Neutrino (Antineutrino)-Nucleus Interactions in the Shallow- and Deep-Inelastic Scattering Regions
NNeutrino(Antineutrino)-Nucleus Interactions inthe Shallow- and Deep-Inelastic ScatteringRegions
M. Sajjad Athar
Department of Physics, Aligarh Muslim University, Aligarh - 202 002, IndiaE-mail: [email protected]
Jorge G. Morf´ın
Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USAE-mail: [email protected]
Abstract. In ν/ν -Nucleon/Nucleus interactions Shallow Inelastic Scattering (SIS) istechnically defined in terms of the four-momentum transfer to the hadronic systemas non-resonant meson production with Q (cid:47) GeV . This non-resonant mesonproduction intermixes with resonant meson production in a regime of similar effectivehadronic mass W of the interaction. As Q grows and surpasses this ≈ GeV limit,non-resonant interactions begin to take place with quarks within the nucleon indicatingthe start of Deep Inelastic Scattering (DIS). To essentially separate this resonantplus non-resonant meson production from DIS quark-fragmented meson production,a cut of 2 GeV in W of the interactions is generally introduced. However, sinceexperimentally mesons from resonance decay cannot be separated from non-resonantproduced mesons, SIS for all practical purposes in this review has been defined asinclusive meson production that includes non-resonant plus resonant meson productionand the interference between them. Experimentally then for W (cid:47) (cid:39) ( M N + M π ) and all Q is here defined as SIS, while for W (cid:39) Q (cid:39) GeV is defined as DIS. The so defined SISand DIS regions have received varying degrees of attention from the community. Whilethe theoretical / phenomenological study of ν -nucleon and ν -nucleus DIS scattering isadvanced, such studies of a large portion of the SIS region, particularly the SIS to DIStransition region, have hardly begun. Experimentally, the SIS and the DIS regions for ν -nucleon scattering have minimal results and only in the experimental study of the ν -nucleus DIS region are there significant results for some nuclei. Since current andfuture neutrino oscillation experiments have contributions from both higher W SIS andDIS kinematic regions and these regions are in need of both considerable theoreticaland experimental study, this review will concentrate on these SIS to DIS transitionand DIS kinematic regions surveying our knowledge and the current challenges. a r X i v : . [ h e p - ph ] J un eutrino(Antineutrino)-Nucleus Interactions in the Shallow- and Deep-Inelastic Scattering Regions PACS numbers: 13.15.+g, 24.10.i, 24.85.+p, 25.30.c
ONTENTS Contents1 Introduction 42 ν l / ¯ ν l -Nucleon Scattering 9 ν l -Nucleon Scattering: Shallow Inelastic Scattering . . . . . . . . . . . . 92.2 ν l -Nucleon Scattering: Deep-Inelastic Scattering . . . . . . . . . . . . . . 162.3 QCD Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 NLO and NNLO Evolutions . . . . . . . . . . . . . . . . . . . . . 202.3.2 Target Mass Correction Effect: . . . . . . . . . . . . . . . . . . . . 212.3.3 Higher Twist Effect: . . . . . . . . . . . . . . . . . . . . . . . . . 22 ν l / ¯ ν l -Nucleus Scattering : Deep-Inelastic Scattering Theory 23 ν l / ¯ ν l -Nucleus Scattering: Shallow Inelastic Scattering Phenomenology 42 ” Effects . . . . 524.3 Neutrino Simulation Efforts in the SIS region . . . . . . . . . . . . . . . . 534.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ν l / ¯ ν l -Nucleus Scattering: Deep-Inelastic Scattering Phenomenology 57 (cid:96) ± A and νA Nuclear Correction Factors . . . . . . . . 755.7 Hadronization of Low Energy ν -A Interactions . . . . . . . . . . . . . . . 795.7.1 The AGKY Hadronization Model . . . . . . . . . . . . . . . . . . 805.7.2 FLUKA: NUNDIS . . . . . . . . . . . . . . . . . . . . . . . . . . 825.8 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 ν/ν Nucleus Scattering . . . . . . . 927.2 Phenomenological Picture of ν/ν
Nucleus Scattering . . . . . . . . . . . . 947.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
The study of neutrino and antineutrino ( ν l / ¯ ν l ) interactions with nuclei covers anextended range of energies from the coherent elastic scattering off nuclei studied byexperiments like CE ν NS [1, 2] to the ultra high energy cosmological (multi-messenger)neutrinos studied by experiments like IceCube [3]. In the energy range of accelerator-based and atmospheric neutrinos, the experimental study of neutrino physics is currentlyfocused on understanding the three flavor ν l oscillation phenomenology in the leptonsector of weak interactions. In particular an accurate measurement of any CP violationas well as determining the mass hierarchy of the three neutrino mass states is the goalof current and future neutrino oscillation experiments.The experimental determination of these important properties depend on accurateknowledge of the energy ( E ν ) of the interacting ν l and the produced particles at theinteraction point. However, due to the weak nature of these interactions, to obtainnecessary statistics ν l oscillation experiments using accelerator and atmospheric ν l / ¯ ν l have been using moderate to heavy nuclear targets like C , O , Ar and F e . Thiscomplicates the precision measurement of these properties since to obtain the initialenergy and produced topology of the interacting neutrino, as opposed to the energyand topology measured in the detectors, model-dependent nuclear corrections, referredto as the ”nuclear model”, must be applied to the interpretation of the data. Thisnuclear model contains the current knowledge of the initial ν l / ¯ ν l - nucleon cross sections,the initial state nuclear medium effects and the final state interactions of the producedhadrons within the nucleus. The introduction of this nuclear model to the interpretationof experimental data is performed by Monte Carlo simulation programs (neutrino eventgenerators) that apply these nuclear effects to the free nucleon interaction cross sections.Note that in this procedure, even before introducing uncertainties associated with thenuclear model [4]-[8], uncertainties are already introduced into the analysis due to thelack of precise knowledge of the ν l nucleon interaction cross sections.In the energy region of ≈ π production and deep inelastic (DIS) scattering processes.These scattering processes are possible via charged(CC) as well as neutral(NC) currentchannels for which the main basic reactions on a free nucleon target are given by: ν l ( k ) + n ( p ) −→ l − ( k (cid:48) ) + p ( p (cid:48) ) , ¯ ν l ( k ) + p ( p ) −→ l + ( k (cid:48) ) + n ( p (cid:48) ) , (cid:41) (CC QE) (1) ν l / ¯ ν l ( k ) + N ( p ) −→ ν l / ¯ ν l ( k (cid:48) ) + N ( p (cid:48) ) (NC elastic) (2) ν l / ¯ ν l ( k ) + N ( p ) −→ l − /l + ( k (cid:48) ) + N ( p (cid:48) ) + mπ ( p π ) (CC resonance)(3) ν l / ¯ ν l ( k ) + N ( p ) −→ ν l / ¯ ν l ( k (cid:48) ) + N ( p (cid:48) ) + mπ ( p π ) (NC resonance)(4) ν l / ¯ ν l ( k ) + N ( p ) −→ l − /l + ( k (cid:48) ) + X ( p (cid:48) ) (CC DIS) (5) ν l / ¯ ν l ( k ) + N ( p ) −→ ν l / ¯ ν l ( k (cid:48) ) + X ( p (cid:48) ) (NC DIS) (6)where the quantities in the parenthesis represent respective momenta carried by theparticles, N represents a proton or neutron, π represents any of the three pion chargestates depending upon charge conservation, m represents number of pions in the finalstate and X represents jet of hadrons in the final state. Besides these productionmodes, kaon, hyperon, eta production and resonance decays to more massive states arealso possible, however, at much reduced rates.In this review, the considered signatures of ν l and ¯ ν l interactions with nucleartargets are exclusively charged current interactions yielding a charged lepton in the finalstate. In addition, we will be concentrating on the higher hadronic effective mass statesthat transition into and are within the deep-inelastic scattering regime. This transitionregion includes higher effective mass resonant and non-resonant single and multi-pionproduction. Although the quasi-elastic [9] interaction and ∆ resonance production [10]are also important in the few GeV region, they are not within the scope of this review.As indicated, when a ν l or ¯ ν l interacts with a nucleon bound in a nuclear target,nuclear medium effects become important. These nuclear medium effects are energydependent and moreover different for each interaction mode. In resonant and non-resonant production nuclear effects of the initial state such as Fermi motion, bindingenergy, Pauli blocking, multi-nucleon correlation effects have to be taken into account.In addition, final state interaction of the produced nucleons and pions within the nucleusare also very important. There are several theoretical calculations of these initial andfinal state nuclear medium effects in inelastic scattering where one pion is produced [11]-[25]. However, as summarized in a recent white paper from NuSTEC [26], thereare much more limited studies of multi- π resonant production and the other shallowinelastic scattering(SIS) processes such as non-resonant π production and the resultinginterference of resonant/non-resonant states in the weak sector.The importance of non-resonant meson production and the resulting interferenceeffects with resonant production is receiving renewed emphasis currently since thereare efforts underway to produce more theoretically-based estimates [27, 28] of theseprocesses rather than the phenomenological approach of extrapolating the DIS crosssections to lower hadronic mass W used in some MC generators [29, 30]. Since it isnot possible to experimentally distinguish resonant from non-resonant pion production,this kinematic regime as both the Q of the non-resonant meson production and W ofthe resonant region increase and transitions into the DIS region can only be studied interms of inclusive production for example, by Morf´ın et al. [7], Melnitchouk et al. [31],Lalakulich et al. [32], Christy et al. [33], and more recently by the Ghent group [34] thathas employed Regge theory to describe this transition region. This then is where low Q non-resonant meson production in the SIS region transitions into higher Q quark-fragmented meson production in the DIS region. The need of improved understanding of ν l / ¯ ν l -nucleus scattering cross sections in this transition region has generated considerableinterest in studying Quark-Hadron Duality (duality) in the weak sector.Duality has been studied in the electroproduction sector for both nucleon andnuclear targets and there is a body of evidence that duality does approximately holdin this sector. The few studies of duality in the weak sector have had to be basedon theoretical models since no high-statistics, precise experimental data is available.These studies have not been encouraging suggesting that increased experiment andbetter modeling is required. This also suggests caution in using the approach of simplyextrapolating the DIS cross sections to lower hadronic mass W used in some MCgenerators to estimate non-resonant π as well as resonant multi- π production in theSIS region. If a form of duality is found to be valid for neutrino scattering, it can beeffectively used to theoretically describe the SIS/DIS transition region of ν l / ¯ ν l - n ucleonscattering.Increasing W and Q of the interaction brings the regime of deep-inelasticscattering. The definition of DIS is based upon the kinematics of the interactionproducts and is primarily defined with Q ≥ . . To further separate resonanceproduced pions from quark-fragmented pions a requirement of W ≥ . σ Fe Aσ D is not unity inthe DIS region. This was surprising as this is the region where the underlying degreesof freedom should be quarks and gluons, while the deviation from unity suggested thatnuclear medium effects were important.In these ( (cid:96) ± -A) DIS interactions the deviation from 1.0 in the ratio of nuclear tonucleon structure functions as a function of x Bjorken ( ≡ x ), reflecting these nuclearmedium effects, have been categorized in four regions: ”shadowing” at lowest-x ( (cid:47) . (cid:47) . (cid:47) .
7) and ”Fermi Motion effect” at high x ( (cid:39) . F W IiA ( x, Q ); i = 1 , , , L ) is important.The nuclear effects in neutrino DIS analyses had been assumed to be the same as forcharged lepton-nucleus ( (cid:96) ± -A) DIS data. However there are now both theoretical andexperimental suggestions that the nuclear effects in the DIS region may be different for ν l / ¯ ν l -nucleus interactions as there are contributions from the axial current in the weaksector and different valence and sea quark contributions for each observable. Therefore,an independent and quantitative understanding of the DIS nuclear medium effects inthe weak sector is required.The historical experimental study of neutrino-nucleus ( ν -A) scattering in the DISregion is summarized in sections 5.1 and 5.2 and began during the bubble chamber era ofthe 1970’s [88, 89, 90, 91, 92]. It continued through the higher-statistics, mainly iron andlead experiments, of the 1990’s using higher energy ν/ν beams such as CDHSW [93],CCFR [94, 95], NuTeV [96]. Currently MINER ν A at Fermilab is dedicated to themeasurement of these cross section to better understand the nuclear medium effectsand has taken data using the medium energy NuMI beam ( < E ν > ∼ ν l as well as ¯ ν l modes with several nuclear targets C , F e and
P b and the largecentral scintillator (CH) tracker. In addition to the dedicated MINER ν A experimentthere are the T2K experiment in Japan as well as the NOvA experiment in the USA,although primarily oscillation experiments, also currently contributing to cross sectionmeasurements. At these lower energies, SIS events dominate however DIS events stillcontribute to the event rates although with more limited kinematic reach. ‡ ‡ For lower energy neutrinos, the MicroBooNE experiment plus the future short baseline near detectorSBND and far detector ICARUS in the Fermilab Booster neutrino beam will be measuring cross sectionsmainly in the E (cid:47)
The method for testing the relevant nuclear models with experimental resultsinvolve the Monte Carlo (MC) generator the experiments employ. At present several MCgenerators have been developed like GiBUU [8], NuWro [22], GENIE [97] and NEUT [98]that are used within the experimental community. These MC generators each havevariations of a nuclear model, plus many other experiment dependent effects that areusually more accurately determined, involved in predicting what a particular experimentshould detect. Comparing the predictions of these generators with the experimentalmeasurements gives an indication of the accuracy of the nuclear model employed.In addition to refining the nuclear model, the MC generator is also a necessarycomponent for determining important experimental parameters such as acceptance,efficiency and systematic errors. To perform all these functions the MC generatorsneed production models for each of the interactions they simulate as well as the nuclearmodel. For resonance production the (often modified) Rein-Sehgal model [99] (R-S) orthe more recent Berger-Sehgal (B-S) model [100] is widely used. However these modelsare limited to single pion production. It is essential to also study the full multi-pionproduction from nucleon resonances in the energy region of 1 - 10 GeV, where variousresonances like P (1232), P (1440), D (1520), S (1535), S (1650), P (1720), etc.contribute. For most MC generators the DIS process is simulated using the Bodek-Yang model [30]. This DIS simulation is then extrapolated down into the SIS region.The extrapolation is supposed to account for all non-resonant processes and resonantmulti- π production.At the planned accelerator-based, long-baseline ν l -oscillation experiments suchas the Deep Underground Neutrino Experiment(DUNE) using an argon target it isexpected that more than 30% of the events would come from the DIS region andmore than 50% of the events would come from the SIS(W ≥ M ∆ ) plus DIS regions.Additionally the atmospheric ν l studies in the proposed Hyper-K experiment (Hyper-Kamiokande), using water target, will also have significant SIS and DIS contributions.It is consequently important to have improved knowledge of nuclear medium effects inthese lower-energy regions and therefore timely to revisit the present status of boththe theoretical/phenomenological and experimental understanding of these scatteringprocesses.To summarize, using the Q − ν plane (Fig.1), one may define the relationshipof the various regions like elastic ( W = M N ), resonance ( M N + M π ≤ W ≤ Q > , W > Q < and W > Q < and W > M N + M π . It is apparent that the resonant and non-resonantpion production with W <
W < Q − ν plane is shown at E ν = 3 GeV (upper panel) and E ν = 7 GeV (lower panel). Asone moves away from the higher W region, where DIS (that deals with the quarks andgluons) is the dominant process to the region of SIS (resonant + nonresonant processeshaving hadrons as a degree of freedom), the boundary between these two regions is notwell defined. In the literature, Q ≥ has been chosen as the lower limit requiredto be interacting with the hadron’s constituents. A kinematic constraint of W ≥ ν l / ¯ ν l - nucleon scattering cross section including the QCDcorrections. In section-3, we describe in short the various phenomenological as well astheoretical efforts to understand nuclear medium effects in weak interaction processesand compare the theoretical results of Aligarh-Valencia group [68]-[70],[75]-[77] withexperimental results. In section-4 we cover the phenomenological and experimentaltreatment of the SIS region including a detailed examination of duality. In section-5we present the phenomenological and experimental treatment of the DIS region. Insection-6 we present a comparison of theoretical and phenomenological (nuclear PDFs)predictions with existing higher-energy experimental results. In section 7 we presentour conclusions on what is needed both theoretically and experimentally to improveour understanding of the physics of the SIS and DIS regions and our predictions forneutrino nucleus interactions with the lower neutrino energy and nuclei relevant forfuture oscillation experiments. ν l / ¯ ν l -Nucleon Scattering ν l -Nucleon Scattering: Shallow Inelastic Scattering For the resonance production process ν l / ¯ ν l ( k ) + N ( p ) → l − /l + ( k (cid:48) ) + R ( p (cid:48) ) (7)the inclusive cross section is given as a sum of the individual contribution from theresonance excitations R , where R = ∆ , N ∗ , etc. This is diagrammatically shown inFig.2. In the above relation, the quantities in the parenthesis are the four momentaof the corresponding particles. The cross section for the resonance excitation of theindividual resonance may be written as: d σd Ω (cid:48) l dE (cid:48) l ∝ A ( p (cid:48) ) (cid:112) ( k · p ) − m l M R L µν W µνR (8)where L µν is the leptonic tensor which is given by L µν = k µ k (cid:48) ν + k ν k (cid:48) µ − k · k (cid:48) g µν ± i(cid:15) µνρσ k ρ k (cid:48) σ (9)0 Figure 1.
Allowed kinematical region for ν l − N scattering in the ( Q , ν ) planefor E ν =3 GeV(top panel) and E ν =7 GeV(bottom panel). The square of the invariantmass is defined as W = M N + 2 M N ν − Q with the nucleon mass M N and the energytransfer ν . The inelasticity is defined as y = νE ν = ( E ν − E l ) E ν and then the forbiddenregion in terms of x and y is defined as x, y / ∈ [0 , x = Q M N ν = 1and, for this review, the SIS region has been practically defined as the region for which M N + M π ≤ W ≤ GeV and Q ≥ Q ≥ GeV and W ≥ GeV , and the Soft DIS region is defined as Q < GeV and W ≥ GeV .Notice the yellow band( M N < W < M N + M π ), where we do not expect anything from ν − N scattering. However, this region becomes important when the scattering takesplace with a nucleon within a nucleus due to the multi-nucleon correlation effect. Figure 2.
Diagrammatic representation of resonance excitations for ν l (¯ ν l ) + N → l − ( l + ) + R , where R represents the different resonancescontributing to the hadronic current. and W µνR is the hadronic tensor corresponding to the N ( p ) excitation of the resonance R ( p (cid:48) ), which may be schematically given as W µνR = (cid:88) (cid:88) (cid:104) R ( p (cid:48) ) | J µ | N ( p ) (cid:105) ∗ (cid:104) R ( p (cid:48) ) | J ν | N ( p ) (cid:105) , (10) A ( p (cid:48) ) = (cid:112) p (cid:48) π Γ( p (cid:48) )( p (cid:48) − M R ) + p (cid:48) Γ ( p (cid:48) ) , (11)where Γ( p (cid:48) ) is the momentum dependent width and M R is the Breit-Wigner mass ofthe resonance. (cid:104) R ( p (cid:48) ) | J µ | N ( p ) (cid:105) corresponds to the transition matrix element for thetransition N ( p ) → R ( p (cid:48) ) induced by the current J µ . The transition matrix elementfor the vector and the axial vector currents are characterized by the various transitionform factors depending upon the spin of the excited resonance R ( p (cid:48) ).For example, in the case of the transition N ( p ) → R ( p (cid:48) ), the general structurefor the hadronic current of spin three-half resonance excitation is determined by thefollowing equation J µ = ¯ ψ ν ( p (cid:48) )Γ νµ u ( p ) , (12)where u ( p ) is the Dirac spinor for nucleon, ψ µ ( p ) is the Rarita-Schwinger spinor for spinthree-half resonance and Γ νµ has the following general structure for the positive(+) andnegative(-) parity states :Γ
32 + νµ = (cid:104) V νµ − A νµ (cid:105) γ ; Γ − νµ = V νµ − A νµ , (13)where V ( A ) is the vector(axial-vector) current for spin three-half resonances, whichare described in terms of C Vi ( C Ai ) transition( N → R ) form factors which are Q dependent.Similarly the hadronic current for the spin resonant state is given by J µ = ¯ u ( p (cid:48) )Γ µ u ( p ) , (14)2where u ( p ) and ¯ u ( p (cid:48) ) are respectively, the Dirac spinor and adjoint Dirac spinor for spin particle and Γ µ is the vertex function which for the positive(+) and negative(-) paritystates are given byΓ + µ = V µ − A µ ; Γ − µ = (cid:104) V µ − A µ (cid:105) γ (15)where V µ represents the vector current and A µ represents the axial vector current. Thesecurrents are parameterized in terms of vector( F i ( Q )( i = 1 , g ( Q )and g ( Q )) form factors.Using the above prescription, the expression for the hadronic current is obtainedand W µνR in Eq.(10) is evaluated, which then is written in a form similar to Eq. 29i.e. in terms of W W IjR , j = 1 −
3. Finally W W IjR is related with the dimensionlessstructure functions F W IjR , following the same analogy as given in Eq. (30), and the crosssection(Eq.28) is evaluated.Besides the resonant terms, non-resonant terms also contribute to the scatteringcross section. They are better known as background terms and play important roleacross the neutrino energy spectrum. These non-resonant background terms havecontributions from the s-, t-, and u- channel Born terms, contact terms, meson inflight term, etc. There are various ways of including these terms like using non-linearsigma model, coupled channel approach, etc. For example, here we will briefly discussthe non-resonant background terms considered by [101] obtained using non-linear sigmamodel. In the case of pion production, the non-resonant background terms involve fivediagrams viz, direct nucleon pole (NP), cross nucleon pole (CNP), contact term (CT),pion pole (PP) and pion in flight (PF) terms (shown in Fig.3), which are calculatedusing a chiral symmetric Lagrangian, obtained in the non-linear sigma model.The contributions from the different non-resonant background terms to the hadroniccurrent are expressed as [16, 101, 102] j µ | NP = A NP ¯ u ( p (cid:48) ) (cid:54) k π γ (cid:54) p + (cid:54) q + M N ( p + q ) − M N + i(cid:15) [ V µN ( q ) − A µN ( q )] u ( p ) ,j µ | CNP = A CP ¯ u ( p (cid:48) ) [ V µN ( q ) − A µN ( q )] (cid:54) p (cid:48) − (cid:54) q + M N ( p (cid:48) − q ) − M N + i(cid:15) (cid:54) k π γ u ( p ) ,j µ | CT = A CT ¯ u ( p (cid:48) ) γ µ (cid:0) g f VCT ( Q ) γ − f ρ (cid:0) ( q − k π ) (cid:1)(cid:1) u ( p ) , (16) j µ | P P = A P P f ρ (cid:0) ( q − k π ) (cid:1) q µ M π + Q ¯ u ( p (cid:48) ) (cid:54) q u ( p ) ,j µ | P F = A P F f P F ( Q ) (2 k π − q ) µ ( k π − q ) − M π M N ¯ u ( p (cid:48) ) γ u ( p ) , where M π is the mass of pion and M N is the nucleon mass. The constant factor A i , i =NP, CNP, CT, PP and PF, are tabulated in Table–1. For details see Refs.[16], [102]-[104].The vector( V µN ( q )) and axial vector( A µN ( q )) currents for the NP and CNP diagrams,in the case of charged current interactions, are calculated neglecting the second class3 Figure 3.
Feynman diagrams contributing to the hadronic current correspondingto W i N → N (cid:48) π ± , , where ( W i ≡ W ± ; i = ± ) for charged current processes and( W i ≡ Z ; i = 0) for neutral current processes with N, N (cid:48) = p or n. First rowrepresents the direct and cross diagrams for the resonance production where R standsfor different resonances, second row represents the nucleon and cross nucleon termswhile the contact and pion pole terms are shown in the third row while the last rowrepresents the pion in flight term. The second, third and fourth rows represent non-resonant pion production. currents and are given by, V µN ( q ) = f V ( Q ) γ µ + f V ( Q ) iσ µν q ν M N (17) A µN ( q ) = (cid:18) g ( Q ) γ µ + g ( Q ) q µ M N (cid:19) γ , (18)where f V , ( Q ) and g , ( Q ) are the vector and axial vector form factors for the nucleons.The isovector form factors viz. f V , ( Q ) are expressed as: f V , ( Q ) = F p , ( Q ) − F n , ( Q ) , (19)where F p,n ( Q ) are the Dirac and F p,n ( Q ) are the Pauli form factors of nucleons. Theseform factors are, in turn, expressed in terms of the experimentally determined electric G p,nE ( Q ) and magnetic G p,nM ( Q ) Sachs form factors.On the other hand, the axial form factor( g ( Q )) is generally taken to be of dipoleform and is given by g ( Q ) = g (0) (cid:20) Q M A (cid:21) − , (20)4Constant term → A (CC ν ) A (CC ¯ ν )Final states → pπ + nπ + pπ nπ − nπ pπ − NP 0 − ig √ f π − ig f π ig f π − ig √ f π CP − ig √ f π ig f π − ig √ f π − ig f π − i √ f π i √ f π if π − i √ f π − if π i √ f π PP i √ f π − i √ f π − if π i √ f π if π − i √ f π PF − ig √ f π ig √ f π ig f π − ig √ f π − ig f π ig √ f π Table 1.
