NNT@UW-20-06Discovery vs.
Precision in Nuclear Physics- A Tale of Three Scales
Gerald A. Miller
Department of Physics, University of Washington, Seattle, WA 98195-1560, USA (Dated: August 18, 2020)At least three length scales are important in gaining a complete understanding of the physics ofnuclei. These are the radius of the nucleus, the average inter-nucleon separation distance, and thesize of the nucleon. The connections between the different scales are examined by using examplesthat demonstrate the direct connection between short-distance and high momentum transfer physicsand also that significant high momentum content of wave functions is inevitable. The nuclear size isconnected via the independent-pair approximation to the nucleon-nucleon separation distance, andthis distance is connected via the concept of virtuality to the EMC effect. An explanation of thelatter is presented in terms of light-front holographic wave functions of QCD. The net result is thatthe three scales are closely related, so that a narrow focus on any given specific range of scales mayprevent an understanding of the fundamental origins of nuclear properties. It is also determinedthat, under certain suitable conditions, experiments are able to measure the momentum dependenceof wave functions.
I. INTRODUCTION
In studying atomic nuclei one encounters threedifferent length scales: the nuclear radius R A ( ≈ d ≈ . r N ≈ .
84 fm. The pion Comptonwave length, 1 /m π = 1.4 fm is close to d , so is nota separate scale. The correlation length associatedwith the Fermi momentum, ≈ π/k F [1] is also of theorder of d .The general modern trend of theorists is to focuson each length scale of a given subject using thetechniques of effective field theory. The main idea(see e.g. [2]) is: if there are parameters that arevery large or very small compared to the physicalquantities (with the same dimension) of interest,one may get a simpler approximate description ofthe physics by setting the small parameters to zeroand the large parameters to infinity. Then the finiteeffects of the large parameters can be included assmall perturbations about the simple approximatestarting point.This scale separation is a common technique(see e.g. [3]) in which physics at large distancesis assumed not to depend on physics at shorterdistances. A famous example is the weak interactionin which the effects of W and Z boson exchangescan be treated as contact (zero-ranged) interactionsat low energies. The general philosophy is that ifone is working at a low mass scale m one doesn’tneed to consider dynamics at a mass scale Λ (cid:29) m . Or in terms of distances: the long distance scale1 /m must be very much greater than the shortdistance scale 1 / Λ. In other words, there must bea large separation of scales for effective field theorytechniques to be maximally efficient. In nuclearphysics the scale separation is not very large–thevalues of relevant distances are not widely separated.In using effective field theory, theorists concentrateon a given range of length scales. A typical proce-dure is to make robust calculations that enable firmpredictions. These are then tested by experiments,and the results may confirm the theories or (morelikely) lead to revision of the theories. Anotherscenario, in which experiment leads, is that anexperiment discovers an unexpected phenomenon,such as the Rutherford’s discovery of the atomicnucleus or the SLAC-MIT discovery of quarks withinthe nucleon [4, 5].The two approaches of the previous paragraphcan be summarized as precision vs discovery. Theeffective field theory approach of working within agiven scale is aptly suited for precision work. Incontrast, discovery of new phenomena is not welltreated by scale separation techniques because newphenomena are often related to discovering a newrelevant scale.I comment on the precision approach. Muchcurrent activity in precision nuclear structure cal-culations is based on using low energy, long lengthscale treatments. These began with interactions,known as V low k , that use renormalization grouptransformations that lower a cutoff in relative a r X i v : . [ nu c l - t h ] A ug momentum to derive NN potentials with vanishingmatrix elements for momenta above the cutoff. Suchinteractions show greatly enhanced convergenceproperties in nuclear few- and many-body systemsfor cutoffs of order Λ = 2 fm − or lower [6–10]. Latercalculations use renormalization group methods tosoften interactions in nuclear systems. This extendsthe range of many computational methods and qual-itatively improves their convergence patterns [11].The similarity renormalization group (SRG) [12–14]does this by systematically evolving Hamiltoniansvia a continuous series of unitary transformationschosen to decouple the high- and low-energy matrixelements of a given interaction [15, 16].However, many conventional NN potentials,feature strong short-range repulsion [17]. This issupported by some lattice gauge QCD calculations[18–23]. The repulsion causes bound states withvery low energies (such as the deuteron) to haveimportant contributions to the binding and otherproperties from high-momentum components.In Ref. [24], the authors calculate cross sectionsfor electron scattering from light nuclei. Theyconclude: “and thus the data confirm the existenceof high-momentum components in the deuteron wavefunction”. The high-momentum components of thedeuteron lead to inclusive electron-scattering crosssection ratios with simple scaling properties [25].That reference finds significant “evidence for thedominance of short-range correlations in nuclei”.Ref. [26] argued that the statement of Ref. [24](and by implication that of Ref. [25]) is not cor-rect because wave functions are not observables.Similarly Ref. [27] argued that nuclear momentumdistributions are not observable. It is certainly truethat wave functions are not observable quantities,but cross sections are observables.There are prominent examples that momentum-space wave functions are closely related to crosssections. Showing that the cross section of the photo-electric effect in hydrogen is proportional to thesquare of the momentum-space ground-state wavefunction of hydrogen is a text-book problem [28, 29].The modern version of the photo-electric effect iscalled Angle Resolved Photoemission Spectroscopy(ARPES) a technique that is well-known, see e.g. Ref. [30], to yield information of about the momen-tum and energy states of electrons in materials. Thestatement that measurements of cross sections canbe used to learn about wave functions violates noprinciples of quantum mechanics. One of the purposes of this paper is to exemplifyhow the use of the impulse approximation simplifiesthe connection between cross sections and wavefunctions for nuclear processes at high momentumtransfer. If the kinematics are correctly chosen theeffects of various processes that are not directlyrelated to wave functions can be minimized [31], sothat in effect measuring cross section measures im-portant properties of wave functions. See Sects. IV,VI, and VII.The principle concern of the present epistle isthat current experiments involving nuclei cover allthe three scales mentioned above. Deep inelasticscattering experiments on nuclei, involving squaresof four momentum transfers ( Q ) between 10 andhundreds of GeV have shown that the quarkproperties (quark distributions) of nucleons boundin nuclei are different than those of free nucleons.This phenomenon is known as the EMC effect; see e.g. the review [32]. The effect is not large, of order10-15%, but is of fundamental interest because itinvolves the influence of nuclear properties on scalesthat resolve the nucleon size. But scales larger thanthe nucleon size are relevant because modificationsof nucleon structure must be caused by interactionswith nearby nucleons. Indeed, after the nucleonsize, the next largest length is the inter-nucleonseparation length, d . This is the scale associatedwith short range correlations between nucleons.Therefore the EMC effect is naturally connectedwith short range correlations between nucleons.But the inter-nucleon separation is not very muchsmaller than that of the nuclear size. This meansthat effects involving the entire nucleus cannot bedisregarded. Such effects are known as mean-fieldeffects in which each nucleon moves in the meanfield provided by other nucleons. Understanding theEMC effect involves understanding physics at allthree length scales.Here is an outline of the remainder of this paper.Sect. II presents a short review of the moderntechnique of softening the nucleon-nucleon interac-tions to simplify calculations of low-energy nuclearproperties. The consequence of this softening is thehardening of the leptonic interactions that probe thesystem. Sect. III is concerned with the largest ofthe three nuclear distance scales–the nuclear radius.This is followed by a discussion of the physics of theseparation between two nucleons in bound states,Sec. IV. The consequent nuclear manifestations arediscussed in Sect. V. This involves understanding theconnection between the physics of short distancesand high momentum. It is shown that the momen-tum dependence of wave functions can in principlebe observed by measuring elastic form factors. Next,Sect. VI discusses the ( e, e (cid:48) p ) reaction as a discov-ery mechanism for the physics of the two-nucleonseparation distance. The concept of virtuality (thedifference between the square of the four-momentumand the square of the mass) as a connection betweenthe scale of the two-nucleon separation-distanceand the nucleon size is introduced in Sec. VII. Theconnection between virtuality and the EMC effectis elucidated in Sect. VIII. Finally, a summary ispresented, Sec. IX.I aim to explain the basic ideas as clearly as possi-ble by using simple examples. There is no intent topresent detailed state-of-the-art calculations. A sep-arate direction, not discussed here, is that precisionnuclear structure calculations can be used in the aidof discovery, such as in the searches for neutrinolessdouble beta decay [33] and/or beyond the standardmodel particles [34]. II. SOFTENED NN POTENTIALS ANDHARDENED INTERACTION OPERATORS
The use of scale separation began with applyingchiral effective field theory to the nucleon-nucleoninteraction [35–37]. This work stimulated manyefforts, see e.g. the reviews [38, 39].Another approach is to use low momentumnucleon-nucleon interactions [6–8, 10, 11, 26, 40].After that came the similarity renormalization group[11–16, 41] which involves a unitary transformationon nucleon-nucleon interactions and the operatorsthat represent observable quantities. The presentsection is intended as a brief review of the latter twotechniques, with emphasis placed on the necessarytransformations of the operators that probe thesystem.Let’s begin by describing a simple cutoff theoryas described by Bogner et al. [6] who found thatthe effective interactions constructed from varioushigh precision nucleon-nucleon interaction models areidentical. Their approach is to obtain the half-offshell T -matrix via the equation T ( k (cid:48) , k ; k ) = V low k ( k (cid:48) , k )+ π P (cid:82) Λ0 V low k ( k (cid:48) ,p ) T ( p,k ; k ) k − p p dp (1) for a single partial wave in which k (cid:48) and k denotethe relative momenta of the outgoing and incomingnucleons, and the mass of the nucleon is taken tobe unity. Furthermore, all momenta are constrainedto lie below the cutoff Λ. A specific formalism wasdeveloped to obtain V low k from the initial bare inter-action V . This construction enforces the conditionthat the half-off-shell T -matrix is independent of thecutoff parameter Λ.As a consequence of the cutoff independence ofthe half-off-shell T -matrix, the interacting scatter-ing eigenstates of the low-momentum Hamiltonian H Λ ≡ H + V low k (where H is the kinetic energy op-erator) are equal to the low-momentum projections ofthe corresponding scattering and bound eigenstates, | Ψ k (cid:105) , | Ψ B (cid:105) of the original Hamiltonian, H + V [42].This means that | χ k (cid:105) = P | Ψ k (cid:105) , with an analogousrelation for bound states, | χ Λ B (cid:105) = P | Ψ B (cid:105) , (2)where P is an projection operator onto states ofrelative momenta less than Λ. The consequencesof the projection operator P in Eq. (2) are studiedbelow.Suppose the system is probed by an interactionoperator, here defined as O . The procedure invokedby