On Spin Content of the Proton
G. Musulmanbekov
Join Institute for Nuclear Research, Dubna
Abstract. It is shown, in the frame of the model proposed by author, that spin of the proton could
be described by the orbital momentum of quark and qluon condensate circulating around valence
quarks.
In this paper we describe the spin of a nucleon as arising from the orbital momentum
of quark and gluon condensate circulating around valence quarks. Considerations are
performed in the frame of so–called Strongly Correlated Quark Model (SCQM) [1].
The ingredients of the model are the following. Single quark of definite color embedded
in vacuum begins to polarize its surrounding that results in formation of quark and gluon
condensate. At the same time it experiences the pressure of the vacuum because of zero
point radiation field or vacuum fluctuations which act the quark tending to destroy the
ordering of the condensate. Suppose that we place the corresponding antiquark in the
vicinity of the first one. Owing to their opposite signs color polarization fields of quark
and antiquark interfere destructively in the overlapped space regions eliminating each
other at most in space around the middle–point between the quarks. This effect leads
to decreasing of condensates density in the same space region and overbalancing of the
vacuum pressure acting on quark and antiquark from outer space regions. As a result
the attractive force between quark and antiquark emerges and quark and antiquark start
to move towards each other. The density of the remaining condensate around quark
(antiquark) is identified with hadronic matter distribution. At maximum displacement
in qq system, that corresponds to small overlapping of polarization fields, hadronic
matter distributions have maximum extent and values. The closer they to each other, the
larger destructive interference effect and the smaller hadronic matter distributions are
around quarks and the larger their kinetic energies. In that way quark and antiquark start
to oscillate around their middle–point. For such interacting qq pair located on X axis
at the distance 2x from each other the total Hamiltonian is
H
mq
mq
1 1 β 212 Vqq2x
(1)
β 2 12
were mq , mq current masses of valence antiquark and quark, β β x their velocity
depending on displacement x and Vqq quark–antiquark potential energy with separation
2x It can be rewritten as
H
mq
1 β 212 U x mq
1 β 212 U x
Hq Hq
CP675, Spin 2002: 15th Int'l. Spin Physics Symposium and Workshop on Polarized Electron
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358
(2)
were U x 12 Vqq 2x is potential energy of quark or antiquark. Quark (antiquark) with
the surrounding cloud (condensate) of quark – antiquark pairs and gluons, or hadronic
matter distribution, forms the constituent quark. It is natural to assume that the potential
energy of quark (antiquark), U x corresponds to the mass MQ of constituent quark:
2U x C1
∞
∞
dz
∞
∞
dy
∞
∞
dx ρ x r 2MQ x
(3)
where C1 is dimensional constant and hadronic matter density distribution, ρ x r is
defined as
ρ x r C2 ϕ x r C2 ϕQ x x y z ϕQ x x y z (4)
Here C2 is a constant, ϕQ and ϕQ are density profiles of the condensates around quark
and antiquark located at distance 2x from each other. We consider by convention the
condensates around quark and antiquark having opposite color charges. They look like
compressive stress and tensile stress (around defects) in solids. Generalization to three–
quark system in baryons is performed according to SU 3 color symmetry: in general,
pair of quarks have coupled representations
33 63
in SU 3color and for quarks within the same baryon only the 3 (antisymmetric) representation occurs. Hence, an antiquark can be replaced by two correspondingly colored
quarks to get color singlet baryon and destructive interference takes place between color
fields of three valence quarks (VQs). Putting aside the mass and charge differences of valence quarks we may say that inside baryon three quarks oscillate along the bisectors of
equilateral triangle. Therefore, keeping in mind that quark and antiquark in mesons and
three quarks in baryons are strongly correlated, we can consider each of them separately
as undergoing oscillatory motion under the potential (3) in 1+1 dimension. Hereinafter
we consider VQ oscillating along X axis and Z axis is perpendicular to the plane
of oscillation XY . Density profiles of condensates around VQs are taken in gaussian
form. It has been shown in papers [2] that the wave packet solutions of time dependent Schrodinger equation for harmonic oscillator move in exactly the same way as
corresponding classical oscillators. These solutions are called ”coherent states”. This
relationship justifies (partly) our semiclassical treatment of quantum objects.
We define the mass of constituent quark at maximum displacement as
MQQ xmax 1
3
m∆ mN
2
360 MeV
where m∆ and mN are masses delta–isobar and nucleon correspondingly. The parameters
of the model, namely, maximum displacement, x max and parameters of 3–D gaussian
function, σxyz for hadronic matter distribution around VQ are chosen to be
xmax 064 f m σxy 024 f m σz 012 f m
359
(5)
They are adjusted by comparison of calculated and experimental values of inelastic cross
sections, σin s and inelastic overlap function Gin s b for pp and pp collisions [3].
