Analysis of the boundaries of the quasi-elastic neutrino-nucleus cross section in the SuSAM* model
I. Ruiz Simo, I.D. Kakorin, V.A. Naumov, K.S. Kuzmin, J.E. Amaro
AAnalysis of the boundaries of the quasi-elastic neutrino-nucleus cross section in theSuSAM* model
I. Ruiz Simo, ∗ I. D. Kakorin, V. A. Naumov, K. S. Kuzmin,
2, 3 and J. E. Amaro Universidad de Granada and Instituto Interuniversitario Carlos I de F´ısica Te´orica y Computacional, E-18071, Granada, Spain Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, RU-141980 Dubna, Russia Institute for Theoretical and Experimental Physics, RU-117259 Moscow, Russia
In this work we obtain the analytical expressions for the boundaries of the charged current quasi-elastic double differential cross section in terms of dimensionless energy and momentum transfers, forthe Relativistic Fermi Gas (RFG) and the Super-Scaling approach with relativistic effective mass(SuSAM*) models, within the scaling formalism. In addition, this new double differential crosssection in the scaling formalism has very good properties to be implemented in the Monte Carlo(MC) neutrino event generator, particularly because its peak is almost flat with the (anti)neutrinoenergy. This makes it especially well-suited for the event generation by the acceptance-rejectionmethod usually used in the neutrino generators. Finally, we analyze the total charged currentquasi-elastic (CCQE) cross section σ ( E ν ) for both models and attribute the enhancement observedin the SuSAM* total cross section to the high-momentum components which are present, in aphenomenological way, in its scaling function, while these are absent in the RFG model. I. INTRODUCTION
The measurement of neutrino/antineutrino-nucleuscross sections is a fundamental topic of research, not onlyin itself because it can provide knowledge on the fun-damental interaction and on the nuclear properties andmodeling, but also for its importance in other specialfields in particle physics such as the mixing of neutrinoflavors, the extraction of the CP-violating phase in thelepton sector and the origin of the asymmetry betweenmatter and antimatter in the Universe. In particular, inthe last years many reviews and works have been dedi-cated to these topics [1–10].The total integrated Charged Current Quasi-elastic(CCQE) neutrino/antineutrino cross section σ CCQE ( E ν )is an important quantity to be known for the neutrinoscattering and oscillation experiments [11–35]. In partic-ular, the knowledge of this observable is crucial for choos-ing of CCQE channel among others to generate appropri-ate final lepton event kinematics in neutrino event gener-ators, that usually use the acceptance-rejection methodto generate the events with a probability distributiongiven by differential cross section.In addition, the importance of a precise knowledge ofthe total CCQE cross section and particularly its ratiobetween the electron and muon neutrinos species is ofgreat importance in order to reduce the systematic uncer-tainties for the determination of the CP violating phasein the lepton sector, as it has been shown in Refs. [36–42].Our aim in this work is to perform a thorough studyof the analytical boundaries of the phase space of theCCQE double differential cross section d σdT µ d cos θ µ for therelativistic Fermi gas (RFG) [43–47] and Super-scaling ∗ [email protected] with relativistic effective mass (SuSAM*) models [48–52]within the scaling formalism [53–57], where the bound-aries are easier to be obtained. To this end, we willstudy the double differential d σdκ dλ CCQE cross section,where κ and λ are the dimensionless momentum and en-ergy transfer variables in the scaling formalism. Thisnew double differential cross section has also the verygood property, for the generation of the final chargedlepton kinematics in the MC event generators, of an al-most flat peak, i.e, very weak dependent on the neu-trino/antineutrino energy. This important feature makesit specially well-suited for the generation of these eventsby the acceptance-rejection method.The paper is organized as follows: In sect. II we re-view in brief the general formalism for the description ofthe CCQE double differential cross section; in sect. IIIwe perform a thorough discussion about the analyticalboundaries of the phase space in the RFG model, latelyextended to the SuSAM* model in sect. IV. In sect. V weshow our main results for the double differential cross sec-tion and the integrated total one, and finally, in sect. VIwe draw our conclusions and outline our future plans orprospects related to the conclusions of the present work. II. GENERAL FORMALISM
In this section, we are going to discuss in brief theelementary ingredients to calculate the double differen-tial CCQE d σdT µ d cos θ µ cross section and its transforma-tion into the easier to work, for our purposes within thescaling formalism, d σdκ dλ cross section. The expressionfor the first double differential cross section is given by a r X i v : . [ h e p - ph ] F e b [51, 58–60]: d σdT µ d cos θ µ = G F cos θ c π k (cid:48) E ν v ( V CC R CC + 2 V CL R CL + V LL R LL + V T R T ± V T (cid:48) R T (cid:48) ) , (1)where G F = 1 . × − MeV − is the Fermi couplingconstant, θ c is the Cabibbo angle (cos θ c = 0 . k (cid:48) isthe value of the final charged lepton momentum, (cid:126)k (cid:48) , E ν is the neutrino/antineutrino energy in the lab frame, and v = ( E ν + T µ + m µ ) − q , with q being the squaredthree-momentum transfer, (cid:126)q , to the nucleus [61]. Finally,it is worth noting that the ± sign in the T (cid:48) contributionof Eq. (1) applies for neutrino and antineutrino CCQEscattering, respectively.The other ingredients appearing in Eq. (1) are the lep-ton kinematic factors V K and the nuclear response func-tions R K , the last ones depending only on the energyand momentum transfer from the leptons to the nucleus, ω and q , respectively. These factors come mainly fromthe contraction of the lepton tensor with the hadron one,and each of them are suitable combinations of the tensorsin a frame where the Z -axis is defined by the direction ofthe three-momentum transfer, (cid:126)q = (cid:126)k − (cid:126)k (cid:48) . Their explicitexpressions can be found, for instance, in Refs. [51, 58–60], and particularly in sect. IIIA and appendices B andC of the recent review [8], where an exhaustive discus-sion and derivation of the response functions and scalingin the RFG model are given.It is quite general that the nuclear response functions R K can be written in factorized form as a product ofan integrated single-nucleon response ( U K or G K in thenomenclature of Ref. [8]) times a scaling function whichdepends on the nuclear model. In this way, the scalingfunction appears as a common factor in all the nuclearresponse functions R K and factorizes in the cross sectiongiven in Eq. (1): d σdT µ d cos θ µ = G F cos θ c π k (cid:48) E ν v ( V CC U CC + 2 V CL U CL + V LL U LL + V T U T ± V T (cid:48) U T (cid:48) ) f scal ( ψ ) , (2)where ψ is the scaling variable and it is, in general, afunction of ω and q . For instance, in the particular caseof the RFG model, its expression is given by f RFG ( ψ ) = 34 (cid:0) − ψ (cid:1) θ (1 − ψ ) , (3)where θ ( x ) is the step function, while in the SuSAM*model its expression is discussed in Sect. IV.However, the differential cross section of Eq. (1) isgiven with respect to the final lepton kinematic variables,its kinetic energy T µ and the cosine of its scattering an-gle with respect to the incident neutrino direction, θ µ . Inthe scaling formalism it is not very difficult to find therelevant boundaries where the differential cross section of Eq. (1) is different from zero (as it will be shown insections III and IV), but using the relevant scaling vari-ables, namely, the dimensionless energy and momentumtransfers, λ = ω/ (2 m N ) and κ = q/ (2 m N ), with m N thenucleon mass.Therefore, in order to inspect the behavior of the dou-ble differential cross section along its phase space andefficiently integrate it to obtain the total CCQE crosssection, it is better to work with the d σdκ dλ cross sec-tion, which can be obtained from that in Eq. (1) usingthe Jacobian transformation from ( T µ , cos θ µ ) variablesto ( κ, λ ) ones: d σdκ dλ = (cid:12)(cid:12)(cid:12)(cid:12) ∂ ( T µ , cos θ µ ) ∂ ( κ, λ ) (cid:12)(cid:12)(cid:12)(cid:12) d σdT µ d cos θ µ = 4 m N qE ν k (cid:48) d σdT µ d cos θ µ , (4)where the Jacobian has been calculated knowing the re-lationships between both sets of independent variables, λ = E ν − T µ − m µ m N (5) κ = (cid:113) E ν + P µ − E ν P µ cos θ µ m N , (6)with E µ = T µ + m µ and P µ ≡ k (cid:48) = (cid:113) E µ − m µ . III. RFG MODEL CASEA. Analytical boundaries in the non-Pauli-blocking(NPB) region
With the definitions of the dimensionless variables inthe scaling formalism, where the electroweak probe trans-fers energy ω and momentum q to the nucleus. λ = ω m N , κ = q m N (7) τ = κ − λ = Q m N ≥ η F = k F m N , (cid:15) F = (cid:113) η F ≥ ψ = (cid:114) (cid:15) − (cid:15) F − λ − τ ) (10)where (cid:15) is defined as: (cid:15) = max (cid:32) κ (cid:114) τ − λ, (cid:15) F − λ (cid:33) , (11)and k F is the Fermi momentum of the nucleus. Thedefinition of (cid:15) given in Eq. (11) represents the minimumenergy of the initial nucleon in units of the nucleon mass, m N , as it can be seen from Eq. (C11) of Ref. [8]. TheRFG model requires that (cid:15) ≤ (cid:15) F . The physical con-straint of τ ≥ κ ≥ λ . Finally, it can be shown thatif κ ≥ η F (which corresponds to the NPB region) then (cid:15) = κ (cid:112) /τ − λ ≥ (cid:15) F [62]. In the NPB region, typ-ically q (cid:38)
500 MeV/c, and the above condition can berewritten as κ (cid:38) / ≤ (cid:15) ≤ (cid:15) F , the scaling variable ψ is restricted to lie between − (cid:15) = 1, i.e, when λ = τ . This condition is equivalent to τ = λ ⇐⇒ κ = λ + λ ⇐⇒ κ = (cid:112) λ ( λ + 1) (12)and hence Eq. (12) corresponds to where the scaling vari-able is always equal to zero, and where the quasi-elasticQE peak appears. For this reason we will call this curvein the ( λ, κ ) plane as κ QE ( λ ) = (cid:112) λ ( λ + 1).The boundaries of the RFG scaling variable ( −
1, +1)are reached when (cid:15) = (cid:15) F as it follows from Eq. (10).Solving the equation (cid:15) = κ (cid:112) /τ − λ = (cid:15) F in theNPB region (corresponding to κ ≥ η F ) we get two differ-ent curves in the ( λ, κ ) plane. One of them, κ NPB+ ( λ ), isalways greater than κ QE ( λ ) and corresponds to ψ = − κ NPB+ ( λ ) > (cid:112) λ ( λ + 1) and this implies λ < τ : κ NPB+ ( λ ) > κ QE ( λ ) ≡ (cid:112) λ ( λ + 1) ≥ ⇐⇒ ( κ NPB+ ( λ )) − λ > λ ⇐⇒ τ NPB+ ( λ ) > λ ⇒ sign( λ − τ NPB+ ( λ )) = − . (13)Therefore, along the curve κ NPB+ ( λ ) the scaling variable isalways equal to −
1. Analogously, there is another curve,solution of (cid:15) = (cid:15) F , called κ NPB − ( λ ), which is always lesserthan κ QE ( λ ), and where (by similar arguments as thoseproven in Eq. (13)) the scaling variable is always equalto +1.The expressions of these two curves, κ NPB+ ( λ ) and κ NPB − ( λ ) are deferred to the next subsections. The im-portant point here is that in the NPB region ( κ ≥ η F ),for a given value of λ , one has to integrate the variable κ between κ NPB − ( λ ) and κ NPB+ ( λ ) to obtain the total CCQEcross section in the RFG model, σ RFG ( E ν ), for a givenneutrino/antineutrino energy E ν . This constraint comesout only from the integrated nucleon kinematics in theimpulse approximation. We will see later that there isan additional constraint coming from the neutrino-leptonkinematics that further constrains the available phasespace.
