Download The spin-dependent structure function

Q)
Q) How well is Unpolarized structure of the nucleon known?
After 40 years DIS experiments, unpolarized structure of the nucleon reasonably well understood.
High x - valence quark dominating
Q) Spin Crisis:
Q) NPM: The “proton spin crisis” was introduced in the late 1980s, when
the EMC-experiment revealed that little or nothing of a proton’s spin
seemed to be carried by its quarks.
The “proton spin crisis” essentially refers to the experimental finding
that very little of the spin of a proton seems to be carried by the quarks
from which it is supposedly built. This was a very curious and unexpected
experimental result of the European Muon Collaboration, EMC [2] (later
confirmed by other experiments), as the whole idea of the original quark
model of Gell-Mann [3] and Zweig [4] was to account for 100 percent of
the hadronic spins, solely in terms of quarks. This original, or “naive”,
quark model also was very successful in explaining and predicting hadron
spectroscopy data.
The naive quark parton model
In the quark-parton model the DIS cross section is the incoherent sum of the scattering
off the point-like quarks and yields the following cross section:
(1.10)
where
is the probability to find a quark in the proton with a fraction
proton momentum. Comparing this expression to eq. 1.9 we find
of the total
(1.11)
The first identity,
, is known as the Callan-Gross relation and is a direct
consequence of the spin-1/2 nature of the quarks. The function
is known as the
parton distribution function. The second identity is also interesting: it predicts that the
cross section only depends on one variable, . This property is called Bjorken scaling.
Approximate scaling is observed in the data at
, but violation of scaling is
observed for lower and higher
. Figure 1.3 shows
versus
measured by various
HERA- and fixed target experiments [2].
Figure 1.3: The structure function
experiments plotted against
broken towards low
measured by ZEUS, H1 and various fixed target
. Bjorken scaling is observed at high , but is gradually
. The line represents a QCD based fit to the data.
Explanation of magnetic moment
The electron is a negatively charged particle with angular momentum. A rotating
electrically charged body in classical electrodynamics causes a magnetic dipole effect
creating magnetic poles of equal magnitude but opposite polarity like a bar magnet. For
magnetic dipoles, the dipole moment points from the magnetic south to the magnetic
north pole. The electron exists in a magnetic field which exerts a torque opposing its
alignment creating a potential energy that depends on its orientation with respect to the
field. The magnetic energy of an electron is approximately twice what it should be in
classical mechanics. The factor of two multiplying the electron spin angular momentum
comes from the fact that it is twice as effective in producing magnetic moment. This
factor is called the electronic spin g-factor. The persistent early spectroscopists, such as
Alfred Lande, worked out a way to calculate the effect of the various directions of
angular momenta. The resulting geometric factor is called the Lande g-factor.
The intrinsic magnetic moment μ of a particle with charge q, mass m, and spin s, is
where the dimensionless quantity g is called the g-factor.
The g-factor is an essential value related to the magnetic moment of the subatomic
particles and corrects for the precession of the angular momentum. One of the triumphs
of the theory of quantum electrodynamics is its accurate prediction of the electron gfactor, which has been experimentally determined to have the value 2.002319... The
value of 2 arises from the Dirac equation, a fundamental equation connecting the
electron's spin with its electromagnetic properties, and the correction of 0.002319...,
called the anomalous magnetic dipole moment of the electron, arises from the electron's
interaction with virtual photons in quantum electrodynamics. Reduction of the Dirac
equation for an electron in a magnetic field to its non-relativistic limit yields the
Schrödinger equation with a correction term which takes account of the interaction of the
electron's intrinsic magnetic moment with the magnetic field giving the correct energy.
The total spin magnetic moment of the electron is
where gs = 2 in Dirac mechanics, but is slightly larger due to Quantum Electrodynamic
effects, μB is the Bohr magneton and s is the electron spin.
The z component of the electron magnetic moment is
where ms is the spin quantum number.
It is important to notice that is a negative constant multiplied by the spin, so the
magnetic moment is antiparallel to the spin angular momentum.
