Nuclear Medium Effects in Electromagnetic and Weak ...ir.amu.ac.in/11250/1/T10098.pdfNuclear Medium...

Nuclear Medium Effects in Electromagnetic and Weak

Structure Functions at moderate Q2

THESISsubmitted for the award of the degree of

Doctor of Philosophyin Physics

byFarhana Zaidi

SupervisorProf. Mohammad Sajjad Athar

Department of PhysicsAligarh Muslim UniversityAligarh-202 002 (India)

2017

Nuclear Medium Effects in Electromagnetic and Weak Structure Functions at moderate Q2

THESISsubmitted for the award of the degree of

Doctor of Philosophyin Physics

byFarhana Zaidi

SupervisorProf. Mohammad Sajjad Athar

Department of PhysicsAligarh Muslim UniversityAligarh202 002 (India)2017

I dedicate this thesis to my late grandmother

&

to my parents

for their love, blessings, patience and support

CANDIDATE’S DECLARATION

I, Farhana Zaidi, Department of Physics certify that the work embodied in

this Ph.D. thesis is my own bonafide work carried out by me under the supervision of

Prof. Mohammad Sajjad Athar at Aligarh Muslim University, Aligarh. The matter

embodied in this Ph.D. thesis has not been submitted for the award of any other

degree.

I declare that I have faithfully acknowledged, given credit to and referred to

the research workers wherever their works have been cited in the text and the body of

the thesis. I further certify that I have not willfully lifted up some other’s work, para,

text, data, result, etc. reported in the journals, books, magazines, reports, dissertations,

thesis, etc., or available at web-sites and included them in this Ph.D. thesis and cited

as my own work.

Date: (Signature of the candidate)

FARHANA ZAIDI (Name of the candidate)

CERTIFICATE FROM THE SUPERVISOR

This is to certify that the above statement made by the candidate is correct to the bestof my/our knowledge.

Signature of the SupervisorName & Designation: Dr. Mohammad Sajjad Athar, ProfessorDepartment: PHYSICS

(Signature of the Chairman of the Department with seal)

COURSE/ COMPREHENSIVE EXAMINATION/ PRE-SUBMISSION SEMINAR COMPLETION CERTIFICATE

This is to certify that Miss. Farhana Zaidi, Department of Physics has

satisfactorily completed the course work/comprehensive examination and pre-

submission seminar requirement which is part of her Ph.D. programme.

Date: ……………. (Signature of the Chairman of the Department)

COPYRIGHT TRANSFER CERTIFICATE

Title of the Thesis: Nuclear Medium Effects in Electromagnetic and WeakStructure Functions at moderate Q2

Candidate’s Name: FARHANA ZAIDI

Copyright Transfer

The undersigned hereby assigns to the Aligarh Muslim University, Aligarh,

copyright that may exist in and for the above thesis submitted for the award of

Ph.D. degree.

(Signature of the Candidate)

Contents

Contents i

Acknowledgements v

List of Figures vii

List of Tables xxi

List of Publications xxiii

1 Introduction 1

2 Lepton-Nucleon Scattering 15

2.1 Deep Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Deep Inelastic Charged Lepton-Nucleon Scattering . . . . . . . . . . 16

i

ii CONTENTS

2.3 Deep Inelastic Charged Current νl/ν̄l-Nucleon Scattering . . . . . . . 25

2.4 QCD Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.1 NLO Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.2 TMC E�ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Lepton-Nucleus Scattering 47

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2 Deep Inelastic Charged Lepton-Nucleus Scattering . . . . . . . . . . 48

3.3 Deep Inelastic νl/ν̄l-Nucleus Scattering . . . . . . . . . . . . . . . . . 71

3.4 Isoscalarity Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.5 Results & Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Nuclear Structure Functions 91


5 Callan-Gross Relation in Nuclei 113


6 Summary & Conclusions 139

APPENDICES 145

CONTENTS iii

A Nucleon & Nuclear PDFs 147

B Dirac Propagator 151

C Pion PDFs Parameterizations 159

Bibliography 165

iv CONTENTS

Acknowledgements

�Oh Allah! Nothing is easy except what You have made easy. If You wish, You canmake the di�cult easy.�

First and foremost, all praise is due to Almighty Allah, the merciful, for Hisblessings and providing me an opportunity and ability to accomplish this task into apractical one.

It is the best opportunity to express my deep gratitude and sincere thanks to mysupervisor Prof. Mohammad Sajjad Athar who has guided me and encouraged methroughout my research career. His guidance helped me to present this thesis in sucha form. His hard-working attitude is always a source of inspiration to me. I amhighly obliged to him. Besides my supervisor, I am very thankful to Prof. S. K. Singhfor his insightful comments and guidance throughout my research work. I would liketo express my heartiest thanks to Dr. I. R. Simo from University of Granada,Spain for the discussions, suggestions and help in completing this work. My sincerethanks goes to the Chairman, Department of Physics, Prof. Tauheed Ahmad forproviding me all the required facilities for the completion of my research work. Ialso acknowledge UGC for providing me �nancial support in the form of MaulanaAzad National Fellowship (F1-17.1/2010/MANF-MUS-UTT-2900) to complete thisresearch work.

I owe my sincere thanks to my labmates Dr. Shikha Chauhan, Dr. Huma Haiderand Dr. Ra� Alam, Miss Faiza Akbar, Miss Atika Fatima, Mr. Zubair A. Dar andMiss Sidra Khan for their help and coordination. My sincere thanks goes to thenon-teaching sta� like Aashish bhai, Naseem bhai and Meraj bhai as well as to themembers of seminar library for their help & support.

v

vi ACKNOWLEDGEMENTS

I am lucky to have some very good friends like Zaheen Haider, Shaista Khan,Anuj Chandra and Dr. Saima Tarannum to whom I want to express my heartfeltthanks for the care, concern, encouragement, help and support. I am also thankfulto my room mates Adiba Ali and Saika Shah for their well wishes and moral support.

It is really pleasant to express my gratitude with great honor and respect to mylate grand mother and to my beloved parents Mrs. & Mr. Syed Fahim Haider fortheir endless love, support and encouragement at the di�erent stages of my life. Aswell as I would like to say a profound thanks to my brothers Zaheen, Azeem, Salman& Ankit and my sisters Huma Zaidi & Khushboo Haider for their well wishes, moralsupport and encouragement. They have always been there for me whenever I was inneed and help me in all the possible ways.

Farhana Zaidi

List of Figures

1.1 Feynman diagram for the electromagnetic interaction induced pro-cesses representing(Left to Right) (i) Elastic scattering process, (ii)Inelastic scattering processes(R≡ resonance; m=1,2,...; Y ≡ Λ,Σ)and (iii) Deep-Inelastic scattering process on free nucleon target (Xrepresents here jet of hadrons in the �nal state). . . . . . . . . . . . . 3

1.2 Double di�erential scattering cross section vs CM energy W (GeV)for the process e + p → e + X at projectile energy of 10 GeV andscattering angle of 6◦ [1]. The elastic peak has been reduced by afactor of 8.5 at W = MN . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Ratio R′ = F2A(x,Q2)

F2D(x,Q2); (A =target nucleus) vs x represents the nuclear

medium e�ects in structure function. Experimental data are takenfrom the Refs. [12,21,40,41,47]. . . . . . . . . . . . . . . . . . . . . 7

1.4 Feynman diagram representing(Left to Right) (i) Quasielastic scatter-ing process, (ii) Resonant scattering processes(pion production, kaonproduction, single hyperon production, eta production....; Y ≡ Λ,Σ),and (iii) Deep-Inelastic scattering process. . . . . . . . . . . . . . . . 10

1.5 Charged current induced neutrino(Left panel) and antineutrino(Rightpanel) scattering cross section per nucleon(for an isoscalar target)divided by neutrino energy are plotted as a function of energy [109]. . 11

vii

viii LIST OF FIGURES

2.1 Feynman diagram for the charged lepton induced deep inelastic scat-tering process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Feynman representation for leptonic and hadronic vertices in the caseof electromagnetic interaction. . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Feynman diagram for the neutrino/antineutrino induced deep inelas-tic scattering process. . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Feynman representation for leptonic and hadronic vertices in the caseof weak interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Diagrammatic representation of (i) Upper panel: the process γ∗q →qg and (ii) Lower panel: the process γ∗g → qq̄ [146]. . . . . . . . . 32

2.6 Results of the free nucleon structure functions obtained using CTEQ6.6PDFs [117] at LO and NLO for Q2 = 2, 10 GeV2. Results are obtainedfor the electromagnetic interaction. . . . . . . . . . . . . . . . . . . . 38

2.7 Results of the free nucleon structure functions obtained using thedi�erent PDFs parameterizations [117�119] for several values of Q2

at NLO. Results are presented for both the electromagnetic and weakinteraction induced processes. . . . . . . . . . . . . . . . . . . . . . . 39

2.8 Results of the weak structure function xFWI3N (x,Q2) for the free nu-

cleon target obtained using the di�erent PDFs parameterizations [117�119] for several values of Q2 at NLO. . . . . . . . . . . . . . . . . . . 40

2.9 Coe�cient function for quarks convoluted over the quark densitiesC

(1)q,2 (x) ⊗ q(x) vs x at the di�erent values of Q2= 2, 5, 10, 20 GeV2.

