A Beautiful Spin
Xiangdong Ji
Department of Physics, University of Maryland, Colleage Park, MD 20742
Abstract. Spin is a beautiful concept that plays an ever important role in modern physics. In this
talk, I start with a discussion of the origin of spin, and then turn to three themes in which spin has
been crucial in subatomic physics: a lab to explore physics beyond the standard model, a tool to
measure physical observables that are hard to obtain otherwise, a probe to unravel nonperturbative
QCD. I conclude with some remarks on a world without spin.
THE ORIGIN OF SPIN
I would like to begin my talk by thanking the organizers for giving me this wonderful
opportunity to talk about spin as a beautiful concept in modern physics. RHIC spin
collaborations have started taking data and the first results have come in. It is a great
occasion to celebrate the fruit of many years of planning and hard work.
What is the origin of spin? As an English word, it existed before the 12th century. As
a fundamental observable of subatomic particles, it was introduced by G. Uhlenbeck and
S. Goudsmit in 1925 (and also Kronig). Many know the story that as graduate students,
the two were not really sure that they did the right thing to introduce the spin degrees
of freedom. They, in fact, at one point tried to withdraw their paper, but failed. Upon
receiving the paper, their advisor, P. Ehrenfrest, made the following remark, "This is a
good idea. Your idea may be wrong, but since both of you are so young without any
reputation, you would not lose anything by making a stupid mistake." [1] Because of
this potential "stupid mistake", one of the most beautiful concepts in modern physics
was born.
It is interesting to note that one of our heros, S. Goudsmit, spent more than two
decades here at Brookhaven during his career. Goudsmit was born in the Hague in
Netherlands, and was educated at the universities of Amsterdam and Leidan, where he
obtained his Ph. D. in 1927. He emigrated to America shortly afterwards, serving as a
professor of physics at the University of Michigan and North Western. Since 1948, he
moved BNL where he remained until his retirement in 1970.
Speaking of BNL, I would like to remind you some of the major events that happened
here which are related to spin physics. In 1957, parity-violation in K-decay (9 — T
puzzle) was discovered at the Cosmotron. Spin turned out to be a key element in many
parity-violation experiments. In the 1980s, a series of experiments were done at AGS,
measuring single-spin asymmetries and hyperon polarizations. More recently, the muon
g — 2 experiment has achieved a record precision of better than one part per million.
The RHIC spin is a whole new and exciting program on QCD spin physics. Finally, the
electron-ion collider, a planned machine with spin as its key feature, may find its home
CP675, Spin 2002:15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
Sources and Polarimeters, edited by Y. L Makdisi, A. U. Luccio, and W. W. MacKay
© 2003 American Institute of Physics 0-7354-0136-5/03/$20.00
3
here at BNL.
Back to the question of the orgin of spin. At present, we believe it is due to the
spacetime symmetry. The 3 + 1 dimensional Minkowski spacetime has symmetries
under continuous spacetime translations and rotations. These transformations form the
well-known Poincare group. Mathematically, it is a contracted SO(3,2) group generated
by ten generators. Four of them, P^, is the energy-momentum vector, and the other six,
/^v, are the generators of Lorentz transformations. The group has two Casimir operators,
allowing defining two universal physical observables for any physical system. The first
one, P^PH, introduces the concept of mass. The second one, W^W^ originated from the
Pauli-Lubanski spin vector W^ ~ ^va^JaRPj^ defines the concept of spin.
The above understanding, however, is not the final story. More revolutions in the
concept of spacetime await us. According to string or M-theory, the 3+1 spacetime is
a result of classical and low-energy approximations. At higher-energy, spacetime is 10
or 11 dimensional. Then an outstanding question is how do we end up seeing only 3+1?
Is it due to compactification, or we live just in a 3+1 brane? It has been known for a long
time that the strong gravitational fluctuations in the spacetime curvature exist at short
distance scales. The meaning of spacetime as we know it ceases to exist there.
The main thesis of my talk is that spin is a beautiful concept that plays a central role in
modern physics. In the following, I will elaborate on this in terms of three major themes:
Theme 1) spin as a laboratory to explore physics beyond the standard model; Theme 2)
spin as a tool to measure physical observables that are hard to obtain otherwise; and
Theme 3) spin as a probe to unravel nonperturbative QCD dynamics.
