CPT, Majorana Fermions, and Particle Physics Beyond the Standard Model Open Access

Abstract

1. Antiparticles

Our first encounter with antiparticles was in 1932, when Anderson discovered positrons in cosmic rays [1, 2]. Their existence had been anticipated by Dirac [3] in interpreting negative energy solutions of the Dirac equation.

Any particle has its own antiparticle. An antiparticle is characterized by possessing the same mass and lifetime as the particle, while additive quantum numbers such as electric charge are opposite in sign. If a particle has no nonzero quantum numbers, it is possible that the particle is its own antiparticle. The photon is the oldest known such particle.

When muons were discovered [4] in cosmic rays in 1937, some of them possessed positive, the others negative electric charges. The two charge states are particles and antiparticles of each other. As for mesons found in the 1940s and 1950s, pions are in three charge states, among which |$\pi ^+$| and |$\pi ^-$| form a particle–antiparticle pair, and |$\pi ^0$| is its own antiparticle. Kaons also have three charge states, but |$K^0$| is distinct from its antiparticle, having a strangeness quantum number of |$+1$|⁠. This was an important finding made by Nakano and Nishijima [5–7] and Gell-Mann [8, 9]. The discovery of antiprotons [10] and antineutrons [11] followed in 1955 and 1956, by accelerator experiments at Bevatron in Berkeley. The energy of the accelerator had been determined for this purpose. A large number of new particles have been discovered since then, but we have no reason to doubt that each particle is accompanied by its antiparticle.

The reason for Dirac to predict (one year before the discovery) the existence of the positron was the well-known Dirac sea, an assumption that the vacuum is the state in which all negative energy states are already filled by the particle (electron). The Pauli principle dictates that any electron with positive energy cannot drop into a negative energy state by emitting a photon etc., so that the existence of negative energy states does not affect the stability of an electron state e.g. at rest. The existence of negative energy states can be perceived when a negative energy electron in the vacuum is excited up to a positive energy state. The absence of a negative energy state, a hole in the vacuum, is seen as a particle having an opposite charge to the electron. This is the antiparticle, the positron.

This picture is totally asymmetrical between particles and antiparticles, and can be applied to fermions only. The charge of the negative energy electrons, the total of which is infinite, is ignored. In relativistic quantum field theories, however, it is not necessary to evoke the Dirac sea. The existence of antiparticles can be shown from general principles for both fermions and bosons. Moreover, the theory is symmetrical between particles and antiparticles. The question of which is the particle and which is the antiparticle makes no sense.

The space-time picture of particle trajectories à la Feynman gives an intuitive reasoning for the existence of antiparticles. It is easier to examine an example. Let us take the nuclear force between a proton and a neutron caused by the exchange of a charged pion. The proton emits |$\pi ^+$| and becomes a neutron; then the neutron absorbs the |$\pi ^+$| and becomes a proton. The propagation of a pion may be described by a Feynman propagator, which does not vanish outside the light cone, despite falling off exponentially. In a different coordinate frame, the timing of the emission and absorption becomes the opposite: the |$\pi ^+$| travels backward in time. According to the interpretation of Stueckelberg [12] and Feynman [13], this should rather be described as |$\pi ^-$| emission by the neutron, followed by its absorption by the proton. In this manner the particle traveling backward in time represents the antiparticle.

2. CPT theorem

In relativistic quantum field theories, the existence of antiparticles is ensured by the CPT theorem, which states that CPT is a symmetry of any relativistic quantum field theory under very general assumptions. There are two kinds of proofs of the CPT theorem. Lüders [14,15] and Pauli [16] gave a constructing proof of the theorem, which essentially examines all possible forms of relativistic local field theory defined by a Lagrangian density, consisting of fields at the same space-time point and being Hermitian, and found they are indeed invariant under the combination of C, P, and T transformations. It is remarkable that their work was done before the suggestion [17] and discovery [18, 19] of parity violation in weak interactions, not to mention of CP violation [20].

Central to their derivation were the transformation properties of spin-1/2 fields. For fields of spin-1/2 particles, CPT transformation of |$\psi (x)$| (in four-component notation) is given by

Consider any bilinear form consisting of two spin-1/2 fields

where f is a complex number and |$\Gamma$| is one of the |$4\times 4$| matrices

It transforms under CPT as

where Fermi statistics is assumed. The sign in Eq. (3) counts the number of gamma matrices in |$\Gamma$|⁠, and is |$-1$| for |$\gamma ^\mu$| and |$\gamma ^\mu \gamma _5$|⁠, |$+1$| for others. Thus the bilinear transforms to its Hermitian conjugate up to a possible minus sign when it is a Lorentz vector.

This matches with the CPT transformation of scalar and vector fields:

If we note that the derivative |$\partial /\partial x^\mu$| changes sign under CPT (since |$x\rightarrow -x$|⁠), it is easy to convince oneself that any Lorentz scalar formed from fields at the same space-time point x is transformed by CPT to its Hermitian conjugate at |$-x$|⁠. As the Lagrangian density |${\cal L}$| is Hermitian as a whole, it transforms as |${\cal L}(x)\rightarrow {\cal L}(-x)$|⁠, and its integral over space-time, the action, is invariant under CPT.

Pauli [16] also extended these considerations to fields with general values of spin. It is convenient to use fields in irreducible representations of the Lorentz group. Using the local isomorphism |$\rm SO(4) \sim SO(3)\otimes SO(3) \sim SU(2)\otimes SU(2)$|⁠, the representations can be characterized by two SU(2) “spin” quantum numbers, |$J_L$| and |$J_R$|⁠. For example, a Dirac field corresponds to the representation |$(1/2,0)\oplus (0,1/2)$| (which corresponds to the left- and right-handed Weyl components), and a vector field corresponds to the (1/2, 1/2) representation. CPT transformation of a general field |$(J_L, J_R)$| is found to be |$(-1)^{2J_L}$| times the conjugated fields with an extra factor of i for fermions. (The factor can alternatively be taken as |$(-1)^{2J_R}$|⁠. The two possibilities are related by a |$2\pi$| rotation.) The reader may convince him/herself that this general rule reproduces the |$\gamma _5$| factor for Dirac fields. Using this transformation property, CPT invariance of a general Lagrangian density can be shown if anticommutation relations for spin half-integer fields are assumed.

