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it was noted many years ago that the form of the @xmath8-meson peak is distorted by the interference between resonant and non - resonant @xmath0 production . for the case of `` elastic '' @xmath1 photoproduction the effect was studied by p.sding in @xcite and s.drell @xcite ( who considered the possibility to produce the pion    beam via the @xmath9 process ) . at high energies the main ( and the only ) source of background is the drell - hiida - deck process @xcite ( see fig . the incoming photon fluctuates into the pion pair and then @xmath10-elastic scattering takes place . thus the amplitude for the background may be written in terms of the pion - proton cross section . recently the difractive elastic production of @xmath1-mesons was measured at hera @xcite both for the cases of photoproduction i.e. @xmath11 and of @xmath12 gev@xmath13 ( the so called deep inelastic scattering , dis , regime ) . it was demonstrated @xcite that the interference with some non - resonant background is indeed needed to describe the distribution over the mass - @xmath14 of @xmath0 pair .    in this paper we present the results of calculation of @xmath15 with correction the numerical error published in fig.6 of @xcite . in sect . 2 the formulae for the @xmath5 background which are valid for the dis region are presented . the expression differs slightly from the sding s one as we take into account the pion form factor and the fact that one pion propagator is off - mass shell . we consider also the absorbtion correction comming from the diagram where both pions ( @xmath16 and @xmath17 ) directly interact with the target proton . in sect . 3 we compute the ratio @xmath15 for a pion pair production in dis . at large @xmath18 gev@xmath13 the background amplitude becomes relatively small , but still not negligible . it changes the ratio of the longitudinal to transverse @xmath8-meson production cross section and leads to the decreasing of @xmath15 value with @xmath19 . the cross section of @xmath1 photo- and electroproduction may be written as : @xmath20 are the resonant and non - resonant parts of the production amplitude , @xmath21 for longitudinal and transverse photons , @xmath22 -*q*@xmath23 is the momentum transfered to the proton and @xmath24 , where @xmath25 and @xmath26 are the azimutal and polar angles between the @xmath16 and the proton direction in the @xmath5 rest frame . the dynamics of vector meson photo- and electroproduction was discussed in the framework of qcd in many papers ( see , e.g. [ 9 - 12 ] ) . however here we will use the simple phenomenological parametrization of the production amplitude because our main aim is the discussion of the interference between resonant and non - resonant contributions . so the amplitude for resonant process @xmath27 ; @xmath28 reads : @xmath29 to take into account the phase space available for the @xmath30 decay we use the width @xmath31 ( with @xmath32 mev and @xmath33 mev  its mass ) ; @xmath34 is the @xmath35-slope of the `` elastic '' @xmath8 production cross section @xmath36 ( at @xmath37 ) and the functions @xmath38 describe the angular distribution of the pions produced through the @xmath8-meson decay : @xmath39 @xmath40 note that for transverse photons with polarization vector @xmath41 one has to replace the last factor @xmath42 in eq . ( 4 ) by the scalar product @xmath43 , where @xmath44 is the unit vector in the pion transverse momentum direction . the amplitude for the non - resonant process @xmath45 is : @xmath46 where @xmath47 is the @xmath35-slope of the elastic @xmath10 cross section , @xmath48 is the pion electromagnetic form factor ( @xmath49 is the virtuality of the incoming photon ) , @xmath50 is the electromagnetic coupling constant and @xmath51  the photon momentum fraction carried by the @xmath17 -meson ; @xmath52 is the total pion - proton cross section . the factor @xmath53 is equal to @xmath54 for longitudinal photons the products @xmath55 are : @xmath56 and @xmath57 , while for the transverse photons we may put ( after averaging ) @xmath58 . expressions ( 5 ) and ( 6 ) are the result of straitforward calculation of the feynman diagram fig . the first term in ( 6 ) comes from the graph fig . 