Resummed small-x and first moment evolution of fragmentation functions in pqcd

Transcription

1 Resummed small- and first moment evolution of fragmentation functions in pqcd Steve Kom University of Liverpool Presented at HP2: High Precision for Hard Processes, Munich 7 Sept 2012

2 Collaborators Talk based on arxiv: (to appear in JHEP) See also Vogt 11 collaborators: Andreas Vogt (Liverpool), Karen Yeats (Simon Fraser)

3 Outline of Topics 1 Introduction 2 D-dimensional structure 3 Results 4 Beyond NNLL: DMS relation 5 Summary and outlook

4 Semi-inclusive e + e annihilation (SIA) Fragmentation functions F h a(,q 2 ) in e + e γ/z/φ(q) h(p)+x ( = 2pq/Q 2 ): 1 σ 0 d 2 σ d dcosθ = 3 8 (1+cos2 θ)f h T sin2 θf h L cosθfh A Factorisation formula (O(1/Q) terms neglected) F h a(,q 2 ) = j=q, q,g 1 dz ( ) ( z c a,j z,α S (Q 2 ) Dj h z,q2) Coefficient functions c a,j known at αs 2 Rijken,van Neerven 96, Mitov,Moch 06 Fragmentation distributions Dj h evolve via time-like splitting functions P T d d lnq 2Dh i (,Q 2 ) = j=q, q,g 1 dz ( z PT ji (z,α S )Dj h z,q2) P T known at α 2 S (NLO) Curci,Furmanski,Petronzio 80, Floratos,Kounnas,Lacaze 81,... NNLO computed via analytic continuation Mitov,Moch,Vogt 06, Moch,Vogt 07, Almasy,Moch,Vogt 11

5 SIA in the small- limit In the small- limit need to resum logarithms of the form: P (n)t gi α n+1 S ln 2n +..., P (n)t qi α n+1 S ln 2n , c (n) T,g αn S ln 2n , c (n) T,q αn S ln 2n , c (n) L,g αn S ln 2n , c (n) L,q αn S ln 2n P T : LL Mueller 81, Bassetto,Ciafaloni,Marchesini,Mueller 82, NLL (MG scheme) Mueller 83 C T(φ),g : LL Mueller 83, Albino,Bolzoni,Kniehl,Kotikov 11 Mellin transform: M [ 1 lnk ] (N) 1 0 dn 1 1 lnk = ( 1)k k! N k+1, N = N 1 Goal: Finite Mellin N = 1 limit (required for multiplicity calculations) at NLO (in MS). Need NLL for c T,i and NNLL for P T ij, c L,i. Resum by analysing D-dimensional structure of unfactorised fragmentation functions. Highest three logs now available, including Higgs echange (φ) in the heavy top limit.

6 pff before factorisation Unfactorised pff in D = 4 2ǫ dimensions (a s = α S /4π): F a,k = C a,i Z T ik, P T = dzt dlnq 2(ZT ) 1, da s dlnq 2 = ǫas +β D=4 Z T epressed in terms of P T : Z T a n s ǫ n : PT 0,β 0; Z T a n s ǫ n+1 : PT 0,β 0,P T 1,β 1;... Z T a n s ǫ 1 : PT n 1 C contains only positive powers of ǫ: C a,i = (1)+ n=1 l=0 a n sǫ l c (n,l) a,i At a n s: ǫ n,..., ǫ 2 : lower order terms. ǫ 1 : n-loop splitting fnc ǫ 0 : n-loop coeff. fnc ǫ 0<k<l : required for order a n+l s NNLO results fi the highest three powers of 1/ǫ to all orders in a s.

7 D-dim. structure of unfactorised observables KLN means all a n sǫ m terms are zero for m > n (zero also for m = n for a = L and off-diagonal terms). Decompose the D-dim. pff with according to # eternal legs Vogt 11. E.g. for a = T,φ: F (n) a,g = 1 ǫ 2n 1 n k=1 1 2kǫ{ } A a,g (k,n) +ǫb a,g (k,n) +ǫ 2 C a,g (k,n) + kǫ = 1 kǫln + (kǫ)2 2 ln 2 + A (k,n) : LL, B (k,n) : NLL, C (k,n) : NNLL etc. (Motivated by phase space structure for SIA in small-: 2 γ/z 2(+k etrapartons) : kǫ 1 other variables. 0 N 2 LO: Matsuura,van Neerven 88, Rijken,vN 95 ) KLN cancellation and fied order results provide constraints to fi A, B...