using Eq. (1) leads to the requirement that O isto be dressed. The transformation corresponding tothe first in the series of three transformations used toderive a V low k that is Hermitian and independent ofenergy [8] is: O → (1 + H P Q E − H QQ ) O (1 + 1 E − H QQ H QP ) , (3)where Q = I − P and H QQ = QHQ , etc. Thisprojection operator procedure maintains the correctvalue of the matrix elements of O , and is sufficientfor present explicative purposes.The key feature of Eq. (3) is that the effects ofany high momentum component ( Q -space) in thewave function that are removed by using Eq. (2) asthe wave function are incorporated in the probe op-erator. Thus, the probe operator must be hardenedby the softening of the two-nucleon potential.The use of V low k to soften the NN potential wasfollowed by renormalization group methods [11]. Thesimilarity renormalization group (SRG) [12–14, 43]achieves softening by evolving Hamiltonians with acontinuous series of unitary transformations chosento decouple the high- and low-energy matrix elementsof a given interaction [15, 16]. Thus H s = U s HU † s = H + V s , (4)with H = H + V ≡ H s =0 , and H is the kineticenergy operator. The generator of the transfor-mation is η s = dU s ds U † s = − η † s and dH s ds = [ η s , H s ],The choice of the anti-Hermitian operator η s as η s = [ H , V s ] has proved to be convenient and is usedhere. The kinetic energy operator is not changed bythe transformation.Ref. [41] correctly emphasized that when usingthe wave functions produced by SRG-evolved inter-actions to calculate other matrix elements of inter-est, the associated unitary transformation of opera-tors must be implemented. See also [44]. The evolu-tion of any operator O ≡ O s =0 is given by the sameunitary transformation used to evolve the Hamilto-nian [13, 26], O s = U s O s =0 U † s , (5)which obeys the general operator SRG equation d O s ds = [[ H , V s ] , O s ] . (6)If implemented without approximation, unitarytransformations preserve matrix elements of theoperators that define observables.The focus here is on the calculation of observables.Consider an operator O , consistent with the bareHamiltonian H = H + V, that probes the system.The applications discussed here involve the interac-tions between a lepton probe and the system. Theoperator flow equation, Eq. (6), is rewritten usingthe Jacobi identity as d O s ds = [ H , [ V s , O s ]] + [ V s , [ O s , H ]] , (7)with the boundary condition O s =0 = O . To illus-trate the main idea, let’s take O to depend only oncoordinate-space operators, and the bare potential tobe local. Then for s = 0 , [ V, O ] = 0, and for a systemin its center of mass[ O , H ] = M r ( ∇ O + 2 ∇ O · ∇ ) (8)[ V, [ O , H ]] = − M r ∇ V · ∇ O (9)with M r the reduced nucleon mass. To first-order in s O s = O − sM r ∇ V · ∇ O , (10) and one sees immediately that the evolution convertsa one-body operator to a two-body operator. Thefactor of M r arises from converting the units hereto those of [41] in which s = 0 . . A term offirst-order in s that arises from the s − dependenceof the potential vanishes here, as shown in theAppendix,To see the explicit effect of hardening of the in-teraction operator, let O be the momentum transferoperator e iλ q · r , (in which the real-valued parameter λ accounts for using the relative coordinate) then O s acquires a factor of q which gets larger as themomentum transfer increases.For an A -nucleon system this evolution procedurewould turn a one-body operator into an A bodyoperator, as explained in Ref. [41].The stage is now set for the discussion of lepton-nucleus scattering in terms of the three scales of nu-clear physics, starting with the largest and proceed-ing to the smallest. III. DISCOVERY OF NON-ZERO NUCLEARSIZES
This Section is concerned with the largest of thethree nuclear scales- the nuclear radius. Thoughsmall on the scale of atomic sizes, the nuclear radiusis large in the present context.Hofstadter, as part of his Nobel-prize winningwork, showed [45, 46] (in first Born approxima-tion) that the electron-nucleus scattering cross sec-tion σ s ( θ ) was proportional to the square of the three-dimensional Fourier transform of the nuclear chargedensity: σ s ( θ ) ∝ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d rρ ( r ) e ı q · r (cid:12)(cid:12)(cid:12)(cid:12) , (11)where ρ ( r ) is the nuclear charge density as afunction of the separation from the center of thenucleus. Relativistic corrections are small for nu-clear targets [47]. The three-dimensional integralappearing in Eq. (11) is defined to be the formfactor F ( q ). Electron scattering, in measuringthe difference between the form factor and unity,showed that the nucleus was not a point charge,as it would have been in a lowest-order effectivefield theory treatment. Importantly, electron scat-tering was one of the main methods to determinethe spatial extent of nuclear charge distributions [48].For large nuclei the density is well-approximated bya Woods-Saxon (Fermi) form ρ ( r ) = ρ e ( r − R ) /a . Fornuclei wth A > ρ = 0 . ZA fm − , r = 1 . A / and a = 0 .
54 fm [48]. The nuclear diffuseness a can be understood as follows. Each nuclear single-particle state falls exponentially with distance awayfrom the nuclear center. Thus the density falls a e − r/a for large r , with a ≈ / / √ M B with B theaverage binding energy at the center of the nucleus B = 16 MeV and M the nucleon mass, a = 0 .
57 fm,which is close to empirical values and close to thesize of the nucleon. The distance scale could insteadbe taken as the surface thickness, t = 4 . a ≈ t is close to the nucleon-nucleon separation distance.Thus the two smallest nuclear size scales enters inunderstanding the largest nuclear radius. This isan example of the principle that all of three nucleardistance scales are connected on a deep level.The remainder of this Section is concerned withunderstanding the role of a , and in examining theeffects of softening the nucleon-nucleon interaction. A. Effects of the Diffuseness
Examining the effects of a is simplified by using thenuclear shape as parameterized by the symmetrizedFermi form [49]: ρ ( r ) = ρ ( ca )( cosh ( ca ) +cosh ( ra )) (12) ρ = πc (cid:16) π a c +1 (cid:17) , (13)which, for large nuclei with c/a (cid:29)
1, is indistinguish-able from the usual Fermi form. The Fourier trans-form of this function yields the nuclear form factorgiven by F ( q ) = ρ π acq sinh πaq ( πa/c coth( πaq ) sin( cq ) − cos( cq )) . (14)The mean-square radius defined by (cid:104) r (cid:105) ≡ (cid:90) d rρ ( r ) r = 15 (3 c + 7 a π ) . (15)Using c = 6 .
38 fm and a = 0 .
535 fm for theGold nucleus [50] as an example, we see that (cid:104) r (cid:105) =28 . with the term proportional to a contribut-ing about 4 fm . Thus the small scale of a contributes about 14% to the mean square radius and about 7%to the rms radius. The small distance scale is im-portant. Another example of importance is that thediffuseness a leads to an exponential fall-off with q :lim q →∞ F ( q ) = e − πaq cos cqq . (16). B. Influence of the Softened Nucleon-NucleonInteraction
Let’s examine the effect of the unitary transforma-tion on the nuclear form factor. Use Eq. (10) withthe probe operator O = e i q · r , taking ( A − /A → r represents the nucleon position operator and q is the momentum transfer. Evaluating the matrixelement of the softened nucleon-nucleon potentialoperator in the nuclear ground states leads, via theHartree-Fock approximation, to a nucleon-nucleus,shell-model interaction which is taken as a localpotential, U ( r ). Such a mean-field potential has theshape of the nuclear density, e.g. Eq. (13), with acentral depth of about 57 MeV [51]. Non-locailityof the mean field is neglected here to simplify thepresentation.One finds from Eq. (10) that
O ≈ (1 − i q · ˆ r sM U (cid:48) ) e i q · r . (17)This first-order change in O is accompanied by afirst-order change in the wave function, so that inprinciple the computed form factor is not modifiedby the unitary transformation.The purpose here is only to illustrate the effect ofthe hardening of the interaction caused by transfor-mations such as those of Eq. (10). Therefore I com-pute the change in the form factor, ∆ F caused byincluding the second term of Eq. (17). This changeis given by∆ F ( q ) = − π ( sM )3 q (cid:82) r drρ ( r ) dUdr j ( qr ) , (18)with value of s = 0 . [41]. A comparison between F ( q ) and F ( q ) + ∆ F ( q ) is made in Fig. 1. The term∆ F is negligible for q < − , but is about a 10%effect for 1.3 fm − and dominates for q > − .If ∆ F is large compared with F it is necessary tocompute higher order terms, so the details wouldchange. Nevertheless, Fig. 1 demonstrates thehardening of the probe interaction that occurs forlarge values of the momentum transfer. F ( q )
This section examines the physics of the two-nucleon separation distance. Bound-state wave func-tions are constructed using simple, two-parametermodels of the S nucleon-nucleon interaction withparameters chosen to reproduce the measured scat-tering length and effective range [52]. As such, theseare low-energy interactions. These simple potentialscontain features such as a hard core or Yukawainteraction that have been parts of more realisticinteractions. The range parameters of that referenceare used here, with the strengths of the potentialadjusted slightly so as to reproduce the value of thebinding energy (2.2 MeV). The different potentialsproduce different bound-state wave functions andmeasurable differences are perceived through thebehavior of the form factors (here the Fouriertransforms of the square of the wave functions). Theimportance of the correction terms in the differencebetween using O s and O is assessed. The scalingproperties of the form factors are also presented inpreparation for use in Sect. V. A. nucleon-nucleon hard core plus exponentialpotential
This potential is defined by having an infinite hardcore at a separation r and an attractive exponentialpotential V ( r ) = − V e − ( r − r ) /a ( V = 1 .
92 fm − )for larger separations. The model is exactly solvable.The values r = 0 . a = 0 .
45 [52] are used.This potential (as others in this section) is a crudemodel for deuteron properties because there is notensor force. The s -state bound state wave function is deter-mined by using the transformation y = 2 aγe − r/ (2 a ) , γ = √ M B , where B is the binding energy and M thenucleon mass, which converts the Schroedinger equa-tion into Bessel’s equation. Then the bound-statewave function is u ( r ) = N J aγ (2 a (cid:112) M V e − r/ a ) , (19)subject to the condition that u ( r ) = 0. The factor N is a normalization constant. One can check the large r limit by using the small argument limit of the Besselfunction ( J ν ( x ) ∼ x ν ) so that lim r →∞ u ( r ) ∝ e − γr ,as expected. The form factor of this model is thebound-state matrix element of the operator O Q = e i Q · r / , (20)in which the probe is defined to act only on one nu-cleon of the two-body system. Then the form factoris given by F ( Q ) = Q (cid:82) ∞ r drr sin ( Q/ r ) u ( r ) , (21)and can be re-expressed in terms of the momentum-space wave function ψ ( k ) given by ψ ( k ) = 1 √ πk (cid:90) ∞ r dr sin kr u ( r ) , (22)with F ( Q ) = (cid:82) d k ψ ( k + ) ψ ( k − ) , (23)and k ± ≡ k ± Q / V low k prescription of Eq. (2)one cuts off the momentum-space wave functionat a relative momentum Λ, with Λ = 2 . − a commonly used value. The aim here is to seehow much of the form factor (as a function of Q )is given by relative momenta that are greater than Λ.The cutoff form factor is then given by F Λ ( Q ) = (cid:90) d kψ ( k + ) ψ ( k − )Θ(Λ − k + )Θ(Λ − k − ) . (24)Using this form factor corresponds to using Eq. (2)for the wave function. Invariance of the form factorwould be obtained if the probe operator were modi-fied according to Eq. (3) or Eq. (10). The purpose incomputing F Λ ( Q ) is only to determine the values ofΛ for which operator modification becomes necessary.Fig. 2 shows the form factor falling asymptoticallyas 1 /Q and modulated by oscillations. Fig. 3 showsthe values of Λ necessary to achieve 5% accuracy inthe form factor as a function of Q . These are greaterthan 2.1 fm − for values of Q > . − , so suchvalues of Q require operator modification. The useof Eq. (10) is not possible because of the hard core ofthe potential. � � � � � � ��� - � �� - � ������������ Q | F ( Q ) | (fm )
95 as afunction of Λ.
