Electromagnetic and Weak Nuclear Structure Functions F 1,2 (x, Q 2 ) in the Intermediate Region of Q 2
H. Haider, F. Zaidi, M. Sajjad Athar, S. K. Singh, I. Ruiz Simo
aa r X i v : . [ nu c l - t h ] F e b Electromagnetic and Weak Nuclear Structure Functions F , ( x , Q ) in the Intermediate Region of Q H. Haider , F. Zaidi , M. Sajjad Athar , S. K. Singh , I. Ruiz Simo Department of Physics, Aligarh Muslim University, Aligarh - 202 002, India, Departamento deF´ısica At´omica, Molecular y Nuclear, and Instituto de F´ısica Te´orica y Computacional Carlos I,Universidad de Granada, Granada 18071, SpainE-mail: [email protected] (Received January 19, 2016)We have studied nuclear structure functions F A ( x , Q ) and F A ( x , Q ) for electromagnetic and weakprocesses in the region of 1 GeV < Q < GeV . The nuclear medium e ff ects arising due to Fermimotion, binding energy, nucleon correlations, mesonic contributions and shadowing e ff ects are takeninto account using a many body field theoretical approach. The calculations are performed in a localdensity approximation using a relativistic nucleon spectral function. The results are compared withthe available experimental data. Implications of nuclear medium e ff ects on the validity of Callan-Gross relation are also discussed. KEYWORDS: nuclear medium effect, structure functions, deep inelastic scattering,Callan-Gross relation
1. Introduction
Recently a better understanding of nuclear medium e ff ects in the deep inelastic scattering re-gion both in electromagnetic(EM) and weak(Weak) interaction induced processes has been empha-sized due to the fact that experiments are being performed using electron beam at JLab [1] and neu-trino / antineutrino beam at the Fermi Lab. The experiments are being performed using several nucleartargets. The di ff erential and total scattering cross sections are expressed in terms of F EM A ( x , Q ) and F EM A ( x , Q ) structure functions for electromagnetic processes and for weak interaction induced pro-cesses in terms of F Weak A ( x , Q ) , F Weak A ( x , Q ) and F Weak A ( x , Q ) structure functions. We have studiednuclear medium e ff ects arising due to Fermi motion, binding energy, nucleon correlations, mesoniccontributions and shadowing e ff ects in these structure functions, in a many body field theoreticalapproach and the calculations are performed in a local density approximation using a relativistic nu-cleon spectral function. The details of the present formalism are given in Ref. [2]. In this paper, wehave compared F EM A ( x , Q ) vs F Weak A ( x , Q ), and F EM A ( x , Q ) vs F Weak A ( x , Q ) structure functions. Theresults for the ratio xF A ( x , Q ) F A ( x , Q ) with nuclear medium e ff ects in carbon has also been presented. Theresults are also compared with some of the available experimental data. For completeness, we arepresenting the formalism in brief. . Formalism The double di ff erential cross section for the reaction ν l / ¯ ν l ( k ) + N ( p ) → l − / l + ( k ′ ) + X ( p ′ ) , l = e − , µ − , in the Lab frame is written as d σ N ν, ¯ ν d Ω ′ dE ′ = G F (2 π ) | k ′ || k | m W q − m W L αβν, ¯ ν W N αβ , (1)where L αβν, ¯ ν = k α k ′ β + k β k ′ α − k . k ′ g αβ ± i ǫ αβρσ k ρ k ′ σ is the leptonic tensor with ( + ve)-ve sign for(anti)neutrino, and W N αβ is the nucleon hadronic tensor expressed in terms of nucleon structure func-tions W Ni ; i = − W N αβ = q α q β q − g αβ ! W N + M N p α − p . qq q α ! p β − p . qq q β ! W N − i M N ǫ αβρσ p ρ q σ W N . (2)In a nuclear medium the expression for the cross section is written as d σ A ν, ¯ ν d Ω ′ dE ′ = G F (2 π ) | k ′ || k | m W q − m W L αβν, ¯ ν W A αβ , (3) W A αβ is the nuclear hadronic tensor defined in terms of nuclear structure functions W Ai ; i = − W A αβ = q α q β q − g αβ ! W A + M A p α − p . qq q α ! p β − p . qq q β ! W A − i M A ǫ αβρσ p ρ q σ W A (4)where M A is mass of the target nucleus. The neutrino-nucleus cross sections are written