Bargmann Invariants, Geometric Phases and Recursive Parametrization with Majorana Fermions
aa r X i v : . [ h e p - ph ] D ec Bargmann Invariants, Geometric Phasesand Recursive Parametrization withMa jorana Fermions
Rohan Pramanick ∗ , Swarup Sangiri and Utpal Sarkar Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
Abstract
A generalized connection between the quantum mechanical Bargmann invariantsand the geometric phases was established for the Dirac fermions. We extend thatformalism for the Majorana fermions by defining proper quantum mechanical rayand Hilbert spaces. We then relate both the Dirac and Majorana type Bargmanninvariants to the rephasing invariant measures of CP violation with the Majorananeutrinos, assuming that the neutrinos have lepton number violating Majoranamasses. We then generalize the recursive parametrization for studying any unitarymatrices to include the Majorana fermions, which could be useful for studying theneutrino mixing matrix.
One of the most interesting problems of the standard model is to understand the originof CP violation. It appears in different forms and was first observed in weak decays ofthe neutral K-mesons. CP violation is also needed to explain why there are more mattercompared to antimatter in the universe [1].CP violation has been studied extensively for the Dirac fermions [2, 3]. All knowncharged quarks and leptons are Dirac particles and their analysis does not have directimplications to the lepton or baryon number violating interactions, including the gener-ation of matter asymmetry of the universe. We thus attempt to generalize some resultsfor the Dirac fermions to models with Majorana fermions like the neutrinos.Although we are yet to infer if there exists any Majorana fermion, many interestingaspects of the Majorana fermions have been pointed out, which may have far reachingconsequences. In particle physics, the masses of the Majorana particles play a crucial roleand can explain the smallness of the neutrino mass naturally [4]. The Majorana neutrinomasses can also explain the baryon asymmetry of the universe [6] and predict the darkmatter and resolve some issues of the dark energy.Considering all these, we intend to study the CP phases for the Majorana fermionsfrom a different angle. There have been some analysis of the Dirac fermions (quarksand leptons [8]) identifying their rephasing invariant measure of quantum mechanicalCP phases to the quantum mechanical Bargmann invariants (BI) [7], which in turn, is ∗ [email protected] elated to the classical geometric phases. Our main aim is to generalize this result for theMajorana fermions and apply our result to the CP violation in the leptonic sector withMajorana neutrinos.The geometric phase was introduced [9] in a cyclic adiabatic quantum mechanicalsystem, where the dynamics is governed by the time-dependent evolution of the statevector in a Hilbert space. These geometric phases has been shown to be related to afamily of quantum mechanical Bargmann invariants (BI). For a physical system, thestate vectors represent Dirac fermions, and the BIs may be identified with rephasinginvariant measures of CP violation. We shall generalize these results to the case whenthe state vectors represent both Dirac and Majorana fermions by defining the ray andthe Hilbert spaces properly. This will introduce additional BIs representing CP violationarising from the Majorana phases and this, in turn, will relate the rephasing invariantmeasures of CP violation for both Dirac and Majorana fermions to the complete sets ofBargmann invariants.We shall first demonstrate how one can define the quantum mechanical Bargmannvariables for the Majorana fermions and relate them with the geometric phases after defin-ing the proper quantum mechanical ray and Hilbert spaces for the Majorana fermions.We then construct the rephasing invariant measures for the Majorana fermions. As ex-pected, compared to the Dirac fermions, there are more number of such CP violatingmeasures arising due to the Majorana phases.We shall then extend our analysis to study the CP violating invariants for the Majo-rana fermions in the formalism of recursive parametrization of unitary matrices. One canstudy the CP phases in the neutrino mixing matrix and the Majorana phases throughthe recursive parametrization of the unitary