Relativistic model of 2p-2h meson exchange currents in (anti)neutrino scattering

arXiv:1604.08423v1 [nucl-th] 28 Apr 2016 Relativistic model of 2p-2h meson exchange currents in (anti)neutrino scattering I. Ruiz Simoa, J.E. Amaroa, M.B. Barbarob,c, A. De Paceb, J.A. Caballerod, T.W. Donnellye, aDepartamento de F´ısica At´omica, Molecular y Nuclear, and Instituto de F´ısica Te´orica y Computacional Carlos I, Universidad de Granada, Granada 18071, Spain bINFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy cDipartimento di Fisica, Universit`a di Torino Via P. Giuria 1, 10125 Torino, Italy dDepartamento de F´ısica At´omica, Molecular y Nuclear, Universidad de Sevilla, Apdo.1065, 41080 Sevilla, Spain and eCenter for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (Dated: April 29, 2016) We develop a model of relativistic, charged meson-exchange currents (MEC) for neutrino-nucleus interactions. The two-body current is the sum of seagull, pion-in-flight, pion-pole and ∆-pole operators. These operators are obtained from the weak pion-production amplitudes for the nucleon derived in the non-linear σ-model together with weak excitation of the ∆(1232) resonance and its subsequent decay into Nπ. With these currents we compute the five 2p-2h response functions contributing to (νl, l−) and (νl, l+) reactions in the relativistic Fermi gas model. The total current is the sum of vector and axial two-body currents. The vector current is related to the electromagnetic MEC operator that contributes to electron scattering. This allows one to check our model by comparison with the results of De Pace et al., Nuclear Physics A 726 (2003) 303. Thus our model is a natural extension of that model to the weak sector with the addition of the axial MEC operator. The dependences of the response functions on several ingredients of the approach are analyzed. Specifically we discuss relativistic effects, quantify the size of the direct-exchange interferences, and the relative importance of the axial versus vector current. PACS numbers: 25.30.Fj; 21.60.Cs; 24.10.Jv I. INTRODUCTION Modern accelerator-based neutrino experiments use nuclear targets to extract neutrino oscillation parameters. The charged current quasielastic (CCQE) process (νl, l) dominates at the typical energies of 1 GeV of the incident neutrino flux, whose reconstruction from the final state requires some assumptions about the initial and final nuclear states. Within the crude approximation that the neutrino interacts with a (bound) nucleon at rest, the neutrino energy can be calculated using only the outgoing lepton kinematics. A recent review of the present understanding of the neutrino-nucleus interaction and its effects on the neutrino energy reconstruction is presented in [1]. Besides, the uncertainties in the neutrino-nucleon interaction, for instance the present limited knowledge of the axial form factor [2, 3], the nuclear many-body effects and the final-state interactions which are not experimentally distinguishable complicate the energy reconstruction. The events in which two or more nucleons are ejected have been suggested to be important, based on several calculations of the QE neutrino cross section [4–10]. The contribution of two-particle-two-hole (2p-2h) excitations are believed to be essential for a proper description of recent neutrino experiments [11–19]. At the intermediate momentum transfers typical of these experiments, relativistic effects have to be taken into account, not only in the kinematics and nuclear wave functions, but also in the current operators. Several calculations of 2p-2h excitations in neutrino scattering have been reported, each of them relying on different assumptions and approximations. In particular, the model in [4, 5] is based on a non-relativistic treatment of meson-exchange currents (MEC) and correlations, with some relativistic corrections added, and the axial MEC contribution estimated from the wellknown vector operator. The model in [7], on the other hand, is relativistic and uses some approximations to compute the momentum-space integrals of the 2p-2h matrix elements; it also neglects some contributions, in particular the direct/exchange interference. Both of the above calculations use the Fermi gas model to compute the 2p-2h matrix elements. Worth mentioning are also some recent efforts towards alternative approaches to the problem, where a factorization ansatz is assumed in order to account for one- and two-body current contributions [20, 21]. Within the super-scaling approach (SuSA) the MEC contribution to the neutrino cross section was estimated [6] from the electromagnetic (em) 2p-2h transverse response model of [22, 23], by neglecting the axial component. This model is fully relativistic, includes all the interference diagrams and evaluates the seven-dimensional integrals without approximations. Our goal in this paper is to extend the model of [22] by including the axial current. This study will be applied to the analysis of neutrino-nucleus scattering processes in a forthcoming publication [24] based on our recent investigation of electron scattering reactions making use of the superscaling approach [25, 26] (see also [27] for details). 2 The relativistic Fermi gas (RFG) is the simplest model that allows a complete and fully relativistic calculation of 2p-2h effects. This requires one to compute the spinisospin traces of all of the many-body diagrams. In electron scattering they involve more than 100,000 terms that were evaluated in [22] with subsequent 7D integrals. To extend this procedure to neutrino scattering implies adding the new axial MEC operator, and computing five nuclear response functions. In addition to the pure vector pieces (V V ) in the squared amplitudes, new axial (AA) and interference (V A) contributions appear, thus increasing the number of traces to compute. The resulting number of terms would make this procedure almost intractable. Therefore, instead of computing analytically the traces with Dirac matrices algebra, here we follow the approach of [28] by computing numerically the spin traces. We first introduce briefly in Sect. II the formalism of neutrino scattering. In Sect. III we provide the expressions for the relativistic MEC matrix elements in the 2p-2h channel. The novel piece is the inclusion of the relativistic axial MEC, deduced from the weak pion production amplitudes of [29]. The expressions of the 2p-2h response functions in terms of the current matrix elements for the separate isospin channels are given in Sect. IV. An important point that deserves clarification concerns the ambiguities between MEC and Delta peak contributions. The ∆ peak is the main contribution to the pion production cross section. Inside the nucleus the ∆ can decay into one nucleon that re-scatters producing twonucleon emission, therefore this channel should be considered part of the 2p-2h channel. In Sect. V we describe how we treat this issue within our approach. In Sect. VI we present and discuss the results for the 2p-2h weak and electromagnetic nuclear response functions for several kinematics and we analyze the importance of the different contributions and ingredients of the model. Finally in Sect. VII we summarize our findings and draw our conclusions. II. FORMALISM OF NEUTRINO SCATTERING This formalism can be applied to both neutrino (νl, l−) and antineutrino (νl, l+) CC reactions in nuclei. The incident and scattered leptons have momenta Kµ= (ǫ, k) and K′µ= (ǫ′,k′), respectively. The four-momentum transfer is Qµ= (ω, q) = (K−K′)µ, where ωis the energy transfer and qis the momentum transfer. We choose the zaxis along the qdirection. The doubledifferential cross section is [30] dσ dΩ′dǫ′=σ0S±,(1) where σ0is a kinematical factor including the weak couplings defined in [30]. The nuclear structure function S± is the linear combination of five response functions S±=e VCC RCC + 2e VCLRCL +e VLLRLL +e VTRT±2e VT′RT′,(2) where the sign of the last term is positive for neutrinos and negative for antineutrinos. The e VK’s are kinematical factors defined in [30]. In this paper we are interested in the five nuclear response functions RCC =W00 (3) RCL =−1 2W03 +W30(4) RLL =W33 (5) RT=W11 +W22 (6) RT′=−i 2W12 −W21.(7) The hadronic tensor Wµν is calculated in a relativistic Fermi gas (RFG) model with Fermi momentum kF. The final states can be excitations of the np-nh kind. Thus the hadronic tensor can be expanded as the sum of oneparticle one-hole (1p-1h) , two-particle two-hole (2p-2h), plus additional channels Wµν =Wµν 1p1h+Wµν 2p2h+··· (8) In the impulse approximation the 1p-1h channel gives the well-known response functions of the RFG [30]. Here we focus on the 2p-2h channel, with two nucleons with momenta p′ 1and p′ 2above the Fermi momentum, p′ i> kF, and two hole states with momenta h1and h2below the Fermi momentum, hi< kF. The spin (isospin) indices are s′ i(t′ i) and si(ti), respectively. The 2p-2h hadronic tensor in the RFG model is proportional to the volume Vof the system, which for symmetric nuclear matter, Z=N=A/2, is V= 3π2Z/k3 F. It is given by Wµν 2p−2h=V (2π)9Zd3p′ 1d3h1d3h2 M4 E1E2E′ 1E′ 2 rµν (p′ 1,p′ 2,h1,h2)δ(E′ 1+E′ 2−E1−E2−ω) Θ(p′ 1, p′ 2, h1, h2),(9) where p′ 2=h1+h2+q−p′ 1is fixed by momentum conservation, Mis the nucleon mass, the energies Eiand E′ iare the on-shell energies of the holes and particles, and Θ(p′ 1, p′ 2, h1, h2)≡θ(p′ 2−kF)θ(p′ 1−kF) ×θ(kF−h1)θ(kF−h2).