2 (N ) = M
(N )
ξ1 (N ) = M
1+x
1+x
ξ(N ) = M
[ψ(z) + γE ][ψ(1 − z) + γE ] = 2ζ2 − ξ(z) − ξ(1 − z)
β(z)[ψ(z) + γE ] = β ′ (z) + β(z) ln(2) − ξ1 (z) + ξ2 (z)
β(z)β(1 − z) = η(z) + η(1 − z)
β 2 (z) = ψ ′ (z) − 2η(z)
Structural Relations
If half-integer arguments in N are allowed
M[Lik (−x)/(x ± 1)](N ) are not independent functions :
1 Lik (x2 )
Lik (x) Lik (x) Lik (−x) Lik (−x)
Lik (−x)
+
+
+
=
→
2k−2 1 − x2
1−x
1+x
1−x
1+x
1−x
• There always exists another IBP relation to express also Lik (−x)/(1 + x)
Li
(−x)
2
(N ) = −S2,−1 (N ) − ln(2)[S2 (N ) − S−2 (N )]
(−1)N M
1+x
1
1
1
− ζ2 S−1 (N ) + ζ3 − ζ2 ln(2)
2
4
2
−Li2 (x) − ln(x) ln(1 − x) + ζ2
3
(−1)N M
(N ) = −S−1,2 (N ) + ζ2 S−1 (N ) − ζ3 + ζ2 ln(2)
1+x
2
S−1,2 (N ) + S2,−1 (N ) = S−1 (N )S2 (N ) + S−3 (N )
13
Structural Relations
Li
(−x)
3
(N ) = −S3,−1 (N ) − ln(2)[S3 (N ) − S−3 (N )]
(−1)(N +1) M
1+x
1
3
1
3
− ζ2 S−2 (N ) + ζ3 S−1 (N ) − ζ22 + ln(2)ζ3
2
4
8
4
S1,2 (1 − x)
19
7
(−1)N M
(N ) = −S−1,3 (N ) + ζ3 S−1 (N ) − ζ22 + ζ3 ln(2)
1+x
40
4
1
S1,2 (1 − x) = −Li3 (x) + log(x)Li2 (x) + log(1 − x) log2 (x) + ζ3
2
S−1,3 (N ) + S3,−1 (N ) = S−1 (N )S3 (N ) + S−4 (N )
• At even w there exists an algebraic relation
i
1h 2
Sw/2,w/2 (N ) =
S (N ) + Sw (N )
2 w/2
which yields an additional relation for Lik (x)/(1 + x).
w = 3:
Li2 (x)
,
→
x±1
ln2 (1 + x)
x±1
14
Double Sums in General
• Applying differential operators one may show :
For N ǫ R double harmonic sums can always be represented
by one basic function for even weight and two basic functions for odd weight.
=⇒
Lik (x)
,
1+x
Lik (x)
1±x
Examples, which reduce :
"
#
ln(x) [S1,2 (1 − x) − ζ3 ] + 3 [S1,3 (1 − x) − ζ4 ]
S2,3 (N ) = M
(N ) + 3ζ4 S1 (N )
x−1
+
"
#
4Li5 (−x) − ln(x)Li4 (−x)
(N )
S−4,−2 (N ) = −M
x−1
+
1
3
21
15
+ ζ2 [S4 (N ) − S−4 (N )] − ζ3 S3 (N ) + ζ4 S2 (N ) − ζ5 S1 (N )
2
2
8
4
S1,3 (1 − x) = −Li4 (x) + log(x)Li3 (x) −
1
1
log2 (x)Li2 (x) − log3 (x) log(1 − x) + ζ4
2
6
15
16
Structural Relations
w = 4; i 6= −1 :
Li3 (x)
,
x+1
S1,2 (x)
x±1
The Mellin transform of
Li3 (x)
x−1
+
reads
M
"
Li3 (x)
x−1
#
+
(N ) =
1
2
(
d
M
dN
"
Li2 (x) + ζ2
x−1
#
(N )
+
−S2,2 (N − 1) + ζ2 S2 (N − 1) + 2ζ3 S1 (N − 1)
and can be traced back to that of (Li2 (x)/(x − 1))+
)
Structural Relations
17
w = 5; i 6= −1 :
Li4 (x)
x±1
Li22 (x)
x−1
w = 6; i 6= −1 :
S1,3 (x)
x+1
S1,3 (x)
x−1
S2,2 (x)
x±1
Li22 (x)
x+1
ln(x)S1,2 (−x) − Li22 (−x)/2
x±1
[Occur in the 3 − loop Wils. Coeff. only]
S3,2 (x)
S2,3 (x)
S1,4 (x)
Li2 (x)Li3 (x)
Li5 (x)
x+1
x±1
x±1
x±1
x±1
S1,2 (x)Li2 (x)
A1 (x)
A2 (x)
A3 (x)
H0,−1,0,1,1 (x)
x+1
x+1
x±1
x+1
x±1
H0,0,−1,0,1 (x)
A1 (−x) + 2S3,2 (−x) − 2S2,2 (−x) ln(x)
x±1
x±1
A1 (−x) + 2S3,2 (−x) − 2S2,2 (−x) ln(x) + Li22 (−x) ln(x)/4 − Li3 (−x)Li2 (−x)
x−1
New numerator functions :
A1 (x) =
Z
0
x
dy 2
Li (y),
y 2
A2 (x) =
Z
0
x
dy
ln(1 − y)S1,2 (y),
y
A3 (x) =
Z
x
0
dy
[Li4 (1 − y) − ζ4 ]
y
18
Harmonic Polylogarithms
• iterated integrals over the alphabet
fa (x) =
H0 (x) =
Z
x
0
dx
,
x
1
,
1−x
1
,
x
H1 (x) =
Ha,~b (x) =
Z
Z
x
0
dx
,
1−x
1
1+x
H−1 (x) =
x
0
dzfa (z)H~b (z)
Z
0
x
dx
1+x
Structural Relations
19
w = 5; i 6= −1 :
Li4 (x)
x±1
Li22 (x)
x−1
w = 6; i 6= −1 :
S1,3 (x)
x+1
S1,3 (x)
x−1
S2,2 (x)
x±1
Li22 (x)
x+1
ln(x)S1,2 (−x) − Li22 (−x)/2
x±1
[Occur in the 3 − loop Wils. Coeff. only]
S3,2 (x)
S2,3 (x)
S1,4 (x)
Li2 (x)Li3 (x)
Li5 (x)
x+1
x±1
x±1
x±1
x±1
S1,2 (x)Li2 (x)
A1 (x)
A2 (x)
A3 (x)
H0,−1,0,1,1 (x)
x+1
x+1
x±1
x+1
x±1
H0,0,−1,0,1 (x)
A1 (−x) + 2S3,2 (−x) − 2S2,2 (−x) ln(x)
x±1
x±1
A1 (−x) + 2S3,2 (−x) − 2S2,2 (−x) ln(x) + Li22 (−x) ln(x)/4 − Li3 (−x)Li2 (−x)
x−1
New numerator functions :
A1 (x) =
Z
0
x
dy 2
Li (y),
y 2
A2 (x) =
Z
0
x
dy
ln(1 − y)S1,2 (y),
y
A3 (x) =
Z
x
0
dy
[Li4 (1 − y) − ζ4 ]
y
20
Representation of Observables
• Unpolarized and Polarized Drell-Yan an Higgs-Boson Production Cross Section O(αs2 ) ,
w = 4 JB and V. Ravindran, Nucl. Phys. B716 (2005) 128.