The second proof of the CPT theorem is the so-called axiomatic one [21–25]. We start from the understanding [26] that knowledge of Wightman functions, vacuum expectation values of products of the fields, of a field theory includes the full information on the theory. Thus we are led to examine the CPT transformation properties of Wightman functions.

A Lorentz group consists of transformations |$x\rightarrow \Lambda x$| that leave the Minkowski metric invariant

in matrix notation. A Lorentz group has four disconnected components, each of which contains the identity, P, T, and PT, respectively. These components are characterized by |$\det \Lambda =\pm 1$| and the sign of |$\Lambda ^0{}_0$| (note that |$|\Lambda ^0{}_0|\ge 1$|⁠), being |$(++)$|⁠, |$(-+)$|⁠, |$(--)$|⁠, and |$(+-)$|⁠. The principle of relativity is the statement that physics laws have the same form in any inertial frames, but we learned in 1957 [18, 19] that the statement is restricted to frames related to each other by continuously connected transformations, namely, only the one of the four components containing identity.

Now an important observation is that the identity component and the PT component can be connected continuously if we extend the Lorentz group to allow complex parameters. In fact, the boost in the x direction

leaves g invariant even when |$\eta$| takes complex values. Putting pure imaginary |$\eta =i\chi$|⁠, one finds

Taking |$\chi =\pi$|⁠, it reduces to |$(t^{\prime }, x^{\prime }) = (-t, -x)$|⁠. Combining this tranformation with a |$\pi$| rotation in the |$yz$| plane, one arrives at PT, total reflection |$x^\mu \rightarrow -x^\mu$|⁠.

For this transformation to be useful, the coordinate space needs to be extended to complex values, and the analyticity of Wightman functions needs to be studied. For simplicity of presentation and to underline essential points, we will look at a two-point function of the form

where the fields are taken to be scalar fields for the moment. (Here we use complex fields and distinguish |$\varphi _1$| and |$\varphi _2$| to highlight the role of charge conjugation. The reader may claim that it is nonzero only for |$\varphi _1=\varphi _2$|⁠, but in general n-point functions all fields can be different from one another. Even for two-point functions, we may regard Eq. (9) as something like |$\langle 0| \nu _e(x_1) \bar{\nu }_\mu (x_2) |0\rangle$| ignoring spin.) |$\Delta (x_1, x_2)$| turns out to be a function of |$x=x_1-x_2$| only, thanks to translation invariance (note |$\varphi (x)= e^{i{\cal P}\cdot x} \varphi (0) e^{-i{\cal P}\cdot x}$|⁠. The vacuum, having |$P^\mu =0$|⁠, is taken to be translation invariant):

which we will write simply as |$\Delta (x)$|⁠. When |$x^\mu$| is extended to complex values |$z^\mu$|⁠, its analyticity property can be seen to be related to the spectrum of the physical states. Inserting a complete set of states, it may be expressed as

The four-momentum |$p^\mu$| in the last line satisfies the physical condition |$p^0 = \sqrt{\vec{p}^2+M^2} \gt |\vec{p}|\ge 0$|⁠. Writing |$z^\mu =x^\mu +iy^\mu$|⁠, the integral on three-momenta can be seen to converge for |$y^0\lt 0$| and |$|y^0|\gt |\vec{y}|$|⁠, i.e. when |$y^\mu$|⁠, the imaginary part of |$z^\mu$|⁠, is inside the backward light cone. This region (with |$x^\mu$| arbitrary) is traditionally called the “backward tube.”

The vacuum expectation value |$\langle 0| \varphi _1(z) \varphi _2^{*}(0) |0\rangle$| itself is a Lorentz invariant quantity, so it is expected to be a function of the only scalar |$z^2$|⁠. In the above, the domain of analyticity is seen to be the backward tube. It is shown that this region can be extended to the union of the region |$z \rightarrow \Lambda _c z$| by analytic continuation, where z is inside the backward tube and |$\Lambda _c$| is a complex Lorentz transformation [27]. This extended region of analyticity (the “extended tube”) contains points where |$z^2$| is real (“Jost points”). This is realized when |$x\cdot y=0$| (then |$z^2=x^2-y^2$|⁠), which implies that |$x^\mu$| is a spacelike four vector.

This result allows the transformation of the Wightman functions under complex Lorentz transformation, keeping their analyticity. Using the Lorentz invariance of the vacuum and the Lorentz transformation properties of the fields |${\cal U}(\Lambda ) \varphi (z) {\cal U}^\dagger (\Lambda ) = S(\Lambda )^{-1} \varphi (\Lambda z)$| (for scalar fields |$S=1$|⁠), and taking |${\cal U}$| as the total reflection, the relation

is obtained. At first glance this looks like a PT invariance relation of Wightman functions, but it is not necessarily so. Especially when |$x^2$| is timelike, the region of analyticity of the two sides has no common region. The right-hand side of Eq. (12) in fact converges when |$y^\mu$| is inside the forward light cone. This in practice means that there can be discontinuity in the limit |$y^\mu \rightarrow \pm 0$|⁠. This is indeed the case for free fields, where the discontinuity is given by the Bessel function.

Now we assume that for spacelike separation the fields commute (anticommute for fermions), not necessarily as operators but at least inside vacuum expectation values (this is called “weak local commutativity”). Then we have

Using translation invariance, the second row may also be written as |$\langle 0| \varphi _2^{*}(z) \varphi _1(0) |0\rangle$|⁠, so we have

Though the equality holds for spacelike |$x^\mu$|⁠, the domain of convergence for the two sides is now common, both in the backward tube. The equality can then be extended to the whole region of z by analytic continuation.

One more twist is needed. When we express the right-hand side of Eq. (14) in terms of |$t^{\prime }=-t$|⁠, it results in the wrong sign for the frequency. (The canonical commutation relation also has a wrong sign with |$\vec{x}^{\prime }=-\vec{x}$| and |$\vec{p}^{\prime }=\vec{p}$|⁠.) As was realized by Wigner, this term should rather be viewed as

for consistent interpretation of time reversal in quantum theory (antiunitarity of transformations including time reversal). So we finally arrive at the relation (translating back in z to the form in the RHS of Eq. (13))

or with the original coordinates

which is the CPT relation for Wightman functions, recalling that the field conjugation corresponds to the exchange of particles and antiparticles.