The values of constant term( A i ) appearing in Eq. 16, where i corresponds tothe nucleon pole(NP), cross nucleon pole(CP), contact term(CT), pion pole(PP) andpion in flight(PF) terms. f π is pion weak decay constant and g is nucleon axial vectorcoupling. where g (0) is the axial charge and is obtained from the quasielastic ν l and ¯ ν l scatteringas well as from the pion electro-production data. We have used g (0) = 1.267 and theaxial dipole mass M A =1.026 GeV, which is the world average value, in the numericalcalculations.The next contribution from the axial part comes from the pseudoscalar form factor g ( Q ), the determination of which is based on Partially Conserved Axial Current(PCAC) and pion pole dominance and is related to g ( Q ) through the relation g ( Q ) = 2 M N g ( Q ) M π + Q . (21)In order to conserve vector current at the weak vertex, the two form factors viz. f P F ( Q ) and f VCT ( Q ) are expressed in terms of the isovector nucleon form factor as [102] f P F ( Q ) = f VCT ( Q ) = 2 f V ( Q ) . (22)The ππN N vertex has the dominant ρ –meson cloud contribution and followingRef. [102], we have introduced ρ − form factor ( f ρ ( Q )) at ππN N vertex and is taken tobe of the monopole form: f ρ ( Q ) = 11 + Q /M ρ ; with M ρ = 0 . . (23)In order to be consistent with the assumption of PCAC, f ρ ( Q ) has also been used withaxial part of the contact term.5The net hadronic current is then written as the sum of non-resonant and resonantcontributions J µ = J NRµ + J Rµ e iφ , (24) φ is the phase factor which tells us how the resonant channels add to the non-resonantcontributions. Generally in numerical calculations φ is taken to be zero, that meansthese two are in the same phase and add up coherently, however, in general this maynot be necessarily true. J NRµ in Eq.24 gets the contribution from non-resonant diagrams shown in Fig. 3 as J NRµ = j µ | NP + j µ | CNP + j µ | CT + j µ | P P + j µ | P F , (25)given in Eq. (16). For all the numerical calculations [101] puts a constraint on W suchthat M N + M π ≤ W ≤ . GeV while evaluating J NRµ ( W ≤ . GeV ), which was dueto the chiral limit. Note this implies that the effect of the non-resonant contributionspresented below are those contributions limited to M N + M π ≤ W ≤ . GeV and donot include any of the additional non-resonant contributions between 1 . GeV ≤ W ≤ . GeV . The non-resonant contribution in this missing W region could be significantsince this contribution, along with any resonance plus interference contributions, mustgrow to transition into the total DIS inelastic cross section at W = 2 GeV. J Rµ has the contribution from spin and spin resonant states with positive ornegative parity i.e. J Rµ = J µ + J µ . For the numerical evaluations they ([101]) tookthe six low lying resonances contributing to one-pion production i.e. J µ R = J µP (1232) + J µP (1440) + J µS (1535) + J µS (1650) + J µD (1520) + J µP (1720) , (26)and the numerical results presented in [101] for the total cross sections are for the threedifferent cases, (i) with no cut on W , an upper limit of W as (ii) 1.4 GeV and (iii)1.6 GeV, while evaluating J Rµ .For example, the authors of [101] found that in the case of ν µ + p → µ − + p + π + induced reaction for a cut of M N + M π ≤ W ≤ . GeV on J NRµ and no cut of W onthe resonance i.e. J Rµ , when the hadronic currents are added coherently(i.e. φ =0), themain contribution to the total scattering cross section comes from P (1232) resonancebetter known as the ∆(1232) resonance and there is no contribution to the p + π + modefrom the higher resonances ( P (1440), D (1520), S (1535), S (1650) and P (1720)).It was also found in the case of p + π + production, that due to the presence of the non-resonant background terms i.e. J µ = J NRµ ( W ≤ . GeV ) + J ∆ µ , there is an increase inthe cross section when compared with the results obtained using ∆(1232) term only inthe hadronic current i.e. J µ = J ∆ µ . This increase is about 12% at E ν µ = 1 GeV whichbecomes 8% at E ν µ = 2 GeV .For ν µ + n → µ − + n + π + as well as ν µ + n → µ − + p + π processes, there arecontributions from the non-resonant background terms as well as other higher resonantterms, although ∆(1232) dominates. The net contribution to the total pion productiondue to the presence of the non-resonant background terms (i.e. J µ = J NRµ ( W ≤ . GeV ) + J ∆ µ ) in ν µ + n → µ − + n + π + reaction results in an increase in the cross section6of about 12% at E ν µ = 1 GeV which becomes 6% at E ν µ = 2 GeV . When other higherresonances are also taken into account i.e. J µ = J NRµ ( W ≤ . GeV ) + J Rµ , where R alsoincludes ∆ as defined in Eq.24, there is a further increase in the n + π + productioncross section by about 40% at E ν µ = 1 GeV which becomes 55% at E ν µ = 2 GeV .While in the case of ν µ + n → µ − + p + π due to the presence of the non-resonantbackground terms the total increase in the p + π production cross section is about26% at E ν µ = 1 GeV which becomes 18% at E ν µ = 2 GeV . Due to the presence ofother higher resonances there is a further increase of about 35% at E ν µ = 1 GeV whichbecomes 40% at E ν µ = 2 GeV .When a cut of W ≤ . GeV (case-ii) or W ≤ . GeV (case-iii) on the center ofmass energy is applied on J Rµ , then the over all cross section decreases. The effect of theseW cuts become apparent at higher neutrino energies where some energy dependence isobserved when also considering higher resonances. For example with n + π + productionat E ν µ = 2 GeV the increase in total cross section, compared to the non-resonant (W ≤ . GeV ) + ∆ cross section, is found to be ∼
55% for a cut W ≤ . GeV and ∼ W ≤ . GeV .It was observed that the inclusion of higher resonant terms lead to a significantincrease in the cross section for ν µ + n → µ − + n + π + as well as ν µ + n → µ − + p + π processes. Furthermore, it was also concluded that contribution from non-resonantbackground terms with W ≤ . GeV decreases with the increase in neutrino energy,while the total scattering cross section increases when other higher resonances wereincluded in their calculations, although the ∆(1232) still dominates. The net increaseincludes the contribution of the interference terms among the resonant and the non-resonant ( W ≤ . GeV ) contributions to the hadronic current.It must be pointed out that in electromagnetic interactions, phase dependence hasbeen studied by a few groups, whereas in the weak interactions there is hardly any suchstudy. ν l -Nucleon Scattering: Deep-Inelastic Scattering The basic process for charged current DIS is given by(Fig.4a) ν l / ¯ ν l ( k ) + N ( p ) → l − /l + ( k (cid:48) ) + X ( p (cid:48) ) , l = e, µ, (27)where a ν l / ¯ ν l interacts with a nucleon( N ), producing a charged lepton( l ) and jet ofhadrons( X ) in the final state. In the above expression k and k (cid:48) are the four momenta ofincoming ν l / ¯ ν l and outgoing lepton respectively; p is the four momentum of the targetnucleon and p (cid:48) is the four momentum of final hadronic state X . This process is mediatedby the exchange of virtual boson W ± having four momentum q (= k − k (cid:48) = p (cid:48) − p ). Thecross section for the inclusive scattering of a ν l / ¯ ν l from a nucleon target is proportionalto the leptonic tensor( L µν ) and the hadronic tensor( W µνN ), where the hadronic tensor isobtained by summing over all the final states (Fig.4b).The double differential scattering cross section evaluated for a nucleon target in its7 Figure 4. (a) Feynman diagrams for the ν l / ¯ ν l induced DIS process. (b) ν l (¯ ν l ) − N inclusive scattering where the summation sign represents the sum over all the hadronicstates such that the cross section( dσ ) for the deep inelastic scattering ∝ L µν W µνN . rest frame is expressed as: d σ W IN d Ω (cid:48) l dE (cid:48) l = G F (2 π ) | k (cid:48) || k | (cid:18) M W q − M W (cid:19) L µν W µνN , (28)where G F is the Fermi coupling constant, Ω (cid:48) l , E (cid:48) l refer to the outgoing lepton and − q = Q with Q ≥
0. The expression of leptonic tensor L µν is given in Eq.9 andthe most general form of the hadronic tensor W µνN in terms of structure functions whichdepend on the scalars q and p.q , is given by W µνN = (cid:18) q µ q ν q − g µν (cid:19) W W I N ( ν, Q ) + W W I N ( ν, Q ) M N (cid:18) p µ − p.qq q µ (cid:19) × (cid:18) p ν − p.qq q ν (cid:19) − i M N (cid:15) µνρσ p ρ q σ W W I N ( ν, Q ) + W W I N ( ν, Q ) M N q µ q ν + W W I N ( ν, Q ) M N ( p µ q ν + q µ p ν ) + iM N ( p µ q ν − q µ p ν ) W W I N ( ν, Q ) , (29)where W W IiN ( ν, Q ); ( i = 1 −
6) are the nucleon structure functions and ν (= k − k (cid:48) ) isthe energy transfer.In the limit m l →
0, the terms depending on W W I N ( ν, Q ), W W I N ( ν, Q ) and W W I N ( ν, Q ) in Eq. 29 do not contribute to the cross section and DIS processesare described by the three nucleon structure functions W W I N ( ν, Q ), W W I N ( ν, Q ) and W W I N ( ν, Q ). Note that when compared to the electromagnetic process there is anadditional structure function W W I N ( ν, Q ) due to parity violation in the case of weakinteractions. When Q and ν become large the structure functions W W IiN ( ν, Q ); ( i =1 −
3) are generally redefined in terms of the dimensionless nucleon structure functions F W IiN ( x ) as: M N W W I N ( ν, Q ) = F W I N ( x ) ,νW W I N ( ν, Q ) = F W I N ( x ) ,νW W I N ( ν, Q ) = F W I N ( x ) . (30)8 F W I ( x ) at the leading order(LO) for ν l and ¯ ν l induced processes on proton and neutrontargets are given by assuming that the CKM matrix is almost unitary in its 2 × F νp = 2 x [ d ( x ) + s ( x ) + ¯ u ( x ) + ¯ c ( x )] , (31 a ) F ¯ νp = 2 x (cid:2) u ( x ) + c ( x ) + ¯ d ( x ) + ¯ s ( x ) (cid:3) (31 b ) F νn = 2 x (cid:2) u ( x ) + s ( x ) + ¯ d ( x ) + ¯ c ( x ) (cid:3) (31 c ) F ¯ νn = 2 x [ d ( x ) + c ( x ) + ¯ u ( x ) + ¯ s ( x )] (31 d )So, for an isoscalar nucleon (N) target assuming s ( x ) = ¯ s ( x ) and c ( x ) = ¯ c ( x ), we maywrite F νN ( x ) = F ¯ νN ( x )= x (cid:2) u ( x ) + ¯ u ( x ) + d ( x ) + ¯ d ( x ) + s ( x ) + ¯ s ( x ) + c ( x ) + ¯ c ( x ) (cid:3) (32)The weak structure function F W I ( x ) at the leading order(LO) for ν l and ¯ ν l interactionson the proton and neutron targets are given by xF νp ( x ) = 2 x [ d ( x ) + s ( x ) − ¯ u ( x ) − ¯ c ( x )] , (33 a ) xF νn ( x ) = 2 x (cid:2) u ( x ) + s ( x ) − ¯ d ( x ) − ¯ c ( x ) (cid:3) , (33 b ) xF ¯ νp ( x ) = 2 x (cid:2) u ( x ) + c ( x ) − ¯ d ( x ) − ¯ s ( x ) (cid:3) , (33 c ) xF ¯ νn ( x ) = 2 x [ d ( x ) + c ( x ) − ¯ u ( x ) − ¯ s ( x )] (33 d )and for an isoscalar nucleon target, F ν/ ¯ νN ( x ) = F ν/ ¯ νp ( x ) + F ν/ ¯ νn ( x )2 (34)The parton distribution functions (PDFs) (defined in Eqs.32 and 34) for the nucleonhave been determined by various groups and they are known in the literature bythe acronyms MRST [105], GRV [106], GJR [107], MSTW [108], ABMP [109],ZEUS [110], HERAPDF [111], NNPDF [112], CTEQ [113], CTEQ-Jefferson Lab(CJ) [114], MMHT [115], etc. In the present work the numerical results are presentedusing CTEQ [113] and MMHT [115] nucleon parton distribution functions.The weak structure function can be compared directly with the electromagneticstructure function F EM ( x ) F EM ( x ) = F ep + F en x (cid:20)
518 ( u ( x ) + ¯ u ( x ) + d ( x )+ ¯ d ( x )) + 19 ( s ( x ) + ¯ s ( x )) + 49 ( c ( x ) + ¯ c ( x )) (cid:21) (35)for an isoscalar nucleon target, by defining the ratio of electromagnetic to weak structurefunctions F EM ( x ) F W I ( x ) = F eN ( x ) F ν/ ¯ νN ( x ) = R EM / WI ( x )= 518 (cid:20) − s ( x ) + ¯ s ( x ) − c ( x ) − ¯ c ( x ) (cid:80) ( q ( x ) + ¯ q ( x )) (cid:21) , (36)9and continue with the assumption s ( x ) = ¯ s ( x ) = c ( x ) = ¯ c ( x ), the above expressionreduces to F EM ( x ) = 518 F W I ( x ) (37)In the quark parton model (QPM), where transverse momentum of partons is consideredto be zero, the longitudinal structure function F L ( x ) is then also 0. In this case, F ( x )is often expressed in terms of F ( x ) using Callan-Gross relation, i.e. F ( x ) = 2 xF ( x ) (38)However, the modified QPM structure functions show a Q dependence and partonspossess a finite value of transverse momentum. Consequently, the longitudinal structurefunction has non-zero value leading to the violation of Callan-Gross relation which hasalso been discussed in the literature [116]-[120].The longitudinal structure function F W IL ( x, Q ) is defined as F W IL ( x, Q ) = (cid:18) M N x Q (cid:19) F W I ( x, Q ) − xF W I ( x, Q ) , (39)where F W I ( x, Q ) is purely transverse in nature while F W I ( x, Q ) is an admixtureof longitudinal and transverse components. The ratio of longitudinal to transversestructure function R W IL ( x, Q ) is given by R W IL ( x, Q ) = F W IL ( x, Q ) F W IT ( x, Q ) = F W IL ( x, Q )2 xF W I ( x, Q ) , = (cid:18) M N x Q (cid:19) F W I ( x, Q )2 xF W I ( x, Q ) − R W IL ( x, Q ) has been measured in the ν l / ¯ ν l scatteringby CCFR experiment [121] in iron as well as several charged-lepton scatteringexperiments [116, 122, 123] have also measured this ratio. In general it is expectedthat this ratio should be A dependent and this dependence will be discussed in the latersections.At low and moderate Q , structure functions show Q dependence, therefore theabove relation becomes: F EM ( x, Q ) = 518 F W I ( x, Q ) (41)Therefore, any deviation of R EM / WI ( x , Q ) = F EM2 ( x , Q ) F WI2 ( x , Q ) from and/or any dependenceon x , Q will give information about the strange and charm quarks distribution functionsin the nucleon.Now, we may write the differential scattering cross section (Eq.28) in terms ofthe dimensionless nucleon structure functions with respect to Bjorken scaling variable x (cid:16) = Q M N ν (cid:17) and the inelasticity y (cid:16) = νE ν = E ν − E l E ν (cid:17) as: d σ W IN dxdy = G F s π (cid:18) M W M W + Q (cid:19) (cid:2) xy F W I N ( x, Q )0+ (cid:18) − y − M N xy E (cid:19) F W I N ( x, Q ) ± xy (cid:16) − y (cid:17) F W I N ( x, Q ) (cid:105) , (42)where the upper/lower sign is for ν l / ¯ ν l and s = ( p + k ) is the center of mass energysquared.In the next subsection, the Q evolution of nucleon structure functions from leadingorder to higher order terms as well as the non-perturbative effects such as target masscorrection and higher twist effects important for low and moderate Q will be discussed. According to the naive parton model (NPM), inthe Bjorken limit structure functions depends only on x , i.e. F N ( x, Q ) −→ [ Q →∞ ,ν →∞ ] x → finite F N ( x ) F N ( x, Q ) −→ [ Q →∞ ,ν →∞ ] x → finite F N ( x )However, in QCD, partons present inside the nucleon may interact among themselvesvia gluon exchange. The incorporation of contribution from gluon emission cause the Q dependence of the nucleon structure functions, i.e. Bjorken scaling is violated. The Q evolution of structure functions is determined by the DGLAP evolution equation [124]which is given by ∂∂lnQ (cid:18) q i ( x, Q ) g ( x, Q ) (cid:19) = α s ( Q )2 π (cid:88) j (cid:90) x dzz (cid:32) P q i q j ( xy , α s ( Q )) P q i g ( xy , α s ( Q )) P gq j ( xy , α s ( Q )) P gg ( xy , α s ( Q )) (cid:33) × (cid:18) q j ( y, Q ) g ( y, Q ) (cid:19) , where α s ( Q ) is the strong coupling constant, q and g are the quark and gluondensity distribution functions, and P ( xy , α s ( Q )) are the splitting functions which areexpanded in power series of α s ( Q ). Now, one may express the nucleon structurefunctions in terms of the convolution of coefficient function ( C f ; ( f = q, g )) withthe density distribution of partons ( f ) inside the nucleon as x − F W Ii ( x ) = (cid:88) f = q,g C ( n ) f ( x ) ⊗ f ( x ) , (43)where i = 2 , , L , superscript n = 0 , , , ... for N ( n ) LO and symbol ⊗ is the Mellinconvolution. To obtain the convolution of coefficient functions with parton densitydistribution, we use the following expression [125] C f ( x ) ⊗ f ( x ) = (cid:90) x C f ( y ) f (cid:18) xy (cid:19) dyy (44)This Mellin convolution turns into simple multiplication in the N-space. The partoncoefficient function are generally expressed as C f ( x, Q ) = C (0) f (cid:124)(cid:123)(cid:122)(cid:125) LO + α s ( Q )2 π C (1) f (cid:124) (cid:123)(cid:122) (cid:125) NLO + (cid:18) α s ( Q )2 π (cid:19) C (2) f (cid:124) (cid:123)(cid:122) (cid:125) NNLO + ... (45)1In the limit of Q → ∞ , the strong coupling constant α s ( Q ) becomes very small andtherefore, the higher order terms such as next-to-leading order (NLO), next-to-next-to-leading order (NNLO), etc., can be neglected in comparison to the leading order(LO)term. But for a finite value of Q , α s ( Q ) is large and next-to-leading order terms givea significant contribution followed by next-to-next-to-leading order term. The details ofthe method to incorporate QCD evolution are given in Refs. [125]-[128]. To calculate thestructure functions, we use the NLO evolution of the parton distribution functions givenin terms of the power expansion in the strong coupling constant α s ( Q ). Following theworks of Vermaseren et al. [129] and van Neerven and Vogt [125], the QCD correctionsat NLO for the evaluation of F ( x ) structure function may be written as x − F ( x ) = C , ns ( x ) ⊗ q ns + (cid:104) e (cid:105) ( C , q ( x ) ⊗ q s + C , g ( x ) ⊗ g ) , (46)and for the evaluation of F ( x ) structure function, it may be written as F ( x ) = C ( x ) ⊗ q v ( x ) , (47)where q s , q ns and q v are respectively the flavor singlet, non-singlet and valence quarkdistributions, C ,q ( x ) and C ,ns ( x ) are singlet and non-singlet coefficient functions forthe quarks, C , g ( x ) is the coefficient function for the gluons and C ( x ) is the coefficientfunction for F ( x ). The coefficient functions are defined in Refs. [125, 129, 130]. (cid:104) e (cid:105) represents the average squared charge which is (cid:104) e (cid:105) = for four flavors of quarks inthe case of EM interaction and (cid:104) e (cid:105) = 1 for the weak interaction channel. The target mass correction (TMC) is a non-perturbative effect, which comes into the picture at lower Q . At finite value of Q ,the mass of the target nucleon and the quark masses modify the Bjorken variable x with the light cone momentum fraction. For the massless quarks, the parton light conemomentum fraction is given by the Nachtmann variable ξ which is related to the Bjorkenvariable x as ξ = 2 x (cid:113) M N x Q . (48)The Nachtmann variable ξ depends only on the hadronic mass and will not havecorrections due to the masses of final state quarks. However, for the massive partons,the Nachtmann variable ξ gets modified to ¯ ξ . These variables ξ and ¯ ξ are related to theBjorken variable as:¯ ξ = ξ (cid:18) m q Q (cid:19) (49)where m q is the quark mass. It is noticeable that the Nachtmann variable corrects theBjorken variable for the effects of hadronic mass while the generalized variable ¯ ξ furthercorrects ξ for the effects of the partonic masses [131].TMC effect is associated with the finite mass of the target nucleon M N and issignificant at low Q and high x ( x M N /Q is large) which is an important region todetermine the distribution of valence quarks. The TMC effect involving powers of 1 /Q F T MC N ( x, Q ) ≈ xξγ F N ( ξ ) (cid:0) r (1 − ξ ) (cid:1) ,F T MC N ( x, Q ) ≈ x ξ γ F N ( ξ ) (cid:0) r (1 − ξ ) (cid:1) ,F T MC N ( x, Q ) ≈ xξγ F N ( ξ ) (1 − r (1 − ξ ) lnξ ) . (50)In the above expressions r = µxξγ , µ = (cid:16) M N Q (cid:17) and γ = (cid:113) M N x Q , respectively. Similar to the TMC effect, there is another non-perturbative effect known as “higher twist(HT) effect” or “dynamical higher twisteffect”. This effect involves the interactions of struck quark with other quarks viathe exchange of gluons and it is suppressed by the power of (cid:16) Q (cid:17) n , where n = 1 , , .... .This effect is also pronounced in the region of low Q and high x like the TMC effectbut negligible for high Q and low x .For lower values of Q , a few GeV or less, non-perturbative phenomena couldbecome important for a precise modeling of cross sections. In the formalism of theoperator product expansion (OPE) [135, 136], unpolarized structure functions can beexpressed in terms of powers of 1 /Q (power corrections): F i ( x, Q ) = F τ =2 i ( x, Q ) + H τ =4 i ( x ) Q + H τ =6 i ( x ) Q + ..... i = 1 , , , (51)where the first term ( τ = 2) is known as the twist-two or leading twist (LT) term,and it corresponds to the scattering off a free quark. This term obeys the Altarelli-Parisi equations and is expressed in terms of PDFs. It is responsible for the evolutionof structure functions via perturbative QCD α s ( Q ) corrections. The HT terms with τ = 4 , qq and qg ), and the HTcorrections spoil the QCD factorization, so one has to consider their impact on the PDFsextracted in the analysis of low- Q data. The coefficients H i ( x ) can only be determinedin QCD as a result of the non-perturbative calculation. However, due to their non-perturbative origin, current models can only provide a qualitative description for suchcontributions. The coefficients H i ( x ) are usually determined via reasonable assumptionsfrom fits to the data [137, 138].Existing information about the dynamical HT terms in lepton-nucleon structurefunctions is scarce and somewhat controversial. Early analyses [139, 140] suggesteda significant HT contribution to the longitudinal structure function F L ( x ). Thesubsequent studies with both charged leptons [141]-[143] and neutrinos [144] raised the3question of a possible dependence on the order of QCD calculation used for the leadingtwist.A recent HT study [145] including both charged lepton and ν l / ¯ ν l DIS data suggestedthat dynamic HT corrections affect the region of Q <
10 GeV and are largelyindependent from the order of the QCD calculation. However, the verification of QHduality at JLab implies a suppression of additional HT terms with respect to the averageDIS behavior, down to low Q ∼ [146] with further details in section 4.2.Furthermore, our formalism suggests that as long as we demand Q ≥ GeV and W ≥ x is replaced by an adhoc scaling variable ξ w and all PDFsare modified by Q -dependent K factors. The free parameters in the ξ w variable and inthe K factors are fitted to existing data.It is worth noting that the transition from the high Q behavior of structurefunctions, well described in terms of perturbative QCD at leading twist, to theasymptotic limit for Q → ν l / ¯ ν l interactions are different with respect to charged leptons, due to the presence of anaxial-vector current dominating the cross sections at low Q and the structure functiondoes not go to zero as Q →
0. The effect of the PCAC [61, 148] in this transition regioncan be formally considered as an additional HT contribution and can be described withphenomenological form factors [63]. In the limit of Q → F T ∝ Q , while in the case of electromagnetic interaction F L ∝ Q and is dominated by the finite PCAC contribution in the weak current. As a result, theratio R L = F L /F T has a very different behavior in ν l / ¯ ν l scattering at small Q values [63]and this fact must be considered in the extraction of weak structure functions from themeasured differential cross-sections. ν l / ¯ ν l -Nucleus Scattering : Deep-Inelastic Scattering Theory After the EMC measurements in the early 1980s [35, 36] and observation made by themhenceforth named as the “EMC effect” that the ratio of F A AF D was not equal to 1.0 and was x dependent, several other experiments were performed by the different collaborationslike SLAC [149], HERMES [150], BCDMS [151, 152], NMC [153, 154], JLab [155], etc.using nuclear targets, both moderate and heavy, for a wide range of Bjorken variable x (0 < x <
1) and four momentum transfer square Q , and the following observationswere concluded from electroproduction experiments: • although the shape of the effect does not change with mass number A , the strengthof the nuclear medium effect increases with the increase in mass number A and • the functional form has very weak dependence on Q .4The results for the nuclear medium effects on mass dependence A were consistent with log ( A ) and average nuclear density [45]. To understand nuclear medium effects onthe structure functions, there are two broad approaches, one is the phenomenologicalapproach involving determination of the effective parton distribution of nucleons withina nucleus, and the other is theoretical approach where dynamics of the nucleons in thenuclear medium is taken into consideration. The phenomenological approach will bepresented in section 4.Theoretically many models have been proposed to study these effects on the basisof nuclear binding, nuclear medium modification including short range correlations innuclei [45]-[46], pion excess in nuclei [48, 50, 56],[78]-[80], multi-quark clusters [81]-[83],dynamical rescaling [84, 85], nuclear shadowing [86, 87], etc. In spite of these efforts, nocomprehensive theoretical/phenomenological understanding of the nuclear modificationsof the bound nucleon across the complete range of x and Q consistent with the presentlyavailable experimental data exists [51],[53]-[55]. In a recent phenomenological studyKalantarians et al. [156] have made a comparison of electromagnetic vs weak nuclearstructure functions ( F EM A ( x, Q ) vs F W I A ( x, Q )) and found out that at low x these twostructure functions are different. Theoretically, there have been very few calculationsto study nuclear medium effects in the weak structure functions and moreover, thereexists limited literature where explicitly a comparative study has been made[74, 77].Therefore, it is highly desirable to make a detailed theoretical as well as experimentalstudies of nuclear medium effects on the weak structure functions and compare theresults with the EM structure functions for a wide range of x and Q for moderate aswell as heavy nuclear targets.For the evaluation of weak nuclear structure functions not much theoretical effortshave been made except that of Kulagin et al. [63] and Athar et al.(Aligarh-Valenciagroup) [67]-[77]. Aligarh-Valencia group [67]-[77] has studied nuclear medium effects inthe structure functions in a microscopic model which uses relativistic nucleon spectralfunction to describe target nucleon momentum distribution incorporating the effects ofFermi motion, binding energy and nucleon correlations in a field theoretical model. Thespectral function that describes the energy and momentum distribution of the nucleonsin nuclei is obtained by using the Lehmann’s representation for the relativistic nucleonpropagator and nuclear many body theory is used to calculate it for an interactingFermi sea in the nuclear matter [157]. A local density approximation is then applied totranslate these results to a finite nucleus. Furthermore, the contributions of the pion andrho meson clouds in a many body field theoretical approach have also been consideredwhich is based on Refs. [56, 158]. In the next subsection, the theoretical approach ofAligarh-Valencia group is discussed.5 It starts with the differential scattering cross section for the charged current inclusive ν l / ¯ ν l -nucleus deep inelastic scattering process ν l / ¯ ν l ( k ) + A ( p A ) → l − /l + ( k (cid:48) ) + X ( p (cid:48) A ) , (52)written in analogy with the charged current ν l (¯ ν l ) − N scattering discussed in section-2,by replacing the hadronic tensor for the nucleon i.e. W µνN in Eq.28 with the nuclearhadronic tensor W µνA : d σ W IA d Ω (cid:48) l dE (cid:48) = G F (2 π ) | k (cid:48) || k | (cid:18) M W M W + Q (cid:19) L W Iµν W µνA . (53)and W µνA is written in terms of the weak nuclear structure functions W W IiA ( ν, Q )( i = 1 , ,
3) as W µνA = (cid:18) q µ q ν q − g µν (cid:19) W W I A ( ν, Q ) + W W I A ( ν, Q ) M A (cid:18) p µA − p A .qq q µ (cid:19) × (cid:18) p νA − p A .qq q ν (cid:19) ± i M A (cid:15) µνρσ p Aρ q σ W W I A ( ν, Q ) , (54)where M A is the mass and p A is the four momentum of the nuclear target and thepositive/negative sign is for the ν l / ¯ ν l . The leptonic tensor in Eq.53 has the same formas given in Eq.9. In the present work, the scattering process has been considered in thelaboratory frame, where target nucleus is at rest( p A = ( p A = M A , p A = 0)). Therefore,one may define p µ A = ( M A , (cid:126) ,x A = Q p A · q = Q p A q = Q A M N q (55)However, the nucleons bound inside the nucleus are not stationary but they arecontinuously moving with finite momentum, i.e. p = ( p , p (cid:54) = 0) and their motioncorresponds to the Fermi motion. These nucleons are thus off shell. If we take themomentum transfer of the bound nucleon along the z -axis such that q µ = ( q , , , q z )then Bjorken variable x N is given by x N = Q p · q = Q p q − p z q z ) (56)These bound nucleons may also interact among themselves via strong interaction andthus various nuclear medium effects are introduced which play important roles in thedifferent regions of the Bjorken variable x . In the following subsections (3.1.1 and 3.1.2),these various nuclear medium effects like Fermi motion, binding, nucleon correlations,isoscalarity correction and meson cloud contribution taken by Aligarh-Valencia groupare discussed in brief.6 To calculate thescattering cross section for a neutrino interacting with a target nucleon in the nuclearmedium to give rise to the process ν l + N → l − + X , we start off with a flux of neutrinoshitting a collection of target nucleons over a given length of time. Now a majority willsimply pass through the target without interacting while a certain fraction will interactwith the target nucleons leaving the pass-through fraction and entering the fractionof neutrinos yielding final state leptons and hadrons. Here we introduce the conceptof ”neutrino self energy” that has a real and imaginary part. The real part modifiesthe lepton mass(it is similar to the delta mass or nucleon mass modified in the nuclearmedium) while the imaginary part is related to this fraction of interacting neutrinos andgives the total number of neutrinos that have participated in the interactions that giverise to the charged leptons and hadrons. The basic ingredients of the model are givenin Appendix A-D.The neutrino self energy (Appendix A) is evaluated corresponding to the diagramshown in Fig.5 (left panel), and the cross section for an element of volume dV in the restframe of the nucleus is related to the probability per unit time (Γ) of the ν l interactingwith a nucleon bound inside a nucleus. Γ dtdS provides probability times a differentialof area ( dS ) which is nothing but the cross section ( dσ ) [56], i.e. dσ = Γ dtds = Γ dtdl dsdl = Γ 1 v dV = Γ E l | k | d r, (57)where v (cid:16) = | k | E l (cid:17) is the velocity of the incoming ν l . The probability per unit time ofthe interaction of ν l with the nucleons in the nuclear medium to give the final state isrelated to the imaginary part of the ν l self energy as [56]: − Γ2 = m ν E ν ( k ) Im Σ( k ) , (58)where Σ( k ) is the neutrino self energy (shown in Fig.5 (left panel)). By using Eq.58 inEq.57, we obtain dσ = − m ν | k | Im Σ( k ) d r (59)Thus to get dσ , we are required to evaluate the imaginary part of neutrino self energy Im Σ( k ) which is obtained by following the Feynman rules: Im Σ( k ) = G F √ m ν (cid:90) d k (cid:48) (2 π ) πE ( k (cid:48) ) θ ( q ) (cid:18) M W Q + M W (cid:19) Im [ L W Iµν Π µν ( q )](60)In the above expression, Π µν ( q ) is the W boson self-energy, which is written in terms ofthe nucleon ( G l ) and meson ( D j ) propagators (depicted in Fig. 5 (right panel)) followingthe Feynman rules and is given byΠ µν ( q ) = (cid:18) G F M W √ (cid:19) × (cid:90) d p (2 π ) G ( p ) (cid:88) X (cid:88) s p ,s l N (cid:89) i =1 (cid:90) d p (cid:48) i (2 π ) (cid:89) l G l ( p (cid:48) l ) × (cid:89) j D j ( p (cid:48) j ) < X | J µ | N >< X | J ν | N > ∗ (2 π ) Figure 5.
Diagrammatic representation of the neutrino self-energy (left panel) andintermediate vector boson W self-energy (right panel). × δ ( k + p − k (cid:48) − N (cid:88) i =1 p (cid:48) i ) , (61)where s p is the spin of the nucleon, s l is the spin of the fermions in X , < X | J µ | N > isthe hadronic current for the initial state nucleon to the final state hadrons, index l, j are respectively, stands for the fermions and for the bosons in the final hadronic state X ,and δ ( k + p − k (cid:48) − (cid:80) Ni =1 p (cid:48) i ) ensures the conservation of four momentum at the vertex.The nucleon propagator G ( p ) inside the nuclear medium provides information about thepropagation of the nucleon from the initial state to the final state or vice versa.The relativistic nucleon propagator G ( p , p ) in a nuclear medium is obtained bystarting with the relativistic free nucleon Dirac propagator G ( p , p ) which is writtenin terms of the contribution from the positive and negative energy components of thenucleon described by the Dirac spinors u ( p ) and v ( p ) [56, 157]. Only the positive energycontributions are retained as the negative energy contributions are suppressed. In theinteracting Fermi sea, the relativistic nucleon propagator is then written in terms of thenucleon self energy Σ N ( p , p ) which is shown in Fig.6. In nuclear many body technique,the quantity that contains all the information on single nucleon properties is the nucleonself energy Σ N ( p , p ). For an interacting Fermi sea the relativistic nucleon propagatoris written in terms of the nucleon self energy and in nuclear matter the interactionis taken into account through Dyson series expansion. Dyson series expansion maybe understood as the quantum field theoretical analogue of the Lippmann-Schwingerequation for the dressed nucleons, which is in principle an infinite series in perturbationtheory. This perturbative expansion is summed in a ladder approximation as G ( p ) = M N E ( p ) (cid:80) r u r ( p )¯ u r ( p ) p − E ( p ) + M N E ( p ) (cid:80) r u r ( p )¯ u r ( p ) p − E ( p ) Σ N ( p , p ) × M N E ( p ) (cid:80) s u s ( p )¯ u s ( p ) p − E ( P ) + ..... = M N E ( p ) (cid:80) r u r ( p )¯ u r ( p ) p − E ( p ) − (cid:80) r ¯ u r ( p )Σ N ( p , p ) u r ( p ) M N E ( p ) (62)The nucleon self energy Σ N ( p , p ) is spin diagonal, i.e., Σ Nαβ ( p , p ) = Σ N ( p , p ) δ αβ , where8 Figure 6.
Diagrammatic representation of neutrino self energy in the nuclear medium. α and β are spinorial indices. The nucleon self energy Σ N ( p ) is obtained following thetechniques of standard many body theory and is taken from Ref. [157, 159] which usesthe nucleon-nucleon scattering cross section and the spin-isospin effective interactionwith random phase approximation(RPA) correlation as inputs. In this approach thereal part of the self energy of nucleon is obtained by means of dispersion relations usingthe expressions for the imaginary part which has been explicitly calculated. The Fockterm, which does not have imaginary part, does not contribute either to Im Σ N ( p , p )or to Re Σ N ( p , p ) through the dispersion relation and its contribution to Σ N ( p , p ) isexplicitly calculated and added to Re Σ N ( p , p ) [157]. The model however misses somecontributions from similar terms of Hartree type which are independent of nucleonmomentum p . This semi-phenomenological model of nucleon self energy is found to bein reasonable agreement with those obtained in sophisticated many body calculationsand has been successfully used in the past to study nuclear medium effects in manyprocesses induced by photons, pions and leptons [160, 161]. The expression for thenucleon self energy in the nuclear matter i.e. Σ N ( p , p ) is taken from Ref. [157], andthe dressed nucleon propagator is expressed as G ( p ) = M N E ( p ) (cid:88) r u r ( p )¯ u r ( p ) (cid:20)(cid:90) µ −∞ dω S h ( ω, p ) p − ω − iη + (cid:90) ∞ µ dω S p ( ω, p ) p − ω + iη (cid:21) , (63) where S h ( ω, p ) and S p ( ω, p ) are the hole and particle spectral functions, respectively. µ = (cid:15) F + M N is the chemical potential, ω = p − M N is the removal energy and η is theinfinitesimal small quantity, i.e. η →
0. The spectral function and its properties havebeen discussed in brief in Appendix-B and Appendix-C, respectively.The cross section (Appendix-D) is then obtained by using Eqs. 59 and 60 : dσ W IA d Ω (cid:48) l dE (cid:48) l = − G F (2 π ) | k (cid:48) || k | (cid:18) M W Q + M W (cid:19) (cid:90) Im ( L µν Π µν ) d r, (64)Now by comparing the above equation with Eqs.53, 61 and 63 the expression ofthe nuclear hadronic tensor for an isospin symmetric nucleus in terms of the nucleonichadronic tensor and spectral function, is obtained as [72] W µνA = 4 (cid:90) d r (cid:90) d p (2 π ) M N E ( p ) (cid:90) µ −∞ dp S h ( p , p , ρ ( r )) W µνN ( p, q ) , (65)where the factor of 4 is for the spin-isospin of nucleon and ρ ( r ) is the charge density ofthe nucleon in the nucleus. In general, nuclear density have various phenomenologicalparameterizations known in the literature as the harmonic oscillator(HO) density,9two parameter Fermi density(2pF), modified harmonic oscillator (MHO) density, etc.The proton density distributions are obtained from the electron-nucleus scatteringexperiments, while the neutron densities are taken from the Hartree-Fock approach [162].Thus the density parameters corresponds to the charge density for proton or equivalentlythe neutron matter density for neutron. Recently at the JLab, PREX and CREXcollaborations [163]-[165] have made efforts to directly measure the neutral weak formfactor of a few nuclei from which the neutron rms radii of nuclei can be obtained. Furtherdevelopment in this area would be of great help to determine precisely the neutron formfactor in nuclei for a broad mass range.For a nonisoscalar nuclear target, the nuclear hadronic tensor is given by W µνA = 2 (cid:88) τ = p,n (cid:90) d r (cid:90) d p (2 π ) M N E ( p ) (cid:90) µ τ −∞ dp S τh ( p , p , ρ τ ( r )) W µνN ( p, q ) , (66)where µ p ( µ n ) is the chemical potential for the proton(neutron). S ph ( ω, p , ρ p ( r )) and S nh ( ω, p , ρ n ( r )) are the hole spectral functions for the proton and neutron, respectively,which provide information about the probability distribution of finding a proton andneutron with removal energy ω and three momentum p inside the nucleus.Now to evaluate the weak dimensionless nuclear structure functions by usingEq.(65), the appropriate components of nucleonic ( W µνN in Eq.29) and nuclear ( W µνA inEq.54) hadronic tensors along the x, y and z axes are chosen. The dimensionless nuclearstructure functions F W IiA ( x, Q )( i = 1 , , W W IiA ( ν, Q ) aredefined as F W I A ( x A , Q ) = M A W W I A ( ν A , Q ) ,F W I A ( x A , Q ) = ν A W W I A ( ν A , Q ) ,F W I A ( x A , Q ) = ν A W W I A ( ν A , Q ) , (67)where the energy transfer ν A = p A · qM A = q .By taking the zz component of the hadronic tensors( W µνN of Eq.29 and W µνA ofEq.54), for a nonisoscalar nuclear target the following expression is obtained [69]: F W I A,N ( x A , Q ) = 2 (cid:88) τ = p,n (cid:90) d r (cid:90) d p (2 π ) M N E N ( p ) (cid:90) µ τ −∞ dp S τh ( p , p , ρ τ ( r )) × (cid:34)(cid:18) Qq z (cid:19) (cid:18) | p | − ( p z ) M N (cid:19) + ( p − p z γ ) M N × (cid:18) p z Q ( p − p z γ ) q q z + 1 (cid:19) (cid:35) (cid:18) M N p − p z γ (cid:19) F W I τ ( x N , Q ) . (68)The choice of xx components of the nucleonic(Eq. 29) and nuclear(Eq. 54) hadronictensors lead to the expression of F W I A,N ( x, Q ) as F W I A,N ( x A , Q ) = 2 (cid:88) τ = p,n AM N (cid:90) d r (cid:90) d p (2 π ) M N E N ( p ) (cid:90) µ τ −∞ dp S τh ( p , p , ρ τ ( r ))0 Figure 7.