• Unpolarized and Polarized Time-like Anomalous Dimensions and Wilson Coefficients
O(αs2 ), w = 4 JB and V. Ravindran, Nucl. Phys. B749 (2006) 1.
• Anomalous Dimensions and Wilson Coefficients O(αs3 ) , w = 5, 6,
from: S. Moch, J. Vermaseren, A. Vogt, Nucl. Phys. B688 (2004) 101; 691 (2004) 129; B724 (2005) 3 → J.B., DESY 07-042
• Polarized and Unpolarized Wilson Coefficients O(αs2 ) , w = 4 J.B. and S. Moch
2(3)
• Polarized and Unpolarized asymptotic Heavy Flavor Wilson Coefficients O(αs ) , w = 4,5,
J.B., A. de Freitas, W. van Neerven, S. Klein, Nucl. Phys. B755 (2006) 272; I. Bierenbaum, J.B., S. Klein, DESY 07-026,
DESY 07-027; DESY-08-029;
• Virtual and soft corrections to Bhabha Scattering O(α2 ) ,
J.B. and S. Klein, arXiv:0706.2426 [hep-ph]
w = 4,
Example: Bhabha s+v
T0
=
248 + 15 N 2 + N 4
2(N
2)(N
S1;1;1;1 (N ) +
1)N (N + 1)(N + 2)
340 + 120 N + 17 N 2 + 18 N 3
31 N 4
2
(N
1)(N + 1)
1344
21
S2 1 1(N )
502 N
; ;
69 N 2
2 N 3 + 57 N 4
S3;1 (N ) +
S4 (N )
1)N (N + 1)(N + 2)
8(N
2)(N
1)N (N + 1)(N + 2)
304
328 N
500 N 2 + 330 N 3
6 N4 + 6 N5
2 N6 + 4 N7
+
2;1 (N )
(N
2)2 (N
1)2 N 2 (N + 1)(N + 2)
48 + 8 N + 6 N 2 + 7 N 3
112
4 N2
4 N4
S2;1 (N ) 1 (N ) +
S3 (N )S1 (N )
+
(N
2)(N
1)N (N + 1)(N + 2)
(N
1)N (N + 1)(N + 2)
1840 + 292 N + 5532 N 2 + 827 N 3
1978 N 4
274 N 5 + 36 N 6 + 19 N 7
22 N 8
+
S1;1;1 (N )
4(N
2)2 (N
1)2 N 2 (N + 1)2 (N + 2)
128
56 N
252 N 2 + 54 N 3 + 177 N 4
91 N 5 + 19 N 6 + 9 N 7
+
S3 (N )
2(N
2)(N
1)2 N 2 (N + 1)2 (N + 2)
4032
2048 N
14200 N 2 + 5036 N 3 + 23610 N 4 + 2521 N 5
12342 N 6
+
S1;1 (N )
4(N
2)3 (N
1)3 N 3 (N + 1)3 (N + 2)
3365 N 7 + 2148 N 8 + 903 N 9 + 14 N 10
167 N 11 + 50 N 12
S1;1 (N )
+
4(N
2)3 (N
1)3 N 3 (N + 1)3 (N + 2)
124 + 16 N + 24 N 2
4 N3
14 N 4
424
118 N + 9 N 2
2 N 3 + 23 N 4
+
S1;1 (N ) (2) +
S2 (N )S1;1 (N )
(N
2)(N
1)N (N + 1)(N + 2)
4(N
2)(N
1)N (N + 1)(N + 2)
224 + 144 N
1216 N 2
56 N 3 + 1786 N 4 + 641 N 5
406 N 6
+
S2 (N )
4(N
2)2 (N
1)3 N 3 (N + 1)3 (N + 2)
+17 N 7
308 N 8 + 141 N 9
56 N 10 + N 11
58 + 21 N + N 2 + 15 N 3 + 10 N 4
+
S2 (N ) +
S2 (N ) (2)
4(N
2)2 (N
1)3 N 3 (N + 1)3 (N + 2)
(N
2)(N
1)N (N + 1)(N + 2)
+
2(N
2)(N
S
S
Example: Bhabha s+v
+
+
232
384 N 2
4(N
560
8(N
26 N
2)(N
576 + 1088 N
17 N 3 + 286 N 4
128 N 5
14 N 6 + N 7
S2 (N )S1 (N )
2)(N
1)2 N 2 (N + 1)2 (N + 2)
31 N 2
10 N 3
33 N 4
S2 (N )2
1)N (N + 1)(N + 2)
3280 N 2
5136 N 3 + 11764 N 4 + 20392 N 5
17385 N 6
30114 N 7
S1 (N )
2)3 (N
1)4 N 4 (N + 1)4 (N + 2)
+5984 N 8 + 17228 N 9
1228 N 10
2754 N 11
112 N 12
8 N 13 + 33 N 14
24 N 15
S1 (N )
+
4(N
2)3 (N
1)4 N 4 (N + 1)4 (N + 2)
56 + 336 N + 522 N 2 + 424 N 3
53 N 4
500 N 5 + 60 N 6 + 28 N 7
5 N8
S1 (N ) (2)
+
2(N
2)2 (N
1)2 N 2 (N + 1)2 (N + 2)
2112 + 608 N + 76 N 2
140 N 3 + 107 N 4
64 + 6 N 2 + N 3
2
S1 (N ) (3) +
(2)
+
(N
2)(N
1)N (N + 1)
10(N
2)(N
1)N (N + 1)(N + 2)
224
136 N + 1688 N 2 + 1290 N 3
1998 N 4
1997 N 5 + 198 N 6
+
(2)
2(N
1)3 N 3 (N
2)2 (N + 2)(N + 1)3
+405 N 7 + 376 N 8
119 N 9 + 56 N 10 + 5 N 11
(2)
+
2(N
1)3 N 3 (N
2)2 (N + 2)(N + 1)3
552 + 144 N + 1654 N 2
370 N 3
361 N 4 + 19 N 5 + 35 N 6
25 N 7
(3)
+
2(N
2)2 (N
1)2 N 2 (N + 1)2
320
64 N
1920 N 2 + 1600 N 3 + 6524 N 4
14872 N 5
19036 N 6 + 31543 N 7
43960 N 8
13935 N 9
+
16(N
1)5 (N + 1)5 (N
2)3 N 5 (N + 2)
+65372 N 10 + 26822 N 11
44576 N 12
9558 N 13 + 9840 N 14 + 339 N 15 + 428 N 16
371 N 17 + 128 N 18
+
16(N
1)5 (N + 1)5 (N
2)3 N 5 (N + 2)
N 4 N 2 + 12
N 4 N 2 + 12
f0;2 + ( 2)
f02;1
+4
(N
2)(N
1)N (N + 1)(N + 2)
(N
2)(N
1)N (N + 1)(N + 2)
+
4(N
z-space: A. Penin, (2005)
=⇒ 3 basic sums only; no alternating sums.
22
5. Factorial Series
23
Consider
Ω(z) =
Z
1
dt t
z−1
ϕ(t);
0
Re(z) > 0,
Ω(z) =
∞
X
k=0
ϕ(1 − t) =
∞
X
ak t k
k=0
ak+1 k!
z(z + 1) . . . (z + k)
• Ω(z) is meromorphic in z ǫ C, obeys a recursion z → z + 1 and has an analytic asymptotic
representation.
• The poles are situated at the non-positive integers.
