Chiral Symmetry in Dirac Equation and its Effects on Neutrino Masses and Dark Matter

The work of Wigner [1] was among the first to highlight the crucial role of group theory in Quantum Field Theory (QFT). He classified the irreducible representations (irreps) of the Poincaré group and identified an elementary particle with an unitary irrep of the group [2]; the explication of single particle states as the irreps of the Poincaré group has done much to validate the physical foundations of modern particle physics (e.g., [3]). As elucidated in previous works [4-7], such group theoretical notions may be utilized to define a fundamental theory as satisfying the following principles: the principle of Poincaré and gauge invariance, the principle of locality, and the principle of least action.

Among the fundamental equations of QFT, the Dirac equation [8] plays a special role because it describes fermionic fields, which include quarks and leptons of the Standard Model of particle physics [3]. The original Dirac method to obtain this equation [9] is often replaced in QFT textbooks by giving the required Lagrangian but without any explanation of its first principle origin (e.g., [10]). Other methods to derive the Dirac equation include the Lorentz transformations of four-component spinors [3] and the Bargmann-Wigner approach [4] that is based on the Poincaré group of the Minkowski metric of QFT. The general structure of the Poincaré group is $\mathcal{P}\ =\ SO(3,1)\otimes_{s}T(3+1)$, where $SO(3,1)$ is a non-invariant Lorentz group of rotations and boosts and $T(3+1)$ an invariant subgroup of spacetime translations, and this structure includes reversal of parity and time [2].

In this paper, we present a novel method that uses the irreps of $T(3+1)$ to derive the Dirac equation with chiral symmetry. There are two main aims of this paper, namely, to derive the Dirac equation with chiral symmetry and discuss its far reaching physical implications. Throughout this paper, we refer to our new equation as the Dirac equation with chiral symmetry (DECS) to make distinction from the commonly used name ’generalized Dirac equation’ in many published papers.

Different generalizations of the Dirac equation (DE) were previously presented and, in general, they can be divided into two cathegories: those that derived the equation and those that just add ad hoc terms to it. More specifically, some obtained equations were used to either unify leptons and quarks [11-13], or account for different masses of three generations of elementary particles [14-16], or extend the mass term to include ad hoc a pseudoscalar mass [17] . In other generalizations, the DE was derived for distances comparable to the Planck length [18], or with the external magnetic field included [19], or even for higher integer and half-integer spins [20], which requires combining the Dirac [8] and Klein-Gordon [21,22] equations. Another generalization of the DE involved changing a phase factor in four-component spinors, which allowed for different masses in the equation [23].

The Dirac equation with chiral symmetry derived in this paper is new and its physical implications are far reaching. First, our derivation of the DECS demonstrates that this equation can be obtained from the eigenvalue equation that represents the condition required by the four-component spinors to transform as one of the irreps of the Poincaré group $\mathcal{P}$ extended by parity. Second, the eigenvalue equation allows deriving either the DE or the DECS. Third, the DE is obtained by factorization of the Klein-Gordon equation if, and only if, a specific choice of chirial basis is selected. Fourth, as compared to the DE, there is an extra mass term in the DECS and its properties allow identifying it with pseudo-scalar mass.

Our formal derivation of the pseudo-scalar mass term and relating it directly to chirial symmetry gives physical justification for the existence of this term and allows us to discuss its physical implications; this makes our approach so different from the previuosly ad hoc addition of pseudo-scalar mass to the DE without neither justifying its physical presence nor its origin [17]. Our obtained results demonstrate that this pseudo-scalar mass in the DECS, its relationship to chiral symmetry and the resulting pseudo-scalar Higgs can be used to explain smallness of neutrino masses [24,25], and also properties Dark Matter (DM) particles [26,27].

The theory presented in this paper emphasizes the distinction between the Lorentz-invariant (but non-conserved) concept of chirality and the non-Lorentz-invariant (but conserved) property of helicity for massive neutrinos. The neutrino considered here is a left-handed Dirac particle.

The paper is organized as follows: the eigenvalue equation for bispinors are given in Section2; the fundamental equation for bispinors with chiral symmetry is derived in Section 3; physical implications of the obtained results are discussed in Section 4; and conclusions are presented in Section 5.