B. Square well potential
The next example is the square well potential witha radius of 2.205 fm [52] and depth 0.157 fm − . Theform factor is shown in Fig. 4. Fig. 5 shows the valuesof Λ necessary to achieve 5% accuracy in the formfactor as a function of Q . Operator modification isfound to be important here for values of Q > . − . The use of Eq. (10) is not possible because thederivatives of the potential are delta functions. C. Exponential potential
The exponential potential is given by the expres-sion V ( r ) = − V e − r/a with a = 0 .
76 fm [52] and � � � � ���� - � ������� Q (fm )
95 as afunction of Λ for square well potential. The rapid rise isdue to a node in the form factor. V = 0 .
779 fm − . The form factor is shown in Fig. 6.One sees that F ( Q ) scales as 1 /Q . Q (fm )
Here V ( r ) = V e − µr /r with µ = 0 .
411 fm − as in[52] and V = 0 .
25. The form factor, as shown Fig. 7,scales as Q − . The function F ( Q )+∆ F ( Q ) (Eq. (26))is shown as the rising curve of Fig. 7. The dramaticchange in the probe operator is caused by the largederivative of the Yukawa potential at short distances.The Fig. 7 again demonstrates the hardening of theprobe interaction. Q F ( Q ) (fm )
Softening of the NN interaction via a unitarytransformation or projection operator procedure re-quires a corresponding transformation of interactionoperators that increases their effects at high mo-mentum transfer. The examples shown indicate thatfor some potentials the effects of transforming theoperator are very important for momentum transfersgreater than about 5 fm − , an important regionfor current experiments that attempt to discovernew phenomena. Furthermore, the transformed op-erators cannot be obtained easily for some potentials.The use of the impulse approximation that involvesusing bare, untransformed operators simplifies the in-terpretation of experiments and therefore seems bestsuited for discovery purposes. V. TWO-NUCLEON SEPARATION INNUCLEI: OBSERVING HIGH MOMENTUMAND SHORT-DISTANCE FEATURES.
The previous Section discusses how high mo-mentum components may arise from interactionsbetween nucleons. The present Section is concernedwith the manifestation of such effects in nuclei,and also one way to observe the relation betweenshort-distance and high momentum physics.Bethe [57] wrote that “Indeed, it is well establishedthat the forces between two nucleons are of shortrange, and of very great strength” and “there arestrong arguments to show that the two-body forcescontinue to exist inside a complex nucleus ”.Brueckner, Eden, and Francis, [58] used a varietyof nuclear reactions to argue that the nuclearwave function contains nucleons with a significantprobability to have high momentum. One par-ticularly telling example is the significant crosssections observed in the ( p, d ) reaction with 95 MeVprotons. The neutron in the nucleus must havehigh momentum comparable to that of the proton,about 420 MeV/c, so that combination with theincident proton allows the deuteron to emerge fromthe nucleus. The only way a bound neutron couldacquire such momentum is via interactions withanother nearby nucleon.Bethe continued “All these processes show thatthe ‘potential’ is fluctuating violently from pointto point in the nucleus, which is compatible withthe assumption that two-body forces continue toact inside the nucleus without much modificcation.”The idea of two strongly interacting nucleons, actingindependently of the other nucleons (the independentpair approximation) is the basis of Bruckner theory[59] which provided a fundamental explanationof how nuclear saturation and the shell model ofnuclei arise from fundamental, hard, short-rangedinteractions of nucleons. This means that thenucleon-nucleon separation distance is related, viathe nucleon-nucleon interaction, to the size of theentire nucleus.One modern implementation of the independentpair approximation is the generalized contact for-malism (GCF) [60]. The GCF is an effective modelthat provides a factorized approximation for theshort-distance (small- r ) and high-momentum (large- k ) components of the nuclear many-body wave func-tion. Its derivation relies on the strong relative inter-action of closely separated nucleons and their weakerinteraction with the residual A − r or relative momentum q ) has been expressed at smallseparation or high momentum as [62]: ρ NN,αA ( r ) = C NN,αA × | ϕ αNN ( r ) | ,n NN,αA ( k ) = C NN,αA × | ϕ αNN ( k ) | , (27) where A denotes the nucleus, N N denotes thenucleon pair being considered ( pn , pp , nn ), and α stands for the nucleon-pair quantum state (spin 0 or1). C NN,αA are nucleus-dependent scaling coefficients,referred to as “nuclear contact terms”, and ϕ αNN are two-body wave functions that are given by thezero-energy solution of the two-body Schr¨odingerequation for the N N pair in the state α . Thefunctions ϕ αNN do not depend on the nucleus, but dodepend on the N N interaction.The authors [60] state that an important fea-ture of the GCF is the equivalence between shortdistance and high momentum, which is built intoEq. (27) by using the same contact terms C NN,αA for both densities. This equivalence is establishedby extracting the contacts separately from thecoordinate- and momentum-space nuclear wavefunctions. The present section is devoted to findinga direct correspondence between short distance andhigh momentum.This analysis uses the zero-energy Lippmann-Schwinger (LS) equation and asymptotic expansionsobtained by integration by parts [64]. The LS equa-tion for scattering at 0 energy is given by ϕ αNN ( k ) = − Mk (cid:90) d r (2 π ) / e − i k · r V ( r ) ϕ αNN ( r ) . (28)If the potential is an approximate delta function incoordinate space, then ϕ αNN ( k ) ∼ k . For other interactions it is useful to express the S -wave, momentum-space wave function as: ψ ( k ) = − M √ πk (cid:90) ∞ dr sin( kr ) V ( r ) u ( r ) , (29)where u ( r ) is the S-state radial wave function andin which the labels N N, α are suppressed. One de-rives expansions for asymptotic values of the mo-menta by replacing the sin( kr ) appearing in the in-tegral of Eq. (29) by − k d cos krdr . Then one can gethigher-order terms by writing cos( kr ) = k d sin krkr .The result, defining K ≡ M √ π , assuming that thepotential is not a delta function, and that V u and itsderivatives exist at r = 0 is: ψ ( k ) = Kk (cid:82) ∞ dr d cos krdr V ( r ) u ( r ) (30)= Kk [ − V (0) u (0) − (cid:82) ∞ dr cos kr ( V u ) (cid:48) ]= Kk V (0) u (0) + Kk ( V u ) (cid:48)(cid:48) (0) + Kk ( V u ) (cid:48)(cid:48)(cid:48)(cid:48) (0) + · · · (31)If the potential is non-local of the form V ( r, r (cid:48) ) the0product V u in Eq. (29) is replaced by V u ( r ) ≡ r (cid:90) ∞ dr (cid:48) V ( r, r (cid:48) ) u ( r (cid:48) ) (32)and the derivatives thereof that appear in Eq. (31)are replaced by derivatives of V u at the origin.One may classify the asymptotic behavior obtainedfrom different classes of potentials. • Class I: The potential is a delta function. Then ψ ( k ) ∼ k as in leading-order pion-less effectivefield theory. Or as in Ref [39] showing thatan approximate delta-function potential arisesfrom treating the iterated effects of the one pionexchange potential. • Class II: u (0) = 0 but V (0) u (0) (cid:54) = 0 . An exam-ple is V ∼ /r and u ( r ) ∼ r for small values of r . In this case, ψ ( k ) ∼ V (0) u (0) k • Class III: u (0) = 0, V (0) u (0) = 0 . An exampleis the exponential potential for which V (0) (cid:54) = 0is finite and u (0) = 0. In this case, ψ ( k ) ∼ V (cid:48) (0) u (cid:48) (0) k • Class IV: The potential has a hard core poten-tial, infinitely repulsive for a distance less thana core radius, r = c . Then using u ( c ) = 0, u (cid:48) ( c ) (cid:54) = 0 and taking the Fourier transform ofthe wave function: ψ ( k ) = − Kk (cid:82) ∞ c dr d cos krdr u ( r ) ∼ Kk sin( kc ) u (cid:48) ( c ) . (33) • Class V:
V u and all of its derivatives vanish atthe origin. This is the square well of range R .Then using the LS equation yields ψ ( k ) ∼ KV (0) k cos( kR ) u ( R ) . (34) • Class VI: Non-local potentials. The quantity ψ ( k ) ∝ lim r → ( V u ( r )) /k unless the limit van-ishes. The Yamaguchi potential [65] V ( r, r (cid:48) ) ∝ e − µr r e − µr (cid:48) r (cid:48) provides an example of ψ ( k ) ∝ k . Apower law fall-off would be obtained even if pre-vious limit did vanish because some non-zeroeven-numbered derivatives of V u at the originmust occur.In each of the first five cases the product of thepotential and wave function at short separationdistances determines the high-momentum behavior of the momentum-space wave function. For non-localpotentials the high-momentum behavior is controlledby V u and/or its derivatives at the origin. Once,again short-distance behavior determines the highmomentum content. Moreover, in each case thereis a power law fall-off with increasing k . Thisslow fall with increasing k means that significanthigh-momentum content can be expected for all ofthe interactions of Classes I through VI.A power-law fall off can be uniquely avoided if thepotential is a function of r . In that case, all of theterms in the series of Eq. (31) would vanish because ofthe vanishing of all odd-number derivatives of V ( r )at the origin. No realistic nucleon-nucleon potentialin current use is a function of r . This means thatsignificant high momentum content can be expected. A. Form factors at high momentum transfer
The previous analysis of zero-energy wave func-tions is also applicable to bound-state wave functions.For a binding energy B the − Mk n factors of Eq. (31)is replaced by − M ( k + MB ) k n − ≈ − Mk n in asymptoticexpansions.An approximate relation between the momentumspace wave function and the elastic form factor canbe obtained using Eq. (23). Ref. [66] argued that thedominant contributions to the integral occur when k = ± Q /
4. Then F ( Q ) ∝ ψ ( Q/ . (35)This result depends on factorizing the momentum de-pendence of the potential, ˜ V from that of the wavefunction, and is denoted the factorization approxi-mation. The procedure is to use the LS equation torepresent the wave functions appearing in Eq. (24).Then Eq. (35) emerges if (cid:90) d k ˜ V ( | Q / − k | ) ψ ( k ) ≈ ˜ V ( Q/ (cid:90) d kψ ( k ) . (36)The integral over d k is the wave function at theorigin of coordinate space.The result Eq. (35) is remarkable. It meansthat under certain conditions, in principle, it ispossible to measure the wave function of a system,or at least its momentum dependence in a specificregime. This means that general statements aboutthe unmeasurable nature of wave functions are not1correct.An (unrealistic) experiment in which one couldattempt to test Eq. (35) is elastic electron scatteringfrom a bb meson. Elastic scattering on the deuteronis complicated by the need to include the effectsof meson exchange currents and corrections tothe non-relativistic treatment [67]. Calculationsof deuteron form factors for momentum transfersgreater than about 7 fm − are not shown in thatreview.Note also that nucleon-nucleon scattering atlaboratory energies less than 350 MeV does not yieldsignificant constraints on ˜ V ( Q/
4) for large values of Q [68]. Large momentum transfer means that largekinetic energy is needed.The following text explains how the differentclasses of potentials discussed here can be or cannotbe manifest by measurements of form factors asexpressed in Eq. (35).Class I: V is a delta function in coordinate space,and therefore a constant in momentum space. Thewave function ψ ( k ) is mainly determined by thepropagator in which k and Q of Eq. (36) of the sameimportance. The factorization argument does notapply.Class II: The Yukawa potential V ( r ) = V e − µr /r .The product V u is well defined as r →
