%Paper: 
%From: shunzo KUMANO <kumanos@himiko.cc.saga-u.ac.jp>
%Date: Sun, 14 Nov 93 12:39:02 +0900

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\begin{flushleft}
\large
{SAGA-HE-48  \hfill September 30, 1993}  \\
\end{flushleft}

\vspace{2.5cm}

\noindent
\Large{{\bf 1)~~  Spin and flavor sum rules based on a parton model}} \\

\vspace{0.5cm}

\noindent
\Large{{\bf 2)~~  SU(2) flavor breaking in antiquark distributions}} \\

\vspace{1.8cm}

\begin{center}

\Large
{S. Kumano}         \\

\vspace{1.3cm}

\Large
{Department of Physics} \\

\vspace{0.2cm}

\Large{Saga University}    \\

\vspace{0.2cm}

\Large
{Saga 840, Japan}         \\

\vspace{2.5cm}

\Large
{Invited and Contributed Talks} \\

{at the XIII International
 Conference on Particles and Nuclei} \\

{Perugia, Italy, June 28 $-$ July 2, 1993} \\

\end{center}


\end{document}


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     \documentstyle[12pt,worldsci]{article}
     \pagestyle{empty}
     \begin{document}
     \title{Spin and Flavor Sum Rules Based on a Parton Model}
%     \author{AUTHOR'S NAME\thanks{Footnotes should be typeset in 9 point
%     roman at the bottom of page where cited}\\
     \author{S. Kumano$^*$
%%%            \thanks{Email: kumanos@himiko.cc.saga-u.ac.jp}
              \\
             {\em Department of Physics, Saga University \\
                  Saga 840, Japan}}
%     \vspace{0.3cm}
%     and \\
%     \vspace{0.3cm}
%     SECOND AUTHOR'S NAME\thanks{Representing the WHATEVER Collaboration.}\\
%     {\em Group, Company, Address, City, State ZIP/Zone, Country}}
     \maketitle
     \setlength{\baselineskip}{2.6ex}
%------------------------------------------------------------------------------
%------------------------------------------------------------------------------

\abstract{We discuss a sum rule of the tensor structure function
          $b_1(x)$ for spin-one hadrons along with the Gottfried sum rule.
          Both sum rules are similar in the sense that
          they are phenomenological ones based on a naive
          parton model. As the Gottfried sum rule became
          useful for studying SU(2)$_{flavor}$
          breaking in antiquark distributions, the $b_1$ sum rule
          could provide important information on sea-quark tensor
          polarization.}
\vspace{0.5cm}

One of the important topics at this conference is the nucleon's spin
dependent structure function $g_1(x)$. For measuring $g_1$ of
``the neutron'', we need polarized deuterons (or $^3$He) as the target.
The structure function $g_1$ exists for hadrons with spin$\geq$1/2.
Because the deuteron spin is one,
other spin dependent structure functions exist.
These are named $b_1$, $b_2$, $b_3$, and $b_4$\cite{MIT}.
In the Bjorken limit, the only relevant structure function
is $b_1$ or equivalently $b_2$.

The structure function $b_1$ can be measured by using
a spin-one target
polarized parallel (and antiparallel) to the lepton beam direction.
The lepton does not have to be polarized.
This structure function has not been measured yet.
It could be, in principle,
measured by the SMC group. However, their experimental accuracy
would not be good enough for measuring tiny quantities such as $b_1$.
There are proposals for measuring $b_1$ at HERA\cite{HERA} and at
the proposed
15 GeV European Electron Facility\cite{KUMANO}.

We discuss a sum rule for the tensor structure function $b_1$
in a parton model\cite{CKSUM} along with the Gottfried sum rule\cite{GOTT}.
It should be noted that the $b_1$ sum rule is not a ``strict'' one
such as those derived by current algebra.
It is rather a phenomenological
sum rule based on a naive parton model.
This is because an assumption for sea-quark tensor polarization
needs to be introduced.
The situation is very similar to the Gottfried sum rule,
where the $SU(2)_{flavor}$ symmetric sea is assumed.
As the Gottfried sum rule became an important
topic for investigating $SU(2)_f$ breaking
in antiquarks, the $b_1$ sum rule could become useful
for studying the tensor polarization in sea quarks.

