Hard lepton-hadron processes in pQCD (I) Inclusive deep-inelastic - PowerPoint PPT Presentation

Generalized double-logarithmic large- x resummation Andreas Vogt (University of Liverpool) mainly with G. Soar, A. Almasy (UoL), S. Moch (DESY), J. Vermaseren (NIKHEF) Hard lepton-hadron processes in higher-order perturbative QCD Large- x/ large- N splitting functions P ik and coefficient functions C a,i ln n (1 − x ) behaviour of DIS, SIA and non-singlet DY physical kernels All-order predictions for C a, ns , fourth-order ln 6 , 5 , 4 (1 − x ) of P ik MV, arXiv: 0902.2342, 0909.2124; SMVV, 0912.0369 p.1

Generalized double-logarithmic large- x resummation Andreas Vogt (University of Liverpool) mainly with G. Soar, A. Almasy (UoL), S. Moch (DESY), J. Vermaseren (NIKHEF) Hard lepton-hadron processes in higher-order perturbative QCD Large- x/ large- N splitting functions P ik and coefficient functions C a,i ln n (1 − x ) behaviour of DIS, SIA and non-singlet DY physical kernels All-order predictions for C a, ns , fourth-order ln 6 , 5 , 4 (1 − x ) of P ik , C L, g Iteration of (next-to) leading-log unfactorized 1 /N structure functions LL resummation of off-diagonal splitting and coefficient functions General D -dimensional structure of large- x DIS and SIA amplitudes Verification and extension to higher logarithmic accuracy for DIS / SIA MV, arXiv: 0902.2342, 0909.2124; SMVV, 0912.0369; A.V., 1005.1606; ASV, 1010.nnnn p.1

Hard lepton-hadron processes in pQCD (I) Inclusive deep-inelastic scattering (DIS), semi-incl. l + l − annihilation (SIA) l Left → right: DIS, q spacelike, Q 2 = − q 2 γ ∗ ( q ) P = ξp , f h i = parton distributions c ai i ( P ) Top → bottom: l + l − , q timelike, Q 2 = q 2 f h p = ξP , fragmentation distributions i h ( p ) Drell-Yan (DY) l + l − production: bottom → top, 2 nd hadron from right ( { . . . } ) Structure functions / normalized cross sections F a : coefficient functions h i i ( µ 2 ) { ⊗ f h ′ F a ( x, Q 2 ) = C a,i { j } ( α s ( µ 2 ) , µ 2 /Q 2 ) ⊗ f h j ( µ 2 ) } ( x ) + O (1 /Q (2) ) Scaling variables: x = Q 2 / (2 p · q ) in DIS etc. µ : renorm. / mass-fact. scale p.2

Hard lepton-hadron processes in pQCD (II) Parton / fragmentation distributions f i : (renorm. group) evolution equations h i d P ( S,T ) d ln µ 2 f i ( ξ, µ 2 ) = ( α s ( µ 2 )) ⊗ f k ( µ 2 ) ( ξ ) ik ⊗ = Mellin convolution. Initial conditions: fits to reference observables Expansion in α s : splitting functions P , coefficient fct’s c a of observables α s P (0) + α 2 s P (1) + α 3 s P (2) + α 4 s P (3) + . . . P = » – α n a c (0) + α s c (1) + α 2 s c (2) + α 3 s c (3) C a = + . . . a a a a s | {z } NLO: first real prediction of size of cross sections NNLO, P (2) , c (2) a : first serious error estimate of pQCD predictions N 3 LO: for high precision ( α s from DIS), slow convergence (Higgs in pp/p ¯ p ) The 2010 frontier: α 4 s /α 3 s for DIS / SIA ( + DY) Baikov, Chetyrkin; MV, . . . p.3

MS splitting functions at large x / large N R 1 0 dx ( x N − 1 {− 1 } ) f ( x ) { + } : M-convolutions → products Mellin trf. f ( N ) = ln n (1 − x ) ( − 1) n +1 ( − 1) n ln n +1 N + . . . , ln n (1 − x ) ln n N + . . . M M = = (1 − x ) + n + 1 N Diagonal splitting functions: no higher-order enhancement at N 0 , N − 1 1 P ( l − 1) qq / gg ( N ) = A ( l ) q / g ln N + B ( l ) q / g + C ( l ) N ln N + . . . , A g = C A /C F A q q / g . . . ; Korchemsky (89); Dokshitzer, Marchesini, Salam (05) p.4