0, becausethen u ( r ) ∝ r . Thus Eq. (31) predicts ψ ( Q ) ∼ /Q and the form factor show in in Fig. 7 also shows a1 /Q behavior.Class III: The exponential potential. In accordwith Eq. (31) the wave function falls as 1 /Q , andso does the form factor shown in Fig. 6.Class IV: Hard core plus exponential. Fig. 2shows oscillations expected from Eq. (33) but factor-ization does not work because the discontinuity of u (cid:48) ( r ) at r = r induces large momentum components.Class V: Square well potential. The factorizationapproximation is not accurate, although oscillationswith period (2 π/R ≈ − are seen. This isbecause condition of Eq. (36) are not maintained dueto oscillations that cause 0’s in ˜ V for large values ofthe argument.In summary, the short distance behavior of the po-tential times the coordinate-space radial wave func- tion determines the high momentum dynamics in allcases. If the factorization approximation of Eq. (36)is valid and the probe operator is well-known themeasurement of the form factor determines the high-momentum behavior of the wave function. VI. THE ( e, e (cid:48) p ) REACTION: DISCOVERY ATTHE NUCLEON-NUCLEON SEPARATIONSCALE
The ( e, e (cid:48) p ) reaction occurs if an electron knocksout a nucleon so that an initial nuclear state of A nucleons is converted to a final nuclear state of A − q and energy ν to a single proton, whichthen leaves the nucleus without interacting withanother nucleon on the way out of the nucleus, seeFig. 10. There are various corrections- final stateinteractions, meson exchange currents etc. However,one can account for such effects by using appropriatekinematics and including the effects of final stateinteractions, see e.g. [31].For high-momentum transfer processes the out-going nucleon has high energy, greater than the350 MeV that is used to constrain nucleon-nucleonpotentials. The softening effects of unitary transfor-mations on nucleon-nucleon potentials requires thatthe potential be Hermitian. No realistic Hermitianpotential applicable for scattering energies greaterthan about 1.5 GeV exists at the present time.This means applying a unitary transformation tosoften the interaction is not practical. Instead,the final state interactions can be treated using theGlauber approximation in which the nucleon-nucleonscattering cross sections are used as input to formthe optical potential [69].If the background effects mentioned above are han-dled correctly, the scattering amplitude is propor-tional to the wave function of the struck bound nu-cleon [70]: M ∝ ψ ( P miss ) . (37)Once again (as in Eq. (35)) the scattering ampli-tude is seen to directly accesses information aboutthe momentum dependence of the wave function.This feature has enabled experimental studies toshow that the high momentum part of the wavefunction is dominated by short-range correlations2 The fact that a nucleon has about 2.5 times larger densitythan the nuclear central density and that nucleons move in thenucleus with about a quarter of the velocity of light opens upthe possibility of large local density fluctuations. These alsolead to large local momentum fluctuations via the uncertaintyprinciple. The strong short-range repulsive force betweennucleons restrains the size of these fluctuations, but since itsrange is smaller than a Fermi, the density and momentumfluctuations in nuclei can still be quite large.The diverse features described indicate that understandingthe broad range of nuclear phenomena requires the use ofmany experimental tools. Since electromagnetic interactionsare well understood and presumably simple, electron scatter-ing has long been used as a tool to investigate different aspectsof nuclear structure. We examine the use of electron scatteringto probe the validity of the single-particle shell model in thenext section.
B. The need for short-range correlations:Beyond the nuclear shell model1. Spectroscopic factors
Data from electroinduced proton knockout reactions onnuclei A ð e;e p Þ provided early evidence for the validity ofthe shell model (Frullani and Mougey, 1984). These studiescomplemented theuse of low-energy nuclear reactions, such as ð d;p Þ and ð p;pp Þ . Later on, more detailed studies usinghigher-energyelectronbeamsexploredthelimitsofthevalidityof the shell model. We next explain how this happened.In the ð e;e p Þ reaction the electron knocks out a nucleon sothat an initial nuclear state j i i of A nucleons is converted to afinal nuclear state j f i of A − nucleons. The reaction can beanalyzed in terms of spectroscopic factors (Macfarlane andFrench, 1960), which are probabilities that all but one of thenucleons will find themselves in the final state. More formally,if one considers a single-particle state of quantum numbers α ,the spectroscopic factor S α is given by the square of theoverlap S α ¼ jh f j b α j i ij , where b α destroys a nucleon. If theindependent particlemodel wereexact,then S α would be unityfor each occupied state α . Thus measuring S α is a useful wayto study the nuclear wave functions and the limitations of theindependent particle model.In the plane wave impulse approximation (PWIA), anelectron transfers a single virtual photon with momentum q and energy ν (sometimes written ω ) to a single proton, thenleaves the nucleus without reinteracting and can thus bedescribed by a plane wave (see Fig. 1).In PWIA the cross section factorizes in the form (Kelly,1996) d σ d ν d Ω e dE miss d Ω p ¼ K σ ep S ð E miss ; p miss Þ ; ð Þ where K ¼ E p p p = ð π Þ is a kinematical factor, E p and p p arethe energy and momentum of the outgoing proton, σ ep is theelectron cross section (De Forest, 1983) for scattering by abound proton, and S is the spectral function, the probability offinding a nucleon in the nucleus with momentum p miss andseparation energy E miss . The missing momentum and missingenergy are given by p miss ¼ q − p p ;E miss ¼ ν − T p − T A − ; ð Þ where T p and T A − are the kinetic energies of the detectedproton and residual (undetected) A − nucleus.However, the knocked-out proton then interacts with othernucleons as it leaves the nucleus; these final state interaction(FSI) effects have been typically calculated either using anopticalmodelatlowmomenta(Kelly,1996)orusingtheeikonalor Glauber approximations at higher momenta (Ryckebusch etal. ,2003;Sargsian etal. ,2005).Calculationswherethewavefunction of the knocked-out proton are distorted by FSI arereferred to as distorted wave impulse approximation (DWIA)calculations. (Note that FSI effects mean that p miss is no longerequaltotheinitialmomentumofthestrucknucleon.)InDWIA,the ð e;e p Þ cross section does not exactly factorize as in thePWIA. However, factorization is a good approximation at Q ≫ p miss andthecrosssectionisapproximatelyproportionaltoadistortedspectralfunction S D (Kelly,1996).NeitherPWIAnorDWIA calculationsconserve currentbecausetheinitialandfinal wave functions of the model calculations are not orthogo-naland because theeffective NN interactions used in the initialand final states are different. [Some models force currentconservation by arbitrarily modifying kinematic variables suchas q μ (De Forest, 1983).] Relativisitic DWIA models weredeveloped by Van Orden and collaborators (Picklesimer, VanOrden, and Wallace, 1985; Picklesimer and Van Orden, 1989)and later elaborated by Udias et al. (1993, 1995, 1999), Kellyand Wallace (1994), and Kelly (1999).Thus, ð e;e p Þ measurements should be sensitive to thespectralfunction,i.e.,tothemomentumandenergydistributionsofnucleonsinthenucleus.Figure2showsthe O ð e;e p Þ crosssection at Q ¼ . GeV and ν ¼ . GeV plotted versusmissingenergyatseveraldifferentmissingmomentaandplottedversus missing momentum for the two p -shell states. There aresharppeaksat E miss ¼ and18MeV,correspondingtoprotonknockout from the p = and p = shells, a broad peak at E miss ≈ MeV corresponding to proton knockout from the s shell (and other processes), and a long tail extending to large E miss ,especiallyatthelargestmissingmomenta.Themomentum FIG. 1.
The A ð e;e p Þ reaction in the plane wave impulseapproximation. A nucleus offour-momentum P A emits a nucleonof four-momentum P miss that absorbs a virtual photon of four-momentum q to make a nucleon of four-momentum P miss þ q ,with ð P miss þ q Þ ¼ M , where M is the nucleon mass. The blobrepresents the in-medium electromagnetic form factors. Hen et al. : Nucleon-nucleon correlations, short-lived … Rev. Mod. Phys., Vol. 89, No. 4, October – December 2017 045002-4
FIG. 8. A nucleus emits a nucleon of four-momentum P miss that absorbs a virtual photon of four-momentum q to make a final-state nucleon of four-momentum P miss + q ,with ( P miss + q ) = M , where M is the nucleon mass. (SRCs) [71]. These are pairs of nucleons with largerelative and individual momenta and smaller center-of-mass (c.m.) momenta, where large is measuredrelative to the typical nuclear Fermi momentum k F ≈
250 MeV / c [32, 72]. At momenta just above k F (300 ≤ k ≤
600 MeV / c), SRCs are dominatedby pn pairs [73–79]. This pn dominance is dueto the tensor part of the nucleon-nucleon ( N N )interaction [80, 81].The presence of nucleon-nucleon short rangedcorrelations in nuclei has many implications forthe internal structure of nucleons bound in nu-clei [32, 82, 83], neutrinoless double beta decaymatrix elements [84–90], nuclear charge radii [91],and the nuclear symmetry energy and neutron starproperties [92].If SRG transformations are applied to thestrong-interaction Hamiltonian, the necessary useof hardened interactions (discussed in Sect. V)in analyzing experiments would complicate theirinterpretation.
VII. VIRTUALITY -A SMALL-DISTANCESCALE
Bound nucleons (of four momentum p ) do notobey the standard Einstein relation p µ p µ = M ,and are said to be off the mass shell. The averagebinding energy is much, much less than the nucleonmass, so the violation of the Einstein relation canbe ignored when computing or understanding manyaverage nuclear properties.If one looks in more detail and examines nucleon- nucleon scattering, one sees that the intermediatenucleons must be off their mass shell. In theBlankenbecler-Sugar [93] and Thompson reduc-tions [94] of the Bethe-Salpeter equation [95] onenucleon emits a meson of 0 energy and non-zero mo-mentum and the other nucleon absorbs the meson.Since the momenta of the nucleons have changed,but their energy hasn’t changed, the intermediatenucleons are off their mass shell. In other reductionsof the Bethe-Salpeter equation [96], one nucleon is onthe mass shell, and the other is not. This means thatthe nuclear wave function, treated relativistically,contains nucleons that are off their mass shell.Such nucleons must undergo interactions beforethey can be observed, and are denoted as virtual.The difference p − M is related to the virtuality [97].Experiments [98–100] using leptonic probes atlarge values of Bjorken x interrogate the virtual-ity of the bound nucleons. To see this, considerthe PWIA situation with ( P miss + q ) = M , let q have the four-momentum ( ν, ⊥ , − (cid:112) ν + Q )) ≈ ( ν, ⊥ , − ( ν + Q ν ), in the Bjorken limit with q = − Q , Q → ∞ , ν → ∞ , and Q /ν finite. Then with q − = q − q ≈ ν (cid:29) q + ≈ − M x , x = Q Mν , one findsthat V ≡ P − M M ≈ − Q M ( P +miss M − . (38)This quantity V , defined here as the virtuality, isgenerally not zero. For example, experiments havebeen done with Q = 3 GeV , P +miss M = 1 . V ≈ − .