If the fields have nonzero spin, Lorentz transformation involves a nontrivial matrix S, which mixes spin components. For total reflection, S just reduces to a sign factor |$(-1)^{2J_L}$|⁠. In addition for fermion fields, exchanging them for spacelike distances provides an extra minus sign. So the conclusion for scalar fields can be applied to fields with arbitrary spin with a possible minus sign.

It is possible to extend these arguments to vacuum expectation values of n fields in a rather straightforward way. We do not discuss the details except for mentioning that the theory of multivariable complex functions needs to be invoked.

To summarize, CPT symmetry can be proved in general field theories, having Lorentz invariance, a physical spectrum, and local commutativity consistent with causality.

Experimental tests of CPT invariance have been performed in a variety of observables (see the section by Pich and Ramsey-Musolf in Ref. [28] entitled “Tests of conservation laws”¹). The most precise verification comes from the difference of |$K^0$| and |$\bar{K}^0$| masses [29] (see also the section by Antonelli et al. in Ref. [28] entitled “CPT invariance tests in neutral kaon decay”²), which is less than |$10^{-18}$| of the mass itself (almost the Planck scale if the breaking effect is linear). Magnetic moments of positive and negative muons are equal at the level of |$10^{-8}$| [30].

3. Relativity and set of states

It is instructive to find out how Poincaré invariance constrains possible set of particle states (not fields!). Consider a particle at rest. In a different (boosted) coordinate frame, the particle has a finite momentum. The principle of relativity then implies in turn that states with finite-momenta have to exist in the original frame. Thus the whole set of particle states with arbitrary three-momenta should be there. Mathematically, the set of states has to form an (irreducible unitary) representation of Poincaré algebra. Now working in the rest frame, spacial rotations do not change the momentum and the state remains at rest. How the state transforms under rotations is prescribed by the representation of SO(3), the group of space rotation, which is the angular momentum states designated by the spin quantum number of the particle. If the spin of the particle is S, the number of states is |$(2S+1)$| times the spacial momentum degrees of freedom (⁠|$\infty ^3$|⁠).

The situation is different if the particle has no mass. The argument above is not applicable since massless states have no rest frame. It can be easily seen that states with any momentum (except 0) can be connected by rotation and boost, forming a set of states. Now instead of the state at rest, we may select a state with one particular momentum as the standard and examine transformations that do not change the momentum. Spacial rotations generally change the momentum of states, but rotation along the momentum vector leaves the momentum unchanged. Angular momentum along the momentum vector can thus be well defined. We call this the “helicity” of the particle, symbolically written as

It can be shown that helicity is invariant under any Lorentz transformation including boost and rotation. (For massive states, we can define the helicity of a particle similarly, but it is only invariant under spacial rotation, and boost along the momentum direction as long as the momentum direction is not reversed. It is not invariant under boost in the transverse directions.) The states with a single particular value of helicity close within themselves under Lorentz transformation. The number of degrees of freedom for massless states is |$1\times \infty ^3$|⁠, as far as Poincaré invariance is concerned.

CPT invariance poses additional constraints on massless states. Considering the transformation properties of momentum and angular momentum under CPT, one can convince oneself that helicity changes sign under CPT. Thus transformation of a particle state with helicity |$\lambda$| is an antiparticle state with helicity |$-\lambda$|⁠. Two states with opposite helicities should exist, except for the case |$\lambda =0$|⁠. The number of degrees of freedom becomes |$2\times \infty ^3$| with this CPT requirement.

4. Charge conjugation and self-conjugate particles

Charge conjugation is the operation of exchanging particles and antiparticles. In terms of fields, charge conjugation is closely related to Hermitian conjugation. Generally, (free) fields can be expanded in terms of plane waves. Their positive frequency parts are associated with annihilation operators of the particle, and negative frequency parts include creation operators of the antiparticle. When the field is conjugated, the negative and positive frequencies are exchanged, and annihilation and creation operators are exchanged at the same time. The Hermitian conjugated field behaves exactly as the antiparticle field: One can see that the conjugated field contains annihilation operators of the antiparticles, in constrast to the original field containing annihilation operators of the particle.

This relation is most simply realized for scalar fields. A real scalar field corresponds to a scalar particle, which is identical to its antiparticle, and a complex scalar field corresponds to a particle, which is different from its antiparticle. Charge conjugation is nothing but Hermitian conjugation.

We note that a complex scalar field can be formed from two real scalar fields that constitute real and imaginary parts of the field. When this is done, the interpretatation of particles and antiparticles changes. In the opposite direction, starting from a complex scalar field, we may impose a reality condition |$\varphi ^{*}(x)=\varphi (x)$| to obtain a real scalar field.

Charge conjugation becomes slightly more complicated for Dirac fields. The complex conjugation of a Dirac field |$\psi ^{*}(x)$| (the symbol |$*$| indicates Hermitian conjugation as an operator, but no transposition as a four-spinor is meant) can play the role of an antiparticle field. The problem is that |$\psi ^{*}$| transforms differently from |$\psi$| under Lorentz transformations. If we impose a reality condition |$\psi ^{*}=\psi$| to obtain a field of self-conjugate spin-1/2 particles, it may work in that particular coordinate frame, but the condition does not hold in different frames.

It is thus convenient if we can find a linear combination of the four components of |$\psi ^{*}$| that transforms in exactly the same way as |$\psi$|⁠. This is possible (thanks to the equivalence of all representations of gamma matrices in four dimensions), and it is usually written as

where C is a constant unitary |$4\times 4$| matrix satisfying

The field |$\psi ^c$| is called the charge conjugation of |$\psi$|⁠.

We do not quote an explicit form of C, as it generally depends on the representation of |$\gamma _\mu$|⁠. Gamma matrices can be any set of four matrices satisfying |$\gamma _\mu \gamma _\nu + \gamma _\nu \gamma _\mu =2g_{\mu \nu }$| and appropriate hermiticity properties (for the Lagrangian to be Hermitian). It is known (from representation theory of finite groups) that any sets satisfying these relations are unitarily equivalent to each other, thus corresponding to a different choice of the basis of the four-component spinor space. The set |$\lbrace -\gamma _\mu ^T\rbrace$| obviously satisfies the same relations as |$\lbrace \gamma _\mu \rbrace$|⁠, and the unitary transformation connecting the two sets is nothing but the matrix C.