In atomic physics, the Bohr magneton (symbol μB) is named after the physicist Niels
Bohr. It is a physical constant of magnetic moment, defined in SI units by
and in Gaussian centimeter-gram-second units by
The nuclear magneton (symbol
defined by:
), is a physical constant of magnetic moment,
where:
is the elementary charge,
is the reduced Planck's constant,
is the proton rest mass
In the SI system of units its value is approximately:
= 5.050 783 24(13) × 10-27 J·T-1
QThe radiation length
provides information about the photon conversion probability
in a specific material.
Radiation length: Scaling variable for the probability of occurrence of bremsstrahlung
pair production, and for the variance of the angle of multiple scattering. The radiation
length is given (in [g/cm**2]) by
Radiation Length , High-energy electrons predominantly lose energy in matter by
bremsstrahlung, and high-energy photons by e + e − pair production. The characteristic
amount of matter traversed for these related interactions is called the radiation length X0,
usually measured in gcm − 2. It is both the mean distance over which a high-energy
electron loses all but 1/e of its energy by bremsstrahlung, and 7/9 of the mean free path
for pair production by a high-energy photon. It is also the appropriate scale length for
describing high-energy electromagnetic cascades.
The radiation length is given, to good approximation, by the expression
Q) Calorimeter:A composite detector using total absorption of particles to measure the
energy and position of incident particles or jets. In the process of absorption showers are
generated by cascades of interactions, hence the occasionally used name shower counter
for a calorimeter.
Calorimeters are usually composed of different parts, custom-built for optimal
performance on different incident particles. Each calorimeter is made of multiple
individual cells, over whose volume the absorbed energy is integrated; cells are aligned to
form towers typically along the direction of the incident particle. The analysis of cells
and towers allows one to measure lateral and longitudinal shower profiles, hence their
arrangement is optimized for this purpose, and usually changes orientation in different
angular regions. Typically, incident electromagnetic particles, viz. electrons and gammas,
are fully absorbed in the electromagnetic calorimeter, which is made of the first (for the
particles) layers of a composite calorimeter; its construction takes advantage of the
comparatively short and concentrated electromagnetic shower shape to measure energy
and position with optimal precision for these particles (which include 's, decaying
electromagnetically). Electromagnetic showers have a shape that fluctuates within
comparatively narrow limits; its overall size scales with the radiation length.
Incident hadrons, on the other hand, may start their showering in the electromagnetic
calorimeter, but will nearly always be absorbed fully only in later layers, i.e. in the
hadronic calorimeter, built precisely for their containment. Hadronic showers have a
widely fluctuating shape; their average extent does not scale with the calorimeter's
interaction length, but is partly determined by the radiation length.
Discrimination, often at the trigger level, between electromagnetic and hadronic showers
is a major criterion for a calorimeter; it is, therefore, important to contain electromagnetic
showers over a short distance, without initiating too many hadronic showers. The critical
quantity to maximize is the ratio
, which is approximately proportional to Z1.3 (see
[Fabjan91]); hence the use of high-Z materials like lead, tungsten, or uranium for
electromagnetic calorimeters.
Calorimeters can also provide signatures for particles that are not absorbed: muons and
neutrinos. Muons do not shower in matter, but their charge leaves an ionization signal,
which can be identified in a calorimeter if the particle is sufficiently isolated (and the
dynamic range of electronics permits), and then can be associated to a track detected in
tracking devices inside the calorimeter, or/and in specific muon chambers (after passing
the calorimeter). Neutrinos, on the other hand, leave no signal in a calorimeter, but their
existence can sometimes be inferred from energy conservation: in a hermetically closed
calorimeter, at least a single sufficiently energetic neutrino, or an unbalanced group of
neutrinos, can be ``observed'' by forming a vector sum of all measured momenta, taking
the observed energy in each calorimeter cell along the direction from the interaction point
to the cell. The precision of such measurements, usually limited to the transverse
direction, requires minimal leakage of energy in all directions, hence a major challenge
for designing a practical calorimeter.