5. Plateaus, kinematically correspondingto to scattering by a pair of nucleons, have beenobserved [71] in this region. Treating highly virtualnucleons requires including relativistic effects. Arecent study is [101].The only way for a nucleon to be so far off the massshell is for it to be interacting strongly with anothernearby nucleon. To see that, consider a configura-tion of two bound nucleons, initially at rest in thenucleus. This is a good approximation for roughly80% of the nuclear wave function. To acquire thelarge missing momentum of the previous paragraph,one nucleon must exchange a boson or bosons withfour-momentum comparable to that of the incidentvirtual photon as shown in Fig. 10. Such a bosonicsystem can only travel a short distance ∆ r betweenthe nucleons with ∆ r ∼ | (cid:126)P miss | . (39)3 ? ~
Thus a highly virtual nucleon gets its virtualityfrom another nearby nucleon which must be closelyseparated. High virtuality is a short-distance phe-nomenon. As such, it serves as an intermediatestep between using nucleonic and quark degrees offreedomRef. [102] attempted to find a difference betweenthe effects of highly virtual nucleons and the effectsof high local density. The simple arguments pre-sented here show that there is a direct connectionbetween high local density and high virtuality. It istherefore not possible to distinguish the two effects.This issue is discussed in more detail in Ref. [103].In evaluating Feynman diagrams the lowest-ordereffects of the non-vanishing of V can be cancelledby propagators and re-organized into low energyconstants See Fig. 10. But understanding the fun-damental origin of virtuality would allow a deeperunderstanding of nuclear physics. ? ⇤
To better understand the connection between vir-tuality and quark degrees of freedom, consider a vir-tual nucleon as a superposition of physical states thatare eigenfunction of the QCD Hamiltonian. Virtualstates with nucleon quantum numbers can be ex-pressed using the completeness of states of QCD: | N ( V ) (cid:105) = n max (cid:88) n =1 c n | N n (cid:105) , (40) in which the states | N n (cid:105) are resonances and alsonucleon-multi-pion states. Each of these states has adetailed underlying structure in terms of quarks andgluons. In exclusive reactions with not very large mo-mentum transfer few states are excited and one mayuse Eq. (40) to describe the physics. However, forhigh energy inclusive reactions of experimental rele-vance one needs many states. In this case a quarkdescription is necessary. VIII. EMC EFFECT- DISCOVERY AT THESMALL NUCLEAR DISTANCE SCALE
The aim of this Section is to exemplify the con-nection between the small-distance scale related tovirtuality and deep inelastic scattering from nuclei.The relation between virtuality and the EMC effecthas been explored previously in Refs. [32, 104–107].Deep inelastic scattering (DIS) on a free nucleontarget was initially expected to observe a set ofresonances and therefore small cross sections forlarge values of three-momentum transfer [4, 5].Instead, the cross sections were large an approximateBjorken scaling was observed. The unambiguousinterpretation is that the nucleon contains quarks.I explain in more detail. For typical DIS kine-matics Q = 100 GeV , x = 0 . , ν ≈
100 GeV, theexpansion of Eq. (40) becomes unwieldy because theabsorption of a virtual photon by free nucleon leadsto a system of mass M X with M X = Q ( x − M X ≈
10 GeV. This high excitation energy tells usthat a huge number of baryon states are involved.Instead it is far more efficient to analyze the crosssections using quark degrees of freedom. Measure-ments determine the quark structure functions q ( x )that are scale and scheme dependent [108]. However,they are well understood and interpreted as momen-tum distributions. Observe again that measurementsof experimental cross sections determine features ofwave functions.Next turn to deep inelastic scattering on nuclei atsimilarly large values of Q . It was initially thoughtthat at such kinematics only very small distances inthe target would be involved [109]. Such distancesare much, much less than the internucleon spacingof ≈ q ( x ) was observed. At high values of x the ratioof the bound to free structure function ratio is less4than one by an amount of only between 10 and 15%,dependent on the nucleus. This effect is known asthe EMC effect [109, 110].That bound structure functions are different thanfree ones is natural in terms of the discussion aboveregarding virtuality and Eq. (40). Bound nucleonsare virtual and the states | N n (cid:105) have different struc-ture functions than the nucleon.Because of the large number of states entering inEq. (40) it is most efficient to use quark degrees offreedom to understand DIS large values of Q . Thenthe free nucleon is regarded as a superposition ofvarious configurations or Fock states, each with adifferent quark-gluon structure.I simplify the discussion using a model inspiredby the QCD physics of color transparency [111–114].The infinite number of quark-gluon configurationsof the proton are treated as two configurations, alarge-sized, blob-like configuration, BLC, consistingof complicated configurations of many quarks andgluons, and a small-sized, point-like configuration,PLC, consisting of 3 quarks. The BLC can bethought of as an object that is similar to a nucleon.The PLC is meant to represent a three-quark systemof small size that is responsible for the high- x behavior of the distribution function. The smallerthe number of quarks, the more likely one can carrya large momentum fraction. The small-sized config-uration (with its small number of qq pairs) is verydifferent than a low lying nucleon excitation. Thistwo-component model is meant to serve as a simpleschematic tool to enable qualitative understanding.When placed in a nucleus, the blob-like configura-tion feels the usual nuclear attraction and its energydecreases. The point-like-configuration feels far lessnuclear-attraction by virtue of color screening [115]in which the effects of gluons emitted by small-sizedconfigurations are cancelled in low-momentumtransfer processes. The nuclear attraction increasesthe energy difference between the BLCs and thePLCs, therefore reducing the PLC probability [111].Reducing the probability of PLCs in the nucleusreduces the quark momenta, in qualitative agreementwith the EMC effect.Working out the consequences of the BLC-PLCmodel enables the connection between the EMC ef-fect and virtuality to be clarified. The Hamiltonianfor a free nucleon in the two-component model can be expressed schematically by the matrix H = (cid:20) E B VV E P (cid:21) , (41)where B represents BLC and P the PLC. The PLC isspatially much smaller than the BLC, so that E P (cid:29) E B . The hard-interaction potential, V , connects thetwo components, causing the eigenstates of H to be | N (cid:105) and | X (cid:105) rather than | B (cid:105) and | P (cid:105) . In lowest-orderperturbation theory, the eigenstates are given by | N (cid:105) = | B (cid:105) + (cid:15) | P (cid:105) , (42) | X (cid:105) = − (cid:15) | B (cid:105) + | P (cid:105) , (43)with (cid:15) = V / ( E B − E P ) (cid:28) . It is natural to assume | V | (cid:28) E P − E B , so that the nucleon is mainly | B (cid:105) andits excited state is mainly | P (cid:105) . The notation | X (cid:105) isused to denote the state that is mainly a PLC, whichdoes not at all resemble a low-lying baryon resonance.The quark structure function is the matrix elementof the operator O DIS that is the imaginary part ofthe virtual-photon- quark Compton scattering am-plitude. This operator acts on a single quark, so that q ( x ) = 11 + (cid:15) (cid:0) (cid:104) B |O DIS | B (cid:105) + (cid:15) (cid:104) P |O DIS | P (cid:105) (cid:1) , (44)in which it is assumed that the single-quark operatordoes not connect the two very different states | B (cid:105) and | P (cid:105) . Furthermore, the condition that the PLCdominates the structure function at large values of x is enforced by defining a function f ( x ) > x increases. In particular,let (cid:104) P |O DIS | P (cid:105) ≡ f ( x ) (cid:104) B |O DIS | B (cid:105) , (45)so that q ( x ) = 11 + (cid:15) (cid:104) B |O DIS | B (cid:105) (1 + (cid:15) f ( x )) . (46)The model quark distributions of [116], based onlight-front holographic QCD, may provide a realiza-tion of the simple relation Eq. (45). These incorpo-rate Regge behavior at small x and inclusive countingrules as x approaches unity and is consistent with DISmeasurements. The model provides quark distribu-tions q τ ( x ) (normalized to unity) as function of τ , thenumber of constituents in the system: q τ ( x ) = Γ ( τ − ) √ π Γ( τ − (cid:0) − w ( x ) (cid:1) τ − w ( x ) − w (cid:48) ( x ) , (47)with w ( x ) = x − x e − a (1 − x ) . The elastic form factorsof this model fall asymptotically as 1 /Q τ , and the5slope of form factors as Q = 0 is proportional to τ .These features mean that an increase in the value of τ corresponds to an increase in effective size. Thefunction q represents a three quark system and isnaturally associated with the PLC.In Eq. (47) the function q τ is normalized to unity.The u and d quark distributions at a scale µ = 1 . ± .
15 GeV are given by u ( x ) = q ( x ) + q ( x ) (48) d ( x ) = q ( x ) , (49)with the u ( x ) and d ( x ) normalized to the flavorcontent of the proton. An excellent reproductionof measured structure functions and elastic formfactors is obtained using only two components andthe flavor-independent parameter a = 0 . ± . q ( x ) /q ( x ) = 1 / (1 − w ( x )) which in-creases monotonically with increasing x , as expectedby the intuition inherent in Eq. (45) with df /dx > q )with becoming more important as the value of x increases. In this model BLC is associated with q , and the PLC component occurs only with upquarks. The relevant combination for a nucleuswith N neutrons and Z protons is proportional to Z ( q ( x ) + q ( x )) + N (3 q ( x ) + 9 q ( x )).Now suppose the nucleon is bound to a nucleus.The nucleon feels an attractive nuclear potential, hererepresented by H , with H = (cid:20) U
00 0 (cid:21) , (50)to represent the idea that only the large-sizedcomponent of the nucleon feels the influence of thenuclear attraction. The treatment of the nuclearinteraction, U , as a number is clearly a simplifi-cation because the interaction necessarily varieswith the relevant kinematics. The present model issimilar to the model of [111], with the importantdifference that the medium effects enter as an am-plitude instead of as a probability. See also Ref. [117].The complete Hamiltonian H = H + H is: H = (cid:20) E B − | U | VV E P (cid:21) , (51)in which the attractive nature of the nuclear bindingpotential is emphasized. Then interactions with the nucleus increase the energy difference between thebare BLC and PLC states and thereby decreases thePLC probability.The medium-modified nucleon and its excitedstate, | N (cid:105) M and | X (cid:105) M , are now (again using first-order perturbation theory) | N (cid:105) M = | B (cid:105) + (cid:15) M | P (cid:105) (52) | X (cid:105) M = − (cid:15) M | B (cid:105) + | P (cid:105) , (53)where (cid:15) M = VE B − | U | − E P = (cid:15) E B − E P E B − | U | − E P (54)and (cid:15) M (cid:15) = E B − E P E B −| U |− E P < (cid:15) M − (cid:15) ≈ | U | E B − E P (55)is relevant for understanding the EMC effect because | N (cid:105) M = | N (cid:105) + ( (cid:15) M − (cid:15) ) (cid:104) P |O DIS | P (cid:105) , (56)and the medium modification of the nucleon isproportional to the interaction with the nucleusrepresented by U .The medium-modified quark distribution function q M ( x ) = (cid:104) N M |O DIS | N M (cid:105) , and is q M ( x ) = q ( x ) +∆ q ( x ) with ∆ q ≈ (cid:15) M − (cid:15) ) (cid:104) N |O DIS | P (cid:105)≈ (cid:15) M − (cid:15) ) (cid:15) (cid:104) P |O DIS | P (cid:105) . (57)in which terms of first-order in ( (cid:15) M − (cid:15) ) kept to rep-resent the small EMC effect. Next use Eq. (45) andEq. (46) to find∆ q ( x ) = 2( (cid:15) M − (cid:15) ) (cid:15) q ( x ) f ( x )1+ (cid:15) f ( x ) ≈ (cid:15) M − (cid:15) ) (cid:15) q ( x ) f ( x ) (58)Note that the product ( (cid:15) M − (cid:15) ) (cid:15) is less than zero,independent of the sign of the interaction V . Thismeans that, at large values of x , the quark structurefunction in the nucleus is less than that of a free nu-cleon, and decreases with increasing x because f ( x )is monotonically increasing with increasing x . Thesefeatures are inherent in the data for values of x < . (cid:15) M − (cid:15) ) ∝ U (viaEq. (55)) to the virtuality. Suppose a photon inter-acts with a virtual nucleon of four-momentum P miss The three-momentum P miss opposes the A − p ≡ P miss = − P A − . The mass of6the on-shell recoiling nucleus is given by M ∗ A − = M A − M + E, where E > A − M V = P − M (59)= ( M A − (cid:113) ( M ∗ A − ) + p ) − p − M (60)which reduces in the non-relativistic limit to M V ≈ − M (cid:18) p M r + E (cid:19) , (61)where the reduced mass M r = M ( A − /A . Thevirtuality, V , is less than 0, and its magnitudeincreases with both the A − U and the virtuality V by using the extensionof the Schroedinger equation to an operator form: p M r + U = − E, (62) so that p M r + E = − U = | U | and via Eq. (55) V = 2 UM = 2( (cid:15) M − (cid:15) )( E P − E B ) M , (63)so that the modification of the nucleon due to thePLC suppression is proportional to its virtuality.Potentially large values of the virtuality greatlyenhance the difference between (cid:15) m and (cid:15) .Recall Eq. (57) and replace ( (cid:15) M − (cid:15) ) therein by itsexpression in terms of V (Eq. (63)) to find q M ( x ) = q ( x ) + ME P − E B V (cid:15) f ( x ) q ( x ) , (64)The conditions that ( (cid:15) M − (cid:15) ) (cid:15) < , V < (cid:15) >
0, which means that
V <
0. The sign of (cid:15) is consistent with the light-frontholographic model for which (cid:15) = 1 / √ df /dx > (cid:15)df /dx >
0. The ratio of structure functions is R ( x ) = q M ( x ) /q ( x ), and dRdx = ME P − E B V (cid:15) dfdx < , (65)as the measurements of the EMC effect have shown.The negative sign is caused by the negative value ofthe virtuality. This expression is only meaningful for x < . TABLE I. EMC effect vs.