Neutrino self energy diagram accounting for neutrino-meson DIS (a) thebound nucleon propagator is substituted with a meson( π or ρ ) propagator (b) byincluding particle-hole (1 p h ), delta-hole (1∆1 h ), 1 p h − h , etc. interactions. × (cid:34) F W I τ ( x N , Q ) M N + (cid:18) p x M N (cid:19) F W I τ ( x N , Q ) ν (cid:35) (69)in the case of nonisoscalar nuclear target.Now by using the xy components of the nucleonic(Eq. 29) and nuclear(Eq. 54)hadronic tensors in Eq. 65, the parity violating nuclear structure function is obtainedas F W I A,N ( x A , Q ) = 2 A (cid:88) τ = p,n (cid:90) d r (cid:90) d p (2 π ) M N E N ( p ) (cid:90) µ τ −∞ dp S τh ( p , p , ρ τ ( r )) × q q z (cid:18) p q z − p z q p · q (cid:19) F W I τ ( x N , Q ) , (70)for a nonisoscalar nuclear target.For an isoscalar target, the factor of 2 in Eqs. 68, 69 and 70, will be replacedby 4 and the contribution will come from the nucleon’s hole spectral function S h ( p , p , ρ ( r )) instead of the individual contribution from proton and neutron targetsin S τh ( p , p , ρ τ ( r )); ( τ = p, n ).The results obtained by using Eqs. 68, 69, and 70 for a nuclear target are labeledas the results with the spectral function(SF) only. Furthermore, the nucleons boundinside the nucleus may interact among themselves via meson exchange such as π, ρ, etc. The interaction of intermediate vector boson with these mesons play an importantrole in the evaluation of nuclear structure functions. Therefore, the mesonic effecthas been incorporated in the Aligarh-Valencia model and is discussed in the next sub-subsection 3.1.2. There are virtual mesons (mainly pion and rho meson) associatedwith each nucleon bound inside the nucleus. This mesonic cloud gets strengthenedby the strong attractive nature of the nucleon-nucleon interaction, which leads to areasonably good probability of interaction of virtual bosons(IVB) with a meson insteadof a nucleon [56, 59, 78, 166]. Although the contribution from the pion cloud is largerthan the contribution from rho-meson cloud, nevertheless, the rho contribution is non-negligible, and both of them are positive in all the range of x . The mesonic contribution1is smaller in lighter nuclei, while it becomes more pronounced in heavier nuclear targetsand dominates in the intermediate region of x (0 . < x < . X but corresponds to themesons arising due to the nuclear medium effects by using a modified meson propagator.These mesons are arising in the nuclear medium through particle-hole (1 p h ), delta-hole(1∆1 h ), 1 p h − h , 2 p − h , etc. interactions as depicted in Fig.7.To evaluate the mesonic structure function F W I A,a ( x, Q ) ( a = π, ρ ) the imaginarypart of the meson propagator is used instead of spectral function, and the expressionfor F W I A,a ( x, Q ) , ( a = π, ρ ) obtained by them [69] is given by: F W I A,a ( x, Q ) = − κ (cid:90) d r (cid:90) d p (2 π ) θ ( p ) δImD a ( p ) 2 M a (cid:18) M a p − p z γ (cid:19) × (cid:20) Q ( q z ) (cid:18) | p | − ( p z ) M a (cid:19) + ( p − p z γ ) M a × (cid:18) p z Q ( p − p z γ ) q q z + 1 (cid:19) (cid:35) F W I a ( x a ) (71)where κ = 1 for pion and κ = 2 for rho meson, x a = − Q p · q , M a is the mass of pion orrho meson. D a ( p ) is the pion or rho meson propagator in the nuclear medium given by D a ( p ) = [ p − p − M a − Π a ( p , p )] − , (72)with Π a ( p , p ) = f M π C ρ F a ( p ) p Π ∗ − f M π V (cid:48) j Π ∗ . (73)In the above expression, C ρ = 1 for pion and C ρ = 3 .
94 for rho meson. F a ( p ) = (Λ a − M a )(Λ a − p ) is the πN N or ρN N form factor, p = p − p , Λ a =1 GeV and f = 1 .
01. For pion (rhomeson), V (cid:48) j is the longitudinal (transverse) part of the spin-isospin interaction and Π ∗ is the irreducible meson self energy that contains the contribution of particle-hole anddelta-hole excitations. Various quark and antiquark PDFs parameterizations for pionsare available in the literature such as given by Conway et al. [167], Martin et al. [105],Sutton et al. [168], Wijesooriya et al. [169], Gluck et al.[170], etc. Aligarh-Valenciagroup have observed [76] that the choice of pionic PDFs parameterization would notmake any significant difference in the event rates. In this work, the parameterization2given by Gluck et al.[170] has been taken into account for pions and for the rho mesonssame PDFs as for the pions have been used.The choice of Λ a = 1 GeV, ( a = π, ρ ) have been fixed by Aligarh-Valenciagroup [68, 72] to describe the nuclear medium effects in electromagnetic nuclear structurefunction F EM A ( x, Q ) necessary to explain the data from JLab and other experimentsperformed using charged lepton scattering from several nuclear targets in the DIS region. Aligarh-Valencia group has taken theshadowing effect into account by following the works of Kulagin and Petti [59, 63]who have used the original Glauber-Gribov multiple scattering theory. In the caseof ν l / ¯ ν l induced DIS processes, they have treated shadowing differently from theprescription applied in the case of electromagnetic structure functions [59, 63], due tothe presence of the axial-vector current in the ν l interactions. The interference betweenthe vector and the axial-vector currents introduces C-odd terms in ν l cross sections,which are described by structure function F W I ( x, Q ), and in their calculation of nuclearcorrections, separate contributions to different structure functions according to their C-parity have been taken into account. This results in a different dependence of nucleareffects on C-parity specially in the nuclear shadowing region. The same prescription hasbeen adopted by the Aligarh-Valencia group. The Aligarh-Valencia group points outthat the inclusion of shadowing effect in the present model is not very comprehensiveand more work is required. A review on the nuclear shadowing in electroweak interactionhas been done in [171]. In the case of heavier nuclear targets, whereneutron number( N = A − Z ) is larger than the proton number( Z ) and theirdensities are also different, isoscalarity corrections become important. As most ofthe neutrino/antineutrino experiments are using heavy nuclear targets( N (cid:54) = Z ),phenomenologically the isoscalarity correction is taken into account by multiplying theexperimental results with a correction factor defined as R IsoA = [ F ν/ ¯ νp + F ν/ ¯ νn ] / ZF ν/ ¯ νp + ( A − Z ) F ν/ ¯ νn ] /A , (74)where F ν/ ¯ νn are the weak structure functions for the proton and the neutron,respectively. Aligarh-Valencia group have applied their model to study the effects of the nuclearmedium on the electromagnetic structure functions [68, 72, 76] as well as the weakstructure functions [67, 69, 70, 77] and have made a comparison between weakand electromagnetic nuclear structure functions for a wide range of x and Q [75].Furthermore an important effect, the isoscalarity correction for the nonisoscalar nuclear3targets has been studied by them (as discussed in 3.1.1 and 3.1.2). They haveapplied their model to study medium effects in extracting sin θ W using the Paschos-Wolfenstein relation [71]. This model has been applied successfully to study the Drell-Yan processes [74] and parity violating asymmetry with nuclear medium effects usingpolarized electron beam( (cid:126)e ) [73].This model describes the nuclear structure functions F W IiA,N ( x A , Q ) ( i = 1 − F W IiN ( x N , Q ),convoluted with the spectral function which takes into account Fermi motion, bindingenergy and nucleon correlation effects followed by the mesonic and shadowing effects.For the evaluation of F W IiN ( x N , Q ) at the leading order(LO), free nucleon PDFs are used.Therefore, their numerical results do not use nuclear PDFs. The results presented in thisreview are obtained using nucleon PDFs of MMHT [115] as well as CTEQ6.6 in the MS-bar scheme [113]. F W IiA,π ( x, Q ) and F W IiA,ρ ( x, Q ) are the structure functions giving pionand rho mesons contribution. In the literature, various pionic PDFs parameterizationsare available and this work uses the pionic PDFs parameterization of Gluck et al. [170]as in Fig. 8. Also for the comparison pion PDFs of Wijesooriya et al. [169] havebeen used. To evaluate the nucleon structure functions in the kinematic region oflow and moderate Q , where the higher order perturbative corrections and the non-perturbative effects become important, PDFs evolution up to NNLO has been performedand included the effects of TMC and higher twist in the numerical calculations. Forthe evolution of nucleon PDFs at the next-to-leading order(NLO) and next-to-next-to-leading order(NNLO) the works of Vermaseren et al. [126] and Moch et al. [130, 172]have been followed. The target mass correction effect has been included following themethod of Schienbein et al. [131]. The dynamical higher twist correction has been takeninto account following the methods of Dasgupta et al. [173] and Stein et al. [174] atNLO.The theoretical results obtained in the Aligarh-Valencia model [67]-[77] arepresented and compared with the experimental data wherever available. The first caseis when the calculations are performed using the spectral function (SF) only and thenthe contribution from meson clouds as well as shadowing effect are taken into accountand this corresponds to the full model (Total) results as quoted by the authors [67]-[77].The expression of total nuclear structure functions with the full theoretical model isgiven by F W IiA ( x, Q ) = F W IiA,N ( x, Q ) + F W IiA,π ( x, Q ) + F W IiA,ρ ( x, Q ) + F W IiA,shd ( x, Q ) , (75)where i = 1 − F W IiA,N ( x, Q ) are the nuclear structure function which has contributionfrom only the spectral function, F W IiA,π/ρ ( x, Q ) take into account mesonic contributions. F W IiA,shd ( x, Q ) has contribution from the shadowing effect which is given by F W IiA,shd ( x, Q ) = δR i ( x, Q ) × F W Ii,N ( x, Q ) , (76)where δR i ( x, Q ) is the shadowing correction factor for which Kulagin and Petti [59] hasbeen followed. In this model, the full expression for the parity violating weak nuclear4 Figure 8. F EM A ( x, Q ) vs x at different values of Q , in C, Al, Fe and Cuwith the full model at NLO and NNLO using MMHT nucleon PDFs [115]. Theresults at NNLO are shown by solid line and at NLO with the HT effect (renormalonapproach [173, 174]) are shown by the dashed-double dotted line using the pionicPDFs parameterization given by Wijesooriya et al. [169] and by the dotted line forthe parameterization of Gluck et al. [170]. The results are also obtained by using thenuclear PDFs parameterization given by nCTEQ group [29] (double-dashed dottedline) and the experimental points are the JLab data [175]. structure function is given by, F W I A ( x, Q ) = F W I A,N ( x, Q ) + F W I A,shd ( x, Q ) . (77)Notice that this structure function has no mesonic contribution and mainly thecontribution to the nucleon structure function comes from the valence quarksdistributions. For F W I A,shd ( x, Q ) similar definition has been used as given in Eq.(76)following the works of Kulagin et al. [59].First the results for the nuclear structure function( F EM A ( x, Q )) in the case ofelectromagnetic interaction have been presented in Fig.8, for the different nuclear targetslike C , Al , F e and Cu [76] at moderate values of Q (1 . ≤ Q ≤ . )and compared with the available experimental results of the JLab [175]. The nucleartargets are treated as isoscalar. For the evaluation of free nucleon structure functions,MMHT [115] parameterization has been used. The numerical results are shown for thefull model using a similar expression as Eq.75 for the electromagnetic nuclear structurefunctions F EM A ( x, Q ) with nucleon structure functions F EM N ( x N , Q ): • at NNLO with mesonic PDFs of Gluck et al. [170] • at NLO with HT effect and mesonic PDFs of Gluck et al. [170] • at NLO with HT effect and mesonic PDFs of Wijesooriya et al. [169]5 Figure 9.
Results are shown for the weak nuclear structure function F W I A ( x, Q )vs x at Q = 2 , GeV , in C, Fe and
Pb for (i) only the spectral function(dashed line), (ii) only the mesonic contribution (dash-dotted line) using Eq.71, (iii) the full calculation (solid line) using Eq.75 as well as (iv) the double-dash-dottedline is the result without the shadowing and antishadowing effects. The numericalcalculations have been performed at NNLO by using the MMHT [115] nucleon PDFsparameterizations.
It may be noticed from Fig.8 that the dependence of different pionic PDFsparameterizations have not much effect on the evaluation of F EM A ( x, Q ). Also theresults obtained show that as long as TMC is applied, NNLO is within a fewpercent of the results obtained at NLO with HT effect. Further details of thisinteresting observation can be found in [176]. In literature, along with the free nucleonPDFs parameterizations, different nuclear PDFs are also available like AT12 [177],nCTEQ15 [29], EPPS16 [43], etc. Also, for the comparison, in this figure, the resultsobtained using nuclear PDFs of nCTEQ group [29] has been shown. It may benoticed that the theoretical results obtained using the full model are reasonably ingood agreement with the nCTEQ results [29] and show a good agreement with theJLab experimental data [175] in the region of intermediate x . However, for x > . Q ≈ GeV they slightly underestimate the experimental results. Since the region ofhigh x and low Q is the transition region of nucleon resonances and DIS, the presenttheoretical results might indeed differ from the experimental data. With the increase in Q , theoretical results show a better agreement with the experimental observations ofJLab [175] in the entire range of x .Turning now to the weak interactions, in Fig. 9, the results are presented for F W I A ( x, Q ) vs x for C , F e and
P b , for isoscalar nuclear targets, at the differentvalues of Q chosen to reflect the current neutrino beam energies. The numerical resultsare obtained first by using the spectral function (dashed line), then we have includedmesonic effect(dash-dotted line) and the final result by including the shadowing and6 Figure 10.
Results are shown for F W I A ( x, Q ) vs x at a fixed Q = 5 GeV , in C, Fe and
Pb for only the spectral function (dashed line) and for the fullcalculation (dotted line) at NLO with HT effect (renormalon approach [173, 174])using MMHT [115] nucleon PDFs parameterizations. Solid line is the result of thefull calculation at NNLO using MMHT PDFs parameterizations [115]. Notice that thecurves for NLO+HT and NNLO are almost the same implying equivalence of the two(NLO+HT and NNLO) for all x . The results at NLO obtained using only the spectralfunction with HT effect are also compared with the corresponding results obtainedusing the CTEQ6.6 [113] nucleon PDFs parameterization in the MS-bar scheme. Allthe nuclear targets are treated as isoscalar. antishadowing effects is shown by the solid line. From the figure, it may be observed thatthe mesonic contributions result in an enhancement in the nuclear structure functionsand is significant in the low and intermediate region of x . Moreover, the effect is morepronounced at low Q and becomes larger with the increase in mass number A . Forexample, in comparison to the total contributions (solid line) in carbon, the mesoniccontribution at x = 0 . x (say x = 0 .
4) the enhancement reduces to 13% and 18% respectivelyand becomes almost negligible for x ≥ . Q = 2 GeV . To depict the coherentnuclear effects(shadowing) which results in suppression of the structure functions at low x , the results without shadowing are shown with the double-dash-dotted line, and itmay be observed that with the increase in mass number of the nuclear target( F e vs P b ), the strength of suppression becomes larger. In Fig.10, we present the results for F W I A ( x, Q ) in three different nuclear targets viz. , F e and
P b . These results areobtained with the spectral function(SF) as well as for the full model using the nucleonPDFs evaluated at NLO with higher twists(HT). To study the dependence of nuclearstructure functions on the nucleon PDFs parameterization the numerical calculations7have been performed by using the MMHT [115] as well as CTEQ6.6 [113] nucleon PDFsparameterizations in the MS-bar scheme. From the figure, it may be observed thatthere is hardly any dependence of F W I A ( x, Q ) on the different choice of nucleon PDFparameterizations. When the results using the full prescription vs spectral function(with MMHT PDFs at NLO including the HT effect) are compared, we find the effectof mesonic contributions are quite significant in the region of present kinematic interest,which increases with the increase in the mass number. Also to observe the effect of PDFsevolution of the nucleon, on the nuclear structure functions the results are presented atNNLO using the full model and compared these results with results obtained at NLOwith HT effect. It may be observed that the results of NLO+HT is the same( < x . For thedetailed discussion, please see the Refs. [76, 77].To study the effect of isoscalarity correction in nonisoscalar nuclear targetslike P b , the Aligarh-Valencia model performs numerical calculations independentlyfor isoscalar nuclear targets by normalizing the spectral function to the number ofnucleons( A ) using the nucleon density parameters and getting the correct bindingenergy (very close to the experimental values) of the nucleons in the nucleus which hasbeen discussed in section-3. Similarly for the nonisoscalar nuclear targets, the spectralfunction is normalized to the proton number( Z ) using the proton density parameters,and the neutron numbers ( A − Z ) using the neutron density parameters. Fig.11 showsthe isoscalarity vs nonisoscalarity effect, where the results are presented at Q = 5 GeV for F W I A ( x, Q ) in Fe and
Pb. In the inset of these figures, the isoscalarity effecthas been explicitly shown by plotting the ratio F Iso A ( x,Q ) F NonIso A ( x,Q ) vs x for the full theoreticalmodel which deviates from unity in the entire range of x . This correction is x as wellas nuclear mass A dependent, and becomes more pronounced with the increase in x aswell as with the increase in the nuclear mass number A .In Fig. 12, the variation of nuclear medium effects in the electromagnetic and weakinteractions has been shown by using different nuclear targets. It should be noticed fromthe figure that the ratio R (cid:48) deviates from unity in the region of low x even for the freenucleon case which implies the non-zero contribution from strange and charm quarksdistributions. However, for x ≥ .
4, where the contribution of strange and charm quarksare almost negligible, the ratio approaches towards unity. Furthermore, if one assumes s = ¯ s and c = ¯ c then in the region of small x , this ratio would be unity for an isoscalarnucleon target following the (cid:0) (cid:1) th -sum rule. One may also observe that for heaviernuclear targets like F e and
P b , this deviation becomes more pronounced. Thisshows that the difference in charm and strange quark distributions could be significantin heavy nuclei. One may also notice that the isoscalarity corrections are different in F EM A ( x, Q ) than in F EM A ( x, Q ) although the difference is small.In Fig.13, the model dependence of the spectral function has been studied by usingthe different spectral functions [56, 63, 157] available in the literature. From the figure,it may be observed that the difference in the results obtained in the low x and low Q region vs Bjorken limit, is within 1% of each other. The results obtained by using the8 Figure 11. F W I A ( x, Q ) vs x at fixed Q = 5 GeV , in Fe and
Pb with the fullcalculation at NLO treating Fe and
Pb as isoscalar (solid line) and nonisoscalar(dotted line) targets. These calculations are performed using CTEQ6.6 [113] nucleonPDFs in the MS-bar scheme. Inset in the figures depicts the ratio of nuclear structurefunctions F Iso A F NonIso A treating Fe and
Pb as isoscalar(Iso) and nonisoscalar(NonIso)nuclear targets using the full model.
Figure 12.
Results for the ratio R (cid:48) = F WIiA ( x,Q ) F EMiA ( x,Q ) ; ( i = 1 ,
2) are obtained with thefull model at NLO in A = C, Fe and
Pb at Q = 5 and 20 GeV by using theCTEQ6.6 nucleon PDFs in the MS-bar scheme [113]. The left figures are for F ( x, Q )and the right are for F ( x, Q ). The numerical results are obtained assuming F e and
P b to be nonisoscalar target nuclei and are compared with the results obtained forthe isoscalar free nucleon target. Figure 13.
Results for F W I A ( x, Q ) ( A = C, Fe and
Pb) vs x are shownusing models given by Marco et al. [56](dotted line) in the Bjorken limit, Kulagin etal. [63](dashed line) and the present model (dashed-double dotted line) in the non-Bjorken limit to observe the model dependence of the spectral function at Q =5 GeV . Numerical results are evaluated at NLO by using CTEQ6.6 [113] nucleonPDFs in the MS-bar scheme. All the nuclear targets are treated to be isoscalar here. Figure 14.
Results are presented for F W I A ( x, Q ) and xF W I A ( x, Q ) vs Q in Feusing the full model at different values of x . The results are obtained by usingCTEQ6.6 nucleon PDFs at NLO in the MS-bar scheme (dotted line), MMHT atNLO(dashed line) and NNLO(solid line). The experimental points are the data fromCDHSW [93] (solid diamond), CCFR [94] (empty triangle) and NuTeV [96] (starsymbol) experiments. In the present case iron is treated as isoscalar nuclear target. Figure 15.
Results of the differential scattering cross section d σdxdy vs y , at different x for ν µ induced reaction on iron target at E ν µ = 65 GeV are shown. The results areobtained by using (i)
CTEQ6.6 [113] nucleon PDFs at NLO in the MS-bar scheme(dotted line), (ii)
MMHT nucleon PDFs [115] at NLO without (dashed line) andwith the HT effect (renormalon approach: dashed-dotted line) as well as at NNLO(solid line). The experimental points are the data from CDHSW [93] and NuTeV [96]experiments. Here iron is treated as isoscalar target. spectral function of Kulagin et al. [63] show small difference even at low x and low Q for the nuclei under consideration and the difference gradually becomes smaller with theincrease in x and Q . Hence, it may be concluded that the nuclear structure functionsshow very little dependence on the choice of spectral function.
For the ν l / ¯ ν l scattering cross sections and structure functions high statisticsmeasurements have been made by CCFR [94], CDHSW [93], and NuTeV [96]experiments by using iron as nuclear target. Experimentally, the extraction of structurefunctions are done by using the differential scattering cross sections measurements.These experiments have been performed in a wide range of ν l / ¯ ν l energies 20 ≤ E ν ≤ GeV . Using Aligarh-Valencia formalism, Haider et al. [69] have studiednuclear modifications for the ν l / ¯ ν l induced processes on iron target and compared theirresults with the available experimental data [93, 94, 96]. The theoretical results inFig.14, obtained by using the full model are compared with the available experimentaldata [93, 94, 96]. These results differ from the experimental data in the region of low x and low Q , however, with the increase in x and Q they are found to be in reasonablygood agreement. It is important to point out that the additional uncertainty(due tonormalization) of ± .
1% has not been included in the NuTeV analysis [96] and theexperimental results also differ among themselves.Moreover, the results obtained by using the CTEQ6.6 [113] (in the MS-barscheme) and MMHT [115] PDFs parameterizations are consistent. The numericalresults evaluated at NNLO using MMHT nucleon PDFs parameterization [115] for1 F W I A ( x, Q ) show a reasonably good agreement with the results evaluated at NLO whilethe results for xF W I A ( x, Q ) differ in the region of low x and low Q . For example, at x = 0 . . ≈ Q = 1 . GeV and 3%(1%)for Q = 5 . GeV . The detailed discussion of the nuclear medium effects for a widerange of x is available in Ref. [69].In Fig.15, the results are presented for d σdxdy vs y at E ν µ = 65 GeV and keeping Q > GeV with the full model. The numerical calculations have been performedby using the CTEQ6.6 [113] nucleon PDFs in the MS-bar scheme at NLO as wellas MMHT [115] nucleon PDFs at NLO without and with the HT effect (renormalonapproach [173, 174]), and at NNLO. One may notice that in the present kinematicalregion, the numerical results obtained for all the cases shown in the figure seems tobe in agreement within a percent. This was expected because the perturbative QCDcorrections have inverse power dependence on Q . The experimental data show a goodagreement with the numerical results except in the region of low x ( ≤ . y ( ≤ . d σ A dxdy ) vs y for different values of x for the incoming beam of energy E = 35 GeVfor ν and ν scattering on Fe and Pb nuclear targets and compare them with theexperimental results from NuTeV in Fe and CHORUS in Pb. These theoretical and thephenomenological results are also compared with the experimental results from NuTeVand CDHSW in Fe at E ν,ν =65 GeV and CHORUS in Pb at E ν,ν =55 GeV.In section-6.2, we have given predictions of the theoretical results using Aligarh-Valencia model and the phenomenological results of nCTEQnu for the differential crosssection vs y in Fe, Pb and Ar at E ν,ν =6.25 GeV as well as at E ν,ν =2.25 GeV in Ar. Thesepredictions may be useful in the analysis of MINERvA experiment being performed usingFe and Pb nuclear targets as well as the proposed DUNE experiment using liquid argonTPC.2 ν l / ¯ ν l -Nucleus Scattering: Shallow Inelastic Scattering Phenomenology Above neutrino quasi-elastic (QE) scattering in effective hadronic mass (W) comesthe resonance region (RES) that starts with the ∆ resonance followed by increasinglyhigher mass resonant states. These resonances sit atop a continuum of non-resonant π production that starts at W = M + m π . This resonant plus non-resonant π productionregion transitions into the deep-inelastic scattering (DIS) region, where interactionsoccur on quarks, at a border kinematically defined for most experiments as W ≥ Q ≥ GeV . The non-resonant pion production under all resonancesis the very intriguing kinematic region referred to technically as the shallow-inelasticscattering (SIS) region. However, since it is not possible to experimentally distinguishresonant from non-resonant pion production or the interference between them, for thisreview SIS is practically defined as the sum of pion production processes contributingto inclusive scattering with W ≤ π production; the non-resonant contribution is much smaller than the∆ and is well controlled by chiral perturbation theory; and the only decay channelthat must be considered is π N. For the region beyond the ∆(1232) up to W (cid:47) π N and ππ N arecomparable and strongly coupled as well as higher mass meson-baryon channels.In addition to the individual pion production model approach that, eventually,must cover both resonance and non-resonance single- and multiple-pion channels, thereis the intriguing alternative treatment of the SIS region by the GiBUU group basedon nuclear transport theory [178]. While the current MC simulation programs treatthe initial interaction and the subsequent final state interaction of produced hadronsindependently, the GiBUU framework attempts to model the full space-time evolution ofparticles from the initial through final state interactions and emphasize that the initialand final state interactions should not be treated independently. Using this frameworkthey have predicted both resonant and non-resonant pion production within the SISregion [179]. The results of the GiBUU model will be presented in the discussion ofduality.An initial anomaly to note is that in some current Monte Carlo (MC) eventsimulators/generators “DIS” is defined as “anything but QE and RES”, instead of theusually expressed kinematic condition on the effective hadronic mass such as
W > Q > . Notice moreover that RES in these simulators is limited to 1 π π production. This MC ”DISdefinition then includes a contribution from the kinematical region Q < , whichis certainly outside of the applicability of the genuine DIS formalism and consequentlyperturbative QCD. Thus the MC definition of DIS contains also part of what we defineas the SIS region. For this review ”DIS” refers to the original kinematical definition ofDIS.This higher-W SIS region between the ∆ resonance and DIS has been quiteintensively studied experimentally in electron/muon-nucleon (e/ µ -N) interactions andsomewhat less thoroughly in e/ µ -nucleus (e/ µ -A) scattering. The studies of e/ µ -Ninteractions in this kinematic region have been used to test the hypothesis of quark-hadron duality (hereafter ”duality”). Duality, as we shall see, relates the averageof inclusive production cross sections in this SIS region to extrapolated results fromthe better known DIS region. To further define the concept of duality, consider thatperturbative QCD is well defined and calculable in terms of asymptotically free quarksand gluons, yet the process of confinement ensures that it is hadrons, pions and protons,that are observed. One speaks the language of quarks/gluons in the DIS region and, asW decreases, transitions to speak the language of hadrons in the SIS region that includesboth resonant and non-resonant pion production. Duality can then be considered as aconceptual experimental bridge between free and confined partons. It is important tonote that the understanding of this SIS region is important for long-baseline oscillationexperiments. As has been mentioned, in the future DUNE experiment [180], more than50% of the interactions will be in these SIS and DIS regions with W above the mass ofthe ∆ resonance. Historically in the 1960s the concept of what was to become ”duality” began with thetotal pion-proton cross sections being compared with Regge fits to higher energy data.It was concluded that low-energy hadronic cross sections on average could be describedby the high-energy behavior. In the 1970’s Poggio, Quinn and Weinberg [181] suggestedthat higher energy inclusive hadronic cross sections, appropriately averaged over anenergy range, should approximately coincide with the cross sections calculated usingquark-gluon perturbation theory. This directly implied that the physics of quarks andgluons could describe the physics of hadrons.Finally, also in the 1970’s, Bloom and Gilman [182] defined duality by comparingthe structure functions obtained from inclusive electron-nucleon DIS scattering withresonance production in similar experiments and the observation that the average overresonances is approximately equal to the leading twist (see 2.3.2) contribution measuredin the DIS region. This seems to be valid in each resonance region individually as wellas in the entire resonance region when the structure functions are summed over higherresonances. That is the DIS scaling curve extrapolated down into the resonance region4passes through the average of the ”peaks and valleys” of the resonant structure. Inthis picture, the resonances can then be considered as a continuing part of the behaviorobserved in DIS. This would suggest there is a connection between the behavior ofresonances and QCD, perhaps even a common origin in terms of a point-like structure forboth resonance and DIS interactions. Along this line it has been conjectured that theremay exist two component duality where the resonance contribution and backgroundcontribution to the structure functions in the resonance excitation region correspondsrespectively to the valence quarks, and the sea quarks contribution in structure functionsin the DIS region [31]. However, these observations are to be verified by modelcalculations as well as by the experimental data when they become available with higherprecision. Currently, the observation of duality in charged-lepton scattering has thefollowing main features [183]: • the resonance region data oscillate around the scaling DIS curve • the resonance data are on an average equivalent to the DIS curve • the resonance region data moves towards the DIS curve with the increase in Q .As more data with better precision become available on inclusive lepton scatteringfrom nucleons and nuclei a verification of QH duality with sufficient accuracy will providea way to describe lepton-nucleon and lepton-nucleus scattering over the entire SIS region.Significantly, if duality does hold for neutrino nucleon interactions, it would be possibleto extrapolate the better-known neutrino DIS structure into the SIS region and givean indication of how well current event simulators are modeling the SIS region. If theapplication of duality to our event generators can help us with this understanding itshould be explored. Duality and Charged-lepton Scattering
By the early 2000’s there was considerableaccumulation of charged lepton DIS studies at multiple laboratories with nucleonstructure functions well measured over a broad range in x, Q , (x Bjorken ≡ x ). Manyexperimental tests had supported the success of QCD and a new examination of dualitywith Jefferson Lab resonant production experiments was begun. An early JeffersonLab measurement (E94-110) [184] showed that global duality was clearly observed for Q ≥ . GeV , as can be seen in Fig. 16, with resonances following the extrapolatedDIS curve.The experimental and theoretical study of duality proceeded relatively smoothlyfor e-N and even for e-A interactions and there are now visual suggestions that dualityholds for F n,p,N , F p , F pL , and F D,C,F e,Au .However, with the much more accurate Jefferson Lab data, it was thought thatthere should be an improved method to test duality precisely. A possible solution isto quantify the degree to which duality is satisfied by defining the ratio of integrals ofstructure functions, over the same ξ interval, from the resonance (RES) region and DISregion. To keep the same ξ interval in the higher W DIS region compared to the lowerW RES region requires a different Q for the RES and DIS regions, thus the indexing of5 Duality in the F Structure Function § Empirically, DIS region is where logarithmic scaling is observed: Q > 5 GeV , W > 4 GeV § Duality:Averaged over W, log scalingobserved to work also for Q > 0.5 GeV , W < 4 GeV § JLab results (E94110):Works quantitatively to better than 10%
Figure 16.