1. Obtaining κ NPB + ( λ ) and κ NPB − ( λ ) One of the easiest ways to obtain the limiting curves κ NPB+ ( λ ) and κ NPB − ( λ ) can be found in Eq. (A.2) of ap-pendix A of Ref. [63] (see also Eqs. (C7)–(C9) of Ref. [8]). This latter equation allows to find the lowest and highest ω limits for fixed q . These two limits can be found fromthe equation (cid:15) = (cid:15) F : ω ± = E k F ± q − E F = (cid:113) ( k F ± q ) + m N − (cid:113) k F + m N . Dividing on both sides of the above equations by 2 m N ,and writing everything in terms of the dimensionless vari-ables given in Eqs. (7) and (9), we obtain the boundariesin λ ± for fixed κ : λ − ( κ ) = (cid:112) ( η F − κ ) + 1 − (cid:15) F or λ + ( κ ) = (cid:112) ( η F + 2 κ ) + 1 − (cid:15) F . (14)The problem with the boundaries given in (14) is thatthey are given as curves λ = λ ( κ ), whereas we need thecurves κ = κ ( λ ). Thus, we have to find the inverse func-tions, i.e, to express κ in terms of λ by inverting Eqs. (14).Then, writing λ + ( κ ) ≡ (cid:112) ( η F + 2 κ ) + 1 − (cid:15) F = λ ,and solving for κ , we obtain the lower bound κ NPB − ( λ ),which is given by κ NPB − ( λ ) = 12 (cid:112) ( (cid:15) F + 2 λ ) − − η F . (15) η F κ λ λ + ( κ ) λ - ( κ ) λ QE ( κ ) FIG. 1: Plot of the two limiting curves λ ± ( κ ) as a func-tion of κ in the RFG model in the NPB region, i.e,for κ ≥ η F (notice that for κ < η F the λ − ( κ ) curvereaches negative values, which are forbidden; this is be-cause it is entering in the Pauli-blocking region). In thisfigure, we have taken η F = 0 . λ QE ( κ ) = − + √ κ corresponds to the inverse of κ QE ( λ ) = κ .In Fig. 1 one can inspect that the inversion of λ − ( κ ) ≡ (cid:112) ( η F − κ ) + 1 − (cid:15) F = λ needs a bit of care becauseit must be solved for κ in the region where λ ≥ λ − ( κ ) = λ ⇐⇒ ( η F − κ ) = (2 λ + (cid:15) F ) − ⇐⇒ | η F − κ | = (cid:112) ( (cid:15) F + 2 λ ) − κ ≥ η F then | η F − κ | = 2 κ − η F and thus Eq. (16)becomes κ NPB+ ( λ ) = 12 (cid:112) ( (cid:15) F + 2 λ ) − η F . (17)Eqs. (15) and (17) are plotted as the thick solid and short-dashed lines of Fig. 2. η F ϵ F λ η F κ κ QE ( λ ) κ - NPB ( λ ) κ + NPB ( λ ) FIG. 2: Same plot as in Fig. 1 but with the ( λ, κ ) axesinterchanged with respect to that figure. Also shown isthe line corresponding to κ = η F (dot-dashed line), andthe abscissa ticks λ = η F and λ = (cid:15) F are marked thicker.
2. Alternative form of obtaining κ NPB + ( λ ) and κ NPB − ( λ ) In the NPB region, it is true that (cid:15) = κ (cid:112) /τ − λ .The maximum value is (cid:15) = (cid:15) F and this last equa-tion defines two curves in the ( λ, κ ) plane. Taking thesquare and using τ = κ − λ , we obtain the followingbi-quadratic equation in κ : κ − (cid:0) λ + φ (cid:1) κ + λ (cid:0) λ + φ + 1 (cid:1) = 0 , (18)where φ = (cid:15) F ( (cid:15) F + 2 λ ) −
1. Now, making a change in avariable t ≡ κ we arrive to a quadratic equation t − (cid:0) λ + φ (cid:1) t + λ (cid:0) λ + φ + 1 (cid:1) = 0 , (19)whose two roots are t ± ( λ ) = 2 λ + φ ± (cid:112) φ − λ λ ≥
0, because it can be written as φ − λ = η F (cid:0) λ + 4 λ(cid:15) F + η F (cid:1) > λ ≥ . (21)This ensures that the roots t ± ( λ ) are real. With this, wecan write Eq. (19) as (cid:2) κ − t + ( λ ) (cid:3) (cid:2) κ − t − ( λ ) (cid:3) = 0 . (22) It is also easy to demonstrate that both roots t ± ( λ ), be-sides being real, are also positive. To this end we firstwrite the negative of the coefficient of t in Eq. (19) as2 λ + φ = 2 λ + 2 λ(cid:15) F + η F > λ ≥ . (23)With this, it is obvious that t + ( λ ) is positive for λ ≥ t − ( λ ) is enough to provethat the square of Eq. (23) is greater than the discrimi-nant given in Eq. (21). This comes from (cid:0) λ + 2 λ(cid:15) F + η F (cid:1) ≥ η F (cid:0) λ + 4 λ(cid:15) F + η F (cid:1) ⇐⇒ λ (cid:0) λ + 2 (cid:15) F λ + (cid:15) F (cid:1) ≥ , (24)which is true if λ ≥
0. With these proofs we can be surethat the four roots of κ in Eq. (22) are all real as well.This means that the boundary in the ( λ, κ )-plane where − ≤ ψ ≤ κ NPB ± ( λ ) = (cid:115) (2 λ + φ ) ± (cid:112) φ − λ , (25)in the NPB region, i.e, for κ ≥ η F .It could seem that these two curves given by Eq. (25)are totally different from those obtained in sect. III A 1and given in Eqs. (15) and (17), but they are actually thesame, and already plotted in Fig. 2. One way of provingthis is raising to the square the κ NPB ± ( λ ) functions ob-tained in sect. III A 1, given by Eqs. (15) and (17); thenan easy but lengthy algebra manipulation can demon-strate that the square of Eq. (17) is equal to t + ( λ ) andthat the square of Eq. (15) is also equal to t − ( λ ), bothjointly given in Eq. (20).In Fig. 3 we show the values taken by the scaling vari-able ψ ( κ ( λ ) , λ ), as a function of λ , along different curvesin the NPB region. It can be seen from Fig. 3 that thescaling variable is zero along the curve κ QE ( λ ), which cor-responds to the λ = τ condition, i.e, the position of thequasi-elastic peak. Along the curves κ NPB ∓ ( λ ), the scal-ing variable always takes its limiting values in the RFGmodel, ψ = ±
1, respectively. These values are shown bythe solid and short-dashed lines in Fig. 3, respectively.Any curve lying in between κ NPB − ( λ ) and κ QE ( λ ), as thedot-dashed one indicates in Fig. 3, has a positive valuefor the scaling variable; while those curves lying in be-tween κ QE ( λ ) and κ NPB+ ( λ ), always have negative valuesfor the scaling variable, as it can be inspected from thedotted line of the same figure. B. Analytical boundaries in the Pauli-blocking(PB) region
Up to now we have been discussing the boundaries inthe ( λ, κ )-plane of the NPB region, where the followingidentity holds true (cid:15) ≡ max (cid:32) κ (cid:114) τ − λ, (cid:15) F − λ (cid:33) = κ (cid:114) τ − λ. (26) ψ ( κ QE ( λ ) , λ ) ψ ( κ - NPB ( λ ) , λ ) ψ ( κ + NPB ( λ ) , λ ) ψ ( κ QE ( λ )+ λ ) ψ ( κ QE ( λ )- λ ) η F ϵ F λ - - - ψ FIG. 3: Plot of the values taken by the scaling variable ψ , defined on Eq. (10), as a function of λ along differentcurves in the ( λ, κ )-plane. The long-dashed line corre-sponds to the case λ = τ , the solid line is for the lowerbound curve given by κ = κ NPB − ( λ ), while the short-dashed line corresponds to the upper bound curve givenby κ = κ NPB+ ( λ ). Two additional curves are shown forcomparison: the dotted line corresponds to a curve lyingin between κ QE ( λ ) and κ NPB+ ( λ ); while the dot-dashedline is for a curve lying in between κ QE ( λ ) and κ NPB − ( λ )(see Fig. 2).As it was stated above, Eq. (26) always holds when κ ≥ η F , but not necessarily when κ < η F . We will seebelow that for λ ≤ κ < η F (since τ ≥ λ, κ )-plane where (cid:15) can be equal to thesecond argument of the maximum function appearing inEq. (26), while there are other regions where (cid:15) is stillequal to the first argument of the maximum function.In order to delimit these boundaries note that κ (cid:114) τ − λ ≤ (cid:15) F − λ ⇐⇒ κ (cid:114) τ ≤ (cid:15) F − λ. (27)The latter inequality defines some region in the ( λ, κ )-plane. As both sides of the inequality given in Eq.(27) are positive (because we are looking for solutionswhere λ < η F < (cid:15) F ), taking the square, substituting τ = κ − λ , and rearranging terms we finish with an-other bi-quadratic inequality for κ : κ − ρκ + (cid:0) λ(cid:15) F − λ (cid:1) ≤ , (28)where ρ = 2 λ − λ(cid:15) F + η F . Performing the usual trick ofsolving the inequality by making the change of variable u ≡ κ , we obtain that the equality holds for u ± ( λ ) = ρ ± (cid:113) ρ − λ(cid:15) F − λ ) . (29) For Eq. (29) to have real solutions, the discriminant mustbe positive at least in the region of λ -values where we areseeking a solution.It is not difficult to write the condition for the discrim-inant to be positive as ρ − (cid:0) λ(cid:15) F − λ (cid:1) ≥ ⇐⇒ (cid:2) ρ + 2 (cid:0) λ(cid:15) F − λ (cid:1)(cid:3) (cid:2) ρ − (cid:0) λ(cid:15) F − λ (cid:1)(cid:3) ≥ ⇐⇒ λ − λ(cid:15) F + η F ≥ . (30)The last equality has two roots for λ . They are λ ± = (cid:15) F ± > . (31)Therefore, we can write the last inequality of Eq. (30) as4( λ − λ + )( λ − λ − ) ≥ . (32)The only meaningful solution to Eq. (32) is that λ ≤ λ − < λ + [64]. The next step is to see if λ − is lesser than η F or not, because if so then the interval in λ where tohave real roots for u ± ( λ ) is further constrained comparedto the interval defined by 0 ≤ λ < η F . It is easy to write λ − as λ − ≡ (cid:15) F −