There exist sets of |$\lbrace \gamma _\mu \rbrace$|⁠, all of which are purely imaginary. These are called Majorana representation. For this representation we can take |$C=-\gamma _0$| (⁠|$C\gamma _0^T=1$|⁠), and then charge conjugation is simply complex conjugation, just as for scalar fields. Indeed it is easy to see that the Dirac equation in this representation can be split into its real and imaginary parts. Unlike the Dirac and chiral representations of the gamma matrices, however, the transformation property under space rotation is cumbersome for the Majorana representation.

The fields |$\psi$| and |$\psi ^c$| transform in the same way under Lorentz transformation and both satisfy the Dirac equation of the same form. So it is legitimate to impose a condition

to obtain the field of the self-conjugate spin-1/2 particle. This condition is called the Majorana condition, and the resulting field is named the Majorana field. Expanding |$\psi$| in plane waves, it can be seen that the antiparticle and particle are identical for Majorana fields. The physical degrees of freedom become half of that of Dirac fields.

5. Spin-1/2 fields and mass

The physical content of a Dirac field has four degrees of freedom (putting the momentum aside), spin up and down, particle, and antiparticle. This is totally appropriate to describe electrons. It is, however, not the minimal spin-1/2 field as has been seen in the previous section. The Majorana field is one possibility of reducing the degrees of freedom to half, but there is a different way to do this.

The matrix |$\gamma _5$|⁠, having the eigenvalues |$\pm 1$|⁠, may be used to split a Dirac field into two parts. Explicitly, left- (right-)handed projection can be performed by multiplying the Dirac spinor by |$(1\mp \gamma _5)/2$|⁠. The result is left-handed and right-handed Weyl (chiral) fields:

However, projecting the Dirac equation |$(i\gamma ^\mu \partial _\mu -m)\psi =0$| into chiral parts we find

which means that the left- and right-handed parts are inter-related to each other. An exception occurs for |$m=0$|⁠, in which case the two decouple. It is then possible to retain one chirality of the Weyl field and discard the other.

As every particle physicist knows well, the Standard Model of fundamental interactions is constructed using Weyl fields as its constituent fields representing quarks and leptons. All these fermions have no mass terms in the Lagrangian, and obtain masses only after spontaneous |$\rm SU(2)\otimes U(1)$| breaking through Yukawa couplings to the Higgs field.

The physical content of Weyl fields is half of that of Dirac fields obviously, but it is notable that the left-handed Weyl fields contain a negative helicity particle and a positive helicity antiparticle. Weyl fields themselves thus break parity and charge conjugation invariance from the outset. This content is consistent with CPT invariance, which is respected by any relativistic field theory. CPT transformation of a negative helicity particle state is indeed a positive helicity antiparticle state.

One may wonder that imposing the Majorana condition and chiral projection at the same time may reduce the physical degrees of freedom further. It turns out that this is not possible. Consideration of CPT invariance in fact shows that two degrees of freedom are minimal for spin-1/2 particles. If a Majorana field |$\psi _M$| is projected to chiral components, each projection no longer satisfies the Majorana condition, and just becomes the usual Weyl field. The left- and right-handed Weyl fields are, however, not independent and are related to each other as follows:

(The interchange of left- and right-handed components may be understood if one remembers the charge conjugation property of Weyl fields.) In turn, starting from a Weyl field |$\psi _L$|⁠, it is possible to construct a Majorana field as

These considerations imply that Weyl and Majorana fields have the same physical content and are equivalent in a sense.

We saw that Weyl fields admit no mass terms. In contrast, Majorana fields can have mass terms, which may seem contradictory. There is actually no contradiction, as the Majorana mass term enters as

from which it can be seen that conserved charge no longer exists. In terms of Weyl fields, the Majorana mass violates the particle number conservation. In the case of neutrinos, lepton number conservation is violated if the neutrino has a Majorana mass.

The Lagrangian density for a free Majorana field |$\psi _M$| has the following form:

This is the same form as that for a Dirac field up to an overall factor of 1/2. This factor is essentially the same as the difference between the Lagrangian of real and complex scalar fields. Rewriting the Lagrangian in terms of the corresponding left-handed Weyl field |$\psi _L$|⁠, we find up to total divergence

The mass term may be rewritten as

which explicitly shows that the Majorana mass violates the fermion number of the Weyl field. Notice that the mass term is compatible with Fermi statistics, as C is an antisymmetric matrix.

In the following we use the shortcut notation

for this Majorana mass term.

In terms of the Majorana field |$\psi _M$|⁠, the mass term |$\bar{\psi }_M\psi _M$| may be rewritten as

From this expression it can also be seen that antisymmetry of C and Fermi statistics go together. Extending this argument to general Majorana bilinears |$\bar{\psi }_M \Gamma \psi _M$|⁠, symmetry of the product |$\Gamma C$| under transposition turns out to be essential. If |$\Gamma C$| is a symmetric matrix, |$\bar{\psi }_M \Gamma \psi _M$|vanishes automatically. This occurs for |$\Gamma =\gamma ^\mu$| and |$\sigma ^{\mu \nu }$|⁠. Absence of the vector bilinear is nothing but the property that Majorana fermions cannot have any charges. Similarly, Majorana fermions cannot have magnetic or electric dipole mements. Other bilinears |$\Gamma =1$| (mass term), |$\gamma _5$|⁠, |$\gamma ^\mu \gamma _5$| are nonzero. Coupling of Majorana fermions to gauge bosons thus is not in general forbidden but is restricted to axial vector coupling. For bilinears of different Majorana fields |$\bar{\psi }_{M1}\Gamma \psi _{M2}$|⁠, these consideration do not apply and vector transition coupling is allowed.

6. Neutrino masses

It is often stated that neutrino masses indicate the existence of physics beyond the Standard Model. Rather, the Standard Model was deliberately constructed to allow no neutrino masses, by not introducing right-handed neutrinos in the model. This came from the experimental observation that the neutrinos from weak decays are always left-handed [31].

There is no evidence that right-handed neutrinos exist, but their expected quantum numbers imply that they are singlets of the Standard Model gauge group and have no gauge interactions. Thus right-handed neutrinos, even if they exist, are hard to produce in particle reactions that we can handle.