The shower development is a statistical process (see Electromagnetic Shower, Hadronic
Shower). This explains why the relative accuracy of energy measurements in calorimeters
improves with increasing energy, according to the empirical formula
where E = energy of incident particle, = standard deviation of energy measurement, and
a and are constants depending on the detector type, e.g. the thickness and characteristics
of active and passive layers. The overall constant includes the systematic errors of the
individual modules. Other, similar formulae, with different energy-dependent terms are in
use; for more details, Energy Resolution in Calorimeters, Compensating Calorimeter.
From the construction point of view, one can distinguish between:
Q1) What is sampling calorimeter? In what way is it different from other types of
calorimeters (if there is any)?
Answer: a) Homogeneous Shower Counters. In homogeneous calorimeters the functions
of passive particle absorption and active signal generation and readout are combined in a
single material. Such materials are almost exclusively used for electromagnetic
calorimeters, e.g. crystals ( Crystal Calorimeter), composite materials (like lead glass,
viz. PbO and SiO2) or, usually for low energy, liquid noble gases (see [Walraff91],
[Fabjan95a]).
b) Heterogeneous Shower Counters (= Sampling Calorimeters). In sampling calorimeters
the functions of passive particle absorption and active signal generation and readout are
separated. This allows optimal choice of absorber materials and a certain freedom in
signal treatment. Heterogeneous calorimeters are mostly built as sandwich counters,
sheets of heavy-material absorber (e.g. lead, iron, uranium) alternating with layers of
active material (e.g. liquid or solid scintillators, or proportional counters). Only the
fraction of the shower energy absorbed in the active material is measured. Hadron
calorimeters, needing considerable depth and width to create and absorb the shower, are
necessarily of the sampling calorimeter type.
Calorimetry is the art of compromising between conflicting requirements; the principal
requirements are usually formulated in terms of resolution in energy, spatial coordinates,
and time, in triggering capabilities, in radiation hardness of the materials used, and in
electronics parameters like dynamic range, and signal extraction (for high-frequency
colliders). In nearly all cases, cost is the most critical limiting parameter. Depending on
the physics goals, the energy range that has to be considered, the accelerator
characteristics, etc., some goals will be favoured over others. The span of possible
solutions for calorimeters is much wider than for tracking devices, and quite ingenious
solutions have been found by imaginative experimental teams over the last 15 years,
since calorimeters became key components of particle detectors.
In practical constructions the ratio of energy loss in the passive and active material
is rather large, typically of the order of 10. Although performance does not strongly
depend on the orientation of active and passive material, their relative thickness
must not vary too much, to ensure an energy resolution independent of direction
and position of showers. Only a few percent of the energy lost in the active layers is
converted into detectable signal. For a discussion, [Fabjan91].
Q2) In general-purpose detectors, an electromagnetic calorimeter (based on
electromagnetic shower due to bremsstrahlung and pair-production because of the
intense electromagnetic field of atomic nuclei when a high energy particle passes
through a medium) is usually followed by a hadronic calorimeter (based on hadronic
showers). In sampling calorimeters, the passive materials have high atomic numbers
(eg. Lead) and they cause the shower to occur and thus play a greater role in the
particle absorption. The particles from shower on the other hand, produce signals
such as light in scintillators to be used for read-out.
Q3) What are the reasons behind anomalous magnetic moments?
Answer: The anomalous magnetic moments of the lepton family are usually
ascribed to their treatment as theoretical points, rather than having actual size,
together with their interactions with the environment. Nucleon anomalous magnetic
moments are believed to be more complicated.
In quantum electrodynamics, the anomalous magnetic moment of a particle is a
contribution of effects of quantum mechanics, expressed by Feynman diagrams with
loops, to the magnetic moment of that particle.
The “Dirac” magnetic moment, corresponding to tree-level Feynman diagrams, can be
calculated from the Dirac equation. It is usually expressed in terms of the g-factor; the
Dirac equation predicts g = 2. For particles such as the electron, this classical result
differs from the observed value by a small fraction of a percent. The difference is the
anomalous magnetic moment, denoted a and defined as
The one-loop contribution to the anomalous magnetic moment of the electron is found by
calculating the vertex function shown in the diagram on the right. The calculation is
relatively straightforward[1] and the one-loop result is:
where α is the fine structure constant. This result was first found by Schwinger in 1948.[2]
As of 1997, the coefficients of the QED formula for the anomalous magnetic moment of
the electron have been calculated through order α4. The QED prediction agrees with the
experimentally measured value to more than 10 significant figures, making the magnetic
moment of the electron the most accurately verified prediction in the history of physics.
