Zhongbo Kang. QCD evolution and resummation for transverse momentum distribution. Theoretical Division, Group T-2 Los Alamos National Laboratory

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1 QCD evolution and resummation for transverse momentum distribution Zhongbo Kang Theoretical Division, Group T-2 Los Alamos National Laboratory QCD Evolution Worshop Jefferson Lab, Newport News, VA

2 Outline: whenever there is a evolution, there is a resummation QCD factorization: collinear.vs. TMD Concepts: evolution, resummation, and their connection Collinear factorization: DGLA evolution = resummation of single logarithms TMD factorization: QCD evolution and resummation Evolution of TMDs = resummation of double logarithms Illustration of unpolarized TMDs Evolution of Sivers function Difference between SIDIS and DY regarding sign change erturbative Y-term henomenology (preliminary) Summary 2

3 QCD factorization: a way to probe hadron structure We want to understand hadron structure in terms of uars and gluons longitudinal momentum distribution: collinear DFs transverse momentum distribution: TMDs To extract information on hadron structure, we send a probe and measure the outcome of the collisions in order to trace bac what s inside hadron from the outcome of the collisions, we rely on QCD factorization QCD factorization collinear factorization: pp h+x at high pt TMD factorization: SIDIS, DY, e + e - h1h2+x They are closely related to each other 3

4 DIS as an example Deep Inelastic Scattering (DIS) σ = e e 2 = X µ All the interesting physics (QCD dynamics) is contained in µ L µν W µν (leptonic tensor) (hadronic tensor) W µν Hadronic tensor in perturbative expansion W µν = Leading order factorization: parton model µ µ µ µ + O xp 2 T Q 2, 2 Q 2 µ 4

5 QCD dynamics beyond leading order Radiative corrections v d A 1 B i 2 1 +i µ? Collinear divergence!!! (from ) i 2 1 i 1 intermediate uar is on-shell t AB v gluon radiation taes place long before the photon-uar interaction a part of DF µ artonic diagram has both long- and short-distance physics 5

6 Factorization: separation of short- from long-distance Systematically remove all the long-distance physics into DFs Q 2 0 d C (0) φ (1) LO + evolution µ µ 2 1 µ = + = 0 d 2 1 µ Q 2 µ 2 d 2 1 µ 2 0 d µ C (1) φ (0) NLO + Q 2 µ 2 d µ d 2 6

7 Scale-dependence of DFs Logarithmic contributions into parton distributions µ C(Q 2 /µ 2 ) φ(x, µ 2 ) Going to even higher orders: QCD resummation of single logs... 1 φ(x, µ 2 ) = α s ln µ2 Λ 2 α s ln µ2 Λ 2 2 7

8 DGLA evolution = resummation of single logs Evolution = Resum all the gluon radiation 1 φ(x, µ 2 ) = φ(x, µ 2 ) - ln µ 2 φ i(x, µ 2 ) = j DGLA Euation = Evolution ernel splitting function ij ( x x ) φ j(x, µ 2 ) By solving the evolution euation, one resums all the single n logarithms of α s ln µ2 Λ 2 8

9 Similar single logs for evolution of twist-3 correlation functions Qiu-Sterman function: first t-moment of Sivers function splitting ernel for unpolarized DFs Another twist-3 correlation function: first t-moment of Boer-Mulders function splitting ernel for transversity Kang-Qiu, arxiv:

10 TMD factorization Example: SIDIS (two scales - Q and t) Ji-Ma-Yuan,

11 Evolution of TMDs follow Collins-Soper evolution Evolution of collinear DFs follow the usual DGLA-type evolution euation, which is euivalent to resum the single-logarithmic contributions to all order α s ln Q2 n µ 2 Evolution of TMDs follow Collins-Soper-type evolution euation, which is euivalent to resum the double-logarithmic contributions to all order, which is usually more difficult α s ln 2 Q 2 n 2 T 11

12 SIDIS cross section in momentum space: in b-space: TMD uar distribution and fragmentation functions contain double logarithms, others contain only single logarithms 12

13 Evolution of TMDs Since one now needs to resum double logarithms, typically it involves two steps: Idilbi-Ji-Ma-Yuan, 2004 Energy evolution of the unpolarized DFs Since it contains double logarithms, the ernel still contains single logarithms µ d dµ K(µ, b) = γ K = µ d G(µ, ζ) dµ Solving these two euations, euivalently one resums the double logs First for the evolution euation of K and G Then feed the solution bac to the energy evolution euation 13

14 The formalism contains all the evolutions Similar for the unpolarized fragmentation function Hard function and Soft function contain only single logs Eventually collect all the terms 14