12 = (cid:112) η F − η F (cid:16)(cid:112) η F + 1 (cid:17) < η F < η F , (33)where the last inequalities hold because 0 < η F < η F < η F . So we can conclude thatthe discriminant of Eq. (29) is positive and u ± ( λ ) arereal roots of Eq. (28) for 0 ≤ λ ≤ λ − < η F . The nextstep is wondering about the sign and magnitude of thenegative of the coefficient of κ in Eq. (28) in the regionof λ -values between 0 and λ − . The reason for this isbecause depending upon the sign and magnitude of it,the roots u ± ( λ ) can be negative and we want them tobe positive because u ± ( λ ) = (cid:2) κ PB ± ( λ ) (cid:3) should be thesquare of real roots of Eq. (28). To this end we set outthe following inequality and seek for their solutions: ρ ≥ . (34)The equality ρ = 0 has two positive roots for λ , whichare λ (cid:48)± = (cid:15) F ± (cid:112) − (cid:15) F (cid:112) η F ± (cid:112) − η F > . (35)We have to compare them with λ − , given in Eq. (31).The reason for this is because if any of the two new roots λ (cid:48)± is lesser than λ − , then the interval in λ where to seekthe boundary of the PB region can be, again, furtherconstrained from the last condition 0 ≤ λ ≤ λ − . It iseasy to see that λ (cid:48) + is clearly greater than λ − : λ (cid:48) + = (cid:15) F + (cid:112) − η F > (cid:15) F > (cid:15) F − ≡ λ − (36)On the other side, it is also straightforward to see that λ (cid:48)− is greater than λ − as well: λ (cid:48)− ≡ (cid:112) η F − (cid:112) − η F η F (cid:112) η F + (cid:112) − η F > η F > λ − , (37)where in the second step we have multiplied and dividedby (cid:112) η F + (cid:112) − η F .Finally, it is worth noting that the inequality (34) canbe rewritten as 2( λ − λ (cid:48) + )( λ − λ (cid:48)− ) ≥ , (38)which is absolutely fulfilled if λ ≤ λ − < λ (cid:48)− < λ (cid:48) + , be-cause then both parentheses in Eq. (38) are negative andtheir product is positive. Having found the most restric-tive region in the λ variable where Eqs. (30) and (34)are simultaneously fulfilled, we can assert that the roots u ± ( λ ) given in Eq. (29) are both real and positive [65].Thus, we can rewrite the inequality given in Eq. (28) as (cid:2) κ − u + ( λ ) (cid:3) (cid:2) κ − u − ( λ ) (cid:3) ≤ (cid:112) u − ( λ ) ≤ κ ≤ (cid:112) u + ( λ ) , where the second step has been possible to be taken be-cause the roots u ± ( λ ) and κ are all positive. Therefore,we can conclude from all this discussion that the regionwhere PB makes (cid:15) to be equal to the second argument, (cid:15) F − λ , of the maximum function displayed in Eq. (26),corresponds to the region κ PB − ( λ ) ≤ κ ≤ κ PB+ ( λ ) in theregion where 0 ≤ λ ≤ λ − with κ PB ± ( λ ) = (cid:118)(cid:117)(cid:117)(cid:116) ρ ± (cid:113) ρ − λ(cid:15) F − λ ) κ PB ± ( λ ) for λ = 0 and λ = λ − . They are κ PB − (0) = 0 , κ PB − ( λ − ) = η F ,κ PB+ (0) = η F , κ PB+ ( λ − ) = η F . The last two equations can be easily found by noticingthat, for λ = λ − = ( (cid:15) F − /
2, the discriminant of u ± ( λ )is exactly zero (see Eq. (32)), and then there is no dif-ference between κ PB+ ( λ − ) and κ PB − ( λ − ). It is also easy tonotice that κ QE ( λ − ) = η F / κ QE ( λ − ) = (cid:112) λ − ( λ − + 1) = (cid:114) (cid:15) F −
14 = η F . (41)This means that for λ = λ − , (cid:15) = (cid:15) F − λ − = 1 and thenthe scaling variable at the point ( λ, κ ) = ( λ − , η F /
2) isexactly 0 (see definition given in Eq. (10)). In Fig. 4 we show the different regions filled with colorsfor κ ≤ η F , where PB effect occurs or not. The shadedregions between κ PB+ ( λ ) and κ = η F , and between κ PB − ( λ )and κ NPB − ( λ ), respectively, correspond to those zones ofthe allowed phase space of the RFG where there is no PB,i.e, where (cid:15) = κ (cid:112) /τ − λ . On the other hand, theshaded regions between κ PB+ ( λ ) and κ QE ( λ ), and betweenthis last curve and the dotted κ PB − ( λ ) one, respectively,correspond to zones where (cid:15) = (cid:15) F − λ , i.e, where thereis PB. It is worth noting that in this region, delimited bythat kind of inverted parabola formed by joining togetherthe dotted and the three-fold dashed curves of Fig. 4, thevariable (cid:15) and, consequently, the scaling variable ψ onlydepend on λ and not at all on κ . The only important issueto select the sign of ψ is whether the points in these re-gions are above the long-dashed thick line correspondingto the curve κ QE ( λ ) (in whose case the scaling variableis negative); or if on the contrary, the points are belowthis line, in whose case the scaling variable is positive.The purpose of Fig. 5 is to highlight, in general, thesmallness of the region of the ( λ , κ )-space where PB playsa role. Note that λ − (cid:28) η F / η F is not visiblein the λ -axis due to units, while in the vertical axis itappears. Also notice that the horizontal straight line κ = η F / ≤ λ ≤ λ − ),just as it happens for the κ PB+ ( λ ) curve. The same can besaid for the straight line κ = η F λ/ (2 λ − ) and the κ PB − ( λ )curve in the filled PB region just below the QE peakposition curve (long-dashed line). The purposes of thesetwo straight lines will be clear in the following discussion.In Fig. 6 we show the values taken by the scaling vari-able ψ in the RFG model along different curves κ = κ ( λ )in the ( λ, κ )-plane in the region where 0 ≤ λ ≤ λ − . Ofcourse, we have shown the limiting boundaries ψ = ± κ NPB ∓ ( λ ), respectively. They corre-spond to the medium-thick solid and dot-dashed horizon-tal straight lines in Fig. 6, respectively; and to the curvesof the same styles in Figs. 4 and 5. Along the long-dashedthick κ QE ( λ ) curve of Figs. 4 and 5, the scaling variableis equal to zero in Fig. 6 because this is the curve where λ = τ and the sign function vanishes (see Eq. (10)).The rest of curves shown, especially in Fig. 5, remainsinside the PB region for 0 ≤ λ ≤ λ − . In this region,remarked by the filled region between κ PB+ ( λ ) and κ PB − ( λ )curves in Figs. 4 and 5, the scaling variable ψ ( κ ( λ ) , λ )does not depend at all on the κ value taken by any pointor curve inside the region, except for the sign of ψ . Thiscan be viewed in different forms. For instance, taking alook at the values taken by ψ along the curves κ PB+ ( λ )(three-fold dashed thick line) and along the straight line κ = η F / κ along the curves are totally different, andstill the scaling variable takes the same values in Fig. 6,i.e, it starts equaling ψ = − λ = 0 because then (cid:15) = (cid:15) F − λ = (cid:15) F and both curves are above the κ = λ - λ η F η F η F κ κ + PB ( λ ) κ - PB ( λ ) κ QE ( λ ) κ = λ κ + NPB ( λ ) κ - NPB ( λ ) FIG. 4: Plot of the ( λ, κ )-plane in the PB region, i.e, for 0 ≤ λ ≤ λ − and 0 < κ < η F . The three-fold dashedthick curve corresponds to the κ PB+ ( λ ) curve, while the dotted thick line is for the κ PB − ( λ ) boundary. All the regionsurrounded by these two curves corresponds to the PB region. We have also displayed the previously shown (inFig. 2) κ NPB ± ( λ ) curves as dot-dashed thin and solid lines, respectively. The curves κ QE ( λ ) and κ = λ are shown aslong-dashed thick and short-dashed thin lines, respectively, as well. κ QE ( λ ) curve, thus having negative values for the scalingvariable. Finally, for λ = λ − , the scaling variable is zeroalong both paths because it is the intersection point withthe κ QE ( λ ) curve (see especially Fig. 4).Something similar occurs along the paths defined by κ PB − ( λ ) (dotted thick line) and κ = η F λ/ (2 λ − ) (medium-dashed thick line), but in this case for positive values ofthe scaling variable, because in this case both paths areentirely in the filled PB region below the κ QE ( λ ) curve ofFig. 5, thus in the region of positive values for the scalingvariable, as it can be seen again in Fig. 6.The final example is a mixed case, a straight line κ = η F / ψ , passes acrossthe κ QE ( λ ) curve, enters in the filled PB region below the κ QE ( λ ) line and, finally, it gets out of the PB region byentering entirely in the NPB region of positive values of ψ . In this case (corresponding to the solid thin line in Fig.6), the initial behavior of the scaling variable is the sameas those corresponding to the other curves lying entirelyin the PB region above the κ QE ( λ ) curve (negative valuesfor ψ ), until the point where κ QE ( λ ) = η F / λ (cid:39) . κ = η F / κ QE ( λ )curve, and it suddenly changes the sign of ψ along thiscrossing point, as it can be seen in Fig. 6 as the vertical solid thin line. Now the values of ψ roam along those ofany curve entirely contained in the PB region below the κ QE ( λ ) curve (corresponding to positive values of ψ ) untilthe new point where κ PB − ( λ ) = η F / λ (cid:39) . κ = η F / (cid:15) is no longer equal to (cid:15) F − λ ,but to κ (cid:112) /τ − λ , and then, while still having posi-tive values, the scaling variable now approaches ψ = +1,what will happen when κ NPB − ( λ ) = η F / λ (cid:39) . > λ − , and therefore out of therange of Fig. 6). C. Analytical boundaries coming from the leptonkinematics
Up to now we have been discussing the boundaries ofthe RFG model which come out from the nuclear dynam-ics point of view by restricting ourselves to the region ofthe phase space in ( λ, κ ) where the scaling function ofthe RFG model is different from zero. This amounts torestrict the values of the scaling variable to be between − κ + PB ( λ ) κ - PB ( λ ) κ QE ( λ ) κ + NPB ( λ ) κ - NPB ( λ ) κ = η F λ - λ λ - η F λ η F η F κ FIG. 5: Same plot as in Fig. 4, but highlighting thesmallness of the region where PB plays a role. No-tice that λ − (cid:28) η F . Also shown a new straight line, κ = η F λ/ (2 λ − ), in short-dashed style, that is entirelycontained in the filled region between the long-dashedand dotted curves, for the range of values 0 ≤ λ ≤ λ − .The purpose of this line and that corresponding to thehorizontal line κ = η F / ω = E ν − E µ ⇐⇒ E µ = E ν − m N λ (42) q = ( (cid:126)k − (cid:126)k (cid:48) ) = E ν + k (cid:48) − E ν k (cid:48) cos θ µ , (43)where E ν is the initial neutrino energy, θ µ is the muonscattering angle with respect to the direction of the in-cident neutrino, and k (cid:48) = (cid:113) E µ − m µ is the final muonmomentum with energy E µ and mass m µ .The minimal muon energy is its mass and from thiscondition we can obtain from Eq. (42) the, in principle,maximum allowed value for λ , λ max = E ν − m µ m N = (cid:15) ν − (cid:101) m µ , (44)where we have introduced “reduced” and dimensionless ψ ( κ + PB ( λ ) , λ ) ψ ( κ - PB ( λ ) , λ ) ψ ( η F λ - λ , λ ) ψ ( κ QE ( λ ) , λ ) ψ ( κ + NPB ( λ ) , λ ) ψ ( κ - NPB ( λ ) , λ ) ψ ( η F , λ ) ψ ( η F , λ ) λ - λ - - ψ FIG. 6: Values taken by the scaling variable ψ ( κ ( λ ) , λ )along the different curves shown in Figs. 4 and 5 in thePB region, i.e, when 0 ≤ λ ≤ λ − . For an exhaustiveexplanation, see the main text.neutrino energy and muon mass variables, defined as (cid:15) ν ≡ E ν m N , (cid:101) m µ ≡ m µ m N . (45)From Eq. (43) we can write that the absolute value ofthe cosine of the muon scattering angle must be lesser orequal to 1: | cos θ µ | (cid:54) ⇐⇒ (cid:12)(cid:12)(cid:12)(cid:12) E ν + k (cid:48) − q E ν k (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) (cid:54) ⇐⇒ − E ν k (cid:48) (cid:54) E ν + k (cid:48) − q (cid:54) E ν k (cid:48) . (46)Notice that Eq. (46) gives two additional inequalities for κ in terms of λ (the variable λ is hidden in k (cid:48) via itsdependence on E µ and the dependence of the latter on λ through Eq. (42)). From the first inequality, and usingthe “reduced” and dimensionless variables, we obtain: q (cid:54) ( E ν + k (cid:48) ) ⇐⇒ q m N (cid:54) E ν m N + (cid:113) ( E ν − m N λ ) − m µ m N ⇐⇒ κ (cid:54) (cid:15) ν + (cid:113) ( (cid:15) ν − λ ) − (cid:101) m µ ≡ κ leptonmax ( λ ) (47)Analogously with the other inequality of expression(46), we obtain the lower bound for κ constrained fromthe lepton kinematics alone: κ (cid:62) (cid:15) ν − (cid:113) ( (cid:15) ν − λ ) − (cid:101) m µ ≡ κ leptonmin ( λ ) . (48)As the PB region (filled domains in Figs. 4 and 5 sur-rounded by the κ PB+ ( λ ) and κ PB − ( λ ) curves) is always con-tained inside the larger region bounded by κ NPB − ( λ ) (cid:54) κ (cid:54) κ NPB+ ( λ ), and the only difference between the PBregion and the NPB one is the dependence of the scal-ing variable with κ and λ , the furthest constrained phasespace for the RFG model is given bymax (cid:16) κ leptonmin , κ NPB − (cid:17) (cid:54) κ (cid:54) min (cid:0) κ leptonmax , κ NPB+ (cid:1) , (49)provided that the maximum on the left-hand side of ex-pression (49) is always smaller than the minimum on theright-hand side of the same expression in the range of λ -values ranging from λ = 0 to λ = λ max , where λ max isgiven in Eq. (44) for a fixed neutrino/antineutrino energy.An inspection to the curves κ leptonmax ( λ ) and κ leptonmin ( λ ),given in Eqs. (47) and (48) respectively, reveals that thefirst curve is a monotonically decreasing one with λ , whilethe second one is a monotonically increasing function of λ . Both curves reach the same value when λ = λ max ,i.e, when the square roots cancel and κ leptonmax ( λ max ) = κ leptonmin ( λ max ) = (cid:15) ν . Thus, as both κ NPB+ ( λ ) and κ NPB − ( λ )are increasing functions of λ (see for example Figs. 2, 4or 5), it is interesting to look for the cutting points be-tween κ leptonmax ( λ ) and κ NPB+ ( λ ) and κ NPB − ( λ ), or between κ leptonmin ( λ ) and κ NPB+ ( λ ) and κ NPB − ( λ ), if any. These cut-ting points will help us to constrain and to understandthe form of the available phase-space in the RFG modelfor a fixed neutrino/antineutrino reduced energy (cid:15) ν .We can start by looking for the λ value where κ NPB+ ( λ ) = κ leptonmax ( λ ). This value can give us the λ point where the minimum function appearing in expres-sion (49) changes from selecting one curve to the otherone. In this case, to obtain the solution for λ , it is betterto use the expression for κ NPB+ ( λ ) given in Eq. (17) ratherthan that given in Eq. (25), although both are equivalent,just because the first one is much simpler to manipulate.To obtain the solution it is necessary to square twice theequation, and we finish with the following second degreeequation for λ after a lengthy algebra manipulation: aλ + bλ + c = 0with a ≡ (cid:15) ν ( (cid:15) F + η F ) ,b ≡ (cid:15) F (cid:0) (cid:101) m µ + (cid:15) ν η F (cid:1) + 2 (cid:15) ν (cid:2) η F + 2 (cid:101) m µ − (cid:15) ν ( (cid:15) F + η F ) (cid:3) ,c ≡ (cid:101) m µ (cid:0) η F − (cid:15) ν η F + (cid:101) m µ (cid:1) . The above equation has two roots. Only the solution withthe positive square root of the second degree equationis positive for some values of (cid:15) ν . The other solution isalways negative and we discard it. The relevant solution, which we call λ ++ , is given by λ ++ = ζ + (2 (cid:15) ν − η F ) − (cid:101) m µ ( (cid:15) F + 2 (cid:15) ν )(1 + 4 ζ + )+ | (cid:15) ν − η F | (cid:113) (cid:101) m µ − (cid:101) m µ + ζ + (cid:0) ζ + − (cid:101) m µ (cid:1) (1 + 4 ζ + ) , (50)where we define ζ + = (cid:15) ν ( (cid:15) F + η F ). In the above equa-tion (50) there is a value for the reduced neutrino energy (cid:15) ν for which λ ++ = 0. This value can be found by equat-ing the numerator of (50) to zero and solving for (cid:15) ν . Thisvalue of (cid:15) ν is precisely that for which the cut point be-tween κ NPB+ ( λ ) and κ leptonmax ( λ ) occurs at λ = 0. Again,a lengthy and tedious algebraic manipulation leaves uswith a second degree equation in the variable (cid:15) ν : a (cid:15) ν + b (cid:15) ν + c = 0with a ≡ η F ( η F + (cid:15) F ) ,b ≡ (cid:0) η F − η F + (cid:15) F ) (cid:0) η F + (cid:101) m µ (cid:1)(cid:1) ,c ≡ − η F − (cid:101) m µ . Only the solution of the second degree equation with thepositive square root is again positive, while the othersolution is always negative and we discard it. The mean-ingful solution is [67] (cid:15) ν + = η F + (cid:101) m µ η F > . (51)Notice that the solution given in the above equation (51)depends both on the reduced final lepton mass and on anuclear property, namely the Fermi momentum (in unitsof the nucleon mass).Also note that, given the behavior of the curves κ leptonmax ( λ ) (which is a monotonically decreasing functionof λ ), and κ NPB+ ( λ ) (which is monotonically increasing),for (cid:101) m µ (cid:54) (cid:15) ν (cid:54) (cid:15) ν + the upper limit of the phase spaceof the QE double differential cross section with respectto final lepton variables in the RFG model is boundedonly by the curve κ leptonmax ( λ ). Or said in other words, if (cid:101) m µ (cid:54) (cid:15) ν (cid:54) (cid:15) ν + , then the minimum function of the right-hand side of inequality (49) is always the curve κ leptonmax ( λ )for all the allowed λ values.On the other hand, the presence of the reduced lep-ton mass in Eq. (51) means that the necessary rangesof neutrino energies to allow the lepton kinematic con-straints to determine by themselves the upper boundaryof the phase space, depends a lot on the kind of neutrinoflavor for charged current processes. For instance, fortau neutrinos and for typical values of Fermi momenta( η F (cid:39) . E ν + = 2 m N (cid:15) ν + (cid:39) . E ν + (cid:39)
240 MeV, which is a quite low0neutrino energy. Of course, these values are completelyrelated to the threshold neutrino energies to produce a τ lepton or a muon in charged-current elastic scatteringwith nucleons, respectively.Now we can look for the cut point between κ leptonmax ( λ )and κ NPB − ( λ ), which will occur for a λ value larger [68]than λ ++ , given in Eq. (50). Again, after a lengthy cal-culation, solving κ NPB − ( λ ) = κ leptonmax ( λ ) for λ , we find tworoots: λ ±− = ζ − (2 (cid:15) ν + η F ) − (cid:101) m µ ( (cid:15) F + 2 (cid:15) ν )(1 + 4 ζ − ) ± (2 (cid:15) ν + η F ) (cid:113) (cid:101) m µ − (cid:101) m µ + ζ − (cid:0) ζ − − (cid:101) m µ (cid:1) (1 + 4 ζ − ) , (52)where we define ζ − = (cid:15) ν ( (cid:15) F − η F ). Both roots are physi-cal (not complex numbers) for some reduced neutrino en-ergy values which depend on the model scaling function.Note that the expression for the first root, λ + − , corre-sponds to the solution λ ++ given in Eq. (50) if one makesthe replacement η F (cid:55)→ − η F , which makes sense becausethe only difference between κ NPB+ ( λ ) (given in Eq. (17)),and κ NPB − ( λ ) (Eq. (15)) is the sign of η F . The other root, λ −− , when it is physical, always corresponds to the cut-ting point between the curves κ NPB − ( λ ) and κ leptonmin ( λ ).This latter solution corresponds to the λ point wherethe maximum function appearing on the left-hand sideof the inequality (49) changes from one of its argumentsto the other one. It could seem striking at first sight thatthis solution appears when we have not used at all the κ leptonmin ( λ ) curve to obtain it, but (as it can be seen inFig. 7) the curves κ leptonmax ( λ ) and κ leptonmin ( λ ) form actuallytwo different branches of the same unique curve, namely,( κ − (cid:15) ν ) = ( (cid:15) ν − λ ) − (cid:101) m µ .In Fig. 7 we show the phase space in the ( λ, κ ) variablesfor two different neutrino energies in the RFG model. Wehave also shown the cutting points between the differentcurves κ leptonmax , min ( λ ) and κ NPB ± ( λ ), which constrain the lep-ton and nucleon kinematics in the RFG model, respec-tively. Note that, because we have shown the plots forthe case of muon neutrinos, m µ = 106 MeV/c , for neu-trino energies close to the muon mass the phase space ismostly constrained by the lepton kinematics (left panel).However, for higher neutrino energies (right panel), theavailable phase space is almost entirely constrained bythe nucleon kinematics in the RFG (thin solid and dot-dashed lines corresponding to the limits of the RFG scal-ing function). In this latter case, lepton kinematics playsa really minor role, except in the region of the endpointin λ , which corresponds to the largest energy transfersto the nucleus (and consequently the least energy carriedby the muon), so one starts to see the effects of the muonmass as if one were in the situation of the left panel. IV. SUSAM* MODEL CASE
