Azimuthal asymmetries in unpolarized SIDIS and Drell-Yan processes

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1 Azimuthl symmetries in unpolrized SIDIS nd Drell-Yn processes Cristin Pisno REF 07 Workshop Resummtion, Evolution, Fctoriztion November Mdrid (Spin In coll. with: A. Bcchett, G. Bozzi, M. Echevrri, A. Prokudin, M. Rdici

2 Unpolrized SIDIS Differentil cross section dσ dφ h dp h { α Q F UU,T + εf UU,L + } ε( + ε cos φ h F cos φ h UU + ε cos(φ h F cos φ h UU l l S φ S P h φ h hdron plne P h lepton plne y x z Azimuthl symmetries A cos φ h UU = cos φ ε( + ε F h UU cos φ h, A cos φ h UU F UU,T + εf UU,L = εf cos φ h UU cos φ h F UU,T + εf UU,L /7

3 Unpolrized SIDIS Scles nd frmeworks Three physicl scles, two theoreticl tools 3/7

4 The structure function F UU,T From high to intermedite q T At leding twist nd order α S in colliner fctoriztion F UU,T given by the convolution of PDFs nd FFs with hrd scttering coefficients F UU,T = α s Q (πz x e d ˆx ( dẑ q x ˆx z ẑ δ T ( ˆx( ẑ Q ˆx ẑ [ ( ( z ( ( f xˆx D C (γ q qg UU,T + f D ẑ xˆx g z ( ( C (γ q gq UU,T + f g z D ẑ xˆx C (γ g q q UU,T ẑ Bcchett, Boer, Diehl, Mulders, JHEP 0808 ( /7

5 The structure function F UU,T From high to intermedite q T Expnsion of delt function for smll q T /Q ( q δ T ( ˆx( ẑ = δ( ˆx δ( ẑ ln Q ˆx Q ˆx ẑ qt + δ( ẑ ( ˆx + ( ẑ q + δ( ˆx + O T Q ln ( ẑ + Q qt Meng, Olness, Soper, PRD 54 ( Extrction of leding behviour for M q T Q F UU,T = q T α s π z x e [ ( Q f (x D (z L qt + f (x ( D Pqq + Dg Pgq (z + ( P qq f + Pqg f g (x D (z 5/7 ( Q L qt = C F ln Q qt 3C F

6 The structure function F UU,T From low to intermedite q T In TMD fctoriztion t leding twist nd order α 0 S F UU,T (x, z, P ht, Q = e x d k T d P T f ( x, k T ; Q D h/ ( z, P T ; Q δ ( zk T P ht + P T 6/7

7 The structure function F UU,T TMD evolution f ( x, b T ; Q π d k e ib T k f ( x, k ; Q f (x, b T ; Q = i=q, q,g ( C/i f i (x, bt, µ b es(µ b,q ( Q Q 0 gk (b T f NP (x, b T Collins, Soper, Stermn, NPB 50 (85 Collins, Foundtions of Perturbtive QCD ( Echevrri, Idilbi, Scimemi, JHEP 07 (0 00 F UU,T (x, z, P ht = π e x ( db T b T J 0 bt P ht /z f (x, bt D (z, bt 0 7/7

8 The structure function F UU,T From low to intermedite q T In the smll-b T region, b T /M, to O(α S with µ b = b 0/b T nd b 0 = e γ E { f (x, b T, Q f π (x, µ b + α ( } S C ( π /i f i (x, µ b e S(b T,Q i ( S(b T, Q = α S 4π C F ln Q b T b 0 3 ln Q b T b 0 + O(α S The colliner PDF f is evolved from the scle µ b to Q using DGLAP eqution f (x; µ b = f (x; Q αs ( Pqq f π + Pqg f g (x ln Q b T b 0 + O(α s We recover the structure function t lowest order in the region M q T Q F UU,T = q T α s π z x e [ ( Q f (x D (z L qt + f (x ( D Pqq + Dg Pgq (z + ( P qq f + Pqg f g (x D (z 8/7

