- Email: [email protected]
Spin physics at RHIC
NUCLEAR PHYSICS A ELSEVIER
Nuclear Physics A638 (1998) 231c 238c
Spin Physics at RHIC Xiangdong Ji a aDepartment of Physics, University of Maryland. College Park, Maryland 20742, USA I discuss physics motivations for the RHIC spin program and highlight some of the key experiments which may lead to a deeper understanding of the internal structure of the nucleon. 1. I n t r o d u c t i o n The topic of my presentation is spin physics at RHIC. After all those talks on heavy-ion collisions, multiplicity distributions, flows and hydrodynamics, I hope you would enjoy this break. My talk consists of three parts. In the first part, I discuss the motivations to study spin physics at RHIC. I understand that the Holy Grail in heavy-ion collisions is the quark-gluon plasma. The goal of the field I am going to describe is to understand the internal structure of the nucleon. In the second part, I talk about the physics issues that are relevant to this goal. Finally, I present some highlights of the RHIC spin experiments. I hope that I can convince you in this short talk that it is worthwhile to make these studies at RHIC. 2. M o t i v a t i o n s The topic of interest in this talk is the quark and gluon structure of the nucleon. The nucleon structure is an important, old, hard, and unsolved problem in the physics of strong interactions. Its importance is quite obvious and I need not elaborate here. It is old because we have been thinking seriously about the problem since Gell-Mann and Zweig proposed the quark model in 1964 [1]. One can get an impressionistic feeling on the degree of difficulty by counting the number of papers appeared on the subject in the last thirty-plus years. A rough estimate would yield an order of 104. Despite these efforts, the nucleon structure is still an unsolved problem. One sign of this is that so far we have no reliable way yet to calculate some of the most fundamental properties of the nucleon, e.g., the magnetic moment. However, over the years we have learned some important lessons. Since I believe these lessons are useful also for people who are looking for the quark-gluon plasma, I would like to share two of the most important ones with you. First, tile nucleon is a far more complicated object than just three valence quarks. From the data obtained with highenergy probes and from the fundamental QCD lagrangian, we know that the nucleon has a rich gluon content and a significant number of quark-antiquark pairs. Furthermore, these constituents interact strongly with the QCD vacuum. Second, one must be careflll when 0375-9474/98/$19 ~O 1998 Elsevier Science B.M All rights reserved. PII S0375-9474(98)00416-3
232c
X. Ji/Nuclear Physics A638 (1998) 231c-238c
comparing model results with data. Models are usually constructed using the so-called "effective degrees of freedom", which have no obvious relation with quarks and gluons in the fundamental lagrangian. On the other hand, experimental data reflect directly the QCD degrees of freedom. For instance, in deep inelastic scattering or elastic nucleon scattering, the electromagnetic probe couples to the current quarks. We would face many "puzzles" if we did not carefully distinguish the difference between them. Indeed, the so-call "spin crisis" is just one of these man-made puzzles [2]. What are the future opportunities in unraveling the structure of the nucleon? With the help of emerging effective algorithms and fast computers, full lattice QCD calculations with respectable precisions may be in sight. Let me mention that there is a strong lattice QCD group here in Tsukuba and it has done many interesting calculations. On the other hand, novel and advanced experimental technologies continue to provide new and high precision probes. We can divide, somewhat arbitrarily, the present generation of experiments into low and high energy categories. In the low-energy frontier, the virtual Z-boson form factor has been measured for the first time at MIT Bates Lab by a team of nuclear experimentalists led by McKeown and Beck [3]. The nucleon electric and magnetic polarizabilities have also been studied actively in recent years [4]. However, what I am going to talk about extensively here is the high-energy frontier. This frontier started with deep inelastic scattering. The key quantities of interest include the light-cone wave functions, the parton distributions, spin-dependent structure functions, quark and gluon correlations, etc. As a theorist, I can't help but mention that the theoretical basis for various high-energy probes of the nucleon structure is factorization theorems first developed by, among others, Sterman, Libby, Politzer, and Collins [5]. These theorems, in a nutshell, provide a QCD justification of Feynman's parton model. A general statement of factorization theorems goes like this: In high-energy processes involving hadrons, it may happen that the cross sections are separable into two factors. One of these is called the hard part, in which only large momenta enter and can be calculated in perturbative QCD. The other is the soft part, in which only nonperturbative hadron structures enter. Schematically, we can write,
