Introduction

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

where F(q2), in literature, is known as the form factor. This accounts for the spatial extent of the scatterer. F(q2) not only tells about the distribution of the charge in space but using it, one can estimate the size of the target particle as well as its charge distribution and density of magnetization. Thus, for an extended charge distribution, the probability amplitude for a point-like scatterer is modified by a form factor.

Physical Significance of the Form Factor

Consider the elastic scattering of a “spinless” electron from a static “spinless” point object having charge. In the Born approximation, where the perturbation is assumed to be weak, the scattering amplitude is written as

where and are the wave functions of the initial and final electron with momentum and, respectively. These waves are assumed to be plane waves such that

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

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

Assuming, which leads to,

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

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

Introduction

We have discussed charged lepton and (anti)neutrino induced deep inelastic scattering (DIS) off free protons in Chapter 13. The interaction cross section and the structure functions of nucleons are modified when the scattering takes place from the nucleons bound inside a nucleus. The reactions are shown in Figure 16.1, where l = e, μ. A is the target nucleus and X is the jet of hadrons are the four momenta of the initial and the final state particles respectively. Historically, the first observations of modifications of the nuclear structure

functions were made by the European Muon Collaboration (EMC) at CERN in 1981–83. The EMC collaboration studied the ratio of structure function F2(x, Q2) per nucleon for iron to deuterium targets, that is in the energy region of 120-280 GeV and its deviation from unity. This effect is known as the EMC effect. Since then, the EMC effect has been confirmed and studied with improved precision in many DIS experiments using electrons, muons [783, 784, 785, 786, 787, 788], neutrinos, and antineutrinos [789, 790, 791, 792, 793, 794] from different nuclear targets as well as in the Drell–Yan processes using proton and pions [795, 796, 797]. Some of these results are presented in Figure 16.2. From the figure, it may be observed that the ratio is different from unity in almost the entire region of the Bjorken scaling variable 0 < x < 1. From these experiments, some general features of the ratio R(x, Q2) may be inferred:

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

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

• • The functional form of R(x, Q2) is relatively independent in the region of high Q2.

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

Contraction of Leptonic Tensors in Electromagnetic Interactions

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

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

The leptonic current is given by

In Eq. (D.2),

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

• • 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 and the Hermitian conjugate are the same thing.

Therefore, instead of, we may write

we can rewrite the aforementioned expression in the component form for an electronic 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 mediated by 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 obtained as

Contraction of Weak Leptonic Tensor with Hadronic Tensor

Contracting the various terms of hadronic tensor with the leptonic tensor, we get

Where

The need for writing a self-contained comprehensive book on the physics of neutrino interactions had been in our minds for a long time, while teaching various graduate courses in high energy physics and nuclear physics and conducting research in the field of neutrino physics at the Aligarh Muslim University. We also realized the need for such a book while attending many topical workshops, conferences, and short-term schools like NuFact, NuInt, NuSTEC, etc., held in the USA, Europe, Japan, and elsewhere in the area of neutrino physics and while responding to questions asked by the young researchers in many formal and informal discussions. The aforementioned scientific events bring together research students and senior scientists working on various aspects of neutrino physics common to nuclear physics, particle physics, and astrophysics, which make the subject interdisciplinary. In recent times, the research activity in the field of neutrino physics, around the world, and its applications in the other areas of 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 the physics of neutrino interactions focusing on introducing the basic mathematical and physical concepts and methods with the help of simple examples to illustrate the calculation of various neutrino processes relevant for applications in particle physics, nuclear physics, and astrophysics, for the benefit of all those interested in learning the subject.

The main aim of the book is to present a pedagogical account of the physics of neutrino interactions, with balance among its theoretical and experimental aspects, for describing various neutrino scattering processes from leptons, nucleons, and nuclei used in studying neutrino properties like its mass, charge, magnetic moment, and the newly discovered phenomenon of neutrino mixing and oscillations. The book is intended primarily for graduate students and young post-doctoral research scientists working in neutrino physics but it can also be used by advanced undergraduates who have some exposure to basic courses in special theory of relativity, quantum mechanics, nuclear physics, particle physics, and are interested in neutrino physics.