Examples:
F5 (z)
=
F5 (z + 1)
=
Asymp. ser. : Li2 (z) →
Li2 (1 − z)
M
(N ) ∼
1+z
Li2 (z)
M
(z)
1+z
1
ψ(z + 1) + γE
−F5 (z) +
ζ2 −
z
z
−Li2 (1 − z) − ln(z) ln(1 − z) + ζ2
1
1
7 1
1 1
73 1
+
−
−
+
...
2N 2
4N 3
24 N 4
3 N5
120 N 6
24
Factorial Series
F13 (z)
F13 (z + 1)
=
=
Asymp. ser. : Li22 (z) →
"
#
2
Li2 (1 − z)
(N ) ∼
M
z−1
+
M
"
Li22 (z)
z−1
+
#
(z)
4ζ3
2ζ2
2S2,1 (z)
2 2
ζ22
+ 2 + 2 S1 (z) +
+ 3 S1 (z) + S2 (z)
−F13 (z) +
z
z
z
z2
z
Li22 (1 − z) + ln2 (z) ln2 (1 − z) + ζ22 + 2Li2 (1 − z) ln(z) ln(1 − z) + . . .
1
7 1
1 1
223 1
7 1
3767 1
38 1
−
+
+
−
−
+
z2
24 z 4
12 z 5
1080 z 6
45 z 7
15120 z 8
105 z 9
14327 1
198 1
138673 1
3263 1
5265804043 1
−
−
+
+
31500 z 10
175 z 11
118800 z 12
693 z 13
1324323000 z 14
143341487 1
25092 1
34809672614 1
13399637 1
−
+
+
−
525525 z 15
8408400 z 16
143 z 17
402026625 z 18
5749693892 1
1
−
+
O
3828825 z 19
z 20
+
25
6. The Basis
w=1
1/(x − 1)+
w=2
ln(1 + x)/(x + 1)
w=3
Li2 (x)/(x ± 1)
w=4
Li3 (x)/(x + 1)
S1,2 (x)/(x ± 1)
w=5
Li4 (x)/(x ± 1)
S1,3 (x)/(x ± 1)
Li22 (x)/(x ± 1)
[ln(x)S1,2 (−x) − Li22 (−x)/2]/(x ± 1)
Li5 (x)/(x + 1)
S1,4 (x)/(x ± 1)
S3,2 (x)/(x ± 1)
Li2 (x)Li3 (x)/(x ± 1)
A1 (x)/(x + 1)
A2 (x)/(x ± 1)
H0,−1,0,1,1(x)/(x ± 1)
H0,0,−1,0,1 (x)/(x ± 1)
w=6
S2,2 (x)/(x ± 1)
S2,3 (x)/(x ± 1)
A3 (x)/(x + 1)
[A1 (−x) + 2S3,2 (−x) − 2S2,2 (−x) ln(x)]
x±1
A1 (−x) + 2S3,2 (−x) − 2S2,2 (−x) ln(x) + Li22 (−x) ln(x)/4 − Li3 (−x)Li2 (−x)
x−1
26
• O(α)
Wilson Coefficients/anom. dim.
#1
• O(α2 )
Anomalous Dimensions
#2
• O(α2 )
Wilson Coefficients
• O(α3 )
Anomalous Dimensions
#≤5
• O(α3 )
Wilson Coefficients
#15
#35
27
7. Conclusions - A
• We considered mathematical structures which determine no scale and single
scale quantities in Quantum Field Theories.
• The former correspond to integrated cross sections, expansion coefficients of
the β–function, or anomalous dimensions at fixed moments, etc.
• The latter correspond to differential scattering cross sections of one variable,
N –dependent anomalous dimensions, coefficient functions, etc.
• The single-scale quantities in Quantum Field Theories to 3 Loop Order
⇔ w = 6 can be represented in a polynomial ring spanned by a few Mellin
transforms of the above basic functions, which are the same for all known
processes. This points to their general nature.
• The basic Mellin transforms are meromorphic functions with single poles at the
non-positive integers.
• The total amount of harmonic sums reduces due to algebraic relations [index
structure], and structural relations N ǫ Q, N ǫ R.
28
• They can be represented in terms of factorial series up to simple “soft
components”. This allows an exact analytic continuation.
• Up to w = 6 physical (pseudo-) observables are free of harmonic sums with
index = {-1}. Up to w = 5 all numerator functions are Nielsen integrals.
29
8. From Moments to Functions in QCD
• Higher order calculations in Quantum Field Theories easily become tedious
due to the larger number of terms and the sophistication of the Feynman
parameter integrals.
• This even applies to Zero Scale and Single Scale Quantities.
• Even more this is the case for higher scale problems.
• While in the latter case the mathematical structure of the solution for the
Feynman Integrals is widely unknown, it is explored to a certain extent for
Zero Scale and Single Scale quantities.
• They can be expressed by rational numbers and certain special numbers as
multiple ζ-values and related quantities.
30
• Single Scale quantities depend on a scale z ∈ [0, 1], with z a ratio of Lorentz
invariants. One may perform a Mellin Transform over z
Z 1
dzz N −1 f (z) = M [f ](N )
0
• Here one assumes N ∈ N, N > 0. Due to this the problem on hand becomes
discrete.
• One may seek a description in terms of difference equations.
• Zero Scale problems are obtained from Single Scale problems treating N as
a fixed integer or considering the limit N → ∞.
• Can one reconstruct the general formula for Single Scale quantities out of a
finite number of fixed moments ? This is possible for recurrent quantities.
• At least up to 3-loop order, presumably to higher orders, single scale
quantities belong to this class.
• Goal : design a general formalism to solve the problem.
31
9. Single Scale Feynman Integrals
as Recurrent Quantities
• Can one reconstruct the general formula for Single Scale quantities out of a
finite number of fixed moments ?
• Polynomials and Nested Harmonic Sums obey recurrence relations, so do
their polynomials.
• Example: Harmonic Sums or linear combinations thereof:
sign(a)N +1
F (N + 1) − F (N ) =
(N + 1)|a|
is solved by Sa (N ); and similarly for deeper nested sums
Sa,~b (N ) =
N
X
(sign(a))k
k=1
.
k |a|
S~b (k)
32
Single Scale Feynman Integrals
as Recurrent Quantities
• Feynman integrals have often a form like
Z 1
Z 1
(−z)N −1 − 1
z N −1 − 1
H~a (z),
H~a (z)
dz
dz
1
−
z
1
+
z
0
0
• This structure leads to recurrences.
• It is very likely that single scale Feynman diagrams do always obey
difference equations.
33
10. Establishing and Solving Recurrences
• One seeks the relation
l
X
k=0
"
d
X
i=0
#
ci,k N i F (N + k) = 0 .
• The corresponding linear system is dense.
• Rational number arithmetics is not applicable for the large systems to be
(3)
determined; C2,q,C 3 would require 11 Tb of memory.
F
• Use arithmetic in finite fields together with Chinese remaindering
=⇒ few Gb of memory
• The linear system approximately minimizes for l ≈ d.
• Join different recurrences found to reduce l to a minimal value.
34
Establishing and Solving Recurrences
• For the solution of the recurrence low degrees are clearly preferred.
• The linear difference equation of order l with polynomial coefficients is
equivalent to a linear system in l variables.
• It is solved in Π − Σ fields.
• Apply advanced symbolic summation methods: telescoping, creative
telescoping and its refinement. Code: sigma.
