We use a relativistic model of meson-exchange currents to compute the
proton-neutron and proton-proton yields in $(e,e')$ scattering from $^{12}$C in
the 2p-2h channel. We compute the response functions and cross section with the
relativistic Fermi gas model for a range of kinematics from intermediate to
high momentum transfers. We find a large contribution of neutron-proton
configurations in the initial state, as compared to proton-proton pairs. The
different emission probabilities of distinct species of nucleon pairs are
produced in our model only by meson-exchange currents, mainly by the $\Delta$
isobar current. We also analyze the effect of the exchange contribution and
show that the direct/exchange interference strongly affects the determination
of the np/pp ratio.Comment: 5 pages, 6 figure

Amaro, J. E.

Barbaro, M. B.

Caballero, J. A.

De Pace, A.

Donnelly, T. W.

Megias, G. D.

Simo, I. Ruiz

English

arXiv

Barbaro, Maria Benedetta

Institutional Research Information System University of Turin

PHYSICAL REVIEW C 94, 054610 (2016)Emission of neutron-proton and proton-proton pairs in electron scatteringinduced by meson-exchange currentsI. Ruiz Simo,1 J. E. Amaro,1 M. B. Barbaro,2,3 A. De Pace,2 J. A. Caballero,4 G. D. Megias,4 and T. W. Donnelly51Departamento de Fı´sica Ato´mica, Molecular y Nuclear, and Instituto de Fı´sica Teo´rica y Computacional Carlos I,Universidad de Granada, Granada 18071, Spain2INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy3Dipartimento di Fisica, Universita` di Torino, Via Pietro Giuria 1, 10125 Torino, Italy4Departamento de Fı´sica Ato´mica, Molecular y Nuclear, Universidad de Sevilla, Apartado 1065, 41080 Sevilla, Spain5Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139, USA(Received 22 June 2016; published 8 November 2016)We use a relativistic model of meson-exchange currents to compute the proton-neutron and proton-protonyields in (e,e′) scattering from 12C in the 2p-2h channel. We compute the response functions and cross sectionwith the relativistic Fermi gas model for a range of kinematics from intermediate- to high-momentum transfers.We find a large contribution of neutron-proton configurations in the initial state, as compared to proton-protonpairs. The different emission probabilities of distinct species of nucleon pairs are produced in our model only bymeson-exchange currents, mainly by the  isobar current. We also analyze the effect of the exchange contributionand show that the direct-exchange interference strongly affects the determination of the np/pp ratio.DOI: 10.1103/PhysRevC.94.054610I. INTRODUCTIONThe process of two-nucleon knockout reactions in electronscattering is thought to involve short-range correlations (SRCs)in nuclei. In this case one expects an excess of neutron-proton(np) pairs over proton-proton (pp) pairs. Experiments on 12Cfor high-momentum transfer and missing momentum havereported a number of np pairs 18 ± 5 times larger than theirpp counterparts [1–3]. More recently the dependence on thenuclear mass number has been studied in Refs. [4–6], wherethe aim of this series of recent investigations at JeffersonLab is to unambiguously determine the short-range propertiesof light nuclei [7–10]. The analysis of these experimentsseems to be in agreement with theoretical predictions ofnucleon and nucleon pair momentum distributions in vari-ational Monte Carlo calculations, where the importance ofthe tensor forces in the ground-state correlations of nucleihas been emphasized [11,12]. Note that in the mentionedelectron scattering experiments the kinematics involved high-momentum transfers, Q2 > 1.4 (GeV/c)2, and thus relativisticcorrections are likely to be important in the theoreticaldescriptions employed.Another source of SRC evidence comes from calculationsof the semi-inclusive electron scattering reaction (e,e′pN ),which however relies on factorization approximations thathave not been fully justified for all the kinematics of interest[13]. While the kinematics of the experiments have beenselected to minimize the contribution from other mecha-nisms that can induce two-particle emission, such as meson-exchange currents (MECs) and isobar excitations [1], thecontribution of MECs cannot be ruled out a priori.Similarly to the electron case, observation of events inneutrino scattering with a pair of energetic protons hasbeen reported in the ArgoNeuT experiment [14]. From theseevents several