Chiral Symmetry in Dirac Equation and its Effects on Neutrino Masses and Dark Matter
aa r X i v : . [ phy s i c s . g e n - ph ] N ov Chiral Symmetry in Dirac Equation and itsEffects on Neutrino Masses and Dark Matter
T.B. Watson and Z. E. Musielak
Department of Physics, The University of Texas at Arlington, Arlington, TX76019, USA
Abstract
Chiral symmetry is included into the Dirac equation using the irreducible repre-sentations of the Poincar´e group. The symmetry introduces the chiral angle thatspecifies the chiral basis. It is shown that the correct identification of these basisallows explaining small masses of neutrinos and predicting a new candidate for DarkMatter massive particle.
The work of Wigner [1] was among the first to highlight the crucial role ofgroup theory in Quantum Field Theory (QFT). He classified the irreduciblerepresentations (irreps) of the Poincar´e group and identified an elementaryparticle with an unitary irrep of the group [2]; the explication of single particlestates as the irreps of the Poincar´e group has done much to validate thephysical foundations of modern particle physics (e.g., [3]). As elucidated inprevious works [4-7], such group theoretical notions may be utilized to definea fundamental theory as satisfying the following principles: the principle ofPoincar´e and gauge invariance, the principle of locality, and the principle ofleast action.Among the fundamental equations of QFT, the Dirac equation [8] plays a spe-cial role because it describes fermionic fields, which include quarks and leptonsof the Standard Model of particle physics [3]. The original Dirac method to ob-tain this equation [9] is often replaced in QFT textbooks by giving the requiredLagrangian but without any explanation of its first principle origin (e.g., [10]).Other methods to derive the Dirac equation include the Lorentz transforma-tions of four-component spinors [3] and the Bargmann-Wigner approach [4]that is based on the Poincar´e group of the Minkowski metric of QFT. Thegeneral structure of the Poincar´e group is P = SO (3 , ⊗ s T (3 + 1), where Preprint submitted to Elsevier 3 November 2020 O (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 includesreversal of parity and time [2].In this paper, we present a novel method that uses the irreps of T (3 + 1) toderive the Dirac equation with chiral symmetry. There are two main aims ofthis paper, namely, to derive the Dirac equation with chiral symmetry anddiscuss its far reaching physical implications. Throughout this paper, we referto our new equation as the Dirac equation with chiral symmetry (DECS) tomake distinction from the commonly used name ’generalized Dirac equation’in many published papers.Different generalizations of the Dirac equation (DE) were previously presentedand, in general, they can be divided into two cathegories: those that derivedthe 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 tothe Planck length [18], or with the external magnetic field included [19], oreven for higher integer and half-integer spins [20], which requires combiningthe Dirac [8] and Klein-Gordon [21,22] equations. Another generalization ofthe DE involved changing a phase factor in four-component spinors, whichallowed for different masses in the equation [23].The Dirac equation with chiral symmetry derived in this paper is new andits physical implications are far reaching. First, our derivation of the DECSdemonstrates that this equation can be obtained from the eigenvalue equa-tion that represents the condition required by the four-component spinors totransform as one of the irreps of the Poincar´e group 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 comparedto the DE, there is an extra mass term in the DECS and its properties allowidentifying it with pseudo-scalar mass.Our formal derivation of the pseudo-scalar mass term and relating it directlyto chirial symmetry gives physical justification for the existence of this termand allows us to discuss its physical implications; this makes our approachso different from the previuosly ad hoc addition of pseudo-scalar mass to theDE without neither justifying its physical presence nor its origin [17]. Ourobtained results demonstrate that this pseudo-scalar mass in the DECS, itsrelationship to chiral symmetry and the resulting pseudo-scalar Higgs can beused to explain smallness of neutrino masses [24,25], and also properties DarkMatter (DM) particles [26,27]. 