Mathematical Structure of QCD Wilson
Coefficients and Anomalous Dimensions
Johannes Blümlein
DESY
1. Introduction
2. x Space Results
3. Multiple Harmonic Sums to Level 6
4. The Length of the Basis
5. The 16th Non–Singlet Moment
6. Phenomenological Results
7. Conclusions
Parton Distributions in the 21st Century, Seattle, WA, November 2004
J. Blümlein
Seattle, WA, November 2004
1
1. Introduction
Consider hard scattering processes in massless field theories:
QCD, QED,
mi → 0
Factorization Theorem Leading Twist:
The cross section σ factorizes as
σ=
X
σk,W ⊗ fk
k
σW
perturbative Wilson Coefficient
f
non-perturbative Parton Density
⊗
Mellin convolution
Z 1
Z
[A ⊗ B] (x) =
dx1
0
1
dx2 δ(x − x1 x2 )A(x1 )B(x2 )
0
M [A ⊗ B] (N ) = M [A] (N ) · M [B] (N )
with the Mellin transform :
Z 1
M [f (x)] (N ) =
dxxN −1 f (x),
Re[N ] > c
0
Observation :
Feynman Amplitudes seem to obey the Mellin Symmetry
i.e. to significantly simplify in Mellin Space
J. Blümlein
Seattle, WA, November 2004
2
2. x Space Results
Usual Starting Point of Higher Order Calculations :
=⇒ Nielsen type Integrals and their Generalization
(−1)n+p+q−1
Sn,p,q (x) =
(n − 1)!p!q!
Z
1
0
dz (n−1)
ln
(z) lnp (1−zx) lnq (1+zx)
z
Special Cases:
Lin (x) = Sn−1,1 (x)
dLi2 (±x)
= − ln(1 ∓ x)
d ln(x)
x
Li0 (x) =
1−x
J. Blümlein
Seattle, WA, November 2004
w=n
w=1
w=0
3
van Neerven, Zijlstra 1992
`
´
(2)
(x) = CF CF − CA /2 ×
2,−
"
(
i
1 + x2 h
2
4 ln (x) − 16 ln(x) ln(1 + x) − 16Li2 (−x) − 8ζ2 ln(1 − x)
1−x
h
i
2
2
+ −2 ln (x) + 20 ln(x) ln(1 + x) − 8 ln (1 + x) + 8Li2 (1 − x) + 16Li2 (−x) − 8 ln(x)
"
!
!#
1−x
1−x
− Li3
−16 ln(1 + x)Li2 (−x) − 8ζ2 ln(1 + x) − 16 Li3 −
1+x
1+x
#
c
−16Li2 (1 − x) + 8S1,2 (1 − x) + 8Li3 (−x) − 16S1,2 (−x) + 8ζ3
h
2
2
+(4 + 20x) ln (x) ln(1 + x) − 2 ln(x) ln (1 + x) − 2ζ2 ln(1 + x) − 4 ln(1 + x)Li2 (−x)
„
i
72 3
8 «
2
+2Li3 (−x) − 4S1,2 (−x) + 2ζ3 + 32 + 32x + 48x −
x +
5
5x2
× [Li2 (−x) + ln(x) ln(1 + x)] + 8(1 + x) [Lis (1 − x) + ln(x) ln(1 − x)] + 16(1 − x) ln(1 − x)
„
1 „
36 3 « 2
8«
2
2
x
ln (x) +
+ −4 − 16x − 24x +
−26 − 106x + 72x −
ln(x)
5
5
x
)
„
1 „
72 3 «
8«
2
2
ζ2 +
x
+ −4 + 20x + 48x −
−162 + 82x + 72x +
5
5
x
(+)
(q,G)
.... several other pages for c2 (x), cG
2 (x), cL
=⇒
77 Functions
(x)
@ 2 Loops
=⇒
partly rather complicated arguments
=⇒
relations are not directly visible ...
The 77 functions do roughly correspond in number to the number
of all possible harmonic sums up to weight w=4: 80.
J. Blümlein
Seattle, WA, November 2004
4
x Space Results
No.
f (z)
1
M [f ](N ) =
δ(1 − z)
2
z
Z
1
0
dzz
N −1
f (z)
1
1
r
N +r
„
3
1
1−z
«
−S1 (N − 1)
+
1
4
(−1)
N −1
1+z
+
[log(2) − S1 (N − 1)]
1 + (−1)N −1
2
5
6
z
z
r
r
(−1)n
n
log (z)
(N + r)n+1
log(1 − z)
−
S1
N −1
2
!
