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<B><FONT FACE="Arial" SIZE=4><P ALIGN="CENTER">THE QCD-ORIENTED VECTOR DOMINANCE MODEL</P>
</B></FONT><FONT SIZE=4><P ALIGN="CENTER">E.V.Bugaev</P>
<P ALIGN="CENTER">Institute for Nuclear Research, Moscow, Russia</P>
<P ALIGN="CENTER">B.V.Mangazeev</P>
<P ALIGN="CENTER">Irkutsk State University, Irkutsk, Russia</P>
<B><P ALIGN="CENTER">Abstract</P>
</B></FONT><P>The total photoabsorption cross section <IMG SRC="Image1.gif" WIDTH=43 HEIGHT=33>is studied using nondiagonal GVDM. Vector meson-nucleon scattering amplitudes are calculated in two-gluon exchange approximation of QCD. The off-diagonal transitions of diffraction dissociation type between different vector mesons and their contributions to the expression for <IMG SRC="Image1.gif" WIDTH=43 HEIGHT=33>are also calculated. It is shown that destructive interference of diagonal and off-diagonal terms is not effective. The main conclusion is that a p -cut-off in <IMG SRC="Image2.gif" WIDTH=81 HEIGHT=30>transition is necessary for obtaining the convergence in the summation over all vector meson contributions.</P>
<OL>

<B><FONT SIZE=4><LI>Introduction</LI>
</B><P>The nondiagonal GVDM [1-3] is based on an assumption about existence of a double mass dispersion relation for the forward Compton scattering amplitude [4]. In particular, in case of transversely polarized virtual photons one has</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image3.gif" WIDTH=423 HEIGHT=75><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)</P>
<P>The spectral function <IMG SRC="Image4.gif" WIDTH=31 HEIGHT=33>is expressed in GVDM through imaginary parts of vector meson-nucleon scattering <IMG SRC="Image5.gif" WIDTH=141 HEIGHT=33>amplitudes</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image6.gif" WIDTH=318 HEIGHT=75><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (2)</P>
<P>It is well known, that GVDM approach, using hadron basis describing, e.g., total photoabsorption cross section is dual to the approach, based on covariant parton model (CPM). In CPM the forward Compton amplitude is determined by a handbag diagram [5] which, in turn, is expressed through off-mass shell parton-hadron amplitudes. Further, the hypothesis is used that parton-nucleon amplitudes are strongly damped when any parton line is far from mass shell. As a result, CPM leads to Bjorken scaling feature for <IMG SRC="Image7.gif" WIDTH=91 HEIGHT=38>, while having a nondiagonal spectral function <IMG SRC="Image8.gif" WIDTH=31 HEIGHT=33>(the correct behavior is reached after cancellations of leading diagonal contributions to <IMG SRC="Image9.gif" WIDTH=31 HEIGHT=33>with negative off-diagonal ones). Analogously, in nondiagonal GVDM, the scaling requires destructive interference between the diagonal (<IMG SRC="Image10.gif" WIDTH=80 HEIGHT=25>) and off-diagonal (<IMG SRC="Image11.gif" WIDTH=85 HEIGHT=26>) transitions.</P>
<P>Nowadays, with advent of QCD, it became clear that the use of hadron basis in VDM-type approaches has some limitations. The process of photoabsorption can be divided on two stages: firstly, the production of <IMG SRC="Image12.gif" WIDTH=31 HEIGHT=30>-pair and, then, the interaction of this pair with target nucleon. One can easily show that the average transverse distance between particles of the pair is inversely proportional to the intrinsic transverse momentum of the pair, <IMG SRC="Image13.gif" WIDTH=120 HEIGHT=36>. Evidently, the condition of vector-meson dominating is</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image14.gif" WIDTH=105 HEIGHT=36><FONT SIZE=4>, or <IMG SRC="Image15.gif" WIDTH=178 HEIGHT=68>.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (3)</P>
