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Neutrino Masses-How to Add Them to the Standard Model

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Neutrino Masses-How to Add Them to the Standard Model he Oscillating Neutrino The Oscillating Neutrino of spatial coordinates) has the property of interchanging the two states eR and eL. Neutrino Masses What about the neutrino? The right-handed neutrino has never been observed, How to add them to the Standard Model and it is not known whether that particle state and the left-handed antineutrino c exist. In the Standard Model, the field ne , which would create those states, is not Stuart Raby and Richard Slansky included. Instead, the neutrino is associated with only two types of ripples (particle states) and is defined by a single field ne: n annihilates a left-handed electron neutrino n or creates a right-handed he Standard Model includes a set of particles—the quarks and leptons e eL electron antineutrino n . —and their interactions. The quarks and leptons are spin-1/2 particles, or weR fermions. They fall into three families that differ only in the masses of the T The left-handed electron neutrino has fermion number N = +1, and the right- member particles. The origin of those masses is one of the greatest unsolved handed electron antineutrino has fermion number N = 21. This description of the mysteries of particle physics. The greatest success of the Standard Model is the neutrino is not invariant under the parity operation. Parity interchanges left-handed description of the forces of nature in terms of local symmetries. The three families and right-handed particles, but we just said that, in the Standard Model, the right- of quarks and leptons transform identically under these local symmetries, and thus handed neutrino does not exist. The left-handedness of the neutrino mimics the they have identical strong, weak, and electromagnetic interactions. left-handedness of the charged-current weak interactions. In other words, the W In the Standard Model, quarks and leptons are assumed to obtain their masses gauge boson, which mediates all weak charge- in the same way that the W and Z0 bosons obtain theirs: through interactions with changing processes, acts only on the fields e Table I. Lepton Charges the mysterious Higgs boson (named the “God Particle” by Leon Lederman). But and n . The interaction with the W transforms Yw before we write down some simple formulas that describe the interactions of quarks e w 1 the left-handed neutrino into the left-handed Q 5 I3 }} and leptons with the Higgs boson, let us define some notation. 2 electron and vice versa (e ↔ n ) or the right- L eL NNParticle States I w Yw Q Defining the Lepton Fields. For every elementary particle, we associate a field handed antineutrino into the right-handed x 3 residing in space and time. Ripples in these fields describe the motions of these positron and vice versa (n ↔ e ). Thus, weR wR 11LeL21/2 21 21 particles. A quantum mechanical description of the fields, which allows one to we say that the fields e and ne, or the particles 11L1nL211/2 210 describe multiparticle systems, makes each field a quantum mechanical operator eL and neL, are a weak isospin doublet under that can create particles out of the ground state—called the vacuum. The act of the weak interactions. 21R e11/2 11 11 creating one or more particles in the vacuum is equivalent to describing a system These lepton fields carry two types of weak wR 21Rn21/2 110 w 1wR2 in which one or more ripples in the fabric of the field move through space-time. charge: The weak isotopic charge I3 couples Let us now discuss the simple system of one family of leptons. To be specific, them to the W and the Z0, and the weak 11R e 022 21 we will call the particles in this family the electron and the electron neutrino. hypercharge Yw couples them to the Z0. (The Z0 R 21Le 012 11 The electron field describes four types of ripples (or particles). We label these four is the neutral gauge boson that mediates neutral- wL 11R n 00 0 types by two quantum charges called fermion number N and handedness, or current weak interactions.) Electric charge Q is R 21LnL 00 0 chirality, Nx. For the electron field, the particle state with fermion number N = +1 related to the two weak charges through the w w w is the electron, and the particle state with N = 21 is the antielectron (or positron). equation Q 5 I3 1 Y /2. Table I lists the weak Each of these states comes as right-handed, Nx = R, and left-handed, Nx =L. charges for the particle states defined by the Handedness is a Lorentz invariant quantity that is related in a nontrivial way to three fields e, n , and ec. Note that the particle states e and e defined by e R wL helicity, the projection of the spin s in the direction of the momentum p. (For a the field ec do not couple to the W and have no weak isotopic charge. The field discussion of handedness versus helicity, see “The Oscillating Neutrino” on page 28.) and the particle states are thus called weak isotopic singlets. However, e and e R wL In relativistic quantum field theory, the right-handed and left-handed electron do carry weak hypercharge and electric charge and therefore couple to the Z0 and the right-handed and left-handed antielectron can be defined in terms of two and the photon. fields denoted by e and ec, where each field is a Weyl two-component left-handed Likewise, the field n c and its neutrino states n and n would be isotopic e R wL spinor. The compositions of the fields are such that singlets with no coupling to the W. But unlike their electron counterparts, they w w must be electrically neutral (Q 5 I3 1 Y /2 5 0), which implies they cannot 0 e annihilates a left-handed electron eL or creates a right-handed have weak hypercharge. Thus, they would not couple to the W, the Z , or the positron e , and photon. Having no interactions and, therefore, not being measurable, they wR are called sterile neutrinos and are not included in the Standard Model. However, ec annihilates a left-handed positron e or creates a right-handed if the left-handed neutrino has mass, it may oscillate into a sterile right-handed wL electron eR. neutrino, a possibility that could be invoked in trying to give consistency to all the data on neutrino oscillations. These fields are complex, and for the action of the Hermitian conjugate fields e† and ec†, just interchange the words annihilate and create above. For example, The Origin of Electron Mass in the Standard Model. What is mass? Mass is e† creates a left-handed electron or annihilates a right-handed positron. Hence, the the inertial energy of a particle. It is the energy a particle has when at rest and the fields e and ec and their complex conjugates can create or annihilate all the possi- measure of the resistance to an applied force according to Newton’s law F = ma. ble excitations of the physical electron. Note that parity (defined as the inversion A massless particle cannot exist at rest; it must always move at the speed of light. 4 Los Alamos Science Number 25 1997 Number 25 1997 Los Alamos Science he Oscillating Neutrino The Oscillating Neutrino A fermion (spin-1/2 particle) with mass has an additional constraint. It must exist for the neutrino that would mirror the mass term for the electron. It would in both right-handed and left-handed states because the only field operators that have the form yield a nonzero mass for fermions are bilinear products of fields that flip the parti- c cle’s handedness. For example, in the two-component notation introduced above, mnne ne . (5) the standard, or Dirac, mass term in the Lagrangian for free electrons is given by* c But, as we said above, the field ne is not included in the Standard Model c mee e . (1) because, so far, weak-interaction experiments have not required it. The neutrino, though, has no electric charge, which makes it possible to write down a mass term This fermion mass operator annihilates a left-handed electron and creates a right- from the existing neutrino field ne with the form handed electron in its place. The mass term does not change the charge of 1 the particle, so we say that it conserves electric charge. Also, because this mass } m n n . (6) 2 n e e term does not change a particle into an antiparticle, we say that it conserves fermion number N. However, the weak isospin symmetry forbids such a mass (Note that mn and mn refer just to the electron neutrinos, but similar masses can operator because it is not an invariant under that symmetry. (The field e is a be defined for the m and t neutrinos.) The mass operator in (6) annihilates a left- member of a weak isotopic doublet, whereas the field ec is a weak isotopic singlet, handed neutrino and creates a right-handed antineutrino, which means that it is a so that the product of the two is not a singlet as it should be to preserve the weak Majorana mass term. Any mass term that changes a particle to an antiparticle is isospin symmetry.) But the electron does have mass. We seem to be in a bind. called a Majorana mass term. In changing a neutrino to an antineutrino, this term The Standard Model solves this problem: the electron and electron neutrino violates fermion number N, changing it by two units.
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