8 D-dim. structure of unfactorised observables (n) LL: F a,g includes terms of the form 1 ln n+δ 1 at all orders ǫ n+δ with δ = 0,1,2,, and is decomposed into n contributions of the form ǫ 2n+1 1 2kǫ A a,g (k,n) = ǫ 2n+1 1[ ] 1 2kǫln (2kǫ)2 ln A a,g (k,n), n 1 (KLN) + 3 (NNLO) constraints The n A a,g (k,n) overconstrained by n + 2 linear eqns. k = 1,2,...,n coefficients are n+1(n) linear eqns for NLL(NNLL) B (l,n) a,g (C (l,n) a,g ). N 2 LO N 2 LL resummation. Quark case similar: ǫ 2n+2 prefactor, one fewer term in sum: F (n) a,q = 1 n 2 1 2(n l)ǫ{ } B (l,n) ǫ 2n 2 a,q +ǫc a,q (l,n) +ǫ 2 D a,q (l,n) + l=0 Also here highest three logs at all orders fied by NNLO results. Obtain NNLL timelike splitting functions and coefficient functions.

9 pff before factorisation Longitudinal case: F (n) L,g = F (n) L,q = 1 ǫ 2n 2 1 ǫ 2n 3 n 1 l=0 n 2 l=0 1 2(n l)ǫ( A (l,n) L,g + ǫb (l,n) L,g +... ), (1) 1 2(n l)ǫ( B (l,n) L,q + ǫc (l,n) L,q +... ) (2) c L known at αs 2 Rijken,van Neerven 96, Mitov,Moch 06 highest two logs. We obtain the highest three logs at NNLO (αs) 3 CHK,Vogt,Yeat 12 via analytic continuation of the physical evolution kernel [ ] Kab(,α T S ) = AC Kab(,α S S ) between the SIA system (F T, F L ) with F L (N) = F L (N)/(α S c (1) L,q (N)) and the DIS counterpart (F 1, F L S ) for which NNLO is known F L,g NNLL (F L,q NNLL 2nd highest log)

10 Results NNLL completely known. F calculated to αs 18 using FORM Vermaseren 00, Tentyukov,Vermaseren 07 Analytical form inferred from the all-order results. CHK,Vogt,Yeat 12 (see also Vogt 11 ) Building blocks: S = (1 4ξ) 1/2, L = ln ( 1 2 (1+S)), ξ = 8C A a s/ N 2 (Also F S 1/2 for the coeff. fcn.), e.g. Pgg(N) T = 1 N(S 4 1) 1 6C a s (11C 2 A A +2C A n f 4C F n f )(S 1 1) Pqq(N) T + other NNLL terms { { Pqq(N) T = 4 C F n f 1 3 C a s A 2ξ }+ (S 1)(L+1)+1 1 C F n f 18 a CA 3 s N ( 11CA 2 +6C A n f 20C F n f ) 1 2 2ξ (S 1+2ξ)+10C 1 2 A ξ (S 1)L (51CA 6C A n f +12C F n f ) (S 1)+(11CA +2C A n f 4C F n f )S 1 L+(5CA 2 } 2C A n f +6C F n f ) 1 ξ (S 1)L2 +(51CA 2 14C A n f +36C F n f )L C T,q (N) = C F n f C 2 A N (F 1 1 FL)+ NNLL terms

11 Results Finite first moments (N = 1): NLO + resummed results. For P T : NLO + NNLL. For C T,φ : NLO + NLL. For C L : NLO + NNLL. E.g. P T gg(n = 1) = (2C A a s) 1/2 1 6C (11C 2 A A +2C A n f +12C F n f )a s ( + 1 [ ζ 144C 2]C 4 A 3 A 140CAn 3 f +4CAn 2 f CAC 2 F n f ) ( Pgq(N=1) T = C F C P T A 80C A C F n 2 f +144C 2 F n 2 f gg(n=1)+ 4 3 P T qg(n=1) = 8 3 n fa s 1 3C 2 A (2C A a 3 s) 1/2 +O(a 2 s), C F n f C A a s )+O(a 2 s). ( 17CAn 2 f 2C A nf 2 +4C F nf 2 ( Pqq(N=1) T = C F C P T A qg(n=1) 4 3 n fa s )+O(as), 2 ) (2C A as) 3 1/2 +O(as) 2, Corresponding NLO+resummed first moments for coefficient functions also calculated CHK,Vogt,Yeats 12

12 Small- gluon-parton splitting functions P T (,α S ) gq P T (,α S ) gg LO NLO -0.1 α S = 0.12, N f = LO + LL NLO + NNLL NB: similar resummed curves: approimate Casimir scaling of P T gi.