Virtuality.Quantity He He C Fe Pb | dRdx | [118]. 0.070 ± ± .
026 0.292 ± ± ± | V M | (MeV) [106] 34.59 69.4 82.28 82.44 92.2 The quantities
M, E P − E B and f ( x ) are inde-pendent of the nucleus, so that the A -dependenceof the EMC effect is determined by the virtuality, V . According to this model, the larger the virtualitythe larger the EMC effect, as measured by the slopeof R ( x ). Table I compares the measurements of theslope with computations of the virtuality. The datafor A=56 is from a mixture of A=56 and A=63. Thetheory for Pb is compared with the data for
Au.The increase of the magnitude of the slope tracks qualitatively well with the corresponding increase ofthe virtuality. A quantitative reproduction of theA-dependence requires a more detailed treatment ofthe separate N and Z dependence as in Ref. [83].Another consequence of this model is that themedium-modified nucleon contains a component thatis an excited state of a free nucleon. The amount ofmodification, (cid:15) M − (cid:15) , which gives a deviation of theEMC ratio from unity, is controlled by the potential7 U and via Eq. (63) the virtuality. A more detailedevaluation of the EMC effect is reserved for anotherpaper. IX. SUMMARY & DISCUSSION
This paper takes a trip through three lengthscales relevant to nuclear physics. These are thenuclear size, the inter-nucleon separation distanceand the nucleon size. Simple examples are used toillustrate the basic underlying features that drive theobservations made at the three different scales. Theintent is to arrive at the realization that all threescales are must be understood to truly understandthe physics of nuclei.Sect. II briefly reviews the currently popularprocedure of softening the interactions betweennucleons, with a focus on the concomitant hardeningof the operators that probe nuclei. A first-orderequation, Eq. (10) is derived to demonstrate that theprobe operators are hardened by the same unitarytransformation that softens the interactions.Sec. III discusses the largest nuclear scale, withthe first point being that momentum transfers higherthan that achieved by Rutherford were needed todiscern the non-zero nature of the nuclear size.Equations (Eq. (17) and Eq. (18)) are derived toestimate the effect of the hardening of the probeoperator, and is used to demonstrate its importancefor momentum transfers, q , greater than about 2fm − .The physics of the nucleon-nucleon separationis explored in Sect. IV by using bound-state wavefunctions produced by four simple models of thenucleon interaction. The high-momentum transfer( q ) scaling of the form factors is exhibited for eachmodel. The values of relative momentum p thatmake important contributions to the form factorare displayed. Increasing the value of q is shown toincrease the values of p that enter. The resultingeffect of the hardening of the probe operator isdisplayed for two of the model interactions, whereagain significant effects of hardening of the operatorare seen for q > − . For other interactions thehardening cannot be computed easily. The role ofthe tensor force in producing high-momentum com-ponents, and in transforming the probe operator, isalso discussed. Current experiments involve transferof high momentum. The interpretation of such experiments is simplified if bare, un-transformedprobe operators can be used.The role of two-nucleon physics in nuclei, asmanifest in the independent pair approximation, isexplored in Sect. V. The modern approach is thegeneralized contact formalism. The high-momentumproperties of 0-energy wave functions enteringthat formalism are examined. The result Eq. (31)demonstrates the explicit connection between short-distance and high-momentum physics. Furthermore,the inevitable power-law falloff indicates that sig-nificant high momentum content must occur. Theconditions necessary for obtaining a direct connec-tion, Eq. (35), between scaling behavior of measuredform factors and the underlying wave functions aredetermined.Sect. VI discusses the ( e, e (cid:48) , p ) reaction as atool for discovery of short-distance physics at thenucleon-nucleon separation scale. Under certainconditions Eq. (37), which directly relates the scat-tering amplitude to the wave function, is valid. Moregenerally, at high momentum transfer, final statenucleons have high energy and undergo differentinteractions than those in the initial state. Thus, insuch situations, it is far simpler to use the impulseapproximation with the fundamental potentials inthe Hamiltonian than to use interactions softened byunitary transformations.The transition from the nucleon-nucleon separa-tion distance to the nucleon size and smaller sizes isbegun in Sect. VII through a discussion of virtuality,Eq. (38). High momentum transfer reactions probehighly virtual nucleons. Nucleons achieve highvirtuality only through strong interactions withclosely separated nucleons, Eq. (39). The internalwave function of such nucleons may be expressed asa superposition of baryonic eigenstates, Eq. (40). Ifthe momentum transfer is large enough many, manystates must be included in the superposition, andit becomes more efficient to use quark degrees offreedom.The role of virtuality in understanding the nuclearmodification of quark structure functions (EMCeffect) is discussed in Sect. VIII. The explicit connec-tion, Eq. (65) is displayed by using a two-component,(point-like/blob-like) model of the nucleon’s quarkdegrees of freedom. The simple model is shownto be consistent with the two-state treatment oflight-front holographic QCD that reproduces freenucleon structure functions and elastic form factors.8In particular, the point-like component is more im-portant relative to the blob-like component at largervalues of x . This model, combined with the conceptof virtuality provides a qualitative explanation ofthe EMC effect. ACKNOWLEDGEMENTS
This work was supported by the U. S. Depart-ment of Energy Office of Science, Office of Nu-clear Physics under Award Number DE-FG02-97ER-41014. I thank S. R. Stroberg and X-D Ji for usefuldiscussions.
X. APPENDIX-DERIVATION OF EQ. (10)
The result, Eq. (10) is stated without treating theterm of first order in s caused by the s -dependenceof the potential. This Appendix shows that the termvanishes for the case of a local, bare potential and alocal operator O .Consider the matrix element M s ≡ (cid:104) Ψ | [ H , [ V s , O ]] | Ψ (cid:105) , (66)which enters in computing elastic form factors. Thegoal here is to show that the term of order s vanishes.To first order in sV s ≈ V + s dVds ( s = 0) = V + s [[ H , V ] , V ] . (67)The double commutator [[ H , V ] , V ] = − M ( ∇ V ) which is function of r . This commutes with O andEq. (10) is obtained. [1] A Bohr and B. R Mottelson, Nuclear Structure:Volume I: Single-Particle Motjon (World ScientificPublishing Co., Singapore, 1998).[2] H. Georgi, “Effective field theory,” Ann. Rev. Nucl.Part. Sci. , 209–252 (1993).[3] Timothy Cohen, “As Scales Become Sepa-rated: Lectures on Effective Field Theory,” PoS TASI2018 , 011 (2019), arXiv:1903.03622 [hep-ph].[4] Elliott D. Bloom et al. , “High-Energy Inelastic ep Scattering at 6-Degrees and 10-Degrees,” Phys.Rev. Lett. , 930–934 (1969).[5] Jerome I. Friedman and Henry W. Kendall, “Deepinelastic electron scattering,” Ann. Rev. Nucl. Part.Sci. , 203–254 (1972).[6] S.K. Bogner, T.T.S. Kuo, A. Schwenk, D.R. En-tem, and R. Machleidt, “Towards a model indepen-dent low momentum nucleon nucleon interaction,”Phys. Lett. B , 265–272 (2003), arXiv:nucl-th/0108041.[7] S.K. Bogner, T.T.S. Kuo, and A. Schwenk, “Modelindependent low momentum nucleon interactionfrom phase shift equivalence,” Phys. Rept. , 1–27 (2003), arXiv:nucl-th/0305035.[8] S.K. Bogner, A. Schwenk, T.T.S. Kuo, and G.E.Brown, “Renormalization group equation for lowmomentum effective nuclear interactions,” (2001),arXiv:nucl-th/0111042.[9] Andreas Nogga, Scott K. Bogner, and AchimSchwenk, “Low-momentum interaction in few-nucleon systems,” Phys. Rev. C , 061002 (2004),arXiv:nucl-th/0405016.[10] S.K. Bogner, A. Schwenk, R.J. Furnstahl, and A. Nogga, “Is nuclear matter perturbative with low-momentum interactions?” Nucl. Phys. A , 59–79(2005), arXiv:nucl-th/0504043.[11] S.K. Bogner, R.J. Furnstahl, and A. Schwenk,“From low-momentum interactions to nuclear struc-ture,” Prog. Part. Nucl. Phys. , 94–147 (2010),arXiv:0912.3688 [nucl-th].[12] Stanislaw D. Glazek and Kenneth G. Wilson,“Renormalization of Hamiltonians,” Phys. Rev. D , 5863–5872 (1993).[13] Sergio Szpigel and Robert J. Perry, “The Similarityrenormalization group,” , 59–81 (2000), arXiv:hep-ph/0009071.