Results are obtained by using Refs. [121,122] and Ref. [123]. . . . . . 41

2.10 Coe�cient function for gluons convoluted over the gluon density C(1)g,2(x)⊗

g(x) vs x at the di�erent values of Q2= 2, 5, 10, 20 GeV2. Resultsare obtained by using Refs. [121,122] and Ref. [123]. . . . . . . . . . . 42

LIST OF FIGURES ix

2.11 Results for the free nucleon structure function FEM2N (x,Q2) vs x at

the di�erent values of Q2= 2, 5, 10, 20 GeV2. Results are obtainedby using Refs. [121,122] and Ref. [123]. . . . . . . . . . . . . . . . . . 43

2.12 Results for the nucleon structure function FEM2N (x,Q2) evaluated at

NLO and at NNLO in the case of electromagnetic interaction fordi�erent Q2 are shown. Results are obtained by using MSTW [118]and CT14 [120] PDFs. . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.13 Results of the free nucleon structure functions showing the e�ect ofTMC at di�erent Q2 for the electromagnetic and weak interactions.Results for the weak interaction are scaled by a factor of 5

18. . . . . . 45

3.1 Feynman diagram showing the deep inelastic scattering processes withbound nucleons for the electromagnetic and weak interactions. . . . . 48

3.2 Results for the Fermi momentum(pFN ) of nucleons vs r are shown forthe di�erent nuclear targets. . . . . . . . . . . . . . . . . . . . . . . . 51

3.3 Feynman diagram representing the lepton self-energy. . . . . . . . . . 53

3.4 Feynman diagram representing the photon self-energy. The imaginarypart is calculated by cutting along the horizontal line and applyingthe Cutkosky rules while putting the particle on mass shell. . . . . . . 55

3.5 Diagrammatic representation of nucleon self-energy in the nuclearmedium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.6 Results for Sh(ω,p) vs ω are shown for (i) p < pF (Top panel) andp > pF (Bottom panel) in various nuclei like

12C, 40Ca, 56Fe, 120Sn and208Pb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.7 Lepton self-energy diagram including particle-hole(1p-1h), delta-hole(1∆-1h), 1p1h-1∆1h, etc. excitations. . . . . . . . . . . . . . . . . . . . . 66

3.8 Diagrammatic representation of the neutrino self-energy. . . . . . . . 72

x LIST OF FIGURES

3.9 Diagrammatic representation of intermediate vector boson W self-energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.10 Results for the nuclear structure functions FEM2A (x,Q2) vs x obtained

with the spectral function only using CTEQ6.6 [117] and MSTW [118]nucleon PDFs parameterizations at Q2 =2, 5 and 10 GeV2. We havetreated iron as isoscalar nuclear target. . . . . . . . . . . . . . . . . . 79

3.11 Results for the nuclear structure functions FEM2A (x,Q2) vs x obtained

using the spectral function with mesonic contribution for di�erentpion PDF parameterizations given by Gluck et al. [124](solid line),Wijesooriya et al. [126](dashed line), Sutton et al. [127](dashed dottedline), Martin et al. [128](cross sign) and Conway et al. [129](dottedline) at Q2 =2, 5 and 10 GeV2. We have treated iron as isoscalarnuclear target. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.12 Results for FEM2A (x,Q2) (A = 12C, 56Fe, 208Pb) vs x are shown using

models given by Marco et al. [61](dotted line), Kulagin et al. [68](dashedline) and the present model(dashed-double dotted line) for the spec-tral function at Q2 =2, 5 and 10 GeV2. Numerical results are evalu-ated at NLO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.13 Results forFEM2A (x,Q

2)

FEM2N (x,Q2)

(A =56Fe, 208Pb) vs x are obtained using models

given by Marco et al. [61](dotted line), Kulagin et al. [68](dashed line)and the present model(dashed-double dotted line) for the spectralfunction at Q2 =10 GeV2. Numerical calculations are performed atNLO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.14 Results are presented for the ratioFEMiCa (x,Q

2)

FEMiC (x,Q2); i = 1, 2 vs x obtained by

using the spectral function(NLO SF) and the full model(NLO Total).Results are also compared with the experimental data of NMC [21]. . 84

3.15 Results are presented at NLO obtained by using the full model(NLO

Total) for the ratioFEM2A (x,Q

2)

FEM2C (x,Q2); A=27Al, 40Ca, 56Fe and 118Sn. . . . . . 84

LIST OF FIGURES xi

3.16 Results of the ratioFEM2A (x,Q

2)

FEM2C (x,Q2)

(A =56Fe(in top panel), 208Pb(in bot-

tom panel)) vs x with (i) the spectral function(SF) only, (ii) thespectral function with mesonic contribution(SF+π + ρ), (iii) withthe full model(Total) treating nuclei to be isoscalar. The results areevaluated at NLO corresponding to the values of x and Q2 as given inRef. [20](Table 6 for 56Fe and Table 8 for 208Pb) in order to comparethem with the experimental data of NMC [20]. . . . . . . . . . . . . . 85

3.17 Results for the % deviation from Isoscalarity (i.e. rEM(WI)(x,Q2) =FEM(WI)2A,Iso (x,Q

2) − FEM(WI)2A,Noniso(x,Q2)

FEM(WI)2A,Iso (x,Q

2)) in 56Fe for the electromagnetic (Top

panel) and weak (Bottom panel) nuclear structure functions at dif-ferent Q2 are shown. Solid(double dashed dotted) line is the result atQ2 = 2 GeV 2, long dashed(dashed double dotted) line is the resultat Q2 = 8 GeV2 and short dashed dotted(dotted) line is the resultat Q2 = 50 GeV2 obtained by using the full model(phenomenologicalprescription using Eqs. 3.91 and 3.93) at NLO. . . . . . . . . . . . . . 87

3.18 Results of the ratio R′ =FEM2A (x,Q

2)

FEM2Fe (x,Q2)

(A =197Au (top panel), 208Pb

(bottom panel)) vs x are obtained (i) Top panel: at a �x value ofQ2 = 5 GeV2 and are compared with the experimental data of SLAC-E139 [12], (ii) Bottom panel: for values of x and Q2 given in Table6 for 56Fe and Table 8 for 208Pb in Ref. [20] and are compared withexperimental data of NMC [20]. The numerical results are obtainedat NLO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.1 Results for FEM2A (x,Q2) vs x are shown at di�erent Q2 in 12C and 27Al.

These results are obtained using the spectral function only(dashed-dotted line) and the full model(dashed line) at LO as well as the re-sults are obtained at NLO for both the spectral function only(dashed-double dotted line) and the full model(solid line). Experimentalpoints are the JLab data [7](bold circles). . . . . . . . . . . . . . . . . 94

xii LIST OF FIGURES

4.2 Results for FEM2A (x,Q2) vs x are shown at di�erent Q2 in 56Fe, 63Cu,

118Sn, 197Au and 208Pb. These results are obtained using the spectralfunction only(dashed-dotted line) and the full model(dashed line) atLO as well as the results are obtained at NLO with spectral functiononly(dashed-double dotted line) and with the full model(solid line).All the nuclei are considered as nonisoscalar. Experimental points arethe JLab data [7](bold circles). . . . . . . . . . . . . . . . . . . . . . 95

4.3 Results of 2xFEM1A (x,Q2) vs x are shown with the spectral function

only and the full model at (i)LO by dashed-dotted line and dashedline, (ii)at NLO by dashed-double dotted line and solid line, respec-tively, in 12C and 27Al at di�erent Q2. Experimental points are theJLab data [7](bold circles). . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4 Results of 2xFEM1A (x,Q2) vs x are shown with the spectral function

only and the full model at (i)LO by dashed-dotted line and dashedline, (ii)at NLO by dashed-double dotted line and solid line, respec-tively, in 56Fe, 63Cu, 118Sn, 197Au and 208Pb at di�erent Q2. All thenuclei are considered as nonisoscalar. Experimental points are theJLab data [7](bold circles). . . . . . . . . . . . . . . . . . . . . . . . . 97

4.5 rEMi (x,Q2) =

FEM,Modifiedi (x,Q2) − FEM,SFi (x,Q

2)

FEM,Modifiedi (x,Q2)

(i = 1, 2) in % vs x, at

Q2 = 2 and 5 GeV2 in 56Fe. Here FEM,SFi (x,Q2) stands for nuclear

structure functions FEMiA (x,Q2) (i = 1, 2) obtained using spectral

function only and FEM,Modifiedi (x,Q2) stands for nuclear structure

functions FEMiA (x,Q2) (i = 1, 2) evaluated (i) with mesonic e�ects

along with the spectral function, and (ii) when shadowing is alsoincluded in (i). The solid line with bold circles(solid line) is the resultobtained for rEM1 (x,Q

2) using case i(ii) and dashed line with boldcircles(dashed line) is the result obtained for rEM2 (x,Q

2) using casei(ii). The numerical calculations are performed at NLO. . . . . . . . . 99

LIST OF FIGURES xiii

4.6 Results for the electromagnetic nuclear structure functions FEM2A (x,Q2)

are shown for A = 56Fe(isoscalar) at Q2 = 2 GeV2(Left panel) andQ2 = 50 GeV2(Right panel). E�ect of the parameters Λπ and Λρ usedin the expressions of πNN and ρNN form factors given in Eqs. 3.59,3.68 respectively, that are varied by 10%(shown by band) from thecentral value(dashed line) is shown. The results are obtained withthe full model at NLO and are compared with the experimental dataof JLab [7](bold circles) and EMC [18](solid square). . . . . . . . . . 100

4.7 Results for FWI2A (x,Q2) vs x at Q2 = 2 and 5 GeV2 in 12C and 208Pb

with the full model(dashed line) at LO and the results with the spec-tral function(dashed-double dotted line) and the full model(solid line)at NLO. 208Pb is treated as isoscalar nuclear target and the results arecompared with the available experimental data of CHORUS [135](starsign). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.8 Results for the weak nuclear structure functions FWI2A (x,Q2) are shown

for A = 56Fe(isoscalar) at Q2 = 2 GeV2(Left panel) and Q2 = 50GeV2(Right panel). E�ect of the parameters Λπ and Λρ used in theexpressions of πNN and ρNN form factors given in Eqs. 3.59, 3.68respectively, that are varied by 10%(shown by band) from the cen-tral value(dashed line) is shown. The results are obtained with thefull model at NLO and are compared with the experimental data ofCCFR(triangle up) [94] and NuTeV(diamond) [96]. . . . . . . . . . . 102

4.9 Results of the electromagnetic(Left panel) and weak(Right panel) nu-clear structure functions in 56Fe(isoscalar) obtained using the spec-tral function(dashed double-dotted line), including mesonic contribu-tion(double dashed-dotted line) and the full model(solid line). Theresults are also presented with CCFR prescription i.e. FEM2A (x,Q

2) =f(x)FEM2N (x,Q

2), where f(x) [161], is given by Eq.3.92(dotted line)and for the free nucleon case when f(x) = 1(dashed-dotted line).These results are presented for di�erent Q2 and are compared withthe available experimental data of JLab(bold circle) [7], EMC(solidsquare) [18], CDHSW(semi solid circle) [93], CCFR(triangle up) [94]and NuTeV(diamond) [96]. . . . . . . . . . . . . . . . . . . . . . . . . 104

xiv LIST OF FIGURES

4.10 Results of the electromagnetic and weak nuclear structure functionsin 56Fe(Isoscalar) are obtained at NLO using the full model(solid lineand dashed line, respectively) and also from CJ12min PDFs [130]for �xed value of strong coupling constant(dotted line and dashed-double dotted line, respectively) as well as for Q2 evolution(dashed-dotted line and double dashed-dotted line, respectively). The re-sults of FWI2A (x,Q