Expression of N(q2) in Terms of Mandelstam Variables

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

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

with

F.2 Expressions of Ah(q2), Bh(q2), and Ch(q2)

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

Expressions of and

The expressions Al (q2), Bl(q2), and Cl (q2) are expressed in terms of the Mandelstam variables and the form factors as:

Introduction

The inelastic scattering processes of (anti)neutrinos from nucleons are relevant in the region starting from the neutrino energy corresponding to the threshold production of a single pion. For neutral current (NC) induced 1π production, this starts at Eth v(⊽) = 144.7 MeV for vl reactions. In the case of charged current (CC) induced 1π production, the threshold energy is higher because of the massive leptons produced in the final state; it corresponds 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 the production of strange mesons (K) and hyperons (Ⲩ) start; both of which are the most relevant inelastic processes in the region of a few GeV. These inelastic processes have been studied very extensively, both theoretically and experimentally, in various reactions induced by photons and electrons which probe the interaction of the electromagnetic vector currents with hadrons in the presence of other strongly interacting particles like mesons and hyperons. In weak processes induced by neutrinos and antineutrinos, the inelastic processes provide a unique opportunity to study the interaction of the weak vector as well as the axial vector currents with hadrons in the presence of strongly interacting particles like mesons and hyperons. Moreover, a study of these weak processes from nucleons and nuclei is of immense topical importance in the context of the present neutrino oscillation experiments being done with the accelerator and atmospheric neutrinos in the energy region of a few GeV. The specific reactions to be studied in the inelastic channels are the various processes induced by the charged and neutral weak currents 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 shown in Figures 11.1(a) and 11.1(b), where vl(⊽l) and l-(l+)

are leptons interacting through the W±(Z) exchanges with the 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 study of various physical phenomena involving electromagnetic radiation. For example, the observations and theoretical explanations of the black body radiation by Planck, the photoelectric effect by Einstein, and the scattering of a photon off an electron by Compton established that electromagnetic radiation can be described in terms of “discrete quanta of energy” called photon, identified as a massless particle of spin 1. Consequently, Maxwell's equations of classical electrodynamics, describing the time evolution of the electric and magnetic fields are interpreted to be the equations of motion of the photon, written in terms of the massless spin 1 electromagnetic field. Later, the quantization of the electromagnetic field was formulated to explain the emission and absorption of radiation in terms of the creation and annihilation of photons during the interaction of the electromagnetic field with the physical systems. The concept of treating photons as quanta of the electromagnetic fields was successful in explaining the physical phenomena induced by the electromagnetic interactions; methods of field quantization were used leading to quantum electrodynamics (QED), the quantum field theory of electromagnetic interactions. The concept was later generalized by Fermi [23, 207] and Yukawa [208, 209] to formulate, respectively, the theory of weak and strong interactions in analogy with the theory of QED.

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

The path of transition from a classical description of fields to a quantum description of fields, requiring the quantization of fields, their equations of motion, propagation, and interaction with matter involves understanding many new concepts and mathematical methods. For this purpose, the Lagrangian formulation for describing the dynamics of particles and their interaction with the fields is found to be suitable.

Relativistic Notation

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

We shall use natural units, in which ћ = c = 1, such that all the physical 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 the quantities expressed in energy units by appropriate powers of the factors ћ, c, and ћ c. For example, mass m = E/c2, momentum p = E/c, length l = ћc/E, and time t = ћ /E, etc.

Metric tensor

In the relativistic framework, space and time are treated on equal footing and the equations of motion for particles are described in terms of space–time coordinates treated as four- component vectors, in a four-dimensional space called Minkowski space, defined by xμ, where μ = 0, 1, 2, 3 and xμ = (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 X direction, the space–time coordinates are related to xμ through

the Lorentz transformation given by:

such that

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

Introduction

We have discussed inelastic processes from free nucleon targets in Chapters 11 and 12. However, most of the early experiments and also the new experiments on inelastic as well as quasielastic reactions induced by (anti)neutrinos use nuclear targets. In this chapter, we will focus on the inelastic 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 stay in the ground state giving almost all the transferred energy in the reaction to the outgoing meson leading to the coherent production of mesons or can be excited and/or broken up leading to the incoherent production of meson. In the subsequent sections, incoherent and coherent pion production from nuclei in the delta dominance model will be discussed with some comments on the inelastic production of kaons and photons.

