Relativistic effects in electromagnetic nuclear responses in the quasi-elastic delta region

arXiv:nucl-th/9905035v1 18 May 1999 Relativistic effects in electromagnetic nuclear responses in the quasi-elastic delta region J.E. Amaro1, M.B. Barbaro2, J.A. Caballero3,4, T.W. Donnelly5and A. Molinari2 1Departamento de F´ısica Moderna, Universidad de Granada, E-18071 Granada, SPAIN 2Dipartimento di Fisica Teorica, Universit`a di Torino and INFN, Sezione di Torino Via P. Giuria 1, 10125 Torino, ITALY 3Departamento de F´ısica At´omica, Molecular y Nuclear Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, SPAIN 4Instituto de Estructura de la Materia, CSIC Serrano 123, E-28006 Madrid, SPAIN 5Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics Massachusetts Institute of Technology Cambridge, MA 02139, USA Abstract A new non-relativistic expansion in terms of the nucleon’s momentum inside nuclear matter of the current for isobar electro-excitation from the nucleon is performed. Being exact with respect to the transferred energy and momentum, this yields new current operators which retain important aspects of relativity not taken into account in the traditional non-relativistic reductions. The transition current thus obtained differs from the leading order of the traditional expansion by simple multiplicative factors. These depend on the momentum and energy transfer and can be easily included together with relativistic kinematics in non-relativistic, many-body models of isobar electro-excitation in nuclei. The merits of the new current are tested by comparing with the unexpanded electromagnetic nuclear responses in the isobar peak computed in a relativistic Fermi gas framework. The sensitivity of the relativistic responses to the isobar’s magnetic, electric and Coulomb form factors and the finite width of the isobar is analyzed. PACS: 25.30.Rw, 14.20.Gk, 24.10.Jv, 24.30.Gd, 13.40.Gp Keywords: Nuclear reactions; Inclusive electron scattering; Delta isobar electro-production. Relativistic Fermi Gas. MIT/CTP#2856 April 1999 1 Introduction The cross section for inclusive electron scattering (e, e′) shows a pronounced peak at an energy transfer ω∼qq2+m2 N−mN, corresponding to the quasi-free interaction with the individual nucleons in the nucleus (here mN= nucleon mass). For high values of the momentum transfer q=|q|and higher energy loss it is possible to produce real pions and the cross section shows another peak dominated by the resonant production of a ∆(1232) at ω∼qq2+m2 ∆−m2 N, where m∆is the ∆ mass [1]. The width of these peaks is related to the Fermi momentum of the nucleons inside the nucleus and, in the case of the ∆-peak, also to the decay width of the ∆ in nuclear matter. Hence for a high enough value of q, these two peaks actually overlap and cannot be separated in inclusive experiments [2, 3]. Thus the response in the region above the quasi-elastic peak contains information about the nucleon’s excited states and their change due to the nuclear medium. Since the electro-excitation of the ∆ requires high energy and momentum transfers, a relativistic treatment of the reaction is needed. Recently, several many-body calculations, both in nuclear matter and finite nuclei, have been performed in this region [4, 5, 6]; all of these calculations are non-relativistic in nature, although some relativistic corrections enter in two of them, including an expansion of the current to order (p/mN)2in [4] and relativistic kinematics in [5]. Clearly some of these corrections are inadequate when one wishes to go to high momentum transfers, q≃1 GeV/c, and there relativistic models such as the ones developed in [7, 8, 9, 10] are more appropriate. However, although these last calculations are fully relativistic, they do not include the full N-∆ vertex. For instance, in the pioneering calculation by Moniz [7] the Peccei Lagrangian was used, which is only appropriate for computing the transverse response for low momentum transfer [11, 12, 13]. In other work [8, 9] a more appropriate M1 magnetic transition current was used, although the electric E2 and Coulomb C2 excitation amplitudes of the ∆ were not included. For years an important program has been pursued to determine more accurately the quadrupole C2 and E2 amplitudes in the ∆ region [14], these being small compared with the dominant dipole M1 amplitude. Using polarized photons, the E2/M1-ratio has been measured to be around -3% at resonance [15]. The C2 amplitude however only appears in electro-production reactions N(e, e′)∆. Values of the ratio C2/M1 around −13% have been reported in H(e, e′π0)pexperiments at Q2= 0.13 (GeV/c)2[16]. However, this value differs with the findings of recent measurements of the transverse-longitudinal asymmetry and proton polarization in H(e, e′p)π0reactions [17]. Hence, for the N→∆ transition our knowledge is still incomplete and has not been possible to undertake a full analysis in the sense of the work by Nozawa and Lee [18] of the effect of the C2 and E2 1 form factors in the nuclear ∆-peak. In this paper we perform a new non-relativistic expansion of the electro-excitation current of the nucleon, providing an extension of our previous expansion of the electromagnetic nucleon current in powers of η=p/mN, in which we retained the full dependence on qand ω[19]. Recently the same procedure has been also applied to meson-exchange currents (MEC) [20]. These currents can be implemented together with relativistic kinematics in standard non-relativistic models of one-particle emission near the quasi-elastic peak. In this paper we apply the same procedure to the ∆ electro-excitation current, which we develop to leading order in η, again retaining the full dependence on the energy and momentum transfers. We perform this expansion for the magnetic transition current M1, which is the dominant one both in the longitudinal and transverse nuclear responses [9]. The resulting current is designed in such a way that it differs from the traditional nonrelativistic limit simply by having (q, ω)-dependent factors which multiply the traditional operators. These corrections, being of leading order in η, are seen to arise mainly from the normalization factor in the Rarita-Schwinger spinor and from the lower-component spinology now included in the effective current operator. The organization of the work is as follows. In sect. 2 we first develop the analytical expressions for the longitudinal and transverse responses in the relativistic Fermi gas (RFG) model by using a ∆-hole approach. We consider the full vertex of Jones and Scadron [12] that includes M1, E2 and C2 ∆-amplitudes. In deriving expressions for these responses we assume a stable ∆ particle and later include its finite width by performing a convolution of these responses with a Lorentz distribution. In sect. 3 we perform the expansion of the magnetic current to leading order in the momentum of the bound nucleon. In sect. 4 we test the validity of the expansion by comparing the exact RFG result with a non-relativistic Fermi gas model, using the new current and relativistic kinematics. Furthermore, we compare several models of the reaction by using the RFG; in particular, we study the differences that arise upon using the Peccei and magnetic Lagrangians, and explore the effects of the E2, C2 multipoles and the finite width of the ∆ on the longitudinal and transverse response functions. Finally in sect. 5 we draw our main conclusions. 2 General formalism We start our discussion by introducing the general formalism, which is based on a relativistic treatment of the nuclear currents entering into the calculation of the response functions. It is well-known [21] that the longitudinal and transverse response functions 2 can be evaluated as components of the nuclear tensor Wµν, namely RL(q, ω) = q2 Q2!2"W00 −ω q(W03 +W30) + ω2 q2W33#=W00 (1) RT(q, ω) = W11 +W22 ,(2) where Qµ= (ω, q) is the space-like four-momentum carried by the virtual photon and the gauge invariance has been exploited in obtaining Eq. (1). We shall compute this nuclear tensor in the RFG framework, where nucleons are assumed to move freely inside the system with relativistic kinematics, hence being on their mass-shell. 