Figure from [146]. Comparison of F p from the series of resonancesmeasured by E94-110 vs the Nachtmann variable ξ (see below) at the indicated Q compared to the extrapolated DIS measurement from the NMC collaboration at 5 GeV Q in the ratios. This method tests local duality within the integrals limits. For perfectlocal quark-hadron duality the value of the ratio would be 1.0. I | ( Q RES , Q DIS ) = (cid:82) ξ max ξ min dξF RESj ( ξ, Q RES ) (cid:82) ξ max ξ min dξF DISj ( ξ, Q DIS ) (78) ξ ( x, Q ) = x √ x M N /Q The integrals use the Nachtmann variable: ξ ( x, Q ) to account for target masseffects (TMC,section 2.3.2) and the integration over the resonance region is definedas typically W min = M N + M π and W max = 2 . Q yields ξ min and ξ max . F RESj is defined theoretically in section 2.1 and experimentally it isdetermined from the total inelastic cross section in the SIS region. Fig. 17 demonstratesthe relationship between ξ, Q and W corresponding to the SIS and DIS regions. Asan example, note that the ξ range of the open red triangles at 3 GeV in Fig. 16 coverthe range 0 . ≤ ξ ≤ .
75 that can now be directly related to a corresponding W range(1 . (cid:47) W ( GeV ) (cid:47) . GeV curve in this Fig. 17.Using this new measure of agreement with quark-hadron duality for eN scattering aGiessen-Ghent collaboration [183] used the GiBUU model [178] that had been shown toreproduce the full range of Jlab e-nucleon resonance results covering the SIS kinematic6 ξ W [GeV] Q2=0.4Q2=1Q2=3Q2=10
FIG. 3: Dependence of the Nachtmann variable ξ on hadronic invariant mass calculated at Q = 0.4, 1, 3 and 10 GeV . The leptonic current is defined as: J µlepton = ¯ u ( k ′ ) γ µ (1 − γ ) u ( k ) . (2)In the RS model the leptonic mass is set to be zero. In this limit q µ J µlepton = 0 . (3)One can introduce the basis of three vectors of length ± q µ : e µL = 1 √ , , − i, ,e µR = 1 √ , − , − i, ,e µS = 1 ! Q ( q, , , ν ) . Correspondingly, the leptonic tensor can be decomposed as: L µν = k µ k ′ ν + k ′ µ k ν − g µν k · k ′ − iε µνκλ k κ k ′ λ = (4)= " α,β ∈ ( S,L,R ) M αβ e µα ( e νβ ) ∗ . (5)When we calculate the contraction of the leptonic tensor with the hadronic tensor W µν = − g µν W + p µ p ν M W − i ϵ µναβ p α q β M W $ , (6)( M is the nucleon mass) we find that L µν W µν = L µνdiag W µν , (7)where L µνdiag = A e µS ( e νS ) ∗ + B e µL ( e νL ) ∗ + C e µR ( e νR ) ∗ . (8) A , B , C are Lorentz scalars which can be evaluated in the LAB frame: A = L µν e µS ( e νS ) ∗ = Q q % (2 E − ν ) − q & , (9) B = L µν e µL ( e νL ) ∗ = Q q (2 E − ν + q ) , (10) C = L µν e µR ( e νR ) ∗ = Q q (2 E − ν − q ) . (11) SIS DIS
Soft DIS
SIS
Figure 17.
Dependence of ξ on W for specific values of Q . The interplay of thesethree variables with the kinematic regions of SIS, DIS and soft DIS are shown .region. They found that, significantly, one m ust include the non-resonant as well asthe resonant contributions to the integral over the SIS region to improve the agreementwith quark-hadron duality as shown in Fig. 18. AQ'= V -vv. \ =0.225 °'S . A 0.525
1 1 >/,'; A1-025 i A J 2.025 - FIGURE 1. l2 Duality for the isoscalar nucleon Fj "^^ structure function calculated within GiBUU model. (Left)
F2^ as a function of ^, for
Q =
Q^ = 10
GeV . (Right) Ratio if^ of the integrated
F2^ in the resonance region to the leading twist functions. ^ correspond to the second (1.40 GeV < W < 1.56 GeV) and the third (1.56 GeV < W < 2.0 GeV) resonance regions. The general picture shows a reasonable agreement with the duality hypothesis. In the right panel of Fig. 1, the ratio of the integrals if^, defined in (3), is shown not only for the whole structure function (resonance + 1-pion background), but also for the resonance contribution separately. For Q^ > if for the resonance contribution only is at the level of 0.85, which is smaller and flatter in Q^ in comparison with the results [6, 15] of the Dortmund group resonance model. The difference is due to the different parameterization of the electromagnetic resonance form factors used in the two models. The background gives a noticeable contribution and brings the ratio up to 0.95. The fact, that it is smaller than 1 is of no surprise, because additional nonresonant contributions like 2- and many-pion background are possible, but not taken into account here. They are the subject of coming investigations. The principal feature of neutrino reactions, stemming from fundamental isospin arguments, is that duality does not hold for proton and neutron targets separately. The interplay between the resonances of different isospins allows for duality to hold with reasonable accuracy for the average over the proton and neutron targets. We expect a similar picture emerges in neutrino reactions with nuclei. For neutrinoproduction, the structure function F2^ and the ratio / j ' ^ are shown in Fig. 2 for the resonance contribution only. The ratio is at the level of 0.7, which is (similar to the electron case) smaller than 0.8, which has been calculated within the Dortmund resonance model [6, 15]. Thus, one would expect a large contribution from the background. The role of the background in neutrino channel is under investigation now. / FIGURE 2. l2 Duality for the isoscalar nucleon Fj '^^ structure function calculated within the GiBUU model. (Left) Fj^'^ as a function of ^, for
Q =
Q^ = 10
GeV . (Right) Ratio / j ' ^ of the integrated Fj^'^ in the resonance region to the leading twist functions.
Figure 18.
Figure from [183] illustrating duality for the isoscalar nucleon structurefunction calculated within the GiBUU electroproduction model. (Left) F eN as afunction of ξ , for values of Q indicated on the spectra, compared with the DIS QCD-fit results for F eN over the same ξ range but at Q = 10 GeV . (Right) Ratio I eN of the integrated F in the resonance region to the integral over this DIS QCD fit tohigh Q data. The Q along the abscissa is the Q involved in computing the limits ξ min = ξ ( W , Q ) and ξ max = ξ ( W , Q ) of the integration of the numerator of I eN . When now considering nuclear as opposed to nucleon targets in e/ µ scattering,the results of duality studies are not as straightforward. When using nuclei, theFermi motion of the bound nucleons within the nucleus serves to average (smear) theproduction of resonances over ξ so that visual evaluation of duality should be moreobvious in nuclei. This concept is supported in Fig. 19 that shows how the resonancesthat are clearly visible for e+p interactions are somewhat less defined in e+D interactionsand essentially smoothed out completely for e+Fe interactions (where the curve hasbeen modified for the EMC effect). The curves for each are the MRST and NMC7fits from the DIS region and, indeed, visual agreement with duality is apparent. Thephenomenon of duality has now been observed in multiple experiments on e-N and e-Ascattering [45, 146, 175, 184],[185]-[198]. x values, it is important tobe able to constrain these effects in a region where other,more exotic, explanations are not expected to contribute.It should be possible to learn more about the EMC effectat large x by taking advantage of the extended scaling ofstructure functions in nuclei [6, 7]. In this paper, we at-tempt to quantify the deviations from perturbative scal-ing at large x , with the goal of improving measurementsof the structure functions and the EMC ratios at large x . II. SCALING OF THE NUCLEAR STRUCTUREFUNCTION
Inspired by a recent series of electron scattering exper-iments in Hall C at Jefferson Lab, we revisit the issuesof scaling in nuclear structure functions and the EMCeffect. The Hall C data are at lower invariant mass W , W = M p + 2 M p ν (1 − x ), and therefore higher x ,than data thus far used to investigate the EMC effect.Most notably, these new data are in the resonance re-gion, W < . In the DIS region, W > ,the Q dependence of the structure functions is predictedby perturbative QCD (pQCD), while additional scalingviolations, target mass corrections and higher twist ef-fects, occur at lower Q and W values. Thus, data inthe resonance region would not naively be expected tomanifest the same EMC effect as data in the deep inelas-tic scaling regime. The effect of the nuclear medium onresonance excitations seems non-trivial, and may involvemuch more than just the modification of quark distribu-tions observed in DIS scattering from nuclei.However, while resonance production may show differ-ent effects from the nuclear environment, there are alsoindications that there is a deeper connection betweeninclusive scattering in the resonance region and in theDIS limit. This connection has been a subject of inter-est for nearly three decades since quark-hadron dualityideas, which successfully described hadron-hadron scat-tering, were first extended to electroproduction. In thelatter, Bloom and Gilman [8] showed that it was possibleto equate the proton resonance region structure function F ( ν, Q ) at low Q to the DIS structure function F ( x )in the high- Q scaling regime, where F is simply theincoherent sum of the quark distribution functions. Forelectron-proton scattering, the resonance structure func-tions have been demonstrated to be equivalent on averageto the DIS scaling strength for all of the spin averagedstructure functions ( F , F , F L ) [9, 10], and for somespin dependent ones ( A ) [11] (for a review of dualitymeasurements, see [12]).The goal of this paper is to quantify quark-hadron du-ality in nuclear structure functions and to determine towhat extent this can be utilized to access poorly under-stood kinematic regimes. While the measurements of du-ality from hydrogen indicate that the resonance structurefunction are on average equivalent to the DIS structure FIG. 1: (Color online) The F structure function per nu-cleon vs ξ for hydrogen (top), deuterium (middle), andiron(bottom). For the hydrogen and deuterium data (0.8 < Q < ), the elastic (quasielastic) data have beenremoved. For the iron data ( Q < ), a cut of W > . is applied to remove the quasielastic peak.The curves are the MRST [13] (solid) and NMC [14] (dashed)parameterizations of the structure functions at Q = 4 GeV ,with a parameterization of the EMC effect [15] applied toproduce the curve for iron. functions, it has been observed that in nuclei, this aver-aging is performed by the Fermi motion of the nucleons,and so the resonance region structure functions yield theDIS limit without any additional averaging [6, 7].Figure 1 shows the structure functions for hydrogen [9],deuterium [16], and iron [7], compared to structure func-tions from MRST [13] and NMC [14] parameterizations.Each set of symbols represents data in a different Q range, with the highest Q curves covering the highest ξ values. Note that the data are plotted as a function ofthe Nachtmann variable, ξ = 2 x/ (1 + ! M x /Q ),rather than x . In the limit of large Q , ξ → x , and so ξ can also be used to represent the quark momentum inthe Bjorken limit. At finite Q , the use of ξ reduces scal-ing violations related to target mass corrections [17]. Thedifference between ξ and x is often ignored in high energyscattering or at low x , but cannot be ignored at large x orlow Q . The goal is to examine ξ -scaling to look for anysignificant scaling violations beyond the known effects ofperturbative evolution and target mass corrections. Ex-amining the scaling in terms of ξ instead of x is only Figure 19.
Figure from [188]. The F structure function per nucleon as a functionof ξ for (top to bottom) ep, eD and eFe. For the H and D data the quasielastic datahas been removed while for the Fe data a cut of W ≥ . GeV has been applied toremove the quasielastic peak. The curves are the MRST and NMC DIS QCD fits withnuclear effect for e Fe applied In contrast to these last comparisons that used data and seem to beclearly consistent with duality, Fig. 20 (left) displays the result of usingtheoretical/phenomenological models, namely the Giessen (GiBUU) model for theresonance plus non-resonance contributions to the F structure functions for a carbontarget. The model predictions for resonance production at several Q values arecompared to experimental data obtained by the BCDMS collaboration [199] in muon-carbon scattering in the DIS region ( Q ≈ − GeV ) that are shown as experimentalpoints connected by smooth curves. Due to the Fermi motion of the target nucleons,the peaks from the various resonance regions, which were clearly seen for the nucleontarget, are hardly distinguishable for the carbon nucleus. The same effect was clearlydemonstrated in Fig. 19.The results for the ratio I eC are shown in Fig. 20 (right). The curve for the isoscalarfree-nucleon case, without including the non-resonant background, is the same as inFig. 18. One can see that the carbon curve obtained by integrating ”from threshold”that takes into account Fermi motion of the nucleons within the carbon nucleus, liesabove the one obtained by integrating ”from 1.1 GeV”. Recall that the flatter the curveis and the closer it gets to one, the higher the accuracy of local duality would be. The8 NUCLEI
Recent electron scattering measurements at JLab have confirmed the validity of the Bloom-Gilman duality for proton, deuterium [2] and iron [3] structure functions. Further experimental efforts are required for neutrino scattering. Among the upcoming neutrino experiments, Minerva[16, 17,18] and SciBooNE[19,20, 21] aim at measurements with carbon, iron and lead nuclei as targets. One of the major issues for nuclear targets is the definition of the nuclear structure functions
FA2 da/dQ^dv is calculated within the GiBUU model. The nucleon is bound in a mean field potential, which is parameterized as a sum of a Skyrme term term depending only on density and a momentum-dependent contribution of Yukawa-type interaction. Eermi motion of the bound nucleon and Pauli blocking are also considered (see [13] for details). Previous work [22] has used the analytical formulas for the nucleon structure functions, presented in [6], and directly apply nuclear effects to them. Nuclear effects are treated within the independent particle shell model, so that each bound nucleon in a nucleus occupies a nuclear shell a with a characteristic binding energy € „ and is described by the bound-state spinor ««. The four-momentum of the bound nucleon can be written as p^ = {mj^ — ea,p), thus the nucleon is off its mass shell. Both the bound-state spinor Ua{p) and the corresponding binding energies are computed in the Hartree approximation to the cr — ft) Walecka-Serot model. As shown in [22], this leads to the following definition of the nuclear structure functions ^2{Q\V)=J^ d'p{2ja+l)na{pW2{Q\v,p' \P\' -PIQ' ^l Pz qz (p • q) (4) In Eig. 3, the results of Ghent and Giessen models for the resonance contribution to the
F2 /A structure functions for a carbon target are shown for several Q^ values. They are compared to experimental data obtained by the BCDMS collaboration [23, 24] in muon-carbon scattering in the DIS region {Q^ - 30 - 50 GeV2). They are shown as experimental points connected by smooth curves. Eor different Q^ values, the experimental curves agree within 5% in most of the B, region, as expected from Bjorken scaling. When investigating duality for a free nucleon, we took the average over free proton and neutron targets, thus considering the isoscalar structure function. Since the carbon nucleus contains an equal number of protons and neutrons, averaging over isospin is performed automatically. Due to the Eermi motion of the target nucleons, the peaks from the various resonance regions, which were clearly seen for the nucleon target, are hardly distinguishable for the carbon nucleus. In general, the curves of the Giessen model are above those of the Gent model, especially (as it would be natural to expect) in the second and the third resonance regions. (0 o I Res: model, different Q DIS: BCDMS collab 30 GeV 0.3 o (0 o o < DIS: BCDSM coll 30 GeV" 50 GeV^ 45 0 0 ^ --•- 0.8
FIGURE 3. (Color online) Resonance curves F | ^/12 as a function of ^, for
Q^ =
As expected from local duality, the resonance structure functions for the various g^ values slide along a curve, whose B, dependence is very similar to the scaling-limit DIS curve. However, for all B, , the resonance curves lie below the experimental DIS data. To quantify this underestimation, we now consider the ratio of the integrals of the resonance (res) and DIS structure functions, determined in Eq. (3) For electron-carbon scattering we choose the data set [24] at 2D/5 = 50 GeV^, because it covers most of the B, region. For nuclear structure functions, as it is explained in [22], the integration limits are to be determined in terms of the effective W variable, experimentally (see, for example, [25]) defined as W^ = m^ + Inif^v — Q^.
For a free nucleon W coincides with the invariant mass W. For a nucleus, it differs from W due to the Fermi motion of bound nucleons, but still gives a reasonable estimation for the invariant mass region involved in the problem. In particular, the resonance curves presented in all figures are plotted in the region from the pion-production threshold up to W = 2 GeV. For a free nucleon, the threshold value for 1-pion production (and thus the threshold value of the resonance region) is Wmin = ^min « 1 • 1 GeV. Bound backward-moving nucleons in a nucleus allow lower W values beyond the free-nucleon limits. The threshold for the structure functions is now defined in terms of v or W, rather than W. Hence, we consider two different cases in choosing the B, integration limits for the ratio (3). First, for a given Q^, we choose the B, limits in the same manner as for a free nucleon: ^min = ^ ( W = 1 . 6 G e V , e 2 ^max = ^ ( W = l . l G e V , e 2 (5) We refer to this choice as integrating "from 1.1 GeV". The integration limits for the DIS curve always correspond to this choice. As a second choice, for each Q^ we integrate the resonance curve from the threshold, that is from as low W as achievable for the nucleus under consideration. This corresponds to the threshold value at higher B, and is referred to as integrating "from threshold". With this choice we guarantee that the extended kinematical regions typical for resonance production from nuclei are taken into account. Since there is no natural threshold for the B,mm, for both choices it is determined from
W = 1.6
GeV, as defined in Eq. (5). The results for the ratio (3) are shown in Fig. 4. The curve for the isoscalar free-nucleon case is the same as in Ref. [6] with the "GRV" parameterization for the DIS structure function. One can see that the carbon curve obtained by integrating "from threshold" lies above the one obtained by integrating "from 1.1 GeV", the difference increasing with
Q^.
This indicates that the threshold region becomes more and more significant, as one can see from Fig. 4. Recall, that the flatter the curve is and the closer it gets to 1, the higher the accuracy of local duality would be. Our calculations for carbon show that in the Ghent model the ratio is slightly lower than the free-nucleon value for both choices of the integration limits. In the Giessen model, the carbon ratio is at the same level as the free nucleon one or even higher. This is mainly due to the fact, that in Giessen model the structure function in second resonance region gets contributions from the 9 resonances, which were not present in Ghent model.
12 12,
C from 1.1 GeV C from threshold free nucleon (Ghent; 0.5
1 1.5 .2 • ''^C from l.i C from threshold free nucleon Q^, GeV^
FIGURE 4. (Color online) Ratio defined in Eq.(3) for the free nucleon (dash-dotted line), and ^^C in Ghent (left) and Giessen (right) models. We consider the under limits determined hyW = ¥2^^ are shown in Fig. 5. As for the electron-carbon results of Fig. 3, the resonance structure is hardly visible for both the Ghent and the Giessen model. The second resonance region is more pronounced in Giessen model because of the high mass resonances taken into account. The resonance structure functions are compared to the experimental data in DIS region obtained by the CCER [26] and NuTeV [27]
Figure 20.
Figure from [183]: (Left) F eC as a function of ξ , for values of Q indicatedon the spectra, compared with the BCDMS data for F eN at the given ξ at Q = 30, 45and 50 GeV . (Right) Ratio I eC of the integrated F in the resonance region withinthe Giessen [179] model to the integral over the DIS QCD fit to BCDMS high Q data. The results are displayed for two choices of the lower limit for the integral ofthe numerator: W = 1.1 GeV (solid line) and ”threshold” that takes into account theFermi motion within the C nucleus (dotted line). For comparison, the ratio I eN forthe free nucleon (dash-dotted line)is shown. GiBUU model in the SIS region emphasizes the importance of initial bound nucleonkinematics as well as non-resonant pion production being included in the calculations.A further rather surprising and significant indication of duality in eA scatteringcan be found in [188] when discussing duality and the ”EMC effect”. The EMC effect,to be covered in the next section, was previously thought to be a phenomena restrictedto purely DIS kinematics. However as shown in Fig. 21 data covering the x-range of theEMC effect measured in the resonance region intermixes seamlessly with EMC effectdata taken in the DIS region. This intriguing result is further demonstration of quark-hadron duality with the nuclear structure functions in the resonance region exhibitingsimilar behavior as in the DIS region. It can also be interpreted as a further suggestionthat to have entered the DIS region is signified by Q no matter what W is involved. Duality and Neutrino Scattering
The experimental study of duality with neutrinos ismuch more restricted since the measurement of resonance production by ν -N interactionsis confined to rather low-statistics data obtained in hydrogen and deuterium bubblechamber experiments from the 70’s and 80’s. Attempting to study duality withexperimental ν -A scattering is also limited due to very limited results above the∆ resonance in the SIS region. A recent NuSTEC workshop (NuSTEC SIS/DISWorkshop) [34] concentrating on this SIS region with neutrino-nucleus interactionsemphasized the considerable problems facing the neutrino community in this transitionregion. Since there are no high-statistics experimental data available across the SISregion, ν -N and ν -A scattering duality studies are by necessity limited to theoreticalmodels. Yet even the theoretical study of ν -N/A duality is sparse with only severalfull studies in the literature [32, 183],[200]-[202]. This is troublesome since modern ν ), requiring W > . to exclude the regionvery close to the quasielastic peak.There are small differences between the analyses ofthe SLAC and JLab data which had to be addressedto make a precise comparison. First, the SLAC andBCDMS ratios were extracted as a function of x ratherthan ξ . Because the conversion from x to ξ depends on Q , we can only compare ratios extracted at fixed Q values. Thus, for E139 we use the “coarse-binned” ra-tios, evaluated at fixed Q , rather than “fine” x binning,which were averaged over the full Q range of the exper-iment. Coulomb corrections were applied in the analy-sis of the JLab data [24], but not the SLAC data. TheSLAC data shown here include Coulomb corrections, de-termined by applying an offset to the incoming and out-going electron energy at the reaction vertex [24], due tothe Coulomb field of the nucleus. The correction fac-tor is < σ n /σ p = (1 − . ξ ).Figure 3 shows the cross section ratio of heavy nucleito deuterium for the previous SLAC E139 [15], E87 [25]and BCDMS [26] DIS measurements, and for the JLabE89-008 [7, 24] data in the resonance region. The sizeand ξ dependence of nuclear modifications in the JLabdata agrees with the higher Q , W data for all targets.Table I shows the ratios extracted from the JLab data.The agreement of the resonance region data with theDIS measurement of the EMC effect, which directly mea-sures the modification of quark distributions in nuclei, isquite striking. There is no a priori reason to expect thatthe nuclear effects in resonance production would be sim-ilar to the effects in scattering from quarks. However, itcan be viewed as a natural consequence of the quantita-tive success of quark-hadron duality [9, 12]. As seen inFig. 1, the structure functions for nuclei show little devi-ation from pQCD, except in the region of the quasielasticpeak (and ∆ resonance at low Q ). As Q increases, thedeviations from pQCD decrease as quasielastic scatteringcontributes a smaller fraction of the cross section. In ret-rospect, given the lack of significant higher twist contri-butions, combined with the fact that any A -independentscaling violations will cancel in the ratio, it is perhaps notsurprising that the resonance EMC ratios are in agree-ment with the DIS measurements.While it is difficult to precisely quantify the highertwist contributions with the present data, we can esti-mate their effect by looking at low W and Q , wherethe higher twist contributions are much larger. At Q ≈ and W ≈ M , the scaling violations (beyondtarget mass corrections) for deuterium are as large as50%, as seen in Fig. 1. However, if one takes the ironand deuterium data from Ref. [7], averages the structurefunction over the ∆ region and then forms the EMC ra-tio, the result differs from the ratio in the DIS region byless than 10%. The decrease in the effect of higher twistcontributions is a combination of the fact that the con-tribution are reduced when averaged over an adequate FIG. 3: (Color online) Ratio of nuclear to deuterium crosssection per nucleon, corrected for neutron excess. The solidcircles are Jefferson lab data taken in the resonance region(1 . < W < . , Q ≈ ). The hollow diamondsare SLAC E139 data, the crosses are the SLAC E87 data, andthe hollow squares are BCDMS data, all in the DIS region.The scale uncertainties for the SLAC (left) and JLab (right)data are shown in the figure. The curves show an updatedversion [27] of the calculations from Ref. [28]. region in W [9, 12], and cancellation between the highertwist contributions in deuterium and iron. The sameprocedure yields 2–3% deviations from the EMC ratioif one looks in the region of the S or P resonances,where the scaling violations in the individual structurefunctions are smaller to begin with.For the ratios in Fig. 3, we expect even smaller highertwist effects because the data is nearly a factor of twohigher in Q and is above the ∆ except for the veryhighest ξ points. At higher Q , the higher twist con-tributions in the individual structure functions becomesmaller, while averaging over the resonance region be-comes less important as the resonances become lessprominent. Thus, we expect that higher twist contri-butions for these data will be smaller than the the 2–3%effect ( <
10% near the ∆) observed on the EMC ratio at Q ≈ . If so, the higher twist corrections willbe small or negligible compared to the large statisticaluncertainty in previous measurements, and this data canbe used to improve our knowledge of the EMC effect atlarge ξ . Figure 21.
Figure from [188] demonstrating the EMC effect in the resonance region.The solid circles are Jefferson Lab data taken in the resonance region (1 . ≤ W ≤ . GeV and Q = 4 GeV ) while all other data points are from DIS experiments.The red curve is a prediction of the EMC effect from reference [202] interaction simulation efforts can not then compare their results with duality predictionsfor ν -N as they do for (cid:96) ± -N interactions for confirmation.An early neutrino nucleon duality study [201] by the Wroclaw group used the Rein-Sehgal model, which, as mentioned, is commonly used in current MC event generatorsfor neutrino nucleon 1- π resonance production. The study suggested that within theoriginal R-S model for ν -N 1- π production across the SIS region, local duality is definitelynot satisfied for neutron targets somewhat better for isoscalar targets and best, althoughnot great, for proton target as shown in Fig. 22. This reflects the fact that resonanceproduction off a proton dominates the resonance region while in the DIS region ν -nscattering dominates the DIS cross section . Other analyses such as [183] observe amuch smaller disagreement of duality for neutrons however it is still a disagreement.Note, this group [201] emphasized that the R-S model treatment of the non-resonantbackground, important for the quantitative evaluation of duality, is not very satisfactory.The significance of this non-resonant pion contribution has also been emphasized bythe previously cited work of the Giessen-Ghent collaboration [183, 178] that examinedduality with ν -N/A scattering as well as e-N/A scattering. Using the GiBUU modelin the resonance region (defined as W < I νN in Equation0 R a t i o Q [GeV ] 10/510/20 FIG. 4: Uncertainties in (41) due to different definitions of Q DIS . Solid line corresponds to (43) and dashed line to (44). RS: Q =0.4 RS: Q =1.0 RS: Q =2.0DIS: Q =10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8F ξ xF ξ F ξ xF FIG. 5: Comparison of the Rein-Sehgal structure functions at Q = 0.4, 1 and 2 GeV with the appropriate scaling functionsat Q DIS =10 GeV . In the first row xF , F and xF structure functions for CC neutrino-proton scattering are plotted. In thesecond row the structure functions for CC neutrino-neutron scattering are shown. In the quantitative analysis we define ratios of two integrals over the resonance region: R ! f, Q R ; g, Q D " = ξ max ξ min dξ f ( ξ, Q R ) ξ max ξ min dξ g ( ξ, Q D ) . (40) Also does not hold for n and p individually when using the Rein-Sehgal Model for n -N Resonances WARNING: R-S model questionable UGent.eps
Similar results in the framework of Rein–Sehgal ModelGraczyk, Juszczak, Sobczyk, Nucl Phys A781 (19 reso-nances included in the model) P ( ) , P ( ) , D ( ) , S ( ) , P ( ) , S ( ) , D ( ) , F ( ) Interplay between the resonances with different isospins:isospin-3/2 resonances give strength to the proton struc-ture functions, while isospin-1/2 resonances contribute tothe neutron structure function only
Olga Lalakulich (Ghent University, Belgium) Duality in Neutrino Reactions NuInt 07 10 / 22 RS: Q =0.4 RS: Q =1.0 RS: Q =2.0DIS: Q =10 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8F ξ xF ξ F ξ xF FIG. 7: Comparison of the Rein-Sehgal structure functions at Q = 0.4, 1 and 2 GeV with the appropriate scaling functionsat Q DIS =10 GeV . In the first row the plots of the xF , F and xF structure functions for CC neutrino-isoscalar targetscattering are presented. In the second row structure functions for NC neutrino-isoscalar target scattering are shown. and we also separate valence and sea quark contributions to the DIS structure functions: F DISj = F j,sea + F j,val . (46)We calculate the following functions: R val ( Q RES , Q DIS ) ≡ R ! F ,res , Q RES ; F ,val , Q DIS " . (47)and R val ( Q RES , Q DIS ) ≡ R ! xF ,res , Q RES ; xF ,val , Q DIS " . (48)
3. NUMERICAL RESULTS AND DISCUSSION
In the numerical analysis we confine ourselves to the case of neutrino interactions and leave out the antineutrinoones.In Figs. 5 – 7 we present a comparison of the scaling structure function with the RS structure functions calculatedat Q RES = 0 . , . The Figs. 5 and 6 correspond to CC and NC reactions respectively with protonstructure functions in the upper row and neutron structure functions below.In the case of the RS model for neutrino-proton CC reaction the ∆ resonance contribution dominates overwhelminglyover other resonances. One can see the typical manifestation of local duality: the sliding of the ∆ peaks (calculatedat different Q RES ) along the scaling function.For neutrino-neutron CC reaction the resonance structure is much richer. The contributions from the ∆ are usuallydominant but those from more massive resonances are also significant. In the figure with the F structure functionthree peaks of comparable size are seen. The DIS contributions dominate over the RS ones in this case. ξ ξ ξ Now for
Neutrinos
NO high-statistics Experimental Data available - turn to Theory When using the Rein-Sehgal Model for n -N Resonances (J. Sobczyk et al.-NuWro) ◆ Comparison to Rein-Sehgal structure functions for n, p and N at Q = 0.4, 1.0 and 2.0 GeV with the LO DIS curve at 10 GeV . ◆ The I integral for the R-S model for resonances off neutron (dotted), proton (solid) and isoscalar (dashed).