2) are scaled by a factor of 518. The results are also

compared with the available experimental data of the JLab(bold cir-cle) [7], CDHSW(semi solid circle) [93], CCFR(triangle up) [94] andNuTeV(diamond) [96]. . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.11 Results of the electromagnetic and weak nuclear structure functionsin 56Fe(Isoscalar) are obtained at NLO using the full model(solid lineand dashed line, respectively) and also from CJ12min PDFs [130]for �xed value of strong coupling constant(dotted line and dashed-double dotted line, respectively) as well as for Q2 evolution(dashed-dotted line and double dashed-dotted line, respectively). The resultsof FWI2A (x,Q

2) are scaled by a factor of 518. The results are also com-

pared with the available experimental data of EMC(solid square) [18]and CDHSW(semi solid circle) [93]. . . . . . . . . . . . . . . . . . . 106

4.12 Results of the electromagnetic and weak nuclear structure functionsin 56Fe(Isoscalar) are obtained at NLO using the full model(solid lineand dashed line, respectively) and also from CJ12min PDFs [130]for �xed value of strong coupling constant(dotted line and dashed-double dotted line, respectively) as well as for Q2 evolution(dashed-dotted line and double dashed-dotted line, respectively). The re-sults of FWI2A (x,Q

2) are scaled by a factor of 518. The results are

also compared with the available experimental data of EMC(solidsquare) [18], CDHSW(semi solid circle) [93], CCFR(triangle up) [94]and NuTeV(diamond) [96]. . . . . . . . . . . . . . . . . . . . . . . . 107

LIST OF FIGURES xv

4.13 Results of the electromagnetic and weak nuclear structure functionsin 56Fe(Isoscalar) obtained at NLO using the full model(solid lineand dashed line, respectively) and also from CJ12min PDFs [130]for �xed value of strong coupling constant(dotted line and dashed-double dotted line, respectively) as well as for Q2 evolution(dashed-dotted line and double dashed-dotted line, respectively). The re-sults of FWI2A (x,Q



4.14 Results of the electromagnetic and weak nuclear structure functionsin 56Fe(Isoscalar) obtained at NLO using the full model(solid lineand dashed line, respectively) and also from CJ12min PDFs [130]for �xed value of strong coupling constant(dotted line and dashed-double dotted line, respectively) as well as for Q2 evolution(dashed-dotted line and double dashed-dotted line, respectively). The re-sults of FWI2A (x,Q



4.15 Results for the ratio R′ =518FWI2A (x,Q

2)

FEM2A (x,Q2)

are obtained by using the full

model at NLO in A = 12C, 27Al, 40Ca, 56Fe, 63Cu, 118Sn and 208Pb atQ2 = 6, 20, and 50 GeV2. The results for the free nucleon case arealso shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.16 Results for the ratio R′ =518FWI2Fe (x,Q

2)

FEM2Fe (x,Q2)

vs Q2 are shown for the di�erent

values of Bjorken scaling variable x in 56Fe. Solid(dashed-dotted) lineis result obtained by using the full model(free nucleon case) at NLO. 112

5.1 Results of the electromagnetic structure function 2xFEM1A (x,Q2); (A =12C)

are presented for two cases: (i) using CGR given in Eq. 5.1 and (ii)using modi�ed form of CGR given in Eq. 5.8. The results are ob-tained for carbon with the spectral function only at NLO for Q2 =2,3, 20 and 50 GeV2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

xvi LIST OF FIGURES

5.2 REM2,A (x,Q2) =

FEM2A (x,Q2)

2xFEM1A (x,Q2)(A =12C, 27Al (isoscalar), 63Cu (nonisoscalar))

vs x showing the deviation from Callan-Gross relation. The resultsare shown using the full model at NLO with a cut on the CM energyW > 1.4 GeV(dotted line) and without any kinematical cut on theCM energy(solid line) at the di�erent values of Q2. These resultsare compared with the results of the free nucleon case obtained byusing the parameterization of Whitlow et al. [131,132](double dashed-dotted line) and the experimental data of the JLab [7](bold circles). . 118

5.3 Results for the ratio of RWI2,A(x,Q2) =

FWI2A (x,Q2)

2xFWI1A (x,Q2)(A =12C(isoscalar),

56Fe and 208Pb (nonisoscalar)) vs x showing the violation of CGR ob-tained using the full model at NLO without(solid line) and with(dottedline) a kinematical cut on the CM energy, W > 1.4 GeV at the di�er-ent values of Q2. These results are compared with the results of thefree nucleon case obtained by using the parameterization of Whitlowet al. [131,132](double dashed-dotted line). . . . . . . . . . . . . . . 120

5.4 Results for the di�erence DEMA (x,Q2) = 2xFEM1A (x,Q

2)−FEM2A (x,Q2)vs x showing the violation of CGR in di�erent nuclei(A=12C, 27Al,63Cu and 197Au) at Q2 = 3.7 GeV2. The results are obtained withthe full model at NLO without applying any cut on the CM energyand are compared with the experimental data of the JLab [7]. . . . . 121

5.5 Results for the di�erence DWIA (x,Q2) = 2xFWI1A (x,Q

2)− FWI2A (x,Q2)vs x showing the violation of CGR in 12C and 56Fe(nonisoscalar)nuclei for Q2 = 2, 5, 10 GeV2 using the full model without(No cut)and with(W > 1.4 GeV) the kinematical cut on the CM energy. Theresults are presented for the weak interaction induced DIS process atNLO. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.6 Results of the longitudinal structure function FEML,A (x,Q2) vs x are

presented at the di�erent values of Q2 in 12C and 27Al. The resultsare obtained without applying any cut on the CM energy for thefull model at LO(dashed line) and NLO(solid line) as well as for thespectral function at NLO(dashed-double dotted line). Experimentalpoints are the JLab data [7](bold circles). . . . . . . . . . . . . . . . 124

LIST OF FIGURES xvii

5.7 Results of the longitudinal structure function FEML,A (x,Q2) vs x are

presented at the di�erent values ofQ2 in various nuclei (A=56Fe, 63Cu,118Sn, 197Au and 208Pb). The results are obtained without applyingany cut on the CM energy for the full model at LO(dashed line) andNLO(solid line) as well as for the spectral function at NLO(dashed-double dotted line). Experimental points are the JLab data [7](boldcircles). Nuclear targets are treated as nonisoscalar in the presentcalculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.8 REML,A (x,Q2) =

FEML,A (x,Q2)

2xFEM1A (x,Q2)(A =12C, 27Al (isoscalar), 56Fe, 63Cu (non-

isoscalar)) vs x are shown at di�erent Q2. The results are presentedfor the full model at NLO(i) without applying any kinematical cut onthe CM energy(solid line) and (ii) with a kinematical cut on the CMenergyW > 1.4 GeV(dotted line). The results are compared with theresults for the free nucleon case obtained using the parameterizationof Whitlow et al. [131, 132](double dashed-dotted line) and with theexperimental data of the JLab(bold circles) [7]. . . . . . . . . . . . . 127

5.9 REML,A (x,Q2) =

FEML,A (x,Q2)

2xFEM1A (x,Q2)

(A =12C and 56Fe) vs Q2 are shown at

di�erent x. The results obtained using the full model at NLO with-out putting any kinematical cut on the CM energy (12C(solid line)and 56Fe(dashed-dotted line)) are compared with the results of theJLab(in 12C and 56Fe) [7], SLAC(in 56Fe) [15] and BCDMS(in 12C) [133].129

5.10 REML,A (x,Q2) =

FEML,A (x,Q2)

2xFEM1A (x,Q2)vs W 2 for 12C, 27Al, 56Fe and 63Cu at dif-

ferent Q2. Numerical results are evaluated at NLO using the fullmodel (solid line) and the spectral function only(dashed-double dot-ted line) without applying any kinematical cut on the CM energy.The results are compared with the results of the free nucleon caseobtained by Whitlow et al. [131,132](double dashed-dotted line) andwith the experimental data of the JLab [7](bold circles). . . . . . . . 130

xviii LIST OF FIGURES

5.11 Numerical results for REML,Sn(x,Q2) − REML,C (x,Q2) vs x are obtained

using the full model without applying any cut on the CM energy(shown by the band) for 4 GeV2 < Q2 < 35 GeV2 at NLO. These re-sults are also compared with the experimental results for RAu(x,Q

2)−RFe(x,Q

2) [13](down triangle), RFe(x,Q2) − RD(x,Q2) [15](circle)

RCa(x,Q2)−RC(x,Q2) [90](diamond) andRSn(x,Q2)−RC(x,Q2) [91](square)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

5.12 REML,A (x,Q2) − REML,C (x,Q2) are shown vs W 2 at di�erent Q2 in 27Al

and 63Cu. The results are obtained using the full model at NLOwithout applying any cut on the CM energy and are compared withthe experimental data of the JLab [7](bold circles) in the range of 2.5GeV2 < Q2 < 4.5 GeV2. . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.13 REML,A (x,Q2) =

FEML,A (x,Q2)

2xFEM1A (x,Q2)vsQ2 in 12C(dashed-dotted line), 63Cu(solid

line) and 197Au(dashed line) at di�erent x. The results are obtainedusing the full model at NLO without any kinematical cut on the CMenergy and are compared with the results for the free nucleon case ob-tained by using the parameterization of Whitlow et al. [131,132](dou-ble dashed-dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.14 Results for REML,A (x,Q2) − REML,D(x,Q2) vs A obtained using the full

model at NLO without applying any cut on the CM energy arepresented for 12C, 56Fe, 63Cu and 197Au in the range of 4 GeV2 <Q2 < 5 GeV2 at x = 0.4(lower band) and x = 0.5(upper band).The results are compared with the experimental results of the SLAC-E140 [173](square and diamond) and the JLab [174](down triangle)corresponding to the x values shown in the �gure. Nuclear targetsare treated as isoscalar. . . . . . . . . . . . . . . . . . . . . . . . . . . 134

5.15 Results for the ratio RWIL,A(x,Q2) =

FWIL,A(x,Q2)

2xFWI1A (x,Q2)(A =56Fe (isoscalar))

vs Q2 are shown with full model without applying any cut on the CMenergy for the weak interaction(solid line) induced DIS process atNLO. The results are compared from the free nucleon case obtainedusing the parameterization given by Whitlow et al. [131, 132](doubledashed-dotted line) and with the experimental data of CCFR [136,137](up triangle). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

LIST OF FIGURES xix

5.16 Results for the ratio RWIL,A(x,Q2) =

FWIL,A(x,Q2)

2xFWI1A (x,Q2)(A =208Pb (isoscalar))

vs Q2 are shown with the full model(solid line) at NLO without apply-ing any cut on the CM energy for the weak interaction induced DISprocess. The results are compared from the free nucleon case obtainedusing the parameterization given by Whitlow et al. [131, 132](dou-ble dashed-dotted line) and with the experimental data of CHO-RUS [135](star sign). . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