In Fig. 8 we show the scaling functions of the two mod-els we discuss in this article. In solid line style the scalingfunction of the RFG model is depicted, whose expressionwas given in Eq. (3). The other three scaling functionsare those of the SuSAM* model, in particular, those ex-tracted in a global fit to the world “QE” electron scat-tering data [69, 70] extracted out from the inclusive datastored in the web site of Ref. [69]. These three scalingfunctions (shown as central, min and max in Fig. 8) wereobtained in Ref. [52] after a selection procedure based onthe scaling hypothesis of the QE data, and the lower andupper scaling functions (min and max in Fig. 8, respec-tively) correspond to the estimation of the uncertaintyor thickness of the super-scaling band where the bulk ofthe QE data tend to accumulate.The functional form of the scaling functions of theSuSAM* model depicted in Fig. 8 is: f SuSAM* ( ψ ) = a e − ( ψ − a a + b e − ( ψ − b b e − ψ − c c , (53)where the parameters a i , b i and c i can be found, for thethree different scaling functions of Fig. 8, in Table I ofRef. [52], corresponding to the set labeled as Band C.It is worth noting the asymmetry shown by the scalingfunctions of the SuSAM* model, which have longer tailstowards positive values of the scaling variable ψ than theyhave for negative ones, in contrast with the symmetricRFG scaling function.In sect. III, and particularly in subsection III A 2, wehave thoroughly discussed the boundaries in the ( λ, κ )-plane where the scaling variable ψ is between − κ NPB ± ( λ )for the RFG model by imposing (cid:15) ≡ κ (cid:112) /τ − λ = (cid:15) F , which is the equivalent condition to ψ = ±
1. All wehave to do now to extend it for the SuSAM* model isto identify extreme values of the scaling variable, namely ψ extr , where we can safely affirm that the SuSAM* scal-ing function is negligible beyond these extreme values,one on the left and the other on the right. Note that,given the asymmetry of the SuSAM* scaling function,these extreme values are not going to be necessarily thesame at the left and at the right of the QE peak position.Let us assume we are in the region where λ > τ , andthe sign function appearing in Eq. (10) is positive. If1 κ maxlepton ( λ ) κ minlepton ( λ ) κ + NPB ( λ ) κ - NPB ( λ ) E λ ++ λ +- λ -- λ η F κ ν =
300 MeV λ ++ λ +- λ -- λ η F κ E ν = FIG. 7: Available phase space in ( λ, κ ) variables in the RFG model for E ν = 300 MeV (left panel) and for E ν = 2000MeV (right panel), shown as the shaded regions for λ ≤ λ + − . Also displayed are the different cut points betweenthe curves constraining the lepton and nucleon kinematics, labeled as in Eqs. (50) and (52). The value of η F hasbeen taken as 0 . k F = 225 MeV/c. Also note that the curves κ leptonmax , min ( λ )(dashed and dotted lines, respectively) are actually two different branches of the same curve.the positive extreme value ( ψ extr ) for the scaling variablein the SuSAM* model is larger than 1, this obviouslymeans that (cid:15) > (cid:15) F . The limiting curve in the ( λ, κ )plane will be obtained when ψ ≡ (cid:113) (cid:15) − (cid:15) F − = ψ extr , where ψ extr has to be chosen properly as a large value wherethe scaling function of the SuSAM* model can be totallyneglected beyond that value. This last equation is totallyequivalent to: (cid:15) ≡ κ (cid:114) τ − λ = 1 + ( (cid:15) F − ψ . (54)The same equation would have been obtained for the case λ < τ , with the negative sign function in Eq. (10), as italso happened in the RFG case. Note that if we choose ψ extr = ±
1, we recover the condition of the RFG, (cid:15) = (cid:15) F , as it should be.To obtain the boundaries of the phase space in the( λ, κ ) plane for the SuSAM* model, note that Eq. (54) isthe same as that for the RFG ( (cid:15) = (cid:15) F ) with the right-hand side replaced by 1 + ( (cid:15) F − ψ instead of (cid:15) F .Consequently, we can take the Eq. (25) and replace anyappearance of (cid:15) F by the new (cid:15) (cid:48) F ≡ (cid:15) F − ψ ,where ψ extr does not have necessarily to be equal for the κ NPB+ ( λ ) (corresponding to negative values of ψ ) and forthe κ NPB − ( λ ) (corresponding to positive values of the scal-ing variable) functions, because of the asymmetry of thescaling functions in the SuSAM* model.The last important point that remains to be shownis that, for the SuSAM* model, it is still true that κ NPB − (0) = 0 and that κ NPB+ (0) > η F ≡ (cid:112) (cid:15) F −
1. This isimportant because mainly the effect of choosing a widerscaling function than that of the RFG is to broaden the available phase space shown in Fig. 7 between the thindot-dashed and solid curves, thus increasing the domainof integration in ( λ, κ ) space and obtaining a larger to-tal cross section for a fixed neutrino/antineutrino energy, σ ( E ν ).The fact that κ NPB+ (0) > η F is totally related to thehigh momentum components, larger than the Fermi mo-mentum, that real nuclei have in its ground state. Thesehigh momentum components, mainly produced by inter-action and short-range correlations (SRC) [71–87], aretotally missing in the RFG model, but not in the phe-nomenological scaling function of the SuSAM* model,which has been obtained from a global fit to selected“QE” electron scattering data from nuclei .It is also well known that the effects of SRC are mainlypresent in the left tail of the scaling function [88–90], i.e,for ψ extr < −
1, which is precisely the left extreme ψ valueto be adequately chosen for the curve κ NPB+ ( λ ). This con-nection between high momentum components and scal-ing violations, and their effects in the total integratedQE neutrino cross section are deferred for a forthcomingstudy.It is straightforward to prove that κ − (0) = t − (0) = 0,either from Eq. (20) or (25), irrespective of the valuetaken by (cid:15) F or (cid:15) (cid:48) F , provided that both are greater than1, as it is the case. For the other demonstration we have,2 f RFG ( ψ ) f SuSAM * central ( ψ ) f SuSAM * min ( ψ ) f SuSAM * max ( ψ ) - - ψ f ( ψ ) FIG. 8: Different scaling functions used in this article asa function of the scaling variable ψ . The well-knownscaling function of the RFG model is shown in solidstyle, while the three SuSAM*-model scaling functions,extracted from a global fit to “QE” electron scatteringdata off nuclei in Ref. [52], are shown as dotted, dot-dashed and dashed lines for the central, the lower andupper bounds, respectively. The scaling functions of theSuSAM* model plotted in this figure correspond to theparameters denoted as Band C in Table I of Ref. [52].from Eq. (20), κ (0) = t + (0) = (cid:15) (cid:48) F − (cid:113) (1 − (cid:15) (cid:48) F ) (cid:15) (cid:48) F − (cid:12)(cid:12) − (cid:15) (cid:48) F (cid:12)(cid:12) (cid:15) (cid:48) F − (cid:0) (cid:15) F − ψ (cid:1) − , (55)where ψ left is the negative value of the scaling variable,lesser than −
1, that one has to take to ensure that thescaling function is negligible beyond that value. Finally,as ψ >
1, it is also true that1 + ( (cid:15) F − ψ > (cid:15) F > ⇒ κ (0) = (cid:0) (cid:15) F − ψ (cid:1) − > (cid:15) F − ≡ η F ⇐⇒ κ + (0) > η F , (56)where the superscripts NPB have been dropped in theabove discussion to shorten the cumbersome notation.In Fig. 9 we show the comparison between the avail-able phase space in the RFG (already shown in Fig. 7)and SuSAM* scaling models, for the same two neutrinoenergies as in Fig. 7. The most remarkable difference isthe enlargement of the phase space, shown as the palershade, in the SuSAM* model. This enlargement is onlyattributable to the tails of the super-scaling function of the SuSAM* model, which are absent in the RFG, be-cause the curves delimiting the boundaries from the lep-ton kinematics constraints are the same, they do not de-pend at all on the scaling function or integrated nucleonkinematics. The main consequence of this enlargementof the phase space will be reflected in a larger total inte-grated cross section. Of course, this increase in the inte-grated cross section will depend on the values attained bythe double differential (with respect to ( λ, κ ) variables)CCQE cross section in the enlarged region. We can en-sure that there is going to be a clear increase, becausein the regions outside the phase space of the RFG, butclose to its boundaries, the differential cross section willstill be substantial because the scaling function of theSuSAM* is truly different from zero for scaling variableslarger than 1 and lesser than −
1, which corresponds topoints lying in the paler shaded regions. In any case, asexpected, when the values of ( λ, κ ) are approaching thetwo-fold thin dashed curve κ NPB+SuSAM* ( λ ) and the long-dashed thin line κ NPB − SuSAM* ( λ ), their contribution to thetotal cross section will be very small because in thesezones of the phase space the SuSAM* scaling functionbecomes negligible.It is worth noting that the SuSAM* boundaries havebeen calculated in Fig. 9 for ψ left = − . ψ right = 6for the upper and lower SuSAM* boundaries, correspond-ing to the two-fold dashed and long-dashed thin curves,respectively. These extreme values of the scaling vari-able for the left and right tails of the SuSAM* super-scaling function have been chosen thinking in the valuesattained by the central SuSAM* function (shown in Fig.8) at them, which amount to roughly a factor 10 − of thevalue of this scaling function at the peak. Also, in Fig. 9we show, in the κ -axis, the value η (cid:48) F ≈ .