Real problems for A with large neutron excess! ⌫ A Interactions: SIS and DIS M. S. Athar and J. G. Morf´ın R a t i o Q [GeV ] 10/510/20 FIG. 4: Uncertainties in (41) due to di erent definitions of Q DIS . Solid line corresponds to (43) and dashed line to (44). RS: Q =0.4 RS: Q =1.0 RS: Q =2.0DIS: Q =10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8F ξ xF ξ F ξ xF FIG. 5: Comparison of the Rein-Sehgal structure functions at Q = 0.4, 1 and 2 GeV with the appropriate scaling functionsat Q DIS =10 GeV . In the first row xF , F and xF structure functions for CC neutrino-proton scattering are plotted. In thesecond row the structure functions for CC neutrino-neutron scattering are shown. In the quantitative analysis we define ratios of two integrals over the resonance region: R f, Q R ; g, Q D = max min d⇠ f ( ⇠, Q R ) max min d⇠ g ( ⇠, Q D ) . (40) Also does not hold for n and p individually when using the Rein-Sehgal Model for n -N Resonances WARNING: R-S model questionable UGent.eps
Similar results in the framework of Rein–Sehgal ModelGraczyk, Juszczak, Sobczyk, Nucl Phys A781 (19 reso-nances included in the model) P ( ) , P ( ) , D ( ) , S ( ) , P ( ) , S ( ) , D ( ) , F ( ) Interplay between the resonances with different isospins:isospin-3/2 resonances give strength to the proton struc-ture functions, while isospin-1/2 resonances contribute tothe neutron structure function only
Olga Lalakulich (Ghent University, Belgium) Duality in Neutrino Reactions NuInt 07 10 / 22 RS: Q =0.4 RS: Q =1.0 RS: Q =2.0DIS: Q =10 0 0.2 0.4 0.6 0.8 1 1.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8F ξ xF ξ F ξ xF FIG. 7: Comparison of the Rein-Sehgal structure functions at Q = 0.4, 1 and 2 GeV with the appropriate scaling functionsat Q DIS =10 GeV . In the first row the plots of the xF , F and xF structure functions for CC neutrino-isoscalar targetscattering are presented. In the second row structure functions for NC neutrino-isoscalar target scattering are shown. and we also separate valence and sea quark contributions to the DIS structure functions: F DISj = F j,sea + F j,val . (46)We calculate the following functions: R val ( Q RES , Q DIS ) ⌘ R F ,res , Q RES ; F ,val , Q DIS . (47)and R val ( Q RES , Q DIS ) ⌘ R xF ,res , Q RES ; xF ,val , Q DIS . (48)
3. NUMERICAL RESULTS AND DISCUSSION
In the numerical analysis we confine ourselves to the case of neutrino interactions and leave out the antineutrinoones.In Figs. 5 – 7 we present a comparison of the scaling structure function with the RS structure functions calculatedat Q RES = 0 . , . The Figs. 5 and 6 correspond to CC and NC reactions respectively with protonstructure functions in the upper row and neutron structure functions below.In the case of the RS model for neutrino-proton CC reaction the resonance contribution dominates overwhelminglyover other resonances. One can see the typical manifestation of local duality: the sliding of the peaks (calculatedat di↵erent Q RES ) along the scaling function.For neutrino-neutron CC reaction the resonance structure is much richer. The contributions from the are usuallydominant but those from more massive resonances are also significant. In the figure with the F structure functionthree peaks of comparable size are seen. The DIS contributions dominate over the RS ones in this case. Figure 6.
Figure from [12]: Comparison of the Rein-Sehgal structure functions at Q = 0.4, 1 and 2 GeV2 with the appropriate DIS scaling functions at Q = 10 GeV . Onthe left s F n vs ⇠ in the middle F p vs ⇠ and on the right F N vs ⇠ . nucleus interactions emphasized the problem facing the neutrino community in thistransition region. Since there are no recent or high-statistics experimental dataavailable, neutrino-nucleon and neutrino-nucleus scattering duality studies are bynecessity theoretical in their nature. Yet even the theoretical study of ⌫ -N/A dualityis sparse with only only several full studies in the literature [10, 11, 12, 5]. Thisis troublesome since modern ⌫ interaction simulation e↵orts can not then comparetheir results with duality predictions for ⌫ A/N as they do for ` ± N interactions forconfirmation.An early study [12] by the Wroclaw group used the Rein-Sehgal model for neutrinonucleon resonance production, which is commonly used in current MC event generators.The study suggested that within the original R-S model for ⌫ -N scattering dualityis definitely not satisfied for neutron targets somewhat better for proton target andbest, although not great, for isoscalar targets but mainly in the vicinity of the (localduality) as shown in Figure 6. This reflects the fact that the ++ o↵ a proton dominatesthe resonance region while in the DIS region ⌫ neutron scattering dominates the crosssection.This group also noted that the R-S model treatment of the non-resonantbackground, important for the quantitative evaluation of duality, is not very satisfactory.For this reason they addressed the idea of two-component duality that was originallyproposed by Harari and Freund [13, 14]. It essentially relates resonance productionof pions with the valence quark component and non-resonant pion production withthe sea quark component of the structure functions. This concept was confirmedvia eN interaction[15] and, as earlier noted and seen in Figure 1, the F structurefunction averaged over resonances at low values of ⇠ ( .
3) behaves like the valencequark contribution to DIS scaling. This suggests the very intriguing concept thatif overall duality is satisfied and the resonance contribution is dual to the valenceDIS contribution, then the non-resonant background could be dual to the sea quarkcontribution. Then this duality could be used to provide a model for non-resonantPage 7 ⌫ A Interactions: SIS and DIS M. S. Athar and J. G. Morf´ınbackground.The conclusions of this extended duality analysis for CC ⌫N interactions is that,as illustrated in Figure 7: for the whole resonance region ( M + m ⇡ W Q . GeV duality is satisfied only for CC proton target reaction and at best tothe 20% level; there is also CC local duality in the vicinity of the resonance for anisoscalar target. F R a t i o : R ES t o D I S Charged Current protonneutronisoscalar 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 F R a t i o : R ES t o D I S Q [GeV ]Neutral Current protonneutronisoscalar FIG. 9: The functions R for di erent targets and reactions. The ratios are calculated for CC and NC structure functions inthe cases of proton (solid lines), neutron (dotted lines) and isoscalar target (dashed lines). increased by a factor of ⇠ .
55 and for proton by ⇠ .
39. The di↵erence is caused by the overwhelming dominanceof the excitation in the case of proton.A characteristic feature of most of the plots of R j ( Q RES ) is a presence of two qualitatively distinct behaviors. For Q RES smaller then ⇠ . the functions R j vary quickly while for larger values of Q RES they become slowlychanging. This seems to correspond to predictions done in [5]. Our statements about the duality will apply only tothe region of Q RES . .In Figs. 9 and 10 the plots of R and R for proton, neutron and isoscalar targets are presented. In the case of CCinteraction the duality is seen on the proton target (accuracy Q DIS , namely Q DIS = 20 GeV makes the values of R , even lower (see Fig. 4).The remaining plots address the question of two component duality. We concentrate on the case of the possibleduality between the resonance and valence quark contributions.In Fig. 11 the plot of R val for the CC interactions is shown. We notice the good duality picture in the case ofproton target but a huge departure from duality in the case of neutron and isoscalar targets. It is worth noting thatthis discrepancy is larger than one shown in Fig. 9 where the general (not two component) notion of duality wasdiscussed. The novel feature is the apparently singular behavior at low Q RES : R val rises quickly in contrast with R falling down when Q RES approaches zero.The explanation of this follows from the Fig. 12 where the region of small Q RES was analyzed in more detail. Wenotice that for Q RES approaching zero the valence quarks scaling function tends to zero while the resonance strengthsremains virtually unchanged.Finally in Fig. 13 the analogous two-component duality analysis is done for R val . The discussion of xF seems tobe favorable for the two-component duality because in the DIS contribution on the isoscalar target there is no seaquark contribution. We remind also that for the CC reaction on the proton the non-resonant contribution is absent. Figure 7.
Figure from [12]: The integral Equation (1.1) for CC interactions in theR-S model for resonances o↵ proton (solid lines), neutron (dotted lines) and isoscalartarget (dashed lines).
Turning back to the analysis of the Giessen-Ghent group [6] that examined dualitywith e N/A scattering. Using the GiBUU model in the resonance region (defined as
W < for larger W,DIS structure functions are much larger than the resonance contribution at lower W .This general conclusion should be kept in mind for consideration of simulation programstreating the SIS region.Quark-hadron duality in the case of neutrino nucleus interactions has been studied,again theoretically, in [16]. The results as in Figure 9 from that reference suggestproblems with applying duality to this process, particularly for non-isotropic nucleisuch as Pb or even Fe or Ar. The Q along the abscissa in Figure 9 is the Q involvedin computing the limits ⇠ min = ⇠ ( W , Q ) and ⇠ max = ⇠ ( W , Q ) of the integration ofthe numerator of I ⌫Fe . Refer to the figure caption for further details of the figure.They observed that the computed resonance contribution to the leptonnucleusstructure functions is qualitatively consistent with the measured DIS structure functions.Page 8 R a t i o Q [GeV ] 10/510/20 FIG. 4: Uncertainties in (41) due to different definitions of Q DIS . Solid line corresponds to (43) and dashed line to (44). RS: Q =0.4 RS: Q =1.0 RS: Q =2.0DIS: Q =10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8F ξ xF ξ F ξ xF FIG. 5: Comparison of the Rein-Sehgal structure functions at Q = 0.4, 1 and 2 GeV with the appropriate scaling functionsat Q DIS =10 GeV . In the first row xF , F and xF structure functions for CC neutrino-proton scattering are plotted. In thesecond row the structure functions for CC neutrino-neutron scattering are shown. In the quantitative analysis we define ratios of two integrals over the resonance region: R ! f, Q R ; g, Q D " = ξ max ξ min dξ f ( ξ, Q R ) ξ max ξ min dξ g ( ξ, Q D ) . (40) x F R a t i o : R ES t o D I S Charged Currentprotonneutronisoscalar 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 x F R a t i o : R ES t o D I S Q [GeV ]Neutral Currentprotonneutronisoscalar FIG. 10: The same as in Fig. 9 but for xF (ratio R ). F R a t i o : R ES t o D I S Q [GeV ]Charged Current, resonant to valenceprotonneutronisoscalar FIG. 11: The plots of functions R val defined in Eq. 47. The computations are performed for the CC reactions for proton (solidline) neutron (dotted line) and isoscalar targets (dashed line). In Fig. 13 we see that two component duality is satisfied within ∼
30% for the proton target but it is absent forneutron and isoscalar targets. We notice also that contrary to what we have seen in the plots for R val now at low Q RES all the curves tend to zero.The explanation of this behavior follows from the Fig. 14. One can see that in the case of xF both the resonanceand valence quark structure functions fall down for Q approaching zero. The behavior of xF is the same as thatdiscussed in [19].We do not present plots exploring the duality between the non-resonant part of the resonance model and the seaquark contribution. No sign of two component duality is seen in this case. F xF Q Q Figure 22.
Figure from [201]: (upper) Comparison of the Rein-Sehgal F structurefunctions vs ξ for neutron, proton and the isoscalar nucleon target at Q = 0.4, 1 and2 GeV2 with the appropriate DIS scaling functions at Q = 10 GeV . (lower) Ratio I ν , of the integrated F (left) and xF (right) in the resonance region to the integralover the DIS LO QCD fit at Q = 10 GeV . (78) using the resonance contribution only (no non-resonance production included),even for the isoscalar nucleon, is about 70% as shown in Fig. 23 consistent with theimportance of correctly accounting for the non-resonant pion contribution. This resultcan be directly compared to the earlier analysis of the Rein-Sehgal model that yieldeda result for the integral of order 50% that did include the Rein-Sehgal estimate of thenon-resonant pion contribution. These results are, obviously, model dependent but ageneral tendency is that for larger W, DIS structure functions are much larger than theresonance contribution at lower W and that the non-resonant contribution cannot beneglected. This general conclusion should be kept in mind for consideration of simulationprograms treating the SIS region.From Fig. 22 and Fig. 23 there is a noticeable decrease at low values of ξ of theintegral ratio below ≈ Q ≤ . GeV . This behavior resembles the fall-off of thevalence quarks ( xF ) and was noted by several studies including [32, 201]. This ledto the idea of two-component duality, which was originally proposed by Harari andFreund [203, 204]. It essentially relates resonance production of pions with the valencequark component and non-resonant pion production with the sea quark componentof the structure functions. This concept was tested via e-N interaction [146] studies1 AQ'= V -vv. \ =0.225 °'S . A 0.525
1 1 > / , ' ; A1-025 i A J 2.025 - FIGURE 1. l2 Duality for the isoscalar nucleon Fj "^^ structure function calculated within GiBUU model. (Left)
F2^ as a function of ^, for
Q =
Q^ = 10
GeV . (Right) Ratio if^ of the integrated
F2^ in the resonance region to the leading twist functions. ^ correspond to the second (1.40 GeV < W < 1.56 GeV) and the third (1.56 GeV < W < 2.0 GeV) resonance regions. The general picture shows a reasonable agreement with the duality hypothesis. In the right panel of Fig. 1, the ratio of the integrals if^, defined in (3), is shown not only for the whole structure function (resonance + 1-pion background), but also for the resonance contribution separately. For Q^ > if for the resonance contribution only is at the level of 0.85, which is smaller and flatter in Q^ in comparison with the results [6, 15] of the Dortmund group resonance model. The difference is due to the different parameterization of the electromagnetic resonance form factors used in the two models. The background gives a noticeable contribution and brings the ratio up to 0.95. The fact, that it is smaller than 1 is of no surprise, because additional nonresonant contributions like 2- and many-pion background are possible, but not taken into account here. They are the subject of coming investigations. The principal feature of neutrino reactions, stemming from fundamental isospin arguments, is that duality does not hold for proton and neutron targets separately. The interplay between the resonances of different isospins allows for duality to hold with reasonable accuracy for the average over the proton and neutron targets. We expect a similar picture emerges in neutrino reactions with nuclei. For neutrinoproduction, the structure function F2^ and the ratio / j ' ^ are shown in Fig. 2 for the resonance contribution only. The ratio is at the level of 0.7, which is (similar to the electron case) smaller than 0.8, which has been calculated within the Dortmund resonance model [6, 15]. Thus, one would expect a large contribution from the background. The role of the background in neutrino channel is under investigation now. / FIGURE 2. l2 Duality for the isoscalar nucleon Fj '^^ structure function calculated within the GiBUU model. (Left) Fj^'^ as a function of ^, for
Q =
Q^ = 10
GeV . (Right) Ratio / j ' ^ of the integrated Fj^'^ in the resonance region to the leading twist functions.
Figure 23.
Figure from [183]: Duality for the isoscalar nucleon F νN structurefunction calculated within the GiBUU model. (Left) F νN in the resonance regionat different Q indicated on the spectra as a function of ξ compared with the leadingtwist parametrizations at Q = 10 GeV . (Right) From Equation (78) the ratio I νN of the integrated F νN in the resonance region to the DIS leading twist functions indicating that the F structure function averaged over resonances at low values of ξ ( (cid:47) SS resonances. The contribution from the latter becomes more significant with increasing Q since its form factors fall off more slowly than the dipole. The contribution of the P (1440) resonance is too small to be seen as a separate peak. The two sets of resonancecurves correspond to the “fast fall-off” (lower curves) and “slow fall-off” (upper curves)scenarios for the axial form factors discussed in Sec. 2.1. The smooth curves are obtainedfrom Eq. (16) using the GRV [31] and CTEQ [32] leading twist parton distributions at Q = 10 GeV , as in Fig. 1. Just as in the case of electron–nucleon scattering, withincreasing Q the resonances slide along the leading twist curve, which is required byduality. As in Fig. 1, we show both the total structure function and the valence-onlycontribution. F ν N ξ valencetotal 0.2 0.5 1.0 2.0GRVCTEQ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.5 1 1.5 2 I ν N Q , GeV valencetotal GRVCTEQ Figure 3: Duality for the neutrino–nucleon F νN structure function. (Left) F νN in theresonance region at several Q values (indicated on the spectra), compared with leadingtwist parameterizations [31, 32] (valence and total) at Q = 10 GeV . (Right) Ratio I νN of the integrated F νN in the resonance region to the leading twist functions [31, 32](valence and total). The upper (lower) resonance curves and the upper (lower) integratedratios correspond to the ”slow” (”fast”) fall-off of the axial form factors.In Fig. 3 (right panel) we show the ratio of the integrals of the neutrino resonance andleading twist structure functions, defined in Eq. (23). The ratio is within ∼ Q ! . and, unlike the corresponding electron–nucleon ratio I eN , doesnot grow appreciably with Q . Again, the two sets of resonance curves correspond tothe “fast fall-off” (lower) and “slow fall-off” (upper) scenarios for the axial form factors.17 Figure 24.
Figure from [32]: Duality for the isoscalar nucleon structure function F νN .(Left) F νN for resonances at Q indicated on the spectra as a function of ξ , comparedwith the leading twist parametrizations (valence and total) at Q = 10 GeV . (Right)From Equation (78) the ratio I νN of the integrated F νN in the resonance region to theleading twist functions. The upper (lower) integrated ratios correspond to different Q behavior of the axial form factors This suggests the very intriguing concept that if overall duality is satisfied and theresonance contribution is dual to the valence DIS contribution, then the non-resonantbackground could be dual to the sea quark contribution. This, in turn, suggests thatduality could be used to guide a model for non-resonant pion production background.Duality in the case of neutrino nucleus interactions has been studied, againtheoretically, in [205]. In particular both the GiBUU and Ghent groups have used theirrespective resonance models to evaluate duality. The main difference in the modelsis that GiBUU [206] uses a resonance model that includes single- and multi- π decaysplus heavier decay states while the Ghent model [207] concentrates on 1 π decays but2extended up to high effective masses using Regge trajectories. They observed as inFig. 25 that the computed integrated resonance strength is about half of the measuredDIS one. Contrary to the free nucleon case, where the ratios I i ( Q ) are at the level of0.8-0.9, they found for nuclei such as Fe ratios of 0.6 for electro-production and 0.4 forneutrino production. This points towards a scale dependence in the role of the nucleareffects. It could suggest that nuclear effects act differently at lower Q (resonanceregime) than at higher Q (DIS regime). In this analysis the contributions of the non-resonant background was ignored. It was stressed that for more detailed investigations ofduality a theoretical or phenomenological model for the non-resonant background acrossthe entire resonance region will be required. The inadequacy of the treatment of non-resonant meson production in current neutrino event simulators has been emphasizedby recent studies [208] that found that the non-resonant background evaluated frombubble chamber data is considerably smaller than the estimates in GENIE. Note thatsubsequent experimental studies have preliminarily suggested that this large reductionin the GENIE non-resonant pion estimate is essentially only for the W-region aroundthe ∆ and the GENIE prediction for the higher-W SIS region may be valid. collaborations. It appears, that the resonance curves slide along the DIS curve, as one would expect from local duality, but lie below the DIS measurements. Hence, the computed structure functions do not average to the DIS curve. The necessary condition for local duality to hold is thus not fulfilled. CD up o NuTeV 7.94
CCFR 7.94 --&----^---->^-- (Ghent; CD up O FIGURE 5. (color online) The computed resonance curves
F2 ^"156 as a function of
E,, calculated within Ghent(Ieft) and Giessen (right) models for
Q^ =
Q^;
2) it is significantly smaller than the one for the free nucleon; 3) h is even lower than the corresponding ratio for electroproduction; 4) h slightly decreases with Q^.
To summarize, within the two models, which implement elementary resonance vertices differently and treat nuclear effects differently, we obtain qualitatively the same effect, that the resonance structure functions are consistently smaller that DIS functions in the same region of Nachtmann variable
B,.
This is not what one would expect from Bloom-Gilman duality. Recall, that in this analysis for nuclei, we included the resonance structure functions, and ignore the background ones. To estimate their contribution and compare the results with the nucleon case would be one of the primary tasks of coming investigation. Further results of the Ghent model are given in [22]. 1.4 1.2 1 0.8 0.6 0.4 0.2 0 ^®Fe from 1.1 GeV '—^
Fe from threshold free nucleon 0.5 Fe from 1.1 Fe from threshold free nucleon (Giessen \ FIGURE 6. (color online) Ratio /^ ^^ defined in Eq. (3) for the free nucleon (dash-dotted line) and Fe calculated within Ghent(left) and Giessen(right) models. For Fe the results are displayed for two choices of the underlimit in the integral: W =\.\
GeV (solid line) and threshold (dotted line). For each of these two choices we have used two sets of DIS data in determining the denominator of Eq. (3). These sets of DIS data are obtained at
Qrijs = 12.59 and 19.95 GeV .
Figure 25.
Figure from [183]: Ratio I νF e for iron calculated within the Ghent [209](left) and Giessen [179](right) models. For Fe the results are displayed for two choicesof the lower limit of the numerator in the integral of Equation (78): W = 1.1 GeV(solid line) and ”threshold” that takes into account the Fermi motion within the Fenucleus (dotted line). For each of these two choices they used two sets of DIS datain determining the denominator of the integral I, one at Q DIS = 12.59 GeV and theother at 19.95 GeV . The ratio I νN for the free nucleon (dash-dotted line) is shownfor comparison However, these studies suggest the need for care in using duality to verify thestrength of contributions of ν -N scattering in the SIS region and, particularly, forconsidering the interpretation of duality with ν -A scattering for nuclei with large excessneutron content. ” Effects In the calculation of the DIS integral for the denominator of Equation (78), the LO orNLO leading twist (see 2.3.3) perturbative QCD fit to high Q data or, if an experimental3measurement of F was used, it was taken from higher- Q measurements. The importantfeature was that no higher twist ”1 / Q ” effects were included in the evaluation of theintegral denominator of the ratio. This being the case, the observation from Fig. 18that the agreement with duality is quite close to complete is a suggestion that there areminimal additional higher twist effects in the DIS data or needed in the DIS theoreticalexpression as long as target mass correction (TMC) are included through the use of ξ Considering these conclusions, it could be possible to learn about possible highertwist effects by observing violations of duality for e/ µ nucleon data at lower Q . Currentneutrino experiments are constrained by their lower-energy neutrino beams to thelower Q edge of the DIS region where possible higher twist effects could then be areal complication of the analysis. On the other hand, improved knowledge of highertwist contributions and how these contributions are exhibited as non-resonant pionproduction could provide a better understanding of the transition from perturbative tonon-perturbative QCD, from the SIS to DIS regions. Improved determination of thehigher-twist effects should then be a goal of current and future analyses.There have been several studies investigating the link between duality and highertwist effects [210]-[214]. In the earlier study [210] the authors emphasize the ability touse duality to determine higher twist contributions from structure function data in theresonance region by using moments (in x) of the structure function F . These highermoments in x emphasizing the contributions from increasingly higher x regions wherehigher twist effects are supposed to be larger.The authors of [211, 212] first examined duality in structure functions [211] andthen used the techniques developed in this study to understand the interplay of dualityand higher twist[212]. This study used a combination of nucleon data from JeffersonLab and SLAC to form the numerator in a ratio of integrals similar to Equation (78).The denominator is taken from dynamical parametrizations coming from free nucleonparton distribution functions. Target-mass effects are then introduced and, in a separatestep, they also include the large-x re-summation effects. These re-summation effects,essentially, reduce the exaggerated Q -dependent suppression of F as x approaches 1.,which, in essence, adds strength to F at large x and increases the integral. The resultsof their study is shown in Fig. 26 and supports their conclusion that with the additionof the TMC and the inclusion of the large-x re-summation there is little space left foradditional (1 / Q ) higher twist effects for Q ≥ . GeV . This is then a quantitativeexercise showing the power of using duality to better understand the need for bothadditional kinematical (TMC) and dynamical higher twist(HT) terms added onto theleading twist perturbative QCD expression for DIS structure functions. For an informative comparison of current simulation efforts in the SIS and DIS regionswith emphasis on NEUT refer to Bronner’s presentation in [34]. It is interesting tonote that there is a common thread among MC event generators in attempts to bridge4
Figure 26.
Modified figure from [212]: Ratio between the integrals of the measuredstructure functions and the calculated ones plotted as a function of Q showing theeffect of adding consecutively the TMC and large-x re-summation to the straightleading twist NLO QCD expression based on PDF fits. the transition from the SIS to DIS regions. As a practical procedure for addressingthis SIS region in contemporary neutrino event generators, such as GENIE, Bodek andYang [215] introduced a model (BY) that is used to bridge the kinematic region betweenthe Delta resonance and DIS. This BY model is also used by GENIE and other eventgenerators to describe the DIS region as well and this application will be consideredin subsequent sections. The model was developed using results from electron-nucleon inelastic scattering cross sections. The model incorporates the GRV98 [106] LO partondistribution functions replacing the variable x with their ξ w scaling variable to includethe effects of dynamic higher twist effects through a modified target mass correction.These modified parton distribution functions are used to describe data at high Q anddown to 0.8 GeV . Below Q = 0.8 GeV they take the GRV98 LO PDFs to get thevalue of F ( x, . GeV ) and multiply it by quark-flavor dependent K factors to reachlower Q and W.The BY model then compares the results obtained with the above procedure withthe expectations of the duality concept as demonstrated with e/ µ - nucleon inelasticscattering (see section 4.1). They find their predictions to be consistent with the averageof charged-lepton nucleon initiated resonance production from the ∆ peak to the startof DIS and therefore consistent with duality. Note that the predictions of this procedureare meant to also include the n on-resonant meson production in this region. The stepsto expand their predictions from e/ µ - nucleon to e/ µ - nucleus , is described in [147]. Inbrief, a model for deuterium nuclear effects is used to produce e/ µ - deuterium from e/ µ - nucleon and then the measured ratio as a function of x of e/ µ - Fe to e/ µ - deuterium isused to predict e/ µ - Fe so that this procedure is then valid for Fe targets only.The BY procedure for ν -N/A scattering is described in [216]. They use the same5GRV98LO, ξ w with K-factor approach as used for charged-lepton scattering but havequite different techniques for evaluating the factors since the axial-vector contribution,involving an axial K-factor, and the additional structure function ( xF ) of neutrinoscattering must be considered. For very high-E ν and high- Q both the vector and axialK-factors are expected to be 1.0 and the expressions for F and xF are straightforward.Since the vector part of F goes to 0 at Q = 0 while the axial component does not, theirapproach to low- Q must account for this difference in the vector and axial componentsof F . They furthermore account for the differences in higher order QCD effects andscaling violations in F and xF at low- Q and end up with expressions for F vector , F axial and xF that they then use to predict neutrino nucleon interactions below theDIS region. Transition from SIS to DIS
The B-Y expressions for this lower-W behavior, which,significantly, includes their estimate of non-resonant meson production, is expected toseamlessly blend with the straightforward expressions for F and xF they predict in theDIS region. They then have mimicked the concept of duality but based the extrapolationfrom the ∆ to DIS on the described components of their model. It is important to notethat this transition region is goverened by non-resonant pion production that, at low Q ,is described in models quoted earlier. This non-resonant pion production then becomesDIS as Q becomes high enough to allow scattering off partons within the nucleus.GENIE, employing the B-Y model, estimates the sum of non-resonant pion as well asmulti-pion resonant production with this extrapolation from DIS. GENIE then usesthe AGKY hadronization model (see 5.7) that for these lower values of W employs theKNO multiplicity model, to predict production of single and multi-pi events. There isunfortunately very limited experimental data to compare with their predictions in thislow- Q , low-W region. However, an indication of a possible problem, an overestimationof the prediction in the region of ∆ production, could be drawn from the earlier quotedrecent studies [208] that found the non-resonant background to be considerably smallerthan the estimates in GENIE that come from the BY model prediction of the averagestrength of the F and xF in this region.Whether this result can be related to the duality approach of extrapolating F and xF from the DIS regime down into the resonant region for neutrino scattering has notbeen explicitly considered. However, we did learn (Section 4.1) that, from all modelsconsidered, such an extrapolation could indeed lead to overestimating this contributionin the lower-W resonant region. That is a smooth extrapolation of the strength in theDIS region tends to overshoot the model predictions for the strength in the SIS region.However an important point to recall is that many of the current models for resonanceproduction include either no or very simple, approximate models for non-resonant pionproduction.6 The Shallow Inelastic Scattering region, particularly the higher-W transition to theDeep-Inelastic Scattering region in ν/ν nucleon/nucleus scattering has been scarcelystudied theoretically or experimentally. In particular the evolution of low- Q non-resonant single pion production to low- Q non-resonant multi-pion production and, as Q increases, DIS pion production needs much more attention. The lack of knowledgeof this region is reflected in the disparity in the current predictions by the community’ssimulation programs as displayed in Fig. 27. SIS DIS
Figure 27.
Figure adapted from Bronner [34] showing a comparison of the predictionsof the community’s then current simulation programs (NEUT 5.4.0, GENIE 2.12.10and NuWro 18.02.1) over the range of W encompassing the SIS and DIS regions. Thepredictions are for a 6 GeV neutrino on Fe.
There have been multiple studies of the ∆ resonance region (W ≤ ν A experiment including somewhat higher Wsingle and multi-pion production ( W ≤ µ -N interactions and ν/ν -N interactions. A brief summary would conclude that: • F ep,en - for e/ µ -N scattering qualitative and quantitative duality i s observed • F νp,νn - for ν/ν -N scattering duality is roughly observed for the average nucleon[(n+p)/2] but duality is n ot observed for neutrons and protons individually. • For electroproduction with nuclei it is a different story. The quantitative evaluationof duality in e-A is not as good as with e-N. • For ν -A interactions it is not clear at all how duality works, particularly with nucleihaving an excess of neutrons.7The challenge of addressing duality with neutrinos is that in general in the SISregion the resonance structure functions for proton are much larger than for neutronsand in the case of deep-inelastic scattering the opposite is the situation. This doessupport the observation that if duality is observed at all with neutrinos it is with theaverage nucleon [(n+p)/2].However there is a more fundamental concern regarding the whole concept oftesting duality experimentally. Can one really test duality if both the ”DIS” and ”SIS”regions are not experimentally accessible at identical kinematics? For example, Fig. 27represents a neutrino energy typical for the MINER ν A experiment and there is verylimited range of W above the 2 GeV DIS cutoff available for any comparison to the SISregion. Furthermore, although there may be limited contributions of higher twist forlower-x and Q structure functions, when including inclusive cross sections over all xand Q leading twist alone may not be sufficient. Thus different extrapolations will giveyou better or worse agreement between the extrapolated ”DIS” part and the measuredSIS part. There is a need for careful consideration of exactly what experimental testscan be made to test duality with neutrino nucleus interactionsThis also strongly suggests that rather than only experimental tests of duality weshould encourage a closer examination of just how well the current neutrino simulationevent generators, GENIE, NEUT and NuWro obey duality in their treatment of thebasic input, ν/ν isoscalar nucleon scattering. ν l / ¯ ν l -Nucleus Scattering: Deep-Inelastic Scattering Phenomenology Neutrino scattering plays an important role in the QCD analysis of deep-inelasticscattering since the weak current has the unique ability to ”taste” only particular quarkflavors resolving the flavor of the nucleon’s constituents: ν interacts with d , s , u and c while the ν interacts with u , c , d and s . This significantly enhances the study of partondistribution functions and complements studies with electromagnetic probes. However,as helpful as this ability of the weak-interaction may be, it should be again emphasizedthat all high-statistic neutrino experiments have had to use heavier nuclear targets.This means the PDFs extracted from these experiments are for nucleons in the nuclearenvironment and are thus nuclear parton distribution functions nPDF. As will be shown,there is considerable difference between these A-dependent nPDFs and the free nucleonPDFs. Furthermore, since the relevant nuclear effects could involve multiple nucleonscattering as in shadowing or scattering from correlated nucleon pairs as possibly in theEMC effect these nPDFs might better be considered nuclear nPDFs and not necessarilythe PDFs of single bound nucleons.Historically the study of the DIS region and first tests of (nuclear) QCD withneutrinos were actually the primary goals of early experiments with higher energyneutrino beams. However, the current focus on neutrino-oscillation studies, with theneed to emphasize lower neutrino energies to maximize oscillations, has led to limitingthe possibility to explore the full DIS region in such experiments. For example, the8future DUNE experiment with the huge statistics expected in the near detectors shouldallow an interesting study of DIS albeit in a rather limited kinematic range suggestingan interesting and necessary study of the non-pQCD / pQCD transition region in thenuclear environment as well as the lower-Q, lower-W DIS region. Neutrino scattering experiments have been studying QCD with DIS for over fourdecades. The early pioneers in these studies were the bubble chambers such as theGargamelle heavy liquid bubble chamber [88] normally filled with heavy freon CF Br ,while the smaller ANL [89] and BNL [90] chambers as well as the much larger BEBC [91]at CERN and the 15’ chamber [92] at FNAL were normally filled with hydrogen ordeuterium and occasionally mixed with heavier nuclei such as Ne or using heavier liquidssuch as propane. With these bubble chambers, initial studies of QCD behavior with theaxial vector current were undertaken by multiple collaborations.These chambers using hydrogen or deuterium targets offered an ideal tool toprobe the structure of the free nucleon and measure the very important fundamentalproduction cross sections essential as input to modern neutrino scattering simulationprograms. Unfortunately, the overall statistics was quite limited and totally insufficientfor contemporary needs such as the vital input for modern event generators GENIE,NEUT and NuWro § .Most of the early studies of QCD with bubble chambers were performed by CERNexperiments. An example of these early CERN studies is the publications [218] thatshowed the results of a combination of the lower energy Gargamelle ( CF Br ) PS runwith the higher energy narrow-band beam BEBC Ne-H exposure. Note this analysiswas performed before the discovery of the DIS x-dependent nuclear effects suggestingthat simply combining the two experiments, using different nuclei, without consideringthese nuclear effects could have been problematic. However with the BEBC run usinga 73% molar Ne/H mix the difference in nuclear effects between the Gargamelle andBEBC runs would have been smaller than the errors on the data. The point of theseearly CERN ν experiments was to perform first measurements of Λ QCD with neutrinosand to better understand the influence of non-perturbative effects such as target massand higher-twist in a quantitative comparison of results with QCD.
