We study the convergence of the series expansions sometimes used in the
analysis of the nuclear effects in Deep Inelastic Scattering (DIS) proccesses
induced by leptons. The recent advances in statistics and quality of the data,
in particular for neutrinos calls for a good control of the theoretical
uncertainties of the models used in the analysis. Using realistic nuclear
spectral functions which include nucleon correlations, we find that the
convergence of the derivative expansions to the full results is poor except at
very low values of $x$

Armesto N

I Ruiz Simó

M J Vicente Vacas

English

arXiv

Crossref

Study of the derivative expansions for the nuclear structure functions

We study the convergence of the series expansions sometimes used in the analysis of the nuclear effects in deep inelastic scattering (DIS) processes induced by leptons. The recent advances in statistics and quality of the data, in particular for neutrinos calls for a good control of the theoretical uncertainties of the models used in the analysis. Using realistic nuclear spectral functions which include nucleon correlations, we find that the convergence of the derivative expansions to the full results is poor except at very low values of x.This work was partially supported by the MEC contract FIS2006-03438 and the EU
Integrated Infrastructure Initiative Hadron Physics Project contract RII3-CT-2004-506078.
IRS acknowledges MEC for additional support.Peer reviewe

Ruiz Simó, Ignacio

Vicente Vacas, Manuel J.