To determine forms of the matrices $X^{\mu}$ and $Y$, the Lorentz transformation must be applied to the RHS and LHS of Eqs. (4) to (3), and this yields

$$\big{(}(\Lambda^{-1}\hat{\Lambda}^{\nu}_{\mu}X^{\mu}\Lambda)\partial_{\nu}+(\Lambda^{-1}Y\Lambda)\big{)}\psi=0\ .$$

(6)

This leads to the necessary conditions for invariance

$$\hat{\Lambda}^{\nu}_{\mu}X^{\mu}=\Lambda X^{\nu}\Lambda^{-1}\qquad\hskip 18.06749pt{\rm and}\hskip 18.06749ptY=\Lambda Y\Lambda^{-1}\ .$$

(7)

Solving these, we find the most general form of our matrix coefficients written in block form

$$X^{\mu}={\begin{bmatrix}0&x_{R}(\sigma^{0}\delta^{\mu}_{0}+\sigma^{k}\delta^{\mu}_{k})\\ x_{L}(\sigma^{0}\delta^{\mu}_{0}-\sigma^{k}\delta^{\mu}_{k})&0\\ \end{bmatrix}}$$

(8)

$$Y={\begin{bmatrix}y_{L}\sigma^{0}&0\\ 0&y_{R}\sigma^{0}\\ \end{bmatrix}}$$

(9)

where $x_{R}$, $x_{L}$, $y_{R}$, and $y_{L}$ are free parameters.

Taking the Dirac $\gamma$ matrices in the Weyl basis

$$\gamma^{0}={\begin{bmatrix}0&\sigma^{0}\\ \sigma^{0}&0\\ \end{bmatrix}}\qquad\gamma^{k}={\begin{bmatrix}0&\sigma^{k}\\ -\sigma^{k}&0\\ \end{bmatrix}}$$

(10)

and identifying the chiral projection operators as

$$P_{L}={\begin{bmatrix}\sigma^{0}&0\\ 0&0\\ \end{bmatrix}}\qquad P_{R}={\begin{bmatrix}0&0\\ 0&\sigma^{0}\\ \end{bmatrix}}$$

(11)

we obtain

$$\big{(}(x_{L}P_{R}+x_{R}P_{L})\gamma^{\mu}\partial_{\mu}+(y_{L}P_{L}+y_{R}P_{R})\big{)}\psi=0\ .$$

(12)

Now, under the assumption that $x_{L}$ and $x_{R}$ are nonzero we are free to multiply from left with $i(x_{L}P_{R}+x_{R}P_{L})^{-1}$, and obtain

$$\bigg{(}i\gamma^{\mu}\partial_{\mu}+i\bigg{(}\frac{y_{L}}{x_{R}}P_{L}+\frac{y_{R}}{x_{L}}P_{R}\bigg{)}\bigg{)}\psi=0\ ,$$

(13)

which shows that there are only two independent degrees of freedom in the derived equation. To identify the physical basis of these degrees, we observe that Eq. (13) gives the following squared-Hamiltonian

$$\mathcal{H}^{2}\psi=\bigg{(}\partial^{k}\partial_{k}-\frac{y_{L}y_{R}}{x_{L}x_{R}}\bigg{)}\psi\ ,$$

(14)

and thus we find the emergence of the propagation mass term

$$m\equiv\pm i\sqrt{\frac{y_{L}y_{R}}{x_{L}x_{R}}}\ .$$

(15)

The restriction of the square of Eq. (15) to positive real numbers is equivalent to the physical restriction of Einstein energy-momentum relationship. Our remaining degree of freedom may be identified with the choice of a chiral basis. Let us define the chiral angle as

$\displaystyle\alpha\equiv-\frac{i}{2}\ln{\bigg{(}\mp\sqrt{\frac{x_{L}y_{L}}{x_{R}y_{R}}}}\bigg{)}\ .$

(16)

Thus, the final compact form of our Dirac equation with chirial symmetry is

$\displaystyle(i\gamma^{\mu}\partial_{\mu}-me^{-2i\alpha\gamma^{5}})\psi=0\ .$

(17)

Since this equation is Poincaré invariant, it is the fundamental equation of physics. This equation reduces to the original Dirac equation when $\alpha=0$.