The current mass of valence quark is taken to be 5 MeV . The behavior of potential (3)
evidently demonstrates the relationship between constituent and current quark states inside a hadron (Fig. 1). At maximum displacement quark is nonrelativistic, constituent
one (VQ surrounded by condensate), since the influence of polarization fields of other
quarks becomes minimal and the VQ possesses the maximal potential energy corresponding to the mass of constituent quark. At the origin of oscillation, x 0 antiquark
and quark in mesons and three quarks in baryons, being close to each other, have maximum kinetic energy and correspondingly minimum potential energy and mass: they
are relativistic, current quarks (bare VQs). This configuration corresponds to so called
”asymptotic freedom”. In the intermediate region there is increasing (decreasing) of constituent quark mass by dressing (undressing) of VQs due to decreasing (increasing) of
the destructive interference effect. This mechanism meets local gauge invariance principle. Indeed, destructive interference of color fields of quark and antiquark in mesons
and three quarks in baryons depending on their displacements can be treated as phase
rotation of wave function of single VQ in color space ψ c on angle θ depending on displacement x of the VQ in coordinate space
ψc x eigθ x ψc x
(6)
Phase rotation,in turn, leads VQ dressing (undressing) by quark and gluon condensate
that corresponds to the transformation of gauge field A µ ϕ 0 0 0
Aµ x Aµ x ∂µ θ x
(7)
Here we drop color indices of Aµ x and consider each quark of specific color separately
as changing its effective color charge, gθ x in color fields of other quarks (antiquark)
due to destructive interference. Thus gauge transformations (6, 7) map internal (isotopic)
space of colored quark onto coordinate space. On the other hand this dynamical picture
of VQ dressing (undressing) corresponds to chiral symmetry breaking (restoration). Due
to this mechanism of VQs oscillations nucleon runs over the states corresponding to the
specific terms of the infinite series of Fock space
B a1 q1q2q3 a2 q1 q2q3qq
The proposed model has some important consequences. Inside hadrons quarks and
accompanying them gluons, as well, are strongly correlated. Nucleons are nonspherical
object: they are flattened along the axis perpendicular to the plane of quarks oscillations.
There are two reasons of that: first, because VQs undergo plane oscillations and second,
owing to the flatness of hadronic matter distributions around VQs (according to (5)).
From the form of the quark potential (Fig.1) one can conclude that dynamics of VQ
corresponds to nonlinear oscillator and VQ with its surrounding can be treated as
nonlinear wave packet. Moreover, our quark – antiquark system turned out to be identical
to, so called, ”breather” solution of (nonlinear) sine–Gordon (SG) equation [4].
So far we dealt with scalar polarization field around VQ. How can one include spin in
the frame of these classical considerations? According to the prevailing belief, the spin
360
0.4
2
b)
F, GeV/fm
U, M, GeV
a)
0.2
0
0
0.5
1
0
1
0
0.5
x, fm
1
x, fm
FIGURE 1. a) Potential energy of valence quark and mass of constituent quark; b) ”Confinement” force.
is quantum feature of microparticle and has no classical analog. However, Belinfante
[5] as early as in 1939 showed that the spin of electron may be regarded as an angular
momentum generated by a circulating flow of energy, or a momentum density, in the
wave field of electron. Furthermore, a comparison between calculations of angular
momentum in the Dirac and electromagnetic fields, performed by Ohanian [6], shows
that the spin of the electron is entirely analogous to the angular momentum carried by
classical circularly polarized wave.
We follow the classical picture of electron spin considered by R. Feynman, and later,
Hiqbie [7, 8]. Let us imagine point electric charge, e, and magnetic bar, µ , settled close
nearby [8]. In surrounding space electric, E, and magnetic, B, fields arise. Although
vectors E and B are constant the circulating flow of energy is created around such a
system which is described by Poynting’s vector
S ε0 c2 E B
(8)
The volume density of angular momentum is given by
L r P
(9)
where P is the volume density of linear momentum defined as
P Sc2 D B r0 r µ qr µ 4π c2ε0r6 q m4π r6
(10)
Here r0 q2e 4πε0me c2 classical radius of electron. Substitution of (12) into (11) gives
L r P r r µ rr µ µ r2
(11)
When we align µ with Z axis and use spherical polar coordinates, z component of L
is
361
Lz µ r2 sin2 θ (12)
s µ qe me (14)
The total angular momentum of the field extending from r a to infinity is given by the
integral
∞
r0 µ
Lz dV 23
(13)
s
a
q
a
eme For a 23r0 we have
which just happens to be the entire spin angular momentum of the electron. Moreover,
Feynman showed that if we extend the classical coulomb field all the way down to 23
of the classical electron radius, r0 the entire mass of the electron is contained in its field.
By analogy we suppose that intersecting chromoelectric, Ech , and chromomagnetic,
Bch , fields create around quark (antiquark) circulating flow of energy, the color analog
of Poynting’s vector
Sch c2 Ech Bch (15)
Spin of quark (antiquark) then is defined by the integral
∞
s
r Ech Bch d 3 r
(16)
a
Obviously, lower limit of integration, a is connected with the size of quark (antiquark).
