The Three-Loop Splitting Functions in QCD: The Non-Singlet Case

Transcription

1 The Three-Loop Splitting Functions in QCD: The Non-Singlet Case Sven-Olaf Moch DESY Zeuthen 1. The Calculation. The Result 3. The Summary in collaboration with J.A.M. Vermaseren and A. Vogt, hep-ph/ Workshop HERA and the LHC, CERN, March 7, 004

2 The Calculation The calculation of splitting functions up to three loops is terribly long and awfully difficult to understand, even in its simpler moments which are, roughly speaking, the beginning and the end.

3 The beginning General structure of the anti-)quark anti-)quark splitting functions constraints from charge conjugation invariance and flavour symmetry P qi q k = P qi q k = δ ik P v qq + P s qq P qi q k = P qi q k = δ ik P v q q + P s q q Non-singlet distributions q ± ns,ik = q i ± q i q k ± q k ) flavour asymmetries q v ns = n f q r q r ) sum of valence distribution of all flavours r=1 Three independently evolving types of distributions q ± ns P ± ns = P v qq ± P v q q q v ns P v ns = P v qq P v q q + n f P s qq P s q q) P ns + P s ns

4 Intermezzo Calculate Mellin moments of splitting functions Z 1 γ ns n)i N) = dx x N 1 P ns n)i x) 0 One-loop Feynman diagrams in total 6 for γ ns 0) / P ns 0) pencil + paper) Two-loop Feynman diagrams in total 57 for γ ns 1),i / P ns 1),i simple computer algebra) Three-loop Feynman diagrams in total 948 for γ ns ),i / P ns ),i leading edge technology computer algebra system FORM Vermaseren 89-04)

5 The end Mellin N-space : harmonic sums S m1,...,m k N) Gonzalez-Arroyo, Lopez, Ynduráin 79 ; Vermaseren 98 ; Blümlein, Kurth 98 N 1 recursive definition S m1,...,m k N) = i=1 i S m m 1,...,m k i) algebra of multiplication S j N)S k N) S { j,k} N) Bjorken x-space : harmonic polylogarithms H m1,...,m k x) Goncharov 98 ; Borwein, Bradley, Broadhurst, Lisonek 99 ; Remiddi, Vermaseren 99 Basic functions of lowest weight H 0 x) = lnx, H 1 x) = ln1 x), H 1 x) = ln1 + x) Higher functions defined by recursion H m1,...,m w x) = Z x 0 dz f m1 z) H m,...,m w z) f 0 x) = 1 x, f 1x) = 1 1 x, f 1x) = x Inverse Mellin transformation of harmonic sums harmonic polylogarithms in x space Unique mapping : H m1,...,m w x)/1 ± x) S n1,...,n w+1 N)

6 The Results Anomalous dimensions in Mellin space One-loop : Gross, Wilczek 73 γ 0) ns N) = C F N + N + )S 1 3 ) Two-loop : Floratos, Ross, Sachrajda 79 ; Gonzalez-Arroyo, Lopez, Ynduráin 79 γ 1)+ ns N) = 4C A C F N + S S S 1 + N + N + ) 18 S 1 + S 1, 11 ) 6 S 1 + 4C F n f S 1 N + N + ) 9 S 1 1 ) 3 S + 4C F 4S 3 + S 1 + S 3 8 ) + N S + S 3 N + N + ) S 1 + 4S 1, + S 1, + S,1 + S 3 γ 1) ns N) = γ 1)+ ns N) + 16C F C F C ) ) A N N + ) S S 3 N + N + )S 1 Compact notation : N ± f N) = f N ± 1), N ±i f N) = f N ± i)