back-to-back nucleon configurations have beenidentified and associated with nuclear mechanisms involvingshort-range correlated np pairs in the nucleus [15]. The SRCexplanation of this excess of back-to-back events in ArgoNeuTis still controversial [16,17].In this work we investigate the relative effects of MECs onthe separate pp and np channels in the inclusive 2p-2h crosssection, without including NN correlations. It is importantto know if the MEC alone can explain, at least partially, theobserved enhancement of the 12C(e,e′np) cross section overthat of the 12C(e,e′pp) cross section, as observed in the data[2]. This is in lieu of a fully reliable relativistic model for the(e,e′pN ) cross section, because such a model is unavailable.II. THE MODELWe have recently developed a fully relativistic model ofmeson-exchange currents in the 2p-2h channel for electronand neutrino scattering [18]. This model is an extension of therelativistic MEC model of Ref. [19] to the weak sector. It hasbeen recently validated by comparing to the 12C(e,e′) inclusivecross section data for a wide kinematical range within thesuperscaling approach [20]. This model describes jointly thequasielastic and inelastic regions using two scaling functionsfitted to reproduce the data, while the 2p-2h MEC contributionproperly fills the dip region in between, resulting in excellentglobal agreement with the data.With this benchmark model we are able to study the separatenp and pp channels in the response functions and cross sectionfor the three (e,e′), (νl,l−), and (ν¯l ,l+) reactions. Although our2p-2h model does not explicitly include NN correlations, theyare implicitly accounted for in the phenomenological scalingfunctions, and thus we cannot isolate the 2p-2h contributionscoming from SRCs in this approach. However, with this modelwe are at least able to provide a precise estimation of the size ofMECs in the separate channels for high-momentum and energytransfers where relativistic effects are important. In this work2469-9985/2016/94(5)/054610(4) 054610-1 ©2016 American Physical SocietyI. RUIZ SIMO et al. PHYSICAL REVIEW C 94, 054610 (2016)we focus on the contributions of pion-in-flight, seagull, and(1232) excitation diagrams of the MECs.We write the inclusive (e,e′) cross section as the product ofthe Mott cross section and a linear combination of longitudinaland transverse response functions:dσddω= σMott[vLRL(q,ω) + vT RT (q,ω)]. (1)The response functions for the momentum transfer q and theenergy transfer ω contain the contribution of all of the nuclearexcitations with energy ω.In the relativistic Fermi gas model the excitations can be1p-1h, 2p-2h, and so on. We describe the particle and holestates as relativistic plane waves with momenta above andbelow the Fermi momentum kF , respectively. In this workwe compute the 2p-2h contributions to the response functionsthat are proportional to the volume V of the system, whichfor symmetric nuclear matter has Z = N = A/2, where V =3π2Z/k3F . They are given byRK2p-2h =V(2π )9∫d3p′1d3h1d3h2m4NE1E2E′1E′2× rK (p′1,p′2,h1,h2)δ(E′1 + E′2 − E1 − E2 − ω)×θ (p′2 − kF )θ (p′1 − kF )θ (kF − h1)θ (kF − h2),(2)for K = L and T , where p′2 = h1 + h2 + q − p′1 is fixed bymomentum conservation, mN is the nucleon mass, and Eiand E′i are the on-shell energies of the holes and particles,respectively. The response functions for the elementary 2p-2h excitation, rK (p′1,p′2,h1,h2), for given initial and finalmomenta, are the sums over spins of the squares of theMEC matrix elements and can be found in Ref. [18] forthe separate np, pp and nn, charge channels. The 2p-2hstates are antisymmetrized and therefore our MEC matrixelements and response functions contain direct and exchangecontributions. The seven-dimensional integral of Eq. (2) iscomputed numerically without approximations by followingthe method of Refs. [21,22]. We refer the reader to Ref. [18]for further details on the model.III. RESULTSIn what follows we show results for the separate np andpp pair emission from 12C. In our model, the nn channelgives the same contribution as the pp one, because both areinduced by the -isobar current. In Figs. 1 and 2 we showthe transverse and longitudinal response functions. For q =600 MeV/c, the np transverse response is around a factorof 12 times larger than the pp one. For q = 1000 MeV/cthis factor gets reduced to ∼ 6. In the longitudinal responsesthe np/pp ratio is further