matrices. We shall extend these analysesand present explicit forms of the rephasing invariant quantities for a few examples whenMajorana fermions are included. A Majorana fermion is the antiparticle of itself. It has two complex components or fourreal components. Two Majorana fermions may combine into a Dirac fermion, dependingon the mass terms. The Lagrangian describing a Majorana fermion can be given as L M = ψ M iγ µ ∂ µ ψ M + m M ψ M ψ cM (1)where ¯ ψ M = ψ † M γ and we work in the Weyl representation, where the γ matrices aredefined as: γ µ = (cid:18) σ µ ¯ σ µ (cid:19) with σ µ = [ I , σ i ]; ¯ σ µ = [ I , − σ i ]where I is a 2 × σ i are the Pauli matrices. In this basis, γ = iγ ◦ γ γ γ = diag ( − I , I ) is diagonal. Defining the charge conjugation as ψ c = − iγ ψ ∗ = − iγ γ ¯ ψ T , (2)the Majorana condition that the Majorana particles are their own antiparticles, can bewritten as ψ cM = λ ∗ ψ M . (3)2ere λ is a complex phase contributing to CP violation, and | λ | = 1. Since the particlesand antiparticles carry opposite quantum numbers or charges under any symmetry group,this condition implies violation of that quantum numbers or the charges. So, chargedleptons or quarks cannot be Majorana particles. We shall thus work with the assumptionthat neutrinos are Majorana particles, while all other charged fermions are Dirac particles.A Dirac fermion has eight real components (or equivalently, four complex components)and may be decomposed into two Majorana fermions. For example, we can consider theleft-handed ( ψ L ) and right-handed ( ψ R ) components of a Dirac fermion ( ψ D ) as twoMajorana fermions ( ψ M and ψ M ) as defined below: ψ M = ψ L + λ ψ cR and ψ M = ψ R + λ ψ cL where ψ L = (1 − γ )2 ψ D ; ψ cR = (1 + γ )2 ψ cD ; ψ R = (1 + γ )2 ψ D and ψ cL = (1 − γ )2 ψ cD . and ψ cR and ψ cL are CP -conjugate states of ψ L and ψ R , respectively.The Majorana fermions ψ M and ψ M satisfy the Majorana conditions: ψ cM = λ ∗ ψ M and ψ cM = λ ∗ ψ M . (4)However, we can always remove an overall phase, so that the relative CP phases remainas independent phases.We can now express the Majorana fields ψ M and ψ M in terms of complex two-component spinors η and χ as: ψ L = 1 − γ ψ M = (cid:18) η (cid:19) ; ψ R = 1 + γ ψ M = (cid:18) χ (cid:19) ; ψ cR = (cid:18) η (cid:19) ; ψ cL = (cid:18) χ (cid:19) . (5)such that ψ M = (cid:18) ηλ ¯ η (cid:19) ; ψ cM = (cid:18) λ ∗ η ¯ η (cid:19) ; ψ M = (cid:18) λ χ ¯ χ (cid:19) ; ψ cM = (cid:18) χλ ∗ ¯ χ (cid:19) ; (6)which satisfies the Majorana condition of equation (3).Any Majorana field can be expressed in terms of the creation and the annihilationoperators as ψ M ( x ) = X p,s r m M ǫ (cid:16) f ps u ps e − ipx + λ ∗ f † ps v ps e ipx (cid:17) , (7)where the energy of the Majorana fermion is ǫ = q p + m M and m M is the mass. The u ps and the v ps spinor operators satisfy the equations of motion( γ µ p µ − m M ) u p = 0; ( γ µ p µ + m M ) v p = 0; (8)and s is the spin.To complete the discussion we shall define the density matrix for the Majoranafermions as [10, 11] ρ M ( ψ ) = ψ c ( ψ c ) † = | ψ c ih ψ c | , (9)3hich satisfies the equation of motion dρ M dt = − i ( Hρ M − ρ M H † ) . (10)This is a key ingredient in this analysis of Majorana fermions, which allows us to definethe smooth parametrized curves C of unit vectors in the Hilbert space H and the corre-sponding free geodesics. The Bargmann invariants are then expressed in terms of thesefree geodesics and the geometric phases can then be defined in terms of the Bargmanninvariants.The density matrices ρ r , corresponding to any unit vector ψ r in the Hilbert space H of Majorana fermion states, are images in the ray space R and any two neighbouringdensity matrices (say, ρ r − to ρ r ) are connected by free geodesics in H . In case of a Diracfermion or Dirac neutrino, the density matrices or the images of the unit vectors in H onto R is defined as ρ D ( ψ ) = ψ ψ † = | ψ ih ψ | . (11)Thus the Dirac density matrix does not correspond to any violation of charge or any con-served quantum number. The density matrix of the Majorana fermions would introducethe Majorana phases into the density matrix giving rise to new sources of CP violation.The Majorana density matrix will provide us with additional Bargmann invariants corre-sponding to the new Majorana CP phases λ that is defined