Cr = / d x a " " dxncrparton(Xl,..., Xn)F(Xl,''" , Xn),
(1)
where aparton is the parton scattering cross section which can be viewed as a generalized probe. F ( x l , ' " , x n ) characterizes the structural information of bound states. Clearly, every hard scattering mechanism provides a probe of the hadron structure. 3. P h y s i c s I s s u e s What are the interesting physical questions that one can address in high-energy scattering? Let us first consider some high-energy observables of the nucleon. Consider a beam of nucleons traveling with momentum p ~ cxD in the lab frame. As far as the leading-order high-energy scattering is concerned, the beam is equivalent to a beam of quarks and gluons with a distribution of momentum. Let us consider the quark beam first. Because of the flavor and charge quantum number, the beam contains u, d, s, ~, d, and $ components if one considers light flavors only. Each
X Ji/Nuclear PhysicsA638 (1998) 231c-238c
233c
of these components, a, can be described by a density matrix,
(q~(x)+Aqa(X)SL 5qa(x)ST) 6qa(x)ST q~(x) -- Aq~(x)SL
'
(2)
where x is the fraction of the nucleon momentum carried by quarks, and SL and ST represent degrees of longitudinal and transverse polarizations of the nucleon, respectively. qa(x) are the familiar unpolarized quark distributions. They have been determined with some accuracy in high-energy experiments [6]. Aq~(x) are the quark helicity distributions, which measure the number of quarks polarized along the helicity of the nucleon minus tile number polarized opposite the helicity of the nucleon. In other words,
Aq~(x) = q+(x) -- q:(x) .
(3)
Some combinations of Aqa(x) have been extracted from polarized deep inelastic scattering data [7]. @a(x) are the quark transversity distributions [8], which measure the number of quarks polarized along the transverse polarization of the nucleon minus the number polarized opposite the transverse polarization of the nucleon,
@o(x) = q~(x) - q2(x) .
(4)
At present, there are no data on the distributions. Now turn to the gluon beam, which can also be described by a density matrix,
( G(x) + AG(x)SL 0
0 a(x) -/\a(X)SL
) "
(5)
Here G(x) is the unpolarized gluon distribution, and has been extracted mainly from deep-inelastic scattering, direct photon production and heavy flavor production in hadronhadron collisions. AG(x) is the gluon helicity distribution and at present is largely unknown [9]. There are no linearly polarized gluons in the nucleon because the angular momentum conservation along the axis of motion forbids such a distribution. Let me remark that the above-described parton distributions are fundamental observables of the nucleon. They encode rich and important structural information and pose serious challenges to any nonperturbative QCD calculation. In recent years, the spin structure of the nucleon has attracted much interest in the nuclear and particle community [2]. To set up some basic notions about the spin structure, consider a nucleon moving in the z direction and polarized in the spin state Jz = 1/2. The total quark spin contribution to J~ can be derived from the quark helicity distributions by summing over quark flavors and integrating over the longitudinal momentum. The resulting quantity is conventionally called AN
A~ = ~_,
/01A%(x)dx.
(6)
a
Similarly the total gluon spin contribution (:an be obtained from the gluon helicity distribution AG(x), ~xa(x) =
dS~a(x)
(7)
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X. Ji/Nuclear Physics A638 (1998) 231c-238c
There are two additional sources of the nucleon spin: the quark and gluon orbital angular momenta, Lq and Lg. Together they account for the intrinsic angular momentum of the nucleon [10] 1
1
= -AS2 + Lq + A G + Lg.
(8)
In 1988, the European Muon Collaboration (EMC) extracted AS from, among other experimental quantities, the polarized deep inelastic scattering data taken at CERN and found that A~; is around 0.12 4-0.17 [2]. This is much smaller than the naive quark model prediction that A ~ = 1. This seemingly inconsistent result was dubbed by some as the "spin crisis." The EMC result has since been confirmed by many other experiments: SMC at CERN, E142, E143, E154, and E155 at SLAC, and HERMES at HERA. Given the fact that the quark spin is not the only source of QCD angular momentum, it is not surprising that it contributes only a part of the 1/2. Still it is an interesting question why A~ is so small. One line of thought is that the sea quarks are strongly polarized in the nucleon and cancel a large part of the valence quark polarization. According to the above definition, AS = Au + Ad + As + Aft + Ad + A~.
