Studies of TMD resummation and evolution

Transcription

1 Studies of TMD resummation and evolution Werner Vogelsang Univ. Tübingen INT, 0/7/014

2 Outline: Resummation for color-singlet processes Contact with TMD evolution Phenomenology Conclusions Earlier work with A. Kulesza, E. Laenen, G. Sterman; J. Nagashima, Y. Koike Work in progress with M. Lambertsen and M. Schlegel

3 Resummation for color-singlet processes

4 Collinear factorization: e.g. Drell-Yan P a,q P b dσ ab dx a dx b f a (x a,µ) f b (x b,µ) dˆσ ab (x a P a,x b P b, α s (µ), µ,, q,...) especially dσ d, dσ d d q partonic cross sections: pcd dˆσ ab = dˆσ (0) ab + α s π dˆσ(1) ab +... sometimes, large (double-)logarithmic corrections to dˆσ (k) ab

5 first example: 4 dσ d = ab dx a dx b f a (x a,µ) f b (x b,µ) ω ab z =, α s(µ), ŝ µ +... LO : q q NLO correction: z 1: ω (1) q q ω (0) q q δ(1 z) α s log(1 z) 1 z

6 yet higher orders:. ω (k) q q α k s log k 1 (1 z) 1 z threshold logarithms for z à 1 real radiation inhibited / exclusive boundary

7 second example: dσ d d q LO : q higher orders:. q dˆσ (k) q q d q α k s dˆσ (0) q q d q δ () (q ) log k 1 (q / ) q q T logarithms + close correspondence with TMD evolution

8 Large logs can be resummed to all orders directly from perturbative diagrams originate from soft / collinear gluon emission CD matrix elements simplify, particularly so for color-singlet processes near threshold, exponentiation of eikonal diagrams Gatherall; Franklin,Taylor; Sterman; for symmetric multi-gluon phase space in the following, Drell-Yan as example. Easily extended to SIDIS, e + e - Sterman, WV

9 total Drell-Yan cross section: δ 1 z j z j = 1 πi C dn e N(1 z P j z j) log k 1 (1 z) 1 z + log k (N) +... q T -differential cross section: q k 1 δ q + j k j = 1 (π) d b e i b (q + P j k j )... log k 1 (q / ) q + log k (b) +... k n both transforms can be taken simultaneously Laenen,Sterman,WV

10 non-abelian exponentiation: Gatherall; Franklin,Taylor; Sterman Berger, Sterman αsl k k αsl k k 1 1+α s L + α sl α s L + α sl exp α s L + α sl α s L + α sl +... α k sl k+1 α k sl k

11 σ eik (N,b) = exp E eik (N,b,ε) E eik (N,b,ε) web function d 4 ε k W(k, (k β)(k β ),µ, α s (µ), ε) e Nk0 / i b k 1 W = C F α s π 1 k δ(k )+O(α s)

12 after performing k + integral and subtraction of collinear div. σ eik (N,b) = exp k < d k [π 1 ε /Γ(1 ε)] k 0 dk W k,k + k,µ, α s, ε e i b k k K 0 N + k ln k + k + i )) (k ln N A q (α s (k )) )1 ε where A q (α s )=C F αs π + αs π CA ( N = Ne γ E ) ζ() 59 T Rn f +... finally, to NLL σ eik (N,b) = exp 0 dk k A q (α s (k )) Nk Nk J 0 (bk ) K 0 + ln

13 σ eik (N,b) = exp 0 dk k A q (α s (k )) Nk Nk J 0 (bk ) K 0 + ln jointly resummed cross section: Laenen,Sterman,WV N b : b N : threshold logs (e.g. b=0) q T logs for the latter case: σ eik (N,b) exp 0 dk k A q (α s (k )) J 0 (bk ) 1 Nk ln

14 0 dk k A q (α s (k )) J 0 (bk ) 1 Nk ln vanishes at b=0 J0 (bk ) 1 cuts off integral at k e γ E b write exponent as /η dk k η Nk A q (α s (k )) ln b e γ E +1 Bozzi, Catani, de Florian, Grazzini; Laenen,Sterman,WV

15 /η dk k Nk A q (α s (k )) ln write as /η dk k A q (α s (k )) ln k + /η dk k + B q (α s (k )) A q (α s (k )) ln N + B q (α s (k )) where B q (α s )= 3 C F α s π + O(α s)