THEME I: SPIN AS A LAB TO EXPLORE PHYSICS BEYOND
STANDARD MODEL
The main research frontier in particle physics today is to discover physics beyond the
standard model. Spin is playing a unique role in this pursuit. There have been many
experiments done, underway, or planned, that exploit the spin degree of freedom to
search for nonstandard-model physics. A partial list includes the muon g — 2 experiment
at BNL, the SLAC El 58 experiment using Moller scattering to measure the eletron weak
charge, the Jefferson Lab Qweak experiment using polarized electron-proton scattering
to measure the weak charge of the proton, the newly proposed neutron electric dipole
moment (EDM) experiment at LANSCE, the TRIG experiment searching for P-even/Todd nuclear interactions, the PSI experiment to measure the polarization of positrons
from muon decay, the RHIC spin experiment measuring parity-asymmetry at highenergy, and so on. I will highlight a few of these here.
The spin of a particle supports the notion of an intrinsic magnetic moment
For a pointlike Dirac particle without interactions, the Lande g-factor is gD = 2. Hence
a = g — 2/2 is a measure of underlying dynamics. The latest result for muon g — 2 from
HIiiiiiiiii^^
HI
HIiiiiiiiiiiii
220
lllllllllllllllllllllllll
200
iiiiiiiiiiiiiiIllllllllllllli
iiiiiiiiiiiiiiiiilliiiiiiiiiiiiii
180
HI
iiiiiiiiiiiiB^^^^^
HI
HI
160
FIGURE 1. New BNL g-2 data.
the experiment here is [2],
cfNL = 11659204(7) (5) x 10 -10
(2)
This yields a world average, a™orld = 11659208(8) x 1010. The lastest theoretical number is afi = 11659182.1(7.2) x 10~10, after taking into account the infamous sign flip
for light-light scattering contribution, and averaging over several latest calculations [3].
The discrepancy is 8a^ = 21(11) x 10~10, about 1.9(7. Taking the above discrepancy
seriously, what are the implications for SUSY? A recent analysis indicates that 1) there
must be at least two sparticles with mass less than 760 GeV, 2) for models with gaugino
mass unification, there must be one sparticle with mass less than 580 GeV, 3) there is no
lower bound on tan/3 [3]. Clearly, pushing for a much better measurement at this point
should go hand in hand with theoretical development in understanding the hadronic
corrections. In particular, there are still large uncertainties in calculating the vacuum
polarization contribution and light-by-light scattering.
For a static particle, the only 3-vector present is the spin. Hence its electric dipole
momentum (EDM) must be d = dlf. Clearly, d is T-odd, and CP-odd in field theory
where the CPT theorem holds. In the standard model, the CP violation in the CKM
matrix leads to a neutron EDM of order 10~31 ecm, which is several orders of magnitude
higher than the current experimental limit, 0.63 x 10~25 ecm. However, the baryon
number asymmetry in the Universe requires additional sources of CP violation which
can contribute significantly to the neutron EDM. Many weak-scale SUSY models predict
the neutron EDM on the order of 10 -25 10 Zl ecm. Therefore, one can expect that
an improvement on the current limit can reveal a lot about the constraints on the these
models. The EDM collaboration at LANSCE has proposed to push the current limit twoorder of magnitude lower, i.e., 10~27 ecm. This is clearly a very exciting development.
The spin of a particle can lead to simple correlations like ?• p in the decay rate and
cross sections. Since it is a pseudo scalar, it is a natural place to look for parity violation.
In fact, C. S. Wu used this to find one of the first evidences of parity violation. It has
been observed recently that this correlation can be used in high-precision measurement
of weak charges. In the standard model, the vector couplings of the electron and proton
to the neutral current Z-boson are
Sy~s£~l/4-sin26^.
(3)
Hence a measurement of the couplings can be translated directly into a measurement
of sin2^ itself. Like ocs(Q2), sin2^ in the minimal subtraction scheme also runs
with the probing scale. Its value has been measured very accurately at the Z-pole,
thanks to experiments at LEP. At low-energy, however, the atomic parity experiment
and neutrino scattering data seem to be inconsistent with the standard model prediction.
A new experiment at SLAC (El58) plans to measure the asymmetry in polarized Moller
scattering ee —> ee, in which the scattering electron is polarized. The experiment will
have a 6% measurement on the (vector) weak charge of the electron at Q2 = 0.03 GeV2.
A similar experiment at JLab (Qweak) plans to measure the asymmetry in polarized ep
elastic scattering. They have planned a 4% measurement on the vector weak charge of
the proton at Q2 = 0.03 GeV2, which will yields a 0.3% accuracy on sin2 0^.