(9)
Combining the deep inelastic scattering data with the hyperon /3-decay data, there is evidence that As + A.~ .~ --0.1. Therefore it would be interesting to make a complete flavor and charge separation in AS. There are also speculations of a large gluon polarization in the nucleon, although, I think, the motivations are less well founded. A large gluon polarization could only deepen the "spin crisis", although not ruled out by QCD. Hence the issue is interesting by itself. At present, there are some inconclusive evidences that AG is about 1 to 2 units of angular momentum [7]. But I think the number could go down in the future. Here I would like to point out that AG is a strongly scale-dependent quantity. Indeed, it can be shown that the gluon polarization increases logarithmically as the probing scale increases; aG(Q)
~ in Q2 -~ ~ .
(10)
One can understand the above result in a simple way [11]. Consider a gluon of helicity +1 with an approximate size of 1/#. If one is to probe inside the gluon at a larger momentum scale, the glnon may be seen as composed of two daughter gluons or a quark-antiquark pair. The two daughter gluons can have helicities (+1, +1) or ( + 1 , - 1 ) . The quarkantiquark pair can have helicities 1/2 and - 1 / 2 for the quark and antiquark, respectively, or vice versa. Only in the first case (both gluons with + l ) , does the total gluon helicity increase by +1. In other cases, it decreases by - 1 . One can use perturbative QCD to calculate the probabilities for various possibilites. The net helicity change is proportional to ~ 11
2 h i _ rio,
3
(11)
which is the coefficient of the leading order QCD beta flmction. It is positive as long as the theory is asymptotically free.
X. Ji/Nuclear Physics A638 (1998) 231~238c
235c
I would like to make some comments about similarities and differences between quark helicity and transversity. As is well known, the total quark helicity is related to the matrix element of the axial current,
(PS I y~ ¢~T"TsOolPS) = A E (2S").
(12)
a
In the naive nonrelativistic quark model, A E = 1. In relativistic quark models like the MIT bag, A E is reduced to about 0.65 due to the p wave contribution from the lower component of the Dirac spinor. In the lattice QCD calculations, A E is fi, rther reduced to about 0.2 ~ 0.3 [12], One can define the total quark transversity
6E -- ~ ~o' dx(Sq~(x) - 5Ft~(x))dx ,
(L:~)
which is related to the matrix element of the tensor current,
(PSI ~ Cao-""75¢olPS) = 5E; (SUP" - 5 " P " ) .
(14)
a
In nonrelativistic quark models, 5E is the same as AE. In relativistic quark models, 6E is reduced, but not as much as it is for AE. For instance, in the MIT bag, 5E is 0.82 [13]. The lattice result by Hatsuda et. al. shows that ~ l a t t = 0.61 [14]. Therefore, it seems that the singlet tensor charge is not as small as the singlet axial charge. This perhaps is not so surprising because the tensor charge is charge-conjugation odd and chiral odd; as such, there is no sea quark contribution and gluon mixing.