62, defined as η (cid:48) F = (cid:113) (cid:15) (cid:48) F − (cid:15) (cid:48) F = 1 + ( (cid:15) F − ψ , as the horizontal thick dot-dashed line.The analytical delimitation of the phase space bound-aries in the ( λ, κ ) variables is important because then,when integrating over the final lepton kinematics in or-der to obtain the total CCQE integrated cross section asa function of the neutrino/antineutrino energy, we canmake the integration procedure as efficient as possible, aswe are evaluating the integrand only where it is differentfrom zero. This is particularly important when the neu-trino energy is really huge, E ν ∼ −
100 GeV, becausethen the contribution of the QE peak is concentrated atsmall values of the energy transfer ω (if compared withthe neutrino energy) and at very forward angles. As forhuge neutrino energies the allowed interval in ω is alsohuge, it is convenient to constrain as much as possible theangular interval (related to κ ) where truly integrating.3 κ maxlepton ( λ ) κ minlepton ( λ ) κ + RFGNPB ( λ ) κ - RFGNPB ( λ ) κ + SuSAM * NPB ( λ ) κ - SuSAM * NPB ( λ ) λ ++ RFG λ +- RFG λ η F η F κ E ν =
300 MeV λ ++ RFG λ +- RFG λ ++ SuSAM * λ +- SuSAM * λ η F η F κ E ν = FIG. 9: Comparison of the available phase spaces in ( λ, κ ) variables in the RFG (darker shade) and SuSAM* (palershade) models for E ν = 300 MeV (left panel) and for E ν = 2000 MeV (right panel), shown as the shaded regions for λ ≤ λ model+ − , in general. Also displayed are the different cut points between the curves constraining the lepton andnucleon kinematics, labeled as in Fig. 7 for the different models. Note that the available phase space gets much moreenlarged in the SuSAM* model, due to the tails of the scaling function. V. RESULTS
In this section, we show the results for the CCQE dou-ble differential d σdT µ d cos θ µ and d σdκ dλ neutrino and antineu-trino cross sections for the RFG and SuSAM* models, aswell as the fully integrated total cross sections in bothmodels. A. Double differential cross sections
In Fig. 10 we show the double differential CCQE d σdκ dλ cross section per neutron for the ( ν µ , µ − ) reaction off C, at incident neutrino energies of 300 MeV, for thetwo models discussed in this work: RFG (left panel) andSuSAM* (right panel). The available phase spaces inthe two models at this neutrino energy are those alreadydepicted in the left panel of Fig. 9. Note that, althoughnot exactly the same, both scales in the two panels arevery similar, as well as the values reached by the crosssection. In Fig. 10 we also show as the short-dashed linethe curve κ = κ QE ( λ ), where λ = τ and ψ = 0, i.e,the curve corresponding to the position of the QE peak,which, as expected, runs over the region of largest crosssection. In solid style, it is also shown the boundary ofthe PB region. As already mentioned in sect. III B, inthe PB region the scaling variable ψ only depends on λ and not on κ . Because of this, the contour lines (curveswith the same value of the cross section) inside the PBregion are almost vertical lines, because the dependenceon κ mainly enters through the lepton kinematic factors V K and the nuclear response functions U K of Eq. (2) andit is very mild at least for the RFG (left panel) model.Notice also that at the boundaries of the PB region, thecontour lines show a sudden change of their direction.This is because at these boundaries the scaling variable ψ starts to sharply depend on κ as well.There is, nevertheless, a remarkable difference betweenthe left and right panels of Fig. 10 in the PB region:in the SuSAM* model (right panel) the color gradientalong vertical lines of constant λ changes abruptly whencrossing the QE curve, especially for small values of λ ;however this effect is totally absent in the left panel, cor-responding to the RFG model. The reason for this isbecause of the properties of the scaling functions in thetwo models. In the RFG, the scaling function given byEq. (3) is an even function of ψ . This means that forconstant λ there is no difference in being above or belowthe QE curve inside the PB region (the only differenceis the sign of the scaling variable, but not the value ofthe scaling function in the RFG model). However, thesituation is very different in the SuSAM* model becauseits scaling function, given by Eq. (53), also depends on ψ and not only on ψ . Hence, a simple change of signin the scaling variable can produce a large difference in4 FIG. 10: Comparison of the density plots for the ν µ CCQE double differential d σdκ dλ cross section per neutron in Cfor the RFG (left panel) and SuSAM* (right panel) models at E ν = 300 MeV. Note that the available phase spacesin the different models are those shown in the left panel of Fig. 9. We show in short-dashed style the curve κ QE ( λ ),where the QE peak is placed; while in solid fashion we also display the boundary of the PB region, already shown inFigs. 4 and 5.the scaling function (see dotted line of Fig. 8), thus in-ducing a sudden change in the value of the cross sectionwhen passing from positive values of the scaling variable(below the short-dashed curve) to negative ones (abovethe same curve) in the PB region. Nonetheless, this ef-fect seems to be quite pronounced only for small valuesof λ in the PB region, and not so perceptible for λ valuescloser to the end point of the PB region, given by λ − inEq. (33).In Figs. 11 and 12 we show the density and contourplots of the double differential CCQE d σdκ dλ cross sectionper neutron for ν µ reactions on C at a fixed neutrinoenergy of 2000 MeV. Figure 11 corresponds to the RFGmodel, while Fig. 12 shows the results for the SuSAM*one. Left panels highlight the region of small values ofenergy transfers λ , i.e, showing clearly the PB region,while right panels in both figures show the full phasespace. At this neutrino energy, the boundary of the phasespace is basically delimited by the curves obtained fromthe nuclear model conditions (limited by imposing thecondition that the scaling function is zero or negligible),and not from the lepton kinematics, as it happened inFig. 10 for smaller neutrino energy. Besides that, thesharp boundaries of the phase space for the RFG model,shown in Fig. 11, are due to the sharp way in which theRFG scaling function goes to zero at ψ = ±
1. However,in Fig. 12, the phase space extends further than for theRFG case just because the SuSAM* scaling function hastails beyond ψ = ±
1. Actually, the phase space of theSuSAM* model would extend even further than what isshown in Fig. 12, but with negligible values of the cross section, already visible in the own figure.In general, the values of the cross section in both mod-els at E ν = 2000 MeV are very similar in the same regionsof the ( λ, κ ) phase space. In Fig. 12, there seems to bea non negligible cross section in the SuSAM* model inregions of the phase space below and close to the lowerboundary of the RFG model, according to its color leg-end. These additional contributions will have a large im-pact in the total integrated CCQE cross section shownlater on in Fig. 17 at E ν = 2000 MeV.Nonetheless, the most important feature of Figs. 11and 12, if compared with Fig. 10 for E ν = 300 MeVof incident neutrino energy, is that the maximum of thedouble differential cross section d σdκ dλ depends very lit-tle on the neutrino energy. Indeed, for E ν = 300 MeVthe maximum of the cross section is around 550 × − cm /neutron, while for E ν = 2000 MeV this maximumis around 450 × − cm /neutron. This remarkablefeature makes this double differential cross section espe-cially well-suited to be used in MC generators to selectthe kinematics of the final lepton events for fixed neutrinoenergy. This is especially relevant for the generators thatuse the acceptance-rejection method to select the events,because using this method it is necessary to normalize thedouble differential cross section to its maximum value. And if this maximum value depends very weakly with theneutrino energy, one can efficiently set a fixed maximumsuitable for all the neutrino energies.