1 ( in which the pomeron couples to the @xmath16 ) and the second one reflects the contribution originated by the @xmath59 interaction . the negative sign of @xmath17 electric charge leads to the minus sign of the second term . we omit here the phases of the amplitudes . in fact , the common phase is inessential for the cross section , and we assume that the relative phase between @xmath60 and @xmath61 is small ( equal to zero ) as in both cases the phase is generated by the same pomeron exchange . the form factor @xmath62 is written to account for the virtuality ( @xmath63 ) of the t - channel ( vertical in fig . 1 ) pion . as in fig . 1 we do not deal with pure elastic pion - proton scattering , the amplitude may be slightly suppressed by the fact that the incoming pion is off - mass shell . to estimate this suppression we include the form factor ( chosen in the pole form ) @xmath64 the same pole form was used for @xmath65 . in the last case the parameter @xmath66 is the mass of the @xmath8-meson  the first resonance on the @xmath8-meson ( i.e. photon ) regge trajectory , but the value of @xmath67 ( in @xmath62 ) is expected to be larger . it should be of the order of mass of the next resonance from the regge @xmath68-meson trajectory ; i.e. it should be the mass of @xmath69 or @xmath70 . thus we put @xmath71 gev@xmath13 . finaly we have to define @xmath72 and @xmath73 . @xmath74 and @xmath75 in these notations @xmath76 and @xmath77 with the pion transverse ( with respect to the proton direction ) momentum @xmath78 ( in the @xmath5 rest frame ) given by expression @xmath79 . note that the positive values of @xmath80 correspond to @xmath81 while the negative ones @xmath82 correspond to @xmath83 .      to account for the screening correction we have to consider the diagram fig . 2 , where both pions interact directly with the target . note that all the rescatterings of one pion ( say @xmath16 in fig . 1 ) are already included into the @xmath10 elastic amplitude . the result may be written in form of eq . ( 5 ) with the new factor @xmath84 instead of the old one @xmath85 : @xmath86 where the second term is the absorptive correction ( fig . 2 ) and @xmath87 is the momentum transfered along the pomeron loop . the factor @xmath88 reflects the contribution of the enhacement graphs with the diffractive exitation of the target proton in intermediate state . in accordance with the hera data @xcite , where the cross section of `` inelastic '' ( i.e. with the proton diffracted ) @xmath8 photoproduction was estimated as @xmath89 we choose @xmath90 . at very large @xmath2 the background amplitude ( 5 ) becomes negligible as , even without the additional form factor ( i.e. at @xmath91 ) , the non - resonance cross section falls down as @xmath92 , one factor @xmath93 comes from the electromagnetic form factor @xmath94 and another one  from the pion propagator ( term - @xmath95 in the denominator of @xmath53 ( see eq.(6 ) ) . ] while experimentally @xcite the @xmath2 behaviour of the elastic @xmath8 cross sections has been found to be @xmath96 , with @xmath97 . nevertheless , numerically at @xmath98 the background as well as interference contributions are still important . the background as well as interference contributions lead to the nontrivial behaviour of the ratio @xmath15 with the two pion mass @xmath99 . in the theoretical formulae which we used the index @xmath100 denotes the polarization of the incoming photon . on the other hand experimentally one measures the @xmath8-meson polarization , fitting the angular distribution of decay pions . to reproduce the procedure we take the flows of initial longitudinal and transverse photons to be equal to each other ( @xmath101 , which is close to hera case ) and reanalyse the sum of cross sections ( @xmath102 ) in a usual way , selecting the constant and the @xmath103 parts . @xmath104 in these terms the density matrix element @xmath105 and @xmath106 namely these last ratios ( 12 ) are presented in fig . 3 for different @xmath2 values . presented in fig . 