4. Highlights of RHIC Spin Experiments Thanks to the novel experimental technology, we expect to have polarized proton beams in tile RHIC rings in the near future. The beams will be 70% polarized and have a luminosity around 2 × 1032. The center-of-mass collision energy v ~ ranges fi-om 50 to 500 GeV. The accelerator can run in many different polarization modes: two beams with longitudinal-longitudinal, transverse-transverse, or longitudinal-transverse polarizations, or one of the beams with transverse or longitudinal polarization. There is a long list of spin experiments that one can do at RHIC to learn about the internal structure of the nucleon. One can measure the gluon helicity distribution AG(x) in direct photon production, jet and 7r° production, and open heavy-flavor production. One can determine the polarized quark and antiquark distributions, Aq(x) and Aq(x), in Drell-Yan collisions and W :e boson production. One can study the quark and antiqnark transversity distributions, 5q(x) and 5O(x), in Drell-Yan collisions and Z-boson production. One can learn about the quark-gluon correlation effects in single-spin processes and transverse-longitudinally polarized collisions. In tile remainder of the talk, l will describe some of these experiments in detail. Theoretically, the most clean process to measure the gluon polarization is direct photon production. Microscopically, there are two parton subprocesses contributing to the production: gluon-quark Compton scattering and quark-antiquark annihilation. The second process does not depend on the gluon polarization. The study by Saito et al. shows that
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X. Ji/Nuclear Physics A638 (1998) 231c-238c
the latter contributes about 10% of the direct photon events at x/~ = 500 GeV [15]. Saito et al. have also calculated the asymmetry as a function of transverse momentum of the photons. With an integrated luminosity of 374 pb -1 and the PHENIX detector, one can measure the direct photon asymmetry at the level of 1 percent at PT = 12 GeV, 3 percent at PT -= 16 GeV, and 5 percent at PT = 22 GeV. Parity-violating W-boson production offers a novel way to measure quark and antiquark helicity distributions [16]. The single-spin parity-violating asymmetries reads A w+
A~(xi)d(x2)
=
Ag-
- Ad(xl)u(x2)
U(Xl)d(x2) -~ d(Xl)U(X2)
'
Ad(xl)~(x2) - ZX~(xl)d(x~) =
(15)
d(x,)ft(x2) + U(Xl)d(x2)
If one is focused in the region of Xl >> x2, then the W + asymmetry reduces to AL W+
Au(x0 -~
u(~i)
(16)
'
which is directly sensitive to the up-quark polarization. On the other hand, in the region of x2 >> Xl, the asymmetry reduces to AW+
Ad(x2) -~
d(~)
(17) '
which determines the anti-down quark polarization density. In a similar fashion, the W asymmetries in different kinematic limits can be used to extract the polarized anti-up and down quark densities. The W-boson detection can be made through its leptonic decay into the # lepton. In the PHENIX detector, the muons with transverse momentum larger than 20 GeV are found mostly from the W - decay. The strongest background comes from charm production and decay. With respective 14% and 4% acceptance rates for W - and W +, there are about 5100 W - events and 5600 W + events in 800 pb -1 of data [15]. With these, the muon asymmetry near x = 0.1 (mainly sensitive to antiquark polarization) and x = 0.3 (mainly sensitive to quark polarization) can be measured with an accuracy better than 5%. 5. C o n c l u s i o n The conclusion of my talk is simple. The RHIC spin project is quite unique in the world. It is economic, it contains rich physics, it has a high rate of success, and there is no competition in sight .... Let's just do it. I thank the organizers of Quark Matter '97 for the invitation; N. Saito for his transparencies on Monte Carlo simulations; and RIKEN for partial financial support. This work is supported in part by funds provided by the U. S. Department of Energy (D.O.E.) under cooperative agreement DOE-FG02-93ER-40762.