2he theory presented in this paper emphasizes the distinction between theLorentz-invariant (but non-conserved) concept of chirality and the non-Lorentz-invariant (but conserved) property of helicity for massive neutrinos. The neu-trino considered here is a left-handed Dirac particle.The paper is organized as follows: the eigenvalue equation for bispinors aregiven in Section2; the fundamental equation for bispinors with chiral symme-try is derived in Section 3; physical implications of the obtained results arediscussed in Section 4; and conclusions are presented in Section 5. The condition that a scalar wavefunction φ transforms as one of the irreps of T (3 + 1) ⊂ P is given by the following eigenvalue equation [6,7] i∂ µ φ = k µ φ , (1)where k µ labels the irreps. To generalize this result to Dirac spinors, calledalso bispinors, we follow Wigner [1,2], who proved that the proper irreps ofspin-1/2 elementary particles are the four-component bispinors ψ for whichthe eigenvalue equation given by Eq. (1) becomes iA µ ∂ µ ψ = A µ k µ ψ , (2)where A µ is an arbitrary constant matrix of 4 ×
4. Defining X µ = − iA µ and Y = A µ k µ , we obtain ( X µ ∂ µ + Y ) ψ = 0 , (3)with X µ and Y to be determined. This is the general condition for the bispinorsto transform as one of the irreps of P and this condition will be now used toderive the DECSfor the four-componet bispinors given by ψ = χ L χ R (4)where χ L and χ R are two component of bispinors. The necessity of couplingthese spinors is understood mathematically as accommodating the sign am-biguity introduced in the construction of the isomorphism between boosts inSO(3,1) and those in SU(2). As a result of this, we find χ L and χ R to trans-form identically under rotations, but oppositely under boosts [3]. We may thuswrite our Lorentz transformation for the bispinors asΛ = Λ L
00 Λ R = exp (cid:16) i~σ · ( ~θ − i~φ )2 (cid:17)
00 exp (cid:16) i~σ · ( ~θ + i~φ )2 (cid:17) (5)3here ~θ and ~φ parameterize our rotations and boosts, respectively, and arerelated to the transformations of four vectors via the four-by-generators ~J and ~K such that ˆΛ = exp i ~J · ~θ + i ~K · ~φ ! . To determine forms of the matrices X µ and Y , the Lorentz transformationmust be applied to the RHS and LHS of Eqs. (4) to (3), and this yields (cid:16) (Λ − ˆΛ νµ X µ Λ) ∂ ν + (Λ − Y Λ) (cid:17) ψ = 0 . (6)This leads to the necessary conditions for invarianceˆΛ νµ X µ = Λ X ν Λ − and Y = Λ Y Λ − . (7)Solving these, we find the most general form of our matrix coefficients writtenin block form X µ = x R ( σ δ µ + σ k δ µk ) x L ( σ δ µ − σ k δ µk ) 0 (8) Y = y L σ y R σ (9)where x R , x L , y R , and y L are free parameters.Taking the Dirac γ matrices in the Weyl basis γ = σ σ γ k = σ k − σ k (10)and identifying the chiral projection operators as P L = σ
00 0 P R = σ (11)4e obtain (cid:16) ( x L P R + x R P L ) γ µ ∂ µ + ( y L P L + y R P R ) (cid:17) ψ = 0 . (12)Now, under the assumption that x L and x R are nonzero we are free to multiplyfrom left with i ( x L P R + x R P L ) − , and obtain iγ µ ∂ µ + i y L x R P L + y R x L P R !! ψ = 0 , (13)which shows that there are only two independent degrees of freedom in thederived equation. To identify the physical basis of these degrees, we observethat Eq. (13) gives the following squared-Hamiltonian H ψ = ∂ k ∂ k − y L y R x L x R ! ψ , (14)and thus we find the emergence of the propagation mass term m ≡ ± i s y L y R x L x R . (15)The restriction of the square of Eq. (15) to positive real numbers is equivalentto the physical restriction of Einstein energy-momentum relationship. Ourremaining degree of freedom may be identified with the choice of a chiralbasis. Let us define the chiral angle as α ≡ − i ∓ s x L y L x R y R ! . (16)Thus, the final compact form of our Dirac equation with chirial symmetry is( iγ µ ∂ µ − me − iαγ ) ψ = 0 . (17)Since this equation is Poincar´e invariant, it is the fundamental equation ofphysics. This equation reduces to the original Dirac equation when α = 0. A consequence of Eq. (17) is made explicit by proffering an alternative, sug-gestive parameterization. Let us define M ≡ − i y R x L + y L x R ! = m cos 2 α , f M ≡ − i y R x L − y L x R ! = − im sin 2 α , (18)5o that Eq. (17) becomes (cid:16) iγ µ ∂ µ − M − f M γ (cid:17) ψ = 0 . (19)This form admits the simultaneous validity of both fundamental scalar andfundamental pseudoscalar mass terms. To better understand these mass terms,consider a global chiral transformation given by ψ → ψ ′ = e iγ β ψ . (20)The Lagrangian of Eq. (20) may be written as L = i ¯ ψγ µ ∂ µ ψ − ¯ ψ ( M + f M γ ) ψ , (21)and under the transformation of (20) this becomes L ′ = i ¯ ψγ µ ∂ µ ψ − ¯ ψ ( M + f M γ ) e − iβγ ψ , (22)so we the effect of (20) is equivalent to the rotation of our mass parameters M ′ i f M ′ = cos 2 β − sin 2 β sin 2 β cos 2 β Mi f M = m cos[2( α + β )] m sin[2( α + β )] (23)This is precisely the transformation required to leave invariant the propaga-tion mass in Eq. (17). Note that for a single field, we may always apply achiral rotation so as to reorient our chiral axes and force the pseudoscalarmass to vanish. However, this is