reduced. However, the longitudinalMEC contribution to the cross section is almost negligiblecompared with the transverse one, because the dominant excitation current is mainly transverse.In Fig. 3 we show the effect of neglecting the direct-exchange interference of the MEC matrix elements. Fornp pairemission it is negligible. On the other hand, for pp emissionppnpq = 600 MeV/cRT[GeV−1]60050040030020010006543210q = 1000 MeV/cω [MeV]RT[GeV−1]100080060040020032.521.510.50FIG. 1. Electromagnetic transverse response function for np andpp pair emission off 12C as a function of ω for two values of q.ppnpq = 600 MeV/cRL[GeV−1]60050040030020010000.250.20.150.10.050q = 1000 MeV/cω [MeV]RL[GeV−1]10008006004002000.50.40.30.20.10FIG. 2. Same as for Fig. 1, but now for the longitudinal responsefunction.054610-2EMISSION OF NEUTRON-PROTON AND PROTON-PROTON . . . PHYSICAL REVIEW C 94, 054610 (2016)np dir.np totalq = 600 MeV/cRT[GeV−1]60050040030020010006543210q = 1000 MeV/c100080060040020043210pp dir.pp totalω [MeV]RT[GeV−1]60050040030020010001.41.210.80.60.40.20ω [MeV]10008006004002000.50.40.30.20.10FIG. 3. Electromagnetic transverse response function showingthe contributions of direct-exchange interference terms to the separatenp and pp channels.it is of the same order as the direct contribution. Thereforeneglecting this interference would reduce the abovementionednp/pp ratios by a factor of 2. This implies that the interferenceis crucial for describing this ratio properly. Although the neteffect of the interference in the 2p-2h is less than 20% andppnppp+np+nnEe = 961 MeV, θ = 37odσ/dωdΩ[nb/MeV]60040020032.521.510.50Ee = 3595 MeV, θ = 16oω [MeV]dσ/dωdΩ[nb/MeV]150010005000.60.50.40.30.20.10FIG. 4. Electromagnetic differential cross section separated intothe different charge contributions.pp (x 12)npEe = 961 MeV, θ = 37odσ/dωdΩ[nb/MeV]60040020032.521.510.50pp (x 6)npEe = 3595 MeV, θ = 16oω [MeV]dσ/dωdΩ[nb/MeV]150010005000.60.50.40.30.20.10FIG. 5. Same as for Fig. 4, but now the pp contribution has beenscaled by a constant factor, as shown in the two panels.sometimes it can be safely disregarded [23,24], this is not thecase for the pp separate contribution. However, up to nowthere have been no calculations of these ratios.In Fig. 4 we show the semi-inclusive double-differentialcross section for pair emission from 12C for two incidentenergies and electron scattering angles. The separate np andpp pair emission channels are also displayed. As in the case ofthe transverse response, the np pair emission clearly dominatesover the pp pair emission. The ratio of np over pp depends onthe kinematics. It is roughly between 12 and 6, as can be seenin Fig. 5, where we have multiplied the pp cross section bya constant factor. It is worth mentioning that the np/pp ratioalso depends on ω, although a constant scaling factor does areasonable job at inter-relating the different channels.In Fig. 6 we show the separate np and pp emission crosssections for the kinematics of the measurement of Refs. [1,2].In this experiment the energy transfer was chosen to beω = 865 MeV, well below the  excitation maximum thatoccurs at about 1.12 GeV. For this reason we only show inFig. 6 the low-energy tail of the cross section below this valueof ω. These kinematics were chosen to minimize the MECcontribution that we are showing here. In fact, for this value ofω the MECs are small, because we are far from the maximumof the  peak. The results of Fig. 6 are the expected MECcontributions to the semi-inclusive (e,e′pN ) cross section inan uncorrelated system. In this case the np/pp ratio is a factor6. This is not sufficient to explain the factor 18 ± 5 found inthe experiment and attributed to SRCs, coming mainly from054610-3I. RUIZ SIMO et al. PHYSICAL REVIEW C 94, 054610 (2016)pp (x 6)npEe = 4621 MeV, θ = 19.5oω [MeV]dσ/dωdΩ[nb/MeV]10009008007006005004000.0080.0070.0060.0050.0040.0030.0020.0010FIG. 6. Same as for Fig. 5, but now for the electron kinematicsof the experimental setup of Ref. [1].the tensor nuclear force. However, it does suggest that anyanalysis where MEC effects are not included must be viewedwith caution. In the MEC case this factor comes roughly fromisospin considerations, but the effect of the direct-exchangeinterference is q dependent and can modify it by a largeextent depending on the kinematics. Note that the kinematicsof the experiment [1] is relativistic, with Q2 = 2 (GeV/c)2and q = 1.65 