Introduction

In the last chapter, we have discussed how to calculate the cross sectionsfor the scattering of two point-like particles. Now the question arises,what happens when an electron interacts with a charge which is distributedin space like the one shown in Figure 10.1. The standard technique tomeasure the charge distribution and get information about the structure ofthe hadron is to measure the differential/total scattering cross sections ofelectron with a hadron and compare it with the cross section of electronscattering with a spinless (J = 0) point target (known asMott scattering cross section). The ratio of these two is generallyexpressed as

where F(q2), in literature, isknown as the form factor. This accounts for the spatial extent of thescatterer. F(q2) not only tellsabout the distribution of the charge in space but using it, one can estimatethe size of the target particle as well as its charge distribution anddensity of magnetization. Thus, for an extended charge distribution, theprobability amplitude for a point-like scatterer is modified by a formfactor.

Physical Significance of the Form Factor

Consider the elastic scattering of a “spinless” electron from astatic “spinless” point object having charge. In the Bornapproximation, where the perturbation is assumed to be weak, the scatteringamplitude is written as

where and are the wave functions of the initial and final electron withmomentum and, respectively. These waves are assumed to be plane waves suchthat

Instead of a point charge distribution, if we assume an extended chargedistribution with normalization, then the potential felt by the electronlocated at is given by

where is the maximum range of the charge distribution. The scatteringamplitude modifies to

Assuming, which leads to,

The term in the square brackets on the right-hand side of Eq. (10.6) is knownas the form factor, which is nothing but the Fourier transform of the chargedensity distribution, given as

In field theory, if we consider the scattering of a spin electron from anexternal electromagnetic field (shown in Figure 10.2), the electromagneticfield in the momentum space is written as

Introduction

We have discussed charged lepton and (anti)neutrino induced deep inelasticscattering (DIS) off free protons in Chapter 13. The interaction crosssection and the structure functions of nucleons are modified when thescattering takes place from the nucleons bound inside a nucleus. Thereactions are shown in Figure 16.1, where l =e, μ. A is thetarget nucleus and X is the jet of hadrons are the fourmomenta of the initial and the final state particles respectively.Historically, the first observations of modifications of the nuclearstructure

functions were made by the European Muon Collaboration (EMC) at CERN in1981–83. The EMC collaboration studied the ratio of structurefunction F2(x,Q2) per nucleon for iron to deuteriumtargets, that is in the energy region of 120-280 GeV and its deviation fromunity. This effect is known as the EMC effect. Since then, the EMC effecthas been confirmed and studied with improved precision in many DISexperiments using electrons, muons [783, 784, 785, 786, 787, 788],neutrinos, and antineutrinos [789, 790, 791, 792, 793, 794] from differentnuclear targets as well as in the Drell–Yan processes using protonand pions [795, 796, 797]. Some of these results are presented in Figure16.2. From the figure, it may be observed that the ratio is different fromunity in almost the entire region of the Bjorken scaling variable 0< x < 1. From these experiments, somegeneral features of the ratio R(x,Q2) may be inferred:

• • The x dependence ofR(x,Q2) has considerable structure, thatis, it is different in different regions of x.

• • The shape of the effect is almost independent ofA.

• • The functional form ofR(x,Q2) is relatively independent in theregion of high Q2.

Generally, the nuclear medium effects manifested through the ratioR(x, Q2)are broadly divided into four regions of x in which thex dependence is attributed to different physicaleffects.

Contraction of Leptonic Tensors in ElectromagneticInteractions

For the scattering discussed in Chapter 9, the transition matrix elementsquared is written as

where is the momentum transfer and the factor of is for the averaging overthe initial electron and muon spins.

The leptonic current is given by

In Eq. (D.2),

• • Adjoint Dirac spinor is a 1 × 4matrix,

• • Dirac spinor (u) is a 4× 1 matrix,

• • γμ is a 4× 4 matrix,

• • Ultimately, we have (1 × 4)(4× 4)(4 × 1) =A, a number,

• • For any number A, the complex conjugate andthe Hermitian conjugate are the same thing.