not a valid procedure in the case of multipar-tite systems whose constituent fields obey Eq. (17) for different chiral anlges.Therefore, we conclude that it is only the nonequivalence of chiral basis thatgives rise to physically observable effects.We found that the chiral rotation of a massive field is equivalent to an al-ternative choice of chiral basis. This, in turn, is an alternative factorizationof the Klein-Gordon equation (i.e. the Einstein energy relationship). Thesenonstandard factorizations necessarily redistribute the fraction of the massavailable to the field between the left- and right-chiral components. Our workpresented in this paper significantly differs from the previous studies of bothchirial symmtery and generalization of Dirac equation.Our main result is first-principle proof of the necessity of specifying the chiralbasis in which the fundamental induced representations of the Poincar´e groupreside. This is not a mere mathematical triviality, but rather the necessaryconsequence of considering fundamental physical symmetries in flat space-time and its anticipated physical implications that are now presented anddiscussed. 6 Physical implications
As demonstrated above, together m and α define the infinite parameter spacein which all first-order Poincare-invariant equations for massive spin-1/2 parti-cles must reside. The simplest and most physically-motivated selection processfor the determination of m and α is obtained by the interpretation of scalarand pseudoscalar mass terms as having their origin in Yukawa couplings torealscalar fields with non-vanishing vacuum expectation values (VEVs).Treating these fields independently and expanding them about the groundstate, we may write the relevant contributions of these coupling to the La-grangian as L Y = − λ ¯ ψφ ψ − λ ¯ ψφ γ ψ ≈ − λ v √ ψψ − λ v √ ψγ ψ , (24)where the v and v are the VEVs of the two fields φ and φ , which are treatedhere as independent. It is easy to restrict this model to encompass a singlefield with a complicated set of coupling parameters. The coupling constants λ , λ and VEVs are related to our parameters m and α via m = s λ v − λ v α = i λ v + λ v λ v − λ v ! , (25)which allows both α and m to be completely fixed by the VEVs of the fields φ and φ and their respective coupling constants. The presented theory highlights the chiral angle as the necessary consequenceof the identification of particle states with the induced irreducible represen-tations of the Poinca.´ere group. This degree of freedom is necessarily presentin all fermions of the Standard Model. However, the non-observation of right-handed neutrinos (left-handed antineutrinos) in Nature permits motivates ap-plications of the theory to neutrinos, which are now described.In the context of Eq. (24), the case of neutrino fields takes on an interestingdimension by offering a potential solution to an open problem. In the Standard7odel [3], right-chiral leptons form singlets that preclude the existence ofright-chiral neutrinos. However, if this condition is enforced, then the neutrinofield ν becomes 12 (1 − γ ) ν = ν
12 (1 + γ ) ν = 0 . (26)Combining these constraints with Eq. (24), we find L Y = − √ ν ( λ v − λ v ) ν . (27)Thus, the neutrino mass in the Dirac Lagrangian appears as m ν = 12 √ λ v − λ v ) . (28)The obtained results demonstrate that the mass of a left-chiral field generatedby coupling to the Standard Model (SM) Higgs field may be suppressed bya non-zero real pseudoscalar coupling to an external scalar field with non-vanishing VEV. This is clearly of potential relevance when considering whythe observed masses of neutrinos vary from those of the other fermions by or-ders of magnitudes. Many different explanations have been offered to accountfor this discrepancy [24,25], but the mechanism presented here is particularlyappealing as it suggests the smallness of the neutrino masses is the neces-sary consequence of the non-existence of right-chiral neutrinos; more detailedcalculations of this phenomenon will be presented elsewhere.Let us point out that the most immediate testable consequence of additionalneutrino-scalar couplings is the necessary modification of elastic scatteringcross sections. While in the absence of interference phenomena it is likely formodifications to the SM predictions to be sub-leading, evidences of devia-tion from SM predictions have already been obtained by the LSND [28] andMiniBooNE [29] collaborations. So far these anomalies have refuted expla-nation. Howevre, additional precision measurements being preformed by theMicroBooNE collaboration and to-be-performed by the DUNE collaborationwill provide further constraints and may–given long enough livetimes–obtainconstraints sufficient to definitively probe the validity of the model proposed. Another