Preliminary plan. 1.Introduction. 2.Inclusive and semi-inclusive DIS (structure functions)

Transcription

1 Preliminary plan 1.Introduction 2.Inclusive and semi-inclusive DIS (structure functions) Basics of collinear PDFs at tree level (definition, gauge link) 3.Basics of collinear PDFs (interpretation) Basics of TMDs at tree level (definition, gauge link, interpretation) 4.Basics of factorization Basics of TMD evolution Phenomenology of unpolarized SIDIS Phenomenology of polarized SIDIS

2 Next lecture May 27, 12:00 PM in F113

3 Quick review of last lecture

4 TMD factorization h Collins, Soper, NPB 193 (81) Ji, Ma, Yuan, PRD 71 (05) q P F UU,T (x, z, Ph,Q 2 2 )=C [ ] f 1 D 1 = H(Q 2,µ 2, ζ, ζ h ) d 2 p T d 2 k T d 2 l T δ (2)( p T k T + l T P h /z ) Hard part x a e 2 a f a 1 (x, p 2 T,µ 2, ζ) D a 1(z, k 2 T,µ 2, ζ h ) U(l 2 T,µ 2, ζζ h ) TMD PDF TMD FF Soft factor

5 High and low transverse momentum

6 SIDIS once again l l S T! S P h! h hadron plane P h lepton plane y x Q = photon virtuality M = hadron mass z P h = hadron transverse momentum q 2 T P 2 h /z 2

7 Low and high transverse momentum AB, D. Boer, M. Diehl, P.J. Mulders, JHEP 08 (08) Low q 2 T Q 2 M 2 Q 2 q 2 T

8 TMD factorization h Collins, Soper, NPB 193 (81) Ji, Ma, Yuan, PRD 71 (05) q P F UU,T (x, z, Ph,Q 2 2 )=C [ ] f 1 D 1 = H(Q 2,µ 2, ζ, ζ h ) d 2 p T d 2 k T d 2 l T δ (2)( p T k T + l T P h /z ) x a e 2 a f a 1 (x, p 2 T,µ 2, ζ) D a 1(z, k 2 T,µ 2, ζ h ) U(l 2 T,µ 2, ζζ h )

9 Low and high transverse momentum High M 2 q 2 T M 2 Q 2 q 2 T

10 Collinear factorization F (x, z, Q 2 )= 1 Q 2 z 2 x a,b f a( x ˆx,µ2 F 1 ) x dˆx ˆx 1 z D b( z ẑ,µ2 F dẑ ( P 2 ẑ δ h (1 ˆx)(1 ẑ) ) Q 2 z2 ˆxẑ ) H ab ( ˆx, ẑ,ln µ2 F Q 2 ) h q P

11 Low and high transverse momentum Low q 2 T Q 2 Intermediate M 2 q 2 T Q 2 High M 2 q 2 T M 2 Q 2 q 2 T

12 Matching F UU,T A q 2 T Must match! M 2 Q 2 q 2 T The leading high-q T part is just the tail of the leading low-q T part Collins, Soper, Sterman, NPB250 (85)

13 Low and high transverse momentum nonperturbative part of TMDs tail of TMDs, calculable with pqcd Low q 2 T Q 2 Intermediate M 2 q 2 T Q 2 High M 2 q 2 T M 2 Q 2 q 2 T

14 Perturbative corrections to TMDs p p l l P Φ q Φ g µν P (a) (b) f q 1 (x, p2 T )= α s 2π 2 1 p 2 T [ L(η 1 ) 2 f q 1 (x) C F f q 1 (x)+( P qq f q 1 + P qg f g 1 ) (x) ], F UU,T = 1 q 2 T α s 2π 2 z 2 Large log, needs resummation a xe 2 a [ ( ) Q f1 a (x) D1(z) a 2 L qt 2 + f1 a (x) ( D1 a P qq + D g 1 P gq) (z) ] + ( P qq f a 1 + P qg f g 1 ) (x) D a 1 (z) where ( ) Q 2 L qt 2 =2C F ln Q2 qt 2 3C F DGLAP splitting functions