A.1 A comparison of the nucleon PDFs obtained using the parameteri-zation of CTEQ [117], GJR [119], MSTW [118] and CT14 [120] atQ2 = 2 GeV2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

A.2 A comparison of the nuclear parton distribution functions obtainedusing the parameterization of nCTEQ15 [113] and CJ12 [130] at Q2 =2 GeV2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

C.1 E�ect of di�erent pion PDFs parameterizations [124,126�129] on thestructure functions FEM2π (x,Q

2) (Top panel) and FWI2π (x,Q2) (Bottom

panel) at Q2 = 2 GeV2. . . . . . . . . . . . . . . . . . . . . . . . . . . 163

xx LIST OF FIGURES

List of Tables

3.1 Di�erent parameters used for the numerical calculations for variousnuclei. For 12C we have used modi�ed harmonic oscillator density(∗ c2is dimensionless) and for 27Al, 40Ca, 56Fe, 63Cu, 118Sn, 197Au and 208Pbnuclei, 2-parameter Fermi density have been used, where superscriptn and p in density parameters(cn,pi ; i=1,2) stand for neutron andproton, respectively. Density parameters are given in units of fm. Thekinetic energy of the nucleon per nucleus(T/A) and binding energy ofthe nucleon per nucleus (B.E/A) for di�erent nuclei are given in MeV. 52

A.1 The kinematic range for the PDFs and number of parton �avors aretabulated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

C.1 Variables of pion PDFs used in di�erent parameterizations [126]. The

variable s ≡ ln ln(Q2/0.204GeV2)µ2LO/0.204GeV

2 with µ2LO = 0.26 GeV2 [124]. . . . . . . . 160

xxi

xxii LIST OF TABLES

List of Publications

International

1. � Nuclear medium e�ects in R = σL/σT",F. Zaidi, H. Haider, M. Sajjad Athar, S. K. Singh and I. Ruiz Simo,Eur. Phys. J. A submitted, (2017); arXiv:1705.09903 [hep-ph].

2. � Nuclear medium e�ects in FEM2A (x,Q2) and FWeak2A (x,Q

2) structure func-tions",H. Haider, F. Zaidi, M. Sajjad Athar, S. K. Singh and I. Ruiz Simo,Nucl. Phys. A 955, 58-78 (2016).

3. �Nuclear medium e�ects in structure functions of nucleon at moderate Q2",H. Haider, F. Zaidi, M. Sajjad Athar, S. K. Singh and I. Ruiz Simo,Nucl. Phys. A 943, 58-82 (2015).

4. �Electron and Muon production cross sections in quasielastic ν(ν̄)-Nucleusscattering for Eν < 1 GeV ",F. Akbar, M. Ra� Alam, M. Sajjad Athar, S. Chauhan, S. K. Singh andF.Zaidi,Int. J. Mod. Phys. E 24, no. 11, 1550079 (2015).

5. �E�ects of nuclear medium on the sum rules in electron and neutrino scatter-ing",F. Zaidi, H. Haider, M. Sajjad Athar, S. K. Singh and I. Ruiz Simo,JPS Conf. Proc. 12, 010055 (2016).

xxiii

xxiv LIST OF PUBLICATIONS

6. �Electromagnetic and weak nuclear structure functions F1,2(x,Q2) in the in-

termediate region of Q2",H. Haider, F. Zaidi, M. Sajjad Athar, S. K. Singh and I. Ruiz Simo,JPS Conf. Proc. 12, 010054 (2016).

7. �Revisiting νµ(ν̄µ) and νe(ν̄e) induced quasielastic scattering from nuclei in sub-GeV energy region",F. Akbar, M. Ra� Alam, M. Sajjad Athar, S. Chauhan, S. K. Singh andF.Zaidi,JPS Conf. Proc. 12, 010053 (2016).

8. �Medium e�ects in the deep-inelastic charged lepton/neutrino-A scattering�,H. Haider, F. Zaidi, M. Sajjad Athar, S. K.Singh and I. Ruiz Simo,To appear in the American Institute of Physics Conference Proceedings (Nu-Fact2016); eprint: arXiv:1611.07167 [nucl-th].

9. �Lepton production cross sections in quasielastic ν/ν̄−A scattering�,M. Sajjad Athar, F. Akbar, M. Ra� Alam, S. Chauhan, S. K. Singh and F.Zaidi,To appear in the American Institute of Physics Conference Proceedings (Nu-Fact2016); eprint: arXiv:1611.07166 [nucl-th].

xxv

National

1. �νl −208 Pb deep inelastic scattering at MINERνA energies",F. Zaidi, H. Haider, M. Sajjad Athar and S. K. Singh,Department of Atomic Energy Symp. on Nucl. Phys. 61, 664 (2016).

2. �Nuclear medium e�ects in the evaluation of Callan Gross relation",F. Zaidi, H. Haider, M. Sajjad Athar and S. K. Singh,Department of Atomic Energy Symp. on Nucl. Phys. 60, 686 (2015).

3. �E�ect of second class currents in the few GeV energy region",F. Akbar, F. Zaidi, S. Chauhan, M. Ra� Alam, M. Sajjad Athar and S. K.Singh,Department of Atomic Energy Symp. on Nucl. Phys. 60, 684 (2015).

4. �Nuclear medium e�ects in the evaluation of GLS and Adlers sum rules",F. Zaidi, H. Haider, M. Sajjad Athar and S. K. Singh,Department of Atomic Energy Symp. on Nucl. Phys. 59, 640 (2014).

5. �E�ect of lepton mass in neutrino induced quasielastic scattering",F. Akbar, F. Zaidi, S. Chauhan and M. Ra� Alam,Department of Atomic Energy Symp. on Nucl. Phys. 59, 660 (2014).

6. �Monte Carlo Generators vs Nuclear Model",S. Chauhan, F. Zaidi, H. Haider, M. Ra� Alam and M. Sajjad Athar,Department of Atomic Energy Symp. on Nucl. Phys. 56, 1104 (2011).

xxvi LIST OF PUBLICATIONS

Chapter 1Introduction

In the standard model of particle physics, leptons, quarks and mediating quantaare the fundamental constituents of matter and their interactions. Hadrons aremade up of quarks and are bounded through gluons. The quantum chromody-namics(QCD) is the theory which describes strong interactions in terms of quarksand gluons with remarkable features of asymptotic freedom at high energies(E) andmomentum transfer square(Q2) and con�nement at low energies and momentumtransfer square. At low E and Q2, the e�ective degrees of freedom to describe thestrong interactions are mesons and nucleons using e�ective Lagrangian motivatedby the symmetry properties of QCD while at high E and Q2, the quark and gluondegrees of freedom are used to describe the strong interactions using perturbativeQCD. Since leptons are point particles, therefore, they are ideal tool to probe thestructure of nucleons. Inclusive lepton-nucleon scattering is an important tool withwhich structure of the nucleon is explored. In inclusive scattering, either usingthe charged lepton beam l± + N → l± + X (l = e, µ) or the neutral lepton beamνl/ν̄l+N → l−/l++X (l = e, µ), the �nal state lepton is observed while the hadronic�nal state X remains undetected. In the scattering processes induced by the chargedleptons and neutrinos/antineutrinos on the nucleons and nuclei, the inclusive crosssections at low and intermediate energies are expressed in terms of the structurefunctions corresponding to the excitations of various resonances like ∆, N∗, etc.,lying in the �rst or higher resonance region, depending on the center of mass(CM)energy W of the �nal hadrons. While at high energies and Q2, the inclusive cross

1

2 CHAPTER 1. INTRODUCTION

sections are expressed in terms of the structure functions corresponding to the deepinelastic scattering(DIS) processes. In the intermediate energy region correspondingto the transition between resonance excitations and DIS, we are yet to �nd a methodbest suited to describe the inclusive lepton scattering. There is no sharp kinematicregion to distinguish the onset of the DIS region from the resonance region but theregion W ≥ 2.0 GeV and Q2 ≥ 1.0 GeV2 is considered to be the safe DIS region. Athigh energy and Q2, the inclusive DIS cross sections are usually expressed in terms ofthe structure functions which are derived in terms of the quark parton distributionfunctions(PDFs) using the methods of perturbative QCD. These structure functionsare experimentally determined from the DIS experiments on nucleon and nucleartargets.

In the case of electromagnetic(EM) interaction, various charged lepton inducedprocesses on free nucleons like elastic, inelastic(IE) which includes single pion pro-duction, multi-pion production, associated particle production, eta production, etc.,and deep inelastic scattering are possible. The basic reaction for these processes aregiven below

l±(k) +N(p) → l±(k′) +N(p′)l±(k) +N(p) → l±(k′) +R(p′)

→ N ′(pN ′) + mπ(pπ)︸︷︷︸m=1,2,3....l±(k) +N(p) → l±(k′) +K(pK) + Y (pY )

l±(k) +N(p) → l±(k′) +N(p′) + η(pη)l±(k) +N(p) → l±(k′) +X(p′),

where l = e, µ; N/N ′ = p, n; R represents the nucleon resonances(∆, N∗, ....), Ystands for hyperon(Σ, Λ) and X represents the jet of hadrons in the �nal state. Thequantities within the brackets correspond to the four momenta of the particles. TheFeynman diagrams corresponding to these processes are shown in Fig. 1.1. Fromthe Feynman diagrams, it may be noticed that the projectile beam emits a virtualphoton(γ∗) which is absorbed by the target nucleon and in the �nal state one obtainsthe charged lepton along with the hadrons. More than 50 years ago, scatteringexperiments were performed at SLAC using electron beam on liquid hydrogen andlater using liquid deuterium targets. The electron beam energies were upto 21 GeV.This was the beginning of the study of deep inelastic processes in electron-nucleon

3

l± l±

γ∗

NX

l± l±

γ∗

N N

l±

l±

RN

nπ

N ′

γ∗

l± l±

γ∗

N Y

K

l± l±

γ∗

N N

η

Figure 1.1: Feynman diagram for the electromagnetic interaction induced processesrepresenting(Left to Right) (i) Elastic scattering process, (ii) Inelastic scatteringprocesses(R≡ resonance; m=1,2,...; Y ≡ Λ,Σ) and (iii) Deep-Inelastic scatteringprocess on free nucleon target (X represents here jet of hadrons in the �nal state).

scattering. In Fig. 1.2, we have shown the di�erential scattering cross section(DCX)for the charged lepton induced processes vs center of mass energy W , which wasmeasured at SLAC [1] in the electron-proton scattering experiment at the beamenergy of E = 10 GeV and scattering angle θ = 6◦. It may be noticed from the�gure that in the region of low CM energy i.e. W ≈ 1 GeV there is a sharp peakwhich corresponds to elastic scattering. Beyond W = 1 GeV nucleon resonancescome into play with broader peaks and with the further increase in W the resultsfor the cross sections are similar to what one obtains in the case of electron-muonscattering, i.e. the cross section becomes independent of W and scales.