15 Final step Once all the logs are resummed, the rest of b-dependent DFs and FFs can be expanded as collinear DFs and FFs dσ dx B dydz h d 2 h = σ 0 F UU F UU = d 2 b (2π) 2 ei b W UU (b, Q, x B,z h ) W UU (b, Q, x B,z h )=e S(b,Q) e 2 C/i f i/a (xb,µ= c b ) D B/j C j/ (z h,µ= c b ) All the large logarithms are resummed to the Sudaov exponential term S(b, Q) = Q 2 dµ 2 A ln(q 2 c 2 /b µ 2 /µ 2 )+B A = A (n) α s 2 π n=1 n 15

16 Evolution of Sivers function The Collins-Soper energy evolution is really for the whole correlator So for Sivers function, it really is f α 1T (x, ) 2 that evolves as a whole in b-space, it is f ( α) 1T (x, b, µ, ζ) = 1 M d 2 e i b α f 1T (x,,µ,ζ) Kang-Xiao-Yuan, RL, 2011 it follows the same energy evolution euation ζ ζ ( α) f 1T (x, b, µ, ζ) =[K(µ, b)+g(µ, ζ)] f ( α) 1T (x, b, µ, ζ) Thus one should get a very similar resummation formalism 16

17 The Sivers effect for SIDIS The resummation formalism (consistent with experimental convention) dσ dx B dydz h d 2 = σ 0 F UU + s sin(φ h φ s )F sin(φ h φ s ) UT h spin-dependent structure function F sin(φ h φ s ) UT = 1 4π 0 db b 2 J 1 ( b)w UT (b, Q, x B,z h ) W UT (b, Q, x B,z h )=e S(b,Q) ( C T /i T i,f )(x B,µ= c b ) (D B/j C j/ )(z h,µ= c b ) only soft-gluonic pole Qiu-Sterman function appears in this part ( C T /i T i,f )(x B,µ)= 1 x B dx x CT /i (x B x,µ)t i,f (x, x, µ) 17

18 Similar form for DY production Unpolarized DY production at low t dσ dq 2 dyd 2 = σ 0 2π 0 db bj 0 ( b)w UU (b, Q, x A,x B ) W UU (b, Q, x A,x B )=e S(b,Q) e 2 (C /i f i/a )(x A,µ= c b ) (C /j f j/b )(x B,µ= c b ) Single transverse spin dependent DY production at low t dσ dq 2 dyd 2 = σ 0 αβ Sβ ˆβ 1 4π 0 db b 2 J 1 ( b)w UT (b, Q, x A,x B ) W UT (b, Q, x A,x B )=e S(b,Q) e 2 ( C T /i T i,f )(x A,µ= c b ) (C /j f j/b )(x B,µ= c b ) 18

19 Comments on SIDIS and DY The only difference comes from so-called coefficient function leading order C T (0) i/j (z,µ = c b )=δ ijδ(1 z) C T (0) i/j (z,µ = c b )= δ ijδ(1 z) DY SIDIS at next-leading-order: well-nown difference due to Q 2 >0 (<0) C T (1) i/j (z,µ = c b )=δ ij 1 + C F 4N c 2 (π2 4)δ(1 z) DY 2 C T (1) i/j (z,µ = c b )= δ ij 1 + C F 4N c 2 ( 4)δ(1 z) SIDIS Thus in the full perturbative QCD region, Sivers between SIDIS and DY is not just a sign: it is interesting to study the conseuence 19

20 Coefficients for Sivers and unpolarized DFs is different When expanded Sivers function in terms of Qiu-Sterman function at small b, only soft-gluon pole contributes to the coefficient function (εterm in dimensional regularization) C T (1) i/j (z,µ = c b )=δ ij 1 + C F 4N c 2 (π2 2 4)δ(1 z) Sivers function C (1) i/j (z,µ = c b )=δ ij CF 2 + C F 2 (π2 2 4)δ(1 z) Unpolarized DFs In b-space, we found a extra collinear divergence which is supposed to absorbed into the evolution of Qiu-Sterman function. We now there is this -N_c term issue. 20

21 What is Y-term? So-far concentrate on the resumed term, which is most relevant when t<<q. When t gets relatively large, the conventional NLO perturbative contribution becomes important Y-term 21

22 Y term can be easily extracted Y term can be easily extracted/derived from existing calculations For the DY production, in the paper of unified picture of Sivers effect erturbative term: Ji-Qiu-Vogelsang-Yuan, RD73, 2006 Asymptotic term (tae the limit t<<q): Y-term = perturbative-term - Asymptotic-term 22