• The solutions are found as linear combinations of rational terms in N
combined with functions, which cannot be further reduced in the Π − Σ
fields. In the present application they turn out all to be harmonic sums
S~b (N ).
• Other or higher order applications may consist of other sums too, which are
uniquely found by the algorithm.
35
11. Application to 3-Loop Anomalous Dimensions
and Wilson coefficients
• We apply the method for the unfolding of the unpolarized anomalous
dimensions and Wilson coefficients up to 3-loop order.
• =⇒ analyze for individual color factors; 141 contributions from 1 − 3 loops
• Input: Moch, Vermaseren, Vogt, 2004/05. The expressions are given in terms of
harmonic sums.
• Calculate the moments (rational numbers) recursively through recursions
for the harmonic sums; MAPLE code.
• Establish the corresponding difference equation by a recurrency finder;
build a difference equation of minimal order possible; test the recurrency.
• Solve the difference equation order by order with the summation package
sigma C. Schneider.; most complicated cases: 4 weeks @ ≤ 10Gb, 2 GHz Proc.
36
C2qq3CF^3
N=3:
#11 digits / #10 digits
Input
-98268084191 / 1166400000
N=500:
#1262 digits / #1256 digits
1641840770424196780953020619176376506284303544481262083057197600746507008493793994
4224110323441591630311482222058287688942209570859151121677307585313995100978363179
2518952817622034037186132846974627021672678012913675099511203807811938593043910803
5044345920218696052588332036355325089998361354226882367322149037631053761764348772
5403810874264968729520075619227285471802419403727207822473765999900236383740315299
2050533601633484348249454757555344664210814111140065475391136798689167410065076749
3578709478683390573977410013520894494463909291327425815766566386397276158317387748
5945471392646089700875157445075073192328542890965462004805711998748144414379386093
9937361798029044425789953726133675199790523770427298500510063464061985840066296071
3372543015648919155964069606994597363886301185067827291937065300754786947063672848
9382081926871078600328628131936766057475970450896556667622163365895808773428119721
5352792131089063577045069693962213061198894057033606068695607123271969726981060056
0115846094360239986233917872260722277322690450132376836253549152130116645670565045
9666945920164586023958060271746606798898861360772333088030741775605546518788793327
2264368297071217405654474375844238250889238538974548421298170425909521742559494728
72017877003947396562261659860366839154407853462338171648227013134266795320251847
/
37
3057444614247225372882570514367358697278130741348282122206492932820352440850471902
7491046962105336645563654873675690796713906565688820365601907263710863954826386081
3227580037879361869941003802807590860358894142891046776447162895908787986423254678
5776778283337231702130612499429819559798501074020676282769289102955679421885795867
1982932998601320344971927374905889934059987271939760212836368619501189238215442366
3805773701929509268157747992859384837403751183019423692868569168206789710047557452
5131217382272060267681480496298975522467614707848639773185909858278799786637303834
1017166676276847525704755493166263297079720470719813623901545811953853986456533543
9994182050551827959988760121168490745476969259468454613431624179198860751513076481
0304734205926703138519418575731315944374873897873646706993620825697218523316375559
4068222004765962715924208526106008109740402380126260947524640509361283802755722132
4856690051525724685919792641506082307567956962328560073471086799287131287564668441
6256698083504233897436484702002471314330803421467773925541151273924985946178771189
2312437162213438137703896064734987157020801413153555435311326719739117599044341913
5922693587373856609594245948237469293148702516714038297077639382332251255360181047
49658623247509112659762997679737527882711111677459300352000000000000000000
N=5114:
#13388 digits / #13381 digits
Table 1: Run parameters for the unfolding of the non-singlet anomalous dimensions
38
PN S,0
number of
order of
degree of
total time
length of
number of
solution
terms needed
recurrence
recurrence
[sec]
recurrence
harm. sums
time [sec]
[kbyte]
a [b]
14
2
3
0.05
0.087
1 [1]
0.55
PN−S,1,C 2
142
5
31
3.32
4.666
6 [10]
7.45
PN−S,1,CA CF
PN−S,1,CF NF
PN+S,1,C 2
F
PN+S,1,CA CF
PN+S,1,CF NF
PN−S,2,C 3
F
PN−S,2,C 3 ζ3
F
PN−S,2,CA C 2
F
PN−S,2,CA C 2 ζ3
F
PN−S,2,C 2 CF
A
PN−S,2,C 2 CF ζ3
A
PN−S,2,CF N 2
F
PN−S,2,C 2 NF
F
PN−S,2,C 2 NF ζ3
F
PN−S,2,CA CF NF
PN−S,2,CA CF NF ζ3
PN+S,2,C 3
F
PN+S,2,C 3 ζ3
F
PN+S,2,CA C 2
F
PN+S,2,CA C 2 ζ3
F
PN+S,2,C 2 CF
A
PN+S,2,C 2 CF ζ3
A
PN+S,2,C 2 NF
F
PN+S,2,C 2 NF ζ3
F
PN+S,2,CA CF NF
PN+S,2,CA CF NF ζ3
PN+S,2,CF N 2
F
PN−S,2,NF dabc
109
4
24
1.91
2.834
6 [7]
6.28
24
2
7
0.13
0.271
2 [2]
0.92
142
5
31
3.35
4.707
6 [10]
7.45
109
4
23
1.88
2.703
6 [7]
6.23
F
24
2
7
0.09
0.271
2 [2]
0.89
1079
16
192
3152.19
529.802
25 [68]
1194.41
48
3
11
0.49
0.643
1 [1]
1.56
974
15
181
1736.08
450.919
25 [62]
1194.41
48
3
11
0.53
0.643
1 [1]
1.53
749
12
147
1004.12
242.892
25 [62]
1100.88
48
3
11
0.56
0.643
1 [1]
1.56
39
2
11
0.31
0.369
3 [3]
1.20
377
8
68
76.34
33.946
12 [24]
72.22
14
2
3
0.12
0.101
1 [1]
0.53
356
7
62
65.25
23.830
12 [20]
52.67
14
2
3
0.12
0.101
1 [1]
0.55
1079
16
192
4713.27
527.094
25[68]
1165.22
48
3
11
0.55
0.643
1[1]
1.562
974
15
178
1715.03
442.031
25[62]
889.047
48
3
11
0.61
0.643
1[1]
1.531
749
12
146
991.22
240.325
25[50]
516.812
48
3
11
0.61
0.643
1[1]
1.593
377
8
69
111.38
33.872
12[24]
71.235
14
2
3
0.15
0.101
1[1]
0.531
307
7
61
48.62
23.808
12[24]
71.235
14
2
3
0.15
0.101
1[1]
0.547
39
2
11
0.40
0.369
3[3]
1.172
39
2
11
0.55
0.369
3 [3]
1.19
39
Table 2: Run parameters for the unfolding of the unpolarized quarkonic Wilson Coefficients for the
structure function F2 (x, Q2 ).