While showing the results of DCX in Fig. 1.2, elastic peak has been reduced by afactor of 8.5 in order to make it comparable with the results for inelastic and deepinelastic processes. The region of 1.4 GeV . W . 3 GeV is of great interest whereboth the DIS and several resonances like P33(1232), P11(1440), D13(1520), S11(1535),S11(1650), P13(1720), etc. contributes. Recently, at JLab [2] scattering experimentswere performed using electron beam on the liquid hydrogen target, in the region of0.2 GeV2 < Q2 < 5.5 GeV2. It has been found that the nucleon resonances havingmasses less than 2 GeV mainly give rise to three distinct enhancements. In theregion of low Q2(Q2 < 1 GeV2), there is contribution from the lowest mass state i.e.P33(1232), above Q

2 = 1 GeV2 there is contribution from the S11(1535) resonancewith some overlapping from the D13(1520) resonant state, and for the large valuesof Q2 dominant contribution comes from the F15(1680) with some more overlappingresonant states [2].


Figure 1.2: Double di�erential scattering cross section vs CM energy W (GeV) forthe process e + p → e + X at projectile energy of 10 GeV and scattering angle of6◦ [1]. The elastic peak has been reduced by a factor of 8.5 at W = MN .

The transition region between the resonance and the DIS processes has beendiscussed in literature, for example, by Melnitchouk et al. [3], Lalakulich et al. [4],Christy et al. [5], Mor�n et al. [6], etc. Moreover, in view of the extensive experimen-tal data on cross section and structure functions for the electromagnetic interactionavailable from JLab [2,7�10], SLAC [11�15], EMC [16�18] and NMC [19�21], a widerange of interest in studying the transition region from the resonance to the deep in-elastic scattering has got generated. This region is also known as quark-hadron(QH)duality region.

The phenomenon of QH duality �rst observed by Bloom and Gilman [22,23] morethan forty years ago in the electron scattering experiments from the proton target atSLAC, provides a connection between the low energy and the high energy descriptionof electron-proton scattering. According to this phenomenon the structure functionsin the low Q2 region of resonance excitation suitably averaged over an energy intervalis the same as the structure function at high Q2 region corresponding to DIS in thesame energy interval. It, therefore, establishes a connection between the quark-gluondescription of certain phenomenon at high Q2 in the region of DIS with the pionnucleon description of the same phenomenon at low Q2 in the region of resonanceexcitations. This seems to be valid in each resonance region individually as well asin the entire resonance region when the structure functions are summed over higher

5

resonances. This is termed as local duality.

When the phenomenon of local QH duality is also observed in the case of highermoments of the structure functions [3], it is termed as global duality. However, theseobservations are to be veri�ed by model calculations as well as by the experimentaldata when they become available with higher precision. Experiments were performednot only for the charged lepton scattering from the free nucleon target but alsofrom the nuclear targets(like carbon, calcium, aluminium, iron, tin, lead etc.), forexample at SLAC [12,15,24], NMC [19,21,25], JLab [7,26,27], E665 [28], EMC [18],etc. The phenomenon of local and global dualities has been observed in the presentexperiments on the electron-nucleon as well as electron-nucleus scattering [2, 7, 10,14, 15, 27, 29�39]. As more data become available on the inclusive lepton scatteringfrom the nucleons and nuclei with better precision, a veri�cation of QH dualitywith su�cient accuracy will provide a way to describe inclusive electron-nucleonscattering in the energy region of a few GeV. Therefore, it is important to understandwhat should be the lower cut o� region of Q2 and ν(energy transferred to hadrontarget) where DIS may be assumed to take care of the resonance contribution andbelow that there is smooth transition up to the contribution from a few low lyingresonances like P33(1232), N

∗(1440), etc.

Before the measurement by european muon collaboration(EMC) at CERN, it wasgenerally assumed that in the DIS region the structure functions of a free nucleonwould be the same as the structure functions of a bound nucleon. During the early1980s, EMC [17] performed scattering experiments using the muon beam in theenergy region of 120-280 GeV from the iron target and compared the results withthe results from the deuterium target. It was found that the ratio of the crosssection σFe

σDis not unity in the DIS region which was surprising as this is the region

where the underlying degrees of freedom should be the quarks and gluons, whilethe deviation from unity suggested that the nuclear medium e�ects were important.Later on, EMC type measurements were performed at SLAC [12], NMC [21, 25],HERMES [40], BCDMS [41, 42], JLab [43], etc. using several other nuclear targetsfor a wide range of Bjorken variable x and Q2. From these experiments it wasconcluded that

# structure functions in nuclear target due to nuclear medium e�ects show adi�erent behaviour from the free nucleon structure functions.

# the nuclear e�ects have x dependence and the e�ect may be divided into


four broad categories viz. shadowing e�ect(0 < x < 0.1), antishadowinge�ect(0.1 < x < 0.2), EMC e�ect(0.2 < x < 0.7) and the e�ect due to theFermi motion(x ≥ 0.7).

# the shape of the EMC e�ect does not change much with the nuclear massnumber A.

# functional form of the ratio σAσD

is relatively Q2 independent and

# the nuclear e�ect on the structure functions increases with the increase in massnumber A.

The four regions of x are brie�y described below

• Shadowing E�ect: In the region of low x, dominant contribution comesfrom the gluons that split into quark-antiquark pairs, known as the sea quarks.In this region because of the overlapping of target partons a suppression is

found in the ratio R′ = F2A(x,Q2)

F2D(x,Q2)and this depletion is known as the shad-

owing e�ect. However, this suppression becomes more pronounced with theincrease in mass number. In other words, because of the multiple scattering ofquarks, the destructive interference of amplitudes give rise to the phenomenonof shadowing. Shadowing e�ect is signi�cant at low x(0 < x < 0.1) and lowQ2 [44].

• Antishadowing E�ect: This is contrary to the shadowing e�ect and con-tributes in the region of 0.1 < x < 0.2. Antishadowing e�ect cause an enhance-

ment in the ratio R′ = F2A(x,Q2)

F2D(x,Q2)of structure functions due to the constructive

interference of amplitudes in the multiple scattering of quarks in the corre-sponding region of x. It is found that antishadowing e�ect has very littlenuclear mass dependence [45].

• EMC E�ect: The ratio of nuclear structure function to the free nucleonstructure function shows a dip in the region of 0.2 < x < 0.7 and this is knownas the EMC e�ect [17].

• Fermi Motion: The bound nucleons inside the nuclear target are neitherstationary nor free but constantly move with a �nite value of Fermi momentum(p ≤ pF ). This motion of nucleons corresponds to the Fermi motion. This is akinematic e�ect and is responsible for the abrupt rise in the ratio of structurefunctions in the region of higher x ≥ 0.7 [46].

7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9X

0.7

0.8

0.9

1

1.1

1.2

F2A

(x,Q

2 ) /

F2D

(x,Q

2 2)

HERMES: 14

N

NMC: 40

Ca

BCDMS: 56

Fe

SLAC: 56

Fe

EMC: 64

Cu

Shadowing

Antishadowing

EMC Effect

Fermi Motion

Figure 1.3: Ratio R′ = F2A(x,Q2)

F2D(x,Q2); (A =target nucleus) vs x represents the nu-

clear medium e�ects in structure function. Experimental data are taken from theRefs. [12, 21,40,41,47].

All these e�ects are shown in Fig. 1.3, where the results for F2A(x,Q2)

F2D(x,Q2)are pre-

sented from di�erent experiments [12, 21, 40, 41, 47]. Phenomenologically [48�52]as well as theoretically [6, 26, 27, 46, 53�88] various attempts have been made tounderstand these nuclear medium e�ects but no model is able to explain the ex-perimental data in the entire region of x and Q2. In the phenomenological stud-ies, e�orts have been made to obtain a nuclear correction factor to the structurefunctions by doing the analysis of the experimental data on the charged lepton-nucleus scattering, neutrino/antineutrino-nucleus scattering, pion-nucleus scatter-ing, proton-nucleus scattering, Drell-Yan processes, etc. In the theoretical studiesmany models have been proposed to study these e�ects on the basis of nuclearbinding, nuclear medium modi�cation including short range correlations in nu-clei [26,27,53�58,60�64,67,68,70,72�76,78�84], pion excess in nuclei [55,57,61,79�81],multi-quark clusters [82�84], dynamical rescaling [85,86], nuclear shadowing [87,88],etc.

In 2009, high precision experiments were performed at the JLab [43] using lightnuclear targets like 3He, 4He, 9Be and 12C and measurements were made in theregion of 0.3 < x < 0.9 and Q2 ≈ 3− 6 GeV2. It was found that there was deviation


from a simple logarithmic A or the dependence on average nuclear density. Theseresults were better explained if one assumes local density or the cluster structure ofthe nucleus. In spite of these e�orts a comprehensive understanding of the nuclearmodi�cations of nucleon structure functions valid for the entire region of x is stilllacking [26,46,58,60].

Recently, at the JLab [7] experiments have been performed in the energy rangeof 2-6 GeV, and precise measurements have been made for the nucleon structurefunctions FEM1N (x,Q

2), FEM2N (x,Q2) and longitudinal structure function FEML,N (x,Q

2)in the region of 1 GeV2 < W 2 < 4 GeV2 corresponding to low and moderate Q2 (0.5GeV2 < Q2 < 4.5 GeV2). These experiments are performed on several nuclear tar-gets like 12C, 27Al, 56Fe and 63Cu. JLab also plans to upgrade electron beam energyto 12 GeV [8, 89] and measure nucleon structure functions at low and moderate Q2

in the region of W relevant for the study of quark-hadron duality in di�erent nucleilike 12C, 63Cu and 197Au. Since most of these experiments are being performed usingnuclear targets, therefore, it is important to understand the nuclear medium e�ectsin the deep inelastic region specially at the low Q2 and ν where these measurementsare planned [9, 89].