[14] S.K. Bogner, R.J. Furnstahl, and R.J. Perry, “Sim-ilarity Renormalization Group for Nucleon-NucleonInteractions,” Phys. Rev. C , 061001 (2007),arXiv:nucl-th/0611045.[15] E.D. Jurgenson, S.K. Bogner, R.J. Furnstahl, andR.J. Perry, “Decoupling in the Similarity Renormal-ization Group for Nucleon-Nucleon Forces,” Phys.Rev. C , 014003 (2008), arXiv:0711.4252 [nucl-th].[16] E.D. Jurgenson, P. Navratil, and R.J. Furnstahl,“Evolution of Nuclear Many-Body Forces withthe Similarity Renormalization Group,” Phys. Rev.Lett. , 082501 (2009), arXiv:0905.1873 [nucl-th].[17] E. Epelbaum, H.-W. Hammer, and Ulf-G. Meißner,“Modern theory of nuclear forces,” Rev. Mod. Phys. , 1773–1825 (2009).[18] N. Ishii, S. Aoki, and T. Hatsuda, “Nuclear forcefrom lattice qcd,” Phys. Rev. Lett. , 022001(2007). [19] Sinya Aoki, Tetsuo Hatsuda, and Noriyoshi Ishii,“Theoretical Foundation of the Nuclear Force inQCD and its applications to Central and Ten-sor Forces in Quenched Lattice QCD Simula-tions,” Prog. Theor. Phys. , 89–128 (2010),arXiv:0909.5585 [hep-lat].[20] Keiko Murano, Noriyoshi Ishii, Sinya Aoki, andTetsuo Hatsuda, “Nucleon-Nucleon Potential andits Non-locality in Lattice QCD,” Prog. Theor.Phys. , 1225–1240 (2011), arXiv:1103.0619 [hep-lat].[21] Takumi Doi et al. , “First results of baryon interac-tions from lattice QCD with physical masses (1) –General overview and two-nucleon forces –,” PoS LATTICE2015 , 086 (2016), arXiv:1512.01610[hep-lat].[22] Sinya Aoki, Takumi Doi, Tetsuo Hatsuda, andNoriyoshi Ishii, “Comment on “relation betweenscattering amplitude and bethe-salpeter wave func-tion in quantum field theory”,” Phys. Rev. D ,038501 (2018).[23] Sinya Aoki, Takumi Doi, and Takumi Iritani, “San-ity check for NN bound states in lattice QCD,” EPJWeb Conf. , 10.1051/epjconf/201817505006,arXiv:1707.08800 [hep-lat].[24] O. Benhar and V.R. Pandharipande, “Scattering ofGeV electrons by light nuclei,” Phys. Rev. C ,2218–2227 (1993).[25] L. L. Frankfurt, M. I. Strikman, D. B. Day, andM. Sargsyan, “Evidence for short-range correlationsfrom high q (e,e’) reactions,” Phys. Rev. C ,2451–2461 (1993).[26] S.K. Bogner, R.J. Furnstahl, R.J. Perry, andA. Schwenk, “Are low-energy nuclear observablessensitive to high-energy phase shifts?” Phys. Lett.B , 488–493 (2007), arXiv:nucl-th/0701013.[27] R.J. Furnstahl and H.W. Hammer, “Are occupationnumbers observable?” Phys. Lett. B , 203–208(2002), arXiv:nucl-th/0108069.[28] Jun John Sakurai and Jim Napolitano, ModernQuantum Mechanics , Quantum physics, quantuminformation and quantum computation (CambridgeUniversity Press, Cambridge, 2017).[29] K Gottfried, , and T-M Yan,
Quantum Mechanics:FUndamentals (Springer-Verlag, New York, 2003).[30] Andrea Damascelli, Zahid Hussain, and Zhi-XunShen, “Angle-resolved photoemission studies of thecuprate superconductors,” Rev. Mod. Phys. ,473–541 (2003).[31] A. Schmidt et al. (CLAS), “Probing the core of thestrong nuclear interaction,” Nature , 540–544(2020), arXiv:2004.11221 [nucl-ex].[32] O. Hen, Gerald. A. Miller, E. Piasetzky, and L.B.Weinstein, “Nucleon-Nucleon Correlations, Short-lived Excitations, and the Quarks Within,” Rev.Mod. Phys. , 045002 (2017), arXiv:1611.09748[nucl-ex].[33] Frank T. Avignone, Steven R. Elliott, andJonathan Engel, “Double beta decay, majorana neutrinos, and neutrino mass,” Rev. Mod. Phys. ,481–516 (2008).[34] Jonathan Kozaczuk, David E. Morrissey, and S.R.Stroberg, “Light axial vector bosons, nuclear tran-sitions, and the Be anomaly,” Phys. Rev. D ,115024 (2017), arXiv:1612.01525 [hep-ph].[35] C. Ordonez and U. van Kolck, “Chiral lagrangiansand nuclear forces,” Phys. Lett. B , 459–464(1992).[36] C. Ordonez, L. Ray, and U. van Kolck, “Nucleon-nucleon potential from an effective chiral La-grangian,” Phys. Rev. Lett. , 1982–1985 (1994).[37] C. Ordonez, L. Ray, and U. van Kolck, “TheTwo nucleon potential from chiral Lagrangians,”Phys. Rev. C , 2086–2105 (1996), arXiv:hep-ph/9511380.[38] Paulo F. Bedaque and Ubirajara van Kolck, “Effec-tive field theory for few nucleon systems,” Ann. Rev.Nucl. Part. Sci. , 339–396 (2002), arXiv:nucl-th/0203055.[39] H.-W. Hammer, Sebastian K¨onig, and U. vanKolck, “Nuclear effective field theory: Status andperspectives,” Rev. Mod. Phys. , 025004 (2020).[40] S.K. Bogner, R.J. Furnstahl, S. Ramanan, andA. Schwenk, “Low-momentum interactions withsmooth cutoffs,” Nucl. Phys. A , 79–103 (2007),arXiv:nucl-th/0609003.[41] E.R. Anderson, S.K. Bogner, R.J. Furnstahl, andR.J. Perry, “Operator Evolution via the Similar-ity Renormalization Group I: The Deuteron,” Phys.Rev. C , 054001 (2010), arXiv:1008.1569 [nucl-th].[42] S.K. Bogner, R.J. Furnstahl, and A. Schwenk,“Comment on ‘Problems in the derivationsof the renormalization group equation for thelow momentum nucleon interactions’,” (2008),arXiv:0806.1365 [nucl-th].[43] Stanislaw D. Glazek and Tomasz Maslowski,“Renormalized Poincare algebra for effective par-ticles in quantum field theory,” Phys. Rev. D ,065011 (2002), arXiv:hep-th/0110185.[44] A.J. Tropiano, S.K. Bogner, and R.J. Furnstahl,“Operator evolution from the similarity renormal-ization group and the Magnus expansion,” (2020),arXiv:2006.11186 [nucl-th].[45] Robert Hofstadter, “Electron scattering and nuclearstructure,” Rev. Mod. Phys. , 214–254 (1956).[46] R. Hofstadter, “Nuclear and nucleon scattering ofhigh-energy electrons,” Ann. Rev. Nucl. Part. Sci. , 231–316 (1957).[47] Gerald A. Miller, “Electromagnetic Form Factorsand Charge Densities From Hadrons to Nuclei,”Phys. Rev. C , 045210 (2009), arXiv:0908.1535[nucl-th].[48] Carlos Bertulani, “Nuclear physics in a nutshell,”Nuclear Physics in a Nutshell, by Carlos A. Bertu-lani. ISBN 978-0-691-12505-3. Published by Prince-ton University Press, Princeton, NJ USA, 2007.(2007). [49] M. Gmitro, S. S. Kamalov, and R. Mach,“Momentum-space second-order optical potentialfor pion-nucleus elastic scattering,” Phys. Rev. C , 1105–1117 (1987).[50] Beat Hahn, D. G. Ravenhall, and Robert Hof-stadter, “High-energy electron scattering and thecharge distributions of selected nuclei,” Phys. Rev. , 1131–1142 (1956).[51] K.S. Krane, Introductory Nuclear Physics (Wiley,NY, 1987).[52] G.E. Brown and A.D. Jackson,
The Nucleon-Nucleon Interaction (North Holland PublishingCompany, 1976).[53] James Lewis Friar, B.F. Gibson, and G.L. Payne,“One Pion Exchange Potential Deuteron ,” Phys.Rev. C , 1084–1086 (1984).[54] Jason R. Cooke and Gerald A. Miller, “Pion - only,chiral light front model of the deuteron,” Phys. Rev.C , 067001 (2002), arXiv:nucl-th/0112076.[55] Norbert Kaiser, S. Fritsch, and W. Weise, “Chiraldynamics and nuclear matter,” Nucl. Phys. A ,255–276 (2002), arXiv:nucl-th/0105057.[56] O. Hen, L.B. Weinstein, E. Piasetzky, G.A. Miller,M.M. Sargsian, and Y. Sagi, “Correlated fermionsin nuclei and ultracold atomic gases,” Phys. Rev. C , 045205 (2015), arXiv:1407.8175 [nucl-ex].[57] H. A. Bethe, “Nuclear many-body problem,” Phys.Rev. , 1353–1390 (1956).[58] K. A. Brueckner, R. J. Eden, and N. C. Francis,“High-energy reactions and the evidence for corre-lations in the nuclear ground-state wave function,”Phys. Rev. , 1445–1455 (1955).[59] K. A. Brueckner, “Two-body forces and nuclear sat-uration. iii. details of the structure of the nucleus,”Phys. Rev. , 1353–1366 (1955).[60] R. Cruz-Torres, D. Lonardoni, R. Weiss, N. Barnea,D.W. Higinbotham, E. Piasetzky, A. Schmidt, L.B.Weinstein, R.B. Wiringa, and O. Hen, “Scaleand Scheme Independence and Position-MomentumEquivalence of Nuclear Short-Range Correlations,”(2019), arXiv:1907.03658 [nucl-th].[61] Ronen Weiss, Betzalel Bazak, and Nir Barnea,“Generalized nuclear contacts and momentum dis-tributions,” Phys. Rev. C , 054311 (2015),arXiv:1503.07047 [nucl-th].[62] R. Weiss, R. Cruz-Torres, N. Barnea, E. Piasetzky,and O. Hen, “The nuclear contacts and short rangecorrelations in nuclei,” Phys. Lett. B , 211–215(2018), arXiv:1612.00923 [nucl-th].[63] E.O. Cohen et al. (CLAS), “Center of Mass Mo-tion of Short-Range Correlated Nucleon Pairs stud-ied via the A ( e, e ? pp ) Reaction,” Phys. Rev. Lett. , 092501 (2018), arXiv:1805.01981 [nucl-ex].[64] A. Erdelyi, Asymptotic Expansions (Dover Publica-tions, Mineola, NY, 1956).[65] Y. Yamaguchi and Y. Yamaguchi, “Photodisinte-gration of the deuteron,” Phys. Rev. , 69–70(1955).[66] Stanley J. Brodsky and G.Peter Lepage, “Exclusive Processes in Quantum Chromodynamics,” Adv.Ser. Direct. High Energy Phys. , 93–240 (1989).[67] L.E. Marcucci, F. Gross, M.T. Pena, M. Piarulli,R. Schiavilla, I. Sick, A. Stadler, J.W. Van Orden,and M. Viviani, “Electromagnetic Structure of Few-Nucleon Ground States,” J. Phys. G , 023002(2016), arXiv:1504.05063 [nucl-th].[68] Gerald A. Miller and Mark Strikman, “Relation be-tween the deuteron form factor at high momentumtransfer and the high energy neutron-proton scat-tering amplitude,” Phys. Rev. C , 044004 (2004).