Towards three-loop QCD corrections to the time-like splitting functions

Transcription

1 Towards three-loop QCD correctons to the tme-lke splttng functons Insttute of Nuclear Physcs (Cracow) Sven-Olaf Moch Unversty of Hamburg Radcor-Loopfest Symposum Los Angeles (USA) 15 Jun 2015

2 Three-loop tme-lke g q splttng functon three-loop aka NNLO aka O(α 3 s ) tme-lke hard scale q 2 > 0 off-dagonal P T qg (x, α s(q 2 )) term

3 Splttng functons n perturbaton theory Splttng Functons unversal quanttes n QCD (process ndependent) govern collnear evoluton n hard scatterng processes wth hadrons n ntal state (a space-lke hard-scale Q 2 = q 2 > 0) d d ln q 2 f h a (x, q 2 ) = 1 x dz ( ) ( z PS ab z, α s(q 2 ) fb h x z, q2) n fnal state (a tme-lke hard-scale Q 2 = q 2 > 0) d d ln q 2 Dh a (x, q 2 ) = 1 x dz ( ) ( z PT ba z, α s(q 2 ) Db h x z, q2) can be computed n perturbaton theory Pba T ( x, αs (q 2 ) ) = α ( s αs ) 2 ( 4π P(0)T (1)T αs ) 3 (2)T ba (x) + P ba (x) + P ba (x) π 4π

4 Splttng Functons at NLO: avalable results axal gauge Prncpal Value Space-lke Curc, Furmansk, Petronzo 80; Ells, Vogelsang 98 Mandelstam-Lebbrandt Bassetto, Henrch, Kunszt, Vogelsang 98 New Prncpal Value OG, Jadach, Skrzypek, Kusna 14 Feynman gauge Floratos, Kounnas, Lacaze 81 Melln space Moch, Vermaseren 99 Tme-lke axal gauge Prncpal Value Furmansk, Petronzo 80 Feynman gauge Floratos, Kounnas, Lacaze 81 Melln space Mtov, Moch 06 analytc contnuaton (Pab S PT ba ) Stratmann, Vogelsang 96; Blumlen, Ravndran, van Neerven 00; Moch, Vogt 07

5 Analytc Contnuaton Technque Analytc contnuaton n energy q 2 +q 2 Drell-Yan-Levy relaton Drell, Levy, Yan 70 Grbov-Lpatov relaton Grbov, Lpatov 72 Relaton between space- and tme-lke fragmentaton functons ( ) 1 F T (x) = x F S x Examples pqq T (0) (x) = pqq S(0) (x) = 1 + x 2 1 x pqg T (0) (x) = pgq S(0) 1 + (1 x)2 (x) = x pgq T (0) (x) = pqg S(0) (x) = x + (1 x) 2 pgg T (0) (x) = pgg S(0) (x) = (1 x + x 2 ) 2 x(1 x) Beware: nave verson of these relatons s not vald beyond LO

6 Splttng Functons at NNLO: avalable results Space-lke Tme-lke Melln space Moch, Vermaseren, Vogt 04 analytc contnuaton (P S ab PT ba ) NNLO non-snglet Mtov, Moch, Vogt 06 NNLO snglet (P T qq, P T gg ) Moch, Vogt 07 NNLO snglet (P T gq, P T qg ) Almasy, Moch, Vogt 11 In summary, these consderatons are stll not suffcent to defntely fx the rght-hand-sde of P T (2) qg. As an estmate of the remanng uncertanty we suggest to use the offset:... Almasy, Moch, Vogt 11

7 Introducton: Summary Three-loop tme-lke P T (2) qg splttng functon s stll mssng and should be calculated drectly. Outlne: 1. The Method mass factorzaton fragmentaton functons for e + e γ, φ partons 2. Fnal-State Integraton (analytc) IBP reducton dfferental equatons for masters choce of the approprate bass for masters boundary condtons

8 Cross-secton at LO q k 0 k 0 q k 0 q k 0 k 0 P T qq, P T gq P T qg, P T gg Unpolarzed dfferental cross-secton 1 σ tot d 2 σ H dx d cos θ = 3 8 (1+cos2 θ) F T (x)+ 3 4 sn2 θ F L (x)+ 3 4 cos θ F A(x) Mass factorzaton relatons Vermaseren, Vogt, Moch 05 F (1) T,g (x) = 1 ɛ P gq (0) (x) + c (1) T,g (x) + ɛ a(1) T,g (x) + ɛ2 b (1) T,g (x) F (1) L,g Knematc varables (x) = c(1) L,g (x) + ɛ a(1) L,g (x) + ɛ2 b (1) L,g (x) x = 2q k0 q 2 q 2 = s > 0 0 < x 1

9 Fragmentaton Functons Fragmentaton functons F T (x, ɛ) = F L (x, ɛ) = kµ 0 kν 0 q k 0 W µν F A (x, ɛ) = ( ) 2 q k0 2 m q 2 W µ µ + kµ 0 kν 0 W µν q k 0 3 q α k β 0 (m 2)(m 3) q 2 ɛ µναβ W µν Hadronc tensor W µν (x, ɛ) = x m 3 dps(n) M µ (n) M ν (n) 4π m = 4 2ɛ dps(n) n-partcle fnal-state phase-space M µ (n) ampltudes for the process γ, φ (q µ ) n partons