(10) The non-trivial part of the calculation is contained in the function rµν (p′ 1,p′ 2,h1,h2), which represents the elementary hadronic tensor for the basic 2p-2h transition, with the given initial and final momenta, summed over 3 spin and isospin projections rµν (p′ 1,p′ 2,h1,h2) = 1 4X s1s2s′ 1s′ 2X t1t2t′ 1t′ 2 jµ(1′,2′,1,2)∗ Ajν(1′,2′,1,2)A. (11) This elementary hadronic tensor is written in terms of the two-body MEC antisymmetrized matrix element jµ(1′,2′,1,2)A(we use the definition given in Eq. (17) of [31]). The factor 1/4 accounts for the antisymmetry of the 2p-2h wave function to avoid double counting. The above sum over isospin combines all the possible charge channels in the final state, corresponding to emission of pp,nn and pn pairs. In our formalism, discussed below, we separate the contributions of these charge states to the response functions. Although we present results for the total contribution, having the possibility to separate the isospin contributions will allow us to apply the formalism to asymmetric nuclei N6=Z. This will be of interest [32] for neutrino experiments based, for instance, on 40Ar, 56Fe or 208Pb. To compute the hadronic tensor in Eq. (9) we took advantage of the symmetry imposed by the choice of having the zaxis along the qdirection. Then the rotational symmetry of the response functions, Eqs. (3–7), around the q direction allows us to integrate over one of the azimuthal angles. We choose φ′ 1= 0 and multiply the responses by a factor 2π. Furthermore, the energy delta function enables analytical integration over p′ 1. The 2p-2h integral is then reduced to 7 dimensions. In this paper we evaluate the resulting 7D integral using the numerical method described in [33, 34]. It is useful to calculate the hadronic tensor expected for rµν = 1 (i.e. that arising from phase-space alone). This was done in [33] in the laboratory frame, and in [34] in the hadronic center-of-mass (CM) system. These results will be modified here when including the effects of the two-body physical current. A related analysis was done in [35, 36] with a pure 2p-2h phase space alone, fitted to the experimental cross section, where the effects of the physics in the tensor rµν were not taken into account. III. ELECTROWEAK MESON-EXCHANGE CURRENTS The evidence for a pion-exchange contribution to the axial current of nuclei is well-known from weak processes such as βdecay, µ-capture or solar proton burning pp → de+νe, generating several theoretical studies [37–39], all of which are focused on low-energy processes where a non-relativistic description is adequate, and where one starts from a non-relativistic current operator. However, for the energies involved in modern neutrino experiments a relativistic approach is mandatory, and accordingly we start with a fully relativistic operator. h1h2 p′ 1p′ 2 Q (a) h1 p′ 1 h2 p′ 2 Q (b) h1 p′ 1 h2 p′ 2 Q (c) h1h2 p′ 1p′ 2 Q (d) h1 p′ 1 h2 p′ 2 Q (e) h1h2 p′ 1p′ 2 Q (f) h1 p′ 1 h2 p′ 2 Q (g) h1h2 p′ 1p′ 2 Q (h) h1 p′ 1 h2 p′ 2 Q (i) FIG. 1: Feynman diagrams of the MEC considered in the present study, including the seagull (a,b), pion-in-flight (c), pion-pole (d,e), and ∆ pole (f–i). In this section we describe our model for relativistic weak meson-exchange currents. We work at tree level including only one-pion exchange. The relativistic electromagnetic MEC has been widely studied previously for intermediate-energy electron scattering and expressions have been given for instance in [22, 28, 40]. To obtain the weak two-body current we need the relativistic axial contribution. To our knowledge this current has not been written explicitly in the literature, so this is one of the novelties of this work. We start from the weak pion production model of [29], based on the non-linear σ-model. We take the pion-production amplitudes from the nucleon given there, and we couple a second nucleonline to the emitted pion. The resulting MEC operator is written as the sum of four contributions, denoted as seagull, pion-in-flight, pion-pole and Delta-pole jµ MEC =jµ sea +jµ π+jµ pole +jµ ∆.