23 henomenological studies Only at small b-region (corresponds to large momentum), one can calculate the relevant coefficients perturbatively. dσ dq 2 dyd 2 = σ 0 2π 0 db bj 0 ( b)w UU (b, Q, x A,x B ) W UU (b, Q, x A,x B )=e S(b,Q) e 2 (C /i f i/a )(x A,µ= c b ) (C /j f j/b )(x B,µ= c b ) However, in order to Fourier transform bac to t-space, we need the whole b-region. Since large b-region will be non-perturbative, we need a non-perturbative input. This part should be universal if QCD factorization holds for the process. 23

24 henomenological study: concentrate on small t part For small t region, we could use the resumed formalism. Don t need to worry about Y-term. Only at small b-region (corresponds to large momentum), one can calculate the relevant coefficients perturbatively. dσ dq 2 dyd 2 = σ 0 2π 0 db bj 0 ( b)w UU (b, Q, x A,x B ) W UU (b, Q, x A,x B )=e S(b,Q) e 2 (C /i f i/a )(x A,µ= c b ) (C /j f j/b )(x B,µ= c b ) However, in order to Fourier transform bac to t-space, we need the whole b-region. Since large b-region will be non-perturbative, we need a non-perturbative input. This part should be universal if QCD factorization holds for the process. 24

25 The parametrizations for the non-perturbative function Different approaches for the non-perturbative functions dσ dq 2 dyd 2 = σ 0 2π 0 db bj 0 ( b)w UU (b, Q, x A,x B ) W pert UU (b, Q, x A,x B )=e S(b,Q) e 2 (C /i f i/a )(x A,µ= c b ) (C /j f j/b )(x B,µ= c b ) arametrize the full b-space function W UU (b, Q, x A,x B )=W pert UU (b, Q, x A,x B )F N (b, Q, x A,x B ) function form (through extrapolation): fitted form directly from experiments: Qiu-Zhang, 2001 Broc-Landry-Nadolsy-Yuan, 2003 W UU (b, Q, x A,x B )=W pert UU (b,q,x A,x B )F N (b, Q, x A,x B ) b = b 1+(b/bmax ) 2 F N (b, Q, x A,x B )=exp Q g 1 (1 + g 3 ln(100x A x B )) + g 2 ln b 2 2Q 0 25

26 The Sudaov nows to A (1) and B (1) For Sivers effect, so far we only calculate A (1) and B (1) for the Sudaov exponent S(b, Q) = Q 2 dµ 2 A ln(q 2 c 2 /b µ 2 /µ 2 )+B A = A (n) α s 2 π n=1 n A (1) = C F B (1) = 3 2 C F They are exactly the same as those in the unpolarized pp collision Thus for consistency, we will also use A (1) and B (1) for the unpolarized DY production As the whole perturbative Sudaov term (up to the order we have calculated) is the same, we will only assume the non-perturbative function is the same for the Sivers effect 26

27 Using A (1) and B (1) still describes data reasonably well E288 and E605 s = 27.4GeV s = 38.8GeV d /dyd 2 p T (pb GeV -2 ) <M<5 5<M<6 6<M<7 7<M<8 d /dyd 2 p T (pb GeV -2 ) <M<8 8<M<9 10.5<M< <M< <M< p T (GeV) p T (GeV) 27

28 Sivers effect of DY production at RHIC Blue curve: bare parton model (using Torino TMD with Gaussian ansatz from SIDIS) Red curve: resummed formalism (using Torino TMD to calculate TF(x, x) as the initial input function, then evolve) gt,f (x, x) = d 2 2 M f 1T (x, 2 ) SIDIS A N s=200 GeV 0< T <1 GeV 4<Q<9 GeV y caution: non-perturbative part could be different for Sivers asymmetry 28

29 Summary QCD factorization is a useful tool to probe and understand hadron structure QCD evolution and resummation is closely related to each other whenever there is an evolution, there is a resummation Scaling violation (QCD evolution of collinear unpolarized DFs) has played a very important role in establishing QCD factorization formalism QCD evolution for collinear twist-3 function and TMDs will be extremely important in understanding hadron structure Evolution maes Sivers asymmetry smaller 29

30 Summary QCD factorization is a useful tool to probe and understand hadron structure QCD evolution and resummation is closely related to each other whenever there is an evolution, there is a resummation Scaling violation (QCD evolution of collinear unpolarized DFs) has played a very important role in establishing QCD factorization formalism QCD evolution for collinear twist-3 function and TMDs will be extremely important in understanding hadron structure Evolution maes Sivers asymmetry smaller Than you 29