Effects of Nuclear Medium on the Sum Rules in Electron and Neutrino Scattering
F. Zaidi, H. Haider, M. Sajjad Athar, S. K. Singh, I. Ruiz Simo
aa r X i v : . [ nu c l - t h ] F e b E ff ects of Nuclear Medium on the Sum Rules in Electronand Neutrino Scattering F. Z aidi , H. H aider , M. Sajjad A thar , S. K. S ingh and I. R uiz S imo Department of Physics, Aligarh Muslim University, Aligarh, 202002, 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)In this work, we study the influence of nuclear medium e ff ects on various parton model sum rules innuclei and compare the results with the free nucleon case. We have used relativistic nucleon spectralfunction to take into account Fermi motion, binding and nucleon correlations. The pion and rhomeson cloud contributions have been incorporated in a microscopic model. The e ff ect of shadowinghas also been considered. KEYWORDS: parton-model sum rules, deep inelastic scattering, nuclear medium effect
1. Introduction
In the deep inelastic scattering, the target nucleon is considered to be a collection of quarks andgluons(described as partons). The deep inelastic cross sections for charged lepton-nucleon scatteringor (anti)neutrino-nucleon scattering are described in terms of nucleon structure functions which de-pend upon the momentum distribution of these quarks and gluons. Using the appropriate relationsbetween these structure functions one can obtain certain relations. These relations are better knownas parton-model sum rules. Some of these sum rules are Gross-Llewellyn Smith sum rule(GLS) [1],Adler sum rule(ASR) [2], Gottfried sum rule(GSR) [3].GLS [1] is defined for an isoscalar nucleon target ’N’ and a symmetric sea as S GLS = Z F ν l N ( x ) dx = . (1)ASR [2] predicts the di ff erence between the quark densities of the neutron and the proton, and isgiven by S AS R = Z dxx [ F ν l n ( x ) − F ν l p ( x )] = . (2)GSR [3] also known as valence isospin sum rule is given by S GS R = Z dxx [ F ep ( x ) − F en ( x )] = + Z dx (¯ u − ¯ d ) (3)where F eN ( x ) is the electromagnetic nucleon structure function and F ν l (¯ ν l ) Ni ( x ); l = e , µ , i = / antineutrino experiments as well as high lu-minosity electron beam experiments it is possible to verify these sum rules. These experiments arebeing done with moderate and heavier nuclear targets. For a nucleus, these sum rules are expressed in erms of nuclear structure functions like F EM A ( x , Q ) and F EM A ( x , Q ) for electromagnetic processesand F Weak A ( x , Q ), F Weak A ( x , Q ) and F Weak A ( x , Q ) for weak interaction induced processes, which getmodified because of the nucleons bound inside the nucleus. In the present work, we have taken intoaccount nuclear medium e ff ects like Fermi motion, binding energy, nucleon correlations, etc., usinga relativistic nucleon spectral function in an interacting Fermi sea and local density approximation isthen applied to obtain the results for finite nuclei. Furthermore, mesonic contributions and shadowinge ff ects have also been taken into account. The results are compared with the free nucleon case as wellas with some of the available experimental data.The details of the present formalism are given in Ref. [4]. We are presenting the formalism inbrief.