in terms ofneutrino self energy Σ ( k ) in the nuclear medium, the expression for which is obtained as [2]: Σ ( k ) = − iG F √ m ν Z d k ′ (2 π ) k ′ − m l + i ǫ m W q − m W L αβ Π αβ ( q ) . (5) Π αβ ( q ) is the W-boson self energy in the nuclear medium which is given in terms of nucleon ( G ) andmeson ( D j ) propagators − i Π αβ ( q ) = ( − ) Z d p (2 π ) iG ( p ) X X X s p , s i n Y i = Z d p ′ i (2 π ) Y l iG l ( p ′ l ) Y j iD j ( p ′ j ) × − G F m W √ h X | J α | N ih X | J β | N i ∗ (2 π ) δ ( q + p − Σ ni = p ′ i ) . (6)Using the expression of W self-energy and neutrino self-energy in the expression of cross section oneobtains [2]: W A αβ = Z d r Z d p (2 π ) Z µ −∞ d p ME ( p ) S h ( p , p , ρ ( r )) W N αβ ( p , q ) , (7)where µ is the chemical potential. The hole spectral function S h takes care of Fermi momentum, Pauliblocking, binding energy and nucleon correlations [3]. W Ni ( x , Q ) and W Ai ( x , Q ) are respectivelyredefined in terms of the dimensionless structure functions F Ni ( x , Q ) and F Ai ( x , Q ) through MW N ( ν, Q ) = F N ( x , Q ); M A W A ( ν, Q ) = F A ( x , Q ) W N ( ν, Q ) = F N ( x , Q ); ν A W A ( ν, Q ) = F A ( x , Q ) ν W N ( ν, Q ) = F N ( x , Q ); ν A W A ( ν, Q ) = F A ( x , Q )The nucleon structure functions are expressed in terms of parton distribution functions(PDFs). Forthe numerical calculations, we have used CTEQ6.6 [4] nucleon PDFs. The evaluations are performedboth at the leading order(LO) and next-to-leading order(NLO). For electromagnetic interaction, wefollow the same procedure, formalism for which is given in accompanying paper by Zaidi et al. [5] inthis proceeding.Expressing W N αβ and W A αβ , in terms of F Ni and F Ai (i = F EM / Weak A ( x A , Q ) = X τ = p , n AM Z d r Z d p (2 π ) ME ( p ) Z µ −∞ d p S τ h ( p , p , ρ τ ( r )) × F EM / Weak ,τ ( x N , Q ) M + p x M F EM / Weak ,τ ( x N , Q ) ν , (8) F EM / Weak A ( x A , Q ) = X τ = p , n Z d r Z d p (2 π ) ME ( p ) Z µ −∞ d p S τ h ( p , p , ρ τ ( r )) × Q q z | p | − p z M + ( p − p z γ ) M p z Q ( p − p z γ ) q q z + ! × Mp − p z γ ! F EM / Weak ,τ ( x , Q ) , (9)where γ = q z q = q + M x Q . The mesonic (pion and rho) cloud contributions are taken intoaccount following the same procedure as for the nucleon, except the fact that now instead of nucleonspectral function, we use meson propagator to describe the meson propagation in the nuclear medium.For this also, we have used microscopic approach by making use of the imaginary part of the mesonpropagators instead of spectral function, and obtain F A ,π ( x ) [3] as F EM / Weak , A ,π ( x π , Q ) = − Z d r Z d p (2 π ) θ ( p ) δ ImD ( p ) 2 m π m π p − p z γ ! × Q q z | p | − p z m π + ( p − p z γ ) m π p z Q ( p − p z γ ) q q z + ! F EM / Weak ,π ( x π )(10)where x π = − Q p · q and D ( p ) is the pion propagator in the nuclear medium given by D ( p ) = [ p − p − m π − Π π ( p , p )] − , (11)where Π π = f / m π F ( p ) p Π ∗ − f / m π V ′ L Π ∗ . (12)Here, F ( p ) = ( Λ − m π ) / ( Λ + p ) is the π NN form factor, Λ= GeV , f = . V ′ L is the longi-tudinal part of the spin-isospin interaction and Π ∗ is the irreducible pion self energy containing thecontribution from particle - hole and delta - hole excitations. For the meson PDFs we have used the arameterization of Gluck et al. [6]. Similarly, the contribution of the ρ -meson cloud to the structurefunction is taken into account [3] F EM / Weak , A ,ρ ( x ρ , Q ) = − Z d r Z d p (2 π ) θ ( p ) δ ImD ( p ) 2 m ρ m ρ p − p z γ ! × Q q z | p | − p z m ρ + ( p − p z γ ) m ρ p z Q ( p − p z γ ) q q z + ! F EM / Weak ,ρ ( x ρ )(13)where x ρ = − Q p · q and D ρ ( p ) is now the ρ -meson propagator in the medium given by: D ρ ( p ) = [ p − p − m ρ − Π ∗ ρ ( p , p )] − , (14)where Π ∗ ρ = f / m ρ C ρ F ρ ( p ) p Π ∗ − f / m ρ V ′ T Π ∗ . (15)In Eq.15, V ′ T is the transverse part of the spin-isospin interaction, C ρ = . F ρ ( p ) = ( Λ ρ − m ρ ) / ( Λ ρ + p ) is the ρ NN form factor, Λ ρ = GeV , f = .