number of
order of
degree of
total time
length of
number of
solution
terms needed
recurrence
recurrence
[sec]
recurrence
harm. sums
time [sec]
[kbyte]
a [b]
(1)
35
3
7
0.26
0.429
2[3]
1.13
C2,q,C 2
(2)
689
11
137
1134.10
177.806
13[39]
258.24
(2)
545
10
121
413.33
127.893
12[35]
178.73
C2,q,C 2 ζ3
(2)
15
2
3
0.27
0.100
1[1]
0.54
C2,q,CA CF ζ3
15
2
3
0.27
0.112
1[1]
0.55
C2,q,NF CF
71
4
16
2.68
1.655
4[10]
3.95
5114
35
938
1.78886 ×106
30394.173
58[289]
0.50924 × 106
284
8
64
31.02
32.363
7 [18]
27.60
19
2
5
0.08
0.163
1 [1]
0.47
C2,q,CF
F
C2,q,CA CF
F
(3)
C2,q,C 3
F
(3)
C2,q,C 3 ζ3
F
(3)
C2,q,C 3 ζ4
F
(3)
C2,q,C 3 ζ5
F
(3)
C2,q,C 2 CA
F
(3)
C2,q,C 2 CA ζ3
F
(3)
C2,q,C 2 CA ζ4
F
(3)
C2,q,C 2 CA ζ5
F
(3)
C2,q,CF C 2
A
(3)
C2,q,CF C 2 ζ3
A
(3)
C2,q,CF C 2 ζ4
A
(3)
C2,q,CF C 2 ζ5
A
(3)
C2,q,C 2 NF
F
(3)
C2,q,C 2 NF ζ3
F
(3)
C2,q,C 2 NF ζ4
F
(3)
C2,q,CF CA NF
(3)
C2,q,CF CA NF ζ3
(3)
C2,q,CF CA NF ζ4
(3)
C2,q,CF N 2
F
(3)
C2,q,CF N 2 ζ3
F
(3)
C2,q,dabc
(3)
C2,q,dabcζ3
(3)
C2,q,dabcζ5
19
2
5
0.08
0.163
1 [1]
0.47
5059
35
930
1.69267 ×106
30122.380
60 [290]
0.47780 ×106
284
8
64
34.00
33.400
7 [18]
28.53
48
3
11
0.32
0.643
1[1]
1.01
19
2
5
0.08
0.167
1 [1]
0.42
4564
33
863
1.38918 ×106
24567.518
60 [258]
0.34941 ×106
284
8
63
26.83
29.918
7 [17]
30.46
48
3
11
0.32
0.643
1 [1]
1.01
19
2
5
0.08
0.175
1 [1]
0.42
1762
20
348
40237.45
2339.516
29 [107]
7548.56
87
4
21
1.94
2.354
3 [5]
2.83
15
2
3
0.07
0.101
1 [1]
0.34
1847
20
360
47661.64
2507.362
29 [111]
7525.89
89
4
24
2.47
2.935
3 [8]
3.19
15
2
3
0.06
0.101
1 [1]
0.34
131
5
30
58.00
5.347
7 [22]
8.97
15
2
3
0.06
0.101
1 [1]
0.38
1199
14
242
6583.27
738.498
14 [62]
841.24
109
4
25
2.33
3.164
2[7]
2.40
8
1
2
0.03
0.041
0[0]
0.10
40
A complicated example
C2,q ∝ CF3 :
• 5114 moments needed. Use a clever way to calculate the input.
• Largest moment: fraction: numerator 13388 digits; denominator 13381 digits.
• CPU time to determine the recurrence: 20.7 days.
- modular prediction of the dimension: 4 h; modular LEQ’s: 5.8 days; modular operator GCDs:
11 days; Chinese Remainder + Rat. Reconstruction: 3.8 days. 140 large primes needed.
31 MB recurrence is established; largest integer: 1227 digits; order: 35; degree: 938
• Solved by sigma within about one week.
• 3 loop anomalous dimensions: much smaller recurrences & shorter computation
times.
=⇒ In practice no method does yet exists to calculate such a high number of
moments.
=⇒ Existence proof of a quite general and powerful automatic
difference-equation solver, standing rather demanding tests.
41
Structure of the Results
• We carry out all algebraic reductions,
J.B. 2003.
• Different color factor contributions lead to the same or nearly the same
amount of sums at a given quantity.
• This points to the fact that the amount of harmonic sums is governed by
topology rather than the fields and color.
• The linear harmonic sum representations by
many more sums than our representation.
Vermaseren et al. 2004/05
require
• There are reductions in the number of sums as 264 −→ 29.
• Further use of structural relations will lead to maximally 35 sums for the
3-loop Wilson coefficients; J.B. arxiv: 0901.0837, arXiv:0901.3106.
42
3n2 + 3n + 2
0
(n) = CF 4S1 −
Pqq
n(n + 1)
"
6
3n + 9n5 + 9n4 − 5n3 − 24n2 − 32n − 24
1,−
− 16S−3
(n) = CF2 −
Pqq
2n3 (n + 1)3
4 (3n2 + 3n + 2)
8(2n + 1)
16
−
16S
− 32S1 + S1
S2
+
+S−2
2
n(n + 1)
n2 (n + 1)2
n(n + 1)
#
16(−1)n
−16S3 + 32S−2,1 +
(n + 1)3
"
268
51n5 + 102n4 + 655n3 + 484n2 + 12n + 144
+ 8S−3 +
S1
+ CA CF −
18n3 (n + 1)2
9
#
8(−1)n
44
8
− S2 + 8S3 − 16S−2,1 −
+S−2 16S1 −
n(n + 1)
3
(n + 1)3
#
"
8
3n4 + 6n3 + 47n2 + 20n − 12 40
− S1 + S2
+ CF NF
9n2 (n + 1)2
9
3
"
6
5
4
3
2
3n + 9n + 9n + 59n + 40n + 32n + 8
1,+
− 16S−3
(n) = CF2 −
Pqq
2n3 (n + 1)3
16
8(2n + 1)
−
16S
− 32S1 + S1
+S−2
2
n(n + 1)
n2 (n + 1)2
#
2
16(−1)n
4 (3n + 3n + 2)
S2 − 16S3 + 32S−2,1 +
+
n(n + 1)
(n + 1)3
"
268
51n5 + 153n4 + 757n3 + 851n2 + 208n − 132
+ 8S−3 +
S1
+ CA CF −
18n2 (n + 1)3
9
#
8
44
8(−1)n
+S−2 16S1 −
− S2 + 8S3 − 16S−2,1 −
n(n + 1)
3
(n + 1)3
#
"
8
3n4 + 6n3 + 47n2 + 20n − 12 40
− S1 + S2
+ CF NF
9n2 (n + 1)2
9
3
(
16 (3n6 + 9n5 + 9n4 + 17n3 + 6n2 + 8n + 2)
64
2
3
2,−
+
− 128S1 S−2
Pqq (n) = CF
n(n + 1)
n3 (n + 1)3
64 (3n2 + 3n − 11) S2
64 (3n2 − n + 1)
− 1408S2 −
+ 1536S3 + 128S−2,1
+ S1
2
2
n (n + 1)
n(n + 1)
!
P1 (n)
16 (3n2 + 3n + 2) S22
− 576S−5
− 5
− 2304S2,1 S−2 −
n(n + 1)
2n (n + 1)5
16 (9n2 + 9n − 26)
+ S−4 −
− 832S1
n(n + 1)
32 (3n2 + 3n + 20) S1 16 (21n2 + 17n + 20)
2
+ S−3 640S1 −
+
−
320S
−
2240S
−2
2
n(n + 1)
n2 (n + 1)2
!