(See precision tests of QED for details.)
The anomalous magnetic moment of the muon is calculated in a similar way; its
measurement provides a precision test of the Standard Model. As of November 2006, the
measurement disagrees with the Standard Model by 3.4 standard deviations[3], suggesting
beyond the Standard Model physics may be having an effect.
Composite particles often have a huge anomalous magnetic moment. This is true for
the proton, which is made up of charged quarks, and the neutron, which has a
magnetic moment even though it is electrically neutral.
Q4) DIS data gave some surprising results such as the original “EMC-Effect”, the
violation of Gottfried sum rule, even giving some hints for quark substructure.
Indication of quark substructure (i.e an even deeper layer of matter): The observed
“excess” of events at extremely high Q2 beyond 15000 GeV-squared
EMC effect: the modification of nucleon structure functions. (2)
EMC effect: quark momentum modified in the nucleus (5).
EMC results: quarks carry only a fraction of the proton’s spin (5)
The violation of the Gottfried sum rule: the apparent breakdown of flavor
symmetry in the quark-antiquark sea of the proton)(2)
The violation of the Gottfried sum rule: flavor asymmetry of the sea established(5)
Unit of Magnetic Moment: Ampere * meter-squared
(or the area of a current loop times the current value gives the magnetic dipole
moment of the current loop.)
Dirac Mag Moment: ‘mu’= eh_cut/2M
Measured nucleon magnetic moments:
Proton: Disagreed with Dirac prediction by 150. (Early 1920s, Stern-Gerlach first
measured the electron mag. Moment by doing measurement on a beam of silver
atoms, thus discovering the electron spin – KJS pg1). Then Stern improved his
instruments and he and his collaborators measured the much smaller proton ‘mu’.
(Analoguous to Mendeleev’s periodic system the correctness of the quark model was
strongly supported by the discovery of the
particle: the
was predicted by
the quark model, but was not discovered yet in 1964. The SLAC data were the first
direct experimental evidence of the existence of quarks. At the time of the
introduction of the periodic system, not all elements had been found, but all ‘holes’
were eventually filled, proving the correctness of the hypothesis. Here we need
something that supports the nucleon substructure, not the one that supports the
quark-model, which is a bit off the line/track/relevance.)
This slow spectator proton may not be detected by CLAS since CLAS has a
momentum threshold of about 300 MeV/c for protons. (i.e. CLAS can detect only
higher momentum protons.) (18, J. Zhang.)
For a new detector one must have a simulation in order to understand the
acceptance and to understand the response of the detector for particle of known
trajectory.
Cherenkov counter (cut) can be used to distinguish between electrons and negative
pions at low momenta. At higher momenta (>3GeV), forward electromagnetic
calorimeter (EC) helps us in making such separation/distinction (as with neutral
particles.)
(18) The Cerenkov Counter provides a reasonable way to separate electrons from
negative pions. This is because a pion has a momentum threshold of 2.5 GeV/c to
produce Cerenkov radiation in the perfluorenutane gas (C4F10), which is used in the
Cerenkov detector in CLAS, while an electron has a momentum threshold of only 9
MeV/c. If the momentum of the scattered electron is less than 3.0 GeV/c, we require
the number of photoelectrons (Nphe) from the Cerenkov detector be larger than 2.5.
Above 3 GeV/c the Cerenkov will detect both pions and electrons, so we require
simply that the number of photoelectrons (Nphe) be greater than 1.0.
At large momenta we use the electromagnetic calorimeter to distinguish electrons
from pions as described below.
In scattering, a differential cross section is defined by the probability to observe a
scattered particle in a given quantum state per solid angle unit, such as within a given
cone of observation, if the target is irradiated by a flux of one particle per surface unit:
To put it another way, it is the rate of scattering events normalized to the beam intensity,
the target density, the length of the beam-target interaction region, the geometrical “size”
of detector, and the efficiency of the detector “counting.”