7 Three-loop : S.M., Vermaseren, Vogt 04 γ )+ ns N) = 16C A C F n f 3 ζ N + 1) 18 S 3 S N + N + ) 1 3 S, + S 1 ζ S 3,1 ) 9 S S S 1, 3 S 4 + S 1,1 5 9 S S 1 3 S 3,1 N + S,1 3 S 3,1 3 S 4 S 1, S S S S 1, 3 S 1,,1 1 3 S 1, 3 1 S 1,3 1 S,1 + 16C F C A ζ 3 + S S 4 4S 4, S 3 + S 3, S 3,1 + 3 S 6S ζ 3 S, 3 + 3S, 4S,,1 + 8S,1, S 1 10S 1, S 1, + 1S 1,,1 + 4S 1,3 4S, 5 S S S S 3 + N + N + ) 3S 1 ζ S 1,1 4S 1,1, + N + N + ) 43 S 1 3S 1, S 1, 3 + 8S 1, 3, S 1, 6S 1,, 9 3 S 1,,1 + 8S 1,1, 3 16S 1,1,,1 4S 1,1, S 1,3 + 4S 1,3,1 + 3S 1,4 + 8S,,1 + S,3 S 3, S 3,1 S 4,1 4S, S, S S 3 N N + ) 3S ζ 3 + 7S,1 3S,1, + S,,1 1 4 S,3 3 S 3, 9 6 S 3, S 4,1 + 1 S, 3 S, 8 + N + 9 S S 8 3 S 4 5 ) 17 S C F nf S S + N + N + ) 9 S S + 1 ) S C F C A 4 ζ S S 4 + 0S 4, S 3 S 3, 31 3 S 3,1 + S 3, 9 S + 18S ζ S, 3 6S, + 8S,,1 8S,1, + 46S 1, S 1, 48S 1,, S 1, S 3 8S 1,3 + S 3, 4S 5 N + N + ) S 1,1 14S 1,1, 4 18 S + 9S, S 4 3S 3, S,1 + N + N + ) 17S 1, S 1, 3 3S 1, 3, S 1, 9S 1 ζ S 1

8 +16S 1,, S 1,,1 S 1,, 36S 1,1, S 1,1,,1 + 8S 1,1, S 1, 4S 1,, S 1,3 8S 1,3,1 11S 1, S, +1S, 3 30S,,1 4S,1, 5S,3 S 4, S, 67 9 S,1 + N N + ) 9S ζ 3 + S, 3 + 4S,,1 1S,1, S,3 +13S 4,1 + 1 S, + 11 S 4 33 S 3, S S, S + N + 8S 3, S 3,1 S 3, + 14S S S ) 4 S C F n f 16 3 ζ S 3, S S S S S 1, 8 3 S 1, S 1, + N + 9 S S 3,1 1 3 S 4 67 N + 1) 36 S 4 3 S, S 3 + N + N + ) S 1 ζ S 1 3 S 1, S 1, 4 3 S 1,, S 1, S 1, 4 3 S 1, S 3 S, S,1 + 1 S 4 3 S, 8 9 S 3 ) + 16C F 3 1S ζ 3 + 9S 4 4S 4,1 4S 3, + 6S 3,1 4S 3, + 3S + 5S 3 1S ζ 3 1S, 3 + 4S,1, 5S 1, 3 + 4S 1, + 48S 1,,1 4S 3, + 67 S 17S 4 + N + N + ) 6S 1 ζ S S 1,1 1S 1,1, + S 1, + 10S, + S,1 + S, S 3,1 3S 5 + N + N + ) 3S 1, 3 S 1, 4 + 3S 1, 3,1 S 1, 8S 1,, 30S 1,,1 6S 1,3 + 4S 1,, + 40S 1,1, 3 48S 1,1,,1 + 8S 1,, + 4S 1,, + 8S 1,3,1 + 4S 1,4 + 8S,,1 + 4S,1, + 4S,,1 + 4S 3,1,1 4S 3, + 8S,1, 6S, 3 S,3 4S 3, 3S, 3S, + 3 S 4 + N N + ) 1S,1, 6S ζ 3 S, 3 + 3S,3 + S 3, 81 4 S,1 + 14S 3,1 5S, 1 S, S + 1 S 3 13S 4,1 + 4S 5 + N + 14S S 87 4 S 3 4S 4,1 4S 5 )