2. Formalism
For the charged lepton induced deep inelastic scattering process ( l ( k ) + N ( p ) → l ( k ′ ) + X ( p ′ ); l = e − , µ − ), the di ff erential scattering cross section is given by d σ N d Ω l dE ′ l = α q | k ′ || k | L µν W µν N , (4)where the hadronic tensor W µν N is defined in terms of nucleon structure functions W Ni (i = W µν N = q µ q ν q − g µν ! W N + p µ N − p N . qq q µ ! p ν N − p N . qq q ν ! W N M (5)with M as the mass of nucleon.For the lepton scattering taking place with a nucleon moving inside the nucleus, the expressionof the cross section is modified as d σ A d Ω l dE ′ l = α q | k ′ || k | L µν W µν A , (6)where W µν A is the nuclear hadronic tensor defined in terms of nuclear hadronic structure functions W Ai (i = W µν A = q µ q ν q − g µν ! W A + p µ A − p A . qq q µ ! p ν A − p A . qq q ν ! W A M A (7)with M A as the mass of nucleus.To get d σ for ( l , l ′ ) scattering on the nucleus, we are required to evaluate imaginary part of leptonself energy Σ ( k ) which is written using Feynman rules as [4] Σ ( k ) = ie Z d q (2 π ) q m L µν k ′ − m + i ǫ Π µν ( q ) , (8)where Π µν ( q ) the photon self energy and L µν is the leptonic tensor L µν = k µ k ′ ν + k ′ µ k ν − k · k ′ g µν ).Now we shall use the imaginary part of the lepton self energy i.e. Im Σ ( k ), to obtain the results for thecross section and for this we apply Cutkosky rules Σ ( k ) → i Im Σ ( k ) , D ( k ′ ) → i θ ( k ′ ) ImD ( k ′ ) Π µν ( q ) → i θ ( q ) Im Π µν ( q ) , G ( p ) → i θ ( p ) ImG ( p ) (9) hich leads to Im Σ ( k ) = e Z d q (2 π ) E l θ ( q ) Im ( Π µν ) 1 q m L µν (10)Notice from Eq. 10, Σ ( k ) contains photon self energy Π µν , which is written in terms of nucleonpropagator G l and meson propagator D j and using Feynman rules this is given by Π µν ( q ) = e Z d p (2 π ) G ( p ) X X X s p , s l Y N i = Z d p ′ i (2 π ) Y l G l ( p ′ l ) Y j D j ( p ′ j ) < X | J µ | H >< X | J ν | H > ∗ (2 π ) δ ( q + p − N X i = p ′ i ) , (11)where s p is the spin of the nucleon, s i is the spin of the fermions in X , < X | J µ | H > is the hadroniccurrent for the initial state nucleon to the final state hadrons, index l runs for fermions and index j runs for bosons in the final hadron state X .The relativistic nucleon propagator G(p) in a nuclear medium is obtained as [6, 7]: G ( p ) = ME ( p ) X r u r ( p )¯ u r ( p ) "Z µ −∞ d ω S h ( ω, p ) p − ω − i η + Z ∞ µ d ω S p ( ω, p ) p − ω + i η , (12)where S h ( ω, p ) and S p ( ω, p ) being the hole and particle spectral functions respectively, which aregiven in Ref. [7].The cross section is then obtained as: d σ A d Ω l dE ′ l = − α q | k ′ || k | π ) L µν Z Im Π µν d r (13)After performing some algebra, the expression of the nuclear hadronic tensor for an isospin sym-metric nucleus in terms of nucleonic hadronic tensor and spectral function, is obtained as [4] W αβ A = Z d r Z d p (2 π ) ME ( p ) Z µ −∞ d p S h ( p , p , ρ ( r )) W αβ N ( p , q ) , . (14)Accordingly the dimensionless nuclear structure functions F Ai = , ( x , Q ), are defined in terms of W Ai = , ( ν, Q ) as F A ( x , Q ) = M A W A ( ν, Q ) F A ( x , Q ) = ν A W A ( ν, Q ) where ν A = p A · qM A = p A q M A = q , p µ A = ( M A , ~
0) and M A is the mass of a nucleus . (15)For weak interaction, we follow the same procedure, formalism for which is given in accompa-nying paper by Haider et al. [5] in this proceeding. For a nonisoscalar nuclear target the expressionfor the dimensionless structure functions F A ( x A , Q ) and F A ( x A , Q ) are obtained as 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 + M p x F EM / Weak ,τ ( x N , Q ) ν . (16) 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 ) , (17)where γ = q z q . F Weak A ( x A , Q ) = Z d r Z d p (2 π ) ME ( ~ p ) "Z µ −∞ d p S ph ( p , p , k F , p ) F p ( x N , Q ) + Z µ −∞ d p S nh ( p , p , k F , n ) F n ( x N , Q ) × p γ − p z ( p − p z γ ) γ ! . (18)The nucleon structure functions F Ni ( x , Q ) (i =
3. Results Q (GeV ) S G L S FreeLO SFLO TotalNLO TotalCCFR (Q =8 GeV )CCFR[10]CCFR (Q =3 GeV )[9] Fe Fig. 1.
Results for GLS sum rule in Fe at both LO and NLO and the results are also compared with CCFRexperimental data [10, 11]. In Fig.1, we have presented the results for the GLS sum rule in the free nucleon evaluated at LOas well as in Fe nucleus, using spectral function of the nucleon. We find that when calculations aredone using spectral function, the value of GLS integral decreases from the free nucleon case whichis around 7 −
8% at all values of Q . When shadowing e ff ects are included following Ref. [12] (notethat there is no mesonic contribution to F N ( x , Q )), S GLS further reduces by ∼
10% at low Q andabout 3 −
4% at Q = − GeV . This is the result of our full calculation at LO. When we evaluatethe results at NLO, S GLS further decreases by ∼
10% at low Q and 6% at Q = − GeV .Similarly in Fig. 2, the results are presented for Adler and Gottfried sum rules. We find that S AS R decreases from free nucleon case when spectral function is used, and the decrease is ∼ − Q (GeV ) S A S R FreeLO SFLO TotalNLO Total Fe Q (GeV ) S G S R FreeLO SFLO TotalNLO Total ` Fe Fig. 2. Left panel :Results for Adler sum rule in Fe at both LO and NLO and the results are also comparedwith the results of free nucleon. Right panel :Results for Gottfried sum rule in Fe at both LO and NLO. in 2 GeV < Q < GeV . When shadowing and mesonic e ff ects are included there is furtherreduction of ∼
10% at low Q and ∼ −
4% at high Q . The evaluation at NLO results in a very smallchange in the Adler sum rule.In the right panel of this figure, we show the results for GSR evaluated with four quark fla-vors(u,d,s,c). We find that the inclusion of spectral function results in an increase of S GS R whichis ∼ −
15% at Q = − GeV which becomes ∼ −
35% at Q = − GeV fromfree nucleon case. However, when shadowing and mesonic e ff ects are taken into account, the net in-crease in S GS R is 7-8% at low Q ( for Q = − GeV ) which becomes 30-32% at higher Q (for Q = − GeV ). There is 2-3% reduction at low Q when the calculations are performed at NLOwhich becomes negligible at high Q .We conclude that there is significant dependence of nuclear medium e ff ects in the sum rulesstudied in this work. Morever, we find that nuclear medium e ff ects lead to Q dependence in thesesum rules. This study may be useful in the future analysis of experiments looking for the validity ofsum rules. References [1] D. J. Gross and C. H. Llewellyn Smith: Nucl. Phys. B (1969) 337.[2] S. L. Adler: Phys. Rev. (1966) 1144.[3] K. Gottfried: Phys. Rev. Lett. (1967) 1174 .[4] H. Haider, F. Zaidi, M. Sajjad Athar, S. K. Singh and I. Ruiz Simo: Nucl. Phys. A (2015) 58 ,H. Haider, I. Ruiz Simo, M. Sajjad Athar and M. J. Vicente Vacas: Phys. Rev. C (2011) 054610,M. Sajjad Athar, I. Ruiz Simo and M. J. Vicente Vacas: Nucl. Phys. A (2011) 29.[5] H. Haider, F. Zaidi, M. Sajjad Athar, S. K. Singh and I. Ruiz Simo, in this Proceeding.[6] E. Marco, E. Oset and P. Fernandez de Cordoba: Nucl. Phys. A (1996) 484.[7] P. Fernandez de Cordoba and E. Oset, Phys. Rev. C (1992) 1697.[8] Pavel M. Nadolsky et al.: Phys. Rev. D (2008) 013004 ; http: // hep.pa.msu.edu / cteq / public.[9] M. Gluck, E. Reya and A. Vogt, Z. Phys. C (1992) 651.[10] W. C. Leung et al. , Phys. Lett. B (1993) 655.[11] J. H. Kim et al. , Phys. Rev. Lett. (1998) 3595.[12] S. A. Kulagin and R. Petti, Phys. Rev. D76