Now we must take into account correlated oscillations of quarks inside hadrons which
lead to periodic changes in values of Ech and Bch . At maximal displacement of quark
from the origin of oscillation Ech and Bch around it are expanded to the maximum
(size of constituent quark) and so is for circulating flow Sch . At minimum displacement
Ech and Bch , owing to destructive interference described above, are confined in small
region around the quark and circulating flow of energy Sch is concentrated in a narrow
shell. For consistency of the model we must demand that spins of quarks are perpendicular to the plane of oscillation. Recalling that entire spin of quark is due total angular
momentum of circulating flow we compare it with circular flow of ideal liquid and apply
hydrodynamic considerations. Equations of hydrodynamics for ideal liquid flow read
∂ξ
∂t
∇ ξ v 0
ξ ∇ v
(17)
(18)
where v is vector of liquid velocity and ξ is its vorticity defining circulation around unit
area. Equation (17) implies angular momentum conservation law applied to liquid flow.
In other words, if σ is the area of vortex field in ideal liquid then
ξ σ vdr const (19)
σ
362
that means the larger area of circulation the less the vorticity and vice versa. In our
case v is velocity of circulation flow depending on the distance from the vortex center.
Following these considerations we assume that the spin of quark is conserved quantity
during oscillation and
s ∝ ξ σ
(20)
Rigorously, the case at hand is the conservation of quark spin module because quarks
inside hadronic system can interact and what is conserved is the total angular momentum
of the system. Thus, according to our approach quark spin is given by circulating flow of
polarized vacuum (sea quarks and gluons) and recalling that entire mass of constituent
quark (antiquark) is contained in it’s color field one can consider quarks as to be
vortices in vacuum. Now gauge field A µ x in (7) contains along with the scalar part,
ϕ , vector components, A, as well. Whereas vortex is singularity (hole) in vacuum the
quantization of spin follows from nonsimple-connectedness of vacuum structure. The
representation of quark spin as created by circulating flow of quark and gluon condensate
gives alternative solution to ”spin crisis” ,[9], that implies the essential deviation of
quantity
∆Σ ∆u ∆d ∆s
(21)
defined as the relative amount of the nucleon spin carried by intrinsic spins of the
quarks, from one. The effect gets clear explanation in proposed picture. In DIS the
transverse size of virtual photon d emitted by incident lepton is given by d 2 ∝ 1Q2 .
Because of potential (irrotational) character of velocity field in (19) contribution to ∆Σ
in specific interaction at random impact parameter depends on if the center of quark
vortex (singularity) falls into the area covered by transverse size of virtual photon. If it
is the circled integral (19) gets nonzero value proportional to spin of valence quark and
this occurrence gives contribution to ∆Σ. Otherwise the integral equals zero and there is
no contribution to (21). This consideration reminds the well known Aharonov – Bohm
effect. Preliminary estimation gives ∆Σ 002 at Q 2 5 GeV c2 . If Q2 0 then
∆Σ 1. Therefore, according to our approach proton spin is defined by spins of valence
quarks, as naive quark model gives , and the (intrinsic) spin of valence quark, in turn, is
defined by orbital momentum of circulating flow of quark and gluon condensate around
it. Irrotational character of the condensate velocity field suggests the zero polarization
of sea quarks. Experimental nonzero value of ∆Σ (0.2–0.3), however, could be the
manifestation of nonzero polarization of sea quarks. To study the contribution of sea
quarks to the spin of nucleons one needs to use microscopic treatment of the mechanism
of condensation (e. g., instanton mechanism).
As noted above, in our scheme nucleons are nonspherical, flattened objects. This
geometrical feature of extended nucleons can lead to observable effects. Since inside
a proton spins of quarks are perpendicular to the plane of their oscillations there should
be the difference between
σ σ and transversal σtot 12 σ σ total cross section in polarized (anti)proton–proton collisions at high energies.
L and σ with the data on σ
Calculated values of σtot σtot
unpolarized proton–
tot for
tot
proton and antiproton–proton collisions in the energy interval s 10 103 GeV
are shown in Fig 2. Details of calculations are given in paper [10]. One can see that
L
longitudinal σtot
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1
2
FIGURE 2. Total cross sections: dot–dashed curve is for longitudinally polarized protons; solid curve
– for unpolarized protons; dotted curve – for transversely polarized protons; data are taken from Review
of Part. Prop.
the differences between these three cross section are significant. The measurements of
these cross sections could be performed at RHIC with longitudinally and transversally
polarized proton–proton beams.
This research was partly supported by the Russian Foundation of Basics Research,
grant 01-07-90144.
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2.
3.
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Singapore, 2000, p. 341–351.
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2001, p. 109–120.
5. F.J. Belinfante, Physica 6, 887 (1939).
6. H.C. Ohanian, Am. J. Phys., 54, 500 (1986).
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2002, p. 339–346.
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