9 γ NS+ N) 0.1 γ NS+ N) / ln N arrows : N in highest term) LO NLO NNLO α S = 0., N f = N N Left : perturbative expansion of anomalous dimension γ + nsn) for four flavours at α s = 0. Right : leading N-dependence for large N with lnn divided out

10 γ ) ns N) = γ )+ ns N) + 16C A C F C F C ) A 367 N + N + ) 18 S 1 + 1S 1 ζ 3 + S 1, + 4S 1, 3 + 8S 1,, S 1,1 16S 1,1, S 5 8S 3,1 S 4 + N N + ) 4S 5 1S ζ 3 4S, 3 8S,, S,1 + 16S,1, + 4S 3, 8S 4, S 3, S S + S, 15 ) 9 S 3 + 4N + 1) 4S, 8S S C F n f C F C ) A 61 N + N + ) 9 S S 1,1 + N N + ) 3 S, S S S 3,1 4 ) 3 S C F C F C ) A N + N + ) 8S 1, 15S 1 1S 1 ζ 3 1S 1, 3 60S 1,1 + 4S 1,1, + 8S 1, + 40S 1S, + 8S,1 + 7S 3 + 1S 3,1 + 6S 5 + N N + ) 1S ζ 3 4S + 1S, 3 + 8S, + 30S,1 4S,1, 4S, 15S 3 38S 3,1 + 4S 3, + 4S 4,1 ) 1S 5 N + 1) 8S 3, + 6S 4

11 Colour structure dabc d abc n c new at three loops γ )s d abc d abc ns N) = 16n f n c 13 + N + N + ) 1 S 1, N + N + ) S 4 + S, S 3,1 5 N + N + ) 3 S S 1, S 1,,1 1 6 S 1,1, 4 S 1,1 3 8 S 1,3 1 4 S, S S S 3,1 3 N + N + ) S 1, 3 S 1,,1 S 1,1, 1 + N 1) 4 S S 5 + N N + ) S, S, S 41 4 S, S 3 1 S 3,1 S,1, S,3 + 1 S 3, 3 4 S 4,1 503 S 1 1 ) S 1, 3 + S 1,,1 S 1,1,