2.1 Nuclear tensor and response functions for inelastic processes We first consider the electro-production of a stable resonance (namely, the ∆ is viewed as a particle on its mass-shell) in the RFG — in this case analytical expressions for the response functions are obtained — and later on we shall include corrections due to inclusion of the decay width of the ∆. The RFG nuclear tensor reads (see, for example, [21]) Wµν =3π2Nm2 N k3 FZdp (2π)3 θ(kF−p) E(p)E∆(p+q)fµν(p,p+q)δ[E∆(p+q)−E(p)−ω],(3) where Nis the number of protons or neutrons in the nucleus. Here mNand m∆are the masses of the struck nucleon and ∆, respectively, and E(p) = qm2 N+p2and E∆(p+q) = qm2 ∆+ (p+q)2are their corresponding energies; kFis the Fermi momentum and fµν the inelastic single-nucleon tensor of the N→∆ transition. Introducing the standard dimensionless variables η=p mN ,ηF=kF mN ,κ=q 2mN ,(4) λ=ω 2mN , τ =κ2−λ2, ε =q1 + η2, µ∆=m∆ mN (5) and performing the angular integration in Eq. (3) we obtain Wµν(κ, λ) = 3N 8mη3 FκZεF ε0 fµν(ε, θ0)dε , (6) where εF=q1 + η2 Fis the Fermi energy and 3 ε0=κs1 τ+ρ2−λρ (7) the minimum energy of the struck nucleon for fixed κand λ, having defined the factor ρ= 1 + 1 4τµ2 ∆−1,(8) which measures the inelasticity of the elementary process. In Eq. (6) the single-nucleon tensor fµν(ε, θ0) contains the angle θbetween ηand κ given via cos θ0=λε −τρ κη ,(9) as required by energy conservation. The condition |cos θ0| ≤ 1 then permits the response of the ∆ to occur only in the range 1 2q(2κ−ηF)2+µ2 ∆−εF≤λ≤1 2q(2κ+ηF)2+µ2 ∆−εF.(10) In analogy with the physics of the quasi-elastic peak [21], it is convenient to introduce a scaling variable ψ∆defined as follows ψ2 ∆(κ, λ) = ε0−1 ξF =1 ξF κs1 τ+ρ2−λρ −1 ,(11) with ξF=εF−1. The physical meaning of the scaling variable may be deduced from the above equation: in terms of dimensionless variables, ξFψ2 ∆is the minimum kinetic energy required to transform a nucleon inside the nucleus into a ∆ when hit by a photon of energy λand momentum κ. It is straightforward to check that, when mN=m∆(hence ρ= 1), the ordinary quasi-elastic scaling variable ψ[21] is recovered. In terms of the scaling variable in Eq. (11) the response region given by Eq. (10) simply reduces to −1≤ψ∆≤1. The energy position λ∆Pof the peak of the ∆ response occurs when ψ∆vanishes, namely for ψ∆= 0 −→ λ∆P=τρ , κ2 ∆P=τ(τρ2+ 1) .(12) 4 We now turn to a consideration of the nucleonic tensor fµν , which we will obtain in the next section for specific N→∆ currents. For any physical process this tensor must comply with Lorentz covariance and current conservation (Qµfµν =fµνQν= 0). The most general unpolarized second-rank tensor consistent with these requirements is [22] fµν(p,p+q) = −w1(τ) gµν −QµQν Q2!+w2(τ)VµVν−i mN w3(τ)εµνρσQρVσ,(13) where w1(τ), w2(τ) and w3(τ) are scalar functions containing the specific dynamics of the process and Vµ=1 mN Pµ−P·Q Q2Qµ!,(14) is a four-vector orthogonal to Qµ,Pµbeing the struck nucleon’s four-momentum. Only the terms involving w1,2(τ) occur in EM interactions, whereas w3(τ) also enters for the full electroweak interaction. Energy conservation via Eq. (9) implies that P·Q/Q2=−ρ/2: hence the longitudinal and transverse components of the single-nucleon tensor read fL=f00 =−κ2 τw1(τ) + (λρ +ε)2w2(τ) (15) and fT=f11 +f22 = 2w1(τ) +  ε2−1− λε −τρ κ!2 w2(τ).(16) Finally, by performing the energy integral in Eq. (6) one gets for the response functions in Eqs. (1,2) the following expressions RL(κ, λ) = 3NξF 8mNη3 Fκ κ2 τh(1 + τρ2)w2(τ)−w1(τ) + w2(τ)D(κ, λ)i(1 −ψ2 ∆)θ(1 −ψ2 ∆) (17) RT(κ, λ) = 3NξF 8mNη3 Fκ[2w1(τ) + w2(τ)D(κ, λ)] (1 −ψ2 ∆)θ(1 −ψ2 ∆),(18) where D(κ, λ)≡τ κ2(λρ + 1)2+ (λρ + 1)(1 + ψ2 ∆)ξF+1 3(1 + ψ2 ∆+ψ4 ∆)ξ2 F−(1 + τρ2) (19) 5 reflects the (modest) Fermi motion of the nucleons. In fact, at the resonance peak (ψ∆= 0) Eq. (19) reduces to D(κ, λ)∆P=ξF+τ 3κ2ξ2 F,(20) and, since ξF≃0.03, yields a small correction. 2.2 Density dependence of the response functions In this subsection we briefly explore the density dependence of the previously deduced responses. This is conveniently achieved by performing an expansion in the parameter ξF. The leading terms of the expansion of RL,T (κ, λ) = 3NξF 8mη3 Fκ(1 −ψ2 ∆)κ2 τR(0) L,T +R(1) L,T ξF+R(2) L,T ξ2 F(21) are given by R(0) L=−w1(τ) + τ κ2(λρ + 1)2w2(τ) (22) R(0) T= 2w1(τ) + −1 + τρ2+τ κ2(λρ + 1)2w2(τ),(23) and the next terms are found to be R(1) L=R(1) T=τ κ2(λρ + 1) (1 + ψ2 ∆)w2(τ) (24) R(2) L=R(2) T=1 3 τ κ2(1 + ψ2 ∆+ψ4 ∆)w2(τ).(25) In performing the ξF→0 limit it is of importance to realize that both responses shrink to the peak where κ2=τ(τρ2+1) and λ=τρ. This constraint requires that D → 0 when ξF→0 (see Eq. (19)) and hence R(0) L(κ, λ;ξF= 0) = −w1(τ) + (1 + τρ2)w2(τ) (26) R(0) T(κ, λ;ξF= 0) = 2w1(τ).(27) Expressions for the ∆ responses in this limit will be given later. In the nucleonic sector one immediately obtains R(0) L=G2 Eand R(0) T= 2τG2 M(cf. [21]). The expressions in Eqs. (17), (18) and (19) are valid for any process involving an initial nucleon which is converted to an on-shell resonance of mass m∆. In particular, the limit ρ= 1 yields the response functions in the quasi-elastic peak region [21]. The specific physical process gives rise to different w1and w2functions, which are evaluated in the next section for the N→∆ transition. 6 2.3 Nucleonic tensor We now evaluate the invariant functions w1and w2relative to the γN →∆ process. These functions, being scalars, can be computed in any reference system and the most convenient one is found to be the rest system of the ∆, where the Rarita-Schwinger spinors take their simplest form. Let us denote with Q∗ µ= (ω∗,q∗) the four-momentum transfer in this system (the corresponding four-vector in the nucleus laboratory frame is Qµ) and let P∗ µ= (E∗,p∗) be the four-momentum of the struck nucleon. The ∆ system is then defined by p∗ ∆= 0 ⇒p∗=−q∗= (0,0,−q∗),(28) having chosen the z-axis in the direction of q∗. The energy conservation condition accordingly reads E∗=m∆−ω∗.(29) Using the energy-momentum relation for the initial nucleon, namely m2 N=E∗2−p∗2=m2 ∆−2m∆ω∗+Q2,(30) we find the value of ω∗in terms of Q2 ω∗=m2 ∆−m2 N+Q2 2m∆ .(31) In this system Vµhas its space components parallel to q∗, while its time component reads V∗ 0=1 mN E∗−P·Q Q2ω∗!=−m∆q∗2 mNQ2.(32) Then we can easily compute the 00 and 11 components of the nucleonic tensor in Eq. (13), obtaining: f∗ 11 =−w1g11 =w1(33) f∗ 00 =−w1 1−ω∗2 Q2!+w2 m2 ∆q∗4 m2 NQ4 =w1 q∗2 Q2+w2 m2 ∆q∗4 m2 NQ4.(34) By inverting the above equations the structure functions in the ∆-system are found to be w1=f∗ 11 (35) w2=m2 NQ4 m2 ∆q∗4 f∗ 00 −q∗2 Q2f∗ 11!.(36) 7 Hence the problem is reduced to computing just two components, namely f∗ 00 and f∗ 11, of the nucleon tensor. With Jones & Scadron [12] we write the transition matrix element for the ∆ excitation as follows h∆|jµ|Ni=uβΓµβu , (37) the most general form of the vertex being Γµβ =C1Γ1 µβ +C2Γ2 µβ +C3Γ3 µβ .(38) In the above the Caare (invariant) form factors and the three couplings Γa µβ are given by∗ Γ1 µβ = (Qβγµ− 6Qgβµ)γ5T+ 3(39) Γ2 µβ = (QβKµ−Q·Kgβµ)γ5T+ 3(40) Γ3 µβ = (QβQµ−Q2gβµ)γ5T+ 3,(41) where Kµ= (Pµ+P∆µ)/2 and T+is the N→∆ isospin transition operator [24]. The N→∆ tensor f∗ µν then reads f∗ µν =4 3 m∆ mNX ss∆ (uλ ∆Γµλu)∗(uβ ∆Γνβu),(42) the factor 4/3 arising from the isospin trace X tt∆ < t|T3|t∆>< t∆|T+ 3|t >=4 3(43) and uβ ∆being the Rarita-Schwinger spinor describing a spin 3/2 particle. In the ∆ rest frame the latter has the simple form u0 ∆(0, s∆) = 0 (44) ui ∆(0, s∆) = X λs′h1 2s′1λ|3 2s∆iei λu∆(0, s′),(45) where the eλ(λ=−1,0,+1) are spherical vectors and u∆(0, s′) =   χ′ s′ 0 (46) ∗Note that the expressions for the transition matrix element of Dufner and Tsai [11] and of Devenish et al. [23] differ from the ones of Jones and Scadron because the latter employ the set of basis vectors (K, Q), whereas Dufner and Devenish use (P∆, P ) and (P∆, Q) respectively. Hence both Γ2and Γ3and the form factors C2and C3of these three sets of authors will be different. 8 Note that the current Jis of order O(1), whereas the charge J0is of order O(η). Thus, one expects the contribution of the N∆ current to be considerably more important for the transverse response. It is important to note that in order to be consistent one should treat the charge and current at the same level and perform an expansion of the current to order O(η). Such program can be carried out by using the techniques developed in [19, 20]. However, for the present case an additional simplification can be made by using the expression of the charge operator to order O(η) in Eq. (96), and taking into account the fact that, before performing the expansion, the original current was gauge-invariant. This invariance property should be valid for all the orders in the expansion (we are not expanding in qor ω). Hence we can relate the longitudinal component of the current with the density, i.e., J·κ=λJ0=−2Gλ√1 + τ′(S†×η)·κ.(98) Hence we can obtain an improved and gauge-invariant current by adding to the expression in Eq. (97) a new piece of order O(η) given by Jgauge =−2Gλ√1 + τ′(S†×η). Of course there could be additional corrections of order O(η) in the transverse current that cannot be fixed by just using the continuity equation. However, as we will show below, the transverse response function computed with this improved current differs from the exact relativistic results by terms only of order O(η2), proving the high quality of this expansion for most applications in nuclear physics. Therefore the new expression of the current that we will consider below is J= 2G√1 + τ′S†×(κ−λη).(99) In order to test the quality of the above expansion of the current we proceed to compute the response functions in a non-relativistic Fermi gas using relativistic kinematics and the new currents in Eqs. (96,99). We begin by computing the analytical expressions and compare with the exact relativistic answer. Results are shown and discussed in the next section. The calculation of the non-relativistic nucleon tensor needed to evaluate the nuclear response functions can be done by performing the following traces fnr 00 =4 3 m∆ mN Tr [J† 0J0]≡wL nrη2 T(100) fnr 11 +fnr 22 =4 3 m∆ mN Tr [J† T·JT]≡2wT nr ,(101) 15 where we have introduced the longitudinal and transverse, non-relativistic structure functions wL nr and wT nr wL nr =4 3 16m∆ 3mN G2(1 + τ′)κ2(102) wT nr =4 3 16m∆ 3mN G2(1 + τ′)(κ2−2λκ·η) + O(η2).(103) In order to compare the relativistic and non-relativistic response functions it is convenient to use the kinematical relations: 1 + τ′=1 2µ∆ (1 + µ∆+ 2τρ) (104) κ2−2λκ·η=τ(1 + τρ2) + O(η2).(105) We can then write, up to first order, the relations wT nr =w1+O(η2) (106) wL nr =κ2 τ w1 1 + τρ2+O(η2) = κ2 τw2+O(η2),(107) where w1and w2are the magnetic relativistic functions given in Eqs. (68,69). The nuclear response functions are given finally by Rnr L=3NξF 8mNη3 Fκθ(1 −ψ2 ∆)(1 −ψ2 ∆)wL nrD(108) =3NξF 8mNη3 Fκθ(1 −ψ2 ∆)(1 −ψ2 ∆)κ2 τhw2+O(η2)iD Rnr T=3NξF 8mNη3 Fκθ(1 −ψ2 ∆)(1 −ψ2 ∆)2wT nr (109) =3NξF 8mNη3 Fκθ(1 −ψ2 ∆)(1 −ψ2 ∆)2 hw1+O(η2)i with Das given by Eq. (19). Although we call these functions “non-relativistic”, actually they contain enough relativistic ingredients to be high-quality approximations to the exact RFG result. In fact, first we use relativistic kinematics, so that the phase space and momentum integrals are done exactly. Second, we consider the new currents expanded to include effects up to order O(η), so that the dynamics of the problem are correct to that order. Comparing these expressions with the exact relativistic responses in Eqs. (18,70), we see that the relative differences between Rnr Land RL, and between Rnr Tand RTare of order O(η2). In the next section we test numerically the quality of the non-relativistic responses of Eqs. (108,109). 16 4 Results In this section we present numerical results for the relativistic response functions in the region of the ∆-peak and check numerically the quality of our new approximation to the N→∆ electromagnetic current. We also investigate the importance of different contributions in the relativistic responses. We shall present results for medium and high momentum transfers, ranging from q= 0.5 to 2 GeV/c, and thus will also be able to analyze the validity of the traditional non-relativistic calculations performed for different values of q. Before starting our analysis it is convenient to check our model with some of the available experimental data [25]. In this way we can fix some of the ingredients that enter in the model, in particular, the magnetic form factor GMof the ∆ and the modification of the response due to the finite ∆ width as a consequence of its later decay into the N-π channel, which we incorporate by performing a convolution with the responses for stable particles. In fig. 1 we show the transverse cross section σTfor the inclusive reaction H(e, e′) from the nucleon. The transverse nuclear cross section is defined by σT= ¯hc 2πα2 ω+Q2 2mN rT,(110) where rT=m∆ E∆2w1is the transverse response of a single nucleon. In this calculation we have included the ∆ width by substituting for the energy-conserving delta function a Lorentzian shape δ(ω+mN−E∆)−→ E∆ m∆ 1 π Γ(s)/2 (√s−m∆)2+ Γ(s)2/4,(111) where s= (mN+ω)2−q2is the invariant mass of the initial photon and nucleon. Results that include a width Γ(s) (which is zero at threshold and equal to the width Γ0= 120 MeV at resonance) are represented by solid lines in fig. 1. The dependence of Γ(s) on the invariant mass is given by [4] Γ(s) = Γ0 m∆ √s p∗ π pres π!3 ,(112) where p∗ πis the momentum of the final pion resulting from the ∆ decay (in the ∆-system) given by p∗ π=1 √s"(s−m2 N−m2 π)2 4−m2 Nm2 π#1/2 (113) 17 and pres πis its value at resonance, obtained from the above expression for √s=m∆. For comparison, we show in fig. 1 with dashed lines results obtained by considering a constant width Γ = Γ0. Other ingredients that enter in the cross section are the ∆ form factors. We use the parameterization GM(Q2) = GM(0)f(Q2) (114) GE(Q2) = GE(0)f(Q2),(115) where we assume that the same dependence in Q2is valid for the electric and magnetic form factors, given by the function [8] f(Q2) = GP E(Q2) 1−Q2 3.5 (GeV/c)2!−1/2 (116) with GP Ethe electric form factor of the proton, for which we use the Galster parameterization (1 + 4.97τ)−2[26]. The above equation reflects the fact that the isobar form factor falls off faster than the proton form factor. Unless otherwise indicated, we take GC= 0 and use the following values [12] of the form factors at the origin GM(0) = 2.97, GE(0) = −0.03.(117) Later on we show the effect on the longitudinal response function introduced by considering a C2 form factor that is different from zero. Our calculation (solid lines) displayed in fig. 1 is slightly below the data for the three values of the momentum transfer Q2=−0.2,−0.3,−0.4 (GeV/c)2, reflecting the fact that we have not included the background contributions of non-resonant pion production, which produce an additional increase of the cross section. In fig. 2 we show results for the nuclear inclusive cross section per nucleon from 12C compared with the experimental data taken from [2, 27]. Dotted lines correspond to the RFG for a stable ∆. A more realistic model of the ∆ peak requires the inclusion of the ∆ width in the cross section, which we show with dashed lines. The nuclear responses including the width, RΓ(q, ω), are computed from the responses R(q, ω, W ) for a stable ∆ with mass Wby a convolution RΓ(q, ω) = ZWmax mN+mπ 1 π Γ(W)/2 (W−m∆)2+ Γ(W)2/4R(q, ω, W)dW , (118) where the integration interval goes from threshold to the maximum value allowed in the Fermi gas model, W2 max = (EF+ω)2−(q−kF)2. The inclusion of the ∆ width produces a broadening of the ∆ peak and correspondingly a decrease of the strength. 18 As an illustration of how one could improve the model in the quasi-elastic-peak region, we also show with dot-dashed lines the quasi-elastic cross section computed with the PWIA model of ref. [28]. Here the mean difference with the RFG model is the inclusion of the momentum distribution of the finite-sized nucleus, which produces the “tails” of the cross section, and the binding energy of the nucleons in the nucleus, which produces a shift to higher energies (in the direction of data). Here for the PWIA calculation we use relativistic kinematics, final states are described as plane waves and the electromagnetic current used contains relativistic corrections to order η. Finally, we show with solid lines the results computed with a hybrid model in which we add the PWIA cross section for the quasi-elastic contribution to the RFG result for the ∆ contribution. As we can see in fig. 2, our results are below the data in the dip and ∆ region. This was expected because other contributions coming mainly from two-nucleon emission and non-resonant pion production (not included in our model) also enter here [1, 4, 10, 29]. However, our intention in this work is not to reproduce the experimental data nor to present a complete model including all of the physical contributions in this energy region, but instead to discuss the effect of different ingredients in the calculation and present a new set of improved currents specifically for excitation of the ∆ peak that now include the relevant relativistic content — these could now straightforwardly be used in standard non-relativistic many-body models with relativistic kinematics. The quality of the new approximation to the relativistic, magnetic ∆ current is shown in fig. 3, where the exact RFG longitudinal and transverse responses using magnetic and electric form factors are displayed with solid lines. Here we show just the ∆ contribution to the responses. In addition we show with dashed lines the responses computed in the non-relativistic Fermi gas model with relativistic kinematics and the new currents in Eqs. (96,99). For comparison we also show with dot-dashed lines results for the nonrelativistic Fermi gas model and relativistic kinematics, but using the traditional nonrelativistic current. The improvement of the description of the relativistic results using our currents is clear from this figure — the solid and dashed lines almost coincide. This proves that our expansion to order O(η) is precise enough to describe the ∆ excitation in nuclei with negligible error for high momentum transfers. In fig. 4 several relativistic effects and ingredients of the calculation are analyzed. Therein we show the longitudinal and transverse responses for q= 0.5,1 and 2 GeV/c. With solid lines we show the ∆ peak computed within our model, while with dashed lines we show the ∆ peak computed using the Peccei Lagrangian [30]. The Peccei Lagrangian only includes the first coupling Γ1 µν given in Eq. (39) with coupling constant C1(0) = 2.5 GeV−1. This value has been chosen so that the transverse response is equal to the one 19 computed with the full Lagrangian in the ∆ peak for q= 500 MeV/c. The importance of using the full vertex in Eq. (38) is clear from this figure. First, although the coupling constants can be chosen so that the transverse responses computed with both Lagrangians are similar for moderate q= 500 MeV/c, they begin to fail for higher q-values. These differences are seen to be most important in the longitudinal response, where the Peccei Lagrangian clearly gives an extremely large result. The reason for this unphysical behavior of the Peccei Lagrangian is that in Eq. (53) there are important cancelations among the C1,C2and C3pieces in the longitudinal channel [11]. Second, as the Peccei Lagrangian is the one usually employed to compute the MEC contribution involving virtual ∆ excitation [1], it is mandatory to use the full Lagrangian — or at least the magnetic piece — if one wants to compute the longitudinal contribution of the MEC in this channel. As reference, in fig. 4 we also show with dot-dashed lines the quasi-elastic peak responses. For high q-values the two peaks overlap and the importance of the ∆ in both responses also increases with q. This is better seen in fig. 5 where we also show the effect on the response functions produced by incorporating the finite ∆ width (solid lines). Dashed lines correspond to results without including the isobar finite width, while dot-dashed lines correspond to the quasi-elastic peak. In the above results no Coulomb form factor has been included. In this case the contribution of the ∆ in the longitudinal response is due to the Fermi motion of the nucleons inside the nucleus [9], the main contribution here coming from the magnetic ∆ excitation, which is zero only for nucleons at rest. This is better seen in Eq. (100) where the longitudinal, magnetic single-nucleon response is seen to be proportional to η2 Tand to the function wnr Lwhich is proportional to κ2(see Eq. (102)), explaining the increase with qof the longitudinal response observed in fig. 5. In the static limit, the longitudinal response becomes proportional to the Coulomb C2 form factor, as shown at the end of sect. 2. In fig. 6 we show the dependence of the longitudinal response on the Coulomb form factor of the ∆. With solid lines we show the ∆ peak without C2 multipoles, while the dashed lines include a Coulomb contribution with form factor GC(Q2) = −0.15GM(Q2). We can see that the longitudinal response is quite sensitive to this form factor, especially for high q. This can be easily understood from the analytical expression of the structure functions in Eqs. (60,61). Only the structure function w2depends on GCand its dependence on the form factors is carried by the quadratic combination Q2 m2 ∆G2 C−G2 M−3G2 E, which is not very sensitive to small values of GC. However looking at RLin Eq. (17), we see that the correction due to the term w2Dis small; hence the main contribution 20 comes from the combination (1 + τρ2)w2−w1. The important point is that, if no C2 term is present, that combination is exactly zero, i.e., (1 + τρ2)w2−w1= 0. Thus, the only contribution to the longitudinal response comes from the higher-order term w2D. This explains why the longitudinal isobar response is so small. On the other hand, if the C2 form factor is nonzero, there will be a contribution from the leading-order term (1 + τρ2)w2−w1which is proportional to G2 C. This term, although small, is of the same order of magnitude as the w2Dpiece. Therefore, we can conclude (at least for values of GCsuch as those assumed here) that a correct treatment of the isobar longitudinal response requires the inclusion of the Coulomb form factor. To finish this section, we have also explored the sensitivity of the response functions to inclusion of the electric E2 form factor, finding that both responses are quite insensitive. The reason is that the E2 contribution always adds incoherently to the M1 form factor in the combination G2 M+ 3G2 E, as seen in Eqs. (60,61). Therefore, the inclusion of a small E2 form factor does not significantly modify any of the responses. In order to obtain appreciable effects due to the electric form factor, one could explore other observables where interferences can occur; for instance, one could analyze the angular distribution of the ∆ emission in the transverse channel [18]. 5 Conclusions In summary, in this paper we have obtained a new expansion of the relativistic ∆ electroexcitation current to first order in η=p/mN, maintaining the exact dependence on the momentum and energy transfers. We have tested this new expansion by performing a calculation using a non-relativistic Fermi gas model together with relativistic kinematics. The resulting longitudinal and transverse responses are found to be very close to the exact result computed with the RFG model. Therefore it is expected that the use of this current will provide a significant improvement when used in more realistic models of inclusive electron scattering from nuclei. We have also performed a comparison between different Lagrangians in the treatment of the ∆ excitation, finding that the Peccei Lagrangian is inappropriate in the longitudinal channel for all of the q-values analyzed. We have studied the contribution of the isobar emission to the longitudinal response, in general finding a small contribution for medium q-values, although increasing significance as qincreases. In our calculations we include the finite width of the ∆ assuming a Lorentzian shape. With regards to the quadrupole amplitudes of the ∆, we have found a large sensitivity of the longitudinal response to inclusion of the Coulomb form factor of the isobar, especially for high q, a fact that could 21 be of importance for the present investigations of the longitudinal nuclear response. On the other hand, both L and T responses are found to be insensitive to the quadrupole E2 form factor. Finally, it is of interest to extend the present studies to the two-particle emission channels which provide important contributions in this energy region. In particular, one expects these ideas to be relevant in the analysis of non-resonant pion production (e, e′pπ) and two-nucleon emission (e, e′2N) reactions, where there is the hope of extracting detailed information on short-range correlations in nuclei. It is clear that, in order to extract ground-state properties from these reactions, high-qvalues are needed, since one wishes to reach a reasonably quasi-free regime in which final-state interaction effects are expected to be minimal. However, for such kinematical conditions, as our present studies indicate, relativistic effects play a role and it is important to have them theoretically under control. 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Rev. 181 (1969) 1902 24 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 200 250 300 350 400 450 500 R L [GeV  1 ] q = 0 : 5 GeV/c 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 400 450 500 550 600 650 700 750 800 R L [GeV  1 ] q = 1 GeV/c 0 0.005 0.01 0.015 0.02 0.025 0.03 1100 1200 1300 1400 1500 1600 R L [GeV  1 ] ! [MeV] q = 2 GeV/c Figure 6