With right-handed neutrinos, the fermion content of the Standard Model becomes symmetric between quarks and leptons, and neutrino masses arise in just the same way as the charge-|$2/3$| quark masses from Yukawa couplings to the Higgs field. The size of the neutrino masses is expected to be of the same order as the corresponding charged lepton, which strongly disagrees with the observation. From direct measurements of tritium beta decay, the electron neutrino mass is less than 0.8 eV [32]. Early cosmology data including CMB constrain the sum of the neutrino masses to be less than around 0.1 eV [33] (see also the section by Lesgourgues and Verde in Ref. [28] entitled “Neutrinos in cosmology”³). This mass range is more than a million times smaller than the mass of the electron, the lightest charged particle.

A clue to understand the smallness of the neutrino masses may be the fact that neutrinos have no electric charge. Unlike charged quarks and leptons, neutrinos can possibly be Majorana fermions. Although lepton number distinguishes neutrinos from antineutrinos, lepton number conservation is based on a global symmetry and is not on a firm ground compared to gauge symmetries. The introduction of neutrino Majorana mass terms to the Standard Model can in fact be done in several ways.

The left-handed neutrino forms an SU(2) doublet with the corresponding charged lepton. Thus the Majorana mass term for the “left-handed” neutrino |$\nu \cdot \nu + {\rm h.c.}$| behaves as |$I=I_3=-Y=1$| in the Standard Model gauge group and is not invariant. If there is an extra triplet Higgs field |$\chi$| with |$I=Y=1$|⁠, gauge invariant Yukawa coupling can be written in the form (for simplicity we do not write the generation degrees of freedom)

where |$\ell _L$| is the doublet left-handed lepton field, h is the triplet Yukawa coupling, and |$\tau _a$| is the gauge SU(2) Pauli matrix. (⁠|$i\tau _2$| is not physically the second component of the Pauli matrix, but the “charge conjugation” matrix of internal SU(2), which makes an SU(2) doublet |$\varphi$| and its complex conjugate |$\varphi ^{*}$| transform in exactly the same way under SU(2) transformation. Thus |$\varphi ^{*} i\tau _2\tau _a\varphi$| is the spin-1 combination formed by the product of two spin 1/2. So it may be better to use different notation such as |$\epsilon$| instead of |$i\tau _2$|⁠.) When the neutral component of the triplet Higgs field develops a vacuum expectation value (vev), the interaction terms (32) produce a neutrino Majorana mass of the order |$h\langle \chi \rangle$|⁠. The smallness of the neutrino masses may be due to the smallness of the Yukawa coupling h and/or the smallness of the triplet vev.

This scenario requires the introduction of a new scalar field, but one may notice that a triplet scalar operator can be formed from the usual doublet Higgs field as

without introducing new fields. An interaction term

may be written down as a gauge invariant interaction. This operator has dimension five and is not renormalizable, as reflected in the negative dimension factor |$1/\Lambda$| in front of the operator. When |${\rm SU(2) \otimes U(1)}_Y$| is broken in the usual way, the interaction (34) gives a Majorana mass to the neutrino

where v is the usual Higgs vev (⁠|$\sim 250$| GeV). If the mass scale |$\Lambda$| is very large, the resulting neutrino Majorana mass becomes tiny.

It is possible to make up a renormalizable “UV completion” of this scenario by introducing right-handed neutrinos. Being gauge singlets, right-handed neutrinos can have gauge invariant Majorana mass terms in the Lagrangian. In addition, Yukawa couplings between the left-handed lepton doublet and the right-handed neutrino can be included to give the usual Dirac mass. The relevant terms in the Lagrangian are

It is convenient to rewrite the right-handed neutrino fields using their conjugate left-handed antineutrino fields |$N_L=(\nu _R)^c$|⁠, so

The neutrino mass terms after |${\rm SU(2) \otimes U(1)}_Y$| breaking form a |$2\times 2$| matrix:

Notice that the right-handed Majorana mass |$M_R$| is independent of the |${\rm SU(2) \otimes U(1)}_Y$| breaking scale (the Higgs mass term) and can be very large, while the Yukawa coupling |$f_\nu$| naturally lies in the same range as the other Yukawa couplings. If so, diagonalizing the matrix gives two mass eigenstates with masses |$\sim M_R$| and |$m_\nu \sim (f_\nu v)^2/M_R$|⁠. The mixing of the two states is of the order |$\sim f_\nu v/M_R$|⁠, so the lighter state essentially consists of the left-handed (doublet) neutrino. This scenario [34,35] is called the “seesaw mechanism.” If one takes |$m_\nu \sim 0.1$| eV and assumes a Dirac mass term of 1 MeV–100 GeV, |$M_R$| ranges from |$10^4$|–|$10^{14}$| GeV.

The Majorana mass term violates lepton number conservation. This effect is essentially invisible at high energies, where the mass can be neglected. The most promising process to detect lepton number violation is neutrinoless double beta decay.

It is known that some nuclei stable against beta decay can decay to the second neighboring nucleus by emitting two electrons and two neutrinos:

where Z is the atomic number (number of protons) of the nucleus and A is the mass number (number of protons and neutrons) of the nucleus. This is the second-order process of beta decay and has a very long lifetime. Around 10 kinds of nuclei have been observed to decay in this way, and the measured lifetimes are in the range of |$10^{18}$|–|$10^{24}$| yr [36].

If neutrinos have Majorana masses and the lepton number is violated, double beta decay without neutrino emission becomes possible:

The matrix element of the process is proportional to a combination of neutrino Majorana masses and neutrino mixing angles/phases. The sum of the energy of the two electrons is monochromatic, in contrast to that in the usual double beta decays. Searches for neutrinoless double beta decay have been undertaken by a number of experiments. The current best limit of |$2.3\times 10^{26}$| yr for this type of decay was obtained for |${}^{136}{\rm Xe}$| by the KamLAND-Zen Collaboration [37]. This limit corresponds to the limit on the “effective” Majorana mass of 36–156 meV, depending on the evaluation uncertainty of the nuclear matrix element.

