5 A resonance with mass M$_{i}$ contributes to the structure function at Bjorken x$_{i}$ = Q $^{2}$/ ( M 2 i − M 2 p + Q $^{2}$). To be in line with measurements in the DIS region the resonance form factors F ( Q $^{2}$, M2 i ) must fall with Q 2 at least as fast as the nucleon dipole form factor. Futher- more, to be consistent with the scaling behavior of deep inelastic structure functions the cut-off in the dipole tran- sition form factors must increase as the mass of the res- onance is increases [30, 42]. Therefore we assume [41] Similar arguments should apply to the transition form factor in the s ( u )-channel nucleon-pole terms. A simplest choice would be to use in Eq. (13) where F p $_{1}$( Q $^{2}$) is the proton Dirac form factor. However, since the nucleon is highly of-mass-shell this assumption might be too naive [40]. Indeed, this prescription un- derestimates the JLAB data for σ$_{T}$ [25] and results in a wrong interference pattern between l/t components. A way to model an intermediate state which is highly off-mass-shell is to increase the Fock space available for the virtual nucleon allowing the latter to excite into res- onances. Similar to the reggeized-exchange, these res- onances with higher masses and spins may lie on the nucleon Regge trajectory or correspond to higher mass states with the same angular momentum as the nucleon. We replace the Born term in the s -channel for π + pro- duction by a sum over all resonance excitations where the value of ξ is a common average cut-off param- eter. This scenario suggests a hardening of the resonance form factors with increasing value of M$_{i}$ [42]. At high energies the level density of resonances ρ ( M 2 i ) is large and we can replace the sum in Eq. (18) over discrete spectrum of resonances by a continuous integral IV. EFFECT OF NUCLEON RESONANCES F ( Q $^{2}$, M2 i ) =   1 1 + ξ Q 2 M 2 i  $^{2}$ , (20) F$_{s}$$_{(}$$_{u}$$_{)}$ ( Q $^{2}$, s ( u )) = F p $_{1}$( Q $^{2}$) , (15) F$_{s}$ ( Q $^{2}$, M$_{p}$ ) s − M 2 p + i 0 + → ∑ i r ( M$_{i}$ ) c ( M$_{i}$ ) F ( Q $^{2}$, M2 i ) s − M 2 i + i 0 $_{+}$, (16) ∑ i → ∞ ∫ M 2 p dM 2 $_{i}$ρ ( M 2 i ) . (21) F p $_{2}$( x$_{B}$, Q $^{2}$) = ( s − M 2 p + Q $^{2}$) r $^{2}$( s )[ F ( Q $^{2}$, s )] $^{2}$ρ ( s ) . (22) F p $_{2}$( ω ′ ) ∝ ( ω ′ − 1) $^{3}$, (23) F$_{u}$ ( Q $^{2}$, M$_{p}$ ) u − M 2 p + i 0 + → ∑ i r ( M$_{i}$ ) c ( M$_{i}$ ) F ( Q $^{2}$, M2 i ) u − M 2 i + i 0 $_{+}$. (17) F ( Q $^{2}$, s ) = ( ω ′ − 1) 2 ξ 2 + O (( $_{ω}$′ − 1) $^{3}$) . (24) ( $^{$_{ω}$′}$− 1) 3 ∝ Q $_{2}$( ω ′ − 1) 4 ξ 4 r $^{2}$( s ) ρ ( s ) . (25) F p $_{2}$( x$_{B}$, Q $^{2}$) = ∑ i ( M 2 i − M 2 p + Q $^{2}$) W ( Q $^{2}$, M$_{i}$ ) δ ( s − M 2 i ) , (18) This is clearly a rough approximation in the resonance region itself, but it makes no difference when we restrict ourselves to the experimental data above the resonance region. Performing the integration over M$_{i}$ yields where the sum runs from the nucleon-pole contribution, M$_{i}$ is the i th resonance mass, r and c are the electro- magnetic and strong couplings, respectively, relative to the lowest lying nucleon state, e.g. r ( M$_{p}$ ) c ( M$_{p}$ ) = 1. For π − production we use a similar expansion over the u -channel contributions The structure function F p 2 can be written in the form of a polynomial in 1 − 1 /ω ′ where ω ′ = 1 + W $^{2}$/Q2 is a Bloom-Gilman variable. As Q $^{2}$/W 2 → ∞ or ω ′ → 1 the leading term yields the Drell-Yan-West behavior In the region of interest for the experiments to be dis- cussed later in this paper the invariant mass is W > ∼ 2 GeV and thus in a region where the DIS regime starts. In order to make a connection to our earlier work in which we modeled the transverse cross section by a partonic subprocess [25, 27] we now invoke duality for the exclu- sive processes treated here. We start with the Bloom- Gilman duality [31] in the local form which shows that the power law behavior of the form fac- tor is related to the suppression of the structure functions in the limit where one quark carries all of the hadron’s momentum. The approximation (23) is supposed to be reasonable down to x$_{B}$ ≃ 0 . 2. Expanding the resonance form factors for ω ′ → 1 the leading term reads The duality relation, Eq. (22), can be written in the form This translates into where x$_{B}$ stands for a Bjorken scaling variable and the deep inelastic structure function F p $_{2}$( x$_{B}$, Q $^{2}$) is expressed as a sum of resonances. In Eq. (18) the hadronic basis is used as a substitute for the quark basis. When Q 2 is large the bulk structure of the resonances becomes less and less important and we are justified when taking the zero-width approximation [41]. W ( Q $^{2}$, M$_{i}$ ) defines the i th resonance contribution to the γ ∗ p forward scattering amplitude; it is essentially the electromagnetic coupling constant r ( M$_{i}$ ) times a resonance form factor F ( Q $^{2}$, M$_{i}$ ) normalized to unity at Q 2 = 0 [42]: Since the level density grows with increasing s , for in- stance, ρ ∝ exp( const × M$_{i}$ ), the coupling strength to W ( Q $^{2}$, M$_{i}$ ) = r $^{2}$( M$_{i}$ )[ F ( Q $^{2}$, M$_{i}$ )] $^{2}$, F (0 , M$_{i}$ ) = 1 . (19) r $^{2}$( s ) ρ ( s ) ∝ 1 Q $^{2}$( ω ′ − 1) = 1 s . (26)