The first higher statistics ν and ¯ ν nucleus measurements were performed by massivenuclear target detectors like CDHS(W) - iron [219] and CHARM/CHARM II -marble/glass [220]. These early experiments were followed by the CCFR [95] andNuTeV [96] - iron experiments and the CHORUS - lead experiment [221, 222]. Asopposed to the high resolution of the earlier low statistics bubble chamber experiments, § There is consideration within the neutrino community to attempt to correct this insufficiency of freenucleon data with an H/D experiment using the high-intensity DUNE LBNF neutrino beam. ν and ¯ ν cross sectionmeasurements.Even the contemporary MINOS oscillation experiment, with the requisite lowenergy ν/ν beams, had to concentrate on total cross section measurements on iron [223]since the rather limited experimental resolution in the measurement of hadron energyresulted in poor x resolution. No extraction of x - Q dependent differential cross sectionswas undertaken.The NOMAD experiment was one of the first modern, finer-grained experimentswith an opportunity for high resolution measurements of exclusive states [224]. HoweverNOMAD has yet to release their measurements of the inclusive cross sections andstructure functions off the various nuclei in their experiment.The latest results come from the MINER ν A experiment that has measured chargedcurrent (CC) ν -A DIS cross sections on polystyrene, graphite, iron and lead targetsboth in the lower energy (LE) NuMI neutrino beam [225, 226], and, more recently, inthe somewhat higher energy (ME) beam, which enabled increased statistics and a widerkinematic range.Without NOMAD results and MINER ν A ME results still pending, the latest high-statistics dedicated studies of QCD using neutrino scattering come from the NuTeV[96] and CCFR [95] experiments off Fe as well as the CHORUS [221, 222] experimentoff Pb. The NuTeV experiment was a direct follow-up of the CCFR experiment usingnearly the same detector as CCFR but with a different neutrino beam and analysismethods. The NuTeV experiment accumulated over 3 million ν and ν events in theenergy range of 20 to 400 GeV off a mainly Fe target. The data were then corrected forQED radiative effects [227] and the charm production threshold. A comparison of theNuTeV differential cross section results with those of CCFR and CDHSW are shown inFig. 28 and Fig. 29 for two different beam energies. The importance of these directlymeasured cross sections as opposed to assumption-based extracted structure functionswill be emphasized in subsequent sections describing the extraction of nuclear partondistributions.A comparison of the NuTeV structure functions F ( x, Q ) and xF ( x, Q ) derivedfrom these cross sections with those from CCFR and CDHSW are shown in Fig. 30. Themain point is that the NuTeV structure function F agrees with CCFR F for values of x ≤ x culminating at x (cid:39) (cid:39)
20% higher than the CCFR result.Although the reason for this difference at high-x was not initially understood, it wasfinally traced to the difference of the magnetic field maps of the two experiments (thatresulted is a shift of the muon energy scales of the two experiments), the different crosssection models used by NuTeV and CCFR and NuTeVs improved muon and hadronenergy smearing models.Providing input on lead targets, the CHORUS detector was comprised of ahigh-resolution lead/scintillator calorimeter coupled with a large acceptance muon0 (E=65 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=65 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=65 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=65 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=65 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=65 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=65 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=65 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=65 GeV) x=0.015
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=65 GeV) x=0.045
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=65 GeV) x=0.125
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=65 GeV) x=0.175
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=65 GeV) x=0.275
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=65 GeV) x=0.35
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=65 GeV) x=0.55
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y
0 0.2 0.4 0.6 0.8 1 (E=65 GeV) x=0.65
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y FIG. 6. Differential cross sections in x bins for neutrinos (left) and anti-neutrinos (right) at E = 65 GeV. Points are NuTeV(filled circles), CCFR (open squares), and CDHSW (crosses). Error bars show statistical and systematic errors in quadrature.Solid curve shows fit to NuTeV data. ( x =0.08, 0.225, 0.45, and 0.75 bins are not shown). Figure 28.
Figure from [96]. Differential cross sections as a function of y in x bins forneutrinos and anti-neutrinos at E = 65 GeV. Points are NuTeV (filled circles), CCFR(open squares), and CDHSW (crosses). Error bars show statistical and systematicerrors in quadrature. Solid curve shows fit to NuTeV data. spectrometer for neutrino interactions in the calorimeter. The experiment used higherpurity sign-selected neutrino and anti-neutrino beams to measure double differentialcross-sections, in different bins of the neutrino energy, with minimal model-dependence.It is these cross sections that were used in the extraction of nuclear parton distributions.From the differential cross sections the structure functions F and x F were extractedand are shown in Fig. 31 and Fig. 32 along with the ν -Fe results of CCFR and CDHSW. That the need for high statistics neutrino experiments resulted in the use of heavynuclear targets eventually introduced significant complications in the attempt to extract free nucleon
PDFs with these neutrino results. The goal of combining the many DIS1 (E=150 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=150 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=150 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=150 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=150 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=150 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=150 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y (E=150 GeV)Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=150 GeV) x=0.015
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=150 GeV) x=0.045
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=150 GeV) x=0.125
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=150 GeV) x=0.175
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=150 GeV) x=0.275
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=150 GeV) x=0.35
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y(E=150 GeV) x=0.55
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y
0 0.2 0.4 0.6 0.8 1 (E=150 GeV) x=0.65
Neutrino Anti-NeutrinoY / E d σ / dxdy ( x - c m / G e V ) Y FIG. 7. Differential cross sections in x bins for neutrinos (left) and anti-neutrinos (right) at E = 150 GeV. Points are NuTeV(filled circles), CCFR (open squares), and CDHSW (crosses). Error bars show statistical and systematic errors in quadrature.Solid curve shows fit to NuTeV data. ( x =0.08, 0.225, 0.45, and 0.75 bins are not shown). Figure 29.
Figure from [96]. Differential cross sections as a function of y in x bins forneutrinos and anti-neutrinos at E = 150 GeV. Points are NuTeV (filled circles), CCFR(open squares), and CDHSW (crosses). Error bars show statistical and systematicerrors in quadrature. Solid curve shows fit to NuTeV data. experimental results on heavy nuclei ranging from C to Pb was thought not to be aproblem in that the PDFs of nucleons in the nuclear environment were assumed to bethe same as the free nucleon. However, this was determined not to be the case withcharged-lepton-nucleus ( (cid:96) ± -A) DIS data that dominated the early study of the nucleareffects in DIS measurements. In the early ’80s, the European Muon Collaboration [35]found that the per-nucleon structure functions F for iron and deuterium were not onlydifferent but also that this difference changed as a function of x . This intriguing resultinitiated an over decade long series of follow-up experiments from [228] up through [229]to investigate the nuclear modifications of this ratio, R [ F (cid:96)A ] = ( F (cid:96)A /A ) / ( F (cid:96)D / A . These experiments establishedthat in the scattering off nucleons within a nucleus in the deep-inelastic region with Q ≥ , the ratio of cross section per nucleon in nuclei to that in deuterium varies2 F ( x , Q ) Q (GeV/c) x=0.015 (X3)x=0.045 (X1.8)x=0.080 (X1.3)x=0.125x=0.175x=0.225x=0.275x=0.35x=0.45x=0.55x=0.65x=0.75 NuTeVCCFRCDHSWNuTeV fit
FIG. 9. NuTeV measurement of F ( x,Q ) (solid circles) compared with previous ν -Fe results; CCFR (open circles) andCDHSW (triangles). The data are corrected to an isoscalar (iron) target and for QED radiative effects as described in the text.The curve show the NuTeV model. x F ( x , Q ) Q (GeV/c) x=0.08 (x40) x=0.015 x=0.045 (x1.2)(x2)(x3.5)(x1.5)(x12)(x6) x=0.125x=0.175 x=0.225x=0.275x=0.35x=0.45x=0.55x=0.65x=0.75 NuTeVCCFR 97CDHSWNuTeV fit
FIG. 10. xF ( x,Q ) NuTeV (solid circles) compared with previous measurements; CCFR97 (open circles) and CDHSW(triangles). The data are corrected to an isoscalar (iron) target and for QED radiative effects as described in the text. Thecurve shows the NuTeV model. Figure 30.
Figure from [96]. NuTeV measurement of F (left) and x F (right)structure functions (solid circles) compared with previous ν -Fe results; CCFR (opencircles) and CDHSW (triangles). The data are on iron and corrected to an isoscalartarget and for QED radiative effects. The curve show the NuTeV model. considerably in the kinematic range from relatively small x ∼ − to large x ∼ . R [ F (cid:96)A ], can be divided into fourregions: • the shadowing region - R [ F (cid:96)A ] ≤ (cid:47) • the antishadowing region - R [ F (cid:96)A ] ≥ (cid:47) x (cid:47) • the EMC effect - R [ F (cid:96)A ] ≤ (cid:47) x (cid:47) • and the Fermi motion region - R [ F (cid:96)A ] ≥ (cid:39) x is the topic of a rigorous review [171]. Nuclearshadowing had been predicted long before it was observed experimentally in lepton-nucleus interactions. Glauber [230] was the first to suggest that a shadowing effectwould be due to successive interactions of the impinging object with nucleons in the3 = this analysis = CCFR = CDHSW 1.0 1.5 2.0 x=0.020 (CDHSW x=0.015)(CCFR x=0.018,0.025) (CCFR x=0.035,0.050) (CCFR x=0.070,0.090) (CCFR x=0.110,0.140) (CCFR x=0.180) (GeV ) F x=0.225 0.7 0.8 0.9 x=0.275 0.5 0.6 0.7 x=0.350 0.3 0.4 0.5 x=0.450 0.2 0.3 x=0.550 0.1 0.20.1 0.2 0.5 1 2 5 10 20 50100200Q (GeV ) F x=0.650 Figure 4: Comparison of our results with measurements from CCFR and CDHSW. The innerbars represent the statistical uncertainties and the outer bars the quadratic sum of statistical and systematicuncertainties. Numerical values of these measurements are available in Ref. [28]. 11
Figure 31.
Figure from [222] CHORUS [222] measurement of F structure functionsoff Pb (solid circles) compared with previous ν Fe results from CCFR (open circles)and CDHSW (triangles). nucleus. On the order of 15 years later Gribov [231] suggested that shadowing could begiven in terms of elementary diffractive scattering cross sections. Then, at the turn ofthe century, Strikman and Frankfurt [232] generalized the ideas of Glauber and Gribovleading to, when combined with the factorization theorem, a QCD leading twist (LT)model, which again incorporates rescattering of intermediate states.In most current models, the origin of the shadowing effect is related to the hadronicfluctuations of the intermediate vector boson. This resolved hadronic componentof the IVB will coherently interact several times with the different nucleons in thenucleus − multiple scattering. These multiple scatters destructively interfere resultingin a reduction of the corresponding cross sections − shadowing. While the basis of theexplanation with multiple scattering models is common, phenomenologically there isconsiderable variation in the details of application from model to model. The hadroniccomponent of the IVB may be given a partonic structure like in the dipole model [233]4 = this analysis = CCFR = CDHSW 0.2 0.4 0.6 x=0.020 (CDHSW x=0.015)(CCFR x=0.018,0.025) (CCFR x=0.035,0.050) (CCFR x=0.070,0.090) (CCFR x=0.110,0.140) (CCFR x=0.180) (GeV ) x F x=0.225 0.6 0.8 x=0.275 0.4 0.6 x=0.350 0.3 0.4 0.5 x=0.450 0.2 0.3 x=0.550 0.1 0.20.1 0.2 0.5 1 2 5 10 20 50100200Q (GeV ) x F x=0.650 Figure 5: Comparison of our results with measurements from CCFR and CDHSW. Theinner bars represent the statistical uncertainties and the outer bars the quadratic sum of statistical andsystematic uncertainties. Numerical values of these measurements are available in Ref. [28].12
Figure 32.
Figure from [222] CHORUS [222] measurement of xF structure functionsoff Pb (solid circles) compared with previous ν Fe results from CCFR (open circles)and CDHSW (triangles). or modeled as a superposition of hadronic states like vector meson dominance, orsome combination of both approaches. The models for shadowing were initiallydeveloped for charged lepton nucleus scattering, thus the vector current. More recentstudies [87, 234, 235] and explicitly [171] based on the dipole model clearly demonstratesthat there is a difference in the shadowing response for the vector and axial vectorcurrents. This is because the electromagnetic and weak interactions take place throughthe interaction of photons and W ± /Z bosons, respectively, with the target hadrons.Considering the large difference in mass, the hadronic fluctuation processes of photonsand W ± /Z bosons could be quite different.An additional difference in shadowing between the electromagnetic and weakprocesses is that sea quarks play an important role in this region of low x . The rolethey play is quite different in the case of the two processes. For example, the seaquark contribution, though small, is not same for F EM ( x, Q ) and F W I ( x, Q ) even atthe free nucleon level and could evolve differently in a nuclear medium. Therefore, a5microscopic understanding of the difference between F EM A ( x, Q ) and F W I A ( x, Q ) will bevery instructive for studying the nuclear medium effects in DIS processes as emphasizedat the NuInt15 [34] workshop.Note that with the well-accepted explanation of shadowing involving hadronicfluctuations of the vector boson into quark-antiquark pairs it is important to emphasize,as mentioned earlier, that the nuclear PDFs associated with the low - x shadowing regionare not necessarily the PDFs of a single bound nucleon but rather of multiple nucleonsin the nuclear environment.The anti-shadowing region is theoretically less well understood but might beexplained by the application of momentum, charge, and/or baryon number sum rules.There is work currently underway to follow up on an earlier study [236] that suggestsanti-shadowing is the constructive interference analog of the shadowing effect. Theseauthors also suggest that anti-shadowing is not universal but rather quark-flavordependent [237], which also suggests the idea of antishadowing is different dependingon the interaction being examined.The modifications at medium x (the so-called “EMC effect”) are still lacking aconvincing, community-accepted explanation, but have often been described as nuclearbinding and medium effects [238]. It has also been shown [188] that this ”EMC effect”persists at lower W in the resonance/transition region albeit at higher Q suggestingthis is not a purely high-W DIS effect. Along this line, there is now growing quantitativeevidence connecting the EMC effect with bound nucleons in short-range correlated(SRC) states [239]. This would suggest this effect is not for all nucleons within a nucleusbut is exhibited only for nucleons bound in multi-nucleon correlated states.With these qualifications, the evidence for nuclear effects in c harged-lepton nucleusscattering can be summarized in Fig. 33, which displays the F F e /F D structure functionratio, as measured by both the SLAC e-A and the BCDMS µ -A collaborations. TheSLAC/NMC curve is the result of an A-independent parametrization fit to calcium(providing measurements in the shadowing region) and iron charged-lepton nucleus DISdata [149, 193, 240].This SLAC/NMC curve has often been used as the standard nuclear correctionfactor (NCF) to convert data from a nuclear target to a free-nucleon target for bothcharged-lepton and neutrino interactions. However concern about the validity ofthe assumption that the NCF was the same for both charged-lepton and neutrinointeractions actually started with a comparison of NuTeV, CCFR and CHORUS resultswith theory/phenomenology predictions based on charged-lepton scattering results.A comparison of the NuTeV results with those of CCFR and the then currentpredictions of the major free-nucleon PDF-fitting collaborations CTEQ and MRST[241],[242] are shown in, Fig. 34 and Fig. 36 and, with emphasis on the F high-xregion, in Fig. 35. The CTEQ and MRST curves (labeled as ”TRVFS” that usedthe MRST2001E parton distribution functions) in Fig. 34 and Fig. 36 are correctedfor nuclear target effects using the Q -independent charged-lepton nuclear correctionfactors [40, 243], target mass effects [133] and QED radiative effects. Fig. 35 emphasizes6 Figure 33.
Figure from [244]. Nuclear correction factor, F F e /F D , as a function of x .The parametrized curve is compared to SLAC and BCDMS data [149, 151, 152, 193,228]. high-x behavior of these neutrino structure functions compared to the charged-leptonderived structure functions by comparing the NuTeV results with the BCDMS and SLACmeasured deuterium structure functions corrected for the measured c harged-lepton FeEMC effect.It is important to emphasize the observation that NuTeV structure functions agreewith the e/ µ -based theoretical calculations for 0 . ≤ x ≤ .
5. However, for x ≤ Q -dependence than the charged-lepton-based theoretical predictions while for 0 . ≤ x ≤ . ≥ smaller nuclear effects compared to charged-lepton scattering (cid:107) . It wasthen not a complete surprise that challenges were found when attempting to combinethese ν ( ν )-Fe results with (cid:96) ± -Fe and then, using (cid:96) ± -A nuclear correction factors, tocombine both with scattering results from free nucleons in global fits. To further testfor this suggested difference in charged-lepton and neutrino NCFs, the nuclear partondistribution functions were extracted independently by the nCTEQ collaboration forcharged-lepton-based and neutrino-based event samples. It is obvious from Fig. 33 that the structure function of nucleons within a nucleus aredifferent from the free nucleon structure functions. Assuming that both free nucleonsand nucleons in nuclei can be described with parton distribution functions (PDFs),this suggests that the PDFs for a nucleon within the nuclear environment (nuclearparton distribution functions - nPDFs) will be different than those of the free nucleon. (cid:107)
From Ref. [96] ”NuTeV perhaps indicates that neutrino scattering favors smaller nuclear effects athigh-x than are found in charged-lepton scattering.” -0.15-0.1-0.05 0 0.05 x=0.015 x=0.045 -0.1-0.05 0 0.05 0.1 x=0.080 x=0.125 -0.1-0.05 0 0.05 0.1 ( F - F T R V FS ) / F T R V FS x=0.175 x=0.225 -0.1-0.05 0 0.05 0.1 x=0.275 x=0.350 -0.1-0.05 0 0.05 0.1 x=0.450 x=0.550 Q (GeV/c) x=0.650
1 10 100 Q (GeV/c) x=0.750 FIG. 11. F ( x, Q ) fractional difference F − F TRV FS F TRV FS with respect to the TRVFS(MRST2001E) model. Data points are NuTeV(solid dots) and CCFR (open circles). Theory curves are ACOTFFS(CTEQ5HQ1) (solid line) and TRVFS(MRST2001E) ± σ (dashed lines). Theory curves are corrected for target mass and nuclear effects. Figure 34.
Figure from [96]. A comparison of the measurements of the F structurefunctions by NuTeV (solid dots) and CCFR (open circles) and the predictions fromthe global PDF fits of the CTEQ collaboration (CTEQ5) [241] (solid line) andTRVFS(MRST2001E) 1 (dashed lines). The results are normalized to the Thorne-Roberts variable-flavor scheme (TRVFS) NLO QCD model that used the MRST2001NLO PDFs [242]. The partonic structure of these nucleons within a nucleus must reflect the nuclearenvironment and, as has been mentioned, in some regions of x can better be consideredas ”effective” nPDFs representing the interaction with multiple nucleons within thenucleus. Consequently, the nucleus cannot simply be considered as an ensemble of Zfree protons PDFs and (A-Z) free neutron PDFs.Currently the analyses of both free nucleons and nucleons within a nuclearenvironment are based on the same factorization theorems [245]-[247] that do notin any way consider the relevant nuclear environment. The PDFs of a free protonare extremely well studied with several global analyses of free proton PDFs regularly8 ! Comparison with Charge Lepton Data for x>0.4 ! • Baseline is NuTeV model fit ! • data points are ! • charge lepton data is corrected for: ! - using CTEQ4D ! - heavy target ! the nuclear correction is dominated ! by SLAC data, which is at lower Q ! than NuTeV in this region ! • NuTeV agrees with charge lepton data for x=0.45. ! • NuTeV is higher than BCDMS(D ), different Q dependence ! - at x=0.55 , at x=0.65, and at x=0.75 ! • NuTeV is higher than SLAC(D ) (bottom 4 plots) ! - at x=0.55 , at x=0.65, and at x=0.75 ! “Perhaps the nuclear correction is smaller for neutrino scattering at high x .” ! Q ! Martin Tzanov
X = 0.45 ! X = 0.55 ! X = 0.65 ! X = 0.75 ! Figure 35.
A further examination of the high-x region of Figure 34 showing thebehavior of the NuTeV structure function F compared to deuterium measurementsfrom BCDMS and SLAC corrected for the measured (charged lepton) EMC effect onFe. updated [115, 137, 248],[249]-[251]. Nuclear PDFs have been determined by severalgroups [38]-[40], [244] using global fits to experimental data that include, mainly, deepinelastic scattering and Drell-Yan lepton pair production on nuclei. However the fitscan also include information from the LHC when nuclear ions are accelerated.Our knowledge of nuclear PDFs is much less advanced than the free nucleon PDFsdue to both theoretical and experimental limitations ¶ . For example, consider that thereis a contribution to nuclear PDFs coming from x ≥ ¶ The discussions and methods of the nCTEQ collaboration as presented in detail in the publicationK. Kovark et al. [29] are the basis for this section and will serve as an example of the process ofdetermining nuclear PDFs. -0.3-0.2-0.1 0 0.1 0.2 0.3 x=0.015 x=0.045 -0.1 0 0.1 x=0.080 x=0.125 -0.1 0 0.1 ( x F - x F T R V FS ) / x F T R V FS x=0.175 x=0.225 -0.1 0 0.1 x=0.275 x=0.350 -0.1 0 0.1 x=0.450 x=0.550 -0.4-0.2 0 0.2 0.4 1 10 100 Q (GeV/c) x=0.650
1 10 100 Q (GeV/c) x=0.750 FIG. 12. xF ( x, Q ) fractional difference xF − xF TRV FS xF TRV FS with respect to the TRVFS(MRST2001E) model. Datapoints are NuTeV (solid dots) and CCFR (open circles). Theory curves are ACOTFFS(CTEQ5HQ1) (solid line) andTRVFS(MRST2001E) ± σ (dashed lines). Theory curves are corrected for target mass and nuclear effects. Figure 36.
Figure from [96]. A comparison of the measurements of the x F structure functions by NuTeV (solid dots) and CCFR (open circles) and the predictionsfrom the global PDF fits of the CTEQ collaboration (CTEQ5) [241] (solid line) andTRVFS(MRST2001E) 1 (dashed lines). The results are normalized to the Thorne-Roberts variable-flavor scheme (TRVFS) NLO QCD model that used the MRST2001NLO PDFs [242]. must be included simultaneously in the fits, the non-trivial nuclear A dependence ofthe PDFs must be considered by including a parmetrization of the A-dependence. Theconstraints on this parametrization are only as strong as the accuracy of the data in thefit. In spite of these challenges, as long as the fit was charged-lepton-based and the moreaccurate ν µ -A DIS data were not used in these fits the existing global nPDF analysesgenerally led to a reasonable description of the data confirming this picture. There areessentially three types of global fits to determine the nPDFs:0 • Those that fit a multiplicative correction factor to apply to the free nucleon PDFs. f ( p/A ) i ( x, Q ) = R i ( x, Q, A ) f free protoni ( x, Q )This method was used by the groups that pioneered the extraction of nPDFs [38]-[40]. • An attempt was made to use a convolution method [37] to isolate the nPDFs. • And finally the method of native nuclear PDFs extracted using the same procedureas the free nucleon PDFs.It is this last method employed by the Nuclear CTEQ Collaboration(nCTEQ)group, a subgroup of the full CTEQ collaboration, that will be used as an exampleto describe the extraction of nPDFs in more detail.In the nCTEQ framework [29], the parton distributions of the nucleus areconstructed as: f ( A,Z ) i ( x, Q ) = ZA f p/Ai ( x, Q ) + A − ZA f n/Ai ( x, Q ) , (79)Isospin symmetry is used to construct the PDFs of a neutron in the nucleus, f n/Ai ( x, Q ),by exchanging up- and down-quark distributions from those of the proton.The parametrization of individual parton distributions are similar in form to thatused in the free proton CTEQ fits [240, 254, 255] and takes the following form at theinput scale Q : xf p/Ai ( x, Q ) = c x c (1 − x ) c e c x (1 + e c x ) c i = u v , d v , g, ¯ u + ¯ d, s + ¯ s, s − ¯ s, ¯ d ( x, Q )¯ u ( x, Q ) = c x c (1 − x ) c + (1 + c x )(1 − x ) c . (80)The input scale is chosen to be the same as for the free proton fits [240, 255],namely Q = 1 . Q to alower value to better reflect the Q range of current neutrino experiments.As in the other available nuclear PDFs [38]-[40], nuclear targets are characterizedby their atomic mass number A . However, in contrast to those groups that derive amultiplicative factor to apply to the free proton PDFs, in the nCTEQ analysis theadditional A dependence is introduced directly to the c -coefficients c k → c k ( A ) in Eq.80. The c k ( A ) are defined such that for A = 1 one recovers the underlying PDFs of a freeproton that are described in [240] and which have the advantage of minimal influencefrom nuclear data. nCTEQ nPDFs for a nucleus A without including νA results as input The datacurrently used in this global fit for nPDFs are from charged lepton DIS, and Drell-Yan lepton pair production experiments and are subject to the following cuts:1 • DIS:
Q >
W > . • DY: M ≥ M is the invariant mass of the produced lepton pair)These cuts are considerably more restrictive than other nuclear PDF analyses with thegoal of limiting the importance of both kinematic and dynamic higher twists in the fit.The results of this nCTEQ fit (labeled nCTEQ15 in the literature) yield the A -dependence of the various nPDF flavors of a proton in nucleus A illustrated in Fig. 37where the central fit predictions for a range of nuclear A values from A = 1 (proton) to A = 208 (lead) are displayed. -3 -2 -1 x x f p / A g -3 -2 -1 x x f p / A s A=1A=4A=9A=12A=27A=40A=56A=84A=119A=131A=197A=208 -3 -2 -1 x x f p / A u v -3 -2 -1 x x f p / A d v -3 -2 -1 x x f p / A u -3 -2 -1 x x f p / A d -3 -2 -1 x x f p / A ¯ u -3 -2 -1 x x f p / A ¯ d Figure 37.
Figure from [29]. The A-dependence of the nCTEQ nuclear proton PDFsat the scale Q = 10 GeV for a range of nuclei from the free proton ( A = 1) to lead( A = 208). Fig. 38 shows the nPDFs ( f p/P b ) for a proton in a lead nucleus at the input scale Q = Q = 1 . x ≤ − and x ≥ . ν/ν -A scattering results. The next section will describe the2 / ( , ) = nCTEQ1510 / ( , ) / ( , ) / ( , ) Figure 38.
Figure from [29]. Results of the nCTEQ fit displaying the actual PDFsfor a proton in lead at the Q scale of Q = 1 . nCTEQ approach to determining the nPDFs for neutrino nucleus scattering. The first attempt at measuring nuclear effects, yielding a nuclear correction factor, with ν was performed by the BEBC bubble chamber experiment from the ratio of neon andhydrogen targets [256] in the mixed Ne-H filling of the chamber. The measurementprovided a suggestion of nuclear shadowing at small x and Q values, however, thelarge associated errors of these lower statistics measurements precluded any carefulcomparison with charged-lepton results. Consequently, in earlier QCD global fits ofnucleon PDFs that attempted to include neutrino nuclear DIS data, the charged-leptonnuclear correction factors (Fig. 33) were simply applied to neutrino nucleus scatteringresults as well.It was immediately noted that these early attempts to include neutrino-nucleusDIS scattering data, corrected with charged-lepton NCFs, introduced such tension3in the shadowing region at low-x in global QCD fits that the low-x neutrino datawas simply excluded in these early CTEQ nucleon PDF global analyses. In morerecent examinations of higher-x parton distribution functions, carried out by the CTEQcollaboration [240, 257], indications began to accumulate that the nuclear correctionfactors for neutrino nucleus scattering not only in the shadowing region could indeed bedifferent than those for charged-lepton nucleus scattering. A conclusion already voicedand quoted by the NuTeV collaborationA study to check these indications was then initiated by the nCTEQ collaborationto extract the neutrino nuclear correction factor F νA ( x, Q ) / F νN ( x, Q ). The sameprocedure used to determine the correction factor for charged lepton nucleus scatteringthat resulted in the SLAC/NMC curve, was used. + To apply this procedure to ν -Ascattering, there were several data sets considered. The earliest is the CDHSW ν -Fedata followed by the CCFR ν -Fe data, the NuTeV ν -Fe data and finally the CHORUS ν -Pb data. The weights of these data sets in the combined fit were dictated by the errorson the data. The NuTeV ν -Fe and CHORUS ν -Pb data had associated full covarianterror treatment of the data, yielding maximal discriminatory power of the data. Theweight of the CDHSW and CCFR data, with their errors calculated via the sum ofthe squares of statistical and systematic errors, when combined with the NuTeV andCHORUS data with their full covariant error matrix for the fit, was greatly reduced.Furthermore, even though both the NuTeV and CHORUS data sets have full covarianterror matrices, the relatively small NuTeV errors with respect to the CHORUS errorsenabled the NuTeV data points to dominate the combined fit.An additional input to the fits was the NuTeV and CCFR di-muon data [258]off Fe, which are sensitive to the strange quark content of the nucleon in the nuclearenvironment of Fe. However, no other data such as charged-lepton nucleus ( (cid:96) ± A) andDY were used. Because the neutrinos alone do not have the power to constrain all of thePDF components, a minimal set of external constraints [259] also had to be employedand some of these external assumptions do indeed affect the behavior of the fit partondistributions at small x - the shadowing region. These include the Callan-Gross relation( F νA = 2 xF νA ) as well as use of the assumption s = s and c = c . In subsequent fitsof neutrino data, the results of the NuTeV analysis [260] of the s- s asymmetry will beincluded.It is important to note that the nCTEQ fit was made directly to the NuTeV andCHORUS measured double differential cross sections in order to extract the set ofnPDFs of the nucleon in the nucleus. The fit did not use the extracted NuTeV andCHORUS structure function results of the average value of F ( x, Q ), which containsall the nuclear-dependent assumptions made to extract them such as, presumably A-dependent, R em ( σ emL /σ emT ) being used instead of R weak ( σ W IL /σ W IT ) and ∆ xF . Theextracted nPDFs were then taken in ratio to the free-nucleon PDFs [240] to form the + It should be apparent that the rather restrictive Q and DIS minimal Q and W cuts from thecharged-lepton-based fits when applied to neutrino scattering results would rule out most contributionsfrom contemporary neutrino nucleus experiments and are thus also being carefully reconsidered. x -1
10 1 ] F e n R [ F =5 GeV QA=56, Z=26 fit A2KPSLAC/NMC HKN07 (NLO) x -1
10 1 ] F e n R [ F =20 GeV QA=56, Z=26 fit A2KPSLAC/NMC HKN07 (NLO) ( a ) ( b ) Figure 39.
Figure from [244]. Nuclear correction factor R for the structure function F in charged current νF e scattering at a) Q = 5 GeV and b) Q = 20 GeV .The solid curve shows the result of the nCTEQ analysis of NuTeV differential crosssections (labeled fit A2), divided by the results obtained with the reference fit (free-proton) PDFs; the uncertainty from the A2 fit is represented by the yellow band.Plotted also are NuTeV data points of the average F to illustrate the consistency ofthe fit with the input points. For comparison the correction factor from the Kulagin–Petti (KP) model [261] (dashed-dot line), from the Hirai, Kumano, Nagai (HKN07) fit[38] (dashed-dotted line), and the SLAC/NMC parametrization, Fig. 33 (dashed line)of the charged-lepton nuclear correction factor are also shown. We compute this for { A = 56 , Z = 26 } . x -1
10 1 ] F e n R [ F =5 GeV QA=56, Z=26 fit A2KPSLAC/NMCHKN07 (NLO) x -1
10 1 ] F e n R [ F =20 GeV QA=56, Z=26 fit A2KPSLAC/NMCHKN07 (NLO) ( a ) ( b ) Figure 40.
The same as in Figure 39 for νF e scattering. individual values of the nuclear correction factor R for a given x and Q . It is alsoimportant to note that these free-nucleon PDFs that were used in the denominator ofthe nuclear correction factors were a special fit to ensure that any data involving nucleartargets was minimally involved. These fits were performed separately for neutrino andanti-neutrino - not the average of both - as shown in Fig. 39 for ν – F e and in Fig. 40 for¯ ν - F e .It was also possible to combine the fitted neutrino nPDFs to form the average of F ( νA ) and F (¯ νA ) for a given x, Q to compare directly with the NuTeV publishedvalues of this quantity. This was also performed by nCTEQ and results can be found5in [262].These studies by nCTEQ [41] have shown a strong indication that there is indeed adifference between the (cid:96) ± A and the νA nuclear correction factors. An analysis by theHKN [263] group also finds some inconsistencies between ν ( ν ) and charged-lepton dataand most recently, a direct comparison [156] of F νF e with F (cid:96) ± F e structure functionsobserved a clear ( ≈ ν ( ν ) and charged lepton scattering off Fefor the structure functions at low x. (cid:96) ± A and νA Nuclear Correction Factors
Certainly there are similarities in the general shape of the nCTEQ νA and theSLAC/NMC (charged-lepton) nuclear correction factors. However the magnitude ofthe effects and the x -region where they apply are quite different. The nCTEQ νA fitsconfirm the earlier impression from the NuTeV collaboration that the size of the nuclearcorrections affecting the NuTeV data are not as strong as those obtained from chargedlepton scattering.The nCTEQ ν -A NCFs are noticeably ”flatter” than the SLAC/NMC curve,especially at lowest and moderate- x where the differences are significant. In the ν case, these differences are smaller but persist across the full x range. The nCTEQcollaboration emphasize that both the charged-lepton-based and neutrino-based resultscome directly from global fits to the data. Other than the assumptions stated earlier,there is no model involved. They further suggest that this difference between the resultsof charged-lepton and neutrino DIS is reflective of the long-standing “tension” betweenthe light-target charged lepton data and the heavy-target neutrino data in the historicalglobal PDF fits [264, 265] particularly at small x. These nCTEQ results further suggestthat the tension is not only between charged-lepton light-target data and neutrino heavy-target data, but also between neutrino and charged-lepton heavy-target data as well. Inother words a difference between charged-lepton ( (cid:96) ± -A) and the neutrino ( ν -A) nuclearcorrection factors when comparing the same A.The general trend is that the anti-shadowing region is shifted to smaller x values,and any decrease at low x is minimal at Q = 5 GeV where shadowing is clearlyobserved in (cid:96) ± -A scattering. The fit to ν -A in the shadowing region gradually approachesthe charged-lepton fit with increasing Q . However, the slope of the fit approaching theshadowing region from higher x, where the NuTeV measured points and the nCTEQ fitare consistently below the charged-lepton Fe fit, make it difficult to reach the degree ofshadowing evidenced in charged-lepton nucleus scattering at even higher Q .There is indeed shadowing observed in ν -A scattering however at lower Q than the5 GeV of the general comparison above. This only heightens the difference between ν -A and (cid:96) ± -A nuclear correction factors. Referring to Fig. 34 it can be clearly seenthat NuTeV and CCFR data favor a significant trend toward increased shadowing as Q decreases down to ≈ . This could suggest significant shadowing in theregime of modern neutrino experiments with their low E ν dominated beams. This point6will be addressed shortly (see 5.8).Concentrating on these interesting differences found by the nCTEQ group, if thenuclear correction factors for the (cid:96) ± -A and ν -A processes are indeed different thereare several far-reaching consequences. For example, what happens to the conceptof ”universal parton distributions”. To maintain the universality of nuclear partondistributions is there an additional term in the factorization ansatz needed to reflect theresponse of the nuclear environment to vector and axial vector probes?Considering these possible significant consequences, the nCTEQ group performeda unified global analysis [41] of the (cid:96) ± -A, DY, and ν -A data to determine if it would bepossible to obtain a “compromise” solution including both (cid:96) ± -A and ν -A data. Theyused a hypothesis-testing criterion based on the χ distribution that can be applied toboth the total χ as well as to the χ of individual data sets. Noting the large differencein the number of involved data points ( (cid:96) ± -A + DY) (708) and the ν -A (3134), theyintroduced a weight (w) applied to the neutrino data sample that allowed adjustmentfor this rather large difference between the samples. With w = 0, only the (cid:96) ± -A + DYwas fit, w = 1 was a straight fit to both the (cid:96) ± -A + DY and the ν -A samples whilew = ∞ was a pure ν -A fit. The results of the fit are displayed in Fig. 41 and thecorresponding w-dependent nuclear parton distribution functions are shown in Fig. 42
274 K. Kovaˇrík et al.
Table 1 continuedID d σ ν A dx dy Experiment µ µ Table 2
Summary table of a family of compromise fits w l ± A χ (/pt) ν A χ (/pt) Total χ (/pt)0 708 638 (0.90) – – 638 (0.90)1 / / ∞ – – 3,134 4,192 (1.33) 4,192 (1.33) x
10 10 1 ] A l R [ F ° w 11/21/70===== a x
10 10 1 ] A ν R [ F ° w 11/21/70===== b Fig. 1
Predictions from the compromise fits for the nuclear correction factors R [ F ℓ Fe ] ≃ F ℓ Fe / F ℓ N (left) and R [ F ν Fe ] ≃ F ν Fe / F ν N (right) as a function of x for Q = . The data points displayed in figure a) are from BCDMS and SLACexperiments [3–5,8,11] and those displayed in figure b) come from the NuTeV experiment [20,21] We first examine the nuclear correction factors R [ F Fe ] ≃ F Fe / F N needed to correct the nuclear datato the free nucleon level. We compute these quantities in the QCD parton model at next-to-leading orderemploying the NPDF fits in Table 2. The x -dependence of R [ F Fe ] is shown in Fig. 1; similar results hold at Q =
20 GeV which we do not present here. The w = ℓ A DIS+DY data, and this agreeswell with the SLAC and BCDMS points [3–5,8,11] displayed in Fig. 1a). However, as we mix in the ν A data,Table 2 shows the χ of the ℓ A data rise from 638 for w = w =
1. Correspondingly, the w = ∞ fit uses only the ν A data, and this agrees well with the data from the NuTeV experiment [20,21] displayed inFig. 1b). Now as we mix in the ℓ A DIS+DY data, we see the χ of the ν A data rise from 4192 for w = ∞ to4710 for w = /
7. Finally, comparing the results obtained with the w = w = ∞ fits one can seethat they predict considerably different x -shapes.The fits with weights w = { , , } interpolate between these two incompatible solutions. As can be seenin Fig. 1a, b, with increasing weight the description of the ℓ Fe data is worsened in favor of a better agreementwith the ν Fe points. This trend clearly demonstrates that the ℓ Fe and the ν Fe data pull in opposite directions.We identify the fits with w = / w = χ given that the fit has N degrees of freedom: P ( χ , N ) = ( χ ) N / − e − χ / N / % ( N / ) . (2)This allows us to define the percentiles ξ p via ! ξ p P ( χ , N ) d χ = p % where p = { , , } . Here, ξ serves as an estimate of the mean of the χ distribution and ξ , for example, gives us the value where thereis only a 10% probability that a fit with χ > ξ genuinely describes the given set of data. In a global PDFfit, the best fit χ value often deviates from the mean value because the data come from different possiblyincompatible experiments having unidentified, unknown errors which are not accounted for in the experimental The details of this definition are outlined in Refs. [16,17]. While we focus on F , we can consider other observables such as { F , F , d σ } in a similar manner. Figure 41.
Figure from [42]. Predictions for the compromise fits for a) (cid:96) ± Fe + DYon the left and b) νF e on the right for the indicated weight w as a function of x at Q = 5 GeV . It was concluded by these authors that it was not possible to accommodate the datafrom ν -A and (cid:96) ± -A DIS by an acceptable combined fit. That is, when investigating theresults in detail, the tension between the (cid:96) ± -Fe and ν -Fe data sets permits no possiblecompromise fit which adequately describes the neutrino DIS data along with the charged-lepton data and, consequently, (cid:96) ± -Fe and ν -Fe have different nuclear correction factors.A compromise solution between ν -A and (cid:96) ± -A data can be found only if thefull correlated systematic errors of the ν -A data are not used and all the statisticaland systematic errors are combined in quadrature thereby neglecting the informationcontained in the correlation matrix. This conclusion underscores the fundamentaldifferences [41] of the nCTEQ analysis with some of the other contemporary analyses [40,266] using different statistical methods. These other analyses suggest the ν -A and (cid:96) ± -ADIS data can be statistically consistent and relates the discrepancies to possible energy-7 Preliminary
Figure 42.
Figure from [34]. Predictions for selected nuclear parton distributions inFe for the indicated weight w as a function of x at Q = 5 GeV . The main comparisonis for the w=0, pure electroproduction and w = infinity, pure neutrino scattering. Theshaded areas are where no appreciable date was available dependent fluctuations of the NuTeV analysis. In particular they cite non-negligibledifferences in the absolute normalization between different neutrino data sets that, theyclaim, are large enough to prevent a tension-free fit to all data simultaneously.On the other hand, a difference between ν -A and (cid:96) ± -A is not completely unexpected,particularly in the shadowing and antishadowing regions, and has previously beendiscussed in the literature [60, 236, 237]. The charged-lepton processes occur(dominantly) via γ -exchange, while the neutrino-nucleus processes occur via W ± -exchange. Since, as was stated, a (simplified) explanation of shadowing is thathadronic fluctuations of the vector boson interact coherently (like a ”pion”) off multiplenucleons in the nucleus and the interactions interfere destructively, the different nuclearshadowing corrections could simply be a consequence of the differing propagation of thehadronic fluctuations of the intermediate bosons (photon, W ) through dense nuclearmatter. Perhaps the shadowing difference is due to the difference in vector bosonmasses, the W-boson is a much more localized probe than the photon. The differencein antishadowing could indeed be a consequence of the quark-flavor dependence ofantishadowing proposed by [237].In particular, theoretical calculations [60] specifically for ν nucleus scatteringsuggest that at small x in the shadowing region the nuclear correction for neutrinos,as opposed to charged leptons, does have a rather strong Q dependence. The standardnuclear correction obtained from a fit to charged lepton data implies a suppression of ≈
10% for iron compared to deuterium independent of Q at x = 0.015. While for x =0.015 reference [60] finds a suppression of 15% at Q = 1 . GeV and a suppression8of 3.4% at Q = 8 . GeV . This predicted effect improves agreement with NuTeV dataat low-x. In addition, this definite Q dependence of the F structure function on Fe atlow x is supported by the predictions of the model of reference [267] shown in Fig. 5 ofthat reference.Furthermore, since the structure functions in neutrino DIS and charged lepton DISare distinct observables with different parton model expressions, it is not surprisingthat the nuclear correction factors would not be exactly the same. What is, however,unexpected is the degree to which the R factors differ between the structure functions F νF e and F (cid:96) ± F e . In particular the lack of evidence for shadowing in neutrino scatteringat Q = 8.0 GeV down to x ∼ .
02 is quite surprising.Should subsequent experimental results and analyses confirm the rather substantialdifference between charged-lepton and neutrino scattering in the shadowing region atlow- Q it is interesting to speculate on the possible cause of the difference. A study ofEMC [35], BCDMS [199] and NMC [154] & data by a Hampton University - JeffersonLaboratory collaboration [268] suggests that anti-shadowing in charged-lepton nucleusscattering may be dominated by the longitudinal structure function F L . As a by-productof this study, their figures hint that shadowing in the data of µ -A scattering is being ledby the transverse cross section with the longitudinal component crossing over into theshadowing region at lower x compared to the transverse.As summarized earlier, in the low- Q region, the neutrino cross section is dominatedby the longitudinal structure function F L via axial-current interactions since F T vanishesas Q → Q demonstrated by ν -A and (cid:96) -A, in addition to the different hadronic fluctuationsin the two interactions, could be due to the different mix of longitudinal and transversecontributions to the cross section of the two processes in this kinematic region.Another hypothesis of what is causing the difference between neutrino and charged-lepton shadowing results comes from Guzey et al. [268] who speculates that at low x,low- Q the neutrino interactions primarily probe the down and strange quarks. This isvery different than the situation with charged-lepton scattering where the contributionfrom down and strange quarks are suppressed by a factor of 1/4 compared to the upand charm. Therefore, the discrepancy between the observed nuclear shadowing in (cid:96) ± -Fetotal cross section at small x and shadowing in total ν -Fe cross section could be causedby the absence of nuclear shadowing of the strange quark nuclear parton distributionsas extracted from the neutrino-nucleus data or even the poor knowledge of the strange-quark distribution in the free-nucleon that affects the neutrino-nucleus ratio more thanthe charged-lepton. These suggestions are not inconsistent with the results shown inFig. 42 that indicate no shadowing of the strange quark for neutrino scattering off Fewith the nCTEQnu nPDFs determined with ν -A scattering data.It is worth repeating to emphasize that this difference in nPDFs depending onwhether extracted from ( ν/ν -A)-based or ( (cid:96) ± -A)-based interactions is a suggestion ofnon-universal nuclear parton distributions. A way to salvage this concept of universal9parton distributions could be to modify factorization to include consideration of thetype of interaction in the nuclear environment. ν -A Interactions Current and particularly the future DUNE long baseline oscillation experiments, haveneutrino energies up to (cid:47)
10 GeV. For such a broad range of neutrino energy, they willhave to use information from the hadronic system in order to estimate the actual E ν ofan event and estimate the backgrounds to their signal topologies. Specific models forquasi-elastic and one-pion resonance production are available. However, for example inthe GENIE simulation program, multi-pion production through resonance decay and allnon-resonant pion production are grouped together under the name GENIE ”DIS” andthe multiplicity of a given event is chosen through models that describe the hadronizationof the initial hadronic component of the interaction. They will then need models thatdescribe the initial state hadronization of the hadronic shower that is then followedby final state interactions of these produced hadrons. A good survey of the currenthadronization models now in use within the community can be found in section sevenof [34].In the DIS region these hadonization models describe the formation of hadronsin inelastic interactions and are characterized by non-perturbative fragmentationfunctions (FF), which in an infinite momentum frame can be interpreted as probabilitydistributions to produce a specific hadron of type h with a fraction z of the longitudinalmomentum of the scattered parton. These universal fragmentation functions can notbe easily calculated but can be determined phenomenologically from the analysis ofhigh-energy scattering data ∗ .Modern event generators often use the LUND string fragmentation model [271, 272],as implemented in the PYTHIA/JETSET [273] packages, to describe the hadronizationprocess. This model results in a chain like production of hadrons with an associated FFproviding the probability that a given ratio z between the hadron energy and the energytransfer is selected. The PYTHIA/JETSET implementation of this LUND model iscontrolled by many free parameters, which can be tuned to describe the data. A detailedstudy of the PYTHIA fragmentation parameters with ν data [274] from proton anddeuterium targets was performed in Ref. [275]. In particular, the various parameter setsdetermined by the HERMES experiment were used within the GENIE event generatorobtaining predictions in agreement with the measured hadron multiplicities.An independent tuning of the JETSET fragmentation parameters was performedin Ref. [276] with NOMAD data from exclusive strange hadron production and inclusivemomentum and angular distributions in ν -C DIS interactions. However, as has beennoted, in ν -nucleus interactions the hadrons originating from the primary interactioncan re-interact inside the nucleus. These final state interactions must, therefore, be ∗ An example of a recent study of pion and kaon FF in e + e − collisions can be found in Ref. [269] whilethe FF for charmed hadrons ( D, D s , Λ c ) in ν l DIS interactions were studied in Ref. [270]. ≈ W < GeV , a better description of the data has been achievedwith a phenomenological description of the hadronization process in which the averagehadron multiplicities are parametrized as linear functions of log W for each channel. ThisKoba-Nielsen-Olesen (KNO) scaling law [277] can then be used to relate the dispersionof the hadron multiplicities at different invariant masses. Both the averaged hadronmultiplicities and the KNO functions are usually tuned from ν bubble chamber data.The challenge faced by the neutrino simulation programs is how to bridge thetransition from the KNO procedure used at low W to the PYTHIA/JETSET LUND-based model at higher W. To do this the GENIE [97] generator uses the hybrid AGKYapproach [278], which has a gradual transition from the KNO hadronization modelto PYTHIA in the region 2 . ≤ W ≤ . W . The NEUT [98] generator has a more abrupttransition for the hadronization process, using KNO for W <
W >
An excellent overview of this topic can befound in [279]. The authors cover the full spectrum of available treatments of this topicas they apply to hadronization in the lower-W kinematic region.The model used by the GENIE simulation program, the AGKY (initials of themain author’s names - Andreopoulos, Gallagher, Kehayias and Yang) hadronizationmodel [278], was developed for the MINOS experiment. The model is split into three W regions shown in Fig. 43 with the AGKY model used to cover the hadronization ofthe GENIE DIS (horizontal hatched curve) in the figure. Also, as mentioned earlier,the so-called DIS region in GENIE extends to the low W resonance region to describenon-resonant pion production as well as resonant multi-pion production in the resonanceregion.At lower W ≤ W increases beyond 2.3 GeV, the AGKY model gradually transitionsfrom this KNO model to PYTHIA [280] which is used for W ≥ W . As1 ) /c (GeV W e v en t s KNO PYTHIATransitionRES DIS
TotalQuasi-elasticResonanceDIS
Figure 43.
From reference [279]. W distribution of ν µ -water target interactions inGENIE showing the quasi-elastic scattering, the resonance interactions, and the DISregion. The W distribution is further split into the three regions, KNO scaling-basedmodel only region, PYTHIA only region, and the transition between the two regionsused in the AGKY model. Hadronization modelling
Fractions of GENIE events generated by each hadronization model: (GeV) ν LAB E f r a c t i on o f a ll e v en t s DIS charm productionSIS/DIS (PYTHIA)SIS/DIS (KNO)Resonance production
Selected W (GeV) f r a c t i on o f i ne l a s t i c e v en t s DIS charm productionSIS/DIS (PYTHIA)SIS/DIS (KNO)Resonance production
C.Andreopoulos (Liverpool/STFC-RAL) GENIE October 13, 2018 7 / 48
Figure 44.
From Andreopoulos presentation in reference [34]. The figure presentsthe division of events coming from the GENIE 1- π resonance model and using theAGKY model to generate events as a function of E ν in GENIE. W increases the fraction of events hadronized using the PYTHIA model increaseswhile the fraction using KNO decreases linearly. PYTHIA is a standard hadronizationtool for higher energy physics experiments used by neutrino interaction generators forhadronization at the relatively higher W region. Whether PYTHIA can be applied tosuch low W and resulting low multiplicities is not at all clear. Refer to [279] for furtherdetails of the KNO and PYTHIA models.The actual results of the application of the AGKY model within GENIE is shownin Fig. 44. It is evident that already with an E ν of ≈ GeV the meson multiplicitiesare coming more from KNO determination than from the GENIE 1- π model. It is2also important to restate that such a procedure suggests that the KNO model is beingused to govern non-resonant pion production rather than the explicit calculation of therelevant theory involved in the process. The FLUKA neutrino event generator is called NUNDISthat describes the neutrino-nucleon interactions from Quasi Elastic through resonanceproduction and into Deep Inelastic Scattering. Hadronization is performed with theFLUKA models based on the LUND string models, for details see Sala’s summary in [34]and[281] from which much of this description has been drawn. They find that for verylow mass situations standard hadronization has to be replaced by what they refer to asa ”phase space explosion”. This treatment has proven to be important for the correctsimulation of single-pion production in neutrino interactions. Although traditionallyassociated only to resonance production, FLUKA finds the DIS contribution to thesingle-pion channel is significant and an important contribution to the one-pi channelin ν -nucleon scattering.Important for FLUKA, and included in GENIE, is the introduction of FormationZone that can be understood as hadrons emerging from an inelastic interactionthat require some time before beginning strong interactions with the surroundingenvironment. This has the effect of allowing certain hadrons to escape any final stateinteractions within the nucleus. Formation Zone is then important to correctly modelhadronic interactions as is illustrated in Fig. 45 that shows the effect on both eventmultiplicities and the momentum spectra of these secondaries when the consideredformation zone is varied. For Formation zone set to 0 - no formation zone - the producedhadrons within the shower can immediately interact within the nucleus thus the averagenumber of hadrons leaving the nucleus is largest and the average momentum of thesehadrons is the smallest. As the formation zone increases more of the hadrons leavethe nucleus without interacting and the average multiplicty decreases with the averagemometum of the hadrons increasing. Although it has been emphasized that neutrino DIS scattering could be a particularlyrich source for for flavor separation in determining free proton parton distributionfunctions, a serious problem in the neutrino community is the very poor state ofknowledge of ν - f ree nucleon interactions. There are presently only low-statistics bubblechamber results from the 1970’s and 1980’s that have relatively large statistical andsystematic errors. This severely limits the influence of neutrino scattering in free nucleonPDFs. That these rather imprecise results are then used as the start of neutrinointeraction simulations by the current community’s event generators is also a matterof real concern. In addition, this also forces the determination of the denominator ofnuclear correction factors for neutrino experiments to use a phenomenological estimateof ν -free nucleon cross sections and structure functions formed from free nucleon PDFs.3 Effect of formation zone, neutrino int.
Total hadron multiplicity Charged hadron spectra
Effect of formation zone, neutrino int.
Total hadron multiplicity Charged hadron spectra
Multiplicity
Figure 45.
From Sala presentation in reference [34]. The figures emphasize therelative change in distributions as a function of formation length with the vertical axisa arbitrary number of events. The resulting dependence of the event multiplicities(left) and particle momentum distributions (right) are from FLUKA for a 10 GeVneutrino on oxygen when the formation length is varied over a wide range.
Turning then to neutrino nucleus scattering, the NuTeV ν -Fe and CHORUS ν -Pbexperiments are the most recent high-statistics DIS experiments that have publisheddouble-differential ν/ν -A scattering cross sections as well as very detailed studies ofsystematic errors. To be able to combine these NuTeV and CHORUS ν -A results withother experiments in global fits of free-nucleon PDFs, a way of converting ν -Fe/Pb to ν -nucleon - nuclear correction factors - had to be determined.Using the results from these experiments, nuclear effects of charged current deepinelastic ν -A scattering were studied by the nCTEQ collaboration in the frame-work ofa χ analysis and, in particular, a set of iron nuclear correction factors for iron structurefunctions was extracted. Comparing these results with structure function correctionfactors for (cid:96) ± -Fe scattering it was determined that the neutrino correction factors differin both shape and magnitude from the correction factors for (cid:96) ± -Fe scattering.This difference, although not unexpected theoretically especially in the shadowingand antishadowing regions, is not universally seen by all groups examining nPDFs ofneutrinos. It is imperative that we carefully consider these contrasting results and gainan understanding of the ν -A nuclear correction factors. The nCTEQ study of the ν -Feand ν -Pb nPDFs provides a foundation for a general investigation that can address thistopic. However the results from a much wider variety of nuclear targets in a neutrinobeam, able to access DIS kinematics, will be needed to definitively answer this question.The MINER ν A neutrino-nucleus scattering experiment at Fermilab [282], acollaboration of high-energy and nuclear physicists, is currently analyzing dataperforming a systematic study of neutrino nucleus interactions. The overall goals ofthe experiment are to measure absolute exclusive and inclusive cross-sections and studynuclear effects in ν - A interactions with He, C, O, Fe and Pb nuclear targets.For QCD oriented studies MINER ν A is pursuing systematic studies of theresonance-to-DIS (SIS) transition region and the lower-Q DIS region. The MINER ν A4experiment has finished both their low- E ν (LE) exposure and their somewhat higherenergy (ME) exposure that yielded a much higher fraction of DIS events with aconsiderably broader kinematic range than the lower energy data and is currently beinganalyzed. MINER ν A used the low-energy (LE) NuMI beam to initiate a first study ofthe DIS cross sections off the MINER ν A suite of nuclear targets and published [226] thecross section ratios of target A to the nominal scintillator (CH) of the main tracker asshown in Fig. 46.These results can be compared to the predictions of nCTEQ nuclear PDF sets,namely the nCTEQnu nPDFs based on neutrino nucleus DIS scattering data. Thepredictions of the extracted neutrino-based nuclear PDFs can be seen in Fig. 47 (left)that shows the predicted ratios using these neutrino-based nCTEQnu nuclear PDFs ata Q of 1.7 GeV . This is roughly the average Q of the lowest x bin and close tothe average of the neighboring x-bin in the cross section ratio of Pb to CH. Fig. 47(right) displays the MINER ν A measured values for the x-dependent cross section ratiosof Pb to CH compared to several current models for this ratio, based on charged-lepton nuclear effects, as well as the predictions of nCTEQnu ( ν -A) nuclear partondistributions. Although this is not the ratio of F as the figure on the left, in this smallx region the contribution of xF is small so the cross section is dominated by F . Inthe lowest x bin. With the data having an approximate Q of 1.8 GeV , the nCTEQnupredictions can be read off the plot to the left. Certainly the associated uncertaintiesare significant, however the measured points do favor the nCTEQnu predictions thatreflect the low-x, low- Q results of the NuTeV, CCFR and CHORUS results. What does MINERvA see? LE DIS Cross Section Ratios – d s /dx.Much improved ME beam ratios soon to be released! The Q distribution within an x bin is essential! ◆ The shape of the data at low x, especially with lead is consistent with nuclear shadowing at
NOT Isoscalar Corrected dx CH σ d : dx Pb σ dRatio of J. Mousseau nCTEQnu – Pb/C 1.7 GeV Joel Mousseau 45
DIS Ratios: dσ /dx ● Our data suggest additional nuclear shadowing in the lowest x bin (0 < x <0.1) than predicted in lead. ● There are some hints of this as well in Iron. ● Lowest x bin is a ~ 2.0 (GeV/c) ● In the EMC region (0.3 < x < 0.75), we see good agreement between data and simulation.
C/CH Fe/CH Pb/CH nCTEQ15nCTEQnu
Figure 46.
Figure from [226]. The ratios of the total DIS cross section on C (left),Fe (center) and Pb (right) to scintillator (CH) as a function of x. Data are drawn aspoints with statistical uncertainty and simulation as lines. The total systematic erroris drawn as a band around the simulation in each histogram. The experimental resultsand simulations are not isoscalar corrected
While these results are suggestive they are certainly not the statistically significantresult needed to resolve this question. It is important that further experimental resultwith well-controlled errors are pursued to determine the neutrino nuclear correctionfactors over a wide range of A. While the MINER ν A experiment is now addressing thisquestion with a somewhat higher beam energy with targets of C, water, Fe and Pb, in5
What does MINERvA see? LE DIS Cross Section Ratios – d s /dx.Much improved ME beam ratios soon to be released! The Q distribution within an x bin is essential! ◆ The shape of the data at low x, especially with lead is consistent with nuclear shadowing at
NOT Isoscalar Corrected dx CH σ d : dx Pb σ dRatio of J. Mousseau nCTEQnu – Pb/C 1.7 GeV Joel Mousseau 45
DIS Ratios: dσ /dx ● Our data suggest additional nuclear shadowing in the lowest x bin (0 < x <0.1) than predicted in lead. ● There are some hints of this as well in Iron. ● Lowest x bin is a ~ 2.0 (GeV/c) ● In the EMC region (0.3 < x < 0.75), we see good agreement between data and simulation.
C/CH Fe/CH Pb/CH nCTEQnu
Figure 47. (left) The x-dependent predictions for the ratios A/C of the structurefunction F at Q = 1.7 GeV using the nuclear parton distributions determinedfrom neutrino scattering – nCTEQnu. (right) As in Fig. 46 the measured DIScross section ratio of Pb/CH as a function of x from MINER ν A (data points) andvarious parametrizations of x- dependent nuclear effects [215, 283, 284] as well asthe predictions based on the nCTEQnu nPDFs. The error bars on the data are thecombined statistical and systematic uncertainties. the near future the much more statistically significant DUNE experiment, if outfittedwith a range of nuclear targets beyond the main Ar of its detectors, can add significantlyto this still open question yielding a thorough A-dependent study of nuclear PDFs andbetter determine the ν -A nuclear correction factor in the DIS region. Perhaps furtherin the future a neutrino factory with very intense and well-known neutrino beams willprovide a direct comparison between nuclear targets and nucleon (liquid hydrogen anddeuterium) targets.Beyond this important comparison of nuclear effects depending on the incominglepton, there are outstanding questions to be resolved for ν/ν -A scattering alone. Thesecan be summarized as main questions to ask subsequent neutrino experiments: • Does the community have the resources to supplement the decades-old bubblechamber measurements of ν/ν -p and ν/ν -n total and differential cross sections withcontemporary high-statistics measurements on free proton and deuteron targets? • In experimentally extracting nuclear structure functions from nuclear cross sections,what nuclear biases are being built in through the assumed R (= σ L /σ T ) and∆( xF )? • What is happening in the region with x ≥ to 1.0 with ν -A interactions and how isthis region to be addressed in global fits to neutrino nPDFs? • When will DIS modeling in generators be updated to reflect the recent nuclearparton distributions and ν -A models available. • At high-x, at what value of Q do the higher-twist contributions become significantafter correcting for target mass effects? • A study of nuclear higher-twist effects is necessary to better understand thetransition region for ν − A interactions.6 • As W decreases and approaches the SIS region, what is the interplay of non-perturbative QCD effects with the approaching resonant/non-resonant region thatgoverns this transition? • Considering the suggested problems with PYTHIA and even KNO at low-W, whenwill the community re-examine hadronization models in current generators to betterdescribe exclusive hadron production at relevant W values? • Considering the importance of ν e interactions for current and future experiments,when will our understanding of the impact of radiative corrections and theirapplicability be improved.7
6. Comparing DIS Theory and DIS Phenomenological Approaches
In the previous sections we have presented both theoretical and phenomenologicalapproaches to describe deep-inelastic scattering. Here we present a direct comparisonof the predictions of these two approaches as well as a comparison of these predictionswith past experimental results. We also present expectations for the DIS contributionsto on-going and future experiments.
The experimental results of the CCFR, NuTeV and CHORUS experiments that can becompared to these two approaches have been presented in 5.2.In Figs. 48 for Fe and 49 for Pb, the theoretical predictions of the Aligarh-Valenciagroup for ν and ν differential cross sections as well as the phenomenological predictionsusing the nCTEQnu nuclear PDFs for ν differential cross sections at E ν = 35 GeV arepresented. The results of Aligarh-Valencia group are shown for the spectral function onlyand using the full model (Eqs.75 and 77) where it can be observed that the mesoniccontributions play important role in the region of x ≤ .