12 Splitting functions in x-space with f0x) = 1 x, f±1x) = 1 1 x. 4.3) A useful short-hand notation is H 0,...,0,±1,0,...,0,±1,...x) = H ±m+1),±n+1),...x). 4.4) }{{}}{{} m n For w 3 the harmonic polylogarithms can be expressed in terms of standard polylogarithms; a complete list can be found in appendix A of Ref. 45. All harmonic polylogarithms of weight w = 4 in this article can be expressed in terms of standard polylogarithms, Nielsen functions 74 or, by means of the defining relation 4.), as one-dimensional integrals over these functions. A FORTRAN program for the functions up to weight w = 4 has been provided in Ref. 75. and For completeness we recall the one- and two-loop non-singlet splitting functions 3, 8 P 0) ns x) = C F pqqx) + 3δ1 x)) 4.5) P ns 1)+ 67 x) = 4C A C F pqqx) 18 ζ H0 + H0,0 + pqq x) ζ + H 1,0 H0, x) + δ1 x) ) 5 3 ζ 3ζ3 4C F n f pqqx) H0 + 1 x) δ1 x) 1 + ) 3 ζ + 4C F pqqx) H1,0 3 4 H0 + H pqq x) ζ + H 1,0 H0,0 1 x) ) H0 H0 1 + x)h0,0 + δ1 x) 8 3ζ + 6ζ3, 4.6) P ns 1) x) = P ns 1)+ x) + 16C F C F C ) A pqq x) ζ + H 1,0 H0,0 1 x) ) 1 + x)h0. 4.7) Here and in Eqs. 4.9) 4.11) we suppress the argument x of the polylogarithms and use pqqx) = 1 x) 1 1 x. 4.8) All divergences for x 1 are understood in the sense of +-distributions. The three-loopsplittingfunctionfor the evolutionof the plus combinationsofquark densities in Eq..), corresponding to the anomalous dimension 3.8) reads P ns ) x) = 16C A C F n f 6 pqqx) ζ 9ζ3 H0 + H0ζ 7H0,0 H0,0, H1,0,0 H pqq x) ζ3 5 3 ζ H,0 H 1ζ 10 H 1,0 H 1,0,0 3 + H 1, + 1 H0ζ H0,0 + H0,0,0 57 H3 + 1 x) ζ H0 1 6 H0,0 H ζ3 5 H,0 11 H0ζ 1 Hζ 5 4 H0,0ζ + 7H 1 4 H,0,0 + 3H H4 + 1 H0,0ζ ζ 8 3 ζ + 17 ζ3 + H,0 19 H H0ζ H0ζ δ1 x) ζ ζ ζ3 5 ) ζ5 + 16C F nf 3 H0,0 + 5 H0,0,0 + 1 H0,0,0, pqqx) H0, H x) H0 δ1 x) ζ + 1 ) 1 9 ζ3 + 16C F n f 5ζ3 3 pqqx) 4H1,0, H0 + H0ζ + 3 H0,0 H0,0, H1, H H,0 H pqq x) 3 ζ 3 ζ3 + H,0 + H 1ζ H 1,0 + H 1,0,0 H 1, 1 H0ζ 5 3 H0,0 H0,0,0 + H x) H0,0 4 3 H1 + 3 H1, H x) H 1,0 3 4 H0 + 1 H0,0,0 + 9 H H0, H δ1 x) ζ 1 30 ζ + 17 ) 6 ζ C F pqqx) 10 ζ H 3,0 + 6H ζ + 1H, 1,0 6H,0, H0 3 H0ζ + H0ζ H0,0 H0,0,0,0 + 8H1, H1ζ3 + 8H1,,0 6H1,0,0 4H1,0,0,0 + 4H1,,0 3H,0 + H,0,0 + 4H,1,0 + 4H, 7 + 4H3,0 + 4H3,1 + H4 + pqq x) ζ 9 ζ3 6H 3,0 + 3H ζ + 8H, 1,0 + 3H,0 6H,0,0 8H, + 6H 1ζ + 36H 1ζ3 + 8H 1,,0 48H 1, 1ζ + 40H 1, 1,0,0 + 48H 1, 1, + 40H 1,0ζ + 3H 1,0,0 H 1,0,0,0 6H 1, 4H 1,,0 3H 1,3 3 H0 3 H0ζ 13H0ζ3 14H0,0ζ 9 H0,0,0 + 6H0,0,0,0 + 6Hζ + 3H3 + H3,0 + 1H4 + 1 x) H 3, H,0,0 + H0,0ζ 3H0,0,0,0 + 35H1 + 6H1ζ H1,0 + 5 H, x) 10 ζ 93 4 ζ 81 ζ3 15H,0 + 30H 1ζ + 1H 1, 1,0 H 1,0 6H 1,0,0 4H 1, H0 8H0ζ H0,0 + 0H0,0, H 3H,0,0 H3,0 + 13H3 4 H4 + 4ζ + 33ζ3 + 4H 3,0 + 10H, H0 + 6H0ζ3 + 19H0ζ 5H0,0 17H0,0,0 9 H H,0 4H3 + δ1 x) 3 ζζ ζ ζ + 17 ) 4 ζ3 15ζ5. 