attractive aspect of this mass-generating mechanism is the naturalinclusion of a dark matter candidate. The evidence for the existence of par-ticle dark matter is strongly supported by the astronomical observations ofgalactic rotational curves and stability of clusters of galaxies [31,32] and whilethere have been numerous theories postulated different elementary particles827,33,34], so far the existence of those particles have not yet been verified ex-perimentally [35-37]. As possible candidates, weakly interacting massive par-ticles (WIMPs), supersymmetric (SUSY) partciles like neutralinos [33], axions[38], and extermely light bosonic particles (ELBPs) suggested by [39,40] andshowed to be physically unacceptable by [41,42].All evidence suggests dark matter couples to ordinary matter primarily throughmass-mass terms, ostensibly gravitational [43,44]. All other couplings are, atmost, sub-leading contributions to DM interactions. Thus, the existence of achiral mass may offer insight into these observations in several ways. First, it ispossible that pseudoscalar mass contribution to the gravitational field differsfrom those of scalar mass terms; an in-depth investigation of that possibilityis beyond the scope of this paper. Second, the results obtained in this workmake also plausible the existence of a massive φ particle that couples min-imally and exclusively to chirally-asymmetric fields (neutrinos) and therebysatisfies many of the required characteristics of dark matter: a minimal non-gravitational coupling with a restricted phase space for decay resulting ina long-lived particle [45]. The contribution of this postulated field to gravi-tational interactions depends on the mass and abundance of the postulatedparticle, both of which cannot be determined by the presented theory.As a last remark, we note a distinguishing characteristic of this model is thewide range of allowable inertial (and thus gravitational) masses consistent withHiggs-portal and direct-detection observations. For massive DM particles, wemust amend the aforementioned exclusivity of interactions to include scalaras well as chiral-asymmetric couplings. Then, we must allow the mass of thepostulated particle to be generated by couplings to both φ (nominally thestandard model Higgs) and itself. This method of mass generation allows forenhancement of the inertial mass and admits a mass parameter that differsnon-trivially from its Higgs-coupling parameter. Thus, the proposed modelmay plausibly span the intermediate parameter space inaccessible to bothELBP and WIMP models. Intriguingly, a non-zero reciprocal coupling between φ and φ may have important implications for the renormalization of theHiggs self-energy in the absence of supersymmetric models. In this paper, a group-theoretical derivation of the most general Poincar´e in-variant Dirac equation with chirial symmetry, which introduces pseudo-scalarmass to the equation, is presented. The derivation is based on the eigenvalueequation that represents the condition required by the four-component spinorsto transform as one of the irreps of the Poincar´e group. It is shown that the chi-ral rotation of a massive field is equivalent to a choice of chiral basis and that9his choice must be specified in order to factorize the Klein-Gordon equationand derive the original Dirac equation.The derived Dirac equation with chirial symmetry has two additional degreesof freedom, which are identified with the propagation mass ( m ) and the chiralangle ( α ). Being a physical parameter, α specifies the chiral basis of the spinorsolutions and therefore its value must be physically justified. The existence ofthe pseudo-scalar mass generating Higgs-like couplings is an interesting phe-nomenon as this pseudo-scalar Higgs when combined with the suppression ofright-chiral neutrinos offers a natural mechanism for producing anomalouslysmall left-chiral neutrino masses. Moreover, this psuedo-scalar Higgs field sat-isfies many of the requirements for a DM candidate, whose existence can beverified experimentally.The derived Dirac equation with chiral symmetry applies to all fermions inthe Standard Model because the introduced chiral angle by the equation ispresent in all particles with spin 1 /
2. Therefore, once interations are includedinto the theory, its further integration with the Standard model will be doneand reported separately.
Acknowledgements
We are grateful to an anonymous referee for providing comments and sugges-tions that allowed us to improve the revised version of this paper. We thankB. Jones for his comments on our first draft of this paper and for bringing toour attention the paper by Leiter and Szamosi (1972). ZEM also thanks for apartial support of this research by the Alexander von Humboldt Foundation.