7. Baryon number of the universe and leptogenesis

We are made of matter, nucleons and electrons. This is the case not only for ourselves: we have so far observed no trace of antimatter existing in the universe, except for tiny amounts in cosmic rays etc., which can be understood to be products of high-energy collisions of normal matter. In the very early universe with extremely high temperatures (⁠|$T\gg 1$| GeV), however, it is expected that matter and antimatter existed essentially in equal amounts. For example, high-energy photons can be converted to any particle–antiparticle pair when energetically possible. When the temperature was reduced, matter and antimatter annihilated each other and a very small amount of matter should have been left. The amount of matter can be calculated from the measurement of cosmological parameters using the cosmic microwave background (CMB) [33] and others. Comparing this with the known amount of 2.7 K CMB photons, we find a baryon-to-photon ratio of

In the very early universe baryons and antibaryons were in thermal equilibrium with other particles including photons. Their number density was of the same order as photons. The asymmetry between baryons and antibaryons in the early universe was therefore of the order of |$10^{-9}$|⁠.

It is known that three ingredients are needed to produce asymmetry between baryons and antibaryons [38]: (1) a baryon number nonconserving process; (2) violation of CP (and C) invariance; and (3) deviation from thermal equilibrium.

We have no evidence for the instability of protons (baryon number nonconservation). Indeed there are no baryon number violating interactions in the Standard Model Lagrangian. It has been shown [39], however, that because only left-handed fermions couple to the SU(2) W boson (the theory is chiral), the quantum effect violates conservation of baryon number current. This is the well-known chiral anomaly. The effect becomes physical only when it is coupled with a nonpertubative gauge field configuration, and is totally negligible at our energy scale. The effect can be substantial when the temperature is above the electroweak scale, and this should be taken into account in tracing the history of the early universe [40]. In addition to baryon number B, lepton number L is also violated. It should be stressed that the difference |$B-L$| is free from anomaly and is conserved.

In the Standard Model, CP violation arises from the complex phase of the Yukawa couplings, as was shown by Kobayashi and Maskawa [41]. Unfortunately its effect on baryon number generation is many orders of magnitude smaller than needed to explain the baryon asymmetry [42].

A scenario called leptogenesis [43] is an attractive alternative to derive the baryon asymmetry. It proceeds in two steps. First, asymmetry in the lepton number is created at a very early stage of the universe (⁠|$T\gg 1$| TeV), when |${\rm SU(2)\otimes U(1)}_Y$| is unbroken. Then, when the temperature of the universe drops to around the electroweak scale, transition from the symmetric universe to the spontaneously broken state occurs (the electroweak phase transition; the Higgs field develops a vacuum expectation value). During the transition, part of the lepton number asymmetry can be transferred to baryon number asymmetry when the universe is temporarily out of thermal equilibrium.

The first stage generation of net lepton number can be related to right-handed Majorana neutrinos with CP violating interaction. The right-handed neutrino (denoted N) can decay into the modes |$N\rightarrow \nu _L H$| and |$N\rightarrow \bar{\nu }_R H$|⁠, because the N Majorana mass itself violates lepton number conservation. If the Yukawa couplings contain CP violation, the branching ratios of these two modes can be unequal. At very high temperatures, however, reverse reactions (formation of N) also proceed at the same rate and no net amount of lepton number is produced. Departure from this state of thermal equilibrium can be realized when the temperature drops to slightly below the N mass. Because the typical energy of particles in the universe becomes insufficient to produce N, the reverse reaction is suppressed, and lepton number can be generated by the asymmetric decay processes. By adjusting the parameters in the model, this scenario can explain the amount of baryon asymmetry [44], as generation mixing of neutrinos is much larger than that of quarks.

8. Grand Unified Theories

A Grand Unified Theory (GUT) is a gauge theory with a large gauge group that contains the three gauge groups of the Standard Model. Moreover, a GUT typically unifies quarks and leptons, putting them in same gauge multiplets. The simplest GUT is based on the group SU(5) [45]. One generation of quarks and leptons, comprising five irreducible representations of |${\rm SU(3)\otimes SU(2)\otimes U(1)}_Y$|⁠, is merged into two SU(5) representations |$5^{*}$| and 10 (⁠|$5^{*}$| is the conjugate representation of the basic 5 representation of SU(5), and 10 is the antisymmetric second-rank tensor representation contained in |$5\otimes 5$|⁠):

where all the Weyl fields are taken to be left-handed. It is not just the quarks and leptons that are in the same representations; the quarks and antiquarks are in the same 10 representation. This shows that baryon number and lepton number cannot be defined in the GUT Lagrangian. Only after the breaking of the SU(5) gauge symmetry does it become possible to distinguish quarks from leptons, and also quarks from antiquarks.

The SU(5) gauge field can be put in a traceless |$5\times 5$| Hermitian matrix. The first diagonal |$3\times 3$| part corresponds to color SU(3), the remaining diagonal |$2\times 2$| weak SU(2). The weak hypercharge |${\rm U}(1)_Y$|⁠, which commutes with SU(3) and SU(2), corresponds to a diagonal matrix proportional to |$(2, 2, 2, -3, -3)$|⁠.

Mathematically, this expresses that the 24D SU(5) adjoint representation may be decomposed into representations of its subgroup |${\rm SU(3)\otimes SU(2)\otimes U(1)}_Y$| as follows:

where the right-hand side denotes the representation in the form |${\rm (SU(3), SU(2))}_Y$|⁠. The first three correspond to gluons and |${\rm SU(2)}$| and |${\rm U(1)}_Y$| gauge bosons. We will touch on the remaining 12 states shortly. Likewise, decompositions of fermion representations are (as can be seen from Eq. (42))

The breaking of SU(5) to |${\rm SU(3)\otimes SU(2)\otimes U(1)}_Y$| can be attained if a scalar field in the adjoint representation 24 develops a nontrivial vacuum. Minimization of a quartic scalar potential gives at most two eigenvalues for the vacuum expectation value, and one possibility is the form proportional to Y, resulting in remaining symmetries of the Standard Model gauge group.

After the GUT symmetry breaking |${\rm SU(5)\rightarrow SU(3)\otimes SU(2)\otimes U(1)}_Y$|⁠, the extra gauge bosons not contained in the Standard Model gauge group develop masses around the GUT scale. The infrared theory containing only “massless” particles (compared to the GUT scale) is the Standard Model, In this stage, separate conservation laws of baryon and lepton numbers emerge as infrared symmetries (barring the nonperturbative effect discussed in the previous section). Couplings of the extra gauge bosons, however, do not conserve baryon and lepton numbers and lead to proton instability. Experimental nonobservation of proton decays [46] constrains the GUT scale to be at least as large as |$10^{16}$| GeV, which coincides with the expectation from the renormalization group evolution of the three gauge couplings [47] (better agreement with supersymmetry).