If the detector is small and sufficiently far from the target, then the geometrical “size” of
the detector is given by:
The integral cross section is the integral of the differential cross section on the whole
sphere of observation (4π steradian):
A cross section is therefore a measure of the effective surface area seen by the impinging
particles, and as such is expressed in units of area. Usual units are the cm2, the barn (1 b =
10−24 cm2) and the corresponding submultiples: the millibarn (1 mb = 10−3 b), the
microbarn (1 μb = 10−6 b), the nanobarn ( 1 nb = 10−9 b), and the picobarn (1 pb = 10−12
b). The cross section of two particles (i.e. observed when the two particles are colliding
with each other) is a measure of the interaction event between the two particles.
[edit] Relation to the S matrix
If the reduced masses and momenta of the colliding system are mi, and mf,
and after the collision respectively, the differential cross section is given by
before
where the on-shell T matrix is defined by
in terms of the S matrix. The δ function is the distribution called the Dirac delta function.
The computation of the S matrix is the main aim of the scattering theory.
[edit] Nuclear physics
Main article: nuclear cross section
In nuclear physics, it is found convenient to express probability of a particular event by a
cross section. Statistically, the centers of the atoms in a thin foil can be considered as
points evenly distributed over a plane. The center of an atomic projectile striking this
plane has geometrically a definite probability of passing within a certain distance r of one
of these points. In fact, if there are n atomic centers in an area A of the plane, this
probability is (nπr2) / A, which is simply the ratio of the aggregate area of circles of
radius r drawn around the points to the whole area. If we think of the atoms as
impenetrable steel discs and the impinging particle as a bullet of negligible diameter, this
ratio is the probability that the bullet will strike a steel disc, i.e., that the atomic projectile
will be stopped by the foil. If it is the fraction of impinging atoms getting through the foil
which is measured, the result can still be expressed in terms of the equivalent stopping
cross section of the atoms. This notion can be extended to any interaction between the
impinging particle and the atoms in the target. For example, the probability that an alpha
particle striking a beryllium target will produce a neutron can be expressed as the
equivalent cross section of beryllium for this type of reaction.
Jump to: navigation, search
In nuclear and particle physics, the concept of a cross section is used to express the
likelihood of interaction between particles.
The term is derived from the purely classical picture of (a large number of) point-like
projectiles directed to an area that includes a solid target. Assuming that an interaction
will occur (with 100% probability) if the projectile hits the solid, and not at all (0%
probability) if it misses, the total interaction probability for the single projectile will be
the ratio of the area of the section of the solid (the cross section) to the total targeted area.
This basic concept is then extended to the cases where the interaction probability in the
targeted area assumes intermediate values - because the target itself is not homogeneous,
or because the interaction is mediated by a non-uniform field.
Sum rule may refer to:




Sum rule in differentiation
Sum rule in integration
Rule of sum, a counting principle in combinatorics
In quantum mechanics, sum rule refers to formulas for transitions between
energy levels, in which the sum of the transition strengths is expressed in a simple
form. Sum rules are used to describe the properties of many physical systems,
including solids, atoms, atomic nuclei, and nuclear constituents such as protons
and neutrons. The sum rules are derived from quite general principles, and are
useful in situations where the behavior of individual energy levels is too complex
to describe by a precise quantum-mechanical theory. In general, sum rules are
derived by using Heisenberg's quantum-mechanical algebra to construct operator
equalities, which are then applied to particles or the energy levels of a system. See
also selection rule.
Measurements of Partial Channels of the GDH Sum Rule (Briscoe,
Strakovsky): The Gerasimov-Drell-Hearn (GDH) sum rule relates the difference in the
total hadronic photo-absorption cross sections for left- and right-handed circularly
polarized photons interacting with longitudinally polarized nucleons to the anomalous
magnetic moment of the nucleon squared divided by m. The GDH sum rule provides an
elegant connection between the nucleon structure functions obtained in high-energy
lepton-scattering experiments and two static properties of the nucleon. Thus, it provides a
bridge between perturbative and nonpertubative QCD. The fundamental interpretation of
the GDH sum rule is that any particle which has a nonzero anomalous magnetic moment
has internal structure and therefore an excitation spectrum. The detailed verification of
the GDH sum rule shows which nucleon resonances play the most significant roles in
this. There is a basic connection between the GDH sum rule which is valid for real
photons and the famous Bjorken sum rule for virtual photons in the limit Q .