16 /η dk k A q (α s (k )) ln k + /η dk k + B q (α s (k )) } standard Sudakov exponent A q (α s (k )) ln N + B q (α s (k )) matches standard CSS result dσ d d q dn πi τ N d b (π) eiq b } α s π P qq N à DGLAP evolution of PDFs from µ = to /η f N q (µ = /η) e S(b,) f N q (µ = /η) can be systematically extended (Y-term, qg contribution, )

17 emphasize: exponent vanishes at b=0 nonperturbative contributions? 0 dk k suggests form b C F π S NP = A q (α s (k )) 0 J 0 (bk ) 1 dk α s (k ) ln g 1 + g log universal M Nk ln Nk + O(b 4 ) b + O(b 4 ) for joint resummation: b + 4N CF π 0 dk α s (k ) ln Nk + O(b 4 )

18 Contact with TMD evolution

19 Ji, Ma, Yuan; Collins; Mert Aybat, Rogers, dσ TMD d d q = q, q hard coefficient dn H q q πi τ N C N d b (π) eiq b f q (N,b,) f q (N,b,) comparison to resummation formula yields f q (N,b,) = exp exp exp 1 1 /η µ F /η dk k g 1 + g log b M dk k A q (α s (k )) ln k + B q (α s (k )) α s (k ) π P N qq f N q (µ F ) probably best for use in phenomenology Sun, Yuan; Echeverria et al.

20 alternatively, relate to f q (N,b, 0 ) : f q (N,b,) = f q (N,b, 0 ) exp exp exp /η 0 0 /η 0 dk k dk k 1 g log b 0 A q (α s (k )) ln k + B q (α s (k )) A q (α s (k )) ln 0 k + B q (α s (k )) exp /η 0 /η 0 dk k α s (k ) π = f q (N,b, 0 ) exp S(b, )+S(b, 0 ) 1 g log exp /η 0 /η 0 dk k α s (k ) π P N qq Mert Aybat, Prokudin, Rogers Anselmino et al. 0 P N qq b coincides with standard result, except that b b0 η +1 e γ E η 0 e γ E +1

21 f q (N,b,) = f q (N,b, 0 ) exp S(b, )+S(b, 0 ) 1 g log b 0 exp /η 0 /η 0 dk k α s (k ) π P N qq at b=0: f q (N,0,) = f q (N,0, 0 ) exp 0 dk k α s (k ) π P N qq DGLAP evolution for k - integrated PDF!

22 Phenomenology

23 NLL expansion of perturbative exponent: 1 /η dk k A q (α s (k )) ln k + B q (α s (k )) β = b 0 α s ()ln η = b 0 α s ()ln = 1 α s (µ) h(0) (β)+h (1) (β) b e γ E +1 h (0) (β) = A(1) q πb 0 [β + ln(1 β)] h (1) (β) = A(1) q b 1 πb 3 0 A() q π b 0 1 ln (1 β)+ β 1 β + ln(1 β) β + ln(1 β) 1 β + B(1) q πb 0 ln(1 β)

24 Treatment of large-b region: b* prescription contour method (b) Collins, Soper, Sterman; Laenen, Sterman, WV ϕ β =1 (b) π 0 db b J 0 (bq T ) f(b) =π 0 db b [ h 1 (bq T,v)+h (bq T,v)] f(b) (h i Hankel functions) parameter free (can be used even w/o Gaussian)

25 In the following, investigate: complex-b method vs b* role of boundary condition at b=0 choice for f q (N,b, 0 ) : Gaussian vs standard resummation

26 f u (x, k, 0 ) exp k /k f CTE u (x, 0 ) Anselmino et al. k =0.5 GeV g =0.68 GeV 0 = 1 GeV b* prescription with b max =0.5 GeV -1, no boundary condition 0 = 1 GeV =.4GeV = 0 GeV

27 b* complex-b =.4GeV = 0 GeV

28 complex-b method w/ g=0.4 GeV

29 complex-b method with k = 0.05 GeV

30 effect of boundary condition at b=0

31 standard resummation : f q (N,b,) = exp 1 /η dk k exp 1 g 1 + g log M /η exp µ F dk k A q (α s (k )) ln α s (k ) π b P N qq f N q (µ F ) k + B q (α s (k )) b* complex-b g 1 =0.065 GeV g =0.68 GeV M = 1 GeV matching region

32 effect of boundary condition at b=0

33 Koike, Nagashima, WV

34 Conclusions:" complex-b method is an alternative to b*, " parameter-free." Will need more detailed studies." role of subleading effects ansatz for f q (N,b, 0 ) : probably standard resummation more useful joint resummation could be relevant in presently relevant kinematic regimes