4.9) The x-space counterpart ofeq. 3.8) forthe evolutionof the minus combinations.) isgivenby P ) ns x) = P )+ ns x) + 16C A C F C F C A ) 134 pqq x) 9 ζ 4ζ 11ζ3 4H 3,0 + 3H ζ + 3 H,0 16H,0,0 3H, + 44 H 1ζ + 48H 1ζ3 64H 1, 1ζ 3 + 3H 1, 1,0,0 + 64H 1, 1, H 1,0 + 44H 1,0ζ + 3 H 1,0,0 1H 1,0,0, H 1, 3H 1,3 3H0 134 H0ζ 16H0ζ3 3 9 H0,0 1H0,0ζ 31 H0,0,0 + 4H0,0,0,0 + 8Hζ x) 3 H 1,0 + 1 H ζ + H H0,0 + δ1 x) ζ ζ + 5 ) 18 ζ C A C F pqqx) ζ3 6 0 ζ H 3,0 3H ζ 14H, 1,0 + 3H,0 + 5H,0,0 4H, H H0ζ H0ζ3 H0,0 4H0,0ζ 4 1 H0,0,0 + 5H0,0,0,0 + 3 H3 4H1ζ3 16H1,, H1,0 H1,0ζ + 31 H1,0,0 + 11H1,0,0,0 + 8H1,1,0,0 8H1,3 + H H Hζ H,0 + 5H,0,0 + H3,0 + pqq x) 4 ζ 67 9 ζ + 31 ζ3 + 5H 3,0 4 3H ζ 4H, 1, H,0 + 1H,0,0 + 30H, 31 3 H 1ζ 4H 1ζ H0 4H 1,,0 + 56H 1, 1ζ 36H 1, 1,0,0 56H 1, 1, 134 H 1,0 4H 1,0ζ H3, H 1, H0,0ζ H0,0,0 5H0,0,0,0 7Hζ 31 6 H3 10H4 + 1 x) H 1,0,0 + 17H 1,0,0,0 + H 1, + H 1,, H0ζ + 9 H0ζ H0, H0,0,0, ζ3 H0ζ3 H 3,0 + H ζ + H, 1,0 3H,0, H0,0,0 H1 7H1ζ H1,0, H1, x) 43 ζ 3ζ + 5 H,0 31H 1ζ 14H 1, 1, H 1,0 + 4H 1, + 3H 1,0, H0ζ + 5H0,0ζ + H0 H0,0 H + Hζ 15H H,0,0 3H4 5ζ 1 ζ + 50ζ3 H 3,0 7H,0 H0ζ H0ζ 9 H0 H0,0ζ H0,0 H0,0,0 4H0,0,0, H + 6H3 + δ1 x) 05 + ζζ ζ ζ ζ ) 45 ζ5 + 16C A C F pqqx) ζ ζ ζ H0 + H 3,0 + 4H, 1, H,0 H,0,0 + H, H0ζ + 4H0ζ3 + H0,0 H,0,0 1 7 H0,0,0,0 + 9H1ζ3 + 6H1,,0 H1,0ζ H1,0,0 3H1,0,0,0 4H1,1,0,0 + 4H1, H0,0, H3 + H4 + pqq x) 18 ζ ζ 11 4 ζ3 H 3,0 + 8H ζ + 11 H,0 4H,0,0 6 3H 1,0,0, H 1ζ + 1H 1ζ3 16H 1, 1ζ + 8H 1, 1,0,0 + 16H 1, 1, H 1,0 8H, + 11H 1,0ζ H 1,0, H 1, 8H 1,3 3 4 H H0ζ 4H0ζ H0,0 3H0,0ζ 31 1 H0,0,0 + H0,0,0,0 + Hζ H3 + H4 + 1 x) H0,0,0,0 + 11H1 H, 1,0 + 1 H 3,0 1 H ζ H,0, H0 + H0ζ H0,0 5 H0,0,0 + H1ζ H1,0, x) 8H 1ζ + 4H 1, 1, H 1,0 5H 1,0,0 6H 1, 13 3 ζ ζ H3 + 8H4 + 1 x) ζ + H 3,0 H ζ 4H, 1,0 10H,0 H0,0 + H,0,0 + H0ζ3 + H0,0ζ H0,0,0,0 + 8H1ζ H x) 3H 1ζ 18ζ 3ζ H 1,0 16H 1,0,0 3H 1, H0 9H0ζ + 5H0,0,0 + 4H H ) ζ ζ3 + 3H0 + 14H0ζ + H0,0,0 16H3 + 16C F n f C F C ) A pqq x) ζ3 0 9 ζ 4 3 H,0 8 3 H 1ζ 40 9 H 1,0 4 3 H 1,0, H 1, + 3 H0ζ H0, H0,0, H3 + 1 x) H x) H0,0 8 3 H 1, H0 4 ) 3 H + 16C F C F C ) A pqq x) 9ζ3 7ζ + 1H 3,0 64H ζ 16H, 1,0 6H,0 + 5H,0,0 + 56H, 1H 1ζ 7H 1ζ3 16H 1,,0 + 96H 1, 1ζ 80H 1, 1,0,0 96H 1, 1, 80H 1,0ζ 6H 1,0,0 + 44H 1,0,0,0 + 1H 1, + 8H 1,,0 + 64H 1,3 + 3H0 + 3H0ζ + 6H0ζ3 + 8H0,0ζ + 9H0,0,0 1H0,0,0,0 1Hζ 6H3 4H3,0 4H4 1 x) H 3,0 + 8H,0,0 + 61H0 + 6H0ζ3 + H0,0ζ 6H0,0,0,0 + 1H1ζ + 60H1 + 8H1, x) 4ζ + 57ζ3 + 10H,0 48H 1ζ 4H 1,0 + 40H 1,0,0 + 48H 1, + 59H0ζ H0,0 35H0,0,0 H 4H,0 44H3 + 8ζ 4ζ3 4H,0 + 4H0 ) 38H0ζ + 14H0,0 16H + 6H0,0,0 + 4H ) Finally the Mellin inversion of γ )s ns N) in Eq. 3.9) leads to the following result for the leading third-order) difference P ns )s x) of the valence and minus splitting functions: P ns )s x) = 16n f d abc dabc nc 1 1 x) ζ 5 4 ζ H 3,0 + H ζ H,0, H3 + H, 1,0 + 3 H0,0ζ 1 H1ζ 3 91 H1,0, H1 + 1 H 1, 1,0 1 + x) 3 H 1ζ H H 1,0 + 1 H 1,0,0 + H 1, 3 H, H0ζ + H0, H Hζ 1 H,0,0 + 3 H x + x) 3H 1ζ + H 1, 1,0 H 1,0,0 H 1, + H1ζ x 5ζ3 H3 + H,0 + 4H0ζ H0,0,0 + H1ζ H0 + ζ3 9 ζ + ζ H0ζ3 H0ζ H0,0ζ H0,0 1 4 H0,0,0 + 1 ) H0,0,0,0 + H,0 H ) Of particular interest is the end-point behaviour of the harmonic polylogarithms at x 0 or x 1, where logarithmic singularities occur. In the limit x 0, the factors lnx are related to trailing zeroes in the index field, whereas in the limit x 1 factors of ln1 x) emerge from 15