Digital.CSIC

arXiv:0807.5055v1  [nucl-th]  31 Jul 2008Study of the derivative expansions for the nuclearstructure functionsI Ruiz Simo´ and M J Vicente VacasDepartamento de F´ısica Teo´rica and IFIC, Centro Mixto Universidad deValencia-CSIC, Institutos de Investigacio´n de Paterna, Aptdo. 22085, E-46071Valencia, Spain.Abstract. We study the convergence of the series expansions sometimes used in theanalysis of the nuclear effects in Deep Inelastic Scattering (DIS) proccesses inducedby leptons. The recent advances in statistics and quality of the data, in particularfor neutrinos calls for a good control of the theoretical uncertainties of the modelsused in the analysis. Using realistic nuclear spectral functions which include nucleoncorrelations, we find that the convergence of the derivative expansions to the full resultsis poor except at very low values of x.PACS numbers: 13.15.+g,13.60.Hb,25.30.-c1. IntroductionThe advent of new and high statistics deeply inelastic neutrino scattering experimentsapart from providing valuable data on F2 and xF3[1, 2] has shown again the importanceof nuclear effects that render difficult the extraction of the nucleon structure functions.Furthermore, the use of nuclear targets is necessary due to the smallness of the crosssections. Qualitatively, the nuclear effects are well known. Shadowing, antishadowing,Fermi motion, binding,... have all been widely studied for charged leptons in the contextof the European Muon Collaboration (EMC) effect. For a review, see [3, 4]. Moreover,it is well known that nuclear effects are substantially different for neutrino reactions dueto the presence of the axial current and the different valence and sea quark contributionsfor each observable [5]. This situation asks for a detailed and quantitative microscopicalunderstanding of the nuclear effects, rather than the parametrizations that have beenused sometimes, like recently by the NuTeV Collaboration [2].One of the basic ingredients in all calculations is the nuclear spectral function.This presents some serious difficulties, as these functions are not so well known tothe precision level reached by current experiments. Therefore, the analysis mightintroduce unwanted model dependences. However, it was soon noticed that under someapproximations the nuclear structure functions could be written as simple expansionson the nucleon structure functions and their derivatives. All the nuclear informationwould then be encoded in the expected values of some nuclear magnitudes, like theStudy of the derivative expansions for the nuclear structure functions 2average kinetic energy of the nucleons or the mean nucleon removal energy. These kindof approximations, both for charged lepton and for neutrino induced reactions, havebeen widely used in the literature [6, 7, 8, 9, 10, 11, 12, 13]. The aim of this paper isto investigate the quality of these expansions and to what extent they could be used inthe analysis of lepton deep inelastic scattering experiments. In order to do that, oncethe formalism is established in the next section, we will make a comparative study forF2,3 and for a few typical nuclear spectral functions used in the literature.2. FormalismThe nuclear structure functions can be written as a convolution of the nuclear spectralfunctions and nucleon structure functions. See for instance [14] and references therein.In the rest frame of the nucleus, the F2 and F3 structure functions areFA2 (x,Q2) =∑τ=p,n∫dǫ d3p(2π)4Pτ (ǫ,p)(1 + γ pzM)γ2(1 + 4p2x′2Q2+ 6x′2p2⊥Q2)F τ2 (x′, Q2), (1)FA3 (x,Q2) =∑τ=p,n∫dǫ d3p(2π)4Pτ (ǫ,p)(1 +pzγM)x′xF τ3 (x′, Q2), (2)where Pp(n)(ǫ,p) is the nuclear spectral function, normalized to the number of protons(neutrons) in the nucleus, and describes the probability of finding a proton (neutron)with momentum p and removal energy ǫ. The four-momentum of the nucleon can bewritten as p = (M + ǫ,p), with ǫ ≤ 0. The z axis is oriented in such a way that q lieson it, p⊥ is the transverse momentum of the nucleon and γ = |q|/q0. Here, x′ is thenatural Bjorken variable for the nucleon in the nuclear medium, i.e. x′ = Q2/(2p · q);while x is the Bjorken variable in the nucleon rest frame, x = Q2/(2Mq0). They arerelated byx′ =xzwhere z = 1 +ǫM+ γpzM. (3)For isoscalar nuclei such as 40Ca, only the isoscalar component of the spectralfunction and the structure