<P>Here, <IMG SRC="Image16.gif" WIDTH=40 HEIGHT=33>is the average transverse radius of a vector meson. So, for consistent use of GVDM concept the cut-off in the <IMG SRC="Image17.gif" WIDTH=31 HEIGHT=33>must be introduced. Correspondingly this leads to appearing of non-GVDM contributions which can be taken into account by the QCD perturbation theory. </P>
<B><LI>Outline of the model</LI>
</B><P>For a calculation of the spectral function one needs hadronic forward scattering amplitudes. We use the Low-Nussinov [6] QCD model of Pomeron in which this amplitudes are calculated in two-gluon exchange approximation. This is a perturbation approach: the non perturbative effects are simulated only by given an effective mass to the gluons and using an effective value of the cuark-gluon coupling constant.</P>
<P>At high energies, due to large lifetime of <IMG SRC="Image12.gif" WIDTH=31 HEIGHT=30>-fluctuations of the vector meson, the transverse size of virtual <IMG SRC="Image12.gif" WIDTH=31 HEIGHT=30>-pair does not change (i.e., is frozen) in the scattering process. Owing to this, the diagonal amplitude can be expressed as an average of the scattering amplitude of eigenstate having definite size of the pair, over <IMG SRC="Image12.gif" WIDTH=31 HEIGHT=30>-wave function of the vector meson. Further one can easily calculate the off-diagonal amplitudes assuming that <IMG SRC="Image18.gif" WIDTH=85 HEIGHT=26>transition is generated by different absorption of various eigenstates [7].</P>
<P>The wave functions and mass spectra of vector mesons which are necessary for the calculation of the spectral function and <IMG SRC="Image19.gif" WIDTH=35 HEIGHT=33>are obtained from the solution of the reduced Bethe-Salpeter equation describing <IMG SRC="Image20.gif" WIDTH=43 HEIGHT=38>-vertex.</P>
<P>As is discussed above, the transverse momentum cut-off must be introduced leading to decrease of high-mass contributions. The present model shows straightforwardly that there is no strong destructive interference, so that without of p -cut-off the integral in (1) would diverge.</P>
<P>As for nonGVDM contribution to <IMG SRC="Image19.gif" WIDTH=35 HEIGHT=33>, it is important to note that due to color transparency property, the cross section of scattering of small-size <IMG SRC="Image12.gif" WIDTH=31 HEIGHT=30>-pair is small, <IMG SRC="Image21.gif" WIDTH=98 HEIGHT=51>[8]. Correspondingly, the direct (perturbative) contribution to <IMG SRC="Image19.gif" WIDTH=35 HEIGHT=33>is also not large and GVDM remains to be useful approach for a prediction of the photoabsorption cross section and related quantities.</P>
<B><LI>Meson-nucleon scattering amplitudes</LI>
</B><P>The straightforward calculation of the elastic meson-nucleon scattering amplitude using two-gluon exchange diagram leads to the expression:</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image22.gif" WIDTH=323 HEIGHT=66></P>
<P ALIGN="CENTER"><IMG SRC="Image23.gif" WIDTH=645 HEIGHT=178><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (4)</P>
<P>Here, <IMG SRC="Image24.gif" WIDTH=85 HEIGHT=43>is the two-particle (<IMG SRC="Image25.gif" WIDTH=31 HEIGHT=30>) wave function of the meson; <IMG SRC="Image26.gif" WIDTH=23 HEIGHT=36>is the transverse distance between </FONT><FONT SIZE=5>q</FONT><FONT SIZE=4> and <IMG SRC="Image27.gif" WIDTH=20 HEIGHT=30>of the meson; the variable y is equal to  - x, here x is a fraction of longitudinal momentum of the meson, carried by one the quarks. The variable y is conjugate to a longitudinal size between </FONT><FONT SIZE=5>q</FONT><FONT SIZE=4> and <IMG SRC="Image27.gif" WIDTH=20 HEIGHT=30>. Further, <IMG SRC="Image28.gif" WIDTH=20 HEIGHT=25>is the gluon effective mass (parameter of the model), <IMG SRC="Image29.gif" WIDTH=21 HEIGHT=33>is a transverse momentum transfer in the scattering, <IMG SRC="Image30.gif" WIDTH=76 HEIGHT=38>. Transverse moments of two exchange gluons are equal to (<IMG SRC="Image31.gif" WIDTH=23 HEIGHT=35>/2<IMG SRC="Image32.gif" WIDTH=53 HEIGHT=36>). At least, the quantity <IMG SRC="Image33.gif" WIDTH=100 HEIGHT=33>is the factor, containing nucleon wave function. Using the simplest assumptions about these wave functions [9] one obtains for the <IMG SRC="Image34.gif" WIDTH=100 HEIGHT=33>the following expression:</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image35.gif" WIDTH=610 HEIGHT=105>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<FONT SIZE=4>(5)</P>