in equation 3. The Majoranaphases would disappear when the neutrinos dont have Majorana masses. Furthermore,if there is only one Majorana particle in any model, an overall phase transformation canremove it. We shall elaborate on these discussions in the next few sections. In this section we shall review the formalism of connecting the Bargmann invariants withthe geometric phases and extend the earlier results by including the Majorana fermions,and hence, the Majorana phases. A connection between the Bargmann invariants and thegeometric phases has been established rigorously for the Dirac fermions [12]. The basicstructure of this formalism relies on the cyclic adiabatic quantum-mechanical evolutionof the state vectors [13]. This has been further generalized to show that the geometricalphases can be related to a family of quantum-mechanical invariants which were proposedby Bargmann [7]. In this section we shall develop the connection between the geometricalphases and the Bargmann invariants for the Majorana particles and discuss how thesestudies can be extended to the CP violation in the lepton sector with Majorana neutrinos.This analysis largely depends on the free geodesics in quantum-mechanical ray andHilbert spaces, as the geometric phases vanish for these geodesics. It has been demon-strated that the generalization of free geodesics to the so-called null phase curves aremore general in establishing a connection between the Bargmann invariants and the ge-ometric phases. These null phase curves are a family of ray and Hilbert space curves,which includes the free geodesics and a large class of other curves, and establish a generalconnection between the Bargmann invariants and the geometric phases. However, fordemonstrating the geometric phases of the Majorana fermions and its connection withthe Bargmann invariants, we shall restrict our discussions to free geodesics only, with theunderstanding that these results are more general.4e start with a Hilbert space of some quantum system of both Dirac and Majoranaparticles H , and construct the associated ray space R with the pure state density matri-ces. The dual space of H will contain both particles and antiparticles for the Majoramafermions, but for the Dirac fermions the dual space of H will contain only the particles.The density matrices for the Dirac fermions are defined in equation 11, where ψ a rep-resents a vector in the Hilbert space H . The inner product of any two Dirac fermionswould then be given by I D = ( ψ a ( s ) , ψ b ( s )) = h ψ a ( s ) | ψ b ( s ) i . (12)The key ingredient for studying a Majorana fermion in this formalism is to enhancethe ray space with the pure state density matrices for the Majorana fermions defined byequation 9 in the previous section. For the Majorana fermions, the charge conjugationof ψ satisfy the Majorana condition of equation 3 and as a result differs from the Diracfermions by the Majorana phase λ . Thus the ray space curves for the Majorana fermionswill differ from that of the Dirac fermions and this would modify the inner product ofany two Hilbert space vectors.We can now write down an inner product of two Majorana fermions ψ i and ψ j in H as I M = (cid:16) ψ ci ( s ) , ψ cj ( s ) (cid:17) = h ψ ci ( s ) | ψ cj ( s ) i . (13)For the Majorana fermions one can write both I D and I M type inner products, but forthe Dirac fermions one can only write I D type inner products.If the geometric phase vanishes for any connected part of the ray space curves, thenthe inner product of two Hilbert space vectors along the lift of such ray space curveswould also vanish. This condition on the inner product of the Hilbert space vectors isalso valid for the free geodesics, which can also link the Bargmann invariants with thegeometric phases. We shall now define the free geodesics in ray and Hilbert spaces, inwhich the geometric phase vanishes and then demonstrate the connection between theBargmann invariants and geometric phases. We shall follow the formalism and notationof [12, 14].Any smooth parametrized curves C of unit vectors in H may then be expressed as C = { ψ ( s ) ∈ H | || ψ ( s ) || = 1 , s ≤ s ≤ s } ⊂ H . (14)The projection of the Hilbert space to the ray space π : H → R . will then provide us the image C r in R : π [ C ] = C r ⊂ R , where the image C r for the Majorana fermions