−
1 − (−1)N −1
2
S1
N −2
2
!
Γ(n + 1)
S1 (N + r)
N +r
7
z
r
2
log (1 − z)
2 (N + r) + S (N + r)
S1
2
3
log (1 − z)
S 3 (N + r) + 3S1 (N + r)S2 (N + r) + 2S3 (N + r)
− 1
N +r
r
8
z
9
» log(1 − z) –
10
» log2(1 − z) –
N +r
1−z
1
+
1−z
11
» log3(1 − z) –
1−z
12
logn(z)
1−z
1
2
S1 (N − 1) + S2 (N − 1)
2
2
−
+
2
3
S1 (N − 1) + S1 (N − 1)S2 (N − 1) + S3 (N − 1)
3
3
h1
i
1
+
3 2
4
S1 (N − 1) + S1 (N − 1)S2 (N − 1)
4
2
3 2
+ S2 (N − 1) + 2S1 (N − 1)S3 (N − 1)
4
3
+ S4 (N − 1)
2
(−1)
n+1
h
i
Γ(n + 1) Sn+1 (N − 1) − ζ(n + 1)
Only single sums.
J. Blümlein
Seattle, WA, November 2004
5
No.
f (z)
M [f ](N )
Li3 (−z)
64
(−1)
1+z
+
1
N −1

h
i
S3,−1 (N − 1) + S3 (N − 1) − S−3 (N − 1) log 2
ζ(2)S−2 (N − 1) −
3
2
4
ff
1 2
3
+ ζ (2) − ζ(3) log 2
8
4
65
1 ˆ
S1 (N )S2 (N ) − ζ(2)S1 (N ) + S3 (N )
N
˜
−S2,1 (N ) + ζ(3)
Li3 (1 − z)
Li3 (1 − z)
66
1
2
ζ(2)S1 (N − 1) + ζ(2)S2 (N − 1)
2
2
2 2
−ζ(3)S1 (N − 1) + ζ (2)
5
»
N −1
S−1,1,2 (N − 1) − ζ(2)S−1,1 (N − 1)
(−1)
„1«
9 2
ζ (2)
+ζ(3)S−1 (N − 1) + Li4
−
2
20
–
1
1
7
2
4
log 2
+ ζ(3) log 2 + ζ(2) log 2 +
8
2
24
−S1,1,2 (N − 1) +
1−z
Li3 (1 − z)
67
1+z
68
1−z
Li3
69
1
1+z
"
−Li3
(−1)N »
!
1+z
!
1−z
−
1+z
−Li3
Li3
−
1−z
N
1
−S−1,2 (N ) − S−2,1 (N ) + S1 (N )S−2 (N )
+S−3 (N )
!
1+z
!#
1−z
1+z
ζ(3)S−1 (N − 1)
–
7
3
1
+ζ(2)S−1 (N ) + ζ(2)S1 (N ) − ζ(3) + ζ(2) log 2
2
8
2
1 »
−S−1,−2 (N ) − S2,1 (N ) + S1 (N )S2 (N ) + S3 (N )
+
N
–
1
21
3
− ζ(2)S−1 (N ) − ζ(2)S1 (N ) +
ζ(3) − ζ(2) log 2
2
8
2

N −1
(−1)
S1,1,−2 (N − 1) − S1,−1,2 (N − 1)
+S−1,1,2 (N − 1) − S−1,−1,−2 (N − 1)
+2ζ(2)S1,−1 (N − 1) +
1
1
2
2
ζ(2)S1 (N − 1) − ζ(2)S−1 (N − 1)
4
4
−ζ(2)S1 (N − 1)S−1 (N − 1) − ζ(2)S−2 (N − 1)
»7
–
3
−
ζ(3) − ζ(2) log 2 S1 (N − 1)
8
2
» 21
–
3
+
ζ(3) − ζ(2) log 2 S−1 (N − 1)
8
2
ff
„1«
19 2
1
1
2
4
−2Li4
+
ζ (2) + ζ(2) log 2 −
log 2
2
40
2
12
2 loop coefficient functions =⇒
Weight w = 4
J. Blümlein
Nested Harmonic Sums of
Seattle, WA, November 2004
6
x Space Results
S−1,−1,−2 (N ) =

ff
1
[F1 (x) + log(1 − x)Li2 (−x)] (N )
(−1)N +1 M
1+x

»
–ff
1
1
+(−1)N +1 M
S1,2 (x2 ) − S1,2 (x) − S1,2 (−x) (N )
1+x 2
1
+ ζ(2) [S−1,1 (N ) − S−1,−1 (N )]
2
»
–
9
3
1
+ ζ(3) − ζ(2) log(2) − log3 (2) S−1 (N )
8
2
6
1
17
7
1
− ζ(2)2 +
ζ(3) log(2) − ζ(2) log2 (2) − log4 (2)
10
8
4
6
with
F1 (x)
=
„
«
„
«
1−x
1−x
S1,2
+ S1,2 (1 − x) − S1,2
2
1+x
„
«
„
«
1
1−x
+S1,2
− ln(2)
1+x
2
„
«
„
«
1 2
1+x
1−x
+ ln (2) ln
− ln(2)Li2
2
2
1+x
F1 (x), although of complicated structure, it reduces completely via
algebraic relations
⇒ Mellin polynomial of simpler objects
These objects can be very complicated integrals.J.B., van Neerven,
Ravindran, Kawamura 2000, 2003
J. Blümlein
Seattle, WA, November 2004
7
3. Multiple Harmonic Sums to Level 6
The simplest example :
1 + x2
2
Pqq (x) =
=
+ ...
1−x +
(1 − x) +
Z
1
0
N
−1
N
−2 Z 1
X
X
1
xN −1
k
dxx = −
=−
= −S1 (N − 1)
dx
(1 − x)+
k
0
k=1
k=0
Alternating sums :
S−1 (N − 1) = (−1)
N −1
–
Z 1
N
−1
X
(−1)k
xN −1
1
dx
=
M
(N ) − ln(2) =
1+x
(1 − x)+
k
0
k=1
»
(Finite for N → ∞.)
General case :
Sa1 ,...,al (N ) =
k1
N
X
(sign(a2 ))k2
(sign(a1 ))k1 X
k1 =1
|a |
k1 1
k2 =1
|a |
k2 2
...
Vermaseren, 1997
All Mellin transforms occurring in massless Field Theories for
1-Parameter Quantities can be represented by Harmonic Sums
(at least to 3–loop order).
J. Blümlein
Seattle, WA, November 2004
8
Algebraic Relations
First relation:
L. Euler, 1775
Sm,n + Sn,m = Sm · Sn + Sm+n ,
m, n > 0
Generalized to alternating sums by
Sm,n + Sn,m
m∧n
= Sm · Sn + Sm∧n ,
= [|m| + |n|] sign(m)sign(n)
Ternary relations: Sita Ramachandra Rao, 1984,
4-ary relation: J.B., Kurth, 1998.
These & other relations hold widely independent
of their Value and Type.
Determined by : • Index Structure
• Multiplication Relation
The Formalism applies as well to the Harmonic Polylogarithms.
Remiddi, Vermaseren, 1999.
Application to QED: T. Riemann et al., 2004
J.B., Comput. Phys. Commun. 159 (2004) 19.
J. Blümlein
Seattle, WA, November 2004
9
Linear Representations of Mellin Transform by Harmonic Sums:
Skw1 ,...,km (N )
M[Fw (x)](N ) =
w=
Pm
i=1
τ 0 , τ 00 < w
|ki |
+P
00
0
Skτ1 ,...,kr , σkτ1 ,...,kp
Weight
P is a polynomial.
w
#
Σ
1
2
2
2
6
8
3
18
26
2 Loop anom. Dimensions
4
54
80
2 Loop Wilson Coefficients
5
162
242
3 Loop anom. Dimensions
6
486
728
3 Loop Wilson Coefficients
2 · 3w−1
3w − 1
J. Blümlein
Seattle, WA, November 2004
10
Shuffle Products
Depth 2:
Sa1 (N ) tt Sa2 (N )
=
Sa1 ,a2 (N ) + Sa2 ,a1 (N )
Depth 3:
Sa1 (N ) tt Sa2 ,a3 (N )
=
Sa1 ,a2 ,a3 (N ) + Sa2 ,a1 ,a3 (N ) + Sa2 ,a3 ,a1 (N )
=
Sa1 ,a2 ,a3 ,a4 (N ) + Sa2 ,a1 ,a3 ,a4 (N ) + Sa2 ,a3 ,a1 ,a4 (N )
+
Sa2 ,a3 ,a4 ,a1 (N )
=
Sa1 ,a2 ,a3 ,a4 (N ) + Sa1 ,a3 ,a2 ,a4 (N ) + Sa1 ,a3 ,a4 ,a2 (N )
+
Sa3 ,a4 ,a1 ,a2 (N ) + Sa3 ,a1 ,a4 ,a2 (N ) + Sa3 ,a1 ,a2 ,a4 (N )
Depth 4:
Sa1 (N ) tt Sa2 ,a3 ,a4 (N )
Sa1 ,a2 (N ) tt Sa3 ,a4 (N )
Depth 5:
Sa1 (N ) tt Sa2 ,a3 ,a4 ,a5 (N )
Sa1 ,a2 (N ) tt Sa3 ,a4 ,a5 (N )
=
Sa1 ,a2 ,a3 ,a4 ,a5 (N ) + Sa2 ,a1 ,a3 ,a4 ,a5 (N )
+
Sa2 ,a3 ,a1 ,a4 ,a5 (N ) + Sa2 ,a3 ,a4 ,a1 ,a5 (N )
+
Sa2 ,a3 ,a4 ,a5 ,a1 (N )
=
Sa1 ,a2 ,a3 ,a4 ,a5 (N ) + Sa1 ,a3 ,a2 ,a4 ,a5 (N )
+
Sa1 ,a3 ,a4 ,a2 ,a5 (N ) + Sa1 ,a3 ,a4 ,a5 ,a2 (N )
+
Sa3 ,a1 ,a2 ,a4 ,a5 (N ) + Sa3 ,a1 ,a4 ,a2 ,a5 (N )
+
Sa3 ,a1 ,a4 ,a5 ,a2 (N ) + Sa3 ,a4 ,a5 ,a1 ,a2 (N )
+
Sa3 ,a4 ,a1 ,a5 ,a2 (N ) + Sa3 ,a4 ,a1 ,a2 ,a5 (N )
Depth 6: .....
J. Blümlein
Seattle, WA, November 2004
11
Algebraic Equations
Depth 2:
Sa1 (N ) tt Sa2 (N ) − Sa1 (N )Sa2 (N ) − Sa1 ∧a2 (N ) = 0
Depth 3:
Sa1 (N ) tt Sa2 ,a3 (N )
−
Sa1 (N )Sa2 ,a3 (N ) − Sa1 ∧a2 ,a3 (N ) − Sa2 ,a1 ∧a3 (N ) = 0
Depth 4:
Sa1 (N ) tt Sa2 ,a3 ,a4 (N )
Sa1 ,a2 (N ) tt Sa3 ,a4 (N )
−
Sa1 (N )Sa2 ,a3 ,a4 (N ) − Sa1 ∧a2 ,a3 ,a4 (N )
−
Sa2 ,a1 ∧a3 ,a4 (N ) − Sa2 ,a3 ,a1 ∧a4 (N ) = 0
−
Sa1 ,a2 (N )Sa3 ,a4 (N ) − Sa1 ,a2 ∧a3 ,a4 (N )
−
Sa1 ,a3 ,a2 ∧a4 (N ) − Sa3 ,a1 ∧a4 ,a2 (N )
−
Sa3 ,a1 ,a2 ∧a4 (N ) − Sa1 ∧a3 ,a2 ,a4 (N )
−
Sa1 ∧a3 ,a4 ,a2 (N ) + Sa1 ∧a3 ,a2 ∧a4 = 0
Depth 5: .....
# Basic Sums = # Permutations - # Independent Equations
J. Blümlein
Seattle, WA, November 2004
12
Some Solution for d = 6
Sa,a,a,a,b,b =
−
1
4
3
Sa Sb,a,a,a,b +
4
1
Sa∧b,a,a,a,b −
1
4
1
Sb,a,a,a,a∧b +
1
12
1
Sa Sa,a,b,b,a
Sa∧a,b,b,a,a −
Sa,b,b,a∧a,a −
Sa,b,b,a,a∧a
12
12
12
1
1
1
1
− Sb,a∧a,a,a,b − Sb,a,a∧a,a,b − Sb,a,a,a∧a,b − Sa,a∧a,a,b,b
4
4
4
4
1
1
1
1
Sa∧a,b,a,b,a
− Sa,a,a∧a,b,b − Sa,b,a∧a,a,b − Sa,b,a,a∧a,b +
4
4
4
12
1
1
1
1
Sa,b,a∧a,b,a − Sa,a,a,b,a∧b − Sa,a,b,a,a∧b +
Sa,a,a∧b,b,a
+
12
4
4
12
1
1
3
+ Sa,a,a∧b,a,b − Sb,b,a,a,a,a + Sb,b,a,a∧a,a − Sa∧a,a,a,b,b
4
4
4
1
1
1
1
+
Sa,b,a,b,a∧a +
Sb,a∧a,a,b,a +
Sb,a,a∧a,b,a +
Sb,a,a,b,a∧a
12
12
12
12
1
1
1
1