01, and Π ∗ is the irreducible rho self energy whichcontains the contribution of particle - hole and delta - hole excitations. We have used the same PDFsfor the ρ meson as for the pions [6].We have also included shadowing e ff ect following the works of Kulagin and Petti [7]. For theshadowing e ff ect which is due to the constructive interference of amplitudes arising from the multiplescattering of quarks inside the nucleus, Glauber-Gribov multiple scattering theory has been used.Shadowing e ff ect is a low x and low Q phenomenon which becomes negligible for high x. We labelthe results of spectral function(SF) with meson cloud contribution and shadowing e ff ect, as the resultsobtained with full prescription(Total). x x F A ( x , Q ) EMWeak(Scaled by 5/18)JLab EM Q =2 GeV C (NLO) x x F A ( x , Q ) EMWeak(Scaled by 5/18)JLab EM Q =1.8 GeV Fe (NLO)
Fig. 1.
Results for the 2 xF A ( x , Q ) in ( A = ) C and Fe nuclei at NLO with full prescription. The experi-mental points are JLab data [1].
3. Results
In Fig.1, the results for 2 xF A ( x , Q ) are shown for electromagnetic and weak interactions incarbon and iron nuclei at a fixed Q with full prescription for nuclear medium e ff ect. The results for2 xF EM A ( x , Q ) are compared with the JLab [1] data. We found that the present results are consistentwith the experimental data. From the figure, one may observe that at low x, EM structure function s slightly di ff erent from weak structure function which is about 1 −
2% for carbon. This di ff erenceincreases with the increase in mass number, for example, in iron it is ∼
4% at x = ff erenceis significant at low x and becomes almost negligible for high x. This di ff erence has also been foundin the free nucleon case which shows a di ff erent distribution of sea quarks for electromagnetic andweak interaction processes. Moreover, the di ff erence in the case of nuclear target becomes a bit largerfrom the free nucleon case due to nuclear medium and nonisoscalarity e ff ects.Similar is the observation for F A ( x , Q ) in the electromagnetic as well as weak structure func-tions as may be observed in Fig.2. In Fig.3 (left panel), we have shown the e ff ect of mesonic cloud x F A ( x , Q ) EMWeak(Scaled by 5/18)JLab EM Q =2 GeV C (NLO) x F A ( x , Q ) EMWeak(Scaled by 5/18)JLab EM Q =1.8 GeV Fe (NLO)
Fig. 2.
Results for F A ( x , Q ) in ( A = ) C and Fe nuclei at NLO with full prescription. The experimentalpoints are JLab data [1]. contribution and shadowing e ff ect on the electromagnetic structure functions F A ( x , Q ) and F A ( x , Q ).For this, we are presenting the results using the expression r i = F Modifiedi ( x , Q ) − F SFi ( x , Q ) F Modifiedi ( x , Q ) , (i = Fe at Q = GeV , where F S Fi ( x , Q ) stands for the results obtained for the nuclear structurefunctions using the spectral function(SF) only while F Modi f iedi ( x , Q ) is the result obtained when weinclude (i) mesonic( π + ρ ) cloud contribution, (ii) mesonic( π + ρ ) cloud contribution and shadowinge ff ect. Mesonic cloud contribution is e ff ective in the intermediate region of x ( x ≤ . ff ect hardly changes this ratio.Furthermore, the e ff ect of mesonic contributions is to increase 2 xF A ( x , Q ) and F A ( x , Q ). Theincrease is larger at small x ( x < . x = .