GeV/c. Thus the relativistic calculation of the2p-2h MEC responses is mandatory. One of the findings ofRefs. [1,2] was that there are very few correlated pp pairs.Therefore we expect that our result for the pp cross sectionwill not change significantly when including SRCs, while thenp cross section should be considerably increased.IV. CONCLUSIONSIn summary, we have computed the semi-inclusive12C(e,e′np) and 12C(e,e′pp) cross sections in the relativisticFermi gas including a fully relativistic model of MECs. Thenp/pp ratio has been quantified and analyzed for severalkinematics. The direct-exchange interference terms are foundto be important, especially in the pp channel. The MECsalone are not able to explain completely the data for thisratio, which is found to be larger than our findings, althoughthey clearly should be expected to play an important rolein determining the np/pp ratio in two-nucleon emissionelectron scattering as well as the related flavor dependence incharge-changing neutrino reactions. Said another way: whilemissing contributions in our model (like SRCs) could beimportant at least for the kinematics of the existing experiment,our results indicate that, to understand in depth the size of sucheffects, the MEC contributions should also be included. In thefuture the relativistic modeling used for the latter could beextended to include correlation currents of the pionic type [25],effective interactions, or, alternatively, correlation operators inthe wave functions.ACKNOWLEDGMENTSThis work was supported by Spanish Direccio´n Generalde Investigacio´n Cientı´fica y Te´cnica and FEDER funds(Grants No. FIS2014-59386-P and No. FIS2014-53448-C2-1),by the Agencia de Innovacio´n y Desarrollo de Andalucı´a(Grants No. FQM225 and No. FQM160), by INFN underthe project MANYBODY, and in part (T.W.D.) by the U.S.Department of Energy under Cooperative Agreement No.DE-FC02-94ER40818. I.R.S. acknowledges support from aJuan de la Cierva-incorporacio´n fellowship from SpanishMINECO. G.D.M. acknowledges support from a Junta deAndalucia fellowship (FQM7632, Proyectos de Excelencia2011). The authors also acknowledge support from “Espacede Structure et de re´actions Nucle´aire The´orique” (ESNT,http://esnt.cea.fr) at CEA-Saclay, where this work was partiallycarried out.[1] R. Shneor et al. (JLab Hall A Collaboration), Phys. Rev. Lett.99, 072501 (2007).[2] R. Subedi et al., Science 320, 1476 (2008).[3] J. Ryckebusch, M. Vanhalst, and W. Cosyn, J. Phys. G: Nucl.Part. Phys. 42, 055104 (2015).[4] O. Hen et al. (CLAS Collaboration), Phys. Lett. B 722, 63(2013).[5] O. Hen et al. (CLAS Collaboration), Science 346, 614 (2014).[6] C. Colle, O. Hen, W. Cosyn, I. Korover, E. Piasetzky, J.Ryckebusch, and L. B. Weinstein, Phys. Rev. C 92, 024604(2015).[7] N. Fomin et al., Phys. Rev. Lett. 108, 092502 (2012).[8] P. Monaghan et al., J. Phys. G: Nucl. Part. Phys. 41, 105109(2014).[9] I. Korover et al., Phys. Rev. Lett. 113, 022501 (2014).[10] K. S. Egiyan et al., Phys. Rev. Lett. 96, 082501 (2006).[11] R. Schiavilla, R. B. Wiringa, S. C. Pieper, and J. Carlson, Phys.Rev. Lett. 98, 132501 (2007).[12] R. B. Wiringa, R. Schiavilla, S. C. Pieper, and J. Carlson, Phys.Rev. C 89, 024305 (2014).[13] C. Colle, W. Cosyn, and J. Ryckebusch, Phys. Rev. C 93, 034608(2016).[14] R. Acciarri et al., Phys. Rev. D 90, 012008 (2014).[15] F. Cavanna, O. Palamara, R. Schiavilla, M. Soderberg, andR. B. Wiringa, arXiv:1501.01983.[16] L. B. Weinstein, O. Hen, and E. Piasetzky, Phys. Rev. C 94,045501 (2016).[17] K. Niewczas and J. T. Sobczyk, Phys. Rev. C 93, 035502 (2016).[18] I. R. Simo, J. E. Amaro, M. B. Barbaro, A. De Pace, J. A.Caballero, and T. W. Donnelly, arXiv:1604.08423.[19] A. De Pace, M. Nardi, W. M. Alberico, T. W. Donnelly, and A.Molinari, Nucl. Phys. A 726, 303 (2003).[20] G. D. Megias, J. E. Amaro, M. B. Barbaro, J. A. Caballero,and T. W. Donnelly, Phys. Rev. D 94, 013012 (2016).[21] I. R. Simo, C. Albertus, J. E. Amaro, M. B. Barbaro, J. A.Caballero, and T. W. Donnelly, Phys. Rev. D 90, 033012 (2014).[22] I. R. Simo, C. Albertus, J. E. Amaro, M. B. Barbaro, J. A.Caballero, and T. W. Donnelly, Phys. Rev. D 90, 053010(2014).[23] R. C. Carrasco, M. J. Vicente Vacas, and E. Oset, Nucl. Phys. A570, 701 (1994).[24] A. Gil, J. Nieves, and E. Oset, Nucl. Phys. A 627, 599 (1997).[25] J. E. Amaro, C. Maieron, M. B. Barbaro, J. A. Caballero, andT. W. Donnelly, Phys. Rev. C 82, 044601 (2010).054610-4

