Comment on "Quasielastic lepton scattering and back-to-back nucleons in the short-time approximation", by S. Pastore et al
aa r X i v : . [ nu c l - t h ] J u l Comment on “Quasielastic lepton scattering and back-to-back nucleonsin the short-time approximation”, by S. Pastore et al.
Omar Benhar ∗ INFN and Department of Physics, Sapienza University, I-00185 Rome, Italy (Dated: July 23, 2020)The article of Pastore et al. , while proposing an interesting and potentially useful approach for thegeneralisation of Quantum Monte Carlo techniques to the treatment of the nuclear electromagneticresponse, features an incorrect and misleading discussion of y -scaling. The response to interactionswith transversely polarised virtual photons receives sizeable contributions from non-scaling pro-cesses, in which the momentum transfer is shared between two nucleons. It follows that, contraryto what is stated by the the authors, y -scaling in the transverse channel is accidental. The work of Pastore et al. [1] can be seen as a first steptowards the implementation of the factorisation scheme,which naturally emerges from the formalism of the im-pulse approximation [2], in the computational frameworkof Quantum Monte Carlo (QMC). In view of the diffi-culties associated with the identification of specific finalstates in the nuclear responses obtained from QMC cal-culations, this is an interesting, and potentially useful,development.The Short Time Approximation (STA) developed bythe authors involves a number of strong simplifying as-sumptions—such as neglect of the energy dependencein the propagator of the spectator system, analysed inRef.[3]—the validity of which will only be fully appraisedin years to come, when the proposed approach will beextended to a broader kinematical range and nuclear tar-gets other than the three- and four-nucleon systems. Thediscussion of scaling in Section IV, on the other hand,comprises incorrect and misleading statements requiringa prompt clarification.The occurrence of y -scaling in electron-nucleus scat-tering—that is, the observation that the target response,which in general depends on both momentum and energytransfer, q and ω , in the limit of large q = | q | can be re-duced to a function of the single variable y = y ( q, ω )[4, 5]—reflects the onset of the kinematical regime inwhich the dominant reaction mechanism is elastic scat-tering off individual nucleons [6].The definition of the scaling variable y follows from theassumption that the momentum transfer is absorbed by only one nucleon. In the absence of Final State Interac-tions (FSI) between the struck nucleon and the specta-tors, conservation of energy in the lab frame entails therelation [7] ω + M A = p m + ( y + q ) (1)+ p ( M A − m + E thr ) + y , where M A and m are the target and nucleon mass,respectively, while E thr denotes the nucleon emission ∗ [email protected] threshold. Equation (1) shows that the scaling vari-able has a straightforward physical interpretation, beingtrivially related to the projection of the momentum ofthe struck nucleon along the direction of the momentumtransfer, k k = k · q /q .The scaling function of a nucleus of mass number Aand charge Z, defined as [7] F ( y ) = lim q →∞ F ( q, y ) (2)is obtained from F ( q, y ) = dσ eA Z dσ ep + (A − Z) dσ en (cid:16) dωdk k (cid:17) , (3)where dσ eA is the measured nuclear cross section,while dσ ep and dσ en are the elastic electron-proton andelectron-neutron cross sections, stripped of the energy-conserving δ -function . Large deviations from the scalingbehaviour, observed at y >
0, arise from processes otherthan elastic scattering, while smaller scaling violations at y < n ( k ). Indeuteron, the relation between scaling function and mo-mentum distribution takes the simple form [8] n ( k ) = − π y dF ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) | y | = k , (4)with k = | k | .In the 1980s, Finn et al. [9] performed the first scal-ing analysis of the carbon responses to interactions withlongitudinally and transversely polarised virtual photons,measured at Saclay [10]. The results of this work revealeda significant excess of strength in the transverse channel,which, however, did not appear to spoil the scaling be-haviour at y <