A consequence of Eq. (17) is made explicit by proffering an alternative, suggestive parameterization. Let us define

$\displaystyle M\equiv-\frac{i}{2}\bigg{(}\frac{y_{R}}{x_{L}}+\frac{y_{L}}{x_{R}}\bigg{)}=m\cos{2\alpha}\ ,\hskip 19.91692pt\widetilde{M}\equiv-\frac{i}{2}\bigg{(}\frac{y_{R}}{x_{L}}-\frac{y_{L}}{x_{R}}\bigg{)}=-im\sin{2\alpha}\ ,$

(18)

so that Eq. (17) becomes

$$\big{(}i\gamma^{\mu}\partial_{\mu}-M-\widetilde{M}\gamma^{5}\big{)}\psi=0\ .$$

(19)

This form admits the simultaneous validity of both fundamental scalar and fundamental pseudoscalar mass terms. To better understand these mass terms, consider a global chiral transformation given by

$$\psi\rightarrow\psi^{\prime}=e^{i\gamma^{5}\beta}\psi\ .$$

(20)

The Lagrangian of Eq. (20) may be written as

$$\mathcal{L}=i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\bar{\psi}(M+\widetilde{M}\gamma^{5})\psi\ ,$$

(21)

and under the transformation of (20) this becomes

$$\mathcal{L^{\prime}}=i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-\bar{\psi}(M+\widetilde{M}\gamma^{5})e^{-2i\beta\gamma^{5}}\psi\ ,$$

(22)

so we the effect of (20) is equivalent to the rotation of our mass parameters

$${\begin{bmatrix}M^{\prime}\\ i\widetilde{M}^{\prime}\\ \end{bmatrix}}={\begin{bmatrix}\cos{2\beta}&-\sin{2\beta}\\ \sin{2\beta}&\cos{2\beta}\\ \end{bmatrix}}{\begin{bmatrix}M\\ i\widetilde{M}\\ \end{bmatrix}}={\begin{bmatrix}m\cos[{2(\alpha+\beta)}]\\ m\sin[{2(\alpha+\beta)}]\\ \end{bmatrix}}$$

(23)

This is precisely the transformation required to leave invariant the propagation mass in Eq. (17). Note that for a single field, we may always apply a chiral rotation so as to reorient our chiral axes and force the pseudoscalar mass to vanish. However, this is not a valid procedure in the case of multipartite systems whose constituent fields obey Eq. (17) for different chiral anlges. Therefore, we conclude that it is only the nonequivalence of chiral basis that gives rise to physically observable effects.

We found that the chiral rotation of a massive field is equivalent to an alternative choice of chiral basis. This, in turn, is an alternative factorization of the Klein-Gordon equation (i.e. the Einstein energy relationship). These nonstandard factorizations necessarily redistribute the fraction of the mass available to the field between the left- and right-chiral components. Our work presented in this paper significantly differs from the previous studies of both chirial symmtery and generalization of Dirac equation.

Our main result is first-principle proof of the necessity of specifying the chiral basis in which the fundamental induced representations of the Poincaré group reside. This is not a mere mathematical triviality, but rather the necessary consequence of considering fundamental physical symmetries in flat space-time and its anticipated physical implications that are now presented and discussed.

In this paper, a group-theoretical derivation of the most general Poincaré invariant Dirac equation with chirial symmetry, which introduces pseudo-scalar mass to the equation, is presented. The derivation is based on the eigenvalue equation that represents the condition required by the four-component spinors to transform as one of the irreps of the Poincaré group. It is shown that the chiral rotation of a massive field is equivalent to a choice of chiral basis and that this choice must be specified in order to factorize the Klein-Gordon equation and derive the original Dirac equation.

The derived Dirac equation with chirial symmetry has two additional degrees of freedom, which are identified with the propagation mass ($m$) and the chiral angle ($\alpha$). Being a physical parameter, $\alpha$ specifies the chiral basis of the spinor solutions and therefore its value must be physically justified. The existence of the pseudo-scalar mass generating Higgs-like couplings is an interesting phenomenon as this pseudo-scalar Higgs when combined with the suppression of right-chiral neutrinos offers a natural mechanism for producing anomalously small left-chiral neutrino masses. Moreover, this psuedo-scalar Higgs field satisfies many of the requirements for a DM candidate, whose existence can be verified experimentally.

The derived Dirac equation with chiral symmetry applies to all fermions in the Standard Model because the introduced chiral angle by the equation is present in all particles with spin $1/2$. Therefore, once interations are included into the theory, its further integration with the Standard model will be done and reported separately.

We are grateful to an anonymous referee for providing comments and suggestions that allowed us to improve the revised version of this paper. We thank B. Jones for his comments on our first draft of this paper and for bringing to our attention the paper by Leiter and Szamosi (1972). ZEM also thanks for a partial support of this research by the Alexander von Humboldt Foundation.