What new physics can one learn about in the weak charge measurements? The discprepancy from the standard model prediction will be sensitive to a new Z-boson of mass
ranging from 0.6 to 1 TeV. It can also detect new contact interactions arising from scalar
Higgs of mass ~ 10 GeV. New physics can also modify the gauge boson propagators,
resulting in new parameters like X, and the future measurements can constraint it to a
level
THEME II: SPIN AS A TOOL TO MEASURE OBSERVABLES
THAT ARE HARD TO OBTAIN OTHERWISE
There are many observables that are interesting in physics but hard to measure experimentally. For instance, we would like to know how strange quarks contribute to the
electromagnetic form factors of the nucleon, but it is hard to isolate just the strange
quarks. The experiments involving spin degrees of freedom can make a significant difference here. Actually, there is a long list of interesting physical observables that can
be measured by exploiting the spin degrees of freedom: The strangeness content of the
nucleon can be measured in polarized eletron-nucleon scattering; the electromagnetic
form factors of the nucleon can be obtained by measuring the recoil polarizations, the
spin-dependent nucleon-nucleon scattering can be used to measure the parity-violating
nucleon forces. The neutron density in a large nucleus is hard to obtained using electromagnetic interactions but can be measured in parity-violating electron scattering.
The sizable strange effects may have been seen in various nucleon matrix elements:
the scalar charge from the n — N sigma term and chiral perturbation calculations of the
baryon octet mass splitting, the proton spin content measured in polarized deep-inelastic
scattering, and the strange quark contribution to the momentum sum rule from neutrino
scattering. On the other hand, the strange vector current defines form factors,
1.0
(8)
0.5
I
0.0
(9)
,(5)
of
-0.5
(7)
(3)
-1.0
-
FIGURE 2.
tions [4, 5].
2
-
1 0
GAe (T=1)
'-.5 -.25 0.0
1
.25
GSM(0.48)
.5
The results on the strange quark form factors from the SAMPLE and HAPPEX collabora-
From these, one obtains the strange magnetic moment JJLS = F2(Q) and strange radius
(r?) — ~6dG|(<2)/^62- R- McKeown was the first to realize that the strange form
factors can be measured in polarized parity-violating eletron-nucleon scattering in which
the interference between the Z and /exchanges is probed [6]. The SAMPLE experiment
at MIT-Bates have measured the electron asymmetries on both proton and deuteron
targets and the result is shown in the left panel of Fig. 2. The strange magnetic moment
seems to be small, and the non-zero axial contribution is inconsistent with the best
theoretical calculation [7]. The HAPPEX experiment at Jlab measured the electron
asymmetry at a different kinematics and the result is a linear constraint on the strange
electric and magnetic form factors [5].
The nucleon electromagnetic form factors tell us detailed information about the charge
and current distributions in the nucleon. Despite many years of effort since the mid
1950s, they are still not well determined. Polarization exepriments are helping in a big
way in the precision measurements! Recently a JLab Hall-A experiment has measured
the polarization transfer in eP-elastic scattering. The ratio of the transverse and longitudinal polarizations yields the ratio of the form factors
G££
*Mp
=
.
2M -tany
(5)
The big surprise of the new data is that the ratio dropped steadily as a function of Q2 [8],
differing from popular expectations. The significance of the result is still under debate.
The same method has been and will be used again to measure the form factors of the
neutron.
The nucleon-nucleon interactions has a parity-violating component when taking
into account the underlying weak-boson exchanges. These interactions have been
parametrized in terms of effective parity-odd meson-nucleon couplings and optimal
values of these couplings have been estimated (DDK) [10]. Many of the couplngs can
30
1.2
20
0.8
10
3
C?0.6
C?ft
0.2
0
<
D Jones [11]
• This work
•-- VMD[32]
• - - PFSA[29]
• SU(6) breaking + CO ff [28]
•-- Soliton[31]
•— SU(6) breaking [28]
•-- COM [26]
1
o
_a
-10
-20
-70
-50
-30
h
-10
x 107
10
30
FIGURE 3. Left: The ratio of the electric to magnetic form factors of the proton from the recoilpolarization measurement [8]. Right: the parity-violating nucleon-nucleon interactions probed in polarized
nucleon-nucleon scattering [9].
only be measured in spin-dependent processes. For example, the parity-violating pionnucleon coupling can be measured in spin-dependent pion photoproduction [11, 12].