(12) The corresponding Feynman diagrams are given in Fig. 1. 4 In this work we do not include the so-called nucleonpole contributions. These can be considered a part of the nucleon correlations that contributes to the nuclear spectral function and final-state interactions (FSI), and are not considered genuine meson-exchange currents. Besides, the corresponding diagrams produce divergences in the quasielastic region and some kind of regularization [28, 41, 42] or subtraction of self-energy diagrams [7, 43] are required. The effect of correlations is taken into account, at least partially, in the one-body cross section by using a spectral function for the nucleon in the medium [44, 45] or alternatively, with the superscaling approach [25, 46]. Each one of the four MEC operators can be decomposed as a sum of vector (V) and axial-vector (A) currents. The vector operators also contribute to electron scattering and are constrained by electromagnetic probes, while the axial ones only appear in weak processes like neutrino scattering. A. Seagull current The weak seagull current depicted in Fig. 1, diagrams (a) and (b), can be written as: jµ sea = (IV)±Jµ sea,(13) where (IV)±= (IV)x±i(IV)y(14) stands for the ±-component of the two-body isovector operator IV=i[τ(1) ×τ(2)] (15) and Jµ sea is the isospin-independent seagull current, given as the sum of Vand Acomponents Jµ sea = (Jµ sea)V+ (Jµ sea)A.(16) They have been derived from the contact term (CT) of the pion neutrino-production amplitudes from [29], and can be written as (Jµ sea)V=f2 πNN m2 π FV 1(Q2)V(s′ 1,s1) πNN (p′ 1,h1) ׯus′ 2(p′ 2)γ5γµus2(h2)−(1 ↔2) (17) (Jµ sea)A=f2 πNN m2 π 1 gA V(s′ 1,s1) πNN (p′ 1,h1)Fρk2 22 ׯus′ 2(p′ 2)γµus2(h2)−(1 ↔2).(18) In these equations: •The coupling constant fπNN /mπcomes from the W±πNN and πNN vertices. In the Acomponent the Goldberger-Treiman relation has been applied to the amplitudes of [29] gA 2fπ =fπNN mπ ,(19) with gA= 1.26 and fπ= 93 MeV being, respectively, the nucleon axial coupling and the pion decay constant. •The same form factors as in [29] are used. FV 1(Q2) is the vector nucleon form factor, and Fρ(k2) accounts for the ρ-meson dominance of the ππNN coupling. •The pion four-momenta kij are the differences between the final and initial nucleon four-momenta in the πNN vertex, i.e., kij =p′ i−hj, i, j = 1,2.(20) •The function V(s′ i,sj) πNN (p′ i,hj) = ¯us′ i(p′ i)γ56kij usj(hj) k2 ij −m2 π (21) accounts for the propagation and subsequent absorption of the exchanged pion and includes the pion propagator and the πNN vertex (see Fig. 1, diagrams (a),(b)). Note that for on-shell nucleons the following relation can be used to simplify the expression of VπNN ¯us′ i(p′ i)γ56kij usj(hj) = −2m¯us′ i(p′ i)γ5usj(hj).(22) •The Cabibbo angle θc, which was present in the amplitudes of [29], has been factorized out from the weak currents, and has been included in the definition of the factor σ0in Eq. (1). •The shorthand notation (1 ↔2) means to interchange the ordering in the labels of the two nucleons’ spins and momenta, but not the isospins. Note that the jµ sea current has to be understood as an operator in isospin space and a matrix element in spin-coordinate space. The symmetrization of the above operator is automatically taken into account due to the antisymmetry of the isovector isospin operator (IV)±under the exchange of (1 ↔2). Finally, note that the electromagnetic current operator can be obtained from the above equations by keeping only the Vcurrent and taking the zcomponent of the isospin operator (IV)±→(IV)z=i[τ(1) ×τ(2)]z.(23) The resulting electromagnetic seagull current is in agreement with previous expressions [22, 28, 47]. B. Pion-in-flight term The weak pion-in-flight current, depicted in diagram (c) of Fig. 1, can be expressed in a similar way to the 5 seagull operator, but with a vanishing axial part: jµ π= (IV)±Jµ π(24) Jµ π= (Jµ π)V+ (Jµ π)A(25) (Jµ π)V=f2 πNN m2 π FV 1(Q2)V(s′ 1,s1) πNN (p′ 1,h1) ×V(s′ 2,s2) πNN (p′ 2,h2) (kµ 11 −kµ 22) (26) (Jµ π)A= 0.(27) Equation (26) reproduces the well-known expression for the pion-in-flight electromagnetic MEC taking the z component of the isospin operator [22, 28, 47]. It corresponds to the so-called PF piece of the pion production amplitudes of [29]. C. Pion-pole term At variance with the pion-in-flight current, the pionpole term (diagrams (d) and (e) of Fig. 1) has only the axial component and therefore it is absent in the electromagnetic case. This new contribution could be considered as the “axial counterpart” of the pion-in-flight term, in the sense that it contains two pion propagators and is proportional to k11 +k22 =Q. The expression for this current is: jµ pole = (IV)±Jµ pole (28) Jµ pole =Jµ poleV+Jµ poleA(29) Jµ poleV= 0 (30) Jµ poleA=f2 πNN m2 π 1 gA Fρk2 11V(s′ 2,s2) πNN (p′ 2,h2) ×Qµ¯us′ 1(p′ 1)6Q us1(h1) Q2−m2 π−(1 ↔2).