15 TMD factorization: b space F UU,T (x, z, b, Q 2 )=x a e 2 a Sudakov form factor [ ] (f1 i C ia )(C aj D j 1 ) e S e S NP collinear PDF and FF calculable with pqcd nonperturbative part of TMDs Intermediate (resummation) Low (nonpert.) High (fixed-order pqcd) F UU,T (x, z, q 2 T,Q 2 )=x a e 2 a d dq 2 T [(f i1 C ia )(C aj D j1 ) e S( 1 e S NP )]

16 Leading-log formula Ellis, Veseli, NPB 511 (98) F UU,T (x, z, q 2 T,Q 2 )=x a e 2 a d dq 2 T [f a1 (x;[q 2T ]) D a1(z;[q 2T ]) e S( 1 e S NP )] S(q 2 T,Q 2 )= Q 2 q 2 T dµ 2 α S (µ 2 ) µ 2 2π 2C F log Q2 µ 2 α s (µ 2 )= 4π β 0 log(µ 2 /Λ 2 )

17 Part 5: Unpolarized Phenomenolgy

18 Experimental access Drell Yan dσ dq 2 T q e 2 q f q 1 (x, p2 T ) f q 1 ( x, p2 T ) Semi inclusive DIS dσ dq 2 T q e 2 q f q 1 (x, p2 T ) D q 1 (z, k2 T ) electron positron annihila5on dσ dq 2 T q e 2 q D q 1 (z, k2 T ) D q 1 ( z, k 2 T )

19 Some studies in Drell-Yan

20 Available studies PARTON INTRINSIC MOTION IN INCLUSIVE M=7-8 GeV M=8-9 GeV M=10-11 GeV M=4-5 GeV M=5-6 GeV M=6-7 GeV M=7-8 GeV Gaussians Ed 3 σ/d 3 q [cm 2 /GeV 2 ] Ed 3 σ/d 3 q [cm 2 /GeV 2 ] D Alesio, Murgia, PRD70 (04) K = 1.6 1/β = 0.95 GeV/c q T [GeV/c] K = 1.8 1/β = 0.8 GeV/c q T [GeV/c] Gaussians + kt resumma5on Landry, Brock, Nadolsky, Yuan, PRD67 (03)

21 Example of resummation effects dσ dq 2 T Q = 5 GeV s = 50 GeV Q = 10 GeV Gaussian + resumma5on Gaussian only q T

22 Nonperturbative part In b space S NP = b2 b 2 1 b 2 = log ( ) Q log (100x A x B ) b max =0.5 GeV data points (Drell-Yan) Brock, Landry, Nadolsky, Yuan, PRD67 (03)

23 Nonperturbative part In b space Kulesza, Stirling, JHEP 12 (03) S NP = b2 b 2 1 b 2 = log ( ) Q log ( ) s / b s = Note: there should be a factor 4 between 1/b and kt Q

24 Nonperturbative part In k T space Kulesza, Stirling, JHEP 12 (03) S NP = q2 T q 2 T q 2 T = log ( ) Q log ( ) s 19.4 q 2 T s = Q

25 Unpolarized SIDIS

26 Unpolarized SIDIS dσ dx dy dφ S dz dφ h dph 2 { = α2 y 2 x y Q 2 F UU,T + ε F UU,L + 2 ε(1 + ε) cos φ h F cos φ h UU 2 (1 ε) + ε cos(2φ h ) F cos 2φ h UU }

27 Azimuth-independent pieces

28 Convolution F UU,T = a e 2 af a 1 D a 1, ( M 2 F UU,L = O Q 2, P h 2 ) Q 2 f D = x B d 2 p T d 2 k T δ (2)( p T k T P h /z ) f a (x B,p 2 T ) D a (z, k 2 T ) f D = x B d 2 p T d 2 k T δ (2)( p T k T P h /z + l T ) f a (x B,p 2 T ) D a (z, k 2 T ) U(l 2 T ) Does not make a big difference if Gaussians are used