X. Ji/Nuclear Physics A638 (1998) 231c-238c
237c,
REFERENCES
1. M. Gell-Mann, Phys. Left. 8 (1964) 241; G. Zweig, CERN preprint Th. 401 and 412, 1964. 2. J. Ashman et al., Nucl. Phys. B238 (1989) 1. 3. B. Mueller et al, Phys. Rev. Lett. 78 (1997) 3824. 4. See for instance, B. E. MacGibbon, G. Garino, M. A. Lucas, A. M. Nathan G. Feldman, B. Dolbilkin, Phys. Rev. C 52 (1995) 2097. 5. For a review and references, see J. Collins, D. E. Soper, and G. Sterman, in Perturbative QCD, ed. by A. H. Muller (World Scientific, Singapore, 1989). 6. See for instance, H. L. Lai, J. Huston, S. Kuhlmann, F. Olness, J. Owens, D. Soper, W. K. Tung, H. Weerts, Phys. Rev. D 55 (1997) 1280. 7. See for instance, G. Altarelli, R. D. Ball, S. Forte, and G. Ridolfi, Nucl. Phys. B496 (1997) 337. 8. J.P. Ralston and D. E. Soper, Nucl. Phys. B152 (1972) 109; X. Artru and M. Mekhfi, Z. Phys. C 45 (1990) 669; R. L. Jafte and X. Ji, Phys. Rev. Lett. 71 (1993) 2547. 9. V. Hughes et al., in Proceedings of the 12th International Symposium on High-Energy Spin Physics, Amsterdam, Holland, 1996 (World Scientific, Singapore, 1997). 10. X. Ji, Phys. Rev. Lett. 78 (1997) 610; R. L. Jaffe and A. Manohar, Nucl. Phys. B337 (1990) 509. 11. X. Ji, J. Tang, and P. Hoodbhoy, Phys. Rev. Lett. 76 (1996) 740. 12. M. Fukugita, Y. Kuramashi, M. Okawa, and A. Ukawa, Phys. Rev. Lett. 75 (1995) 2092; S. J. Dong, J.-F. Lagae, K. F. Liu, Phys. Rev. Lett. 75 (1995) 2096; M. GSckeler et. al., Phys. Rev. D 53 (1996) 2317. 13. It. He and X. Ji, Phys. Rev. D 52 (1995) 2960. 14. S. Aoki, M. Doui, T. Hatsuda, and Y. Kuramashi, Phys. ReV. D 56 (1997) 433. 15. N. Saito, private communication. 16. J. Softer and C. Bourrely, Phys. Lett. B314 (1993) 132.
Nuclear Physics A638 (1998) 231c 238c
Spin Physics at RHIC Xiangdong Ji a aDepartment of Physics, University of Maryland. College Park, Maryland 20742, USA I discuss physics motivations for the RHIC spin program and highlight some of the key experiments which may lead to a deeper understanding of the internal structure of the nucleon. 1. I n t r o d u c t i o n The topic of my presentation is spin physics at RHIC. After all those talks on heavy-ion collisions, multiplicity distributions, flows and hydrodynamics, I hope you would enjoy this break. My talk consists of three parts. In the first part, I discuss the motivations to study spin physics at RHIC. I understand that the Holy Grail in heavy-ion collisions is the quark-gluon plasma. The goal of the field I am going to describe is to understand the internal structure of the nucleon. In the second part, I talk about the physics issues that are relevant to this goal. Finally, I present some highlights of the RHIC spin experiments. I hope that I can convince you in this short talk that it is worthwhile to make these studies at RHIC. 2. M o t i v a t i o n s The topic of interest in this talk is the quark and gluon structure of the nucleon. The nucleon structure is an important, old, hard, and unsolved problem in the physics of strong interactions. Its importance is quite obvious and I need not elaborate here. It is old because we have been thinking seriously about the problem since Gell-Mann and Zweig proposed the quark model in 1964 [1]. One can get an impressionistic feeling on the degree of difficulty by counting the number of papers appeared on the subject in the last thirty-plus years. A rough estimate would yield an order of 104. Despite these efforts, the nucleon structure is still an unsolved problem. One sign of this is that so far we have no reliable way yet to calculate some of the most fundamental properties of the nucleon, e.g., the magnetic moment. However, over the years we have learned some important lessons. Since I believe these lessons are useful also for people who are looking for the quark-gluon plasma, I would like to share two of the most important ones with you. First, tile nucleon is a far more complicated object than just three valence quarks. From the data obtained with highenergy probes and from the fundamental QCD lagrangian, we know that the nucleon has a rich gluon content and a significant number of quark-antiquark pairs. Furthermore, these constituents interact strongly with the QCD vacuum. Second, one must be careflll when 0375-9474/98/$19 ~O 1998 Elsevier Science B.M All rights reserved. PII S0375-9474(98)00416-3
232c
X. Ji/Nuclear Physics A638 (1998) 231c-238c