<P>Here, <IMG SRC="Image36.gif" WIDTH=51 HEIGHT=51>is the mean square radius of the nucleon.</P>
<P>One can rewrite Eq.(4) for the elastic scattering of <IMG SRC="Image37.gif" WIDTH=33 HEIGHT=33>meson on nucleon in the form</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image38.gif" WIDTH=243 HEIGHT=40><FONT SIZE=4>,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (6)</P>
<P>where <IMG SRC="Image39.gif" WIDTH=85 HEIGHT=33>is the scattering amplitude for the eigenstate, i.e. for the <IMG SRC="Image25.gif" WIDTH=31 HEIGHT=30>-pair with the transverse size <IMG SRC="Image40.gif" WIDTH=26 HEIGHT=33>. The brackets in Eq. (6) mean averaging over the wave function.</P>
<P>It is convenient to work in the plane of impact parameters:</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image41.gif" WIDTH=265 HEIGHT=40><FONT SIZE=4>,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(7)</P>
<P>where</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image42.gif" WIDTH=363 HEIGHT=61><FONT SIZE=4>.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (8)</P>
<P>Calculating <IMG SRC="Image43.gif" WIDTH=98 HEIGHT=33>for each vector meson one can parameterize it in the Regge-type form:</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image44.gif" WIDTH=293 HEIGHT=93><FONT SIZE=4>. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(9)</P>
<P>Here, index 2G means two-gluon exchange; <IMG SRC="Image45.gif" WIDTH=46 HEIGHT=33>is the total cross section of <IMG SRC="Image46.gif" WIDTH=53 HEIGHT=33>-interaction and <IMG SRC="Image47.gif" WIDTH=48 HEIGHT=33>is the slope of diffraction cone in elastic<IMG SRC="Image46.gif" WIDTH=53 HEIGHT=33>-scattering.</P>
<P>To take into account the energy dependence of scattering amplitudes the expression (9) must be modified. We use for this aim the two-pole parameterization [10]:</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image48.gif" WIDTH=628 HEIGHT=116><FONT SIZE=4>.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (10)</P>
<P>Here <IMG SRC="Image49.gif" WIDTH=180 HEIGHT=33>, <IMG SRC="Image50.gif" WIDTH=73 HEIGHT=33>and R are constants (<IMG SRC="Image51.gif" WIDTH=123 HEIGHT=33>, where<IMG SRC="Image52.gif" WIDTH=33 HEIGHT=33> is intercept of the F-trajectory, <IMG SRC="Image53.gif" WIDTH=38 HEIGHT=33>is the trajectory slope). It is assumed here, for simplicity, that the slopes of diffraction cones are the same for 2G- and F-terms. If R&gt;&gt;1, the relation contribution of the F-term at low energies is small and Eq.(9) is valid.</P>
<P>Returning to the calculation of forward scattering amplitudes, one has</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image54.gif" WIDTH=15 HEIGHT=30><IMG SRC="Image55.gif" WIDTH=231 HEIGHT=40><FONT SIZE=4>.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(11)</P>
<P>Unitarized amplitude accounting the rescatterings of <IMG SRC="Image12.gif" WIDTH=31 HEIGHT=30>-pair is given by the formula</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image56.gif" WIDTH=376 HEIGHT=123><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (12)</P>
<P>Finally, for off-diagonal amplitudes one has, evidently (<IMG SRC="Image57.gif" WIDTH=63 HEIGHT=25>):</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image58.gif" WIDTH=353 HEIGHT=48><FONT SIZE=4>.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(13)</P>