The validity of the GDH sum rule for the proton has been satisfactorily established. In the
case of the neutron, the experimental limits are still much too large; this is due
substantially to the large uncertainty in the 2 photoproduction cross section, especially
in n  n in the photon energy range 500 to 1500 MeV. Most of the required
experimental tools to remedy this, namely, high-quality circularly polarized taggedphoton beams and longitudinally polarized H2 and 3He targets are already available at
MAMI up to 800 MeV and will be available up to 1.5 GeV in 2005. The Crystal Ball can
provide the crucial 4 solid-angle multiphoton detector. An important byproduct of the
 photoproduction data with polarized beams and targets is the help in the unraveling
of the N* and * resonance spectra, since the polarization data have enhanced sensitivity
to small multipoles.
From the thesis (BjorkenSumRule_Thesis – See the folder Talks):
“The Bjorken Sum Rule and the Strong Coupling” Diplomarbeit
zur Erlangung des wissenschaftlichen Grades
Diplom-Physiker
vorgelegt von
Anke Knauf
geboren in Meißen
__
Institut f¨ur Theoretische Physik
Fachrichtung Physik
Fakult¨at Mathematik und Naturwissenschaften
der Technischen Universit¨at Dresden
2002
The BSR in the Naive Parton Model
Deep inelastic phenomena have been extensively studied for about a quarter of a century,
both experimentally and theoretically. The main purposes of this vast physics program
were first to elucidate the internal structure of the nucleon and later on to test
perturbative Quantum Chromodynamics (QCD). At the end of the 60’s, the first
measurements on charged lepton deep inelastic scattering (DIS) at the Stanford
Linear Accelerator (SLAC) have shown that the nucleon is made of hard pointlike
objects [3], which was the first evidence for the existence of sub-nuclear particles
called partons. The observed cross sections turned out to be independent
of the momentum transfer Q2 and obeyed the scaling behavior predicted by
Bjorken [4], whose physical picture was given by Feynman [1] in terms of the
Quark Parton Model (QPM). The explanation of scaling was based on the fact
that a hard collision among partons takes place in such a short time, so that
partons behave as if they were free particles. This requirement of an interaction
that becomes weak (at high energies) led to the consideration of non-Abelian
gauge field theories, which posess the crucial property of asymptotic freedom,
and to propose QCD as the fundamental theory for strong interactions [5]. It is
now well established that QCD is an asymptotically free theory and that the
strong coupling αs(Q2) becomes small when Q2 is large, i.e. at short distances.
However, scale invariance is broken because of quantum corrections and
therefore one expects scaling violations which are calculable in perturbative
QCD. Although the naive Quark Parton Model, describing the nucleon as
composed of three quarks, explains many features of protons and neutrons (built
up from two up- and one down quark or vice versa, respectively) it was facing a
serious problem when the Electron Muon Collaboration (EMC) published its
results in 1988 [30]. That was the start of the so-called “spin crisis” since
it became obvious that the nucleon is more than the sum of parts, in the sense
that three quarks alone do not account for the nucleon spin. In the following
years many experiments followed, determined to solve the question how the
nucleon’s spin is distributed among its constituents. In 1970 Feynmann and
Bjorken developed the so-called Parton Model [1, 2] that describes nucleons
as a loose bound assemblage of partons. Deep inelastic scattering off nucleons
can be replaced by elastic scattering off a parton in this model. The Parton Model
has suffered modifications due to the fact that also gluons and so-called “sea
quarks” (virtually generated quarks) are present. The so-called QCD improved
parton model explains many features of nucleons experimentalists encounter.