Other experiments include measuring asymmetries and polarizations of /-rays in nuclear transitions, asymmetries in polarized proton-proton elastic scattering and thermal
neutron-nuclei scattering, and the neutron spin rotation through nuclei matter. A good
summary of these experiments can be found in [13]. Shown on the right panel of Fig.3
is the constraint from parity-violating nucleon-nucleon scattering on the p and (0 meson
couplings, together with the latest result from TRIUMF [9].
THEME III: SPIN AS A PROBE TO UNRAVEL
NONPERTURBATIVE QCD DYNAMICS
To find how nonperturbative QCD works is one of the greatest challenges in theoretical
physics. At present, besides numerically solving the theory on a spacetime lattice, another approach to gain a better understanding is to learn from experimental data. Spin offers a unique way to probe into the internal structure of the nucleon and nonperturbative
QCD dynamics. Over the years and in particular after the EMC experiment on polarized
deep-inelastic scattering [14], QCD spin physics has evolved into a large subfield. A partial list of important topics would include the spin structure functions of the nucleon—g l
and #2, the gluon polarization Ag(;t), the quark transversity distribution dq(x), the single
spin asymmetries, the generalized parton distributions, the semi-inclusive deep-inelastic
scattering, the GDH sum rule and its generalizations, etc. All of these subjects will be
discussed and debated extensively at this conference. Due to time limitation, I cannot go
through all of them in this introductory talk.
Let's start with the spin structure of the nucleon, as it has been the major interest in the
field for more than a decade. According to QCD, we can write down a decomposition of
the nucleon spin as [15]
+ Jq(V) •
(6)
This decomposition is independent of gauge fixing but depends on the resolution scale
jU2. After a decade of careful experimental study following the original EMC experiment, we are now quite certain that the total quark spin AE = 8u + 8d + 8s contributes
only about 30% of the nucleon spin [16], contrary to the prediction of the naive quark
model. The total quark contribution/^ = 1/2AE+Z^, spin plus orbital, has been shown
to be related to a sum rule of the generalized parton distributions [15]. The total gluon
angular momentum Jg contains the gluon helicity contribution Ag which can be measured in high-energy scattering.
It has been speculated that the small size of AE = A Ev + AE5 results from a cancellation between the valence and sea contributions. While inclusive DIS cannot separate
valence from sea, semi-inclusive processes can [17]. A surprise from the recent SMC
and HERMES experiments is that the data do not show the expected large negatively
polarized sea [18]! The data are not very accurate yet at this point, and a better precision is needed to make a firm conclusion. The validity of factorization might also be a
concern. Let me point out that a measurement of strange sea contribution As can be a
useful test of SU(3) flavor symmetry. Fortunately, the RHIC spin can help here. Through
W-boson production, the sea quark polarization can be measured rather cleanly [19].
Gluon polarization Ag(;t) is a well-defined quantity theoretically (gauge invariant),
just like the quark helicity distribution,
(7)
where L is a light-cone gauge link. Although its role in the spin decomposition is
not transparent, Ag(;t) can be interpreted as the gluon helicity in the light-cone gauge
A+ = 0. So far, some information about Ag has been obtained from NLO fit to the
gj (jc) structure function data [16], and large pt hadron production by HERMES [20]. An
interesting question here is if Ag is as large as some preliminary results indicate, why do
the gluons contribute so much to the spin of the nucleon? After all, the nucleon spin is
h/2 and the quarks do not seem to contribute much more than/z/2.
To make an accurate measurement of the gluon distribution is clearly a high priority
for the future high-energy spin physics. The gluon polarization can be measured from
direct photon production at RHIC, jet and high-P, hadron production at RHIC and
COMPASS, open charm and heavy quark production at SLAC, COMPASS, and RHIC,
and Q2 evolution, for example, at EIC. There are many talks at this conference that
are devoted to this topic. One important question one should keep asking is: are the
perturbative mechanisms clean? Eventually, are we going to get a good determination of
the integral?
Generalized parton distributions (GPDs) have become a very popular subject in theory and experiment for several years now [21, 22, 23]. These studies were motivated by
finding the spin structure of the nucleon [15]! It is a new type of parton "distributions"
which contains much more information than any other nucleon observable considered
<-" 0.6
0.4
0.4
0.3
0.2
0.2
0
0.1
-0.2
< 0
-0.4
-0.1
-0.6
-0.2
1
2
3
-0.3
4> (rad)
-0.4
0
50
100 150 200 250 300 350
FIGURE 4. The single-spin asymmetry from HERMES (left) and CLAS (right) collaborations.
so far. In various limits, they reduce to elastic form factors and Feynman parton distributions. They depend on three variables: the parton momentum fraction jt, the form-factor
resolution t = —q2, and the skewness %. For % = 0, the GPDs have a nice interpretation
in the impact parameter space [24].