function have to be accounted for. When this is done (1)and (2), these read asFA2 (x,Q2) = A∫dǫd3p(2π)4P0(ǫ,p)(1 + γ pzM)γ2(1 + 4p2x′2Q2+ 6x′2p2⊥Q2)FN2(xz,Q2), (4)FA3 (x,Q2) = A∫dǫd3p(2π)4P0(ǫ,p)(1 +pzγM)x′xFN3(xz,Q2), (5)where P0(ǫ,p), which is the isoscalar part of the nuclear spectral function, is nownormalized to unity and we perform the calculations for the nuclear structure functionsaveraged over neutrinos and antineutrinos, i.e., we only consider the symmetric ν + ν¯combination [14, 15].In a nucleus, the expected values of 〈ǫ〉Mand〈p2〉M2averaged with the nuclear spectralfunction are much smaller than unity. Thus, z ≈ 1 and x′ ≈ x. Under these assumptions,Study of the derivative expansions for the nuclear structure functions 3we can perform a Taylor expansion of the integrands in expressions (4) and (5) aroundz = 1, keeping terms up to order ǫ/M and p2/M2. In this way we will be able to takeout of the integral the structure functions and their derivatives and we will be left withexpected values of the removal energy ǫ and momentum squared p2. This statementis true if the nuclear spectral functions only depend on the modulus of the momentum|p| (as it is in the case of the nuclear spectral functions we will consider) and not uponits direction. Under this assumption we can drop the expected values of pz (or anyother component of the momentum) and ǫ pz because they are identically zero due tosymmetry considerations.We will begin with the structure function FA2 (x,Q2). After performing the Taylorexpansion, keeping terms up to order ǫ/M and p2/M2, and dropping those terms whichgo with 〈pz〉 /M or 〈ǫ pz〉 /M2 (because they are identically zero as stated above), weobtainFA2 (x,Q2)A≃ FN2(x,Q2) [1 +(γ2 − 1) 〈p2〉3M2]− x∂FN2 (x,Q2)∂x[〈ǫ〉M+(2− γ2) 〈p2〉3M2]+(x∂FN2 (x,Q2)∂x+x22∂2FN2 (x,Q2)∂x2)γ2〈p2〉3M2, (6)where 〈O〉 is the expected value of the operator O(ǫ,p) averaged with the isoscalarspectral function P0(ǫ,p):〈O〉 =∫dǫ d3p(2π)4P0(ǫ,p)O(ǫ,p). (7)We have also used the fact that, with a nuclear spectral function which depends onlyon the modulus of the momentum, the expected value of momentum squared is sharedequally among every squared component, i.e: 〈p2i 〉 =〈p2〉3. In the Bjorken limit, (γ → 1),(6) coincides with the expansions used by Frankfurt et al. [6] and Ciofi degli Atti et al.[7].If we do the same for the nuclear structure function FA3 (x,Q2), we obtainFA3 (x,Q2)A≃ FN3(x,Q2)−〈ǫ〉M{FN3(x,Q2)+ x∂FN3 (x,Q2)∂x}+〈p2〉3M2{(γ2 − 1) [FN3 + x∂FN3∂x]+ γ2[x∂FN3∂x+x22∂2FN3∂x2]}, (8)a similar result to that Kulagin [10]. To allow for an easier comparison, in the aboveexpressions, γ2 can be rewritten asγ2 = 1 +4M2x2Q2. (9)Then, the only difference between (8) and that of [10] is that, for simplicity, we donot consider the off-shell dependence in the nucleon structure function FN3 . In ourcalculation, the only source of off-shell dependence is through the nuclear spectralfunction P0. Thus, we do not have the term ∂p2FN3 that appears in [10] .For our study we have selected the 40Ca nucleus, which is isoscalar and already willshow important medium effects. The free nucleon structure functions have been takenStudy of the derivative expansions for the nuclear structure functions 40.2 0.4 0.6x0.911.11.21.3R 2(x)(I) Full(II) Full(III) FullFigure 1. R2 =F2AF2ratio for 40Ca at Q2 = 20 GeV2 with the nuclear spectralfunctions (I), (II) and (III) described in the text.from [16] and we have chosen several different nuclear spectral functions. The first one(labelled I) is a phenomenological model of the spectral function which has a mean fieldpart and high-momentum components coming from NN-correlations and it is describedin Kulagin et al. [15] where it was used in a global study of nuclear structure functions.The second one (labelled II), that also contains correlations is taken from [17] where ithas been tested in the calculation of several electron scattering observables. The thirdspectral function is taken from the semiphenomenological model based on a many bodycalculation and the local density approximation that is described in [18] (labelled III).Finally, we also consider the simple mean field spectral function of [12](labelled IV),which was used in the study of the EMC effect.3. ResultsIn figure 1, we show the results of the ratio R2 =F2AF2for the nuclear spectral functions(I), (II) and (III). This