In the parton model, $b_1(x)$ is given by
$
b_1 (x) =
%%%          {\displaystyle \sum_i}
            \sum_i
            e_i^2~
            [~\delta q_i(x)+ \delta \bar q_i(x)~]   ,
$
where the tensor polarization $\delta q_i$ is defined by
$
\delta q_i (x) =  {q_\uparrow ^0} _i (x)
          - {1 \over 2}
          [ {q_{\uparrow i} ^{+1} } (x) + {q_{\uparrow i} ^{-1}} (x) ]
           ~=~  {1 \over 2} [{q_i ^0}  (x)
                            -{q_{i} ^{+1} } (x)]
$
with the target polarization states, $0$ and $\pm 1$.
Using valence-quark distributions in the deuteron,
$u_v^D= u_v^p+ u_v^n= u_v+ d_v$ and
$d_v^D= d_v^p+ d_v^n= d_v+ u_v$,
we obtain the integral of $b_1(x)$ over $x$ as
$
I(b_1^D)\equiv
%%%       {\displaystyle \int}
       \int
       dx  b_1^D (x)
        = {5 \over 9}
%%%       {\displaystyle \int}
       \int
       dx [\delta u_v(x)+\delta d_v(x)]
      +{1 \over {9}} \delta Q_{sea}^D,
$
where
$
\delta Q_{sea}^D=
%%%         {\displaystyle \int}
          \int
          dx
          [8 \delta \bar u (x) +2 \delta \bar d (x)
           +\delta s (x) +\delta \bar s (x)]^D.
$
We try to relate the integral to
the elastic amplitude
$
\Gamma _{H,H} ~=~ <p,H ~|~ J_0 (0) ~|~ p,H>  .
$
First, we calculate this amplitude in an infinite
momentum frame in order to use a quark-parton picture.
The amplitude is then described
in terms of quark distributions in the hadron as
$
\Gamma_{H,H} =
%%%           {\displaystyle \sum_i e_i \int }
               \sum_i e_i \int
                dx
                        [ q_{\uparrow i}^{H} (x)
                         +q_{\downarrow i}^{H} (x)
                         -\bar q_{\uparrow i}^{H} (x)
                         -\bar q_{\downarrow i}^{H} (x)].
$
The tensor combination of the amplitudes is
$
{1 \over 2 }~ [ \Gamma _{00} -
                       {1 \over 2}( \Gamma _{11} +\Gamma_{-1-1})]
          = {1 \over 3}
%%%         {\displaystyle \int}
             \int
            dx
          ~[~\delta u_v (x)+\delta d_v (x)~] .
$
The right hand side is identical to the first term
in the integral of $b_1$, so that the integral becomes
$
I(b_1^D)
         ={5 \over 6} [ \Gamma_{00} - {1 \over 2}
                       (\Gamma_{11}+\Gamma_{-1-1})]
           +{1 \over 9} \delta Q_{sea} .
$
Macroscopically, these amplitudes can be expressed
in terms of charge and quadrupole form factors of
the hadron. However, the tensor combination is given only by
the quadrupole term:
$ [\Gamma_{00}-{1 \over 2} (\Gamma_{11}+\Gamma_{-1-1}) ]/2
           = {\displaystyle \lim_{t \rightarrow 0} }
               -{t /{(4 M^2) }} F_Q(t) $,
where $M$ is the hadron mass and
$F_Q(t)$ is the electric quadrupole form factor
measured in the unit of $e/M^2$.
Using these equations, we finally obtain the integral
as\cite{CKSUM}