10 NLO contrbutons to P T gq Transverse Fragmentaton Functon F (2) T = Mass Factorzaton Vermaseren, Vogt, Moch 05 F (2) T = + 1 { 1 2 P(0) g P (0) q ɛ 2 { c (2) g P (0) g a (1) β0p(0) gq } + ɛ } 1 { 1 ɛ { a (2) g P (0) g b (1) } 2 P(1) gq + P(0) g c (1) }

11 NNLO contrbutons to P T gq Transverse Fragmentaton Functon F (3) T = Mass Factorzaton Vermaseren, Vogt, Moch 05 F (3) T = 1 { 1 ɛ 3 6 P(0) g P (0) j P (0) jq β0p(0) g P (0) q { ɛ 2 6 P(0) g P (1) q P(1) g P (0) q 1 { 1 ɛ 3 P(2) gq P(1) g c (1) + P (0) g c (2) { + c (3) g P (0) g a (2) 1 2 P(1) g a (1) β2 0 P(0) gq } β1p(0) gq P(0) g P (0) j c (1) j 1 2 P(0) g P (0) P(0) g P (0) j b (1) j j a (1) j ( 1 + β 0 3 P(1) gq + 1 ) } 2 P(0) g c (1) } 1 2 β0p(0) g a (1) } β0p(0) g b (1)

12 NNLO contrbutons to P T gq Pure-vrtual contrbutons contan overall δ(1 x) factor. We do not consder such contrbutons. Can be extracted from Garland, Gehrmann, Glover, Koukoutsaks, Remdd 01 Calculated by Duhr, Gehrman, Jaquer [hep-ph] One-loop helcty ampltudes by Bern, Dxon, Kosower 97 Fnal-state ntegraton s of NLO complexty smple. Contrbuton s known from analytcal contnuaton by Almasy, Moch, Vogt 11 Unknown!

13 Fnal-state ntegraton The man challenge of the calculaton s n-partcle fnal-state ntegraton: dps(n) = n 1 =0 ( d m k δ + (k 2 ) δ x 2q k ) 0 q 2 n 1 δ(q j=0 k j ) NLO (n = 4) NNLO (n = 5) We attack such ntegrals wth IBP denttes and dfferental equatons Do not mss Methods Sessons on Thursday and Frday afternoons

14 Integraton-by-Parts Identtes QGRAF FORM trace of gamma matrces ndex contracton color traces partal fractonng Mathematca analyze symmetres splt by topologes 8 x 8 dagrams 495 ntegrals 210 x 8 dagrams ntegrals 48 x 48 dagrams ntegrals LteRed fnd IBP reducton rules fnd masters 10 h 8 masters 100 h 83 masters >1000 h 80 masters We consder to swtch to Reduze2 for NNLO calculatons snce LteRed can not solve some sectors.

15 Dfferental Equatons for Masters general representaton f x = n a j (x, ɛ) f j (x, ɛ) j=1 can be easly solved (matrx multplcaton and HPL ntegraton) as ɛ-seres when a j (x, ɛ) = 0 for ɛ 0 n HPLs when alphabet s {x, 1 x, 1 + x} partcular example: a j (x, ɛ) at NLO for γ q qgg (2ɛ 1)(2x 1) (1 x)x ɛ 2 (1 x)x 3ɛ 1 x (2ɛ 1) ɛ 2 0 6ɛ 1 2 (1 x)x(x+1) x+1 x+1 ( ) (2ɛ 1)(3ɛ 1) 2(6ɛ 1) 2ɛ x 2 +3x 2 x 2 x(x+1) (1 x)x(x+1) (2ɛ 1) ( ɛx 2 (x+1) ) ( ) 2 x 2 +4x+1 2(2ɛ 1)(x 1) 2(6ɛ 1)(x 1) 4 x 2 +1 ɛ 2 (1 x)x 3 (x+1) 3 ɛx 3 (x+1) 2 x 2 (x+1) 3 x 2 (x+1) 3 (2ɛ+1)(2x+1) x(x+1) (2ɛ 1) 2ɛ ɛ(1 x)x ( ) 1 x ɛ 2 (1 x) 3 x 3 2(2ɛ 1)(x 2) (x+1) ɛ(1 x) 2 x 3 2(6ɛ 1) 4 x 2 +1 (1 x)x 2 (x+1) (1 x) 2 x 2 0 4ɛ (2ɛ+1)(2x 1) (x+1) (1 x) 2 x (1 x)x (2ɛ+1)(2x 1) (1 x)x