We will see thatthis efficiency would not be so attainable if one uses thedouble differential cross section d σdT µ d cos θ µ , given in Eq.(2), instead of d σdκ dλ , just because the maximum of the5 FIG. 11: Density plots for the ν µ double differential CCQE d σdκ dλ cross section per neutron in C in the RFG modelat E ν = 2000 MeV. The left panel highlights the PB region, while the right one shows the full phase space regioncorresponding to those already shown in the right panels of Figs. 7 and 9. Lines have the same meaning as in Fig. 10. FIG. 12: Same as Fig. 11 but for the SuSAM* model. Short-dashed and solid lines have the same meaning as in Figs.10 and 11. Notice, however, in the left panel, the dot-dashed lines that correspond to the upper and lower boundariesof the RFG model, i.e, the same boundaries shown in the left panel of Fig. 11.former depends very strongly on the neutrino energy.Indeed, if we inspect Fig. 13, where the CCQE ν µ -double differential cross section d σdT µ d cos θ µ per neutronhas been plotted for three different neutrino energies inthe SuSAM* model, we can conclude two main things:First, the height of the peak of this cross section isstrongly growing with the neutrino energy, as it can beseen from the values taken in the graduated color scale.Second, the larger the neutrino energy is, the more con-centrated the bulk of the cross section is in a smaller re- gion of the phase space, although this last conclusion canget overshadowed by the differences in the figure scales.Moreover, the gradient of the cross section grows stronglywith the neutrino energy for this differential cross sec-tion (larger variations of the cross section in a smallerregion of the phase space, which makes the contour linesof constant cross section to appear closer and closer asthe neutrino energy increases). Note, in particular, thatthis behavior is very striking in the bottom panel of Fig.13, i.e, for E ν = 20 GeV. In this latter panel, in the bot-6 FIG. 13: Comparison of the density plots for the CCQE ν µ -double differential cross section N d σdT µ d cos θ µ off C in theSuSAM* model for three different neutrino energies, highlighting the PB region, where the bulk of the cross sectionis concentrated. In the top left panel we show the density plot for E ν = 300 MeV; in the top right panel we displaythat for E ν = 2000 MeV; while in the bottom panel the plot for E ν = 20 GeV is shown as well. Curves on the plothave the same meaning as they had in Figs. 10–12. The white hole in the bottom left corner of the bottom panelmeans that in that region the cross section is reaching values larger than the maximum shown in its scale.tom left corner, there is a white hole which means thatthere, the double differential cross section is much largerthan the maximum value shown in its scale.These conclusions should be compared with those ofthe d σdκ dλ cross section per neutron shown in Figs. 10–12,where they were the opposite, i.e, the peak of the crosssection was almost flat with the neutrino energy (we havealso checked that this statement is also true for E ν = 20GeV, although not shown in any figure), and the vari-ation of the cross section over the phase space is muchsofter. These two special features make the double differ- ential cross section d σdκ dλ much more suitable to generatethe final lepton events in any MC generator, speciallythose which use the acceptance-rejection method. In-deed, the event generation consists in the following steps: • randomly select a point in kinematic phase space,e.g. λ and κ ; • randomly choose uniformly distributed variable t inrange (0,1); • accept event if d σdκ dλ | λ = λ ,κ = κ > t max (cid:16) d σdκ dλ (cid:17) andreject otherwise.7For this method to be efficient, the differential cross sec-tion has to be as flat as possible for all neutrino ener-gies. First, it allows maximum search algorithm to bemore efficient (seeking the maximum more accurately in ashorter time). Second, the fewer attempts to select kine-matic variables that are rejected, the faster the eventsare generated.In Figs. 14 and 15 we show the CCQE ¯ ν µ -induced dou-ble differential cross section d σdκ dλ per proton for two dif-ferent antineutrino energies, respectively. In the left pan-els we display the density plots for the RFG model, whilein the right ones we show those for the SuSAM* model.The main conclusion that can be drawn from these fig-ures if compared with the corresponding ones for the neu-trino case is that, as expected, the antineutrino cross sec-tions are smaller than their neutrino counterparts. Thisis especially clear in Fig. 14, if compared with Fig. 10,because the values in the scales of the figure for antineu-trinos are roughly half of the values shown in Fig. 10,and the regions where the maximum values are reachedin Fig. 14 are clearly smaller in size than those of Fig. 10,despite the fact that the available phase space is exactlythe same.The comparison can be less clear for the case of an-tineutrinos of E ¯ ν = 2000 MeV (Fig. 15) if one comparesthe corresponding model with the left panels of Figs. 11and 12, because in this case the color scales reach similarvalues, although a bit smaller for antineutrinos. How-ever, one can notice that the number of contour lines ofconstant cross section that enter completely inside theshown phase space (this is the same area of phase spaceshown in the left panels of Figs. 11 and 12) is larger inFig. 15 (8 contour lines out of 10) than it was for theneutrino case (6 contour lines out of 10). This meansthat, even although the color scales could be consideredsimilar, the contour lines for the antineutrino case appearmore concentrated in the same region of phase space thantheir neutrino counterparts. Thus, we can conclude thatlarger cross sections extend far beyond the same phasespace shown in Fig. 15 for the neutrino case than forthe antineutrino one, yielding a larger CCQE total crosssection for neutrinos than for antineutrinos. B. Total integrated cross section
In this section we discuss the integrated CCQE crosssections both for neutrinos and antineutrinos off C,when integration over the ( λ, κ ) phase space is carriedout.First of all, we want to point out a thorough descrip-tion of how the ( λ, κ ) phase space behaves as the neutrinoenergy increases. Notice that the curves κ NPB ± ( λ ), eitherthose described by Eq. (25), or by Eqs. (15) and (17)(which are actually the same expressions, as explained insect. III A 2), do not depend at all on the reduced neu-trino energy (cid:15) ν . Therefore these boundaries are alwaysthe same irrespective of the values taken by the neutrino energy. The dependence on the neutrino energy is in thecurves κ leptonmax , min ( λ ) given by Eqs. (47) and (48).At low neutrino energies, the phase space is completelybounded by the final lepton kinematics, i.e, by the curves κ leptonmax , min ( λ ) solely. This is the case, for instance, of theright panels of Figs. 10 and 14. In this case, there are nocutting points between the two lepton kinematic branchesand between the upper and lower κ NPB ± ( λ ) curves. Thissame effect would occur in the left panels of Figs. 10 and14, corresponding to the RFG model, but at a neutrinoenergy lower than 300 MeV, because in the RFG the κ NPB ± ( λ ) curves are squeezed with respect to those of theSuSAM* model. This can be seen, for instance, in thetop left panel of Fig. 16 for the RFG.As the neutrino energy increases, the cutting pointsbetween the curves κ leptonmax , min ( λ ) and κ NPB ± ( λ ) start to ap-pear. One of these cuts has been already discussed insect. III C, labelled as λ ++ and given in Eq. (50). Theother two additional cuts that can occur are those givenby Eq. (52). However, there is a reduced neutrino en-ergy (cid:15) ν for which λ + − = λ −− . This happens when theradicand of Eq. (52) is zero. Thus, we can find the (cid:15) ν value for this to happen by equating the radicand of Eq.(52) to zero and solving the second degree equation for (cid:15) ν . The result is ˜ (cid:15) ν ± = (cid:101) m µ ( (cid:101) m µ ± (cid:15) (cid:48) F − η (cid:48) F , (57)where (cid:15) (cid:48) F ≡ (cid:15) F − ψ and η (cid:48) F = (cid:112) (cid:15) (cid:48) F −
1. Ofcourse, if one wants to recover the results of the RFG,one substitutes ψ right = 1.In principle, the ˜ (cid:15) ν − solution can be ruled out for elec-tron and muon neutrinos because it is negative [91], but˜ (cid:15) ν + is positive and must be considered. At this reducedneutrino energy ˜ (cid:15) ν + , there is a single and tangent cutbetween the curves κ leptonmin ( λ ) and κ NPB − ( λ ), as it can beseen in the left panel of Fig. 16.If the neutrino energy continues increasing, the two λ -cuts given by Eq. (52) are different, but still both cutsoccur between the κ leptonmin ( λ ) and κ NPB − ( λ ) curves (as itcan be observed in the top right panel of Fig. 16), until ahigher neutrino energy (˜ (cid:15) ν ) is reached, at which λ + − = λ max , with λ max given by Eq. (44). In this range of valuesfor (cid:15) ν ∈ [˜ (cid:15) ν + , ˜ (cid:15) ν ], the range of integration in λ still runsfrom λ ∈ [0 , λ max ].To find ˜ (cid:15) ν one could equate λ + − = λ max and tryto solve it for (cid:15) ν , but this is very difficult because theequation turns out to be a third degree equation in (cid:15) ν .Nonetheless, there is a very easy way to obtain this valueof ˜ (cid:15) ν : we can equate κ leptonmax ( λ max ) = κ NPB − ( λ max ) andsolve it for (cid:15) ν . Given that κ leptonmax ( λ max ) = (cid:15) ν , we cantake Eq. (15) for κ NPB − ( λ max ) and solve the equation for (cid:15) ν . The result is straightforward˜ (cid:15) ν = (cid:101) m µ ( (cid:15) (cid:48) F − (cid:101) m µ ) (cid:15) (cid:48) F − η (cid:48) F − (cid:101) m µ . (58)8 FIG. 14: Comparison of the density and contour plots for the ¯ ν µ CCQE double differential d σdκ dλ cross section perproton in C for the RFG (left panel) and SuSAM* (right panel) models at E ¯ ν = 300 MeV. Lines have the samemeaning as in Fig. 10. Note that now the cross section is roughly half than that for the neutrino case, but also muchsmaller along other regions of the whole phase space because of the minus sign in Eq. (2), which applies for CCQEantineutrino scattering. Notice as well about the difference this minus sign makes in the contour lines of constantdouble differential cross section. FIG. 15: Same as Fig. 14 but for E ¯ ν = 2000 MeV. The left panel (RFG model) should be compared with the leftpanel of Fig. 11, while the right one (SuSAM* model) should be compared with the left one of Fig. 12. These plotshighlight the PB region and do not show the full phase space. Note again that the double differential cross section issmaller for CCQE antineutrino scattering than it is for neutrino case. The dot-dashed lines in the right panel are theboundaries of the RFG model shown in the left panel.This is the situation shown in the bottom left panel ofFig. 16. And now, we can ensure that for (cid:15) ν > ˜ (cid:15) ν , the λ + − cut given by Eq. (52) is lesser than λ max , but now itis a cut between the curve κ NPB − ( λ ) and the upper branchof the lepton kinematics boundary, κ leptonmax ( λ ). This can be observed in the bottom right panel of Fig. 16.Now, the integration range in the λ variable is fur-ther constrained to be λ ∈ [0 , λ + − ] where λ + − < λ max (only valid when (cid:15) ν > ˜ (cid:15) ν ). Thus we can integrate thedouble differential cross section d σdκ dλ in the region of the9 κ maxlepton ( λ ) κ minlepton ( λ ) κ - NPB ( λ ) λ +- λ ϵ ˜ ν + κ E ν = λ +- λ -- λ ϵ ν κ E ν =