Therefore, instead of, we may write

we can rewrite the aforementioned expression in the component form for anelectronic tensor as:

where we have used the trace properties,

The trace of an odd number of gamma matrices is zero. Similarly,

Using Eqs. (D.3) and (D.4), we get

Contraction of Leptonic Tensors in the Case of Weak Interactions

For the scattering discussed in Chapter 9, where the interaction is mediatedby a W boson, the transition matrix element squared is expressed as

where the factor of is for the averaging over the initial muon spin.

Leptonic tensor,

Therefore,

We can rewrite this expression in the component form as:

Similarly, for the muonic tensor,

Using Eqs (D.8) and (D.9), the transition matrix element squared is obtainedas

Contraction of Weak Leptonic Tensor with Hadronic Tensor

Contracting the various terms of hadronic tensor with the leptonic tensor, weget

Where

The need for writing a self-contained comprehensive book on the physics ofneutrino interactions had been in our minds for a long time, while teachingvarious graduate courses in high energy physics and nuclear physics andconducting research in the field of neutrino physics at the Aligarh MuslimUniversity. We also realized the need for such a book while attending manytopical workshops, conferences, and short-term schools like NuFact, NuInt,NuSTEC, etc., held in the USA, Europe, Japan, and elsewhere in the area ofneutrino physics and while responding to questions asked by the youngresearchers in many formal and informal discussions. The aforementionedscientific events bring together research students and senior scientistsworking on various aspects of neutrino physics common to nuclear physics,particle physics, and astrophysics, which make the subjectinterdisciplinary. In recent times, the research activity in the field ofneutrino physics, around the world, and its applications in the other areasof physics has attracted a large number of students to this field. It was,therefore, felt that this is an appropriate time to write a book on thephysics of neutrino interactions focusing on introducing the basicmathematical and physical concepts and methods with the help of simpleexamples to illustrate the calculation of various neutrino processesrelevant for applications in particle physics, nuclear physics, andastrophysics, for the benefit of all those interested in learning thesubject.

The main aim of the book is to present a pedagogical account of the physicsof neutrino interactions, with balance among its theoretical andexperimental aspects, for describing various neutrino scattering processesfrom leptons, nucleons, and nuclei used in studying neutrino properties likeits mass, charge, magnetic moment, and the newly discovered phenomenon ofneutrino mixing and oscillations. The book is intended primarily forgraduate students and young post-doctoral research scientists working inneutrino physics but it can also be used by advanced undergraduates who havesome exposure to basic courses in special theory of relativity, quantummechanics, nuclear physics, particle physics, and are interested in neutrinophysics.

Expression of N(q2) inTerms of Mandelstam Variables

The expression of N(q2) isexpressed in terms of the Mandelstam variables and the form factors as:

where (+)− sign represents the (anti)neutrino inducedscattering and the Mandelstam variables are defined as

with

F.2 Expressions ofAh(q2),Bh(q2),andCh(q2)

The expressions Ah(q2), Bh(q2), and Ch(q2) are expressed in terms of theMandelstam variables and the form factors as:

Expressions of and

The expressions Al(q2),Bl(q2), andCl (q2) areexpressed in terms of the Mandelstam variables and the form factors as:

Introduction

The inelastic scattering processes of (anti)neutrinos from nucleons arerelevant in the region starting from the neutrino energy corresponding tothe threshold production of a single pion. For neutral current (NC) induced1π production, this starts at Ethv(⊽) = 144.7 MeV forvl reactions. In the case of charged current(CC) induced 1π production, the threshold energy ishigher because of the massive leptons produced in the final state; itcorresponds to Ev (⊽)≥ 150.5 MeV (277.4 MeV) for ve(nm) reactions. As the neutrino energy increases,inelastic processes of multiple pion production, viz.,2π, 3π, etc., and theproduction of strange mesons (K) and hyperons(Ⲩ) start; both of which are the most relevantinelastic processes in the region of a few GeV. These inelastic processeshave been studied very extensively, both theoretically and experimentally,in various reactions induced by photons and electrons which probe theinteraction of the electromagnetic vector currents with hadrons in thepresence of other strongly interacting particles like mesons and hyperons.In weak processes induced by neutrinos and antineutrinos, the inelasticprocesses provide a unique opportunity to study the interaction of the weakvector as well as the axial vector currents with hadrons in the presence ofstrongly interacting particles like mesons and hyperons. Moreover, a studyof these weak processes from nucleons and nuclei is of immense topicalimportance in the context of the present neutrino oscillation experimentsbeing done with the accelerator and atmospheric neutrinos in the energyregion of a few GeV. The specific reactions to be studied in the inelasticchannels are the various processes induced by the charged and neutral weakcurrents of neutrinos and antineutrinos, given in Table 11.1.