Sb,a∧a,b,a,a −
Sb,a,b,a∧a,a + Sb,b,a∧a,a,a + Sb,b,a,a,a∧a
−
12
12
4
4
1
1
1
1
− Sa∧a,a,b,a,b − Sa,a∧a,b,a,b − Sa,a,b,a∧a,b +
Sa∧a,a,b,b,a
4
4
4
12
1
1
1
1
Sa,a∧a,b,b,a +
Sa,a,b,b,a∧a − Sa∧a,b,a,a,b −
Sb,a,b,a,a∧a
+
12
12
4
12
1
1
1
1
Sa∧b,a,a,b,a +
Sb,a,a,a∧b,a −
Sa∧b,a,b,a,a + Sa∧b,b,a,a,a
+
12
12
12
4
1
1
3
1
+ Sb,a∧b,a,a,a −
Sb,a,a∧b,a,a + Sa,a,a,a∧b,b +
Sa,a,b,a∧b,a
4
12
4
12
1
1
1
3
Sa,a∧b,a,b,a +
Sa,b,a,a∧b,a
+ Sa,a∧b,a,a,b − Sa,b,a,a,a∧b +
4
4
12
12
1
1
1
1
−
Sa,a∧b,b,a,a −
Sa,b,a∧b,a,a − Sa Sa,a,a,b,b −
Sa Sa,b,b,a,a
12
12
4
12
1
1
1
1
+
Sa Sa,b,a,b,a −
Sa Sb,a,b,a,a +
Sa Sb,a,a,b,a − Sa Sa,b,a,a,b
12
12
12
4
1
1
+Sb Sa,a,a,a,b + Sa Sb,b,a,a,a − Sa Sa,a,b,a,b
4
4
+Sa,a,a,a,b∧b −
J. Blümlein
Seattle, WA, November 2004
13
Depth d = 3
Index Set
Number
Dep. Sums of Depth 3
min. Weight
Fraction of
fund. Sums
{a, a, a}
1
1
3
0
{a, a, b}
3
2
3
1/3
{a, b, c}
6
4
4
1/3
Depth d = 4
Index Set
Number
Dep. Sums of Depth 4
min. Weight
Fraction of
fund. Sums
{a, a, a, a}
1
1
4
0
{a, a, a, b}
4
3
4
1/4
{a, a, b, b}
6
5
4
1/6
{a, a, b, c}
12
9
5
1/4
{a, b, c, d}
24
18
6
1/4
Depth d = 6
Index Set
Number
Dep. Sums of Depth 6
min. Weight
Fraction of
fund. Sums
{a, a, a, a, a, a}
1
1
6
0
{a, a, a, a, a, b}
6
5
6
1/6
{a, a, a, a, b, b}
15
13
6
2/15
{a, a, a, b, b, b}
20
17
6
3/20
{a, a, a, a, b, c}
30
25
7
1/6
{a, a, a, b, b, c}
60
50
7
1/6
{a, a, b, b, c, c}
90
76
8
7/45
{a, a, a, b, c, d}
120
100
8
1/6
{a, a, b, b, c, d}
180
150
8
1/6
{a, a, b, c, d, e}
360
300
10
1/6
{a, b, c, d, e, f }
720
600
12
1/6
J. Blümlein
Seattle, WA, November 2004
14
Theory of Words
Can we count the Basis in simpler way ? =⇒ YES.
Free Algebras and Elements of the Theory of Codes
=⇒ Particle Physics
Only the multiplication relation
and the Index structure matters
A = {a, b, c, d, . . .}
Alphabet
a < b < c < d < ...
A∗ (A)
ordered
Set of all words W
W = a1 · a2 · a27 . . . a532 ≡ concatenation product (nc)
W = p · x · s p = prefix; s = suffix
Definition:
A Lyndon word is smaller than any of its suffixes.
Theorem:[Radford, 1979]
The shuffle algebra KhAi is freely generated by the Lyndon words.
I.e. the number of Lyndon words yields the number of basic
elements.
Examples :
{a, a, . . . , a, b} = aaa . . . ab
n
a0 s : nbasic /nall = 1/n
J. Blümlein
1 Lyndon word for these sets
n ≡ depth of the sums
Seattle, WA, November 2004
15
{a, a, a, b, b, b}
aaabbb, aababb, aabbab
3 Lyndon words
nbasic /nall = 3/20 < 1/6. Symmetries lead to a smaller fraction.