comparing model results with data. Models are usually constructed using the so-called "effective degrees of freedom", which have no obvious relation with quarks and gluons in the fundamental lagrangian. On the other hand, experimental data reflect directly the QCD degrees of freedom. For instance, in deep inelastic scattering or elastic nucleon scattering, the electromagnetic probe couples to the current quarks. We would face many "puzzles" if we did not carefully distinguish the difference between them. Indeed, the so-call "spin crisis" is just one of these man-made puzzles [2]. What are the future opportunities in unraveling the structure of the nucleon? With the help of emerging effective algorithms and fast computers, full lattice QCD calculations with respectable precisions may be in sight. Let me mention that there is a strong lattice QCD group here in Tsukuba and it has done many interesting calculations. On the other hand, novel and advanced experimental technologies continue to provide new and high precision probes. We can divide, somewhat arbitrarily, the present generation of experiments into low and high energy categories. In the low-energy frontier, the virtual Z-boson form factor has been measured for the first time at MIT Bates Lab by a team of nuclear experimentalists led by McKeown and Beck [3]. The nucleon electric and magnetic polarizabilities have also been studied actively in recent years [4]. However, what I am going to talk about extensively here is the high-energy frontier. This frontier started with deep inelastic scattering. The key quantities of interest include the light-cone wave functions, the parton distributions, spin-dependent structure functions, quark and gluon correlations, etc. As a theorist, I can't help but mention that the theoretical basis for various high-energy probes of the nucleon structure is factorization theorems first developed by, among others, Sterman, Libby, Politzer, and Collins [5]. These theorems, in a nutshell, provide a QCD justification of Feynman's parton model. A general statement of factorization theorems goes like this: In high-energy processes involving hadrons, it may happen that the cross sections are separable into two factors. One of these is called the hard part, in which only large momenta enter and can be calculated in perturbative QCD. The other is the soft part, in which only nonperturbative hadron structures enter. Schematically, we can write,
Cr = / d x a " " dxncrparton(Xl,..., Xn)F(Xl,''" , Xn),
(1)
where aparton is the parton scattering cross section which can be viewed as a generalized probe. F ( x l , ' " , x n ) characterizes the structural information of bound states. Clearly, every hard scattering mechanism provides a probe of the hadron structure. 3. P h y s i c s I s s u e s What are the interesting physical questions that one can address in high-energy scattering? Let us first consider some high-energy observables of the nucleon. Consider a beam of nucleons traveling with momentum p ~ cxD in the lab frame. As far as the leading-order high-energy scattering is concerned, the beam is equivalent to a beam of quarks and gluons with a distribution of momentum. Let us consider the quark beam first. Because of the flavor and charge quantum number, the beam contains u, d, s, ~, d, and $ components if one considers light flavors only. Each
X Ji/Nuclear PhysicsA638 (1998) 231c-238c
233c
of these components, a, can be described by a density matrix,
(q~(x)+Aqa(X)SL 5qa(x)ST) 6qa(x)ST q~(x) -- Aq~(x)SL
'
(2)
where x is the fraction of the nucleon momentum carried by quarks, and SL and ST represent degrees of longitudinal and transverse polarizations of the nucleon, respectively. qa(x) are the familiar unpolarized quark distributions. They have been determined with some accuracy in high-energy experiments [6]. Aq~(x) are the quark helicity distributions, which measure the number of quarks polarized along the helicity of the nucleon minus tile number polarized opposite the helicity of the nucleon. In other words,
Aq~(x) = q+(x) -- q:(x) .
(3)
Some combinations of Aqa(x) have been extracted from polarized deep inelastic scattering data [7]. @a(x) are the quark transversity distributions [8], which measure the number of quarks polarized along the transverse polarization of the nucleon minus the number polarized opposite the transverse polarization of the nucleon,
@o(x) = q~(x) - q2(x) .
(4)
At present, there are no data on the distributions. Now turn to the gluon beam, which can also be described by a density matrix,
( G(x) + AG(x)SL 0
0 a(x) -/\a(X)SL
) "
(5)
Here G(x) is the unpolarized gluon distribution, and has been extracted mainly from deep-inelastic scattering, direct photon production and heavy flavor production in hadronhadron collisions. AG(x) is the gluon helicity distribution and at present is largely unknown [9]. There are no linearly polarized gluons in the nucleon because the angular momentum conservation along the axis of motion forbids such a distribution. Let me remark that the above-described parton distributions are fundamental observables of the nucleon. They encode rich and important structural information and pose serious challenges to any nonperturbative QCD calculation. In recent years, the spin structure of the nucleon has attracted much interest in the nuclear and particle community [2]. To set up some basic notions about the spin structure, consider a nucleon moving in the z direction and polarized in the spin state Jz = 1/2. The total quark spin contribution to J~ can be derived from the quark helicity distributions by summing over quark flavors and integrating over the longitudinal momentum. The resulting quantity is conventionally called AN
A~ = ~_,
/01A%(x)dx.