<P>The imaginary parts of just these T-amplitudes are to be used in the expression (2) for the spectral function.</P>
<B><LI>Mass spectrum and wave functions of vector mesons</LI>
</B><P>In the initial form, the formulas for hadron scattering matrix elements contain <IMG SRC="Image59.gif" WIDTH=41 HEIGHT=38>-vertex functions expressed through Bete-Salpeter <IMG SRC="Image60.gif" WIDTH=31 HEIGHT=30>-wave functions of the hadrons (we use the constituent quark model of hadrons). These wave functions can be found by approximate solution of Bete-Salpeter equation using the null-plane formalism [11] and additional angular condition [12]. The resulting wave functions depend on one variable <IMG SRC="Image61.gif" WIDTH=30 HEIGHT=40>and are solutions of simple equation (<IMG SRC="Image62.gif" WIDTH=195 HEIGHT=40>:</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image63.gif" WIDTH=456 HEIGHT=85><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(14)</P>
<P>Here, M and <IMG SRC="Image64.gif" WIDTH=36 HEIGHT=38>are masses of the meson and quark, respectively, <IMG SRC="Image65.gif" WIDTH=46 HEIGHT=45>and <IMG SRC="Image66.gif" WIDTH=35 HEIGHT=40>are parameters. From formal point of view this is the equation for a wave function of a particle moving in three-dimensional oscillator potential (we assumed here that the confining <IMG SRC="Image67.gif" WIDTH=55 HEIGHT=30>-interaction is of harmonic oscillator type).</P>
<P>The solutions of Eq.(14) are well known. The mass spectrum of the mesons with L=0 (e.g., mesons of <IMG SRC="Image68.gif" WIDTH=18 HEIGHT=25>-family) is</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image69.gif" WIDTH=456 HEIGHT=83><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(15)</P>
<P>Here, <IMG SRC="Image70.gif" WIDTH=18 HEIGHT=31>is given by the definition</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image71.gif" WIDTH=163 HEIGHT=45><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (16)</P>
<P>One can see from Eq.(15) that, if <IMG SRC="Image70.gif" WIDTH=18 HEIGHT=31>and <IMG SRC="Image72.gif" WIDTH=35 HEIGHT=40>are constants, the mass spectrum has the form</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image73.gif" WIDTH=126 HEIGHT=40><FONT SIZE=4>,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(17)</P>
<P>i.e., the spectrum is equidistant for square masses.</P>
<P>&nbsp;</P>
<B><P ALIGN="JUSTIFY"><LI>Cut-off factor</LI></P>
</B><P>One can show that the invariant mass of produced <IMG SRC="Image74.gif" WIDTH=31 HEIGHT=30>-pair is connected with <IMG SRC="Image75.gif" WIDTH=33 HEIGHT=40>and x by simple relation,</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image76.gif" WIDTH=213 HEIGHT=45><FONT SIZE=4>,&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(18)</P>
<P>We will assume, for simplicity, that a meson is formed only if <IMG SRC="Image77.gif" WIDTH=126 HEIGHT=33>and this value <IMG SRC="Image78.gif" WIDTH=70 HEIGHT=33>is the same for all vector mesons (irrespectively on their mass). In this case the relation of phase volume part of produced pair with <IMG SRC="Image77.gif" WIDTH=126 HEIGHT=33>to total phase volume is equal to (for large <IMG SRC="Image79.gif" WIDTH=51 HEIGHT=38>)</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image80.gif" WIDTH=250 HEIGHT=45><FONT SIZE=4>.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(19)</P>
<P>All hadronic amplitudes contained in the expression (2) for the spectral function must be multiplied on these cutting factors (assuming that <IMG SRC="Image81.gif" WIDTH=120 HEIGHT=46>). The value of <IMG SRC="Image78.gif" WIDTH=70 HEIGHT=33>is a parameter of model.</P>
<P>&nbsp;</P>
<B><LI>Results of the calculations and conclusions</LI></OL>