Is there a general Counting Relation ?
E. Witt, 1937
(n/d)!
1X
,
µ(d)
ln (n1 , . . . , nq ) =
n
(n1 /d)! . . . (nq /d)!
X
i
d|ni
µ(k)
ni = n
Möbius function
2nd Witt formula.
The Length of the Basis is a function mainly of
the Depth.
l6 ({a, a, a, b, b, b})
=
n6 ({a, a, a, b, b, b})
=
Weight
# Sums
»
–
1
6!
2!
µ(1)
+ µ(3)
=3
6
3!3!
1!1!
6!
= 20
2!3!
Cum. #
# Basic
Cum. #
Cum.
Sums
Sums
Basic Sums
Fraction
1
2
2
0
0
0.0
2
6
8
1
1
0.1250
3
18
26
6
7
0.2692
4
54
80
16
23
0.2875
5
162
242
46
69
0.2851
6
486
728
114
183
0.2513
⇑ 2nd Witt formula
J. Blümlein
Seattle, WA, November 2004
16
Structural Relations
Seek for further Reduction:
Relations using the Value of the Objects [DESY 04-064]
Use Relations like:
Li2 (x)
1 Li2 (x2 )
Li2 (x)
Li2 (−x)
Li2 (−x)
=
+
+
+
2 1 − x2
1−x
1+x
1−x
1+x
1
M
4
"„
Li4 (x)
1−x
« #„
+
N
2
«
=
"„
« #
Li4 (−x)
M
(N ) + M
(N )
1−x
+
+
–
»
–
»
Li4 (−x)
Li4 (x)
(N ) + M
(N )
+M
1+x
1+x
3
41
1
ζ(5) .
+ ζ(4) ln(2) − ζ(2)ζ(3) +
8
4
128
Li4 (x)
1−x
« #
"„
and similar ones.
Apply Symmetries among Mellin-Transforms of Nielsen Integrals.
Since all harmonic sums are meromorphic functions for N C since they may
be represented by Factorial Series Derivatives are not essentially new functions.
M[lnk (x)f (x)](N ) =
∂k
M[f (x)](N )
∂N k
Further Reduction due to the Structure of Feynman Amplitudes
The Lord is mercy, after all!
J. Blümlein
Seattle, WA, November 2004
17
Analytic Continuation
The Harmonic Sums and Mellin Transforms have to be represented
such, that the outer summation index can be analytically
continued to N C [J.B., Comput.Phys.Commun.133 (2000) 76.]
• Use precise, adaptive Representations in analytic Form
• Refer to the Representation through Factorial Series etc.
• The Residue Theorem is used to get back to x space
J. Blümlein
Seattle, WA, November 2004
18
4. The Length of the Basis
The anomalous dimensions and Wilson coefficients for mi = 0 can
be expressed in terms of multiple harmonic sums to 3–loop order.
What are the irreducible functions behind this representation ?
We will not count Euler’s Γ-function neither all derivations of the
functions occurring.
The final set of functions:
Trivial functions:
S±k (N )
−→
ψ (k−1) (N + 1)
For w = 1, 2 no non–trivial functions contribute to the anomalous
dimensions and Wilson coefficients.
Non–trivial functions:
N = 3 : Two–Loop anomalous dimensions
Li2 (x)
M
(N )
1+x
Yndurain et al., 1980
N = 4 : Two–Loop Wilson Coefficients
ln(1 + x)
Li2 (x)
S1,2 (x)
M
(N ), M
(N ), M
(N )
1+x
1−x
1±x
J.B., S. Moch, 2003,
J.B., V. Ravindran, 2004.
J. Blümlein
Seattle, WA, November 2004
19
N = 5 : Three–Loop Anomalous Dimensions
S1,3 (x)
Li4 (x)
(N ), M
(N ),
M
1±x
1+x
S2,2 (x)
S2,2 (−x) − Li22 (−x)/2
M
(N ), M
(N ),
1±x
1±x
2
Li2 (x)
M
(N )
1+x
J.B., S. Moch, 2004.
Essentially 14 Functions seem to rule the single scale processes of
massless QCD.
This is a rather small number if compared to the number of
possible harmonic sums 3w − 1.
J. Blümlein
Seattle, WA, November 2004
20
5. The 16th Moment of the
3-Loop Non–Singlet
Anomalous Dimension of F2,L (x, Q2 )
Seek for another, blind check of the complete calculation of the
NS–anomalous dimension by Moch, Vermaseren, and Vogt, 2004.