13 Easy-to-use parametrization Combine exact limits for x 0 and x 1 with smooth fit for intermediate x fit quality better than one per mille Notation : end-point logarithms L 0 = lnx), L 1 = ln1 x), +-distribution D 0 = 1/1 x) + P )+ ns x) = D δ1 x) L x x 5.1 x 3 + L 0 L L L L /7 L /81 L0 4 + n f D δ1 x) 510/81 L x x + n f x xl L 0 L L 0 608/81 L0 64/7 L0 3 D 0 51/16 + 3ζ 3 5ζ ) δ1 x) + x1 x) 1 L 0 3/ L 0 + 5) + 1 ) + 1 x)6 + 11/ L 0 + 3/4 L0) 64/81 )

14 P ) ns x) = D δ1 x) L x x 433. x 3 + L 0 L L L L /9 L /81 L n f D δ1 x) 510/81 L x x + n f x xl L 0 L L 0 316/81 L0 56/81 L0 3 D 0 51/16 + 3ζ 3 5ζ ) δ1 x) + x1 x) 1 L 0 3/ L 0 + 5) + 1 ) + 1 x)6 + 11/ L 0 + 3/4 L0) 64/81 ) P )s ns x) = nf L x x) x 43.1 x x 3 1 x ) + L 0 L L L L0 + 40/7 L0 4 L0 3