In the weak sector, the requirement of the understanding of the nuclear mediume�ects is many fold. First of all, it must be pointed out that neutrino has importanceover charged lepton as it has an ability to interact with the particular quark �avorswhich would help to understand the partons distribution inside the target nucleon.Hence, precise determination of weak structure functions (FWIiA (x,Q

2); i = 1, 2, 3, L)is required. The weak interaction has vector - axial vector nature, therefore, in theregion of low Q2 the longitudinal structure function having axial vector compo-nent gives �nite contribution due to the axial part which is related to the partialconservation of axial vector current(PCAC). While in the case of electromagneticinteraction longitudinal structure function is zero as it has only vector part whichvanishes due to the conservation of vector current(CVC). Therefore, it is not ap-propriate to determine the weak structure functions using the measurements of theratio of longitudinal to transverse structure functions REML,A (x,Q

2) from the chargedlepton scattering. It hints towards an another important aspect that the mediummodi�cations for weak structure functions may be di�erent than the electromagneticstructure functions.

The nuclear dependence of REML,A (x,Q2) has also been a topic of much exper-

imental investigation in many experiments [12, 15, 90, 91] but no signi�cant nu-

9

clear dependence has been reported. The earlier experiments performed using neu-trino/antineutrino beam in the high energy region have used either light target orheavy nuclear targets [92�96] while present generation neutrino/antineutrino experi-ments are using medium and heavy nuclear targets [97�100]. The experiments usingneutrino/antineutrino beam require either high luminosity or gigantic targets be-cause neutrino/antineutrino interact very weakly. Experiments are being performedusing νl/ν̄l beams produced by the proton accelerators where the protons are bom-barded on some target material and pions and kaons, are collimated to travel alonga straight path and later they decay to neutrinos. Therefore, neutrino beams havewide energy range as compared to the charged lepton beams which are essentiallymonochromatic. The neutrino event rates are a convolution of energy dependentneutrino �ux, energy dependent cross section, and the energy dependent nucleare�ects. One of the major source of systematic errors in all these neutrino oscillationexperiments is due to the uncertainty in the neutrino-nucleus cross section arisingfrom the nuclear medium e�ects(NME) [6,101�103]. It is estimated that the lack ofwell understood neutrino/antineutrino-nucleus cross section leads to about 20-25%systematic uncertainty in event rates. In the few GeV energy region(1 < E < 3GeV) the contribution to the cross section comes from quasielastic(QE) scatter-ing, inelastic scattering as well as deep inelastic scattering processes. The MonteCarlo(MC) generators which are currently being used by the di�erent experimentersfor analyzing the data are GENIE [104], NuWro [105], NEUT [106], etc. GENIE usesBodek and Yang corrections [107,108] to take into account nuclear modi�cations inthe DIS region. Recently, MINERνA has measured cross section ratios in carbon,iron and lead nuclear targets [97] and the results for the ratios

[dσdx

]Fe/[dσdx

]CH

and[dσdx

]Pb/[dσdx

]CH

are not consistent with the predictions from the GENIE MC gen-erator. Therefore, it is important to study the nuclear medium e�ects in the DISprocesses in the energy region of a few GeV.

The basic reactions for neutrino/antineutrino scattering from the nucleon targetare as follows

νl(k)/ν̄l(k) +N(p) → l−(k′)/l+(k′) +N ′(p′)νl(k)/ν̄l(k) +N(p) → l−(k′)/l+(k′) +B(p′) ; B ≡ π(p′), K(p′)

ν̄l(k) +N(p) → l+(k′) + Y (p′)νl(k)/ν̄l(k) +N(p) → l−(k′)/l+(k′) + η(pη) +N ′(p′)νl(k)/ν̄l(k) +N(p) → l−(k′)/l+(k′) +X(p′),


νl/ν̄ll−/l+

W+/W−

N N ′

π,K, .....

N N ′

νl/ν̄ll−/l+

W+/W−

ν̄l l+

W−

N Y

νl/ν̄l l−/l+

W+/W−

N N ′

η

νl/ν̄ll−/l+

W+/W−

NX

Figure 1.4: Feynman diagram representing(Left to Right) (i) Quasielastic scatteringprocess, (ii) Resonant scattering processes(pion production, kaon production, singlehyperon production, eta production....; Y ≡ Λ,Σ), and (iii) Deep-Inelastic scatteringprocess.

The Feynman diagram for the above processes are shown in Fig. 1.4. For weakinteraction, charged-current induced processes are mediated by W+/W− which is avector boson having mass of 80.4 GeV/c2.

In Fig. 1.5, the total scattering cross sections for the neutrino and antineutrinoinduced processes are shown against the energy of the projectile beam and are com-pared with the results of earlier experiments [109]. From the �gure, it may beseen that QE process gives dominant contribution to the scattering cross section inthe energy region of less than 1 GeV, however, in the region of 1 GeV < E < 2GeV resonance processes make signi�cant contribution and beyond E > 3 GeVDIS dominates. Furthermore, it is important to emphasize that the intermediateenergy region(1 GeV < Eν < 3 GeV) where νl/ν̄l scattering processes lying be-tween the inelastic production of resonance excitation and onset of DIS process,known as transition region is not very well described theoretically either by thee�ective Lagrangian approach or by the DIS process (as it is considered low in en-ergy for application of quark parton model). The experimental results have largesystematic uncertainties which is due to the lack of the precise knowledge of theneutrinos interaction with matter, true energy of neutrinos, uncertainty in the �uxand limited detector e�ciency. Presently, many experiments have been performedwith improved knowledge of neutrino interactions and more detector accuracy likeT2K, MiniBooNE, MicroBooNE, SciBooNE, MINOS and many more are planned,for example, MINOS+, ArgoNEUT, MINERνA, NOνA, DUNE, etc. using di�erentnuclear targets like 12C, 16O, 40Ar, 56Fe and 208Pb.

11

Figure 1.5: Charged current induced neutrino(Left panel) and antineutrino(Rightpanel) scattering cross section per nucleon(for an isoscalar target) divided by neu-trino energy are plotted as a function of energy [109].

There are many theoretical calculations which take into account the nuclearmedium e�ects in QE scattering, one pion production process but there are veryfew calculations where nuclear medium e�ects in the DIS region for weak inter-action have been studied [64, 68, 73, 75, 110]. In the precision era of neutrino os-cillation physics it is important to understand the interplay of various nucleare�ects like Fermi motion, nuclear binding, nucleon correlations, meson exchangecurrents, nuclear shadowing, o� shell e�ects, etc. in the DIS process in the dif-ferent regions of Bjorken x and Q2. In the literature there also exists some phe-nomenological approaches where the nuclear modi�cations of structure functionshave been parameterized in terms of nuclear parton distributions functions (nPDFs),which are conventionally extracted from global QCD �ts to nuclear data mainlyfrom the charged lepton scattering experiments in the DIS region, proton-nucleusDrell-Yan (DY) data, and data from heavy ion collisions at colliders. Some ofthe analysis also include neutrino/antineutrino-nucleus scattering data. Recentlyneutrino/antineutrino-nucleus scattering data have been independently analyzed tounderstand nuclear medium e�ects in the weak sector. While analyzing these datatwo di�erent approaches are being followed. At the basic level, one takes a given setof free proton PDFs and convolute it with the nuclear correction factor RAi (x,Q0)


obtained from the global QCD �ts. This is done independently for each parton�avor. This approach has been adopted by Hirai et al.(HKN) [49, 111], Eskola etal.(EPS09 and EPPS16) [50, 112], DeFlorian et al.(DSSZ) [51], etc. In the anotherapproach, which is mainly adopted by the nCTEQ group [113], nuclear parton dis-tribution functions �t is performed independent of any nucleon PDFs. Furthermore,the nCTEQ group has made independent analysis of the electromagnetic and weaknuclear structure functions and found them to be di�erent, unlike the assumptionsmade by Hirai et al.(HKN) [49,111], Eskola et al.(EPS09 and EPPS16) [50,112] andothers, who take the same nuclear correction factor.

In this thesis, we have studied the nuclear medium e�ects in the DIS regioncorresponding to the Q2 and x relevant for the ongoing measurements both in theelectromagnetic sector as well as the weak sector. This study has been performedwith a microscopic model which uses relativistic nucleon spectral function to de-scribe target nucleon momentum distribution incorporating Fermi motion, bindingenergy e�ects and nucleon correlations in a �eld theoretical model. The spectralfunction that describes the energy and momentum distribution of the nucleons innuclei is obtained by using the Lehmann's representation for the relativistic nucleonpropagator and nuclear many body theory is used to calculate it for an interactingFermi sea in the nuclear matter [114]. A local density approximation is then appliedto translate these results to a �nite nucleus. Furthermore, we have considered thecontributions of the pion and rho mesons in many body �eld theoretical approachbased on Refs. [61, 115]. Due to the fact that JLab data [7, 8] have been taken ina region of relatively low and moderate Q2 we have not assumed the Bjorken limit.We have incorporated target mass correction (TMC) following Ref. [116]. TMC ise�ective at moderate Q2 and high x. The calculations are performed at the lead-ing order(LO) as well as next-to-the leading order (NLO). The nucleon PDFs havebeen taken from the works of CTEQ group [117]. To understand the role of partondistribution functions in the evolution of structure functions, we have used di�erentPDFs parameterizations like CTEQ6.6 [117], MSTW [118], GRV [119], CT14 [120]and compared the results. The NLO evolution of the deep inelastic structure func-tions has been taken from the works of Vermaseren et al. [121] and van Neerven andVogt [122]. Furthermore, we have studied the NLO evolution using the approachgiven by Furmanski et al. [123] and compared the results of structure functions withthe results obtained using the approach given in Refs. [121,122] as well as we brie�ydiscuss the structure functions at NNLO. In the case of pions we have taken thepionic parton distribution functions given by Gluck et al. [124, 125]. We have alsodiscussed the e�ect of other pion PDFs available in the literature [126�129]. For the

13

rho mesons, we have applied the same PDFs as for the pions as in Ref. [61]. Wehave also considered the e�ect of shadowing following Ref. [64].

In the chapter 2, we have studied the charged lepton-nucleon and neutrino/anti-neutrino-nucleon deep inelastic scattering processes and discussed the electromag-netic and weak structure functions of nucleon. We have studied

# the e�ect of di�erent PDFs parameterizations on the nucleon structure func-tions.

# the e�ect of next-to-leading order terms on the structure functions and madea comparison with the results at the leading order.

# two di�erent approaches of NLO evolution and the results for the structurefunctions are compared.

# the e�ect of target mass corrections on the structure functions.

In the chapter 3, the charged lepton-nucleus and neutrino/antineutrino-nucleusDIS processes have been discussed in detail. We have evaluated the nuclear structurefunctions both in the electromagnetic and weak interaction processes by taking

# the nuclear medium e�ects like Fermi motion, binding energy, nucleon corre-lations through the use of spectral function. For the numerical calculations,we have used CTEQ6.6 nucleon PDFs [117].