2
48
(2n
−
n
+
1)
128S
96(5n
+ 3)S1
64S2
−2
n
+ (−1) −
+
+
−
(n + 1)5
(n + 1)3
(n + 1)4
(n + 1)3
4 (13n4 + 26n3 + 13n2 − 16n − 20) S3 16 (15n2 + 15n + 2) S4
−
− 192S5 − 832S−4,1
n2 (n + 1)2
n(n + 1)
896S−3,1
32 (15n2 + 11n + 16) S−2,1
32 (3n2 + 3n + 1)
+
+ 1152S−3,2 + S12 −
− 768S−2,1 −
n(n + 1)
n3 (n + 1)3
n2 (n + 1)2
2 (3n6 + 9n5 + 9n4 + 19n3 + 12n2 − 4n − 16)
+ S2
+ 64S3 + 2176S−2,1
n3 (n + 1)3
2
2
64 (3n + 3n − 2) S3,1
32 (3n + 3n − 26) S2,−2
− 1472S3,−2 +
+ 192S3,2 + 192S4,1
+
n(n + 1)
n(n + 1)
+
+ 2304S−3,1,1 + 512S−2,1,−2 +
384 (n2 + n − 4) S−2,1,1
64(2n + 1)S2
+ S1 64S22 − 2
n(n + 1)
n (n + 1)2
4 (22n6 + 186n5 + 167n4 − 40n3 − 115n2 − 120n − 44)
− 192S3 + 64S4 − 1792S−3,1
n4 (n + 1)4
!
192 (n2 + n − 4) S−2,1
−
+ 1664S2,−2 + 256S3,1 + 3072S−2,1,1 + 2304S−2,2,1 + 2304S2,1,−2
n(n + 1)
+
− 384S3,1,1 − 4608S−2,1,1,1
#)
"
3
24 (5n4 + 10n3 + 9n2 + 4n + 4)
+
CF3 − CF2 CA ζ3 −
−
192S
−2
2
n2 (n + 1)2
(
16 (3n2 + 3n + 8)
2
256S1 −
S−2
+ CA CF2
n(n + 1)
"
8 (81n6 + 243n5 − 229n4 − 389n3 − 130n2 + 228n + 72) 32 (31n2 + 31n − 81) S2
+
+ −
9n3 (n + 1)3
3n(n + 1)
#
4
3
2
32 (134n + 268n + 215n + 45n + 54)
S−2
−
1792S
−
192S
+
2688S
+ S1 1728S2 −
3
−2,1
2,1
9n2 (n + 1)2
8 (97n2 + 97n − 210)
P2 (n)
176 2
+ 672S−5 + S−4
S −
+ 1120S1
+
3 2 36n5 (n + 1)5
3n(n + 1)
16 (31n2 + 31n + 108) S1 8 (268n4 + 536n3 + 811n2 + 507n + 450)
−
3n(n + 1)
9n2 (n + 1)2
!
8 (382n2 + 41n − 161)
256S−2
16(127n + 121)S1
−
−
+ 480S−2 + 2656S2 + (−1)n
9(n + 1)5
(n + 1)3
3(n + 1)4
!
4
3
2
2
32S2
8 (385n + 770n + 427n + 6n − 126) S3 8 (151n + 151n − 30) S4
+
−
+
(n + 1)3
9n2 (n + 1)2
3n(n + 1)
+ S−3 −576S12 +
+ 384S5 + 864S−4,1 −
960S−3,1
− 1344S−3,2
n(n + 1)
43
2 (453n5 + 906n4 + 1325n3 + 488n2 − 120n + 144)
−
32S
−
2624S
3
−2,1
9n3 (n + 1)2
16 (268n4 + 536n3 + 625n2 + 321n + 414) S−2,1
+ S12 (128S3 + 896S−2,1)
+
9n2 (n + 1)2
16 (31n2 + 31n − 174) S2,−2
32 (29n2 + 29n − 24) S3,1
−
+ 1824S3,−2 −
− 384S3,2 − 384S4,1
3n(n + 1)
3n(n + 1)
+ S2
− 2688S−3,1,1 − 768S−2,1,−2 + S1
− 1536S−2,1,1,1
8 (135n6 + 731n5 + 245n4 − 617n3 − 395n2 − 309n − 144)
−
9n4 (n + 1)4
32 (31n2 + 31n − 12) S3
32 (31n2 + 31n − 84) S−2,1
2144
S2 +
+ 160S4 + 1920S−3,1 +
9
3n(n + 1)
3n(n + 1)
!
2
64 (31n + 31n − 84) S−2,1,1
− 2688S−2,2,1 − 2688S2,1,−2
− 1856S2,−2 − 512S3,1 − 3584S−2,1,1 −
3n(n + 1)
)
−
"
2
24 (n + n + 2)
2
+
− 96S1 S−2
n(n + 1)
6
5
4
3
+
CF NF2
"
51n6 + 153n5 + 57n4 + 35n3 + 96n2 + 16n − 24 16
80
16
− S1 − S2 + S3
27n3 (n + 1)3
27
27
9
+
CF2 NF
"
−
+
+
+
−
+
16 (11n2 + 11n + 24) S1 8 (134n4 + 268n3 + 311n2 + 177n + 135)
+
3n(n + 1)
9n2 (n + 1)2
!
4 (389n4 + 778n3 + 398n2 + 9n − 81) S3 8 (55n2 + 55n − 24) S4
− 160S−2 − 768S2 +
−
9n2 (n + 1)2
3n(n + 1)
+ S−3 128S12 −
+
256S−3,1
+ 384S−3,2 + S12 (−64S3 − 256S−2,1 )
n(n + 1)
16 (134n4 + 268n3 + 245n2 + 111n + 135) S−2,1
4172
−
+ S2 768S−2,1 −
2
2
9n (n + 1)
27
32 (11n2 + 11n − 12) S3,1
16 (11n2 + 11n − 48) S2,−2
− 544S3,−2 +
+
3n(n + 1)
3n(n + 1)
64 (11n2 + 11n − 24) S−2,1,1
+ 192S3,2 + 192S4,1 + 768S−3,1,1 + 256S−2,1,−2 +
3n(n + 1)
+
− 160S5 − 224S−4,1 +
+ S1
−
+
8 (11n2 + 11n − 8) S3
32 (11n2 + 11n − 24) S−2,1
− 128S4 − 512S−3,1 −
n(n + 1)
3n(n + 1)
!
+ 512S2,−2 + 256S3,1 + 1024S−2,1,1
+ 768S−2,2,1 + 768S2,1,−2, − 384S3,1,1
2,+
Pqq
#
32 2 4 (15n4 + 30n3 + 79n2 + 16n − 24) S2
S −
3 2
9n2 (n + 1)2
207n8 + 828n7 + 1443n6 + 1123n5 − 38n4 − 779n3 − 632n2 + 120 128
−
S−4
9n4 (n + 1)4
3
64S1
128(4n + 1)
32 (10n2 + 10n + 3) 64
− S1 + (−1)n
−
S−3
9n(n + 1)
3
3(n + 1)3
9(n + 1)4
32 (16n2 + 10n − 3) 640
128
16 (29n2 + 29n + 12) S3 128
S−2 −
+
S1 −
S2 +
−
S4
9n2 (n + 1)2
9
3
9n(n + 1)
3
5
4
3
2
128
128
2 (165n + 330n + 165n + 160n − 16n − 96) 320
+
S
−
S
−
S
S1 −
2
3
−2,1
9n3 (n + 1)2
9
3
3
#
64 (10n2 + 10n − 3) S−2,1 64
64
256
+ S2,−2 + S3,1 +
S−2,1,1
9n(n + 1)
3
3
3
#
"
8 (3n2 + 3n + 2)
CF2 − CF CA NF ζ3 32S1 −
n(n + 1)
"
2 (270 n7 + 810n6 − 463n5 − 1392n4 − 211n3 − 206n2 − 156n + 144)
CA CF NF −
27n4 (n + 1)3
2
32
16 (10n + 10n + 3)
32S1
64(4n + 1)
64
S−4 + S−3
S1 −
−
+ (−1)n
3
3
9n(n + 1)
9(n + 1)4
3(n + 1)3
2
2
1336
64
16 (16n + 10n − 3) 320
8 (14n + 14n + 3) S3 80
−
S2 + S−2
S1 + S2 −
+ S4
27
9n2 (n + 1)2
9
3
3n(n + 1)
3
4 (209n6 + 627n5 + 627n4 + 281n3 + 36n2 + 36n + 18)
32 (10n2 + 10n − 3) S−2,1
+ S1 −
9n(n + 1)
27n3 (n + 1)3
!