0. As a consequence, the analysis led Here, the elementary electron-nucleon cross sections, which ex-plicitly depend on the nucleon momentum, k , and and removalenergy, E , are evaluated at | k | = | y | and E = E thr . to the determination of two distinct q -independent func-tions, F L ( y ) and F T ( y ), even though the interpretation of F T ( y ) as a scaling function cannot be reconciled with thepresence of contributions arising from non-scaling pro-cesses, driven by two-nucleon currents. More recently,similar results have been obtained from the analysis ofthe longitudinal and transverse responses of light nu-clei [11].Pastore et al. fail to introduce the reader to the con-cept of scaling and the interpretation of the scaling vari-able. They limit themselves to assert that, because theresults of their Green’s Function Monte Carlo (GFMC)calculations reproduce the experimental data, they ob-viously scale . Regretfully, this statement is meaninglessand misleading In Section IV, the authors go to considerable lengthsto explain the mechanism leading to q -independence ofthe GFMC response functions. However, their conclu-sion that y -scaling is preserved even in the presence ofa mechanism other that single-nucleon knock out doesnot take into account the fact that q -independence and y -scaling are distinct properties, and do not necessarilyimply one another.Accidental y -scaling—that is, scaling in the presenceof non-scaling mechanisms, such as FSI, giving rise tosizeable q -independent contributions to the nuclear re- sponse—is long known to occur in a variety of processes,ranging from electron-nucleus scattering [13] to neutronscattering from liquid helium [14]. Obviously, when scal-ing is accidental, the interpretation of both the scalingvariable and the scaling function discussed above is nolonger applicable.Processes involving two nucleon currents do not scale in the variable y , because the momentum transfer isshared between two nucleons, and conservation of en-ergy cannot be written as in Eq. (1). It follows that y -scaling in the transverse channel is, in fact, accidental.As correctly noted by the authors of Ref. [9], a meaning-ful scaling function, providing information on initial-statedynamics, can only be obtained from the analysis of thelongitudinal response.On the constructive side, it should be noted that, afterremoval of the excess transverse strength arising formprocesses involving two-nucleon currents, the longitudi-nal and transverse responses obtained by Pastore et al may be employed to perform a fully consistent studyof the universality of the scaling function. The resultsof such a study would be valuable for the ongoingefforts to exploit y -scaling as a tool for the analysis ofthe signals detected by accelerator-based searches ofneutrino oscillations [15]. [1] S. Pastore, J. Carlson, S. Gandolfi, R. Schiavilla, andR.B. Wiringa, Phys. Rev. C , 044612 (2020).[2] N. Rocco, A. Lovato, and O. Benhar, Phys. Rev. Lett , 192501 (2016).[3] O, Benhar, A, Fabrocini, and S. Fantoni, Phys. Rev. Lett. , 052501 (2001).[4] I. Sick, D. Day and J.S. McCarthy, Phys. Rev. Lett. ,871 (1980).[5] B.D. Day et al. , Phys. Rev. Lett. , 427 (1989).[6] G. B. West, Phys. Rep. , 263 (1975).[7] C. Ciofi degli Atti, E. Pace, and G. Salm`e, Phys. Rev C , 1155 (1991). [8] C. Ciofi degli Atti, E. Pace, and G. Salm`e, Phys. Rev C , 1208 (1987).[9] J.M. Finn, R.W. Lourie, and B.H. Cottmann, Phys. Rev.C , 2230 (1984) .[10] P. Barreau et al. , Nucl. Phys. A , 515 (1983).[11] J. Carlson, J. Jourdan, R. Schiavilla, and I. Sick, Phys.Rev. C , 024002 (2002).[12] T.W. Donnelly and I. Sick, Phys. Rev. C , 065502(1999).[13] O. Benhar, Phys. Rev. Lett. , 3130 (1999).[14] J.J. Weinstein and J.W. Negele, Phys. Rev. Lett. ,1016 (1992).[15] G.D. Megias et al. , Phys. Rev. D , 093004 (2016). The reader is, in fact, also misled into believing that Fig. 3 ofthe paper of Carlson et al. [11] show longitudinal and transverse response functions, while the quantities displayed are actually thefunctions F L ( y ) and F T ( yy