No SU(5) invariant mass terms are allowed for the fermions, since no combinations of |$5^{*}\otimes 10$|⁠, |$10\otimes 10$|⁠, or |$5^{*}\otimes 5^{*}$| contain singlets (in contrast to |$5^{*}\otimes 5$| and |$10\otimes 10^{*}$| appearing in the kinetic terms). This should be the case since SU(5) contains the Standard Model gauge group. Otherwise the fermions could have gained masses of the order of the GUT scale. The Higgs field in the Standard Model can most simply be put in the basic 5 representation of SU(5). The Yukawa couplings to generate quark and lepton masses are in the invariant combinations

The former generates down quark and charged lepton masses, and the latter is responsible for up quark masses. It can be seen that the latter interaction is of Majorana type, combining the 10 representation with itself.

There is one global symmetry respected by these Yukawa interactions. If no Yukawa interactions exist, the theory is symmetric under phase rotations of each of the three fields |$5^{*}_L$|⁠, |$10_L$|⁠, and |$5_H$|⁠. The two Yukawa interactions break two of the three symmetries. The remaining U(1) has the following charge X:

Although this |${\rm U(1)}_X$| is broken when the Standard Model Higgs field in |$5_H$| develops a vacuum expectation value, the combination |$X+4Y$| remains unbroken. The combination |$(X+4Y)/5$| turns out to be identical with |$B-L$|⁠, the difference of baryon and lepton number.

The SU(5) GUT group can be elevated to a larger group SO(10) [48], which has rank five, one rank larger than SU(5) and |${\rm SU(3)\otimes SU(2)\otimes U(1)}_Y$|⁠. A single generation of quarks and leptons are unified into a single representation 16, which is a spinor representation of SO(10). This representation is a complex representation and is not equivalent to its conjugate |$16^{*}$|⁠.

The relation between SU(5) and SO(10) group can be understood from the decomposition of the fermion representations

There is an extra SU(5) singlet in 16. This field is neutral under the Standard Model gauge group and can be naturally interpreted as the (charge conjugated) right-handed neutrinos. The X charge given above, with the assignment |$X(1_L)=5$|⁠, becomes a gauge symmetry in SO(10), as the fifth diagonal generator: |${\rm SO(10)} \supset {\rm SU(5)}\otimes {\rm U(1)}_X$|⁠. The fermion masses arise from the product

Higgs fields giving quark and lepton masses need to be in a representation on the right-hand side of Eq. (49). The Yukawa couplings are again in Majorana-type form. The SU(5) |$5_H$| and its conjugate |$5^{*}_H$| are in the vector 10 representation of SO(10): |$10 = 5 \oplus 5^{*}$|⁠.

9. Supersymmetry

The principle of relativity states that physics laws are the same in any inertial coordinate frame. This means that the theory is invariant under the Poincaré group (inhomogeneous Lorentz group), whose generators are the energy–momentum four vector |$P^\mu$| and a Lorentz generator |$M^{\mu \nu }$| consisting of boost and angular momentum operators. Conserved quantum numbers such as electric charge Q are generators of internal symmetries that commute with the Poincaré group, and are Lorentz scalar. The question of whether other kinds of symmetry can exist is negatively answered by the Coleman–Mandula theorem [49]. With the assumption of nontrivial interactions (scattering matrix |$S\ne 1$|⁠) and massive particle states, vector charges or higher-rank tensor charges other than the Poincaré generators are not allowed.

If one extends the concept of symmetry to include operations relating bosons to fermions, with anticommuting (Grassmann) parameters, new kinds of symmetry become possible. The corresponding charges are restricted to being a spinor (having spin-1/2) [50], and the algebra of the spinor charges provide an extension of the Poincaré algebra. This is called supersymmetry [51]. In its minimal version (with one kind of spinor charge, |$N=1$|⁠), the spinor charge forms a Majorana spinor with its anticommutator given by

or equivalently

The Hamiltonian can be read off from these relations:

The last expression explicitly shows that H is a nonnegative operator. Supersymmetry is thus interwoven with Poincaré generators to form a super-Poincaré algebra. This may be regarded as the only possible nontrivial extension of inhomogeneous Lorentz symmetry.

The spectrum of the states in theories having these symmetries has special characteristics. To see the essence, it is useful to consider supersymmetry in |$0{+}1$| dimensions, with the algebraic relation

Starting from a bosonic state |$|B\rangle$|⁠, operating Q to the state gives a fermionic state |$|F\rangle =Q|B\rangle$|⁠. If we apply Q again, we find

The resulting state is nothing but the original bosonic state. Therefore the states form a pair made of a bosonic and a fermionic state. We can also see that the energy eigenvalues are the same for these two states by further operating Q the third time. When the first state is fermionic, the same conclusion is also obtained. An exception to this pairing can occur if |$Q|B\rangle =0$|⁠. Then the state |$|B\rangle$| has zero energy (easily seen by applying Q again). Thus physical states should always form a boson–fermion pair, except for zero-energy states like the vacuum.

This situation also holds in higher space dimensions, and the numbers of bosonic and fermionic states have to be equal. In our |$3{+}1$| dimensions, the representation of the states depends on whether the particles are massless or massive. For the massless case, states having helicities |$\lambda$| and |$\lambda +1/2$| form a pair. The corresponding field multiplets containing the lowest spin states consist of a complex scalar field and a Weyl spinor field. The degrees of freedom for bosons and fermions are both two. This multiplet, called a chiral supermultiplet, can also be written in terms of two real scalar fields and a Majorana field. Next comes a pair of spin 1/2 and spin 1. The helicities of the states are |$\pm 1/2$| and |$\pm 1$|⁠; the degrees of freedom are both two. The existence of both positive and negative helicities is the consequence of CPT invariance. The massless spin-1 state is necessarily a gauge boson for consistent field theory description, and forms an adjoint representation of the gauge group. The spin-1/2 fermion is a Majorana fermion and should also be in the adjoint representation.