#
64
32
64
128
+ 16S3 + S−2,1 − S2,−2 − S3,1 −
S−2,1,1
3
3
3
3
"
16 (3n6 + 9n5 + 9n4 + n3 + 2n2 + 4n + 2)
64
2
3
+
− 128S1 S−2
= CF
n(n + 1)
n3 (n + 1)3
2
2
64 (3n + 7n + 5)
64 (3n + 3n − 11) S2
+ S1 −
− 1408S2 −
+ 1536S3 + 128S−2,1
n2 (n + 1)2
n(n + 1)
+
2 (245n8 + 980n7 + 1542n6 + 1524n5 + 851n4 + 100n3 + 36n2 + 22n − 6)
3n4 (n + 1)4
#
+
2
8 (27n + 81n − 155n − 271n − 92n + 78n + 27)
9n3 (n + 1)3
16 (134n4 + 268n3 + 188n2 + 54n + 45)
32 (11n2 + 11n − 24) S2
+ S1
− 512S2 −
+ 512S3
9n2 (n + 1)2
3n(n + 1)
!
2
8 (35n + 35n − 66)
P3 (n)
− 192S−5 + S−4 −
− 352S1
+ 64S−2,1 − 768S2,1 S−2 +
5
5
108n (n + 1)
3n(n + 1)
16 (82n2 + 17n − 47)
96S−2
16(41n + 47)S1
n
+ (−1) −
+
+
9(n + 1)5
(n + 1)3
3(n + 1)4
+ CA2 CF
12 (5n4 + 10n3 + 9n2 − 4n − 4)
− 96S−2
−
n2 (n + 1)2
CA2 CF ζ3
+
+ 768S3,1,1 + 5376S−2,1,1,1
"
#
44
−
+
+
+
−
+
+
+
!
16 (3n2 + 3n + 2) S22
P4 (n)
2304S2,1 S−2 −
− 5
− 576S−5
n(n + 1)
2n (n + 1)5
16 (9n2 + 9n − 26)
32 (3n2 + 3n + 20) S1
S−4 −
− 832S1 + S−3 640S12 −
n(n + 1)
n(n + 1)
!
16 (9n2 + 5n + 8)
16 (2n2 + 11n + 1)
128S−2
− 320S−2 − 2240S2 + (−1)n
+
n2 (n + 1)2
(n + 1)5
(n + 1)3
!
64S2
4 (13n4 + 26n3 + 13n2 − 16n − 20) S3
96(5n + 3)S1
−
+
4
3
(n + 1)
(n + 1)
n2 (n + 1)2
896S−3,1
16 (15n2 + 15n + 2) S4
− 192S5 − 832S−4,1 +
+ 1152S−3,2
n(n + 1)
n(n + 1)
32 (3n2 + 3n + 1)
32 (3n2 − n + 4) S−2,1
S12 −
− 768S−2,1 −
n3 (n + 1)3
n2 (n + 1)2
6
5
4
3
2
2 (3n + 9n + 9n + 83n + 76n + 60n + 16)
+
64S
+
2176S
S2
3
−2,1
n3 (n + 1)3
32 (3n2 + 3n − 26) S2,−2
64 (3n2 + 3n − 2) S3,1
− 1472S3,−2 +
+ 192S3,2 + 192S4,1
n(n + 1)
n(n + 1)
+ 2304S−3,1,1 + 512S−2,1,−2 +
384 (n2 + n − 4) S−2,1,1
64(2n + 1)S2
+ S1 64S22 − 2
n(n + 1)
n (n + 1)2
4 (22n6 − 54n5 + 23n4 + 88n3 + 197n2 + 160n + 52)
− 192S3 + 64S4 − 1792S−3,1
n4 (n + 1)4
!
192 (n2 + n − 4) S−2,1
+ 1664S2,−2 + 256S3,1 + 3072S−2,1,1 + 2304S−2,2,1
−
n(n + 1)
#
+
+ 2304S2,1,−2 − 384S3,1,1 − 4608S−2,1,1,1
"
24 (5n4 + 10n3 + n2 − 4n − 4)
− 192S−2
n2 (n + 1)2
(
16 (3n2 + 3n + 8)
2
S−2
256S1 −
+ CA CF2
n(n + 1)
+ CF3 ζ3 −
5
4
3
2
#
2
16 (31n2 + 31n + 108) S1 8 (268n4 + 536n3 + 487n2 + 183n + 126)
−
3n(n + 1)
9n2 (n + 1)2
!
+ (−1)n
8(346n − 125)
256S−2
16(103n + 73)S1
32S2
−
−
+
9(n + 1)4
(n + 1)3
3(n + 1)4
(n + 1)3
8 (385n4 + 770n3 + 427n2 + 6n − 126) S3 8 (151n2 + 151n − 30) S4
+
+ 384S5
9n2 (n + 1)2
3n(n + 1)
960S−3,1
2 (453n5 + 1359n4 + 2231n3 + 1525n2 + 80n − 264)
− 1344S−3,2 + S2
n(n + 1)
9n2 (n + 1)3
!
4
3
2
16 (268n + 536n + 301n − 3n + 90) S−2,1
+ S12 (128S3 + 896S−2,1 )
− 32S3 − 2624S−2,1 +
9n2 (n + 1)2
+ 864S−4,1 −
−
16 (31n2 + 31n − 174) S2,−2
32 (29n2 + 29n − 24) S3,1
+ 1824S3,−2 −
− 384S3,2 − 384S4,1
3n(n + 1)
3n(n + 1)
− 2688S−3,1,1 − 768S−2,1,−2 + S1 −
8 (135n6 − 649n5 − 1039n4 − 569n3 + 487n2 + 621n + 216)
9n4 (n + 1)4
32 (31n2 + 31n − 12) S3
32 (31n2 + 31n − 84) S−2,1
2144
S2 +
+ 160S4 + 1920S−3,1 +
9
3n(n + 1)
3n(n + 1)
!
64 (31n2 + 31n − 84) S−2,1,1,n
− 1856S2,−2 − 512S3,1 − 3584S−2,1,1 −
− 2688S−2,2,1
3n(n + 1)
#
−
− 2688S2,1,−2 + 768S3,1,1 + 5376S−2,1,1,1
+
+
"
#
36 (5n4 + 10n3 + n2 − 4n − 4)
+
288S
−2
n2 (n + 1)2
2
8 (27n6 + 81n5 − 209n4 − 595n3 − 272n2 − 48n − 9)
24 (n + n + 2)
2
− 96S1 S−2
+
CA2 CF
n(n + 1)
9n3 (n + 1)3
4
3
2
16 (134n + 268n + 116n − 18n − 27)
32 (11n2 + 11n − 24) S2
S1
− 512S2 −
+ 512S3
9n2 (n + 1)2
3n(n + 1)
!