\vspace{0.2cm}
\centerline
{$
\displaystyle{
 \int dx~ b_1(x)
         =
             {\displaystyle \lim_{t\rightarrow 0}}
              -{5 \over 3} {t \over {4M^2}} F_Q (t)
           +{1 \over 9} \delta Q_{sea}
{}~~.
}
$}
\vspace{0.2cm}
\indent
This equation is very similar to the Gottfried sum rule.
If the sea is not $SU(2)_f$ symmetric,
the Gottfried sum rule is modified as

\vspace{0.2cm}
\centerline
{$
\displaystyle{
 \int {{dx} \over x}~ [F_2^p(x)-F_2^n(x)]
         = {1 \over 3} + {2 \over 3}
 \int dx ~[\bar u(x)-\bar d(x)]
{}~~.
}
$}
\vspace{0.2cm}

\noindent
As we have the $SU(2)_f$ symmetric sea ($\bar U-\bar D=0$)
in a naive parton model,
the tensor polarization for sea quarks should vanish
($\delta Q _{sea}=0$) in the naive model.
Hence, we call the following equation a sum rule on the same
level with the Gottfried sum rule:

\vspace{0.2cm}
\centerline
{$
\displaystyle{
   \int dx~ b_1(x)
           =   \lim_{t\rightarrow 0}
              -{5 \over 3} {t \over {4M^2}} F_Q (t)
=0~~~.
}
$}
\vspace{0.2cm}
\noindent
The physics meaning of this sum rule is that
the number of valence quarks does not depend
on spin.
As the breaking of the Gottfried sum rule became
an interesting topic recently for studying
the $SU(2)_{flavor}$ breaking $\bar u-\bar d$, it is
worth while investigating a possible mechanism to produce
the tensor polarization $\delta Q_{sea}$, which
breaks the sum rule.
The proposed 15 GeV European Electron Facility is
an appropriate one for measuring $b_1$\cite{KUMANO}.

\vspace{0.5cm}
\noindent
* Email: kumanos@himiko.cc.saga-u.ac.jp

$~~~~~~$

\begin{thebibliography}{90}
\bibitem{MIT}     P. Hoodbhoy, R. L. Jaffe, and A. Manohar,
                           Nucl. Phys. \underline{B312}, 571 (1989).
\vspace{0.18cm}
\bibitem{HERA}    HERMES collaboration, research proposal at HERA (1989).
\vspace{0.18cm}
\bibitem{KUMANO}  S. Kumano, Mainz report MKPH-T-93-03.
\vspace{0.18cm}
\bibitem{CKSUM}   F. E. Close and S. Kumano,
                           Phys. Rev. \underline{D42}, 2377 (1990).
\vspace{0.18cm}
\bibitem{GOTT}    K. Gottfried,
                           Phys. Rev. Lett. \underline{18}, 1174 (1967).
\end{thebibliography}


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     \documentstyle[12pt,worldsci]{article}
     \pagestyle{empty}
     \begin{document}
     \title{SU(2) Flavor Breaking in Antiquark Distributions}
%     \author{AUTHOR'S NAME\thanks{Footnotes should be typeset in 9 point
%     roman at the bottom of page where cited}\\
     \author{S. Kumano$^*$
%%%\thanks{Email: kumanos@himiko.cc.saga-u.ac.jp}
             \\
             {\em Department of Physics, Saga University \\
                  Saga 840, Japan}}
%     \vspace{0.3cm}
%     and \\
%     \vspace{0.3cm}
%     SECOND AUTHOR'S NAME\thanks{Representing the WHATEVER Collaboration.}\\
%     {\em Group, Company, Address, City, State ZIP/Zone, Country}}
     \maketitle
     \setlength{\baselineskip}{2.6ex}
%------------------------------------------------------------------------------
%------------------------------------------------------------------------------