MS splitting functions at large x / large N R 1 0 dx ( x N − 1 {− 1 } ) f ( x ) { + } : M-convolutions → products Mellin trf. f ( N ) = ln n (1 − x ) ( − 1) n +1 ( − 1) n ln n +1 N + . . . , ln n (1 − x ) ln n N + . . . M M = = (1 − x ) + n + 1 N Diagonal splitting functions: no higher-order enhancement at N 0 , N − 1 1 P ( l − 1) qq / gg ( N ) = A ( l ) q / g ln N + B ( l ) q / g + C ( l ) N ln N + . . . , A g = C A /C F A q q / g . . . ; Korchemsky (89); Dokshitzer, Marchesini, Salam (05) Off-diagonal: double-log behaviour, colour structure with C F = C A − C F A C − 1 P ( l ) − 1 P ( l ) N ln 2 l N # C l 1 gq / n = qg f A F F f ) C l − 1 N ln 2 l − 1 N ( # C 1 + F + # C F + # n + . . . A A F Double logs ln n N , l +1 ≤ n ≤ 2 l vanish for C F = C A ( → SUSY case) Aim: obtain, at least, these (next-to) leading terms to all orders l in α s p.4

MS coefficient functions at large x / large N dσ q¯ 1 q ‘Diagonal’ [ O (1) ] coeff. fct’s for F 2 , 3 ,φ in DIS, F T ,A,φ in SIA, F DY = σ 0 dQ 2 2 , q /φ, g /... = # ln 2 l N + . . . + N − 1 (# ln 2 l − 1 N + . . . ) + . . . C ( l ) N 0 parts: threshold exponentiation Sterman (87); Catani, Trentadue (89); . . . Exponents known to next-to-next-to-next-to-leading log (N 3 LL) accuracy - mod. A (4) ⇒ highest seven (DIS), six (SIA, DY, Higgs prod.) coefficients known to all orders DIS: MVV (05), DY / Higgs prod.: MV (05); Laenen, Magnea (05); Idilbi, Ji, Ma, Yuan (05) ( + more papers, esp. using SCET, from 2006), SIA: Blümlein, Ravindran (06); MV (09) p.5

MS coefficient functions at large x / large N dσ q¯ 1 q ‘Diagonal’ [ O (1) ] coeff. fct’s for F 2 , 3 ,φ in DIS, F T ,A,φ in SIA, F DY = σ 0 dQ 2 2 , q /φ, g /... = # ln 2 l N + . . . + N − 1 (# ln 2 l − 1 N + . . . ) + . . . C ( l ) N 0 parts: threshold exponentiation Sterman (87); Catani, Trentadue (89); . . . Exponents known to next-to-next-to-next-to-leading log (N 3 LL) accuracy - mod. A (4) ⇒ highest seven (DIS), six (SIA, DY, Higgs prod.) coefficients known to all orders DIS: MVV (05), DY / Higgs prod.: MV (05); Laenen, Magnea (05); Idilbi, Ji, Ma, Yuan (05) ( + more papers, esp. using SCET, from 2006), SIA: Blümlein, Ravindran (06); MV (09) ‘Off-diagonal’ [ O ( α s ) ] quantities: leading N − 1 double logarithms C ( l ) φ, q / 2 , g /... = N − 1 (# ln 2 l − 1 N + # ln 2 l − 2 N + . . . ) + . . . Longitudinal DIS / SIA structure functions [ recall: l = order in α s – 1] C ( l ) C ( l ) L , q = N − 1 (# ln 2 l N + . . . ) + . . . , L , g = N − 2 (# ln 2 l N + . . . ) + . . . Aim: predict highest N − 1 [ N − 2 for C L , g ] double logarithms to all orders p.5

Non-singlet and singlet physical kernels Eliminate parton densities from scaling violations of observables ( µ = Q ) X dF d C a l +1 = KF ≡ K l F = d ln Q 2 q + CP q s d ln Q 2 l =0 “ ” β ( a s ) dC C − 1 + [ C, P ] C − 1 + P = F da s p.6