15 The large x-limit : x 1 Large x-limit identical for P )±,v P )±,v x 1 x) = A 3 1 x) + + B 3 δ1 x) + C 3 ln1 x) + O1) Result for A 3 is new important for threshold resummation in soft/collinear limit one-loop A 1 = 4C F ) 67 two-loop A = 8C F C A 18 ζ 5 9 C Fn f 45 three-loop A 3 = 16C F CA ζ ζ ) 5 ζ + 16C Fn f 55 ) 4 + ζ C F C A n f ζ 7 3 ζ 3 ) + 16C F n f 1 7 ) Surprising relation for subleading logarithms C 1 = 0, C = 4C F A 1, C 3 = 8C F A

16 The small x-limit : x 0 Terms up to ln k x occur at order α k+1 s. P )i x 0 x) = Di 0 ln 4 x + D i 1 ln 3 x + D i ln x + D i 3 lnx + O1) P )+ ns : D + 0 = 3 C 3 F = P ) ns : D 0 = C F C A + 4C FC A 10 3 C 3 F = D + 1 = n f D 1 = n f D + = n f n f D = n f n f D + 3 = n f n f D 3 = n f n f Coefficient of ln 4 x in agreement with prediction Kirschner, Lipatov 83 ; Blümlein, Vogt 96 New colour structure in P )s ns D s 0 = n f, D s 1 =.9696 n f, D s = n f, D s 3 = n f

17 x) P ) x) +,0 exact N = x) P ) x) +,0 Lx NLx N Lx N 3 Lx exact x Left : comparison of exact result with estimates from fixed moments for n f -independent term van Neerven, Vogt 00 Right : comparison of exact result with the small-x approximations for n f -independent term x

18 x) P ) x),0 exact N = x) P ) x),0 Lx NLx N Lx N 3 Lx exact x Left : comparison of exact result with estimates from fixed moments for n f -independent term van Neerven, Vogt 00 Right : comparison of exact result with the small-x approximations for n f -independent term x

19 100 P ) x) S, P ) x) S, exact N = exact Lx NLx N Lx N 3 Lx x Left : comparison of exact result with estimates from fixed moments van Neerven, Vogt 00 Right : comparison of exact result with the small-x approximations x

20 Numerical implications for non-singlet distributions Perturbative expansion of logarithmic scale derivative d lnq i ns/d lnµ f d d lnµ f q i nsx,µ f ) = n=0 αs µ f ) 4π ) n+1 P n)i ns x) q i nsx,µ f ) NNLO corrections are generally rather small, even for large values of α s Parametrization of non-singlet quark distribution xq i ns = x a 1 x) b size of relative corrections weakly dependent on power b increase at small x with increasing power a

21 d ln q + / d ln µ f NS µ r = µ f LO NLO NNLO NLO/LO NNLO/NLO α S = 0., N f = x x Perturbative expansion of logarithmic scale derivative d lnq + ns/d lnµ f Non-singlet distribution xq + ns = x x) 3 at scale µ r = µ f with α s µ 0 ) = 0.

22 0.1 0 d ln q / d ln µ f NS 1.3 µ r = µ f NLO/LO q v NNLO/NLO -0.3 LO -0.4 NLO NNLO 1 q α S = 0., N f = x Perturbative expansion of logarithmic scale derivative d lnq,v ns /d lnµ f Non-singlet distribution xq,v ns = x x) 3 at scale µ r = µ f with α s µ 0 ) = 0. x

23 d ln q + / d ln µ f NS -0.1 NLO NNLO x = x = x = x = x = x = µ r / µ f µ r / µ f NLO and NNLO dependence of q + ns d lnq + ns/d lnµ f on renormalization scale µ r

24 The Summary Complete NNLO non-singlet evolution for deep-inelastic structure functions F,F 3 milestone in calculation of QCD perturbative corrections NNLO offers high precision stable evolution corrections generally rather small Outlook Complete NNLO singlet evolution coming soon, stay tuned