     \abstract{Pionic contributions to the antiquark distribution
               $\bar u-\bar d$ and violation of the Gottfried sum rule
               are discussed.
               We find that the contributions account
               for about half of the discrepancy found by the
               New Muon Collaboration between their result and the sum rule.
               In order to distinguish
               various theoretical explanations, we need independent
               experimental tests, for example, by using Drell-Yan processes.
               As an application of mesonic calculations,
               the difference between strange and antistrange quark
               distributions in the nucleon is briefly discussed.}
     \vspace{0.5cm}

Although the Gottfried sum rule was derived a long time ago,\cite{GOTT}
it was not possible to test it experimentally until recently.
The experimental result
obtained by the New Muon Collaboration (NMC) is practically
the first one in testing the sum rule.
The NMC measured proton and ``neutron" structure functions at very
small $x$, and they obtained a significant deviation
from the Gottfried sum rule:\cite{NMC}

\vspace{0.2cm}
\centerline
{$
\displaystyle{
\int_{0.004}^{0.8}{{dx} \over x} [F_2^{\mu p}(x)-F_2^{\mu n}(x)]=
0.227 \pm 0.007 \pm 0.014
}$}
\vspace{0.2cm}

\noindent
The integral is given in the parton model by

\vspace{0.2cm}
\centerline
{$
\displaystyle{ S_G\equiv
\int_0^1 {{dx} \over x} [F_2^{\mu p}(x)-F_2^{\mu n}(x)]=
{1 \over 3} + {2 \over 3} \int_0^1 dx [\bar u(x)-\bar d(x)]
{}~~,
}$}
\vspace{0.2cm}
\noindent
where isospin symmetry ($u_p=d_n\equiv u$, $d_p=u_n\equiv d$,
                        $\bar u_p=\bar d_n\equiv \bar u$,
                        $\bar d_p=\bar u_n\equiv \bar d$)
is assumed for quark distributions
in the proton and neutron.
If the antiquark distributions are SU(2)$_{flavor}$ symmetric,
the second term vanishes and we obtain the Gottfried sum rule
($S_G=1/3$).
As it is obvious by comparing the above equations,
the NMC result suggests
that the antiquark distribution $\bar u(x)$
in the nucleon is different from $\bar d(x)$.
We have been using $SU(2)_{flavor}$ symmetric
sea-quark distributions in parametrizations
of parton distributions. Therefore, if
the $SU(2)_f$ breaking suggested by the NMC is right,
we need much more serious consideration on flavor distributions
in the nucleon. In this sense, the NMC finding
is very important in parton physics.
However, even if the sum rule violation
is confirmed in future, QCD itself is not in danger.
This is because the sum rule is not a rigorous one
derived from QCD, but it is obtained
in the naive parton model with the $SU(2)_f$ symmetric
sea ($\bar u=\bar d$).

Theoretical situation for explaining the data
is not very clear at this stage.\cite{KUMANO,OTHERS,LIU,DY}
Although the NMC data implies an $SU(2)_f$ breaking
sea $\bar u \ne \bar d$,
it is also possible to explain the data simply by
contributions of
valence-quark distributions at very small $x(<0.004)$.
For example, the HMRS parametrization
predicts 30\% contribution from the $x<0.004$ region,
so that it is consistent with
$\int_{0.004} dx/x[F_2^{\mu p}(x)-F_2^{\mu n}(x)]$
obtained by the NMC.
(However, the NMC obtained $S_G=0.240\pm 0.016$
 by extrapolating their data and it is inconsistent
 with the HMRS value $S_G=1/3$.)
On the other hand, several theoretical models with
explicit $SU(2)_f$ breaking effects have been proposed.
These models include the Pauli principle,
diquark models, and mesonic (pionic) models\cite{KUMANO,OTHERS}.
In the Pauli blocking model, a $\bar d$ excess
over $\bar u$ is expected because
creation of a $u \bar u$ pair
is suppressed relative to $d \bar d$
due to the extra valence up quark in the proton.
Details of these models are given in
original papers listed in Refs. 3$-$6.
In the following, we discuss
the pionic contributions to
the $SU(2)_f$ breaking distribution $\bar u(x)-\bar d(x)$
in Ref. 3.