5. In comparing the twoapproaches for ν , the nCTEQnu-based results are somewhat lower than the theoreticalprediction at the lowest- x presented while the results of the two approaches are inreasonable agreement with each other in the region of higher x.Both approaches are compared with the limited experimental results from NuTeVand CHORUS [221, 222] experiments at E ν = 35 GeV. In general, for ν the resultsobtained with both the full theoretical model and using the nCTEQnu nuclear PDFsare below the experimental results for ν -Pb at higher x and the full theoretical modelpredictions are above the experimental results for ν in the lowest x bin for both nuclei.A comparison of the differential cross sections for Fe at E ν = 65 GeV with thenPDFs labeled nCTEQnu as well as the theoretical predictions of the Aligarh-Valenciagroup based on both CTEQ and MMHT nucleon PDFs can be found in Fig. 50. Boththe approaches are compared with the measured ν -Fe cross sections from the CDSHWand NuTeV experiments. A first observation is that there is little difference in the fulltheoretical prediction based on either CTEQ or MMHT nucleon PDFs. It is also clearthat the low-x, low-y (= low- Q ) and medium-x behavior of the NuTeV and CDHSWmeasurements tend to favor the phenomenological nPDF (nCTEQnu) results ratherthan the theoretical approach based on applying nuclear effects to nucleon PDF basedstructure functions. This observations is not surprising since the NuTeV results wereused in the fit to determine the nCTEQnu nPDFs.We draw different conclusions from the comparison of CHORUS ν -Pb results at E ν = 55 GeV with theoretical and phenomenological predictions in Fig. 51. At low-x thedata are consistent with the theoretical approach and above the nPDF predictions. Atmid- and high-x both the theoretical and the nPDF approaches agree and the CHORUSdata lie above both. The better fit of the nPDF results to Fe compared to Pb is not8 Figure 48.
Differential cross section vs y for different values of x for the incomingbeam of energy E = 35 GeV for ν -Fe DIS (top row) and ν -Fe DIS (bottom row).Theoretical predictions are shown with the spectral function only (dashed line) andwith the full model (solid line) at NNLO. In the inset the effects of an additionalkinematical cut of W ≥ ν -Fe with Q ≥ . GeV . Solid circles with error bars are the limited experimentaldata points of NuTeV at this lower energy. Figure 49.
Differential cross section for ν -Pb DIS (top row) and ν -Pb DIS (bottomrow) for the incoming beam of energy E = 35 GeV. Lines representing the theoreticaland nCTEQ nPDF approaches have the same meaning as in Fig. 48. Solid circles arethe experimental data points of CHORUS. surprising since the quite small errors on the NuTeV data insured that Fe results woulddominate the global fit.9 Figure 50.
Results of the differential scattering cross section vs y, at different x for ν -Fe (treated as an isoscalar target) at E ν = 65 GeV. The theoretical results are obtainedfor iron by using (i) CTEQ 6.6 nucleon PDFs at NLO in the MS-bar scheme (dottedline), (ii) MMHT nucleon PDFs at NLO (solid line). The blue dash-dotted line is theresult from nCTEQnu nPDFs with Q ≥ . GeV . The experimental points are thedata from CDHSW and NuTeV experiments. Figure 51.
Differential scattering cross section for ν -Pb (treated as an isoscalar target)at E ν = 55 GeV. The lines representing the theoretical and nPDF approaches havethe same meaning as in Fig. 50. Solid circles are the data points from the CHORUSexperiment. Of course, the on-going and future neutrino cross section and oscillation experimentsare not using ν and ν beams with the high energies of past experiments. In light of this0we include our predictions of what on-going cross section experiments and the futureDUNE oscillation experiment might expect as DIS contributions to their statistics.Assuming a 6.25 GeV neutrino beam, the average energy of the MINER ν A MEbeam, and a Q ≥ . GeV cut, Fig. 52 and Fig. 53 show the expected crosssections from the Aligarh-Valencia theoretical calculations and the CTEQ neutrino-based (nCTEQnu) nuclear PDFs for Fe and Pb respectively. For both nuclei, in themid-x region from ≈ ≈ Q ).As y decreases, the nPDF approach predicts lower cross sections than the theoreticalapproach. For high-x ( (cid:39) (cid:47) E ν . This alsosupports the observation that the x-dependent nuclear effects for larger nuclei, such asthe three here considered nuclei, have a rather weak A-dependence. The actual ratiosof Fe/Ar and Pb/Ar in this analysis differ by less than 3 % over the entire allowable xand y kinematic plane.Fig. 55 illustrates the much more restricted DIS contribution expected with 2.25GeV neutrinos. The Q (cid:61) . GeV cut restricts lower-x contributions at this energyand further restricts lower-y contributions at a given x. Over the kinematic regionsallowed, there are obvious differences in the predictions of the two approaches that aresimilar to the observations drawn for the E ν = 6.25 GeV Ar example.Note that for the predictions of the nCTEQnu nuclear PDFs at 6.25 GeV there isan x-y region corresponding to Q ≥ GeV however lower than the Q = 1.69 GeV of the nCTEQnu DGLAP expansion. This region requires an extrapolation that hasbeen performed with the technique provided by the LHAPDF library [285]. Althoughthe low x - low y NuTeV and CCFR data in Fig. 50 support a downward trend of thecross section, the lower-y behavior at a given x is coming mainly from this extrapolationbelow Q .Future global fits of neutrino-nucleus results should consider the well-known lowerrange of neutrino energies required for current neutrino experiments. The future fitsshould then take into account the observation of the current theoretical study indicatingthat, with inclusion of the TMC, any required dynamical higher twist is minimal. Thisshould allow the introduction of a lower Q and lower Q cut on the included data thanthat used in current analyses.1 Figure 52.
Predictions for the differential scattering cross section vs y, at differentvalues of x for 6.25 GeV ν -Fe treated as an isoscalar target. The results are obtainedwith a Q ≥ . GeV cut by the Aligarh-Valencia model using CTEQ 6.6 nucleonPDFs at NLO in the MS-bar scheme (solid line). The nCTEQnu nuclear PDFs basedprediction is the blue dash-dotted line. Figure 53.
Predictions of the differential scattering cross section vs y at differentvalues of x for 6.25 GeV ν -Pb treated as an isoscalar target. The solid and dash-dotted lines in this figure have the same meaning as in Fig.52.
7. Conclusions
In this review we have examined the higher-W SIS region and the kinematically definedDIS region. We have found in both the SIS and DIS regions considerable need for furthertheoretical and experimental efforts to better understand these regions. We summarizehere the main conclusions of our study.2
Figure 54.
Prediction of the differential scattering cross section vs y at differentvalues of x for 6.25 GeV ν -Ar. The lines in this figure have the same meaning as inFig.52. Figure 55.
Prediction of the differential scattering cross section vs y at differentvalues of x for 2.25 GeV ν -Ar. The lines in this figure have the same meaning as inFig.52. ν/ν Nucleus Scattering
We have studied nuclear medium effects in the structure functions F Ai ( x, Q ), i=1-3,using Aligarh-Valencia model and obtained the differential scattering cross sections,in ν , ¯ ν scattering from several nuclear targets like C, Ar, Fe and Pb. Starting withfree nucleons, using several free nucleon PDF sets, the medium effects were includedusing many body field theoretical technique to describe the spectral function of thenucleon in the nuclear medium. The local density approximation has been appliedto translate results from nuclear matter to nuclei of finite size. The effect of Fermimotion, binding energy, nucleon correlations as well as the effect of mesonic( π and ρ )contributions in F Ai ( x, Q ), i=1-2 and shadowing have been taken into account leading to3a dynamical(nonstatic) treatment of the nucleon and the mesons in the nuclear medium.This study has been performed for a wide range of x and Q . In general, incomparison to the results obtained for the free nucleon case, we find that the use ofthe spectral function results in the reduction of the nuclear structure functions (andconsequently the differential cross sections) in the intermediate region of x and anenhancement (mainly due to Fermi motion effect) at high x , These results are Q dependent with the effect more pronounced at low Q and A dependent with thesuppression in the intermediate region of x and the enhancement at high x increasingwith the the mass number A . Furthermore, the inclusion of mesonic contributions resultsin an enhancement in the nuclear structure functions in the low and intermediate regionof x with the enhancement mainly due to pionic rather than rho meson effects. Thesemesonic contributions are suppressed with an increase in x and Q and are observed tobe more pronounced with the increase in mass number A as there are more nucleonsand the probability of interactions among nucleons via meson exchange increases. Theeffect of shadowing is included resulting in a reduction in the nuclear structure functionsat low x that increases with increased A .The nuclear medium effects are found to be significant in the evaluation of nuclearstructure functions F W I A ( x, Q ), F W I A ( x, Q ) and F W I A ( x, Q ). In the free nucleon case wehave shown that the difference in the nucleon structure functions F W IiN ( x, Q ) ( i = 1 , < F W I N ( x, Q )at low x and low Q which becomes smaller with the increase in Q . In the case ofnucleons bound inside a nucleus, the HT corrections are even further suppressed dueto the presence of nuclear medium effects. Consequently, the results for ν/ ¯ ν − A DISprocesses which are evaluated at NNLO have almost negligible difference from the resultsobtained at NLO with HT effect. Thus we conclude that as long as TMC effects areapplied, the effect of the dynamical higher twist (HT) in nuclei is small in comparisonto the free nucleon case and the results obtained at NNLO are very close to the resultsobtained at NLO with HT(within a percent).We find that the nuclear-medium effects are different in F A ( x, Q ), F A ( x, Q ) and F A ( x, Q ) structure functions and are more pronounced in the ¯ ν − A reaction channelthan in the case of ν − A scattering. This can be observed in F A ( x, Q ), describingthe behavior of valence quarks, where the mesonic contributions are absent and in thebehavior of the Callan-Gross relation F A ( x,Q )2 xF A ( x,Q ) , which is observed to become violatedat low x .The correction due to the excess of neutrons over protons (isoscalarity effect) issignificantly large for the lead nucleus, for example, 5% at low x and 15% at high x , whilein argon nucleus it is ∼
2% at low x and ∼
4% at high x . Significantly, we have foundthat the nuclear medium effects are different in electromagnetic and weak interactionchannels especially for the nonisoscalar nuclear targets. The contribution of strangeand charm quarks is found to be different for the electromagnetic and weak interactioninduced processes off free nucleon target which also gets modified differently for the4heavy nuclear targets. Furthermore, we have observed that the isoscalarity corrections,significant even at high Q , and are not the same in F W I A ( x, Q ) and F W I A ( x, Q ).As presented in section 6, the full theoretical model shows reasonable agreementwith the experimental data of CCFR, CDHSW, NuTeV and CHORUS data in themid x and high Q regions. However in the low-x (shadowing) region and the high-x(EMC) region the agreement of the predicted differential scattering cross sections withthe NuTeV and CHORUS data is not as good.It is apparent that in the precision era of neutrino oscillation physics, it is necessaryto address differences in predictions compared to the few existing experimental results.Suggesting a need for more measurements of nuclear effects in a wide range of A , usingneutrino and antineutrino beams in a broad kinematic range of x and Q . ν/ν Nucleus ScatteringShallow Inelastic Scattering
It should now be quite obvious that the higher-W SISregion in both neutrino nucleon ν -N and neutrino nucleus ν -A scattering is unexploredexperimentally and essentially so theoretically. Fig.27 starkly presents the difference inthe simulations of this kinematic region. In increasing W from the ∆ there are onlya few ν -N resonance models that treat more than 1- π production and it is clear thatmulti- π production can be significant in this high-W region. As far as non-resonantproduction is concerned there are several models available for single- π non-resonantproduction including the recent efforts of [28] and references therein. However modelsof non-resonant two- π or more production are not available. Certainly the carefulunderstanding of how SIS non-resonant π production smoothly transforms into DISpion production is crucial for this transition region and has not been carefully addressedtheoretically or experimentally.Approaching the SIS region from the higher-W DIS region there is no well-definedsharp boundary between the two. Q ≥ GeV is chosen as the minimum Q neededto be interacting with quarks within the nucleon and W ≥ Q and W is the kinematicregion where non-perturbative QCD effects come into serious consideration. A topicvery much neglected in ν nucleon/nucleus physics. Is there a change in the relativestrength of SIS and DIS cross sections at this transition? Is there not a theoreticalconnection that can be made between increasing W non-resonant pion production andnon-perturbative QCD effects? This is, of course, the goal of the application of duality.Duality is a concept that supposedly allows phenomena in the DIS region toapproximate activity in the SIS region. Although duality has been quite thoroughlytested in electroproduction experiments. It cannot presently be tested in the samemanner in ν -N and ν -A scattering due to an obvious lack of experimental data. However,from the model-dependent studies that have been made of ν -N scattering it appears5that duality might be better applied to ν -isoscalar N scattering and not for ν -p or ν -nscattering individually.The many open challenges for this kinematic region can then be summarized as: • A need for much increased experimental investigation of the higher-W kinematicregion for single and multi- π production. • A need for models of resonant multi- π production up to and through the transitioninto the DIS region. • A need for models of non-resonant multi- π production and a better understandingof how non-resonant single and multi- π production in the SIS region transitionsinto DIS single and multi- π production. • A much more thorough investigation of non-perturbative QCD effects and how theycan be mapped onto non-resonant π production in the SIS to DIS transition regionis required. • A better understanding of how duality can help address some of these previouslisted challenges would be helpful. The managers of the various simulation programsshould check whether their simulations of the SIS and DIS regions for the averagenucleon (n+p)/2 are reasonably consistent with the current expectations of duality.
Deep Inelastic Scattering
In contrast to the SIS region, there have been severalexperimental and many phenomenological studies of the DIS region for both ν -Nand ν -A scattering. In the DIS region perturbative QCD plus factorization allows aphenomenological approach to the extraction of the parton distribution functions ofboth the free nucleon (PDFs) and nucleons bound in the nuclear environment (nPDFs)where nuclear medium effects are significant. While the free nucleon PDFs have beenextracted via global fits by many groups, far fewer attempts have been made to extractthe nPDFs of nucleons within a nucleus.Among the groups concentrating on these nPDFs the nCTEQ group has found adifference in the nPDFs extracted from a global fit using (cid:96) ± -A scattering and thoseextracted from a fit using ν ( ν )-A scattering based on the experimental results ofCCFR, NuTeV and CHORUS. The difference is most evident in comparing the nuclearcorrection factors as a function of x for ν ( ν )-A and (cid:96) ± -A based analyses. The differenceis significant in both location and intensity of the expected nuclear effects of shadowing,antishadowing and the EMC effect. Other groups fitting nPDFs based on DIS neutrinoscattering use different techniques than nCTEQ and are able to find compatible fitsincluding both (cid:96) ± -A and ν -A.It is significant to note that the kinematic regions showing the largest differencebetween nCTEQ ν ( ν )-A based and (cid:96) ± -A based analyses are also the regions withthe largest differences between the theoretical and nPDF results summarized in thispaper. Particularly the nPDF predicted stronger suppression of the cross section inthe low- Q , low-x shadowing region and the elevated cross section in the EMC region,both directly reflecting the quoted experimental results, emphasize these differences.6When comparing the nCTEQ ν ( ν )-A to (cid:96) ± -A based analyses, these differences canbe attributed to the differences of the weak compared to EM interactions. However,the theoretical considerations of ν ( ν )-A DIS, summarized in this paper, includesthe accepted theoretical considerations of the weak interaction of these two regionsin the calculations so the differences here are intriguing. Since the most recentconsiderations of shadowing and the EMC effect in ν ( ν )-A DIS interactions presentedin the phenomenological section are still speculative, they have not yet been included inthe theoretical treatment of these two regions presented here but could indeed providean explanation of the differences.The differences in the (cid:96) ± -A based and ν -A based results could suggest interestingconsequences. In particular for the low-x region, there are many theoretical and nowexperimental indications that shadowing is a quite different process in (cid:96) ± -A and ν -Ainteractions. The theoretical indications are based on the presence of the axial vectorcurrent and the considerably more massive IVB involved in neutrino scattering. Ifthis fundamental difference does exist, it would follow that there should not be thesame universal nPDFs describing this low-x region for (cid:96) ± -A and ν ( ν )-A analyses unless,for example, a term is incorporated perhaps in the factorization that accounts for IVB-dependent phenomena in the nuclear environment. A resolution of these disagreements isessential for proper simulation of DIS scattering in current and future neutrino oscillationexperiments. On-going and next generation oscillation experiments like T2K, NOvA, DUNE andHyperK as well as experiments using atmospheric neutrino such as IceCube [286],JUNO [287] and INO [288] are expected to provide valuable information aboutneutrino properties in the roughly 1-10 GeV neutrino energy region. Significantly,precise measurements of these properties can only be achieved by reducing systematicuncertainties. Currently, considering the target material of these experiments, a largeportion of these uncertainties is due to the lack of precise cross sections and, mostimportantly, nuclear effects in ν (¯ ν )-nucleus scattering. For NOvA and DUNE as well asatmospheric neutrino oscillation studies of SuperK, HyperK, IceCube, JUNO and INOa reasonable or even major fraction of events come from the higher-W shallow inelasticand deep inelastic scattering regions. This review has highlighted the many currentconcerns and challenges, both theoretical and experimental, in these regions.Therefore, it is important to much improve the nuclear model that covers these tworegions, which includes the understanding of nucleon dynamics in the nuclear medium,the resulting hadron production in ν (¯ ν )-nucleon induced processes as well as the role offinal state interactions within the nucleus. To improve this model in the SIS and DISregions will take the dedicated efforts of theorists and experimentalists working togetherwith neutrino event simulation experts. In particular a significant enhancement in themeasurement of fundamental ν (¯ ν )-nucleon scattering as well as precision measurements7of ν (¯ ν ) scattering off a variety of nuclear targets in the SIS and DIS regions would bewelcome. The community and relevant funding agencies should recognize this essentialcollaborative effort and provide the support necessary for the experiments to reach theirstated precision goals.
8. Acknowledgements
We most gratefully acknowledge the invaluable assistance of G. Caceres Vera, H. Haider,I. Ruiz Simo, A. Kusina and F. Zaidi. We also appreciate the many informativediscussions we had with our NuSTEC colleagues. M. S. A. is thankful to Department ofScience and Technology (DST), Government of India for providing financial assistanceunder Grant No. EMR/2016/002285. J. G. M. has been supported by Fermi ResearchAlliance, LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department ofEnergy, Office of Science, Office of High Energy Physics.
9. AppendicesA. Neutrino self-energy
When a neutrino interacts with a potential provided by a nucleus (in the presentscenario), then the interaction in the language of many body field theory can beunderstood as the modification of the fermion two points function represented by thediagrams shown in Fig.56.
Figure 56.
Representation of neutrino self energy.
Figure 57. (Top) Free field fermion propagator, (Bottom) The term that constitutesto neutrino self energy, − i Σ( k ) = (cid:90) d k (cid:48) (2 π ) (cid:18) − ig √ γ µ (1 − γ ) (cid:19) i ( (cid:54) k (cid:48) + m l ) k (cid:48) − m l + i(cid:15) × (cid:18) − ig √ γ ν (1 − γ ) (cid:19) − ig µν ( k − k (cid:48) ) − M W + i(cid:15) (81)Notice that Σ has real and imaginary parts. The imaginary part of the neutrino selfenergy accounts for the depletion of the initial neutrinos flux out of the non-interactingchannel, into the quasielastic or the inelastic channels.By using the Feynman rules the neutrino self-energy corresponding to Fig.5 iswritten as − i Σ( k ) = (cid:90) d q (2 π ) (cid:16) ¯ u ν ( k ) − ig √ γ µ (1 − γ ) × i ( (cid:54) k (cid:48) + m l ) k (cid:48) − m l + i(cid:15) − ig √ γ ν (1 − γ ) u ν ( k ) (cid:17) × (cid:16) − ig µρ q − M W (cid:17) ( − i Π ρσ ( q ) ) (cid:16) − ig σν q − M W (cid:17) (82)which after simplification modifies to − i Σ( k ) = g M W (cid:90) d q (2 π ) T r { ( (cid:54) k + m ν ) γ µ (1 − γ )( (cid:54) k (cid:48) + m l ) γ ν (1 − γ ) }× Π µν ( q )2 m ν ( k (cid:48) − m l + i(cid:15) ) (cid:16) M W q − M W (cid:17) Now by using the following relations g M W = G F √ d q = d k (cid:48) ; (cid:88) r u r ( k )¯ u r ( k ) = (cid:54) k + m ν m ν and the trace properties, neutrino self-energy is further simplified toΣ( k ) = − iG F √ (cid:90) d q (2 π ) L W Iµν m ν k (cid:48) − m l + i(cid:15) ) (cid:18) M W Q + M W (cid:19) Π µν ( q ) , (83)To obtain the imaginary part of neutrino self-energy which is required to evaluate thescattering cross section, Cutkosky rules are applied:Σ( k ) → iIm Σ( k ); Lepton self-energyΠ µν ( q ) → iθ ( q ) Im Π µν ( q ); W boson self-energyIt gives 2 iIm Σ( k ) = − iG F √ m ν (cid:90) d q (2 π ) iIm (cid:16) k (cid:48) − m l + i(cid:15) ) (cid:17) iθ ( q ) (cid:18) M W Q + M W (cid:19) × Im [ L W Iµν Π µν ( q )] . Using Sokhotski-Plemelj theorem and equating the imaginary terms on both sides, onemay write Im (cid:18) k (cid:48) − m l + i(cid:15) (cid:19) = − π E ( k (cid:48) ) , f or E ( k (cid:48) ) > k (cid:48) (84)9where the energy transfer is q = k − k (cid:48) = k − E ( k − q ). Using the property of thestep function, the imaginary part of neutrino self-energy may be written as ⇒ Im Σ( k ) = G F √ m ν (cid:90) d q (2 π ) πE ( k (cid:48) ) θ ( q ) (cid:18) M W Q + M W (cid:19) Im [ L W Iµν Π µν ( q )] . (85) B. Nucleon spectral function
The relativistic free nucleon Dirac propagator G ( p , p ) is given by G ( p , p ) = 1 (cid:54) p − M N + i(cid:15) = (cid:54) p + M N ( p − M N + i(cid:15) ) (86)which may be rewritten in terms of both positive and negative energy states as G ( p , p ) = M N E N ( p ) (cid:26) (cid:80) r u r ( p )¯ u r ( p ) p − E N ( p ) + i(cid:15) + (cid:80) r v r ( − p )¯ v r ( − p ) p + E N ( p ) − i(cid:15) (cid:27) , (87)where E N ( p ) = (cid:112) | p | + M N is the relativistic energy of an on shell nucleon. As it hasbeen already mentioned in section3.1.1 that negative energy components are suppressedthan the positive energy components, therefore, only first term will contribute. Hence, G ( p , p ) = M N E N ( p ) (cid:88) r u r ( p )¯ u r ( p ) (cid:20) − n ( p ) p − E N ( p ) + i(cid:15) + n ( p ) p − E N ( p ) − i(cid:15) (cid:21) where n ( p ) is the occupation number of the nucleons in the Fermi sea, n ( p ) = for p ≤ p F N while n ( p ) = for p > p F N . Using the following relation: (cid:88) r u r ( p )¯ u r ( p ) = (cid:54) p + M N M N the aforementioned expression for nucleon propagator modifies to G ( p , p ) = (cid:54) p + M N p − M N + i(cid:15) + 2 iπθ ( p ) δ ( p − M N ) n ( p )( (cid:54) p + M N ) (88)In the interacting Fermi sea, the relativistic nucleon propagator is written using Dysonseries expansion (shown in Fig.6) in terms of nucleon self energy Σ N ( p , p ). Thisperturbative expansion is summed in a ladder approximation as G ( p ) = G ( p ) + G ( p )Σ N ( p ) G ( p ) + G ( p )Σ N ( p ) G ( p )Σ N ( p ) G ( p ) + ....... One may notice that the aforementioned equation is a geometric progression series andusing Eq.87, one may write Eq.62 as: G ( p ) = M N E N ( p ) (cid:88) r u r ( p )¯ u r ( p ) p − E N ( p ) − ¯ u r ( p )Σ N ( p , p ) u r ( p ) M N E N ( p ) This expression contains nucleon self energy in the denominator which is a complexquantity, i.e. Σ N ( p , p ) = Re { Σ N ( p , p ) } + iIm { Σ N ( p , p ) } (89)00Using this definition in Eq.62, the dressed nucleon propagator may be rewritten as G ( p ) = M N E N ( p ) (cid:88) r u r ( p )¯ u r ( p ) × (cid:34) { p − E N ( p ) − M N E N ( p ) Re (Σ N ) } + i { M N E N ( p ) Im (Σ N ) }{ p − E N ( p ) − M N E N ( p ) Re (Σ N ) } + { M N E N ( p ) Im (Σ N ) } (cid:35) (90)The use of nucleon Green functions in terms of their spectral functions offers a preciseway to account for Fermi motion and binding energy. Basically spectral functions areused to describe the momentum distribution of nucleons in the nucleus. Therefore, todetermine the spectral functions of particle and hole let us define (cid:90) µ −∞ dω S h ( ω, p ) p − ω − iη + (cid:90) + ∞ µ dω S p ( ω, p ) p − ω + iη = P (cid:90) µ −∞ dω S h ( ω, p ) p − ω + iπ (cid:90) µ −∞ dωS h ( ω, p ) δ ( p − ω ) + P (cid:90) + ∞ µ dω S p ( ω, p ) p − ω − iπ (cid:90) ∞ µ dωS p ( ω, p ) δ ( p − ω ) , One may write with the help of Eq.90 P (cid:90) µ −∞ dω S h ( ω, p ) p − ω + P (cid:90) + ∞ µ dω S p ( ω, p ) p − ω + iπS h ( p , p ) θ ( µ − p ) − iπ × S p ( p , p ) θ ( p − µ ) = (cid:34) { p − E N ( p ) − M N E N ( p ) Re (Σ N ) } + i { M N E N ( p ) Im (Σ N ) }{ p − E N ( p ) − M N E N ( p ) Re (Σ N ) } + { M N E N ( p ) Im (Σ N ) } (cid:35) On comparing imaginary parts on both sides, we obtain S h ( p , p ) = 1 π M N E N ( p ) Im Σ N ( p − E N ( p ) − M N E N ( p ) Re Σ N ) + ( M N E N ( p ) Im Σ N ) ; for p ≤ µS p ( p , p ) = − π M N E N ( p ) Im Σ N ( p − E N ( p ) − M N E N ( p ) Re Σ N ) + ( M N E N ( p ) Im Σ N ) ; for p > µ Using the above two equations in Eq.(90), the dressed nucleon propagator is obtainedin terms of the particle and hole spectral functions as: G ( p , p ) = M N E N ( p ) (cid:88) r u r ( p )¯ u ( p ) (cid:20)(cid:90) µ −∞ S h ( p , p ) dω ( p − ω − iη ) + (cid:90) ∞ µ S p ( p , p ) dω ( p − ω + iη ) (cid:21) C. Properties of spectral function
The hole and particle spectral functions fulfill the following relations, (cid:90) µ −∞ dp S h ( p , p ) = n ( p ) (cid:90) ∞ µ dp S p ( p , p ) = 1 − n ( p )and thus the spectral functions obey the following sum rule (cid:90) µ −∞ dp S h ( p , p ) + (cid:90) ∞ µ dp S p ( p , p ) = 1 (91)01In the absence of interactions (i.e. Σ N ( p ) = 0), the nucleon energy p is the freerelativistic energy E ( p ) and the dressed propagator G ( p ) reduces to the free propagator G ( p ) then S h ( p , p ) = S p ( p , p ) = δ ( p − E N ( p )) (92)which leads to (cid:90) µ −∞ dp S h ( p , p ) = (cid:90) µ −∞ dp δ ( p − E N ( p )) = (cid:40) µ > E N ( p )0 if µ < E N ( p ) (cid:90) ∞ µ dp S p ( p , p ) = (cid:90) ∞ µ dp δ ( p − E N ( p )) = (cid:40) µ < E N ( p )0 if µ > E N ( p )If E N ( p ) is the total relativistic energy, then chemical potential µ must incorporate thenucleon mass M N : µ = M N + (cid:15) F (93)This definition leads to a constant shift in the integration variable p such as: p = ω + M N ; ⇒ ω = p − M N . (94)Now the integration of hole and particle spectral functions will be modified to (cid:90) µ −∞ dω S h ( ω, p ) = (cid:90) µ − M N −∞ dω δ ( ω + M N − E N ( p ))= (cid:40) µ − M N > E N ( p ) − M N ⇒ (cid:15) F > (cid:15) ( p )0 if µ − M N < E N ( p ) − M N ⇒ (cid:15) F < (cid:15) ( p ) (cid:90) ∞ µ dω S p ( ω, p ) = (cid:90) ∞ µ − M N dω δ ( ω + M N − E N ( p ))= (cid:40) µ − M N < E N ( p ) − M N ⇒ (cid:15) F < (cid:15) ( p )0 if µ − M N > E N ( p ) − M N ⇒ (cid:15) F > (cid:15) ( p )where (cid:15) ( p ) = E N ( p ) − M N is the nucleon kinetic energy. The behavior of hole spectralfunction vs removal energy ω is shown in Fig. 58 for C , F e and
P b . From thefigure, one may notice that for p < p F , spectral function has a sharp and narrowdistribution similar to the delta function while for p > p F , the distribution has a widerange though very small in magnitude. Furthermore, it may be noticed that the holespectral function has a smaller magnitude for heavier nuclear targets which is becauseof the enhancement in the probability of interaction among the nucleons. D. Local Density Approximation
In the local density approximation, Fermi momentum is not fixed but depends upon theinteraction point ( r ) in the nucleus and is related to the nuclear density as p F ( r ) = (cid:18) π ρ ( r )2 (cid:19) / . (95)02 Figure 58.
Results for S h ( ω, p ) vs ω are shown for (i) p < p F (Left panel) and p > p F (Right panel) in various nuclei like C, Fe and
Pb.
Thus the Fermi momentum of the nucleon is not a constant number unlike the globalFermi gas model. In the global Fermi gas model p F is taken to be a constant value like,221 MeV for C, 251 MeV for Ca, etc. In the local density approximation, the freelepton-nucleon cross section is folded over the density of the nucleons in the nucleusand integrated over the whole volume of the nucleus. The differential scattering crosssection is then given by dσ A = (cid:90) d r ρ ( r ) dσ N (96)In a symmetric nuclear matter, each nucleon occupies a volume of (2 π (cid:126) ) . However,because of the two possible spin orientations of the nucleon, each unit cell in theconfiguration space is occupied by the two nucleons. Therefore, the number of nucleonsin a certain volume is given by ( (cid:126) = 1 in natural units) N = 2 V (cid:90) p F d p (2 π ) , (97) ⇒ ρ = NV = 2 (cid:90) p F d p (2 π ) n ( p , r ) , (98)where n ( p , r ) is the occupation number of a nucleon lying within the Fermi sea suchthat n ( p , r ) = (cid:40) p ≤ p F p > p F (99)In the present model, the spectral functions of proton and neutron are respectively thefunction of local Fermi momentum p Fp,n ( r ) = (cid:2) π ρ p ( n ) ( r ) (cid:3) / for proton and neutron inthe nucleus. The proton and neutron densities ρ p ( n ) ( r ) are related to ρ ( r ) as [72, 75] ρ p ( r ) = ZA ρ ( r )03 ρ n ( r ) = ( A − Z ) A ρ ( r )The equivalent normalization to Eq.(101) is written as2 (cid:90) d p (2 π ) (cid:90) µ −∞ S h ( ω, p, p Fp,n ( r )) dω = ρ p,n ( r ) , (100)These spectral functions are normalized individually for the proton ( Z ) and neutron( N = A − Z ) numbers in a nuclear target.2 (cid:90) d r (cid:90) d p (2 π ) (cid:90) µ p −∞ S ph ( ω, p , ρ p ( r )) dω = Z , (cid:90) d r (cid:90) d p (2 π ) (cid:90) µ n −∞ S nh ( ω, p , ρ n ( r )) dω = N , where factor 2 is due to the two possible spin projections of nucleon. Through the holespectral function ( S h ( p , p , ρ ( r ))), we incorporate the effects of Fermi motion, Pauliblocking and nucleon correlations. The spectral function is properly normalized andchecked by obtaining the correct baryon number and binding energy for a given nucleussuch that 4 (cid:90) d p (2 π ) (cid:90) µ −∞ S h ( ω, p , p F ( r )) dω = ρ ( r ) (101)or equivalently (cid:90) d r (cid:90) d p (2 π ) (cid:90) µ −∞ S h ( ω, p , p F ( r )) dω = A. (102)Since we are not looking at the final state particles, therefore, we will consider onlyhole spectral function. The normalization of the hole spectral function is ensured byobtaining the baryon number ( A ) of a given nucleus and the binding energy of thesame nucleus. Furthermore, for the nonisoscalar nuclear target the spectral function isnormalized to the proton and neutron numbers separately.04
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