Sivers Func,on in the Quasi- Classical Approxima,on

Transcription

1 Sivers Func,on in the Quasi- Classical Approxima,on Yuri Kovchegov The Ohio State University (based on arxiv: [hep- ph] with MaP Sievert)

2 Outline Preview/Summary STSA sign- flip in SIDIS vs DY: diagramma,c interpreta,on? Quasi- classical picture of STSA genera,on: Quasi- classical STSA: the OAM and transversity channels Space-,me picture and physical/diagramma,c interpreta,on of sign- flip Outlook: can calculate any TMD in the quasi- classical picture.

3 Preview/Summary

4 Single Transverse Spin Asymmetry Consider transversely polarized proton scapering on an unpolarized proton or nucleus. Single Transverse Spin Asymmetry (STSA) is defined by A N (k) d " d # d 2 kdy d 2 kdy d " d 2 kdy + d # d 2 kdy = d " d 2 kdy (k) d " d 2 kdy ( k) d( ) d " d 2 kdy (k)+ d " d 2 kdy ( k) 2 d unp

5 Classical picture of STSA in Drell- Yan Think of a transversely polarized proton as a rota,ng disk with the axis perpendicular to the collision axis The proton is not transparent: it has some amount of screening/ shadowing (e.g. gray disk, black disk, etc.) Incoming an,- quark (in DY) is more likely to interact near the front of the proton: hence, due to the rota,on, the outgoing virtual photon is more likely to be produced le#- of- beam, thus genera,ng STSA.

6 Classical picture of STSA in SIDIS DiPo for SIDIS: except now the incoming virtual photon is more likely to interact near the back of the proton, in order for the produced quark to be able to escape out of the proton remnants. Owing to the same rota,on, the outgoing quark is more likely to be produced right- of- beam, thus genera,ng STSA in SIDIS with the opposite sign compared to STSA in DY!

7 Summary It appears that STSA can be easily interpreted as a combina,on of OAM and some amount of shadowing. Sign- flip between STSA and DY has a very simple interpreta,on in this framework too.

8 In search of a diagramma,c interpreta,on of STSA sign- flip

9 Defini,ons TMDs are defined through the correlator ij(x, k; P, S) Z dx d 2 x? 2(2 ) 3 e i ( 1 2 xp + x x k) hp, S j(0) U i (x + =0,x,x) P, Si with the gauge links U SIDIS = V 0 [+1, 0] V x[+1,x ] U DY = V 0 [0, 1] V x [x, 1] The correlator is decomposed in terms of quark TMDs as ij(x, k; P, S) = M 2 P + 1 M h 1T (x, k T ) i µ apple f 1 (x, k T ) P M + 1 M 2 f? 1 T (x, k T ) µ µ P k? S? 5 S µ? P 1 M 2 h? 1s(x, k) i µ 1 M g 1s(x, k) P 5 k µ? P + h? 1 (x, k T ) µ k µ? P M 2 5 ij

10 STSA spin- flip between SIDIS and DY Operatorially everything is clear (Collins 2002): under T- reversal Wilson lines go from future- poin,ng (SIDIS) to past- poin,ng (DY), while the Sivers func,on, being T- odd, changes sign, such that f?a 1T (x, k T ) SIDIS = f?a 1T (x, k T ) DY Diagramma,cally things are not so simple

11 What generates STSA p p To obtain STSA need χ χ transverse polariza,on (χ) dependence (comes with a factor of i ) ū (p) u (p) a + i b, a,b real However, cross sec,on has to be real ū (p) u (p) A 1 A 2 +c.c=(a + i b) A 1 A 2 +(a i b) A 1 A 2 =( independent) + i b(a 1 A 2 A 1 A 2 ) such that we also need A 1 A 2 Efremov, Teryaev 82 a complex phase difference between the amplitude (A 1 ) and the cc amplitude (A 2 ) to cancel the i from χ- dependence (from Qiu and Sterman, early 90 s)

12 STSA IN SIDIS Single-spin asymmetry e Leading-Twist Sivers Effect i ~S p ~q ~p q Pseudo-T-Odd S e proton!* quark Light-Front Wavefunction S and P- Waves current quark jet final state interaction spectator system A06 QCD S- and P- Coulomb Phases --Wilson Line Lensing Effect Sign reversal in Drell Yan Brodsky, Hwang, Schmidt 02; Collins 02 To generate STSA need a final state interac,on (the blob above). In TMD factoriza,on this is usually absorbed into the polarized proton TMD and is referred to as the ini,al- state effect, and hence iden,fied with the Sivers effect.

13 STSA in SIDIS STSA arises from the interference diagrams between Born- level and the one- rescapering graphs: Spin- dependence comes from the vertex. The phase is generated by an extra rescapering, which gives the amplitude an Im part represented by the second cut. Brodsky, Hwang, Schmidt 02; (see also Brodsky, Hwang, YK, Schmidt, Sievert 13)

14 STSA in Drell- Yan Here we also need interference between the Born level amplitude and a one- rescapering correc,on to it. The DY STSA is also caused by two essen,al ingredients: (i) spin dependence from the quark- proton vertex and (ii) phase due to the extra cut (intermediate state) in the amplitude. Note that the 2 nd cuts in SIDIS and DY are different! S,ll at large Q 2 the two STSAs are equal up to a sign reversal: A DY N = A SIDIS N (Collins 02; Brodsky, Hwang, Schmidt 02; same+ YK, Sievert 13).

15 Side- by- side: SIDIS and DY The second cuts in the SIDIS and DY STSA diagrams look very different: in SIDIS we cut the s- channel quark and the di- quark, while in DY one has to cut the s- channel an,- quark and the exchanged quark. Diagramma,cally the sign- flip is not obvious. (Math works, but can we see why diagramma,cally?)

16 Quasi- Classics

17 Rules of the game We want to evaluate STSA in the Glauber- Mueller (GM)/ McLerran- Venugopalan (MV) quasi- classical approxima,on. This approxima,on is valid for a large nucleus or a dense proton. I will talk about the nucleus simply because the nuclear atomic number A allows for a controlled approxima,on: quasi- classical physics resums powers of 2 s A 1/3 Note that the GM/MV quasi- classics is not just a model, but is a valid QCD asympto,cs in the high- energy & large- A limit.

18 Quark Produc,on Start with inclusive classical quark produc,on cross sec,on in SIDIS. The kinema,cs is standard: qγ x k s Q 2? 2 b x d The result is +A!q+X d 2 kdy Z dp + d 2 pdb =A 2(2 ) 3 Z d 2 k 0 (2 ) 2 e i (k Z d 2 xd 2 yw k 0 ) (x y) Wigner distribu6on p, b, x + y 2 dˆ +N!q+X d 2 k 0 dy LO cross sect (p, q) D x y [+1,b ] Wilson lines

19 Wilson lines Here D x y [+1,b ]= 1 h i Tr V x [+1,b ] V y [+1,b ] N c is the quark dipole scapering S- matrix with V x [b,a ] P exp deno,ng Wilson lines ig 2 a bz 3 dx A + (x + 7 =0,x,x) 5 In the quasi- classical approxima,on D xy is real!

20 Dipole scapering in the GM/MV limit The dipole S- matrix D x y [+1,b ]= 1 h i Tr V x [+1,b ] V y [+1,b ] N c taken in the symmetrized form in the GM/MV approxima,on is (note all real) S xy [+1,b ]=exp apple 1 4 x y 2 Q 2 s S xy = D xy + D yx 2 x + y R (b) 2 b 2R (b) ln 1 x y This corresponds to mul,ple rescaperings as shown here: x x y

21 Quasi- Classical STSA in SIDIS To generate STSA we need spin- dependence and a complex phase. They may come from three sources: Sivers func,ons of the (polarized) nucleons: transversity channel Orbital rota,on due to OAM (just gives real OAM- dependence) Going beyond classical approxima,on in D xy by including extra rescaperings per nucleon (the odderon): this gives a phase, but is A- suppressed. Hence we will drop it. qγ x qγ x k k b x b x OAM Channel Transversity Channel

22 Quasi- Classical STSA in SIDIS When the dust seples one gets Z dp ẑ (J k) f 1T?A + d 2 pdb ( x, k T )=M A 2(2 ) 3 d 2 xd 2 y d2 k 0 (2 ) 2 e i (k ixp (x y) AWunp OAM p +,p,b, x + y f1 N (x, kt 0 ) ẑ S k 0 W symm trans p +,p,b, x + y m N 2 J = L+S = total spin of the nucleus (analogue of proton spin) L = net OAM of the nucleons (analogue of quark and gluon OAM) S = net spin of all nucleons (analogue of net spin of quarks and gluons) S xy = dipole rescapering S- matrix k 0 ) (x y) f?n 1T (x, k 0 T ) S xy [+1,b ] W = Wigner distribu,ons of nucleons, unp = unpolarized (may have <S>=0), trans = transversity, with W (symm OAM ) (p, b) 1 2 W (p, b) ± (p! p)

23 LO TMDs f1 N and f 1T?N are the LO unpolarized quark TMD (in a nucleon) and the nucleon Sivers func,on: A They can be arbitrary non- perturba,ve objects: we can always run them through the quasi- classical grinder. ;) Resul,ng quasi- classical Sivers func,on can be used as ini,al condi,on for evolu,on equa,ons (e.g. CSS for s ~Q 2 or small- x evolu,on if s>>q 2 ). B

24 OAM and Transversity channels The final formula is a sum of the contribu,ons of the OAM and Transversity (Sivers func,on density) channels: Z dp ẑ (J k) f 1T?A + d 2 pdb ( x, k T )=M A 2(2 ) 3 d 2 xd 2 y d2 k 0 (2 ) 2 e i (k ixp (x y) AWunp OAM p +,p,b, x + y f1 N (x, kt 0 ) ẑ S k 0 W symm trans p +,p,b, x + y m N 2 Transversity channel k 0 ) (x y) OAM channel f?n 1T (x, k 0 T ) S xy [+1,b ] The non- perurba,ve input comes though the Wigner func,ons and LO TMDs: the dipole scapering is perturba,ve due to Q s >> Λ QCD.

25 Rigid Rotator Model To get the feel for what the result is like, our main formula can be evaluated using a rigid- rotator nucleus with simple (powers of k T ) models of LO TMDs. The result is (for k T not much larger than Q s ) f 1T?A ( x, k T )= Z d 2 b m N N c s C F p max R apple 4 xp max (b) C 1 e k2 T /Q2 s (b) +2 k 2 T k2 T Q 2 s(b) Ei OAM channel kt 2 Q 2 s(b) + s m N C 2 e k2 T /Q2 s (b) Transversity channel Note a new func,onal form for Sivers func,on due to the OAM channel (not a Gaussian).

26 Rigid Rotator Model The OAM contribu,on in this model (!) calcula,on is shown below (arbitrary units). It changes sign as a func,on of k T. 1.0 A f 1 T k TêQ s

27 Rigid Rotator Model The transversity (Sivers density) contribu,on comes out to be a simple Gaussian (in this calcula,on); units are again arbitrary: 1.0 A f 1 T k TêQ s

28 STSA at Large- k T For k T >> Q s we get f?a apple 4 s m N xc 1 1T ( x, k T ) kt Q s = S J 3 kt 6 apple 4 s m N xc 1 = p max R 3 kt 6 ln k2 T 2 Z OAM channel ln k2 T 2 At high- k T the transversity (Sivers density) channel dominates over the OAM one. Z d 2 bt(b) p max (b) Q 2 s(b)+af?n 1T ( x, k T ) d 2 bt(b) p max (b) Q 2 s(b)+ A 2 s m 2 N C 2 k 4 T ln k2 T 2 Transversity channel However, the OAM channel dominates over a fairly broad range (if p T ~ m N ). k T < Q s p s

29 Quasi- Classical STSA in DY The DY process in the quasi- classical approxima,on looks as follows: k x q γ b x Note that ordering interac,ons along the x - direc,on makes the reversal of the Wilson line direc,on between SIDIS and DY explicit.

30 Origin of Sign Reversal f?a 1T (x, k T ) SIDIS = f?a 1T (x, k T ) DY In the transversity (Sivers density) channel the origin of the sign reversal is simple: the (LO) Sivers func,on of the nucleon changes sign, and mul,ple rescaperings do not affect this. In the OAM channel the reversal ul,mately happens for the simple reasons described in the beginning:

31 Summary We have calculated Sivers func,on in the quasi- classical approxima,on. Valid for a nucleus or a dense proton. The result can serve as an ini,al condi,on for QCD evolu,on (CSS for s ~Q 2 or small- x evolu,on for s>>q 2 ). New func,onal form(s) of ini,al condi,ons could be used in phenomenology and may perhaps help build MCs. In general, given a guess for proton Wigner distribu,on one can construct new ini,al condi,ons for Sivers func,on evolu,on. The asymmetry is generated through two channels: Sivers density (conven,onal, BHS- type) and OAM channel. In the OAM channel the asymmetry is due to a combina,on of OAM and shadowing: could this be a general trend?

32 Summary OAM channel allows for a simple classical- physics interpreta,on of the asymmetry and of the sign reversal between STSA in SIDIS and in DY! Our technique applies to other TMDs: all of them can be (and are being) calculated in the quasi- classical limit.