6 of our previous paper @xcite were wrong due to a misprint in the code . ] we have to recall , that the ratio @xmath107 given by the last expression of eq . ( 12 ) strictly speaking is not identical to the ratio of the longitudinal to transverse photon cross sections @xmath4 . due to the form factor @xmath62 the background pion angular distribution ( corresponding to the non - resonant amplitude ( 5 ) , ( 6 ) ) are more complicated than the trivial @xmath108 ( for @xmath109 ) and @xmath110 ( for @xmath111 ) . for example , even in the case of the transverse photon polarization vector ( i.e. @xmath109 ) at very large @xmath2 the non - resonant contribution @xmath112 reveals the peaks in the forward and backward ( @xmath113 ) directions instead of a pure @xmath114 behaviour . that is why in fig . 3 we present not the theoretical @xmath4 ratio , but the values of @xmath115 which are more close to the ratios measured experimentally . one can see that they depend both on @xmath2 and @xmath14 . note that for the resonant amplitude @xmath60 in fig . 3 we have fixed @xmath116 independently on the mass @xmath14 and @xmath2 . on the other hand for the non - resonant production ( @xmath112 ) , neglecting for simplicity the form factor ( i.e. @xmath117 ) and the absorptive corrections , one has @xmath118 .    at large @xmath119 10 gev@xmath13 it gives @xmath120 . at last we have to take into account the interference , which is positive ( constructive ) at small @xmath121 and destructive at @xmath122 . thus the final curves 3 and 4 fall down with @xmath14 giving @xmath123 at @xmath124 gev and @xmath125 at @xmath126 gev . of course at very large @xmath2 the background amplitude dies out but simultaneously the value of @xmath127 increases with @xmath2 . therefore the curves 3 and 4 ( which correspondent to @xmath2 = 10 gev@xmath13 and 30 gev@xmath13 ) are rather close to each other .    in the region of not too large @xmath128 gev@xmath13 the non - resonant amplitude is compartible with the resonant one at @xmath129 gev . in particular , for @xmath130 gev@xmath13 ( curve 1 ) the destructive interference at @xmath131 gev cancel the main part of the transverse cross section , leading to a very large value of @xmath107 ( @xmath132 ) . this effect depends on the @xmath2 value . say , the maximum of the peak of @xmath107 is at @xmath133 0.03 gev@xmath13 for @xmath14 = 1.15 gev and at @xmath133 0.3 gev@xmath13 for @xmath14 = 1.25 gev . however one must not take the presented predictions in fig . 3 too seriously for large @xmath14 . the interference with the next ( heavier ) resonances ( say , @xmath134 ) was not taken into account in fig . 3 but at @xmath135 gev their contribution can be significant that can change the results . we presented the results of @xmath15 ratio calculations for @xmath8-meson electroproduction with accounting for the background to elastic mechanism . the role of @xmath8-meson  background interference decreases with @xmath2 however it is not negligible even at @xmath136 gev@xmath13 .            fig . 3 . the ratio @xmath15 in electroproduction process as a function of pion pair mass at @xmath137 1 gev@xmath13 ( curve 1 ) , 4 gev@xmath13 ( curve 2 ) , 10 gev@xmath13 ( curve 3 ) and 30 gev@xmath13 ( curves 4 ) with ( solid curve ) and without ( dashed curve ) form factor .    99 sding . ( 1966 ) 702 . lett . 5 ( 1967 ) 278 . ( 1964 ) 169 . m.derrick et al . ( zeus coll . ) c69 ( 1995 ) 39 . m.derrick et al . ( zeus coll . ) b356 ( 1995 ) 601 . s.aid et al . ( h1 coll . ) b463 ( 1996 ) 3 . s.aid et al . ( h1 coll . ) b468 ( 1996 ) 3 . m.g.ryskin and yu.m.shabelski . hep - ph/9701407 . s.j.brodsky et al . d50 ( 1994)3134 . b.z.kopeliovich , j.nemchick and n.n.nikolaev . b324 ( 1994 ) 177 . a.donnachie and p.v.landshoff . b348 ( 1995 ) 213 . a.d.martin , m.g.ryskin and t.teubner . preprint dtp/96/64 ; hep - ph/9609448 . s.aid et al . ( h1 coll . ) c69 ( 1995 ) 27 .
the background due to the direct diffractive dissociation of the photon into the @xmath0-pair to the `` elastic '' diffractive @xmath1-meson production in electron - proton collisions is calculated . at large @xmath2 the interference between resonant and non - resonant 
 @xmath3 production changes the @xmath4 ratio with the mass of the @xmath5 ( i.e. @xmath1-meson ) state . 
 1.5 truecm    @xmath6 * in the * @xmath1 * -meson diffractive electroproproduction . * 
 +    m.g.ryskin and yu.m.shabelski +    petersburg nuclear physics institute , + gatchina , st.petersburg 188350 russia +    e - mail : @xmath7 ryskin@thd.pnpi.spb.ru + e - mail : @xmath7 shabel@vxdesy.desy.de +