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   <TITLE>About Anomalous Magnetic Moment and Structure of Nucleon</TITLE>
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<CENTER><B><FONT SIZE=+2>Can Nucleon Tell Where the Couplings Run?</FONT></B></CENTER>

<CENTER><FONT SIZE=-1>A.V.Kopylov (Kopylov@al20.inr.troitsk.ru)</FONT></CENTER>

<CENTER><FONT SIZE=-1>Institute for Nuclear Research of RAS, Moscow, Russia,</FONT></CENTER>

<CENTER><FONT SIZE=-1>117312, Prospect of 60<SUP>th</SUP> Anniversary of
October Revolution 7a</FONT></CENTER>


<P>In the world of particles neutron and proton play a very special role
being the main building blocks of matter. In some sense they are very similar
in other - very different. They are similar because they are just different
members of one isospin multiplet - nucleon and their masses are very close:
the difference of masses is only about 0.1% of the mass, this is not observed
for other baryons. And they are really very different: one is a charged
particle, another one has no charge; one is stable, another one is not
and so on. The new era of quarks introduced by Zweig and Gell-Mann announced
that their quark contents are uud and udd and made the situation really
intriguing. It looks like we should just compare the observables for neutron
and proton one by one and this will give us all But we have three quarks
and only two particles. The idea proposed by Ida and Kobayashi [1] and
Lingenfeld and Tassie [2] that nucleon is composed of quark and a diquark
gave a balance: two particles and two constituents. Now, the question is:
if to make comparison, step by step, of the properties of nucleons exploiting
the fact that their masses are very close, and trying to use only the well
fixed observables, what one can gain from this comparison? One attempt
to do this was described in Ref.3. If to assume that magnetic moment of
a nucleon can be expressed as the sum of Dirac magnetic moments of different
quarks
<CENTER><B><FONT FACE="Symbol">m</FONT></B> = <FONT FACE="Symbol">m</FONT><SUB>N</SUB>.<B><FONT FACE="Symbol">S</FONT></B>e<SUB>i</SUB>g<SUB>i</SUB><B>J</B><SUB>i</SUB>
(1)</CENTER>


<P>here e<SUB>i</SUB>, g<SUB>i</SUB> and <B>J</B><SUB>i</SUB> are the charge,
factor of Lande and moment of a constituent quark, <FONT FACE="Symbol">m</FONT><SUB>N</SUB>
<FONT FACE="Symbol">-</FONT> is a nuclear magneton, then one can write
the system of two equations: for magnetic moments of neutron and proton
as different members of one isospin multiplet. The expression (1) assumes
that the only representative mass parameter here is a mass of a nucleon
which determines the overall scale. This supposition is quite natural for
confined quarks.

<P>So if to take into account that diquark can exist as a pair of like
quarks and as a pair of different quarks two systems of equations can be
written:

<P>Case I: n = u + dd p = d + uu

<P>&nbsp;
<CENTER><IMG SRC="coup1.gif" HEIGHT=87 WIDTH=174></CENTER>

<CENTER>&nbsp;</CENTER>


<P>From here it follows that: g<SUB>1</SUB>J<SUB>1</SUB> <FONT FACE="Symbol">=</FONT>
-1.033 and g<SUB>2</SUB>J<SUB>2</SUB> <FONT FACE="Symbol">=1.836</FONT>

<P>The fact that g<SUB>1</SUB>J<SUB>1</SUB> turned out to be very close
to 1 means that the single quark acts as elementary particle placed at
the center of nucleon, some extra of 0.033 may be assumed here as a correction
similar to the Schwinger correction to magnetic moment of electron.

<P>Case II. The similar equations can be written for the case when diquark
<B>ud</B> is in a state <SUP>3</SUP>S<SUB>1</SUB>, only here g<SUB>1</SUB>J<SUB>1</SUB>
is the value assigned to the single quarks, constituents of a diquark <B>ud</B>:

<P>n <FONT FACE="Symbol">=</FONT> ud + d p <FONT FACE="Symbol">=</FONT>
ud + u
<BR>&nbsp;
<CENTER><IMG SRC="coup2.gif" HEIGHT=87 WIDTH=178></CENTER>

<CENTER>&nbsp;</CENTER>
To satisfy this system one should take: g<SUB>1</SUB>J<SUB>1</SUB> <FONT FACE="Symbol">=</FONT>
-1.033 and g<SUB>2</SUB>J<SUB>2</SUB> <FONT FACE="Symbol">=4.7</FONT>.
The fact that g<SUB>1</SUB>J<SUB>1</SUB> turned out to be close to 1 (with
the same correction) agrees perfectly with the assumption <SUP>3</SUP>S<SUB>1</SUB>.
Generally we have two more states in this case: one with a spin of a peripheral
quark up, and another one - with a spin down.

<P>The argument in favor that the model presented here is realistic is
that one finds agreement with the measured magnetic moments of both nucleons
if quarks at the center of nucleon act as elementary particles, this is
observed for all three states. Qualitatively this conforms the parton model
of the nucleon and <I>x</I>-scaling behavior in deep-inelastic lepton scattering.
A notable thing is also that if to consider a center of nucleon as the
place of the highest priority, the couple of mixed quarks is stronger
then a single quark which is stronger then a couple of like quarks. This
explains why there is no place for any symmetrization in this scheme. Figure
1 of Ref.1 illustrates these states. The question why these states can
be the stationary states is discussed in Ref. 4. Of course the arguments
presented here are very speculative but if these states are real and will
be found [5,6], this can be important in view of their possible application
in practice.

<P>Another question: the correction factor of 0.033 obtained here, does
it contain some significant piece of information? It appears to be very
similar to the Schwinger correction which is known to be <FONT FACE="Symbol">a</FONT>+o(<FONT FACE="Symbol">a</FONT>),
here <FONT FACE="Symbol">a=</FONT>e<FONT FACE="Symbol"><SUP>2</SUP>/</FONT>hc.
The wondrous thing is that this correction 0.033 is very close by magnitude
to the universal coupling, i.e. the value where all the couplings unite
around <FONT FACE="Symbol">a<SUP>-1</SUP>=30</FONT>.<FONT FACE="Symbol">
</FONT>Certainly, we have no reasons to claim this, it is just our fantasy
based on a free comparison of different values. But what will happen if
the value where all the couplings unite is changed, will it not produce
the change in the observables of a nucleon? This turnover probably does
not look like a great surprise because the undisturbed nucleon is a stationary
system which should be described by fixed coupling. By contrast the scattering
on quarks is a dynamic process having a certain scale, hence it should
be described by running coupling taken at this scale. The case of <FONT FACE="Symbol">a<SUP>-1</SUP>=30
</FONT>corresponds to nearly constant value of <FONT FACE="Symbol">a</FONT><SUB>2</SUB>
what generally is not excluded because it is expected for example in some
of the realistic string models [7].

<P><B>References</B>.
<OL>
<LI>
Ida M. and Kobayashi R. <I>Prog.Theor.Phys.</I> <B>36</B>, 846 (1966).</LI>

<LI>
D.B.Lichtenberg and L.J.Tassie, <I>Phys.Rev</I>. <B>155</B>, 1601 (1967)</LI>

<LI>
A.V. Kopylov , .</LI>

<LI>
A.V. Kopylov , </LI>

<LI>
A.V. Kopylov , .</LI>

<LI>
A.V. Kopylov , </LI>

<LI>
K.R.Dienes, <I>Phys. Rep</I>. <B>287,</B> 448-525 (1997)</LI>
</OL>

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