(6)
a
Similarly the total gluon spin contribution (:an be obtained from the gluon helicity distribution AG(x), ~xa(x) =
dS~a(x)
(7)
234c
X. Ji/Nuclear Physics A638 (1998) 231c-238c
There are two additional sources of the nucleon spin: the quark and gluon orbital angular momenta, Lq and Lg. Together they account for the intrinsic angular momentum of the nucleon [10] 1
1
= -AS2 + Lq + A G + Lg.
(8)
In 1988, the European Muon Collaboration (EMC) extracted AS from, among other experimental quantities, the polarized deep inelastic scattering data taken at CERN and found that A~; is around 0.12 4-0.17 [2]. This is much smaller than the naive quark model prediction that A ~ = 1. This seemingly inconsistent result was dubbed by some as the "spin crisis." The EMC result has since been confirmed by many other experiments: SMC at CERN, E142, E143, E154, and E155 at SLAC, and HERMES at HERA. Given the fact that the quark spin is not the only source of QCD angular momentum, it is not surprising that it contributes only a part of the 1/2. Still it is an interesting question why A~ is so small. One line of thought is that the sea quarks are strongly polarized in the nucleon and cancel a large part of the valence quark polarization. According to the above definition, AS = Au + Ad + As + Aft + Ad + A~.
(9)
Combining the deep inelastic scattering data with the hyperon /3-decay data, there is evidence that As + A.~ .~ --0.1. Therefore it would be interesting to make a complete flavor and charge separation in AS. There are also speculations of a large gluon polarization in the nucleon, although, I think, the motivations are less well founded. A large gluon polarization could only deepen the "spin crisis", although not ruled out by QCD. Hence the issue is interesting by itself. At present, there are some inconclusive evidences that AG is about 1 to 2 units of angular momentum [7]. But I think the number could go down in the future. Here I would like to point out that AG is a strongly scale-dependent quantity. Indeed, it can be shown that the gluon polarization increases logarithmically as the probing scale increases; aG(Q)
~ in Q2 -~ ~ .
(10)
One can understand the above result in a simple way [11]. Consider a gluon of helicity +1 with an approximate size of 1/#. If one is to probe inside the gluon at a larger momentum scale, the glnon may be seen as composed of two daughter gluons or a quark-antiquark pair. The two daughter gluons can have helicities (+1, +1) or ( + 1 , - 1 ) . The quarkantiquark pair can have helicities 1/2 and - 1 / 2 for the quark and antiquark, respectively, or vice versa. Only in the first case (both gluons with + l ) , does the total gluon helicity increase by +1. In other cases, it decreases by - 1 . One can use perturbative QCD to calculate the probabilities for various possibilites. The net helicity change is proportional to ~ 11
2 h i _ rio,
3
(11)
which is the coefficient of the leading order QCD beta flmction. It is positive as long as the theory is asymptotically free.
X. Ji/Nuclear Physics A638 (1998) 231~238c
235c
I would like to make some comments about similarities and differences between quark helicity and transversity. As is well known, the total quark helicity is related to the matrix element of the axial current,
(PS I y~ ¢~T"TsOolPS) = A E (2S").
(12)
a
In the naive nonrelativistic quark model, A E = 1. In relativistic quark models like the MIT bag, A E is reduced to about 0.65 due to the p wave contribution from the lower component of the Dirac spinor. In the lattice QCD calculations, A E is fi, rther reduced to about 0.2 ~ 0.3 [12], One can define the total quark transversity
6E -- ~ ~o' dx(Sq~(x) - 5Ft~(x))dx ,
(L:~)
which is related to the matrix element of the tensor current,
(PSI ~ Cao-""75¢olPS) = 5E; (SUP" - 5 " P " ) .