</B><P>We calculated the contribution to <IMG SRC="Image82.gif" WIDTH=136 HEIGHT=40>from mesons of the <IMG SRC="Image68.gif" WIDTH=18 HEIGHT=25>-family (<IMG SRC="Image68.gif" WIDTH=18 HEIGHT=25>,<IMG SRC="Image83.gif" WIDTH=23 HEIGHT=31>,<IMG SRC="Image84.gif" WIDTH=30 HEIGHT=31> etc). Mass spectrum of the mesons is obtained from Eq.(15) if masses of any two family members are known. For <IMG SRC="Image85.gif" WIDTH=110 HEIGHT=38>and <IMG SRC="Image86.gif" WIDTH=110 HEIGHT=38>one obtains from the system of equations</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image87.gif" WIDTH=345 HEIGHT=135><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(20)</P>
<P>the following values of the parameters (<IMG SRC="Image88.gif" WIDTH=95 HEIGHT=38>):</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image89.gif" WIDTH=345 HEIGHT=40></P>
<FONT SIZE=4><P>From here one obtains the mass spectrum</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image90.gif" WIDTH=205 HEIGHT=45><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(21)</P>
<P>and the value <IMG SRC="Image91.gif" WIDTH=93 HEIGHT=33>which does not contradict with the data.</P>
<P>For normalization of the scattering amplitudes we use the relation from the additive quark model,</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image92.gif" WIDTH=221 HEIGHT=61><FONT SIZE=4>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(22)</P>
<P>And two-pole parameterization of Eq.(10) for the <IMG SRC="Image93.gif" WIDTH=31 HEIGHT=25>-amplitudes. Corresponding parameters for <IMG SRC="Image94.gif" WIDTH=31 HEIGHT=25>-amplitude, found from comparison with the data, are:</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image95.gif" WIDTH=141 HEIGHT=38><FONT SIZE=4>; R=10; <IMG SRC="Image96.gif" WIDTH=95 HEIGHT=33>; <IMG SRC="Image97.gif" WIDTH=106 HEIGHT=33></P>
<P>Further, knowing <IMG SRC="Image98.gif" WIDTH=96 HEIGHT=33>we perform in Eqs.(12,13) the changes:</P>
</FONT><P ALIGN="CENTER"><IMG SRC="Image99.gif" WIDTH=371 HEIGHT=70></P>
<FONT SIZE=4><P>For the calculation of <IMG SRC="Image100.gif" WIDTH=40 HEIGHT=33>, <IMG SRC="Image101.gif" WIDTH=43 HEIGHT=33>we use the value <IMG SRC="Image102.gif" WIDTH=78 HEIGHT=33>; this choice leads to more or less acceptable predictions of effective slopes for known mesons.</P>
<P>The calculation showed that, at least in the present model, the cancellations of diagonal and off-diagonal amplitudes are not effective (in particular, amplitudes <IMG SRC="Image103.gif" WIDTH=80 HEIGHT=33>have the same sign as <IMG SRC="Image104.gif" WIDTH=80 HEIGHT=33>and the same order of magnitude). It is clear that, in this case, the use of the <IMG SRC="Image105.gif" WIDTH=33 HEIGHT=33>-cutting procedures is essential for the convergence of the summation over vector mesons. From the comparison with data on <IMG SRC="Image106.gif" WIDTH=148 HEIGHT=38>we obtained (adding <IMG SRC="Image107.gif" WIDTH=46 HEIGHT=25>-contributions without their families) that</P><DIR>
<DIR>

</FONT><P ALIGN="CENTER"><IMG SRC="Image108.gif" WIDTH=183 HEIGHT=33></P></DIR>
</DIR>

<FONT SIZE=4><P>One should note, in conclusion, that probably this physical result depends rather strongly on the model of hadrons used here (constituent quark picture, confining potential of harmonic oscillator type). In principle al required information about hadrons can be taken from experiment (this is an advantage of using hadron basis in VDM approaches) but this information must be very detailed because the off-diagonal transitions can, in principle, be rather large.</P>
<B><P>References</P>
<OL>

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<LI>P.Landshoff et al, Nucl.Phys B28, 225 (1971)</LI>
<LI>F.Low, Phys.Rev. D12, 163; S.Nussinov, Phys.RevLett. 34, 1286 (1975)</LI>
<LI>M.Good and W.Walker, Phys.Rev. 120, 1857 (1960)</LI>
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