P6 (N)
8 (35n2 + 35n − 66)
64S−2,1 − 768S2,1 S−2 +
−
192S
+
S
−
−
352S
−5
−4
1
108n3 (n + 1)5
3n(n + 1)
2
96S
16(29n
+
23)S
16
(91n
+
80n
−
29)
−2
1
+
+
(−1)n −
9(n + 1)5
(n + 1)3
3(n + 1)4
+ CA CF2 ζ3
+
−
+ S−3 −576S12 +
−
+
8 (81n + 243n − 337n − 1181n − 526n − 60) 32 (31n + 31n − 81) S2
+
9n2 (n + 1)3
3n(n + 1)
!
4
3
2
32 (134n + 268n + 89n − 81n − 72)
−
1792S
−
192S
+
2688S
S−2
+ S1 1728S2 −
3
−2,1
2,1
9n2 (n + 1)2
8 (97n2 + 97n − 210)
P5 (n)
176 2
S −
+ 672S−5 + S−4
+ 1120S1
+
3 2 36n4 (n + 1)4
3n(n + 1)
+
+ 480S−2 + 2656S2
16 (11n2 + 11n + 24) S1 8 (134n4 + 268n3 + 203n2 + 69n + 27)
+
3n(n + 1)
9n2 (n + 1)2
!
4 (389n4 + 778n3 + 398n2 + 9n − 81) S3 8 (55n2 + 55n − 24) S4
−
− 160S−2 − 768S2 +
9n2 (n + 1)2
3n(n + 1)
+ S−3 128S12 −
256S−3,1
+ 384S−3,2 + S12 (−64S3 − 256S−2,1 )
n(n + 1)
16 (134n4 + 268n3 + 137n2 + 3n + 27) S−2,1
4172
−
+ S2 768S−2,1 −
2
2
9n (n + 1)
27
32 (11n2 + 11n − 12) S3,1
16 (11n2 + 11n − 48) S2,−2
− 544S3,−2 +
+ 192S3,2
+
3n(n + 1)
3n(n + 1)
− 160S5 − 224S−4,1 +
45
+ 192S4,1 + 768S−3,1,1 + 256S−2,1,−2 +
+ S1
−
−
64 (11n2 + 11n − 24) S−2,1,1
3n(n + 1)
2 (245n8 + 980n7 + 1542n6 + 964n5 + 211n4 − 60n3 + 156n2 + 222n + 90)
3n4 (n + 1)4
8 (11n2 + 11n − 8) S3
− 128S4 − 512S−3,1
n(n + 1)
32 (11n2 + 11n − 24) S−2,1
+ 512S2,−2 + 256S3,1 + 1024S−2,1,1
3n(n + 1)
)
!
+ 768S−2,2,1
+ 768S2,1,−2 − 384S3,1,1 − 1536S−2,1,1,1
"
+ CA2 CF ζ3 −
+
+
+
+
−
+
+
+
+
+
12 (5n4 + 10n3 + n2 − 4n − 4)
− 96S−2
n2 (n + 1)2
#
(
32 2 4 (15n4 + 30n3 + 79n2 + 16n − 24) S2
P7 (n)
128
S −
+ 4
−
S−4
3 2
9n2 (n + 1)2
9n (n + 1)4
3
2
32 (10n + 10n + 3) 64
128(4n + 1)
64S1
S−3
− S1 + (−1)n
−
9n(n + 1)
3
3(n + 1)3
9(n + 1)4
128
16 (29n2 + 29n + 12) S3 128
32 (16n2 + 10n − 3) 640
+
S1 −
S2 +
−
S4
S−2 −
9n2 (n + 1)2
9
3
9n(n + 1)
3
128
128
2 (165n5 + 495n4 + 495n3 + 517n2 + 336n + 80) 320
+
S2 −
S3 −
S−2,1
S1 −
2
3
9n (n + 1)
9
3
3
)
64
256
64 (10n2 + 10n − 3) S−2,1 64
+ S2,−2 + S3,1 +
S−2,1,1
9n(n + 1)
3
3
3
#
"
8 (3n2 + 3n + 2)
CF2 NF ζ3 32S1 −
n(n + 1)
"
#
6
5
51n + 153n + 57n4 + 35n3 + 96n2 + 16n − 24 16
80
16
CF NF2
−
S
−
S
+
S
1
2
3
27n3 (n + 1)3
27
27
9
"
7
6
5
4
3
2
2 (270n + 1080n + 383n − 979n − 571n + 507n + 106n − 132)
CA CF NF −
27n3 (n + 1)4
2
32
16 (10n + 10n + 3)
64(4n + 1)
32S1
64
S−4 + S−3
S1 −
+ (−1)n
−
3
3
9n(n + 1)
9(n + 1)4
3(n + 1)3
1336
16 (16n2 + 10n − 3) 320
64
8 (14n2 + 14n + 3) S3 80
S2 + S−2
−
S1 + S2 −
+ S4
27
9n2 (n + 1)2
9
3
3n(n + 1)
3
CF2 NF
−
4 (209n6 + 627n5 + 627n4 + 137n3 − 108n2 − 108n − 54)
32 (10n2 + 10n − 3) S−2,1
+ S1 −
9n(n + 1)
27n3 (n + 1)3
!
#
64
128
32
64
S−2,1,1
+ 16S3 + S−2,1 − S2,−2 − S 3 , 1 −
3
3
3
3
"
#
8 (3n2 + 3n + 2)
+ CA CF NF ζ3
− 32S1
n(n + 1)
+
2,−,dabc
Pqq
"
P8 (n)
4 (n2 + n + 2) S−3
P9 (n)S1
dabc dabc
NF − 5
+
− 4
=
Nc
3n (n + 1)5 (n + 2)3
n2 (n + 1)2
3n (n + 1)4 (n + 2)3
2
+ S−2
+ (−1)n
8S1 (n2 + n + 2)
4 (n6 + 3n5 − 8n4 − 21n3 − 23n2 − 12n − 4)
−
−
2
2
(n − 1)n (n + 1) (n + 2)
(n − 1)n3 (n + 1)3 (n + 2)
16 (5n6 + 29n5 + 78n4 + 118n3 + 114n2 + 72n + 16) S1
3(n − 1)n2 (n + 1)3 (n + 2)3
4 (13n8 + 74n7 + 179n6 + 314n5 + 644n4 + 1000n3 + 816n2 + 352n + 64)
3(n − 1)n3 (n + 1)4 (n + 2)3
#
2 (n2 + n + 2) S3 8 (n2 + n + 2) S−2,1
−
−
n2 (n + 1)2
n2 (n + 1)2
−
!
!
46
12. Conclusions - B
• We established a general algorithm to calculate the exact expression for
single scale quantities from a finite (suitably large) number of moments
(zero scale quantities).
• The latter ones are much more easily calculable.
• We applied the method to the anomalous dimensions and Wilson
coefficients up to 3-loop order.
• To solve 3-loop problems this way is not possible at present, since the
number of required moments is too large for the methods available.
• We attempted to solve the quantities for all color projections at once.
This problem is too voluminous.
• Yet we showed that giant difference equations [order 35; degree ∼ 1000] can
be reliably and fast established and solved unconditionally for advanced
problems in Quantum Field Theory.