We consider processes $p \to \pi^+ +n$, $\pi^0 +p$,
$\pi^+ +\Delta^0$, $\pi^0 +\Delta^+$, and $\pi^- +\Delta^{++}$,
where the virtual photon interacts with the pion.
Assuming $SU(2)_f$ symmetry in the pion sea, the
$\bar u -\bar d$ distributions in the pion are given by
$(\bar u-\bar d)_{\pi^+} = - V_\pi$,
$(\bar u-\bar d)_{\pi^0} = 0      $, and
$(\bar u-\bar d)_{\pi^-} = + V_\pi$,
where $V_\pi$ is the valence-quark distribution in the pion.
By including isospin coefficients at the
$\pi$NN/$\pi$N$\Delta$ vertices,
$| \widetilde \phi^*_{\pi^+} \cdot \widetilde \tau |^2 =2  $,
$| \widetilde \phi^*_{\pi^+} \cdot \widetilde  T   |^2 =1/3$, and
$| \widetilde \phi^*_{\pi^-} \cdot \widetilde  T   |^2 =1$,
the isospin times $(\bar u-\bar d)_\pi$ factors are
$-2   V_\pi$ for the $\pi$NN process and
$+2/3 V_\pi$ for $\pi$N$\Delta$. In this way, we find that
the $\pi$NN contribution to $S_G$ is negative and is partly cancelled by
a positive contribution from $\pi$N$\Delta$.
Integrating the pionic contributions over $x$,
we obtain

%%\vspace{0.2cm}
\centerline
{$\displaystyle{
\Delta S_G ={2 \over 3} \int dx ~[\bar u(x)-\bar d(x)]_\pi =-0.04  ~~~,
}$}
\vspace{0.2cm}

\noindent
which accounts for about half of
the discrepancy found by the NMC.
The assumption of $SU(2)_f$ symmetry in the pion sea
may seem contradictory to the $SU(2)_f$ breaking physics,
which we investigate in the nucleon. However, it is not
a problem as long as $x$ is not small ($x>0.1$).
Because of the $1/x$ factor in the integral
$\int (dx/x)x[\bar u-\bar d]$, there is a significant contribution
from the small $x$ region ($x<0.05$). This contribution is roughly
40\%; hence, the above numerical calculation should be
considered a rough estimate.
Although we discussed only $\pi$NN and $\pi$N$\Delta$ processes,
other processes should be investigated.\cite{OTHERS}
In spite of these issues,
it is encouraging that the mesonic model gives
a reasonable value for
the magnitude obtained by the NMC.

In order to distinguish different models, we need
accurate experimental data of $F_2^{\mu p}(x)-F_2^{\mu n}(x)$
in the small $x(<0.004)$ region, neutrino data for the proton
and the deuteron targets, or Drell-Yan data.
Fortunately, the Drell-Yan experiment has been approved
at the Fermi Lab\cite{DY}, so the situation should become
clearer in a few years.
Detailed theoretical and experimental investigations have to be
done in future for clarifying the situation.