9 Azimuthl symmetries in SIDIS 9/7

10 Azimuthl symmetries in SIDIS Fourier trnsformed expressions New expressions for the structure functions (TMD fctoriztion ssumed F cos φ h UU = MM [( h Q B xĥ Ĥ ( M h M F cos φ h UU = MM h B [ ĥ ( Ĥ ( f D ( z [ x + M4 Q B ( MMh x f ( D + M h ĥ ( M ( f 3 D H z Bcchett et l., JHEP 070 ( Wei, Song, Chen, Ling, PRD 95 ( B n [ f D = π e x db T b n+ ( T Jn bt P ht /z f (x, b T D (z, b T 0 f (n (x, b T = n! ( M b T n f (x, b T D(n (z, b T = n! ( M h b T n D(z, b T QCD equtions of motion led to the reltions 0/7 xf = x f + f xf3 = x f 3 + f D z = D D z

11 Azimuthl symmetries in SIDIS Wndzur Wilczek pproximtion Wndzur Wilczek pproximtion, the pure twist-3 functions with tilde re neglected: xf f, xf 3 f The whole leding term of the low-q T expnsion of the colliner result F cos φ h UU = Qq T F cos φ h UU = Q α s π z α s π z ( Q x e f (x D (z L qt +... ( Q x e f (x D (z L qt +... ( Q L qt = C F ln Q qt 3C F /7

12 Azimuthl symmetries in SIDIS From low to intermedite q T We know the function xf t order α S ( x f ( (x, bt = α S [C M bt 4π F ln Q bt b0 3 f (x + ( P i f i (x Similrly, we ssume D ( (z, b T = zm h b T Chen, M, PLB 768 ( [ ( α S 4π C F ln Q bt b0 3 D (D (z + i P i (z p p Ph q Ph g µν l l l l P Φ q Φ g P µν k k ( (b ( (b We reproduce the right leding high-p T expressions of xf nd D Bcchett, Boer, Diehl, Mulders, JHEP 0808 ( /7

13 Azimuthl symmetries in SIDIS From low to intermedite q T We recover the leding power behvior t smll q T /Q of the colliner LO results F cos φ h UU = Qq T α s π z x e [ ( Q f (x D (z L qt + f (x ( D P qq + D g gq P (z + ( P qq f + P qg f g (x D (z Assuming for xf 3 n expression similr to xf : F cos φ h UU = Q α s π z x e [ ( Q f (x D (z L qt + f (x ( D P qq + Dg gq P (z + ( P qq f + P qg f g (x D (z Bcchett, Boer, Diehl, Mulders, JHEP 0808 ( /7

14 Azimuthl symmetries in Drell-Yn 4/7

15 Azimuthl symmetries in Drell-Yn Gottfried-Jckson frme [ F cos φ UU GJ = M Q BDY ( x f (x, bt f ā (x, bt f (x, bt x ( ā f (x, bt B DY ā n [ f ĝ π e 0 Lu, Schmidt, PRD 84 ( db T b n+ ( T Jn bt q T f (x, b T ĝ ā(x, b T + ( l P ^x φ P l θ ^z lepton plne (cm In greement with the known result in the region M q T Q Boer, Vogelsng, PRD 74 ( [ ( F cos φ UU GJ = αs π e Q L Qq T q f (x f ā (x + (P qq f q (x f ā (x T +(P qg f g (x f ā (x + f (x (P + qq f q (x + f (x (P + qg f g (x 5/7

16 Azimuthl symmetries in Drell-Yn Collins-Soper frme F cos φ UU CS = M Q BDY {[x f ( (x, b T + x f ( (x, b T f ā (x, b T [ } f (x, bt ( ā x f (x, bt + x ( ā f (x, bt Lu, Schmidt, PRD 84 ( P l P ^x φ l θ ^z lepton plne (cm In greement with the known result in the region M q T Q Boer, Vogelsng, PRD 74 ( F cos φ UU CS = αs [ e f πqq (x ( P qq f q (x + f (x ( P qg f g (x T ( P qq f q (x f ā (x ( P qg f g (x f ā (x 6/7 Logrithmic term bsent lso in Wndzur-Wilczek pproximtion

17 Conclusions Description of SIDIS nd DY with mesured q T involves two frmeworks: TMD fctoriztion t low q T nd colliner fctoriztion t high-q T In the region M q T Q both re pplicble. Depending on the observbles the leding terms in the two pproches my coincide Assuming TMD fctoriztion t higher twist-3 level, cos φ symmetries in SIDIS nd DY mtch the result t high q T Twist 4 contributions nd cos φ symmetries need further investigtion Comprison with experiment expected in the ner future 7/7