(14)
a
In nonrelativistic quark models, 5E is the same as AE. In relativistic quark models, 6E is reduced, but not as much as it is for AE. For instance, in the MIT bag, 5E is 0.82 [13]. The lattice result by Hatsuda et. al. shows that ~ l a t t = 0.61 [14]. Therefore, it seems that the singlet tensor charge is not as small as the singlet axial charge. This perhaps is not so surprising because the tensor charge is charge-conjugation odd and chiral odd; as such, there is no sea quark contribution and gluon mixing.
4. Highlights of RHIC Spin Experiments Thanks to the novel experimental technology, we expect to have polarized proton beams in tile RHIC rings in the near future. The beams will be 70% polarized and have a luminosity around 2 × 1032. The center-of-mass collision energy v ~ ranges fi-om 50 to 500 GeV. The accelerator can run in many different polarization modes: two beams with longitudinal-longitudinal, transverse-transverse, or longitudinal-transverse polarizations, or one of the beams with transverse or longitudinal polarization. There is a long list of spin experiments that one can do at RHIC to learn about the internal structure of the nucleon. One can measure the gluon helicity distribution AG(x) in direct photon production, jet and 7r° production, and open heavy-flavor production. One can determine the polarized quark and antiquark distributions, Aq(x) and Aq(x), in Drell-Yan collisions and W :e boson production. One can study the quark and antiqnark transversity distributions, 5q(x) and 5O(x), in Drell-Yan collisions and Z-boson production. One can learn about the quark-gluon correlation effects in single-spin processes and transverse-longitudinally polarized collisions. In tile remainder of the talk, l will describe some of these experiments in detail. Theoretically, the most clean process to measure the gluon polarization is direct photon production. Microscopically, there are two parton subprocesses contributing to the production: gluon-quark Compton scattering and quark-antiquark annihilation. The second process does not depend on the gluon polarization. The study by Saito et al. shows that
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the latter contributes about 10% of the direct photon events at x/~ = 500 GeV [15]. Saito et al. have also calculated the asymmetry as a function of transverse momentum of the photons. With an integrated luminosity of 374 pb -1 and the PHENIX detector, one can measure the direct photon asymmetry at the level of 1 percent at PT = 12 GeV, 3 percent at PT -= 16 GeV, and 5 percent at PT = 22 GeV. Parity-violating W-boson production offers a novel way to measure quark and antiquark helicity distributions [16]. The single-spin parity-violating asymmetries reads A w+
A~(xi)d(x2)
=
Ag-
- Ad(xl)u(x2)
U(Xl)d(x2) -~ d(Xl)U(X2)
'
Ad(xl)~(x2) - ZX~(xl)d(x~) =
(15)
d(x,)ft(x2) + U(Xl)d(x2)
If one is focused in the region of Xl >> x2, then the W + asymmetry reduces to AL W+
Au(x0 -~
u(~i)
(16)
'
which is directly sensitive to the up-quark polarization. On the other hand, in the region of x2 >> Xl, the asymmetry reduces to AW+
Ad(x2) -~
d(~)
(17) '
which determines the anti-down quark polarization density. In a similar fashion, the W asymmetries in different kinematic limits can be used to extract the polarized anti-up and down quark densities. The W-boson detection can be made through its leptonic decay into the # lepton. In the PHENIX detector, the muons with transverse momentum larger than 20 GeV are found mostly from the W - decay. The strongest background comes from charm production and decay. With respective 14% and 4% acceptance rates for W - and W +, there are about 5100 W - events and 5600 W + events in 800 pb -1 of data [15]. With these, the muon asymmetry near x = 0.1 (mainly sensitive to antiquark polarization) and x = 0.3 (mainly sensitive to quark polarization) can be measured with an accuracy better than 5%. 5. C o n c l u s i o n The conclusion of my talk is simple. The RHIC spin project is quite unique in the world. It is economic, it contains rich physics, it has a high rate of success, and there is no competition in sight .... Let's just do it. I thank the organizers of Quark Matter '97 for the invitation; N. Saito for his transparencies on Monte Carlo simulations; and RIKEN for partial financial support. This work is supported in part by funds provided by the U. S. Department of Energy (D.O.E.) under cooperative agreement DOE-FG02-93ER-40762.
X. Ji/Nuclear Physics A638 (1998) 231c-238c
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