gives us some idea of the uncertainties related to these functions.The differences are small, even when the spectral funtions have been obtained withdiverse methods and are in fact quite different if one studies in detail their energy andmomentum dependence. However, the expected values of the mean removal energy 〈ǫ〉and the mean kinetic energy per nucleon 〈T 〉 =〈p22M〉are quite similar, as can be seenin Table 1. In particular, spectral functions (I) and (III) that have quite close expectedStudy of the derivative expansions for the nuclear structure functions 5Table 1. Expected values of the nucleon removal and kinetic energies for the nuclearspectral functions of Kulagin et al. [15] (I), Ankowski et al. [17] (II) and Fernandez deCordoba et al. [18] (III).Spectral Function (I) (II) (III)〈ǫ〉 (MeV) -49 -40 -47〈T 〉 (MeV) 30.1 26.2 28.80.2 0.4 0.6x0.911.11.21.31.4R 2(x)(I) Expansion(I) Full(II) Expansion+0.10(II) Full+0.10(III) Expansion+0.20(III) Full+0.20Figure 2. R2 =F2AF2ratio for 40Ca at Q2 = 20 GeV2. Comparison of the full resultswith the approximation of (6) for the nuclear spectral functions (I), (II) and (III).values also produce very similar ratios, whereas (II) which has an appreciably smallerbinding energy gives slightly larger values for the ratio.In figure 2, we compare the full results for the same ratio (4) whith those obtainedmaking use of the approximation of (6). The use of this expansion has been assumedto be a good approximation for x ≤ 0.5 [7, 8, 9]. Obviously, the series expansion agreeswell at low values of x with the full results. However, in the three cases produces lowervalues for the ratio at intermediate x showing a maximum deviation of a 3-4% aroundx = 0.5 − 0.6. This region, with a dip in the ratio, is dominated by the mean removalenergy (or equivalently the binding energy) per nucleon. Although this could look asmall error, we should remark that it means increasing the deviation due to nucleareffects from the value 1 by around a 30% . At higher x’s, where Fermi motion of thenucleons provides the dominant effect, the series expansions grow faster than the fullresults and become larger for x & 0.65. Thus, we find that for typical nuclear spectralfunctions the convergence of the series expansion is not so good except at very low x,Study of the derivative expansions for the nuclear structure functions 60.2 0.4 0.6x0.911.11.21.31.4R 3(x)(I) Expansion(I) Full(II) Expansion+0.10(II) Full+0.10(III) Expansion+0.20(III) Full+0.20Figure 3. R3 =F3AF3ratio for 40Ca at Q2 = 20 GeV2. Comparison of the full resultswith the approximation of (8) for the nuclear spectral functions (I), (II) and (III).where in any case other effects not considered here, like shadowing play a major role.The results for the R3 =F3AF3ratio are shown in figure 3. The full model for R3is given by (5) and the series expansion by (8). As it was the case for R2, there isa dip region dominated by the binding energy and the Fermi motion of the nucleonsproduces the large rising at high values of x. The comparison of the full results andthe series expansions shows the same features as for R2. The expansions systematicallyunderestimate the ratios at intermediate values of x and overestimate them for x > 0.7.This overestimation of the effect of the Fermi motion was already discussed in [10].There, it was also claimed that in the limit of high Q2 and for heavy nuclei the expansionshould be a good approxiamtion up to x . 0.75. However, after studying R2 and R3at different Q2 values, apart from the one shown in this paper, we have found that formedium nuclei the convergence of the series is only good at very low x where othernuclear effects are very relevant.In order to obtain a better convergence one would need to reach a higher order in theexpansions. This implies also the sensitivity to nuclear expected values of higher powersof the nucleons momenta as < (p/M)4 >. One should notice that due to correlationsthese expected values are not negligible [20, 19]. The three spectral functions consideredabove contain high momentum components and one may expect any expansion up toorder p2 to fail to have a good convergence. To test this point, we have also calculated R2for the case of a simpler mean field spectral function which does not incorporate nucleonStudy of the derivative expansions for the