The first four reactions in Table 11.1 are strangeness conserving (∆S= 0) reactions and the last one is a strangeness changing (∆S = 1)reaction. The generic Feynman diagrams describing these reactions are shownin Figures 11.1(a) and 11.1(b), wherevl(⊽l) andl-(l+)

are leptons interacting through theW±(Z) exchanges withthe nucleon (N) producing the final nucleon(N') and hyperons(Ⲩ) and mesons like pions(π) and kaons (K).

Introduction

The concept of associating particles with fields originated during the studyof various physical phenomena involving electromagnetic radiation. Forexample, the observations and theoretical explanations of the black bodyradiation by Planck, the photoelectric effect by Einstein, and thescattering of a photon off an electron by Compton established thatelectromagnetic radiation can be described in terms of “discretequanta of energy” called photon, identified as a massless particle ofspin 1. Consequently, Maxwell's equations of classicalelectrodynamics, describing the time evolution of the electric and magneticfields are interpreted to be the equations of motion of the photon, writtenin terms of the massless spin 1 electromagnetic field. Later, thequantization of the electromagnetic field was formulated to explain theemission and absorption of radiation in terms of the creation andannihilation of photons during the interaction of the electromagnetic fieldwith the physical systems. The concept of treating photons as quanta of theelectromagnetic fields was successful in explaining the physical phenomenainduced by the electromagnetic interactions; methods of field quantizationwere used leading to quantum electrodynamics (QED), the quantum field theoryof electromagnetic interactions. The concept was later generalized by Fermi[23, 207] and Yukawa [208, 209] to formulate, respectively, the theory ofweak and strong interactions in analogy with the theory of QED.

In order to describe QED, the quantum field theory of electromagnetic fieldsand their interaction with matter, in terms of the massless spin 1 fieldscorresponding to photons, the equations of motion of should be fullyrelativistic. This requires the reformulation of classical equations ofmotion for the fields to obtain the quantum equations of motion for thefields and find their solutions, in case of free fields as well as fieldsinteracting with matter. This is generally done using perturbation theoryfor which a relativistically covariant perturbation theory is required.

The path of transition from a classical description of fields to a quantumdescription of fields, requiring the quantization of fields, their equationsof motion, propagation, and interaction with matter involves understandingmany new concepts and mathematical methods. For this purpose, the Lagrangianformulation for describing the dynamics of particles and their interactionwith the fields is found to be suitable.

Relativistic Notation

Neutrinos are neutral particles of spin; they are completely relativistic inthe massless limit. In order to describe neutrinos and their interactions,we need a relativistic theory of spin 1 particles. The appropriate frameworkto describe the elementary particles in general and the neutrinos inparticular, is relativistic quantum mechanics and quantum field theory. Inthis chapter and in the next two chapters, we present the essentials ofthese topics required to understand the physics of the weak interactions ofneutrinos and other particles of spin 0, 1, and.

We shall use natural units, in which ћ = c = 1, such that all thephysical quantities like mass, energy, momentum, length, time, force, etc.are expressed in terms of energy. In natural units:

The original physical quantities can be retrieved by multiplying thequantities expressed in energy units by appropriate powers of the factorsћ, c, and ћc. For example, mass m =E/c2, momentump = E/c, lengthl = ћc/E, andtime t = ћ /E,etc.

Metric tensor

In the relativistic framework, space and time are treated on equal footingand the equations of motion for particles are described in terms ofspace–time coordinates treated as four- component vectors, in afour-dimensional space called Minkowski space, defined byxμ, whereμ = 0, 1, 2, 3 andxμ =(x0' = t,x1 = x,x2 = y,x3 = z')in any inertial frame, say S. In another inertial frame,say, whichis moving with a velocity in the positive Xdirection, the space–time coordinates are related toxμ through

the Lorentz transformation given by:

such that

remains invariant under Lorentz transformations. For this reason, thequantity is called the length of the four-component vectorxμ in analogy with the length ofan ordinary vector, that is, which is invariant under rotation inthree-dimensional Euclidean space. Therefore, the Lorentz transformationsshown in Eq. (2.1) are equivalent to a rotation in a four-dimensionalMinkowski space in which the quantity defined as, remains invariant, thatis, it transforms as a scalar quantity under the Lorentz transformation.This is similar to a rotation in the three-dimensional Euclidean space inwhich the length of an ordinary vector, defined as remains invariant, thatis, transforms as a scalar under rotation.