The calculation was started far before the complete calculation was
completed and is based on the MINCER algorithm used before by
Larin, Noguiera, van Ritbergen, Vermaseren and Retey, 1994–2000
Moment
CPU time [days]
gµν
Pµ Pν
2
0.002567
0.002190
4
0.012562
0.020027
6
0.057144
0.059320
8
0.303415
0.332731
10
1.108047
1.219046
.
.
.
16
236.236
327.542
J.B., J. Vermaseren, hep-ph/0411111
Set-up used: 1 XEON-Dual PC 3 GHz (Drittmittel) and 1 XEON-Dual
2.6 GHz PC (borrowed from AMANDA) linked to a 4.2 Tbyte RAID
system; partly 1 64-bit OPTERON 4Gbyte (dual)
Results after 564 CPU days
Many Thanks to: P. Wegner, S. Wiesand, U. Gensch & C. Spiering for
supporting this project.
J. Blümlein
Seattle, WA, November 2004
21
⇒
DESY-Z
urgently needs a Parallel PC-Facility ⇐
for FORM Formula Manipulation
CPU Time/sec
Run-time statistics :
10 6
10 5
10 4
10 3
10 2
10
1
10
10
10
-1
-2
-3
50
100
150
200
250
300
350
Diagram
Several diagrams stayed in the CPU for 25–40 days each.
J. Blümlein
Seattle, WA, November 2004
22
Results :
(0)
γ16 =
(1)
64419601
CF
6126120
21546159166129889
1176525373840303
C F NF +
CF CA
112588038763200
484994628518400
3689024452928781382877
−
CF 2
459818557352009856000
γ16 = −
(2)
γ16 =
«
„
58552930270652300886778705063429867
59290512768143
ζ3 −
CF 3
1563722760600
3451337970612452534317096673280000
!
15018421824060388659436559
64419601
+ −
−
ζ 3 C F C A NF
579371382263532418560000
765765
!
1670423728083984207878825467
59290512768143
2
+
+
ζ 3 CF CA
6488959481351563087872000
3127445521200
5559466349834573157251
2
C F NF
2069183508084044352000
„
«
59290512768143
1229794646000775781127856064477
2
+ −
−
ζ 3 CF CA
30335885575318557435801600000
1042481840400
„
«
71543599677985155342551355451
64419601
+ −
+
ζ 3 C F 2 NF
938967886855098206346240000
765765
−
Agreement with : Moch, Vermaseren, Vogt, hep-ph/0403192.
J. Blümlein
Seattle, WA, November 2004
23
C
NS,16
(x, as )
2
=
4047739719
190590400
CF as
44426674163044428879366970127 24439538
+
321931846921747956461568000
+
17918308408498294222783087
255255
113298677
−
59422705873182812160000
−
1021020
!
143568372761907472111177
2
C F NF a s
2758911344112059136000
59290512768143
+
3127445521200
+
+
+
+
+
27643576
21879
ζ3
!
!
CF
2
CF CA
ζ5
3036813397599509725084677293842505976559161689
8034458016040775933421647863403347968000000
!
3
ζ3 C F
595674040206012768000
1494341926940450865387403
59290512768143
6254891042400
+
ζ4 −
ζ3
ζ4 +
262865377883475726558800935515033190333
56646805852503848671021043712000000
!
47187263
15355050469171482313
2
ζ5 −
ζ3 C F C A
51051
4991403051835200
!
7227384935999670312318789884999
64419601
2
+
ζ3 C F N F
76056398835262954714045440000
20675655
7750026627118768752845091760890051465242741
1652500620329242273431025887166464000000
−
−
−
2849482004138921491531
ζ3 +
983963
6741167121672984000
!
59290512768143
2
ζ4 CF ca +
2084963680800
21879
−
ζ5
552298563960959
4021001384400
4073207241348493196152222079933557529
+
3529777469944553728278848870400000
+
598788865585667
1850495446800
−
ζ3 −
64419601
1531530
404620041803598919078721740800000
+
hei
+
3248429831350704000
64419601
1531530
ζ4
!
CF
2
NF
ζ4
582811634921542995647179358698536547
„ 705894258514655486993
ζ3
!
C F C A NF
38404365803
1533061530
ζ3 −
14560
51
!