# the mesonic contribution also has been taken into account in a �eld theoreticalapproach by introducing dressed meson propagators instead of the spectralfunction. For pion PDFs we have used the parameterizations of Gluck etal. [124], Wijesooriya et al. [126], Sutton et al. [127], Martin et al. [128] andConway et al. [129]. Furthermore, shadowing e�ect has also been incorporatedfollowing the works of Kulagin and Petti [64].

# the nonisoscalarity e�ect for the heavy nuclear targets have been explicitlyconsidered and comparisons are made if one treat the nucleus to be isoscalar.

Using the formalism developed in the chapters 2 and 3, we have presented theresults in chapter 4


# the results for the electromagnetic and weak nuclear structure functions viz.FEM1A (x,Q

2), FEM2A (x,Q2) and FWI2A (x,Q

2) and made a comparative study ofthese structure functions.

# the numerical results at the leading order(LO) as well as at the next-to-leadingorder(NLO) for a wide range of x and Q2.

# the di�erence between the nucleon PDFs [117] and the nuclear PDFs [130]parameterizations.

# the numerical results for the structure functions and compared the resultswith the experimental data of JLab [7], SLAC [11, 12, 15, 26], EMC [18],CDHSW [93], CCFR [94] and NuTeV [96] experiments.

In the chapter 5, we have studied

# the validity of Callan-Gross relation in the nuclear medium as well as thetransition region between resonances and deep inelastic processes. This maygive information about the phenomenon of quark-hadron duality.

# the results forR2A(x,Q2) =F2A(x,Q

2)2xF1A(x,Q2)

, FL,A(x,Q2) =

(1 +

4M2Nx2

Q2

)F2A(x,Q

2)−2xF1A(x,Q

2) and RL,A(x,Q2) =

FL,A(x,Q2)

2xF1A(x,Q2)using the formalism presented in

chapters 2 and 3. These results are compared with the parameterization ofWhitlow et al. [131,132] for the free nucleon.

# the numerical results for RL,A(x,Q2) are also compared with the available ex-perimental data of JLab [7], SLAC [15], BCDMS [133], CDHSW [134], CHO-RUS [135] and CCFR [136, 137]. Furthermore, the results for R2A(x,Q

2) andFL,A(x,Q

2) are compared with the experimental data of the JLab [7].

In the chapter 6, we summarize the �ndings of this thesis and discuss the futureplan.

Chapter 2Lepton-Nucleon Scattering

2.1 Deep Inelastic Scattering

In this chapter, we study the nucleon structure functions for the electromagnetic(EM)and weak interactions(WI) induced inclusive deep inelastic scattering processes. Foran inclusive process, we observe the �nal state lepton regardless of the �nal statehadronic system. As the hadronic system contains number of particles interactingamong themselves via strong interaction, therefore, it would be di�cult to under-stand the dynamics of this state. However, through structure functions of nucleonthat gives information about the parton probability density, one may determine thescattering cross section without knowing the details of �nal state hadronic system.For the numerical calculations we use the parton distribution functions(PDFs) pre-scribed by the CTEQ group [117] as well as compared the results with the resultsobtained by using MSTW [118], GJR [119] and CT14 [120] parameterizations. Fur-thermore, we incorporate the target mass correction(TMC) e�ect which is importantat high x and moderate Q2 regions. To incorporate TMC e�ect we have followed theprescription of Schienbein et al. [116]. The numerical evolution of structure func-tions have been made at the next-to-leading order(NLO) following the approach ofVermaseren et al. [121] and van Neerven and Vogt [122]. We have also performedthe NLO evolution following another approach given by Furmanski et al. [123] andcompare the obtained results with the results of earlier approach. We will discuss

15

16 CHAPTER 2. LEPTON-NUCLEON SCATTERING

l∓(k)

l∓(k′)

γ∗(q)

N(p)X(p′)

qqq

Figure 2.1: Feynman diagram for the charged lepton induced deep inelastic scatter-ing process.

the formalism of the charged lepton induced deep inelastic scattering in section 2.2and neutrino/antineutrino induced DIS in section 2.3 on free nucleon target. In sec-tion 2.4, NLO corrections and TMC e�ect are discussed in brief. In section 2.5, wepresent the results of free nucleon structure functions for both the electromagneticand weak interaction induced processes showing their variations for the di�erentPDFs parameterizations, target mass correction e�ect and NLO evolution using thetwo di�erent approaches. Furthermore, we have studied the e�ect of NNLO termsfollowing Ref. [122] on the nucleon structure functions for which the results are alsopresented.

2.2 Deep Inelastic Charged Lepton-Nucleon Scat-

tering

The basic reaction for the charged lepton induced deep inelastic scattering processis given by

l∓(k) +N(p)→ l∓(k′) +X(p′) ; l = e, µ, (2.1)

which is shown in Fig. 2.1. The incoming lepton beam of four momentum k(= (E,k))when interacts with the target nucleon with four momentum p(= (EN ,p)) in thelaboratory frame, it produces a jet of hadrons X along with a lepton in the �nalstate having four momenta p′(= (E ′N ,p

′)) and k′(= (E ′,k′)), respectively. This

2.2. DEEP INELASTIC CHARGED LEPTON-NUCLEON SCATTERING 17

l∓(k)

l∓(k′)

γ∗(q)−ie

γ∗(q)

N(p)X(p′)

qqq

−ie

Figure 2.2: Feynman representation for leptonic and hadronic vertices in the caseof electromagnetic interaction.

process occurs via the exchange of virtual photon(γ∗) carrying momentum transferq which is de�ned as

q = k − k′ = p′ − p

q2 = k2 + k′2 − 2k · k′ ' −4EE ′sin2(θ

2

); neglecting lepton mass,

where θ is the lepton scattering angle in the laboratory frame. For l∓ − N deepinelastic scattering process, the di�erential scattering cross section is written as [138]

d2σEMNdΩ′dE ′

=1

16π2|k′||k| |M|

2, (2.2)

where dΩ′ is the solid angle and |M|2 is the invariant matrix element square whichis given by

|M|2 = e4

q4Lµν W

µνN , (2.3)

In the above expression, e =√

4πα is the coupling strength of the electromagneticinteraction at the leptonic and hadronic vertices(shown in Fig. 2.2) with �ne struc-ture constant α(= 1

137). As the matrix element is a Lorentz invariant quantity,

therefore, both the leptonic and hadronic tensors must be Lorentz invariant. Beingthe point like particles, the interaction of leptons is well understood while due to


the extended structure of the nucleon, hadronic vertex is unknown. In Eq. 2.3, thespin averaged leptonic tensor Lµν which describes the interaction of participatinglepton with the intermediate vector boson is given by

Lµν = 2(kµk′ν + k

′µkν − k · k′gµν), (2.4)

and W µνN is the hadronic tensor that we de�ne to parameterize our ignorance aboutthe hadronic vertex. Hadronic tensor must be Lorentz invariant and generally de-�ned in terms of nucleon structure functions WEMiN (ν,Q

2); (i = 1 − 6) with metrictensor gµν , four momentum p and four momentum transfer q as

W µνN = −gµν WEM1N (ν,Q2) +pµpν

M2NWEM2N (ν,Q

2)

−i�µνλσ pλqσ2M2N

WEM3N (ν,Q2) +

qµqν

M2NWEM4N (ν,Q

2)

+(pµqν + pνqµ)

M2NWEM5N (ν,Q

2) + i(pµqν − pνqµ)

M2NWEM6N (ν,Q

2) , (2.5)

where MN is the mass of target nucleon. The nucleon structure functions arethe functions of energy transfer ν = k0 − k′0 and four momentum transfer squareQ2 = −q2 ≥ 0. As the nucleon structure functionWEM3N (ν,Q2) arises due to the par-ity violation, therefore, the term with WEM3N (ν,Q

2) would not contribute in the elec-tromagnetic interaction. Furthermore, for unpolarized scattering, the antisymmet-ric term that is related to WEM6N (ν,Q

2) vanishes when contracted with the leptonictensor which has symmetric terms only. Therefore, while contracting the leptonictensor(Eq. 2.4) with the hadronic tensor(Eq. 2.5), the terms with nucleon structurefunctionsWEM3N (ν,Q

2) andWEM6N (ν,Q2) becomes zero which may be mathematically

expressed as:

Lµν

[i�µνλσ

pλqσ2M2N

]−→ 0 ,

Lµν

[i(pµqν − pνqµ)

M2N

]−→ 0. (2.6)

Moreover, on applying the conservation of vector current(CVC) at the hadronicvertex, i.e.

qµ WµνN = 0,


WEM4N (ν,Q2) and WEM5N (ν,Q

2) may be expressed in terms of WEMiN (ν,Q2); i = 1, 2,

i.e.

WEM4N (ν,Q2) =

M2Nq2

WEM1N (ν,Q2) +

(p · qq2

)2WEM2N (ν,Q

2), (2.7)

WEM5N (ν,Q2) =

−p · qq2

WEM2N (ν,Q2). (2.8)

Therefore, one is left with the two independent structure functions, namelyWEM1N (ν,Q2)

and WEM2N (ν,Q2) and the expression of W µνN is written as

WµνN =

(qµqν

q2− gµν

)WEM1N (ν,Q

2) +

(pµ − p.q

q2qµ)(

pν − p.qq2

qν)WEM2N (ν,Q

2)

M2N(2.9)

WEMiN (ν,Q2); i = 1, 2 are in general expressed in terms of the dimensionless nucleon

structure functions

MNWEM1N (ν,Q

2) = FEM1N (x,Q2),

νWEM2N (ν,Q2) = FEM2N (x,Q

2),

(2.10)where x is the fraction of the incoming nucleon momentum carried by the partonand is given by

x =Q2

2MNν=

Q2

2MN(E − E ′).

However, in the in�nite momentum frame, where energy transfer and square of fourmomentum transfer is very large, i.e.

ν →∞ ; Q2 →∞,

with �nite value of the ratio Q2

2MNν, these dimensionless nucleon structure functions

become the function of dimensionless variable x known as Bjorken variable i.e.