Introduction

We have discussed inelastic processes from free nucleon targets in Chapters11 and 12. However, most of the early experiments and also the newexperiments on inelastic as well as quasielastic reactions induced by(anti)neutrinos use nuclear targets. In this chapter, we will focus on theinelastic process of producing mesons and photons from the nuclear targets.When inelastic processes like

(discussed in Chapter 12) take place inside a nucleus, the nucleus can stayin the ground state giving almost all the transferred energy in the reactionto the outgoing meson leading to the coherent production of mesons or can beexcited and/or broken up leading to the incoherent production of meson. Inthe subsequent sections, incoherent and coherent pion production from nucleiin the delta dominance model will be discussed with some comments on theinelastic production of kaons and photons.

The first experiments on inelastic scattering of (anti)neutrinos from nucleiwere done at CERN in the early 1960s using heavy liquid bubble chambers(HLBC) filled with propane, freon and with spark chambers, and at ANL/BNLwith spark and bubble chambers. The importance of nuclear medium effects inthe analysis of these experiments was realized and discussed in the contextof inelastic as well as quasielastic reactions. Some of the mesons, mostlypions, produced in the inelastic reactions could be absorbed in the parentnucleus giving rise to ‘pionless’ lepton events in chargedcurrent (CC) induced reactions enhancing the yield of quasielastic eventsand reducing the yield of ‘pionic’ events as compared to thetheoretical predictions for these reactions from free nucleon targets(Chapter 12). While some theoretical calculations were made to estimate,quantitatively, the effect of nuclear medium effects in quasielasticreactions (see Chapter 14), no serious efforts were made in the case ofinelastic reactions. Subsequently, many experiments were done at CERN [696,697, 698, 699, 700], SKAT [701, 702, 703, 704], FNAL [705, 706], and CHARM[707], using propane, propane– freon, neon, marble, and aluminiumtargets; only a qualitative description of the nuclear medium effects wasused. Most of these experiments studied the incoherent production of pionswhile some of them also studied the coherent production of pions [698, 699,700, 703, 704, 706, 708, 709] from nuclei.

Introduction

The history of the phenomenological theory of neutrino interactions and weakinteractions in general, begins with the attempts to understand the physicsof nuclear radiation known as β-rays. These highlypenetrating and ionizing component of the radiation discovered by Becquerel[9] in 1896 were subsequently established to be electrons by doing manyexperiments in which their properties like charge, mass, and energy werestudied [8]. Since the energy of these β-rayelectrons was found to be in the range of a few MeV, they were believed tobe of nuclear origin in the light of the basic structure of the nucleusknown at that time [4]. It was assumed that the electrons are emitted in anuclear process called β-decay in which a nucleus inthe initial state goes to a final state by emitting an electron. The energydistribution of the β-ray electrons was found to becontinuous lying between me, the mass of theelectron, and a maximum energy Emax corresponding to theavailable energy in the nuclear β-decay, that is,Emax = Ei − Ef , where Ei andE f are the energies of the initial andfinal nuclear states. A typical continuous energy distribution for theelectrons from the β-decay of RaE is shown in Figure1.1 of Chapter 1. It was first thought that the electrons in theβ-decay process were emitted with a fixed energyEmax and suffered random losses in theirenergy due to secondary interactions with nuclear constituents as theytraveled through the nucleus before being observed leading to a continuousenergy distribution. However, the calorimetric heat measurements performedby Ellis et al. [15] and confirmed later by Meitner et al. [16] in theβ-decays of RaE, established that the electronsemitted in the process of the nuclear β-decay havean intrinsically continuous energy distribution. The continuous energydistribution of the electrons from the β-decay poseda difficult problem toward its theoretical interpretation in the context ofthe contemporary model of the nuclear structure and seemed to violate thelaw of conservation of energy.