Emission of neutron-proton and proton-proton pairs in electron scattering induced by meson-exchange currents

We use a relativistic model of meson-exchange currents to compute the proton-neutron and proton-proton yields in (e,e′) scattering from [superscript 12]C in the 2p-2h channel. We compute the response functions and cross section with the relativistic Fermi gas model for a range of kinematics from intermediate- to high-momentum transfers. We find a large contribution of neutron-proton configurations in the initial state, as compared to proton-proton pairs. The different emission probabilities of distinct species of nucleon pairs are produced in our model only by meson-exchange currents, mainly by the Δ isobar current. We also analyze the effect of the exchange contribution and show that the direct-exchange interference strongly affects the determination of the np/pp ratio.Spain. Dirección General de Investigación Científica y TécnicaFonds Europeen de Developpement Economique et Regional (Grants FIS2014-59386-P and FIS2014-53448-C2-1)Agencia de Innovacion y Desarrollo de Andalucia (Grants FQM225 and FQM160)Istituto nazionale di fisica nucleare (INFN) (Project MANYBODY)United States. Dept. of Energy (Cooperative Agreement DE-FC02-94ER40818

Donnelly, T William

DSpace@MIT

I. Ruiz Simo

J. E. Amaro

M. B. Barbaro

A. De Pace

J. A. Caballero

G. D. Megias

T. W. Donnelly

Crossref

We use a relativistic model of meson-exchange currents to compute the proton-neutron and proton-proton
yields in (e,e
 ) scattering from 12C in the 2p-2h channel. We compute the response functions and cross section
with the relativistic Fermi gas model for a range of kinematics from intermediate- to high-momentum transfers.
We find a large contribution of neutron-proton configurations in the initial state, as compared to proton-proton
pairs. The different emission probabilities of distinct species of nucleon pairs are produced in our model only by
meson-exchange currents, mainly by the isobar current.We also analyze the effect of the exchange contribution
and show that the direct-exchange interference strongly affects the determination of the np/pp ratio.Dirección General de Investigación Científica y Técnica y FEDER FIS2014-59386-P FIS2014-53448-C2-1Agencia de Innovación y Desarrollo de Andalucía FQM225  FQM16

Ruiz Simo, I.

Amaro Soriano, José Enrique

Pace, Arturo de

Caballero Carretero, Juan Antonio

Megías Vázquez, Guillermo Daniel

idUS. Depósito de Investigación Universidad de Sevilla

Emission of neutron-proton and proton-proton pairs in electron
  scattering induced by meson-exchange currents

Emission of neutron-proton and proton-proton pairs in electron scattering induced by meson-exchange currents

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Institutional Research Information System University of Turin

DSpace@MIT

Crossref

idUS. Depósito de Investigación Universidad de Sevilla