Non-singlet and singlet physical kernels Eliminate parton densities from scaling violations of observables ( µ = Q ) X dF d C a l +1 = KF ≡ K l F = d ln Q 2 q + CP q s d ln Q 2 l =0 “ ” β ( a s ) dC C − 1 + [ C, P ] C − 1 + P = F da s dσ 1 q ¯ q Non-singlet: F = F 2 , 3 ,φ and F L in DIS, F T ,A,φ and F L in SIA, F DY = dQ 2 σ 0 Singlet: a) F = ( F 2 , F φ ) with large- m top Higgs-exchange DIS Furmanski, Petronzio (81); . . . Coefficient functions for F φ to order α 2 s /α 3 s Daleo, Gehrmann-De Ridder, Gehrmann, Luisoni; SMVV (09) F L = F L /a s c (0) b) F = ( F 2 , b F L ) with b Catani (96); Blümlein et al. (00) L,q p.6

Non-singlet and singlet physical kernels Eliminate parton densities from scaling violations of observables ( µ = Q ) X dF d C a l +1 = KF ≡ K l F = d ln Q 2 q + CP q s d ln Q 2 l =0 “ ” β ( a s ) dC C − 1 + [ C, P ] C − 1 + P = F da s dσ 1 q ¯ q Non-singlet: F = F 2 , 3 ,φ and F L in DIS, F T ,A,φ and F L in SIA, F DY = dQ 2 σ 0 Singlet: a) F = ( F 2 , F φ ) with large- m top Higgs-exchange DIS Furmanski, Petronzio (81); . . . Coefficient functions for F φ to order α 2 s /α 3 s Daleo, Gehrmann-De Ridder, Gehrmann, Luisoni; SMVV (09) F L = F L /a s c (0) b) F = ( F 2 , b F L ) with b Catani (96); Blümlein et al. (00) L,q NNLO / N 3 LO: all physical kernels K above single-log enhanced at large N Conjecture: double-log contributions also vanish at all higher orders in α s p.6

Non-singlet evolution kernels and predictions DIS / SIA a � = L leading-logarithmic kernels, with p qq ( x ) = 2 / (1 − x ) + − 1 − x K a, 0 ( x ) = 2 C F p qq ( x ) ˆ ˜ F β 0 ∓ 8 C 2 K a, 1 ( x ) = ln (1 − x ) p qq ( x ) − 2 C F ln x ˆ ˜ F β 0 ln x + O (ln 2 x ) K a, 2 ( x ) = ln 2 (1 − x ) p qq ( x ) F β 2 0 ± 12 C 2 2 C ˆ ˜ 0 ln x + O (ln 2 x ) K a, 3 ( x ) = ln 3 (1 − x ) p qq ( x ) F β 3 0 ∓ 44 / 3 C 2 F β 2 − 2 C ˆ ˜ 0 ln x + O (ln 2 x ) K a, 4 ( x ) = ln 4 (1 − x ) p qq ( x ) F β 4 0 ± ξ K 4 C 2 F β 3 2 C First term: leading large n f , all orders via C 2 of Mankiewicz, Maul, Stein (97) p.7

Non-singlet evolution kernels and predictions DIS / SIA a � = L leading-logarithmic kernels, with p qq ( x ) = 2 / (1 − x ) + − 1 − x K a, 0 ( x ) = 2 C F p qq ( x ) ˆ ˜ F β 0 ∓ 8 C 2 K a, 1 ( x ) = ln (1 − x ) p qq ( x ) − 2 C F ln x ˆ ˜ F β 0 ln x + O (ln 2 x ) K a, 2 ( x ) = ln 2 (1 − x ) p qq ( x ) F β 2 0 ± 12 C 2 2 C ˆ ˜ 0 ln x + O (ln 2 x ) K a, 3 ( x ) = ln 3 (1 − x ) p qq ( x ) F β 3 0 ∓ 44 / 3 C 2 F β 2 − 2 C ˆ ˜ 0 ln x + O (ln 2 x ) K a, 4 ( x ) = ln 4 (1 − x ) p qq ( x ) F β 4 0 ± ξ K 4 C 2 F β 3 2 C First term: leading large n f , all orders via C 2 of Mankiewicz, Maul, Stein (97) Conjecture ⇒ coefficients of highest three logs from fourth order in α s , ln 7 , 6 , 5 (1 − x ) at order α 4 for F 1 , 2 , 3 in DIS and F T , I , A in SIA etc s Leading terms: K 1 = K 2 , K T = K I [ total (‘integrated’) fragmentation fct.] ⇒ also three logarithms for space- and timelike F L : ln 6 , 5 , 4 (1 − x ) at α 4 s etc Alternative derivation: physical kernels for F L , agreement non-trivial check p.7