GPDs can be measured in real experiments, such as deeply-virtual Compton [25, 26]
scattering and exclusive meson production [27]. Deeply virtual Compton scattering is an
exclusive production of high-energy photons in deep-inelastic kinematics in which the
single quark scattering dominates—it is a Compton scattering on a single quark! If one
compares this to deep-inelastic scattering, DVCS is a noninvasive surgery on the proton
in which the structural information is obtained without breaking it into pieces. The Zeus
and HI collaborations at HERA were first to report the DVCS type of events. At low
lepton energy, however, it is difficult to measure the DVCS process directly. Rather,
one can isolate the interference between the DVCS and Bethe-Heitler amplitudes. The
HERMES collaboration at DES Y [28] and CLAS at JLab [29] have measured the photon
production asymmetry, and they are found to be consistent with the dominance of the
DVCS amplitude. Exclusive meson productions are also sensitive to GPDs. Different
flavors and spins of mesons help to disintangle the contributions of different GPDs.
The quark transversity distribution is one of the three twist-two quark distributions
describing the state of a quark beam in a nucleon beam [30, 31, 32, 33]. Contrary
to what's commonly believed, it is not a transverse-spin distribution because a fastmoving quark cannot be in a transverse-spin eigenstate. In a non-relativistic quark
model, however, the transversity distribution is the same as the helicity distribution
and a quark can be in the transverse-spin eigenstates [34, 35]. The knowledge of the
distribution allows access to the tensor charge of the nucleon which is useful, among
others, in the neutron EDM calculation.
To measure the transversity distribution is very difficult! It is a chirally-odd quantity,
requires another chiral-odd observable present in a hard process. It vanishes asymptotically at a large resolution scale. The most clean process is the Drell-Yan scattering with
10
transversely polarized protons. One can measure it in inclusive jet production and direct
7 production. It can also be accessd through chiral-odd quark fragmentation functions.
In most cases, however, there are other competing processes that one must isolate. A
review of measuring the transversity distribution can be found in [36].
Finally, I would like to briefly mention transverse single-spin asymmetries. These
asymmetries have been observed in (single transversely polarized) proton-proton scattering and more recently in electron-proton scattering. The hyperon polarization in unpolarized proton-proton scattering is a closely related phenomenon. A good reference
about the subject is a recent review article by Barone and Ratcliffe [37] (see also [38]).
The asymmetry is time-reversal-"odd" and arises only when there are initial and final
state interaction phases. Moreover, it requires viable mechanisms for helicity flips, such
as Collins fragmentation functions[39], Sivers distributions [40, 41, 42, 43], or twistthree correlations[44,45,46]. Recently, it was shown by Belitsky, Yuan, and myself that
in the light-cone gauge, a gauge link at spatial infinity is crucial to have a nonvanishing
Collins and Sivers functions [47,48].
CLOSING REMARKS
In non-relativistic quantum mechanics, spin is considered an extra degree of freedom
that one can put in by hand. In relativistic theories, however, spin is an essential part of a
theory, as we have mentioned in the begining of the talk. In particular, the spin-statistics
theorem followed from causality has profound implications about the fundamental structure of matter and the stability of many-body systems, even in the non-relativistic limit.
For example, the atomic structure which lays the foundation for chemistry and biology
critically depends on the operation of the Pauli principle. The supersymmetric theories,
which have recently become very popular in particle physics, assume a symmetry between fermions and bosons or different spin states. The importance of spin is neatly
summarized in the preface to S. Tomonaga's book "The Story of Spin," by T. Oka who
wrote that spin, "is a mysterious beast, and yet its practical effect prevail the whole of
science. The existence of spin, and the statistics associated with it, is the most subtle and
ingenious design of Nature— without it the whole universe would collapse." [1]
To conclude the talk, I have compiled a top ten list of disasters in a world without
spin; for fun:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Chemistry would have been messed up.
You can't get rid of the glare from the sunlight.
You can't have an MRI if you've gotten brain cancer.
Quarks aren't compelled to have color.
Neutron stars would never be stable.
Weak interactions can't be chiral.
There wouldn't be any gauge theories.
You can't play around with supersymmetry.
You would be stuck in 26-dimensions.
Spin doctors in Washington, DC would be jobless!
11
ACKNOWLEDGMENTS
This work is supported partly by the DOE grant DE-FG-02-93ER-40762.
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