The first experiments on inelastic scattering of (anti)neutrinos from nuclei were done at CERN in the early 1960s using heavy liquid bubble chambers (HLBC) filled with propane, freon and with spark chambers, and at ANL/BNL with spark and bubble chambers. The importance of nuclear medium effects in the analysis of these experiments was realized and discussed in the context of inelastic as well as quasielastic reactions. Some of the mesons, mostly pions, produced in the inelastic reactions could be absorbed in the parent nucleus giving rise to ‘pionless’ lepton events in charged current (CC) induced reactions enhancing the yield of quasielastic events and reducing the yield of ‘pionic’ events as compared to the theoretical 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 quasielastic reactions (see Chapter 14), no serious efforts were made in the case of inelastic 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 aluminium targets; only a qualitative description of the nuclear medium effects was used. Most of these experiments studied the incoherent production of pions while 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 weak interactions in general, begins with the attempts to understand the physics of nuclear radiation known as β-rays. These highly penetrating and ionizing component of the radiation discovered by Becquerel [9] in 1896 were subsequently established to be electrons by doing many experiments in which their properties like charge, mass, and energy were studied [8]. Since the energy of these β-ray electrons was found to be in the range of a few MeV, they were believed to be of nuclear origin in the light of the basic structure of the nucleus known at that time [4]. It was assumed that the electrons are emitted in a nuclear process called β-decay in which a nucleus in the initial state goes to a final state by emitting an electron. The energy distribution of the β-ray electrons was found to be continuous lying between me, the mass of the electron, and a maximum energy Emax corresponding to the available energy in the nuclear β-decay, that is, Emax = Ei − E f , where Ei and E f are the energies of the initial and final nuclear states. A typical continuous energy distribution for the electrons from the β-decay of RaE is shown in Figure 1.1 of Chapter 1. It was first thought that the electrons in the β-decay process were emitted with a fixed energy Emax and suffered random losses in their energy due to secondary interactions with nuclear constituents as they traveled through the nucleus before being observed leading to a continuous energy distribution. However, the calorimetric heat measurements performed by Ellis et al. [15] and confirmed later by Meitner et al. [16] in the β-decays of RaE, established that the electrons emitted in the process of the nuclear β-decay have an intrinsically continuous energy distribution. The continuous energy distribution of the electrons from the β-decay posed a difficult problem toward its theoretical interpretation in the context of the contemporary model of the nuclear structure and seemed to violate the law of conservation of energy.

Cabibbo Theory, SU(3) Symmetry, and Weak N–Y Transition Form Factors

For the ∆S = 0 processes,

and for the |∆S| = 1 processes,

the matrix elements of the vector (Vμ) and the axial vector (Aμ) currents between a nucleon or a hyperon and a nucleon N = n, p are written as:

and

where and are the masses of the nucleon and hyperon, respectively. and are the vector, weak magnetic and induced scalar N − Y transition form factors and and are the axial vector, induced tensor (or weak electric), and induced pseudoscalar form factors, respectively.

In the Cabibbo theory, the weak vector (Vμ) and the axial vector (Aμ) currents corresponding to the ∆S = 0 and ∆S = 1 hadronic currents whose matrix elements are defined between the states are assumed to belong to the octet representation of SU(3).

Accordingly, they are defined as:

Where are the generators of flavor SU(3) and is are the well-known Gell–Mann matrices written as

The generators obey the following algebra of SU(3) generators

are the structure constants, and are antisymmetric and symmetric,

respectively, under the interchange of any two indices. These are obtained using the λi given in Eq. (B.9) and have been tabulated in Table B.1.

From the property of the SU(3) group, it follows that there are three corresponding SU(2) subgroups of SU(3) which must be invariant under the interchange of quark pairs ud, ds, and us respectively, if the group is invariant under the interchange of u, d, and s quarks. Each of these SU(2) subgroups has raising and lowering operators. One of them is SU(2)I , generated by the generators (λ1, λ2, λ3) to be identified with the isospin operators (I1, I2, I3) in the isospin space. For example, I± of isospin space is given by

The other two are defined as SU(2)U and SU(2)V generated by the generators , respectively, in the U-spin and V-spin space with (d s) and (u s) forming the basic doublet representation of SU(2)U and SU(2)V .