Still, the question how much of the nucleon’s spin is actually carried by which
constituent has not been satisfactorily resolved yet. Another aspect of high
energy scattering experiments has been to verify certain sum rules, as those
derived by Bjorken [6] and Ellis–Jaffe [7]. Though the latter seems to be violated
[8, 9, 10], the Bjorken Sum Rule has been confirmed within 8% [12]. It relates the
first moment of the longitudinal polarized structure function g1 to a constant, the
axial vector coupling. A whole lot of new possibilities open up and new ways of
determining fundamental constants may be gone when one assumes the validity
of this sum rule. One of the ultimate important constants for high energy physics
is the strong coupling constant αs. Since the Bjorken Sum Rule relates a quantity
that can be principally measured to a perturbative series in αs one can
extract a value of αs from the measured Bjorken Sum. Previous attempts to do so
[11, 12] usually neglected some of the theoretical obstacles one is facing, as
there are the contributions from higher twist matrix elements, the unknown smallx behaviour of the structure functions or the fact that experimentalists are using
some constant value for αs in order to determine the Bjorken Sum. Here I will take
all of those into account and moreover give a statement which aspects have the
biggest influence on the determination of αs. On the other hand one can ask the
question what predictions for the theoretical unknowns one can make once αs is
fixed and the Bjorken Sum Rule fulfilled. That way one can extract the higher
twist contributions that cannot be derived from first principles or the presently
much discussed small-x behaviour of structure functions which is also not clear
from theory. The latter one turns out to be the largest uncertainty in determining
the Bjorken Sum. That is because one uses a perturbative QCD procedure in
next–to–leading order (NLO) to evolve all data points to one common scale Q2.
This procedure cannot take effects into account that are beyond NLO in QCD.
We will see, that those effects arise especially for small x and prevent the
extraction of αs from the Bjorken Sum Rule with high precision. This unfortunate
outcome may be used to derive a new prediction for the small-x behaviour if one
requires the data to fulfill the Bjorken Sum Rule. This imposes some special
small–x behaviour since the data constrains the structure function g1 quite well in
the low–x regime.
--
The spin-dependent structure function
This page under construction
The spin-dependent structure function g1 is given by the following equation,
.
Bjorken sum rule
The Bjorken sum rule is written as follows using the axial and vector coupling constants
GA and Gv,
.


J.D.Bjorken, Phys Rev. 148 1467 (1966); Phys. Rev. D1 1376 (1970).
J.Kodaira et al., Phys. Rev. D20 627 (1979); Nucl. Phys. B165 129 (1980).
Ellis-Jaffe sum rule
Accoring to the Ellis and Jaffe, another sum rule is deriven from the quark light-cone
algebra with the assumption that strange sea quarks do not contribute to the asymmetry,
.

J. Ellis and R.L.Jaffe, Phys. Rev. D9 (1974) 1444; D 10 (1974) 1669(E).
Recent measurements and analyses of the Deep Inelastic Scattering of polarized muons or
electrons from the polarized nucleons by the EMC [1], SLAC [2] and SMC [3] groups
have shown the deviations from the Ellis-Jaffe sum rule of the experiment data. This
means that the spin of the nucleon (= 1/2) cannot be reproduced from the sum of the spin
of quarks in nucleon: this is called spin crisis .
(Click here to get some plots for g1 integrations from SLAC E143.)



[1] J. Ashman et al.,Phys. Lett. 206 364 (1988); Nucl. Phys. B328 1 (1989)
[2] D.L.Anthony et al., Phys. Rev. lett. 71 959 (1993).
[3] B. Adeva et al., Phys. Lett. B302 533 (1993); Phys. Lett. B320 400 (1994).
To solve Spin Crisis
The possible explanation for the spin crisis is thought to be due to the contributions to the
nucelon spin from



Spin of the sea quarks,
Polarization of the gluons,
Orbital angular momentum (L) of the quarks in nucleon, etc.
PHENIX/Spin collabration is planning to carry out several experiments with polarized
proton-proton collider at the enegy of = 50-500 GeV at RHIC to extract the helicity
distributions of quarks and anti-quarks in nucleon and to investigate gluon polarization,
via