13 Small- quark-parton splitting functions P T (,α S ) qq P T (,α S ) qg LO NLO NLO + NNLL α S = 0.12, N f = NB: similar resummed curves: approimate Casimir scaling of P T qi.

14 Small- coefficient functions for gluons c (,α S ) T,g 0.6 c (,α S ) L,g NLO NLO + NLL α S = 0.12, N f = LO LO + LL NLO + NNLL

15 Small- coefficient functions for quarks c (,α S ) T,q 0.06 c (,α S ) L,q NLO NLO + NLL LO NLO + NNLL α S = 0.12, N f =

16 Analytical results in -space Non-log parts (ie. the S(N) s) for splitting functions can be epressed in terms of Bessel functions of the first kind, J 1(z), J 2(z), z = (32C A a s) 1/2 ln 1, e.g. { } Pgg T +Pqq T = 4C A a s + 8 NNLL 3 (11C A 2 +2C A n f 4C F n f )as 2 ln 1 2 z J1(z) { } (26C Fn f 23C A n f )as C (11C 2 A A +2C A n f 4C F n f ) 2 as 3 ln z J1(z) ( ) C [134 72ζ A 2]CA 4 +23CAn 3 f 48CAC 2 F n f +4C A C F nf 2 8CF 2 nf 2 asln z J 2(z) 2 NB: hint of second resummation? ( ) NB: J n(z) 2 πz cos z π 2 (n+ 1 2 ) for z. Log parts (L(S)) as convolutions of J 0: 1 1 ( d N 2 0 ln J0(2 aln) 1 ) ( ) =ln a (N 1) 2 NB: L(S) do not appear in physical kernels scheme dependent feature. Coefficient functions as convolutions of generalised hypergeometric functions (see also Albino,Bolzoni,Kniehl,Kotikov 11 )

17 Beyond NNLL Recap of what we already know: P T gi: NNLL; P T qi: N 3 LL C a,g: NNLL; C a,q: N 3 LL (closed form at NNLL known) for a = T,φ C L,i : NNLL i = q,g (C L,q at N 3 LL also possible from physical kernel) Pgg T at CF =0 N3 LL and N 4 LL using DMS relation Dokshitzer,Marchesini,Salam 05 : lnq 2 f σ(n,q 2 ) = P σ(n)f σ(n,q 2 ) = P u(n +σ lnq 2)f σ(n,q 2 ) ( ) σ n n 1 Pu P σ(n) = P u(n)+ n! N n 1 N [Pu(N)]n σ = +1: time-like. σ = 1: space-like. P u: universal splitting fnc n=1 Time-Space difference given by lower order quantities.

18 Beyond NNLL P S single log enhanced BFKL. At LL: ( ) Pgg(N) S = C 4 Aα S LL π N 1 + 2ζ CA α S 3 π N 4 + O(αS 6 N 6 ). Recall: Pgg(N) T O(α S N 1 )+O(α 2 N S 3 )+O(α 3 N S 5 )+O(α 4 N S 7 )+ LL P T gg CF =0 at LL,NLL,NNLL completely fied without any space-like input. P T gg CF =0 at N 3 LL requires P S (3) gg P T gg CF =0 at N 4 LL requires P S (3) gg at LL (known). at NLL (not known in MS). Also leads to additional cross checks with N 2 LL results obtained from D-dim structure. See CHK,Vogt,Yeats 12 for the analytic form of P T gg CF =0 and corresponding first moments at N 3 LL and at N 4 LL (up to one parameter)

19 Summary and outlook D-dimensional structure of unfactorised SIA fragmentation functions NNLO highest three logs. NLO + resummed (finite N=1 limit) time-like splitting and coeff. fnc. Results towards NNLO + resummed (require N 4 LL) partially available. Application to small- DIS CHK,Vogt and large- SIA Almasy,LoPresti,Vogt : soon. See also large- DIS Almasy,Soar,Vogt 10