145 MeV λ +- λ -- λ ϵ ˜ ν κ E ν = λ +- λ -- λ ϵ ν κ - NPB ( λ +- ) κ E ν = FIG. 16: Plot of the phase space for the RFG model at four different neutrino energies, where different situationsarise. In the top left panel, the reduced neutrino energy is given by ˜ (cid:15) ν + in Eq. (57), and the cut between the curves κ leptonmin ( λ ) and κ NPB − ( λ ) is sole and tangent. However, when the neutrino energy increases a bit (top right panel), thetwo different solutions λ ±− given by Eq. (52) appear, first as cuts between the curves κ leptonmin ( λ ) and κ NPB − ( λ ). Forhigher neutrino energies, as that shown in the bottom left panel, the λ + − cut given by Eq. (52) occurs exactly at λ = λ max , and this happens for the reduced neutrino energy ˜ (cid:15) ν given in Eq. (58). Finally, in the bottom right panelwe show the situation for a bit larger neutrino energy. In this case, the λ + − solution given by Eq. (52) correspondsto a cut point between the curves κ NPB − ( λ ) and κ leptonmax ( λ ).phase space where it is truly different from zero, thusmaking the integration algorithm the most efficient aspossible. The integrated total CCQE cross section cannow be written σ ( E ν ) = (cid:90) λ u dλ (cid:90) κ u ( λ ) κ d ( λ ) dκ d σdκ dλ ( E ν ) , (59) where λ u = (cid:26) λ max if (cid:15) ν (cid:54) ˜ (cid:15) ν ,λ + − if (cid:15) ν > ˜ (cid:15) ν ; (60) κ d ( λ ) = max (cid:16) κ NPB − ( λ ) , κ leptonmin ( λ ) (cid:17) , (61) κ u ( λ ) = min (cid:0) κ NPB+ ( λ ) , κ leptonmax ( λ ) (cid:1) . (62)In Fig. 17 we show the results for the total CCQE inte-grated cross section off C for the two models discussedin this work: RFG (solid line) and SuSAM* (short-0dashed line). The left panel is for muon neutrino scat-tering, while the right one corresponds to muon antineu-trino. We have displayed the uncertainty band of theSuSAM* model, taken as the area between the predic-tions for the total cross sections obtained by taking the f max , minSuSAM* ( ψ ) scaling functions depicted in Fig. 8, insteadof taking the f centralSuSAM* ( ψ ) scaling function of the samefigure, which is the one we have used throughout this ar-ticle. To compare with another important and relevantscaling model, already incorporated in GENIE [92], theSuSAv2-MEC model of Ref. [93], we have plotted thecurve of this model in Fig. 17 in dot-dashed style as well.The main conclusion that can be drawn by comparingthe curves of the three different scaling models (RFG,SuSAM* and SuSAv2-MEC) is that all of them lie in-side the uncertainty band of the SuSAM* model. It istrue that this band is very large, but not so large if com-pared with the experimental uncertainties, which it iseven truer for the antineutrino total cross section (rightpanel of Fig. 17), where one can see that the theoreticaluncertainty of the SuSAM* model is of the same order asthe error bars of the experimental points. In fact, the cen-tral prediction of the SuSAM* model is in between thoseof the RFG and the SuSAv2-MEC, and as observed inFig. 17, it passes closer to both sets of the experimentaldata shown in the figure. These are the results of Mini-BooNE [20–22] (for intermediate neutrino energies andfor the new technique based on kaon decay at rest [20])and NOMAD [94] (for the high neutrino energy range)experiments. It is worth another remark: the SuSAM*predictions (band and central curve) are solely based onthe super-scaling properties of the selected “QE” electronscattering data out of the total inclusive ( e, e (cid:48) ) data, inthe global fit carried out in Ref. [52], and no CCQE neu-trino scattering parameter has been fitted at all. Thesame can be said for the SuSAv2-MEC model, which isbased on another scaling function [95] and with the con-tribution of the weak charged meson-exchange currents(MEC) calculated in Ref. [96]. The SuSAv2-MEC modeldescribes the MiniBooNE data very well, but systemati-cally overestimates the cross section at the NOMAD en-ergies. In fact, this could point to a conflict between theMEC contribution used in this model and the NOMADdata.It is not the purpose of this work to discuss thediscrepancies between both sets of data shown in Fig.17, because the experimental collaborations recognize intheir works [21] that the experiments use different de-tector technologies and assume different topologies indefining CCQE events. And, in addition, the neu-trino/antineutrino energy drawn in the abscissa axes ofFig. 17 is the true neutrino energy, while in the exper-iments the energy is the reconstructed one (except forthe kaon decay at rest technique), which assumes an ed-ucated guess to obtain it from the measured final leptonkinematic variables via an unfolding procedure. In fact,the problems related to the reconstruction of the neu-trino energy have been addressed in a series of articles [97–108].A warning is in order here: the total integrated CCQEcross sections shown in Fig. 17 have been obtained withthe set of parameters ( k F = 212 MeV/c and M ∗ = m ∗ N /m N = 0 .
83) for C obtained in the global fit to“QE” electron scattering data of Ref. [52], and given inTable II of the same reference. We did not try to ad-just these and any other parameters of the model (e.g.,axial mass of the nucleon) to the neutrino data. Thisis important to be stressed, because in the previous fig-ures and formulae of this work, we have utterly used thevalues of k F = 225 MeV/c and M ∗ = 1 for the Fermi mo-mentum and the relativistic effective mass, respectively.We used these values in the calculations of the previ-ous figures because we did not want to bother the readerwith additional complications related to the underlyingWalecka model [109, 110] (see also Refs. [111, 112]) inwhich the SuSAM* approach is based. It is also worthwarning the reader that in this work we have used theusual dipole axial-vector form factor with an axial mass of M A = 1 .
032 GeV, and the set of vector form factors takenfrom the Galster parametrization given in Ref. [113].The Walecka model was the first relativistic, many-body, quantum-field theory model that exhibited satu-ration in nuclear matter. The Relativistic Mean Field(RMF) version of the model for nuclear matter has con-stant scalar and time-like vector potentials, associated tothe expectation values of the scalar and time-like compo-nent of the vector fields (the σ − ω model). However, fornuclear matter, the RMF version is exactly solvable andthe dynamic nucleon fields can be expanded as plane-wave solutions as in a free theory, because the dynamicsdue to the scalar and vector potentials is hidden in a shiftof the nucleon mass and the energy. Thus, the effect ofthe condensed value of the scalar field is to shift the massof the nucleon, reducing it. What we have done in previ-ous works [48–52] is to use this underlying well-foundedtheory to phenomenologically adjust the relativistic ef-fective mass for several nuclear species, assuming thatthe “QE” electron scattering data scale within an uncer-tainty band, that has been also estimated; and that these“QE” electron scattering data can be selected from thewhole inclusive data by means of a density criterion.It is worth noting that all the formulae appearing inthis work can be translated to the real SuSAM* model byjust changing the value of the Fermi momentum and thefree nucleon mass m N → m ∗ N , where m ∗ N is the value ofthe relativistic effective mass. Of course, these changesaffect the values of η F , (cid:15) F , (cid:15) (cid:48) F , η (cid:48) F . . . , but the form of theequations obtained in this work remains the same. Ofcourse, what also does not change at all is the form ofthe scaling functions shown in Fig. 8. What changes isthe value of the scaling variable ψ for a given kinematics( ω , q ), but not the form of the scaling function.After this relevant warning for the reader, what canalso be stressed from the inspection of Fig. 17 is the ef-fect of nuclear correlations in the integrated cross section.The RFG model does not contain nuclear correlations,1 σ ( x10 - c m / n e u t r on ) E ν (GeV) SuSAM * bandRFGSuSAM * centralSuSAv2-MECMiniBooNENOMADMiniBooNE monoenergetic 0 2 4 6 8 10 12 14 0.10 1 10 100 σ ( x10 - c m / p r o t on ) E ν− (GeV) SuSAM * bandRFGSuSAM * centralSuSAv2-MECMiniBooNENOMAD FIG. 17: Plot of the total CCQE cross section σ ( E ν ) (normalized per interacting nucleon) as a function of theneutrino/antineutrino energy E ν (¯ ν ) for the two models discussed in this work: RFG (solid line) and SuSAM* (short-dashed line). In the left panel the neutrino total cross section per neutron off C is displayed along with theexperimental measurements of MiniBooNE [20, 22] and NOMAD [94]. In the right panel, we show the same for theantineutrino total cross section per proton, compared with the measurements of MiniBooNE [21] and NOMAD [94]collaborations. For both models, RFG and SuSAM*, the nucleon relativistic effective mass m ∗ N = 0 . m N has beentaken, as well as a Fermi momentum of k F = 212 MeV/c, accordingly to the global fit to “QE” electron scatteringdata performed in Ref. [52]. Additionally, in dot-dashed style, it is also shown the SuSAv2-MEC model prediction,which has been taken from Ref. [93].not either high momentum components in its nuclearground state. However, the SuSAM* model does con-tain them phenomenologically, because its scaling func-tion has been fitted to a selected sample of “QE” elec-tron scattering data extracted from the inclusive ( e, e (cid:48) )reaction data from a large list of different nuclear targets(see Ref. [52]). Therefore, the SuSAM* contains high mo-mentum components in its nuclear model, although phe-nomenologically. In fact, the tails of the SuSAM* scalingfunction, that extend beyond ψ = ±
1, partially accountfor these high momentum components, producing the en-larging of the available phase space if compared with theRFG (see in particular the right panel of Fig. 9, where thecut of the upper boundary of the SuSAM* model withthe κ -axis occurs at κ = η (cid:48) F , which can be considered asplaying the role of an effective higher Fermi momentum).One can also compare the left panels of Figs. 11 and 12, orthe left and right panels of Fig. 15 for CCQE antineutrinoscattering, where one can observe that for the SuSAM*model, beyond the boundaries of the RFG denoted bythe dot-dashed lines, there is still a significant region ofthe phase space with a non-negligible contribution to thedouble differential cross section. This is mainly responsi-ble of the enhancement observed in the total cross sectionin both panels of Fig. 17 for the SuSAM* model with re-spect to the RFG. VI. CONCLUSIONS
In this work we have thoroughly analyzed the analyt-ical boundaries of the phase space for the CCQE doubledifferential cross section d σdκ dλ within the scaling formal-ism for the RFG model, where these boundaries can bemore easily obtained. This allows to perform the integra-tion of this double differential cross section only in theregion where it is truly different from zero, thus mak-ing the integration algorithm as efficient as possible. Wehave also easily extended the formalism to accommodatethe SuSAM* model as well, taking into account the tailsof the scaling function.We have analyzed these double differential cross sec-tions for CCQE muon neutrino and antineutrino scatter-ing off C at several neutrino/antineutrino energies as abenchmark. Our results show that the d σdκ dλ cross sectionhas very good properties to be implemented in the MCneutrino event generators, basically because of two mainreasons: it is quite flat regardless of the neutrino energy,and it has a significant contribution in a larger regionof the available phase space if compared with the usual d σdT µ d cos θ µ cross section, especially at very high neutrinoenergies E ν (¯ ν ) (cid:38) −
20 GeV. We think these featuresof the d σdκ dλ cross section make it especially well-suitedto be used to generate events in any MC generator thatuses the acceptance-rejection method.Finally, we have used the analytical boundaries ob-2tained in this work to integrate the double differentialcross section in order to obtain the CCQE total inte-grated σ ( E ν (¯ ν ) ) cross section for the two models studiedin this work: RFG and SuSAM*. The effect of the tailsof the phenomenological SuSAM* scaling function, thatpartially account for nuclear correlations in the model,are directly responsible of an enhancement of about a17–18% for intermediate and high neutrino energies. Thesame conclusion can be drawn for CCQE antineutrinoscattering with roughly the same enhancement in per-centage.Future works can be done based on the findingsof this study. In particular, we are working on the study of how nuclear correlations can be approximatelyand phenomenologically incorporated in the RFG model. VII. ACKNOWLEDGEMENTS
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