FEM1N (x,Q2)

Q2→∞, ν→∞−−−−−−−−→x→finite

FEM1N (x)

FEM2N (x,Q2)

Q2→∞, ν→∞−−−−−−−−→x→finite

FEM2N (x)

(2.11)


and satisfy the well known Callan-Gross Relation(CGR) [139]

2xFEM1N (x) = FEM2N (x). (2.12)

These dimensionless nucleon structure functions are written in terms of the par-ton distribution functions which are de�ned as momentum distribution function ofpartons within the nucleon. In other words, the probability density of �nding aparton carrying momentum fraction x is known by the term PDFs. For the chargedlepton-nucleon scattering, when the virtual photon interacts with the quark, for thatshort duration quark behaves like a free particle. Therefore, the cross section for thecharged lepton-nucleon scattering process would be the incoherent sum of interac-tion probabilities of photon-quark scattering. Hence, one may de�ne the structurefunctions in terms of parton distribution functions having momentum fraction x as

FEM2N (x) = 2xFEM1N (x) =

∑i

e2i x [qi(x) + q̄i(x)] , (2.13)

where index i runs over the �avor of quarks, ei is the charge of corresponding quarkor antiquark and xqi(x)/xq̄i(x) is the probability of �nding a quark/antiquark insidethe nucleon carrying a momentum fraction x of the momentum of target nucleon.For the electromagnetic interaction, PDFs for a proton and neutron target are givenby [140]

F ep2 (x) =4 x

9(uv(x) + us(x) + ūs(x) + cs(x) + c̄s(x) + ....)

+x

9

(dv(x) + ds(x) + d̄s(x) + ss(x) + s̄s(x) + ....

)F en2 (x) =

x

9(uv(x) + us(x) + ūs(x) + ss(x) + s̄s(x) + ....)

+4 x

9

(dv(x) + ds(x) + d̄s(x) + cs(x) + c̄s(x) + ....

),

where uv(x) = u(x) − us(x) and dv(x) = d(x) − ds(x) are the valence quarksand the subscript s stands for the sea quarks. If we consider only four �avorsof quarks/antiquarks for an isoscalar target, where N = (p+n)

2, the nucleon structure

function will take the form

F eN2 (x) =5

18x

[u(x) + ūs(x) + d(x) + d̄s(x) +

8

5{cs(x) + c̄s(x)}

+2

5{ss(x) + s̄s(x)}

]. (2.14)


As a result of intensive experimental collaboration to understand DIS process viz.at SLAC, CERN, Fermilab, DESY, etc., parton distribution functions has been ob-tained for a wide kinematical range of x and Q2. The parton distribution functionsfor the nucleon are also proposed by various phenomenological groups like Coor-dinated Theoretical-Experimental Project on QCD(CTEQ) [117]; Martin, Stirling,Thorne and Watt(MSTW) [118]; Gluck, Jimenez and Reya(GJR) [119]; Gluck, Reyaand Vogt(GRV) [141]; Jimenez and Reya(JR) [142]; Alekhin, Blumlein, Moch andPlacakyte (ABMP) [143]; and HERA [144] etc. These PDFs are obtained by usingthe data from the deep inelastic lepton-nucleon scattering, proton-proton scattering,proton-nucleus scattering, pion-nucleus scattering etc. processes. A few groups hasalso proposed nuclear PDFs, for example, CTEQ collaboration recently proposed thenuclear parton distribution functions nCTEQ15 [113] and CTEQ-Je�erson Lab(CJ)Collaboration proposed CJ12 and CJ15 by using the global QCD �ts [130,145]. Wehave discussed some of the nucleon and nuclear PDFs in Appendix A.

Now, by using Eqs. 2.3, 2.4, 2.5 and 2.10 in Eq. 2.2, the di�erential scatteringcross section is obtained in terms of the dimensionless nucleon structure functionsas


=4α2E ′2 cos2

(θ2

)q4MNν

[2νFEM1N (x,Q

2) tan2(θ

2

)+MNF

EM2N (x,Q

2)

], (2.15)

The di�erential scattering cross section given in Eq. 2.15 may also be expressed,with respect to the Bjorken variable x and inelasticity y = p·q

p·k =νEwhich signi�es

that how much energy is transferred from lepton to the target nucleon, by using therelation

dΩ′dE ′ =2πMNEy

E ′dx dy , (2.16)

which results

d2σEMNdxdy

=8MNEπα

2

q4

{xy2FEM1N (x,Q

2) +

(1− y − xyMN

2E

)FEM2N (x,Q

2)

}.(2.17)

The dimensionless nucleon structure functions FEM1N (x,Q2) and FEM2N (x,Q

2) are alsorelated to the photoabsorption cross sections. In the case of charged lepton-nucleon


DIS process that occurs via the exchange of virtual photon, it becomes importantto keep in mind that the virtual photon has polarization states which should alsobe taken into account. Since, the virtual photon has both longitudinal and trans-verse polarization states, therefore, the total scattering cross section is consideredto be the sum of longitudinally and transversely polarized virtual photon absorp-tion cross section. However, in the case of real photons that have only transversepolarization, the contribution from longitudinal polarization would be zero. Hencethe total scattering cross section for the interaction of virtual polarized photon withthe unpolarized proton target is given by [140]

σλ =4π2α

K�λµ�

λν

∗W µνN , (2.18)

where �λµ is the photon polarization vector with helicity λ = 0 for the longitudinalstate, λ = ±1 for transverse states, respectively. The factor K is de�ned as theenergy required to create the �nal hadronic state X and is given by

K =W 2 −M2N

2MN,

with

W =√W 2 =

√(p+ q)2 =

√M2N −Q2 + 2MN(E − E ′)

as the invariant mass of that state and the hadronic tensor W µνN is given by Eq. 2.9.Since virtual photon has longitudinal as well as transverse polarization states withhelicity 0 and ±1, respectively, therefore by using Eq. 2.18, we may write the trans-verse and longitudinal scattering cross sections in terms of the nucleon structurefunctions as [140]

σEMT (x,Q2) =

σ(λ = +1) + σ(λ = −1)2

=4π2α

KMNFEM1N (x,Q

2) (2.19)

σEML (x,Q2) = σ(λ = 0) =

4π2α

KMN

[(1 +

ν2

Q2

)MNν

FEM2N (x, Q2)

− FEM1N (x, Q2)]

(2.20)


Hence, the di�erential scattering cross section given in Eq. 2.15 is written as


=

(αE ′

4π2Q2MNE(1− �)

)[σEMT (x,Q

2) + �σEML (x,Q2)]

=

(αE ′

4π2Q2MNE(1− �)

)σEMT (x,Q

2)[1 + �REML,N (x,Q

2)], (2.21)

where

REML,N (x,Q2) =

σEML (x,Q2)

σEMT (x,Q2)

(2.22)

The above expression of REML,N (x,Q2) is the ratio that de�nes the probability of

longitudinally polarized to transversely polarized photon absorption. The virtualphoton polarization parameter � is given by

� =

[1 + 2

(1 +

ν2

Q2

)tan2

θ

2

]−1=

1− y − M2Nx2y2Q2

1− y + y22

+M2Nx

2y2

Q2

(2.23)

with the limits 0 < � < 1. Moreover, by using Eqs. 2.19 and 2.20, one may write thenucleon structure functions FEM1N (x, Q

2) and FEM2N (x, Q2) in terms of σEML (x,Q

2)and σEMT (x,Q

2) as

FEM1N (x, Q2) =

KMN4π2α

σEMT (x,Q2)

FEM2N (x, Q2) =

Kν

4π2α

(1 +

ν2

Q2

)−1[σEML (x,Q

2) + σEMT (x,Q2)] (2.24)

and it may be noticed that FEM1N (x,Q2) has contribution only from transversely

polarized photons while the contribution to FEM2N (x,Q2) comes from both longi-

tudinally as well as transversely polarized photons. Therefore, in order to obtainthe contribution from the longitudinal part only, one may de�ne the longitudinal


structure function FEML,N (x,Q2) as

FEML,N (x,Q2) =

2xν(1− x)MN4π2α

σEML (x,Q2),

FEML,N (x,Q2) = γ2FEM2N (x,Q

2)− 2xFEM1N (x,Q2), (2.25)

where γ2 =(

1 +4M2Nx

2

Q2

). However, FEML,N (x,Q

2) is zero in the naive parton model

because photons with the longitudinal polarization do not couple to the quarkshaving spin 1

2. But in the case of QCD improved parton model because of the quark-

quark interaction via gluon, quark may couple with the longitudinally polarizedphotons that results a direct relationship between FEML,N (x,Q

2) and gluon distributionin the target nucleon. Therefore, the longitudinal structure function has signi�cantimportance in order to evolve the gluon distribution. A separate determination ofthe longitudinal and the transverse structure functions are very di�cult.

The ratio REML,N (x,Q2) de�ned in Eq. 2.22 may be rede�ned in terms of structure

functions as

REML,N (x,Q2) =

σEML (x,Q2)

σEMT (x,Q2)

=FEML,N (x,Q

2)

2xFEM1,N (x,Q2)

REML,N (x,Q2) =

(1 +

4M2Nx2

Q2

)REM2,N (x,Q

2)− 1 (2.26)

with

REM2,N (x,Q2) =

FEM2N (x,Q2)

2xFEM1N (x,Q2). (2.27)

Assuming CGR in Eq. 2.26, the ratio would be

REML,N (x,Q2) =

4M2Nx2

Q2(2.28)

which is referred as �CGR limit� which tends to zero as Q2 goes to in�nity, i.e.

REML,N (x,Q2)→ 0 as Q2 →∞. (2.29)

2.3. DEEP INELASTIC CHARGED CURRENT νL/ν̄L-NUCLEON SCATTERING 25

νl/ν̄l(k)

l−/l+(k′)

W+/W−(q)

N(p)X(p′)

qqq

Figure 2.3: Feynman diagram for the neutrino/antineutrino induced deep inelasticscattering process.

A �nite value of the ratio REML,N (x,Q2) was measured in electron scattering experi-

ments at SLAC [132] which is an evidence of spin 1/2 nature of partons. Anotherpossibility of REML,N (x,Q

2) to be nonzero is that quarks may have a �nite value of thetransverse momentum component originated from the QCD.

Till now we have discussed the formalism only for the charged lepton-nucleondeep inelastic scattering process. In the next section, we will discuss the formalismfor the neutrino/antineutrino induced DIS scattering process from the free nucleontarget.

2.3 Deep Inelastic Charged Current νl/ν̄l-Nucleon

Scattering

The charged current deep inelastic scattering process for the weak interaction(shownin Fig. 2.3)

νl/ν̄l(k) +N(p)→ l−/l+(k′) +X(p′); l = e, µ (2.30)takes place via the exchange of vector bosonW+/W− when neutrinos/antineutrinosbeam interact with a target nucleon(N) in the laboratory frame and emits a lepton(l)in the �nal state with a jet of hadrons(X). For the above reaction, the quantiti

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