2, 12% at x = . x ( x > . ∼
2% at x = .
5. The increment in the results of F A ( x , Q ) is morethan in the results of 2 xF A ( x , Q ) over an entire range of x . We have also studied this ratio in thecase of weak interaction and found that the results almost overlap with the results obtained for EMinteraction.It may be observed that when shadowing e ff ects are included there is net reduction in the result.This is because the shadowing and the mesonic e ff ects tend to cancel each other specially in theregion of small x ( x < . F A ( x , Q ) in iron at Q = GeV and compared them with some of the experimental data [8–14] available for F EM A ( x , Q ) and F Weak A ( x , Q ). One may observe that theoretically F EM A ( x , Q ) liesbelow F Weak A ( x , Q ) over the entire region of x . Furthermore, we observe that explicitly our model inagreement with the experimental data of CCFR [8], EMC139 [10,11], EMC140 [12] and NuTeV [14]experiments.To quantify our results in Fig.4, we present the results for the ratio of xF A ( x , Q ) F A ( x , Q ) in carbonand compare it with the JLab data [1]. We have found that the ratio of xF A ( x , Q ) F A ( x , Q ) is di ff erent than .1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x [ ( F i A M od i f i e d ( x , Q ) - F i A SF ( x , Q ) ) / F i A M od i f i e d ( x , Q ) ] % F F F π+ρ F π+ρ Fe (NLO)Q = 5 GeV x F A ( x , Q ) F (Q =6 GeV )F (Q =6 GeV , scaled by 5/18)F EMCF
CDHSWF
E140F
CCFRF
NuTeVF
E139 < 8 GeV Fe (NLO)
Fig. 3. Left panel: r i vs x in ( A = ) Fe at Q = GeV for EM interaction at NLO; Right panel:
Resultsfor F A ( x , Q ) in iron for EM and weak interactions at NLO with full prescription. The experimental pointsare data from Refs. [8–14]. It may be noted that the experimental data points lying below the theoretical curvesare from the older experiments which have measured F EM A ( x , Q ) [13] and F Weak A ( x , Q ) [9]. unity, i.e. the results presented here give a microscopic description of deviation from Callan-Grossrelation( xF A ( x , Q ) F A ( x , Q ) =
1) due to nuclear medium e ff ects. From the figure, one may observe that theratio of EM and weak structure functions overlap each other. It would be interesting to make similarstudies in the case of MINERvA experiment. x x F A ( x , Q ) / F A ( x , Q ) EM SFEM SF+ π+ρ
EM TotalEM JLabWeak Total Q =3 GeV C (NLO)
Fig. 4.
Ratio of structure functions showing the violation of Callan-Gross relation at nuclear level. The ex-perimental points are JLab data [1]. Dashed line corresponds to the Callan-Gross relation xF A ( x , Q ) F A ( x , Q ) = References [1] V. Mamyan, arXiv:1202.1457 [nucl-ex].[2] H. Haider, F. Zaidi, M. Sajjad Athar, S. K. Singh and I. Ruiz Simo, Nucl. Phys. A (2015) 58, H. Haider,I. R. Simo, M. S. Athar and M. J. V. Vacas, Phys. Rev. C (2011) 054610, M. Sajjad Athar, I. Ruiz Simoand M. J. Vicente Vacas, Nucl. Phys. A (2011)29.[3] E. Marco, E. Oset and P. Fernandez de Cordoba, Nucl. Phys. A , (1996) 484.[4] Pavel M. Nadolsky et al., Phys. Rev. D (2008) 013004; http: // hep.pa.msu.edu / cteq / public.[5] F. Zaidi, H. Haider, M. Sajjad Athar, S. K. Singh and I. Ruiz Simo, in this Proceeding.[6] M. Gluck, E. Reya and A. Vogt, Z. Phys. C (1992) 651.[7] S. A. Kulagin and R. Petti, Phys. Rev. D (2007) 094023.[8] E. Oltman et al. , Z. Phys. C (1992) 51.[9] J. P. Berge et al. , Z. Phys. C (1991) 187. 610] L.W. Whitlow, SLAC-357
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