(31) Note the similarity with the axial part of the seagull current because it has the same form factor and it contains a factor 1/gA. Since it is proportional to Qµ, this current only contributes to the longitudinal and time components of the hadronic tensor. D. ∆(1232) term The ∆-pole terms correspond in Fig. 1 to diagrams (f,g) for the forward and (h,i) for the backward ∆ propagations, respectively. We start from the ∆-pole and the crossed-∆-pole pion-production amplitudes of [29]. Attaching a second nucleon which absorbs the pion, we obtain the following currents jµ ∆=jµ ∆,forw +jµ ∆,back (32) jµ ∆,forw =−f∗fπNN m2 π √3Uforwt′ 1t′ 2;t1t2V(s′ 2,s2) πNN (p′ 2,h2) ×kα 22 ¯us′ 1(p′ 1)Gαβ(h1+Q)Γβµ(h1, Q)us1(h1) + (1 ↔2) (33) jµ ∆,back =−f∗fπNN m2 π √3Ubackt′ 1t′ 2;t1t2V(s′ 2,s2) πNN (p′ 2,h2) ×kβ 22 ¯us′ 1(p′ 1)ˆ Γµα(p′ 1, Q)Gαβ(p′ 1−Q)us1(h1) + (1 ↔2).(34) The meaning of the different quantities in these equations is as follows: •The πN∆ coupling constant is denoted f∗= 2.13. •The ∆-propagator Gαβ(P) is described by the Rarita-Schwinger propagator of a spin 3/2 particle Gαβ(P) = Pαβ(P) P2−M2 ∆+iM∆Γ∆(P2),(35) where Pαβ is the projector over spin-3 2, Pαβ (P) = −(6P+M∆)gαβ −1 3γαγβ−2 3 PαPβ M2 ∆ +1 3 Pαγβ−Pβγα M∆(36) and M∆and Γ∆stand for the ∆(1232) resonance mass and width, respectively. •In the forward piece, we have introduced the weak N→∆ transition vertex written as the sum of vector and axial-vector vertices Γβµ(P, Q) = Γβµ V(P, Q) + Γβµ A(P, Q) (37) Γβµ V(P, Q) = CV 3 Mgβµ 6Q−Qβγµ +CV 4 M2gβµQ·P∆−QβPµ ∆(38) +CV 5 M2gβµQ·P−QβPµ+CV 6gβµγ5 Γβµ A(P, Q) = CA 3 Mgβµ 6Q−Qβγµ +CA 4 M2gβµQ·P∆−QβPµ ∆ +CA 5gβµ +CA 6 M2QβQµ,(39) with P∆=P+Q. The symbols CV,A i(i= 3−6) in the above equations stand for the Q2-dependent vector and axial-vector form factors. 6 •In the backward term we use instead the ∆ →N transition vertex given by ˆ Γµα(P′, Q) = γ0[Γαµ(P′,−Q)]†γ0.(40) •The quantities (Uforw)t′ 1t′ 2;t1t2and (Uback)t′ 1t′ 2;t1t2 are the matrix elements of the following forward and backward isospin operators Uforw =TiT†+1⊗τi(41) Uback =T+1 T† i⊗τi,(42) where T+1 is the spherical component of the isovector transition operator 3 2→1 2, normalized as 3 2, t∆(T†)λ1 2, tN=C1 2,1,3 2tN, λ, t∆(43) for λ=±1,0. E. Isospin structure of MEC The isospin dependence of the ∆ current is more complex than the other operators (seagull, pion-in-flight, and pion-pole). However, it is possible to expand the Uforw,back operators as linear combinations of the three basic isospin matrices τ(1),τ(2), and IV. This is a consequence of the following basic property of the isospin transition operators in cartesian coordinates: TiT† j=2 3δij −i 3ǫijk τk.(44) From this relation it follows that √3Uforw =1 √6−2τ+(2) + (IV)+(45) √3Uback =1 √6−2τ+(2) −(IV)+(46) Analogously, in the (1 ↔2) terms of Eqs. (33,34) the isospin operators have to be modified by making the change √3Uforw (1↔2) −−−→ 1 √6−2τ+(1) −(IV)+(47) √3Uback (1↔2) −−−→ 1 √6−2τ+(1) + (IV)+,(48) where we have made use of the antisymmetry property of the isovector operator IV=i(τ(1) ×τ(2)) under the interchange (1 ↔2). Substituting these relations in Eqs. (33,34), it is clear that the ∆-current operator can be written as the sum of three currents, each one characterized by a specific isospin dependence jµ ∆=τ+(1) Jµ ∆1(1′,2′; 1,2) + τ+(2) Jµ ∆2(1′,2′; 1,2) + (IV)+Jµ ∆3(1′,2′; 1,2),(49) where the three functions Jµ ∆i(1′,2′; 1,2) depend only on spins and momenta. This expression for neutrinos can be applied to antineutrinos by taking the (−) component of the isospin operators. In the same way, for electron scattering one should take the zcomponent of the isospin operators and keep only the Vpart of the current. The resulting electromagnetic ∆ current is in agreement with previous expressions [28]. From Eqs. (13, 24, 28, 49) we note that the total CC MEC for neutrino scattering can be written as jµ MEC =τ+(1) Jµ 1(1′2′; 1 2) + τ+(2) Jµ 2(1′2′; 1 2) + (IV)+Jµ 3(1′2′; 1 2),(50) where Jµ 1=Jµ ∆1 (51) Jµ 2=Jµ ∆2 (52) Jµ 3=Jµ sea +Jµ π+Jµ pole +Jµ ∆3.(53) This explicitly shows that the CC MEC operators transform as irreducible vectors in isospin space, implying in particular that the final 2p-2h nuclear states must have T= 1 for isoscalar nuclei (T= 0). Expression (50) will be useful in obtaining the response functions for the separate charge channels because the action of the three operators τ+(1), τ+(2), and (IV)+can be computed directly (see Appendix A). IV. ELECTROWEAK RESPONSE FUNCTIONS In the previous section we presented the expressions for the currents in our fully relativistic model of MEC. These currents were derived from the pion production amplitudes of [29]. In this section we give the explicit expressions for the weak response functions in the different 2p-2h charge channels. A. (νl, l−)responses CC neutrino scattering can induce two possible 2p-2h transitions: np →pp and nn →np. In the first case, the pp emission channel, the diagonal components of the hadronic tensor are of the type Wµµ pp =1 2X Z|hpp|jµ MEC (1′2′; 12) −jµ MEC (2′1′; 12)|npi|2,(54) where for brevity we have defined a symbol implying an integration over momenta and a sum over nucleon spins X Zf(1′2′; 12) ≡V (2π)9Zd3p′ 1d3h1d3h2 M4 E1E2E′ 1E′ 2 Θ(p′ 1, p′ 2, h1, h2)X s1s2s′ 1s′ 2 f(1′2′; 12) δ(E′ 1+E′ 2−E1−E2−ω),(55) 7 where f(1′2′; 12) is any function depending on the momenta and spins of the final 2p-2h states. Note that in the second line of Eq. (54) we have exchanged the momenta and spins of the final protons. Here we do not apply the general Eq. (11) which provides the total elementary hadronic tensor including all the charge channels. Using the expansion in Eq. (50) we get Wµµ pp =1 2X Zhpp|τ(1) +Jµ 1(1′2′; 12) + τ(2) +Jµ 2(1′2′; 12) +(IV)+Jµ 3(1′2′; 12) −τ(1) +Jµ 1(2′1′; 12) −τ(2) +Jµ 2(2′1′; 12) −(IV)+Jµ 3(2′1′; 12)|npi 2.(56) The isospin matrix elements can be computed from Eqs. (A2–A4) of Appendix A, resulting in Wµµ pp = 2 X Z|Jµ 1(1′2′; 12) + Jµ 3(1′2′; 12) −Jµ 1(2′1′; 12) −Jµ 3(2′1′; 12)|2.(57) Notice that this is written as the square of direct minus exchange matrix elements of the following “effective current” for pp-emission with neutrinos Jµ pp =Jµ 1+Jµ 3.(58) Changing variables 1′↔2′in the final state, it can be demonstrated that the contribution of the square of the exchange and direct parts are equal. Thus we obtain Wµµ pp = 4 X ZnJµ pp(1′2′; 12)2 −Re Jµ pp(1′2′; 12)∗Jµ pp(2′1′; 12)o.(59) The first term is usually called the “direct” contribution, and the second one is the “exchange” contribution, actually being the interference between the direct and exchange matrix elements. The exchange contributions to the 2p-2h cross section have not been included in the existing models of neutrino scattering [4, 7], whereas in this work we include them. In Fig. 2 we show a many-body diagrammatic representation of the direct and exchange contributions. The np emission case can be obtained in a similar way, the only difference being that now the exchanged particles should be the two initial neutrons. We obtain Wµµ np = 4 X ZnJµ np(1′2′; 12)2 −Re Jµ np(1′2′; 12)∗Jµ np(1′2′; 21)o,(60) where the effective current for np emission with neutrinos has been defined Jµ np =Jµ 2+Jµ 3.(61) W+(a) W+(b) W+(c) W+(d) FIG. 2: Some contributions of 2p-2h to the virtual W+selfenergy, or polarization propagator Πµν . The response functions considered in this work are related to the imaginary part of the polarization propagator, Im Πµν . The circle stands for the elementary model for W+N→πN of [29] without the nucleon-pole diagrams. Diagrams (a,b) represent the direct contribution. Diagrams (c,d) are the exchange contributions. final state ν ν e pp J1+J3×J1+J2 np J2+J3J1−J3−J1+J2 2J3 nn ×J2−J3−J1−J2 TABLE I: Effective currents for two-nucleon emission that appear in the different charge channels for (anti) neutrinos and electrons (see text). The above equations allow one to compute the diagonal hadronic tensor components appearing in the LL, CC and Tresponses. To compute the CL and T′responses the non-diagonal hadronic tensor components are necessary. They are computed in a similar way, resulting in Wµν pp = 4 X ZnJµ pp(1′2′; 12)∗Jν pp(1′2′; 12) −Jµ pp(1′2′; 12)∗Jν pp(2′1′; 12)o(62) Wµν np = 4 X ZnJµ np(1′2′; 12)∗Jν np(1′2′; 12) −Jµ np(1′2′; 12)∗Jν np(1′2′; 21)o.(63) 8 B. (¯νl, l+)responses In the case of antineutrinos the allowed charge 2p-2h channels are np →nn and pp →np. The corresponding formulae are obtained following the lines of the previous section, by taking the matrix elements of the (−) isospin components of the MEC. The results are similar to Eqs. (59,60,62,63), by using the effective currents given in Table I. C. (e, e′)responses In the case of electron scattering the three charge channels are all active. We take the matrix elements of the z-component of the MEC in isospin space. In the pp and nn cases the effective currents are given also in Table I. For np emission with electrons two effective currents appear, namely Wµµ pp =1 2X ZnJµ pp(1′2′; 12)2 −Re Jµ pp(1′2′; 12)∗Jµ pp(1′2′; 21)o(64) Wµµ nn =Wµµ pp (65) Wµµ np =X ZnJµ np1(1′2′; 12)2+Jµ np2(1′2′; 12)2 +2Re Jµ np1(1′2′; 12)∗Jµ np2(2′1′; 12)o,(66) where the two effective currents are Jµ np1=−Jµ 1+Jµ 2(67) Jµ np2= 2Jµ 3.(68) These are summarized in fourth column of Table I. V. TREATMENT OF THE ∆CURRENT In this section we provide the details of the treatment of the ∆ current in our model and compare with other approaches. The relativistic ∆ current contribution to the electromagnetic RTresponse was first computed in [40] and [22]. These authors started with the Peccei lagrangian for the γN∆ interaction [48]. This introduces a difference with respect to the vector interaction given in Eq. (38). The Peccei vertex only includes the O(1/M) term that should correspond to the CV 3/M term of Eq. (38). There is still another difference between the two approaches because the Peccei vertex includes a contraction with the tensor Θµν =gµν −1 4γµγν.(69) This tensor takes into account possible off-shellness effects of the virtual ∆ [49]. In the case the ∆ is on-shell, this is reduced to gµν because of the properties of Rarita- Schwinger spinors. We have verified that upon multiplying the above tensor by the first term of Eq. (38) the ∆ current of [22] is reproduced. This is a consequence of the identity Θβν(gµ ν6Q−Qνγµ) = 1 2(Qµγβ−γµ6Qγβ).(70) The resulting current coincides with the one given in [22] (notice that there is a relative minus sign with respect to Eq. (35) in the definition of the ∆ propagator in that reference). We have checked numerically that the inclusion of Θµν has a negligible effect on the transverse response at the kinematics relevant for this work, and therefore it will not be included in the calculations. Therefore in this work we are using exactly the same operator as in [22] for the ∆ vector current, corresponding to the term CV 3/M in Eq. (38). We neglect the terms CV 4and CV 5(CV 6= 0 by conservation of vector current) that are expected to give much smaller contributions because they are supressed by O(P/M). To be consistent, in the axial part we only include the leading contribution of Eq. (39), proportional to CA 5and neglect the other terms. In what follows we discuss the important point concerning the ambiguity related to the theoretical separation between 2p-2h and ∆-peak contributions. The ∆ peak is the main contribution to the pion production cross section. But inside the nucleus the ∆ can also decay into one nucleon that re-scatters producing twonucleon emission without pions. Therefore this decay of the ∆ should be considered part of the 2p-2h channel. Let us consider the diagram of Fig. 3. This diagram is implicitly included in our calculation, being one of the contributions to diagram (a) of Fig. 2. Thus, it can be considered to contribute to the 2p-2h responses and/or to the ∆ peak. In fact this diagram contains one self-energy insertion contributing to dressing the ∆ propagator. As a consequence, there is no unique way of separating the ∆ emission from the 2p-2h channels because ∆ emission already includes 2p-2h decays inside the nucleus. Hence, the MEC contribution given by Eq. (9), while being purely 2p-2h, also contributes to some extent to the ∆ peak and, conversely, any calculation of the ∆ peak including a dressed ∆ propagator would include implicitly some contribution from the diagram of Fig. 3. It is a matter of choice in building some specific model whether this contribution is regarded as part of the 2p-2h or ∆ peak responses. Here we include it in the 2p-2h response. The previous discussion should make clear that a comparison between models of MEC that use different prescriptions for the treatment of the ∆-pole makes no sense, as long as they contain different admixtures of ∆ emission. In other words, it is only the total cross section that is meaningful and worthwhile to compare. Before providing reliable predictions for neutrino scattering, any model must be validated by confronting it with quasielastic electron scattering data. Thus the val- 9 W+ FIG. 3: Direct term of the MEC with excitation of an intermediate ∆ decaying into a 2p-1h state. idation of any prescription for the MEC contribution requires one to compute the total (e, e′) cross section with a model that includes also both the quasielastic and inelastic contributions. The validation of our prescription for electron scattering has been recently performed in [26], where we have shown that the experimental world-data for 12C can be nicely reproduced within the super-scaling approach [27] using the MEC of [22]. For other models this necessary test has yet to be performed systematically. VI. RESULTS In this section we present results for the five 2p-2h response functions of neutrino scattering, as a function of (q, ω). This assumes that the energy of the incident neutrino is known. We study the dependence of the results on several ingredients of the model. First we validate the relativistic currents for low energy and momentum transfer by comparison with the electromagnetic transverse response function computed in the non-relativistic limit. The relativistic and nonrelativistic responses should coincide in this limit. We follow the semi-analytical method of [47] (see also [22, 41]) to compute the non-relativistic 2p-2h transverse response function in electron scattering. The comparison with the relativistic calculation also allows one to evaluate the size of the relativistic corrections. The nonrelativistic model is described in Appendix B. In Fig. 4 we show the electromagnetic Tresponse for low momentum transfer from q= 100 to 400 MeV/c and mass number A= 56. The value of the Fermi momentum is chosen to be kF=q/2. This is so because the nonrelativistic limit requires that all of the initial and final momenta go simultaneously to zero, and kFshould also be reduced accordingly. Another reason to reduce kF in this non-relativistic test is that for q < 2kFPauli blocking may reduce considerably the response function non rel rel q= 100 MeV/c kF=q/2 RT em [GeV]−1 100806040200 0.25 0.2 0.15 0.1 0.05 0 q= 200 MeV/c RT em [GeV]−1 200150100500 2 1.5 1 0.5 0 q= 300 MeV/c RT em [GeV]−1 300250200150100500 20 15 10 5 0 q= 400 MeV/c ω[MeV] RT em [GeV]−1 400350300250200150100500 50 40 30 20 10 0 FIG. 4: Electromagnetic transverse response function for 2p- 2h for low momentum qand kF=q/2. Here we take A= 56. and the comparison cannot be made. In the figure we see that for q= 100 MeV/c the relativistic and non-relativistic results are the same, and they start to differ only around q= 200 MeV/c. The difference is due mainly to the ∆ propagator, that in one case is considered constant, and in the other case has the relativistic energy-momentum dependence. In fact, for these low values of q, the maximum of the relativistic response appears around ω∼M∆−M≃300 MeV, which is the minimum energy needed to produce the ∆ excitation for a nucleon at rest. 16 where mF≡mπ/kFand x2 T=x2−q2 F+x2−y2 2qF2 .(B9) This non-relativistic result coincides with [47] for the seagull current except for the last plus sign in the last term of Eq. (B8) that in the cited reference is a multiplication sign (see Eq. (2.24) of [47]). The pion-in-flight response and its interference with the seagull current are rT π(x, y) = 2f2 πNN m2 π FV 1216 k2 F x2y2x2 T (x2+m2 F)2(y2+m2 F)2 (B10) rT sea,π(x, y) = −2f2 πNN m2 π FV 1216 k2 F x2 T (x2+m2 F)(y2+m2 F) ×x2 (x2+m2 F)+y2 (y2+m2 F).(B11) This result coincides with [47], except for the x2-term in the interference, which was missing in Eq. (2.24) of [47]. In the case of the ∆ current various schemes are typically adopted in going to the non-relativistic limit and thus in deriving the non-relativistic limit of the ∆ current. We take the approach described in [53], where the non-relativistic reduction of the ∆ current reads J∆=i 6 CV 3f∗fπNN Mm2 π k2·σ(2) m2 π+k2 2hBτ(2) zk2 −A 2[τ(1) ×τ(2)]zσ(1) ×k2×q +(1 ↔2),(B12) where A=8 3(M∆−M)and B= 2A. The pure ∆ response can be written as rT ∆=4 k2 F2f2 πNN m2 π2q2 Fa2k4 Fx2(x2+x2 L) (x2+m2 F)2 +y2(y2+y2 L) (y2+m2 F)2+2q2 Fx2 T (x2+m2 F)(y2+m2 F) + 2q2 Fb2k4 Fx2 Tx2 (x2+m2 F)2+y2 (y2+m2 F)2, (B13) where we have defined a=1 2 CV 3 M f∗ 6fπNN A(B14) and b= 2a. The a, b factors notation is similar to that used in [47], while the A, B factors are used in [22]. Note that there are several definitions for Aand Bin the literature which arise from different approximations in deriving the non-relativistic limit of the ∆ current, already discussed in [22]. 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