Conventions and references Double-log enhancement: two additional - PowerPoint PPT Presentation

Resummation of large- x and small- x double logarithms in deep-inelastic scattering & semi-inclusive annihilation Andreas Vogt (University of Liverpool) partly with G. Soar, A. Almasy (UoL), S. Moch (DESY), J. Vermaseren (NIKHEF) Splitting and coefficient functions and their endpoint behaviour 4 th -order / all-order large- x logs from physical evolution kernels Large- x & small- x via unfactorized D -dim. structure functions Galileo Galilei Institute, Florence, 08-09-11 p.1

Conventions and references Double-log enhancement: two additional logs L per additional order in α s ˛ ∼ L ℓ 0 ( # L 2 n + # L 2 n − 1 + # L 2 n − 2 + . . . ) + . . . ˛ Q α n s LL NLL NNLL LL, NLL, . . . : leading logarithms, next-to-leading logarithms, . . . Counting of a resummation, cf. small- x , not of a (stronger) exponentiation, cf. soft gluons: NNLL resummation ⇔ (re-expanded) NLL exponentiation p.2

Conventions and references Double-log enhancement: two additional logs L per additional order in α s ˛ ∼ L ℓ 0 ( # L 2 n + # L 2 n − 1 + # L 2 n − 2 + . . . ) + . . . ˛ Q α n s LL NLL NNLL LL, NLL, . . . : leading logarithms, next-to-leading logarithms, . . . Counting of a resummation, cf. small- x , not of a (stronger) exponentiation, cf. soft gluons: NNLL resummation ⇔ (re-expanded) NLL exponentiation Non-singlet NNLL (NLL for DY) resummation from physical kernels MV, arXiv:0902.2342 (JHEP), arXiv:0909.2124 (JHEP) Singlet NNLL for fourth-order splitting functions and F L in DIS SMVV, arXiv:0912.0369 (NPB), arXiv:1008.0952 (proc. LL 2010) Large- x resummation of splitting & coefficient functions in DIS and SIA ∗ A.V., arXiv:1005.1606 (PLB); ASV, arXiv:1012.3352 (JHEP); ∗ to appear Small- x resummation of splitting & coefficient functions in SIA and DIS ∗ A.V., arXiv:1108.2993 (JHEP); ∗ to appear p.2

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 ( { . . . } ) p.3

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 ( { . . . } ) DIS and SIA structure functions, DY cross section 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.3

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 incalculable in perturbative QCD. ⇒ predictions: fit-analyses of reference processes, universality of f i ( ξ, µ 2 ) p.4

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 incalculable in perturbative QCD. ⇒ predictions: fit-analyses of reference processes, universality of f i ( ξ, µ 2 ) 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 p.4

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 incalculable in perturbative QCD. ⇒ predictions: fit-analyses of reference processes, universality of f i ( ξ, µ 2 ) 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 New: P (2) T now (almost) completely known AMV, arXiv:1107.2263 (NPB) ik 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) p.5

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 A F 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) 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 = dQ 2 σ 0 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, SIA), six (DY, Higgs prod.) coefficients known to all orders DIS: MVV (05), DY / Higgs prod.: MV (05); Laenen, Magnea (05); Idilbi, Ji, Ma, Yuan (05) ( + SCET papers, from 06), SIA: Blümlein, Ravindran (06); MV, arXiv:0908.2746 (PLB) p.6

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 = dQ 2 σ 0 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, SIA), six (DY, Higgs prod.) coefficients known to all orders DIS: MVV (05), DY / Higgs prod.: MV (05); Laenen, Magnea (05); Idilbi, Ji, Ma, Yuan (05) ( + SCET papers, from 06), SIA: Blümlein, Ravindran (06); MV, arXiv:0908.2746 (PLB) ‘Off-diagonal’ [ O ( α s ) ] quantities: leading N − 1 double logarithms C ( ℓ ) φ, q / 2 , g /... = N − 1 (# ln 2 ℓ − 1 N + . . . ) + . . . Longitudinal DIS / SIA structure functions [ convention: 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 + . . . ) + . . . p.6

Flavour singlet – non-singlet decomposition δ ik P v qq + P s P q i q k = P ¯ = Quark-quark splitting functions: q i ¯ q k qq δ ik P v q + P s P q i ¯ q k = P ¯ = q i q k q¯ q¯ q P v P s qq , P s q : α 2 P v q : α 2 P s q � = P s qq : α 3 qq = O ( α s ) q¯ s q¯ s q¯ s Three types of difference (non-singlet) combinations: P ± ns = P v qq ± P v q , P v q¯ ns Evolution of gluon and flavour-singlet quark distributions g and q s „ « „ « „ « q s = P n d q s P qq P qg q s f ⊗ r =1 ( q r + ¯ q r ) , = d ln µ 2 g P gq P gg g P qq = P + f ( P s qq + P s qq ) ≡ P + ns + n ns + P ps with (ps = ‘pure singlet’) ¯ Quark coefficient fct’s: analogous decomposition C a, q { ¯ q } = C a, ns + C a, ps p.7

Second- and third-order N -space C 2 , ns in DIS 8 20 c 2,2 (N) c 2,3 (N) 6 15 all N 0 all N 0 + all N − 1 + all N − 1 4 10 exact exact 2 5 0 0 n f = 4 ( ∗ 1/160) n f = 4 ( ∗ 1/2000) -2 -5 0 5 10 15 20 0 5 10 15 20 N N N − 1 terms relevant over full range shown, O ( N − 2 ) sizeable only at N < 5 Sum of N − 1 ln k N looks almost constant: half of maximum only at N ≃ 150 p.8

Second-order C T in SIA and C DY in N -space 100 20 c T,2 (N) c DY,2 (N) 80 all N 0 all N 0 15 + all N − 1 + all N − 1 60 exact exact 10 40 5 20 0 n f = 5 ( ∗ 1/160) n f = 5 ( ∗ 1/160) 0 0 5 10 15 20 0 5 10 15 20 N N DIS → SIA → DY : increase of the N 0 terms, N − 1 corrections less important p.9