[69] R.J. Glauber and G. Matthiae, “High-energy scat-tering of protons by nuclei,” Nucl. Phys. B , 135–157 (1970).[70] J.D. Walecka, Theoretical nuclear and subnuclearphysics (Oxford University Press, Oxford, 1995).[71] Nadia Fomin, Douglas Higinbotham, MisakSargsian, and Patricia Solvignon, “New Re-sults on Short-Range Correlations in Nuclei,”Ann. Rev. Nucl. Part. Sci. , 129–159 (2017),arXiv:1708.08581 [nucl-th].[72] Claudio Ciofi degli Atti, “In-medium short-rangedynamics of nucleons: Recent theoretical and exper-imental advances,” Phys. Rept. , 1–85 (2015).[73] A. Tang et al. , “n-p short range correlations from(p,2p + n) measurements,” Phys. Rev. Lett. ,042301 (2003), arXiv:nucl-ex/0206003.[74] E. Piasetzky, M. Sargsian, L. Frankfurt, M. Strik-man, and J.W. Watson, “Evidence for thestrong dominance of proton-neutron correlationsin nuclei,” Phys. Rev. Lett. , 162504 (2006),arXiv:nucl-th/0604012.[75] R. Subedi et al. , “Probing Cold Dense Nu-clear Matter,” Science , 1476–1478 (2008),arXiv:0908.1514 [nucl-ex].[76] I. Korover et al. (Lab Hall A), “Probing the Repul-sive Core of the Nucleon-Nucleon Interaction via the He ( e, e ? pN ) Triple-Coincidence Reaction,” Phys.Rev. Lett. , 022501 (2014), arXiv:1401.6138[nucl-ex].[77] O. Hen et al. , “Momentum sharing in imbal-anced Fermi systems,” Science , 614–617 (2014),arXiv:1412.0138 [nucl-ex].[78] M. Duer et al. (CLAS), “Probing high-momentumprotons and neutrons in neutron-rich nuclei,” Na-ture , 617–621 (2018).[79] M. Duer et al. (CLAS), “Direct Observation ofProton-Neutron Short-Range Correlation Domi-nance in Heavy Nuclei,” Phys. Rev. Lett. ,172502 (2019), arXiv:1810.05343 [nucl-ex].[80] R. Schiavilla, Robert B. Wiringa, Steven C. Pieper,and J. Carlson, “Tensor Forces and the Ground-State Structure of Nuclei,” Phys. Rev. Lett. ,132501 (2007), arXiv:nucl-th/0611037.[81] M. Alvioli, C. Ciofi degli Atti, and H. Morita,“Proton-neutron and proton-proton correlations inmedium-weight nuclei and the role of the tensorforce,” Phys. Rev. Lett. , 162503 (2008).[82] Or Hen, D.W. Higinbotham, Gerald A. Miller, Eli Piasetzky, and Lawrence B. Weinstein, “TheEMC Effect and High Momentum Nucleons in Nu-clei,” Int. J. Mod. Phys. E , 1330017 (2013),arXiv:1304.2813 [nucl-th].[83] B. Schmookler et al. (CLAS), “Modified structureof protons and neutrons in correlated pairs,” Nature , 354–358 (2019), arXiv:2004.12065 [nucl-ex].[84] Markus Kortelainen and Jouni Suhonen, “Improvedshort-range correlations and 0 neutrino beta betanuclear matrix elements of Ge-76 and Se-82,” Phys.Rev. C , 051303 (2007), arXiv:0705.0469 [nucl-th].[85] Markus Kortelainen and Jouni Suhonen, “Nuclearmatrix elements of neutrinoless double beta decaywith improved short-range correlations,” Phys. Rev.C , 024315 (2007), arXiv:0708.0115 [nucl-th].[86] J. Menendez, A. Poves, E. Caurier, and F. Nowacki,“Disassembling the Nuclear Matrix Elements of theNeutrinoless beta beta Decay,” Nucl. Phys. A ,139–151 (2009), arXiv:0801.3760 [nucl-th].[87] Fedor Simkovic, Amand Faessler, Herbert Muther,Vadim Rodin, and Markus Stauf, “The 0 nu bb-decay nuclear matrix elements with self-consistentshort-range correlations,” Phys. Rev. C , 055501(2009), arXiv:0902.0331 [nucl-th].[88] Omar Benhar, Riccardo Biondi, and Enrico Sper-anza, “Short-range correlation effects on the nuclearmatrix element of neutrinoless double- β decay,”Phys. Rev. C , 065504 (2014), arXiv:1401.2030[nucl-th].[89] Reynier Cruz-Torres, Axel Schmidt, Gerald A.Miller, Lawrence B. Weinstein, Nir Barnea, RonenWeiss, Eliezer Piasetzky, and O. Hen, “Short rangecorrelations and the isospin dependence of nuclearcorrelation functions,” Phys. Lett. B , 304–308(2018), arXiv:1710.07966 [nucl-th].[90] X.B. Wang, A.C. Hayes, J. Carlson, G.X. Dong,E. Mereghetti, S. Pastore, and R.B. Wiringa,“Comparison between Variational Monte Carlo andShell Model Calculations of Neutrinoless DoubleBeta Decay Matrix Elements in Light Nuclei,”Phys. Lett. B , 134974 (2019), arXiv:1906.06662[nucl-th].[91] Gerald.A. Miller, A. Beck, S. May-Tal Beck, L.B.Weinstein, E. Piasetzky, and O. Hen, “CanLong-Range Nuclear Properties Be Influenced ByShort Range Interactions? A chiral dynamicsestimate,” Phys. Lett. B , 360–364 (2019),arXiv:1805.12099 [nucl-th].[92] Bao-An Li, Bao-Jun Cai, Lie-Wen Chen, andJun Xu, “Nucleon Effective Masses in Neutron-RichMatter,” Prog. Part. Nucl. Phys. , 29–119 (2018),arXiv:1801.01213 [nucl-th].[93] R. Blankenbecler and R. Sugar, “Linear integralequations for relativistic multichannel scattering,”Phys. Rev. , 1051–1059 (1966).[94] R.H. Thompson, “Three-dimensional bethe-salpeter equation applied to the nucleon-nucleoninteraction,” Phys. Rev. D , 110–117 (1970). [95] E.E. Salpeter and H.A. Bethe, “A Relativistic equa-tion for bound state problems,” Phys. Rev. ,1232–1242 (1951).[96] Franz Gross, “Three-dimensional covariant integralequations for low-energy systems,” Phys. Rev. ,1448–1462 (1969).[97] Gerald A. Miller, “Confinement in Nuclei and theExpanding Proton,” Phys. Rev. Lett. , 232003(2019), arXiv:1907.00110 [nucl-th].[98] K.S. Egiyan et al. (CLAS), “Observation of nu-clear scaling in the A(e, e-prime) reaction at x(B)greater than 1,” Phys. Rev. C , 014313 (2003),arXiv:nucl-ex/0301008.[99] K.S. Egiyan et al. (CLAS), “Measurement of 2-and 3-nucleon short range correlation probabilitiesin nuclei,” Phys. Rev. Lett. , 082501 (2006),arXiv:nucl-ex/0508026.[100] N. Fomin et al. , “New measurements of high-momentum nucleons and short-range structuresin nuclei,” Phys. Rev. Lett. , 092502 (2012),arXiv:1107.3583 [nucl-ex].[101] R. Weiss, A.W. Denniston, J.R. Pybus, O. Hen,E. Piasetzky, A. Schmidt, L.B. Weinstein, andN. Barnea, “Study of inclusive electron scatteringscaling using the generalized contact formalism,”(2020), arXiv:2005.01621 [nucl-th].[102] J. Arrington and N. Fomin, “Searching for flavordependence in nuclear quark behavior,” Phys. Rev.Lett. , 042501 (2019).[103] O. Hen, F. Hauenstein, D.W. Higinbotham, G.A.Miller, E. Piasetzky, A. Schmidt, E.P. Segarra,M. Strikman, and L.B. Weinstein, “Comment on”Searching for flavor dependence in nuclear quarkbehavior”,” (2019), arXiv:1905.02172 [nucl-ex].[104] W. Melnitchouk, Andreas W. Schreiber, and An-thony William Thomas, “Deep inelastic scatteringfrom off-shell nucleons,” Phys. Rev. D , 1183–1198 (1994), arXiv:nucl-th/9311008.[105] Sergey A. Kulagin and R. Petti, “Global study ofnuclear structure functions,” Nucl. Phys. A ,126–187 (2006), arXiv:hep-ph/0412425.[106] C. Ciofi degli Atti, L.L. Frankfurt, L.P. Kaptari,and M.I. Strikman, “On the dependence of the wavefunction of a bound nucleon on its momentum andthe EMC effect,” Phys. Rev. C , 055206 (2007),arXiv:0706.2937 [nucl-th].[107] E.P. Segarra, J.R. Pybus, F. Hauenstein, D.W. Hig-inbotham, G.A. Miller, E. Piasetzky, A. Schmidt,M. Strikman, L.B. Weinstein, and O. Hen,“Short-Range Correlations and the Nuclear EMCEffect in Deuterium and Helium-3,” (2020),arXiv:2006.10249 [hep-ph].[108] M. Tanabashi et al. (Particle Data Group), “Re-view of Particle Physics,” Phys. Rev. D , 030001(2018).[109] J.J. Aubert et al. (European Muon), “The ratio ofthe nucleon structure functions F n for iron anddeuterium,” Phys. Lett. B , 275–278 (1983).[110] J. Gomez et al. , “Measurement of the A-dependence of deep inelastic electron scattering,” Phys. Rev. D , 4348–4372 (1994).[111] L.L. Frankfurt and M.I. Strikman, “Point-Like Con-figurations ,” Nucl. Phys. B , 143–176 (1985).[112] Stanley J. Brodsky and G.F. de Teramond, “SpinCorrelations, QCD Color Transparency and HeavyQuark Thresholds in Proton Proton Scattering,”Phys. Rev. Lett. , 1924 (1988).[113] John P. Ralston and Bernard Pire, “FluctuatingProton Size and Oscillating Nuclear Transparency,”Phys. Rev. Lett. , 1823 (1988).[114] B.K. Jennings and G.A. Miller, “Color transparencyin (p, p p) reactions,” Phys. Lett. B , 7–13(1993), arXiv:hep-ph/9305317.[115] L.L. Frankfurt, G.A. Miller, and M. Strikman,“The Geometrical color optics of coherent high- energy processes,” Ann. Rev. Nucl. Part. Sci. ,501–560 (1994), arXiv:hep-ph/9407274.[116] Guy F. de Teramond, Tianbo Liu, Raza Sab-bir Sufian, Hans Gunter Dosch, Stanley J. Brod-sky, and Alexandre Deur (HLFHS), “Universalityof Generalized Parton Distributions in Light-FrontHolographic QCD,” Phys. Rev. Lett. , 182001(2018), arXiv:1801.09154 [hep-ph].[117] M.R. Frank, B.K. Jennings, and G.A. Miller, “TheRole of color neutrality in nuclear physics: Modifi-cations of nucleonic wave functions,” Phys. Rev. C , 920–935 (1996), arXiv:nucl-th/9509030.[118] L.B. Weinstein, E. Piasetzky, D.W. Higinbotham,J. Gomez, O. Hen, and R. Shneor, “Short RangeCorrelations and the EMC Effect,” Phys. Rev. Lett.106