Introduction

In 1951, Lyman, Hanson, and Scott [541] were the first to observe elastic electron scattering from different nuclei using a 15.7 MeV beam obtained with the help of the betatron accelerator facility at Illinois. Further extensive studies using electron beams started with the development of the Mark III linear accelerator (LINAC) in 1953 at the High Energy Physics Laboratory (HEPL), Stanford. Hofstadter, Fechter, and McIntyre studied the effect of electron scattering (Ee ~ 125 − 150 MeV) on various nuclear targets and concluded that these nuclei have finite charge distribution. With increased energy (550 MeV) available for scattering, the first evidence of elastic scattering from the proton was observed by Chambers and Hofstadter at HEPL in 1956 [542], using a polyethylene target. Assuming that proton had an exponential density distribution, they found the r.m.s. (root mean squared) radius of the proton to be about 0.8 fm. Later, Yearian and Hofstadter performed electron scattering experiments on deuteron targets and determined the magnetic moment of the neutron [543]. At that time, the investigation of the structure of the proton and neutron was a major objective of HEPL, which was upgraded to achieve electron beams of energies up to 20 GeV; today, HEPL is known as the Stanford Linear Accelerator Center (SLAC) [544]. By the 1960s, it became possible to perform both elastic and inelastic scattering experiments at high energies and for a wide range of four-momentum transfer squared (Q2). Thus, by the end of the 1960s, nuclear physics entered the ‘deep inelastic scattering’ (DIS) era, when experiments with 20–40 times higher energies were being performed at SLAC; it became possible to probe the hadron. DIS is the scattering of charged leptons/neutrinos from hadrons in the kinematic region of very high Q2 and energy transfer ν. During the late 1960s, experiments by MIT-SLAC collaboration, led by Taylor, Kendall, and Friedman [544] confirmed the scaling phenomenon in the deep inelastic region, which was theoretically predicted by Bjorken [545, 546]. They received the 1990 Nobel Prize in Physics for the experiments. These experimental results confirmed that, in the scaling region, the constituents of protons behave like free particles called partons. Charged partons are identified as quarks and neutral partons are identified as gluons. Later, many experiments using electron and muon beams were performed at CERN, DESY, Fermilab, etc.

Cross Section

Consider a two body scattering process with four momenta. There are N particles in the final state with four momenta.

The general expression for the cross section is given by

where

In the laboratory frame, where, say, particle “b” is at rest (i.e. vb = 0, Eb = mb)

va = c = 1, if the incident particle is relativistic, that is, Ea ≫ ma.

In the center of mass frame, where particles “a” and “b” approach each other from exactly opposite direction, that is, θab = 180o, with the same magnitude of three momenta, that is, such that

where is the center of mass energy.

More conveniently, this is also written as

Two body scattering

For a reaction, where “a” and “b” are particles in the initial state and “1” and “2” are particles in the final state, that is,

the general expression for the differential scattering cross section is given by

In any experiment, one observes either particle “1” or particle “2”; therefore, the kinematical quantities of the particle which is not to be observed are fixed by doing phase space integration. For example, if particle “2” is not to be observed, then

which gives the constraint on where is the three momentum transfer and, which results in

Integrating over the energy of particle “1”, using the delta function integration property, we Get

Thus,

Which result in,

If the scattering takes place in the lab frame, and, it result in

If the scattering takes place in the center of mass frame, where, is the center of mass energy, we write

Energy distribution of the outgoing particle “1”

Here, we evaluate energy distribution in the lab frame.

Historical Introduction to Neutrinos

Neutrino hypothesis

Beginning with the neutrino hypothesis proposed by Pauli in 1930, the story of the neutrino has been an amazing one [1]. It all started with a letter written by Pauli to the participants of a nuclear physics conference in Tubingen, Germany, on December 4, 1930 [1], in which he proposed the existence of a new neutral weakly interacting particle of spin and called it “neutron” as a “verzweifelten Ausweg” (desperate remedy), to explain the two outstanding problems in contemporary nuclear physics which posed major difficulties with respect to the scientists’ theoretical interpretations. These two problems were related with the puzzle of energy conservation in β-decays of nuclei [2, 3], discovered by Chadwick in 1914 [4], and anomalies in understanding the spin–statistics relation in the case of 14N and 6Li nuclei within the context of the nuclear structure model that was prevalent in the early decades of the twentieth century [5, 6] in which electrons and protons were considered to be nuclear constituents.

This proposed ‘verzweifelten Ausweg’ was considered so tentative by Pauli himself that he postponed its scientific publication by almost three years. Today, neutrinos, starting from being a mere theoretical idea of an undetectable particle, are known to be the most abundant particles in the universe after photons, being present almost everywhere with a number density of approximately 330/cm3 pan universe. The history of the progress of our understanding of the physics of neutrinos is full of surprises; neutrinos continue to challenge our expectations regarding the validity of certain symmetry principles and conservation laws in particle physics. The study of neutrinos and their interaction with matter has made many important contributions to our present knowledge of physics, which are highlighted by the fact that ten Nobel Prizes have been awarded for physics discoveries in topics either directly in the field of neutrino physics or in the topics in which the role of neutrino physics has been very crucial.

In this chapter, we will provide a historical introduction to the development of our understanding of neutrinos and their properties as they have emerged from the theoretical and experimental studies made over the last 90 years.