Our pionic calculations could be applied to other
flavor distributions,
for example, strange-quark distributions in the nucleon.\cite{HARRACH}
It has been assumed that the strange-quark distribution $s(x)$
is equal to the antistrange distribution $\bar s(x)$.
Because there is no net strangeness in the nucleon,
the integral $\int dx [s(x)-\bar s(x)]$
has to vanish. However, it does not mean that $s(x)$ is
equal to $\bar s(x)$. If we consider the mechanism of
a nucleon splitting into a kaon and a hyperon, two distributions
could be different. The kaon mass is about 500 MeV, so the
virtual kaon is distributed in the length scale 1/(500 MeV)=0.4 fm.
On the other hand, the hyperon is heavier and is distributed
in 1/(1200 MeV)=0.2 fm.
Constituent quark structures in the splitting processes are
$p(uud)\rightarrow K^+(u\bar s) \Lambda (uds)$,
$p\rightarrow K^+(u\bar s) \Sigma^0(uds)$,
and $p\rightarrow K^0(d\bar s) \Sigma^+(uus)$.
In these expressions, we note that the antistrange quark
is in the kaon and that the strange is in the hyperon.
Considering these quark configurations and the hadron masses,
we expect that the antistrange-quark distribution $\bar s(x)$
is softer than the strange distribution $s(x)$.
At this stage, there is no experimental indication
of the difference. However, if accurate experimental
data are taken,\cite{HARRACH} it could be possible to investigate
strange and antistrange quark distributions in the nucleon in detail.

\vspace{0.5cm}
\noindent
* Email: kumanos@himiko.cc.saga-u.ac.jp

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\begin{thebibliography}{90}
\bibitem{GOTT} K. Gottfried, Phys. Rev. \underline{18}, 1174 (1967).
\vspace{0.18cm}
\bibitem{NMC} NMC collaboration (P. Amaudruz et al.) ,
                       Phys. Rev. Lett. \underline{66}, 2712 (1991);
              NMC (A. Arvidson et al),
              paper \#6.18 at this conference.
\vspace{0.18cm}
\bibitem{KUMANO} S. Kumano, Phys. Rev. \underline{D43}, 59 \& 3067 (1991);
                 S. Kumano and J. T. Londergan,
                            Phys. Rev. \underline{D44}, 717 (1991).
\vspace{0.18cm}
\bibitem{OTHERS} Papers based on similar mesonic models are
                 listed in the following. For other theoretical works, see
                 reference sections of listed papers.
                 E. M. Henley and G. A. Miller,
                             Phys. Lett. \underline{B251}, 453 (1990);
                 A. I. Signal, A. W. Schreiber, and A. W. Thomas,
                             Mod. Phys. Lett. \underline{A6}, 271 (1991);
                 W. Melitchnouk, A. W. Thomas, and A. I. Signal,
                             Z. Phys. \underline {A340}, 85 (1991);
                 W.-Y. P. Hwang and J. Speth,
                             Phys. Rev. \underline{D46}, 1198 (1992);
                 S. Szczurek and J. Speth,
                             Nucl. Phys. \underline{A555}, 249 (1993);
                             paper \#6.14 at this conference;
                 W.-Y. P. Hwang, paper \#3.19;
                 M. Wakamatsu, Phys. Rev. \underline{D44}, R2631 (1991);
                                          \underline{D46}, 3762 (1992);
                 E. J. Eichten et al.'s paper in Ref. 6.
\vspace{0.18cm}
\bibitem{LIU}  Other contributions to this conference:
               K. F. Liu and S. J. Dong, preprint UK/93-02 (paper \#3.20);
               S. Koretune, paper \#9.26.
               See also recent investigations:
                   B. Q. Ma, A. Sch\"afer, and W. Greiner,
                            Phys. Rev. \underline{D47}, 51 (1993);
                   R. D. Ball and S. Forte, preprint OUTP-93-18P.
\vspace{0.18cm}
\bibitem{DY}  E772 and E866 collaborations (C. N. Brown et al.),
              paper \#6.30 at this conference.
          For recent progress on the Drell-Yan process,
          see E. J. Eichten, I. Hinchliffe, and C. Quigg,
                    Phys. Rev. \underline{D45}, 2269 (1992);
                               \underline{D47}, R747 (1993);
              W.-Y. P. Hwang, G. T. Garvey, J. M. Moss, and J.-C. Peng,
                    Phys. Rev. \underline{D47}, 2697 (1993).
\vspace{0.18cm}
\bibitem{HARRACH} D. von Harrach, personal communication (1992).
\end{thebibliography}
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