29 Fragmentation functions For the favored functions for the unfavored functions D u π+ 1 = D d π + 1 = D d π 1 = Dū π 1, D f 1 D1 u K+ = Dū K 1, D1 fd D s K+ 1 = D1 s K D1 f Dū π+ 1 = D1 d π+ = D d π 1 = D1 u π D1, d D s π+ 1 = D s π+ 1 = D s π 1 = D s π 1 D df 1, Dū K+ 1 = D d K + 1 = D1 d K+ = D d K 1 = D1 d K = D1 u K D1 dd, D s K+ 1 = D s K 1 D d 1.

30 Various combinations F p/π+ UU,T (x, z, P 2 h ) = F p/π UU,T (x, z, P 2 h ) = F n/π+ UU,T (x, z, P 2 h ) = F n/π UU,T (x, z, P 2 h ) = F p/k+ UU,T (x, z, P 2 h ) =4f u 1 D fd 1 + F p/k UU,T (x, z, P 2 h ) =4f ū1 D fd 1 + F n/k+ UU,T (x, z, P 2 h ) =4f d 1 D fd 1 + F n/k UU,T (x, z, P h ) 2 =4f d 1 D1 fd + ) ( ) ( ) (4 f1 u + f d 1 D1 f + 4 f ū1 + f1 d D1 d + f1 s + f1 s ( ) ) ( ) 4 f ū1 + f1 d D1 f + (4 f1 u + f d 1 D1 d + f1 s + f1 s ( ) ( ) ( ) 4 f1 d + f ū1 D1 f + 4 f d 1 + f1 u D1 d + f1 s + f1 s ( ) ( ) ( ) 4 f d 1 + f1 u D1 f + 4 f1 d + f ū1 D1 d + f1 s + f1 s ) (4 f ū1 + f1 d + f d 1 (4 f u 1 + f d 1 + f d 1 ) ( ) 4 f d 1 + f1 u + f ū1 ( ) 4 f1 d + f1 u + f ū1 D df 1, D df 1, D df 1 D df 1, D dd 1 + f s 1 D f 1 + f s 1 D d 1, D dd 1 + f s 1 D f 1 + f s 1 D d 1, D dd 1 + f s 1 D f 1 + f s 1 D d 1, D1 dd + f1 s D1 f + f1 s D1 d

31 Valence and pions only F p/π+ UU,T (x, z, P 2 h ) =4f u 1 D f 1 + f d 1 D d 1, F p/π UU,T (x, z, P 2 h ) =f d 1 D f 1 +4f u 1 D d 1, F n/π+ UU,T (x, z, P 2 h ) =4f d 1 D f 1 + f u 1 D d 1, F n/π UU,T (x, z, P 2 h ) =f u 1 D f 1 +4f d 1 D d 1

32 Gaussian ansatz f a 1 (x, p 2 T )= f a 1 (x) πρ 2 a e p2 T /ρ2 a, D a 1 (z, k 2 T )= Da 1(z) πσ 2 a e z2 k 2 T /σ2 a f a 1 D a 1 = 1 2 π(z 2 ρ 2 a + σa) 2 e P h /(z2 ρ 2 a +σ2 a ) With Gaussian soft factor f a 1 D a 1 = 1 2 π(z 2 ρ 2 a + σa 2 + τ 2 ) e P h /(z2 ρ 2 a +σ2 a +τ 2 )

33 Interesting ratio 5 p Π n Π σ 2 f = σ 2 d =0.3 GeV 2 f u 1 /f d D d 1/D f Ρ u 2 0.3, Ρ d Ρ u 2 0.1, Ρ d P ht

34 Hall-C results JLab Hall C, Mkrtchyan et al., PLB665 (08)