Massive multiplets are different from massless multiplets in general. This may be anticipated from the difference that we have seen for the Poincaré algebra. An exception is the lowest spin case, spin 0 and spin 1/2. The smallest supermultiplet consists of two real scalars and a Majorana fermion. If the fermion is charged, two complex scalars and a Dirac fermion make a massive supermultiplet, so the states must be doubled. A supermultiplet containing massive spin-1 states is more complicated. A massive spin-1 state has three helicity degrees of freedom and this should match the fermionic degrees of freedom. One possibility is a supermultiplet containing one spin-0 state, two spin-1/2 states, and one spin-1 state (bosonic: |$1+3$|⁠; fermionic: |$2\times 2$|⁠), which may be realized when the gauge symmetry is spontaneously broken by the vacuum expectation value of a scalar Higgs field. Another possibility is one spin-1/2 state, two spin-1 states, and one spin-3/2 state.

The equalness of bosonic and fermionic degrees of freedom has a profound consequence in field theories that possess supersymmetry. A free field is equivalent to an infinite number of harmonic oscillators. The ground state of each oscillator has a zero-point energy of |$\hbar \omega /2$| and the total vacuum energy diverges. Usually the infinity is just subtracted in defining the energy of the vacuum. The situation is different in supersymmetric field theories. Each mode of fermions, unlike those of bosons, has a zero-point energy of |$-\hbar \omega /2$| due to the anticommuting property. When the bosonic and fermionic degrees of freedom coincide with each other, as in supersymmetric theories, the zero-point energies exactly cancel and the vacuum energy vanishes without subtraction. The absolute value of energy, therefore, is well defined in supersymmetric theories.

10. Minimal supersymmetric standard model

Cancellation between contributions from bosons and fermions in supersymmetric theory is not limited to the vacuum energy. It has been invoked to resolve an uncomfortable situation related to the high-energy structure of the Standard Model. The one-loop correction to the Higgs self-energy has quadratic divergence, which is absent in the self-energies of fermions and gauge bosons. When we cut off the momentum integral over intermediate states around |$\Lambda$|⁠, the one-loop self-energy is of the order of |$\lambda \Lambda ^2/16\pi ^2$|⁠, where |$\lambda$| is the Higgs self-coupling. The cutoff |$\Lambda$| should be larger than the limit under which scale the Standard Model is valid. If we expect the Standard Model to be valid up to the extremely large scale suggested by grand unification, the one-loop quadratic contribution becomes many order of magnitude larger than the physical Higgs mass term. The Bare Higgs mass must be fine tuned extremely well (something like 30 orders of magnitude) to cancel the loop contribution and leave the tiny physical |$m_H^2$|⁠.

This conceptual problem of fine tuning can be avoided if there is supersymmetry, because of cancellations of bosonic and fermionic contributions in the loop. Cancellations occur not only in the vacuum energy (quartic divergence) but also in the quadratically divergent piece, leaving a logarithmically divergent piece that is not problematic. Of course, particles forming supersymmetric pairs do not exist in the Standard Model, nor in the list of particles that we have found so far. To supersymmetrize the Standard Model requires the introduction of “superpartners” for all particles in the Standard Model, doubling the number of degrees of freedom. In addition, two Higgs doublets are necessary to produce the masses of all quarks and leptons due to constraints on the interactions imposed by supersymmetry.

The resulting theory is called the minimal supersymmetric standard model (MSSM) [52,53]. It contains supersymmetric partners of every particle in the Standard Model, quarks and leptons, gauge bosons, and Higgs bosons. These are named squarks and sleptons, gauginos, and higgsinos. These particles have spin that differs by 1/2 from the original particle. The squarks and sleptons (sfermions) are scalars, and gauginos and higgsinos have spin 1/2. The internal quantum numbers of these superpartners are equal to those of the original particles.

If the theory is totally supersymmetric, supersymmetry transformations commute with the Hamiltonian, and supersymmetric pairs have equal masses. This is not the case in nature, so supersymmetry must be broken. Spontaneous breaking of supersymmetry may be attractive, but does not work in MSSM, so explicit supersymmetry breaking terms are added to the Lagrangian. These breaking terms are restricted to the so-called “soft” breaking, which does not generate quadratic divergence and upset the cancellation in the Higgs self-energy. This soft breaking is supposed to be the consequence of spontaneous breaking in a “hidden” sector contained in the whole theory, transmitted to our MSSM sector by some mechanism. A number of such possibilities of “mediation” have been proposed. In addition, the masses of the superpartners should not be too large compared with the weak scale (⁠|$m_W$| or |$m_H$|⁠), since the quadratic cutoff |$\Lambda$| is replaced with the mass of superpartners in the loop integral.

Among the Standard Model particles, gauge bosons (except |$W^\pm$|⁠) are self-conjugate particles. These gauge bosons, gluinos, photons, Z have spin-1/2 superpartners that are self-conjugate, Majorana fermions. Superpartners of the neutral Higgs bosons are also Majorana fermions. After the spontaneous breaking of the gauge |${\rm SU(2)\otimes U(1)}_Y$| symmetry due to the Higgs vacuum expectation values (the two Higgs fields should both develop vacuum expectation values), mixing arises between the neutral SU(2) and |${\rm U(1)}_Y$| gauginos and two higgsinos, giving four mass eigenstates in all. These mixed states are called neutralinos and all are Majorana particles. The lightest of these neutralinos is stable in most models, where discrete symmetry |$R=(-1)^{3B+L+2J}$| is present. All the known particles have |$R=+$| and the superpartners have |$R=-$|⁠, and the lightest neutralino is typically the lightest R-odd particle. The lightest neutralino interacts with matter only weakly like neutrinos, and is expected to have a mass around the weak scale. It is an attractive candidate for dark matter particles in the universe.

Extensive searches have been made at high-energy colliders to discover various superpartners directly. Unfortunately, no trace of supersymmetry has so far been found. Proposals of possible patterns of supersymmetric particle masses compatible with present experimental limits have been made. On the other hand, superstring theory, which unites all known fundamental interactions including gravity, needs supersymmetry for theoretical consistency, and we expect that supersymmetry appears at some energy scale, if not at the weak scale.

Motivations for supersymmetry have been rather theoretical. In contrast, we do have observational needs for physics beyond the Standard Model. These include neutrino masses, dark matter, dark energy, the baryon number of the universe, and inflation in the very early universe. We have seen that at least three of these can be related to Majorana fermions.

Funding

Open Access funding: SCOAP³.

Footnotes

References