« d2
3
abc
ζ5
NF a s
Nc
NF
3 X
ef
hei =
NF
k=1
J. Blümlein
Seattle, WA, November 2004
24
C
NS,16
(x, as )
L
=
4
17
"
CF as
29393927457809
+ −
−
+
44659922042736
48
17
"
−
ζ3 C F C A +
39360
17
ζ5 −
CF
2
−
39366889
39054015
55969347000169
8209544493150
96
CF CA +
196256899828170631
133698296031300
17
ζ3 C F
2
#
as
2
ζ3
7508281821276771498126447290110919
13647898235438852429242598400000
+
C F NF
!
CF
3
296045501010133565322039207159677
936620467137960460830374400000
!
2253147763389895
40160
2
ζ5 +
ζ3 C A C F
−
17
1188429298056
+
3529137346321170453160463
−
136796020812222932160000
+ −
1634895686765221
2673965920626
+
+
44651224
765765
ζ3
!
NF C F
2
ζ3
1460792499427100139493280371
8256042197255336964480000
!
895967716232
2
2
ζ5 C A C F +
C F NF
17
209134250325
10240
+ −
4495805144658565385501573689
57792295380787358751360000
!
43594330672
+
ζ3 C A N F C F
1249937325
„ 1798450729620489619601
28854977192
−
ζ3
+hei −
18272417801347710000
547521975
#
« d2
2560
3
abc N
+
ζ5
F as
17
Nc
J. Blümlein
Seattle, WA, November 2004
25
6. Phenomenological Results
S. Moch, J. Vermaseren, A. Vogt: non-singlet: hep-ph/0403192 :
0.105
0.3
γNS+(N)
γNS+(N) / ln N
0.1
0.2
arrows : N → ∞
(in highest term)
0.095
0.09
0.1
LO
NLO
0.085
NNLO
0
0.08
αS = 0.2, Nf = 4
-0.1
0.075
0
5
10
15
0
5
10
N
15
N
A new contribution @ 3 loops :
100
(2)
(2)
30000
PS,1 (x)
PS,1 (x)
exact
50
exact
20000
N = 1...13
Lx
NLx
2
N Lx
3
N Lx
0
10000
-50
0
0
0.2
0.4
0.6
x
J. Blümlein
0.8
1
10
-5
10
-4
Seattle, WA, November 2004
10
-3
x
10
-2
10
-1
1
26
S. Moch, J. Vermaseren, A. Vogt: non-singlet: hep-ph/0403192 :
Slope of the NS+ distribution
0.1
1.3
d ln qNS / d ln µf2
+
0
µr = µf
1.2
-0.1
-0.2
NLO/LO
1.1
NNLO/NLO
LO
-0.3
NLO
1
NNLO
-0.4
αS = 0.2, Nf = 4
-0.5
0
0.2
0.4
0.6
0.8
1
x
0.9
-5
10
10
-4
10
-3
x
Slope of the NS− distribution
10
-2
10
-1
1
0.1
1.3
−
d ln qNS / d ln µf2
0
µr = µf
1.2
-0.1
NLO/LO
NNLO/NLO
-0.2
1.1
qv
LO
-0.3
NLO
NNLO
-0.4
q
1
−
αS = 0.2, Nf = 4
-0.5
0
0.2
0.4
0.6
x
J. Blümlein
0.8
1
10
-5
10
-4
Seattle, WA, November 2004
10
-3
x
10
-2
10
-1
1
27
7. Conclusions
• Mellin space expressions of anomalous dimensions and Wilson
coefficients are of much simpler structure than the x–space
results.
• Index-based algebraic relations of harmonic sums, structural
relations of Mellin–transforms of Nielsen–integrals and the
specific structure of Feynman amplitudes cause this reduction.
• All algebraic relations are derived in explicit form up to weight
w = 6 and apply to harmonic sums, harmonic polylogarithms
and all other objects in the corresponding equivalence class.
The structural relations are worked out up to w = 4 and the
anomalous dimensions for w = 5.
• The number of multiple harmonic sums of weight w is
2 · 3w−1 . The number of the harmonic sums after algebraic
reduction is given by the Witt formula(e) yielding a reduction
to ≈ 1/4. Further reductions result from structural relations.
• The number of non–trivial basic functions for w ≤ 5 which are
needed to express the known anomalous dimensions and the
(space– and time–like) Wilson coefficients for mi = 0 is given
by the sequence :
{0, 0, 1, 4, 8, ...}
to compared to
{2, 6, 18,54,162,486, ...}
J. Blümlein
Seattle, WA, November 2004
28
• An independent calculation of the 16th moment of the
non–singlet structure function F2,L (x, Q2 ) shows agreement
with the complete calculation and upcoming results for the
anomalous dimension and the Wilson coefficients.
J. Blümlein
Seattle, WA, November 2004
29