16 Master Algorthm make system regular as ɛ 0 Algorthm I (comng soon) make system zero-dagonal Algorthm II make system zero-trangular Algorthm III solve system (see Henn 13) fnd boundary condtons (2ɛ 1)(2x 1) (1 x)x ɛ 2 (1 x)x 3ɛ 1 x (2ɛ 1) ɛ 2 0 6ɛ 1 2 (1 x)x(x+1) x+1 x+1 ( ) (2ɛ 1)(3ɛ 1) 2(6ɛ 1) 2ɛ x 2 +3x 2 x 2 x(x+1) (1 x)x(x+1) (2ɛ 1) ( ɛx 2 (x+1) ) ( ) 2 x 2 +4x+1 2(2ɛ 1)(x 1) 2(6ɛ 1)(x 1) 4 x 2 +1 ɛ 2 (1 x)x 3 (x+1) 3 ɛx 3 (x+1) 2 x 2 (x+1) 3 x 2 (x+1) 3 (2ɛ+1)(2x+1) x(x+1) (2ɛ 1) 2ɛ ɛ(1 x)x ( ) 1 x ɛ 2 (1 x) 3 x 3 2(2ɛ 1)(x 2) (x+1) ɛ(1 x) 2 x 3 2(6ɛ 1) 4 x 2 +1 (1 x)x 2 (x+1) (1 x) 2 x 2 0 4ɛ (2ɛ+1)(2x 1) (x+1) (1 x) 2 x (1 x)x (2ɛ+1)(2x 1) (1 x)x

17 Algorthm I: fnte bass algorthm s output (2ɛ 1)(2x 1) (1 x)x 3ɛ 2 1 3ɛ (1 x)x x (2ɛ 1) 1 6ɛ 2ɛ 0 (x 1)x(x+1) 1+x x ( (2ɛ 1) x 2 ɛ 2(6ɛ 1) (x+1) x 2 2ɛ x 2 ) +3x ( x(x+1) (x 1)x(x+1) 2ɛ x 2 ) +4x+1 2ɛ 2 (x 1) 2ɛ(x 1) 4ɛ 2 ( x 2 ) +1 (1 x)x 3 (x+1) 3 x 3 (x+1) 2 x 2 (x+1) 3 x 2 (x+1) 3 (2ɛ+1)(2x+1) x(x+1) ɛ ɛ (x 1)x x ɛ (x 1) 3 x 3 2ɛ2 (x 2) 4ɛ (x+1) (x 1) 2 x 3 2ɛ 2( x 2 ) +1 (x 1)x 2 (x+1) (x 1) 2 x 2 0 4ɛ (2ɛ+1)(2x 1) (x+1) x(1 x) 2 0 (1 x)x (2ɛ+1)(2x 1) (1 x)x algorthm s output as ɛ 0 2x (x 1)x 2 1 (x 1)x x (1 x)x(x+1) 0 1 x x (x+1) x(x+1) x+1 x(x+1) x 1 0 (1 x)x x 1 (1 x)x new bass g such that g (x, ɛ) = ɛ n f (x, ɛ) where n are ntegers automated soon

18 Algorthm II: zero-dagonal bass algorthm s nput (ɛ 0 lmt) 2x (x 1)x 2 1 (x 1)x x (1 x)x(x+1) 0 1 x x (x+1) x(x+1) x+1 x(x+1) x 1 0 (1 x)x x 1 (1 x)x algorthm s output (ɛ 0 lmt) 2 (x+1) (x 1) x(x+1) 1 x 2 x new bass g such that g (x, ɛ) = b (x, ɛ)f (x, ɛ) where b (x, ɛ) = exp ( ) dx a (x, ɛ) b 1(x, ɛ) = 1 x(1 x) b 2(x, ɛ) = 1 x b 3(x, ɛ) = 1 1+x...

19 Algorthm III: zero-trangular bass algorthm s nput (ɛ 0 lmt) 2 (x+1) (x 1) x(x+1) 1 x 2 x algorthm s output (ɛ 0 lmt) new bass h such that 1 h (x, ɛ) = c j (x, ɛ) g j (x, ɛ) + g (x, ɛ) where c j (x, ɛ) = dx j=1 j+1 k 1 h 3(x, ɛ) = 2 g1(x, ɛ) + g3(x, ɛ)... 1+x c k (x, ɛ) a kj (x, ɛ) + a j (x, ɛ)

20 Boundary condtons smple master ntegral (green functons are known) f (x, ɛ) = C (0) f (0) (x) + ɛ C (1) f (1) (x) +... correspondng nclusve master F (ɛ) = 1 boundary condtons are found as coeffcents A (j) 0 dx f (x, ɛ) = A (0) + ɛ A (1) +... C (j) = 1 0 A (j) dx f (j) (x) are non-trval to calculate at NNLO

21 To be contnued...

22 Summary Done: Fragmentaton Functons generator (QGRAF + FORM + LteRed + Mathematca) nput: process up to NNLO, e.g. γ q qq qg output: fragmentaton functon n terms of masters Algorthm to fnd masters (Mathematca) ntegral bass choce solutons for masters boundary condtons fnder (manual) In progress: IBP rules wth Reduze Automated boundary condtons fnder Inclusve ntegrals at NNLO