nuclear structure functions 70.2 0.4 0.6 0.8x0.911.11.21.3R 2(x)(IV) Full(IV) ExpansionFigure 4. R2 =F2AF2ratio for 40Ca. Comparison of the full results with theapproximation of (6) for the nuclear spectral functions (IV).correlations and does not have those high momentum components [12]. This functionwas used to analyse the A dependence of the position and magnitude of the dip. Here,we have used the same parton distribution functions as in the original reference [12].The results are shown in figure 4. In this case, there is an almost perfect agreementbetween the full results and the expansion.In summary, we have studied the quality of some series expansions commonly usedto incorporate approximately the nuclear effects in the analysis of DIS processes. Wehave found that for realistic enough nuclear spectral functions, that include nucleoncorrelations and have high momentum components, the convergence of the series to thefull result is poor except at very low values of x, where in fact, other nuclear effects,like shadowing or antishadowing, are more relevant. At high x values, in the Fermimotion region, the expansions clearly overestimate the full result. This was known andexpected. However, even at relatively low x values, where the expansions were suppossedto provide a good approximation, we have found that they systematically underestimatethe value of F2 and F3, artificially increasing the size of the nuclear corrections.AcknowledgmentsThis work was partially supported by the MEC contract FIS2006-03438 and the EUIntegrated Infrastructure Initiative Hadron Physics Project contract RII3-CT- 2004-506078. I.R.S. acknowledges MEC for additional support.Study of the derivative expansions for the nuclear structure functions 8[1] J. M. Conrad, M. H. Shaevitz and T. Bolton, Rev. Mod. Phys. 70 (1998) 1341[arXiv:hep-ex/9707015].[2] M. Tzanov et al. [NuTeV Collaboration], Phys. Rev. D 74 (2006) 012008 [arXiv:hep-ex/0509010].[3] D. F. Geesaman, K. Saito and A. W. Thomas, Ann. Rev. Nucl. Part. Sci. 45 (1995) 337.[4] N. Armesto, J. Phys. G 32 (2006) R367 [arXiv:hep-ph/0604108].[5] I. Schienbein, J. Y. Yu, C. Keppel, J. G. Morfin, F. Olness and J. F. Owens, Phys. Rev. D 77(2008) 054013 [arXiv:0710.4897 [hep-ph]].[6] L. L. Frankfurt and M. I. Strikman, Phys. Lett. B 183 (1987) 254.[7] C. Ciofi Degli Atti and S. Liuti, Phys. Lett. B 225 (1989) 215.[8] C. Ciofi degli Atti and S. Liuti, Phys. Rev. C 41 (1990) 1100.[9] C. Ciofi degli Atti and S. Liuti, Phys. Rev. C 44 (1991) 1269.[10] S. A. Kulagin, Nucl. Phys. A 640 (1998) 435 [arXiv:nucl-th/9801039].[11] G. A. Miller and J. R. Smith, Phys. Rev. C 65 (2002) 015211 [Erratum-ibid. C 66 (2002) 049903][arXiv:nucl-th/0107026].[12] J. R. Smith and G. A. Miller, Phys. Rev. C 65 (2002) 055206 [arXiv:nucl-th/0202016].[13] C. Ciofi degli Atti, L. L. Frankfurt, L. P. Kaptari and M. I. Strikman, Phys. Rev. C 76 (2007)055206 [arXiv:0706.2937 [nucl-th]].[14] S. A. Kulagin and R. Petti, Phys. Rev. D 76 (2007) 094023 [arXiv:hep-ph/0703033].[15] S. A. Kulagin and R. Petti, Nucl. Phys. A 765 (2006) 126 [arXiv:hep-ph/0412425].[16] M. Gluck, E. Reya and A. Vogt, Eur. Phys. J. C 5 (1998) 461 [arXiv:hep-ph/9806404].[17] A. M. Ankowski and J. T. Sobczyk, Phys. Rev. C 77 (2008) 044311 [arXiv:0711.2031 [nucl-th]].[18] P. Fernandez de Cordoba and E. Oset, Phys. Rev. C 46 (1992) 1697.[19] C. Ciofi degli Atti and S. Simula, Phys. Rev. C 53 (1996) 1689 [arXiv:nucl-th/9507024].[20] C. Ciofi degli Atti, S. Simula, L. L. Frankfurt and M. I. Strikman, Phys. Rev. C 44 (1991) 7.

Journal of Physics G Nuclear and Particle Physics

We study the convergence of the series expansions sometimes used in the analysis of the nuclear effects in deep inelastic scattering (DIS) processes induced by leptons. The recent advances in statistics and quality of the data, in particular for neutrinos calls for a good control of the theoretical uncertainties of the models used in the analysis. Using realistic nuclear spectral functions which include nucleon correlations, we find that the convergence of the derivative expansions to the full results is poor except at very low values of x

Study of the derivative expansions for the nuclear structure functions

Abstract

Similar works

Full text

Available Versions

Crossref

Digital.CSIC

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura