1. Proton Spin Puzzle: 20 years later Hai-Yang Cheng Academia Sinica, Taipei Deep inelastic scattering Proton spin puzzle Theoretical progress CYCU, June 26, 2007

2. 2 Non-relativistic SU(6) QM ⇒ proton spin comes from constitutent quark’s spin.  U=4/3,  D=-1/3, so that  U+  D=1. However, this model predicts g A =5/3, while g A =1.258 experimentally Relativistic QM ⇒ quark spin + orbital angular momentum =  Q+ L q  ½(0.65+0.35) How to explore the proton’s spin content ? It can be studied in deep inelastic scattering (DIS)

3. 3 Deep Inelastic Scattering DIS process l+p→l+X was first studied by Friedman, Kendall, Taylor (’67-’69) at SLAC Unpolarized structure functions: F 1 (x,Q 2 ), F 2 (x,Q 2 ) F 1 (x) = ½ ∑e i 2 [q i + (x)+q i - (x)] = ½ ∑e i 2 q i (x) x: fraction of proton’s momentum carried by the struck quark, 0<x<1 Structure functions ⇒ (i)3 valence quarks (ii) sea quarks (iii) half of proton’s momentum carried by gluons k’ e(E,p) e’(E’,p’) ** N  q + : q - :

4. 4 Polarized DIS Consider polarized DIS process: l   +p  →l+X and measure asymmetry  q(x,Q 2 )= g 1 p (x)= ½∑e i 2 [q i + (x)-q i - (x)]= ½∑e i 2  q i (x) In general,  q=  q v +  q s. In absence of sea polarization  1 0 g 1 p (x)dx=½∑e q 2  q=½(4/9  u v +1/9  d v ) neutron  decay ⇒  u v -  d v =1.2695±0.0029 hyperon  decay ⇒  u v +  d v =0.585±0.025  u v = 0.93±0.02,  d v = -0.34±0.02,  1 0 g 1 p (x)dx  0.18 first derived by Ellis & Jaffe in 1974

5. 5 SLAC (’76,’83) covers the range 0.1<x<0.7  1 0 g 1 p (x)dx = 0.094±0.016 Extrapolation to the unmeasured x region ⇒  1 0 g 1 p (x)dx=0.17±0.05, consistent with Ellis-Jaffe sum rule EMC (European Muon Collaboration, 87-89), 0.01 =10.7 GeV 2  0.7 0.1 g 1 p (x)dx = 0.090 ± 0.015  0.1 0.01 g 1 p (x)dx = 0.030 ± 0.016 Hence,  1 0 g 1 p (x)dx = 0.126 ± 0.018 Lower than EJ sum rule expectation ⇒ importance of sea polarization

6. 6 Solving the three equations for  q  u-  d = 1.2695±0.0029,  u+  d-2  s = 0.585±0.025 yields  u = 0.77±0.06,  d = -0.49±0.06,  s = -0.15±0.06  ∑≡  u+  d+  s = 0.14±0.18 Two surprises: strange sea polarization is sizable & negative very little of the proton spin is carried by quarks ⇒ Proton Spin Crisis

7. 7 The so-called “proton spin crisis” is not pertinent since the proton helicity content explored in the DIS experiment is, strictly speaking, defined in the infinite momentum frame in terms of QCD current quarks and gluons, whereas the spin structure of the proton in the proton rest frame is referred to the constituent quarks. ….. It is not sensible to compare apple with orange. What trigged by the EMC experiment is the “proton helicity decomposition puzzle” rather than the “proton spin crisis” HYC, hep-ph/0002157  q(  momentum frame)   q QM (rest frame)

8. 8 Experimental Progress  1 =  1 0 g 1 (x)dx x has been pushed down to O(10 -3 - 10 -4 )

9. 9 COMPASS, HERMES  u+  d+  s=0.33±0.06 ⇒  u = 0.84±0.02  d = -0.43±0.02  s = -0.09±0.02

10. 10 Sea quark polarization The result for  s is very different from the inclusive DIS plus SU(3) symmetry analysis! HERMES result from Semi-inclusive DIS Airapetian et al, PRL 92 (2004) 012005

11. 11 Anomalous gluon interpretation Consider QCD corrections to order  s : Efremov, Teryaev; Altarelli, Ross; Carlitz, Collins, Muller (’88) Anomalous gluon contribution (  s /2  )  G arises from photon-gluon scattering. Since  G(Q 2 )  lnQ 2 and  s (Q 2 )  (lnQ 2 ) -1 ⇒  s (Q 2 )  G(Q 2 ) is conserved and doesn’t vanish in Q 2 →  limit  G(Q 2 ) is accumulated with increasing Q 2 from (a) from (b) Why is this QCD correction so special ?

12. 12 QCD corrections imply that If  G is positive and large enough, one can have  s  0 and  u+  d  0.60 ⇒ proton spin problem is resolved provided that  G  (2  /  s )(0.09)  1.7 ⇒ L q + G also increases with lnQ 2 with fine tuning This anomalous gluon interpretation became very popular after 1988 Historical remarks: 1.Photon-gluon box diagram was first computed by Kodaira (’80) 2.In 1982 Chi-Sing Lam & Bing-An Li obtained anomalous gluon contribution to  1 p and identify  G with 3.The same box diagram was also computed by Ratcliffe (’83) using dimensional regularization

13. 13 Operator Product Expansion moments of structure function=  1 0 x n-1 F(x)dx = ∑ C n (q) =short-distance  long-distance No twist-2, spin-1 gauge-invariant local gluonic operator for first moment OPE ⇒ Gluons do not contribute to  1 p ! One needs sea quark polarization to account for experiment (Jaffe, Manohar ’89) How to achieve  s  -0.09 ? Sea polarization (for massless quarks) cannot be induced perturbatively from hard gluons (helicity conservation ⇒  s=0 for massless quarks) J  5 has anomalous dimension at 2-loop (Kodaira ’79) ⇒  q is Q 2 dependent, against intuition

14. 14 A hot debate between anomalous gluon & sea quark interpretations before 1995 ! anomalous gluon sea quark Efremov, Teryaev Altarelli, Ross Carlitz, Collins, Muller Soffer, Perparata Strirling Roberts Ball, Forte Gluck, Reya, Vogelsang Lampe Mankiewicz Gehrmann …. Anselmino, Efremov, Leader [Phys. Rep, 261, 1 (1995)] Jaffe, Manohar Bodwin, Qiu Ellis, Karlinear Bass, Thomas …

15. 15 Factorization scheme dependence It was realized by Bodwin, Qiu (’90) and by Manohar (’90) that hard gluonic contribution to  1 p is a matter of convention used for defining  q Consider polarized photon-gluon cross section 1.Its hard part contributes to  C G and soft part to  q s. This decomposition depends on the choice of factorization scheme 2.It has an axial QCD anomaly that breaks down chiral symmetry fact. scheme dependent Int. J. Mod. Phys. A11, 5109 (1996)

16. 16 Photon-gluon box diagram is u.v. finite.  C G is indep of choice of IR & collinear regulators, but depends on u.v. regulator of  q/G (x)=  q G (x) Polarized triangle diagram has axial anomaly ⇒ If u.v. cutoff respects gauge symmetry but breaks chiral symmetry ⇒  q G  0 CI anomaly GI Axial anomaly resides at k  2 →   q G convolutes with  G to become  q s HYC(’95) Muller, Teryaev (’97)

17. 17 Two extreme schemes of interest (HYC, ’95) gauge-invariant (GI) scheme (or MS scheme) -- Axial anomaly is at soft part, i.e.  q G, which is non-vanishing due to chiral symmetry breaking and  1 0  C G (x)=0 (but  G  0) -- Sea polarization is mainly induced by gluons via axial anomaly chiral-invariant (CI) scheme (or “jet”, “parton-model”, “k T cut-off’, “Adler-Bardeen” scheme) Axial anomaly is at hard part, i.e.  C G, while  q s induced by hard gluon vanishes  Hard gluonic contribution to  g 1 p is matter of factorization convention used for defining  q  It is necessary to specify the factorization scheme for data analysis

18. 18 In retrospect, the dispute among the anomalous gluon and sea-quark explanations…before 1996 is considerably unfortunate and annoying since the fact that g 1 p (x) is independent of the definition of the quark spin density and hence the choice of the factorization scheme due to the axial- anomaly ambiguity is presumably well known to all the practitioners in the field, especially to those QCD experts working in the area. hep-ph/0002157 My conclusion:

19. 19 How to probe gluon polarization ? DIS via scaling violation in g 1 (x,Q 2 ) photon or jet or heavy quark production in polarized pp collider, lepton- proton collider or lepton-proton fixed target RHIC (at BNL): via direct high-p T prompt   production 2 jet production HERMES (at DESY): via open charm production COMPASS (at CERN): via open charm production

20. 20 Q-evolution in inclusive spin structure function g 1 (x,Q 2 ) NLO splitting functions  P ij are available in ’95 van Neerven, Mertig, Zijlstra ⇒ A complete & consistent NLO analysis of g 1 data is possible Most analyses are done in MS scheme (GI)  u v (x),  d v (x) are fairly constrained Sea distribution is poorly constrained  G(x) is almost completely undetermined

21. 21 Two leading-hadron production in semi- inclusive DIS Large  G  2-3 ruled out by data But  G  ½ (gluons carries 100% of nucleon spin) still possible

22. 22 RHIC:The First Polarised pp Collider

23. 23  production in polarized pp collision at RHIC Jet production in polarized pp collision at RHIC √s=200 GeV

24. 24 Calculating  G &  G(x) in models Jaffe (’95) gave a pioneering estimate of  G in NR & bag models and obtained a negative  G Barone et al. (’98) pointed out additional one-body contribution that partially cancels two-body one ⇒ positive  G Ji et al. (’06) computed  G(x) in QM and obtained  G  0.34

25. 25 Lattice QCD Can lattice QCD shed some light on the protn spin content ? Sea polarization from disconnected insertion ⇒  u s =  d s =  s = -0.12±0.01

26. 26 Orbital angular momentum Orbital angular momentum can be inferred from lattice result for J q =0.30±0.07 (Mathur et al. 2000) ⇒ L q  0.10±0.06 for  ∑  0.25 At Q 2 → , Ji, Tang & Hoodbhoy found (’96) Analogous to the nucleon’s momentum partition: half of the proton’s momentum is carried by gluons for n f =6

27. 27 Conclusions  ∑ & L q are factorization scheme dependent, but not J q DIS data ⇒  ∑ GI  0.30,  s GI  -0.08  G(x) &  q s (x) are weakly constrained SIDIS & RHIC data imply a small  G At Q 2 → , J q =0.26, J G =0.24 Lattice QCD ⇒ J q =0.30 ± 0.07 What do we learn in past 20 years about the proton helicity decomposition ?