follows from the definition of the purestate density matrices for the Majorana fermions: C r = { ρ M ( s ) = ψ c ( s ) ψ c ( s ) † | s ≤ s ≤ s } . (15)Thus any curve of unit vectors C in the Hilbert space H ( C ⊂ H ) is a lift of the image C r in the ray space R ( C r ⊂ R ). It is apparent that the end points should satisfy theboundary condition that they are not orthonormal( ψ c ( s ) , ψ c ( s )) = 0 (16)5nd ψ ( s ) , ψ c ( s ) , ρ M ( s ) and ρ D ( s ) are smooth curves, satisfying certain smoothness con-ditions [14].We shall now consider the horizontal lift of the curve C ( h ) r ⊂ R , such that the vectors ψ c ( h ) ( s ) along this lift satisfy ψ c ( h ) ( s ) , dds ψ c ( h ) ( s ) ! = 0 . (17)This immediately implies vanishing of the dynamical phase for any curve C r ⊂ R , alongthe horizontal lift: φ dyn [ C ] = Im Z s s ds ψ c ( h ) ( s ) , dds ψ c ( h ) ( s ) ! = 0 . (18)The geometric phase is thus given by φ g = arg ( ψ c ( s ) , ψ c ( s ) ) . (19)One can then define the free geodesics in H and R and show that the geometric phasevanishes along the free geodesics [12] φ g [free geodesics ∈ R ] = 0 , (20)and relate the geometric phases to the Bargmann variables. A more general analysisutilizing the null phase curves can also establish these relations [12], but for our purposewe shall directly move to the final result.Bargmann invariants (BIs) were developed for the Dirac fermions, and the relationshipwith the geometric phase utilized the definition of the inner product ( I D ) and the densitymatrices ( ρ D ) for the Dirac fermions [7]. To include the Majorana particles and extendthe applicability of the Bargmann invariants, the Hilbert space H for the Dirac fermionsis enhanced to incorporate the antiparticles, and hence, the Majorana phases, as definedin equation 3. The corresponding ray space is also modified by the definition 9 of thedensity matrix ( ρ M ), and hence, the inner products involving the Majorana fermions( I M ). Accordingly we can write down two types of the Bargmann invariants (BI) forthe Majorana fermions, one (∆ D ) containing only I D type inner products, and the other(∆ M ) containing both I D and I M type inner products.We first present a BI with Dirac fermions. Since h ψ a | ψ b i = h ψ b | ψ a i ∗ , all second orderBIs are real and the corresponding geometric phase vanishes. We thus present a thirdorder BI with Dirac fermions:∆ D ( ψ , ψ , ψ ) = ( ψ , ψ ) ( ψ , ψ ) ( ψ , ψ )= Tr h ρ D ( ψ ) ρ D ( ψ ) ρ D ( ψ ) i = Tr h ( ψ ψ † ) ( ψ ψ † ) ( ψ ψ † ) i . (21)This is an example of a third order Bargmann invariant defined with three mutuallynonorthogonal vectors ψ i ∈ H| i = 1 , , ρ Di = ψ i ψ i † ∈ R| i = 1 , ,
3. Any fourth or higher order BIs may be reduced to third orderBIs. It is straightforward to generalize this definition to m -th order Bargmann invariants∆ Dm ( ψ , ψ , · · · , ψ m ) = ( ψ , ψ ) , ( ψ , ψ ) , · · · , ( ψ m , ψ )= Tr h ρ D ( ψ ) ρ D ( ψ ) · · · ρ D ( ψ m ) i = Tr h ψ ψ † ψ ψ † · · · ψ m ψ m † i . (22)6rom the properties of the free geodesics (equation 20), we can write down the relationshipbetween the Bargmann invariants and the geometric phases for an m -vertex closed loop( P m ) as, φ g [ P m ] = − arg ∆ Dm ( ψ , ψ , · · · , ψ m ) , (23)where P m = m − vertex closed loop ∈ R , with ρ D → ρ D → ρ D → . . . → ρ Dm → ρ D , being connected by free geodesics.We now present an example of the third order BIs with Majorana fermions. TheseBIs are possible only when the Hilbert space H incudes the antiparticles ( ψ c ) and boththe density matrices ρ D and ρ M appear in the definition of the BIs:∆ M ( ψ , ψ c , ψ c ) = ( ψ , ψ c ) ( ψ c , ψ c ) ( ψ c , ψ )= Tr h ρ D ( ψ ) ρ M ( ψ c ) ρ M ( ψ c ) i = Tr h ψ ψ † ψ ψ c † ψ ψ c † i . (24)both ψ i and ψ ci enter in the definition of the BIs and the Majorana phases enter in thedefinition of the BIs through the density matrices ρ M ( ψ ci ). The consequences of theMajorana type BIs (∆ Mi ) including the Majorana phases will become clear when we shallrelate them to the rephasing invariant measures of CP violation in the leptonic sectorwith Majorana neutrinos in the next section. The smallness of the neutrino mass can be explained in simple extensions of the standardmodel without any fine tuning or introducing arbitrarily small parameters, by consideringthe neutrinos to be Majorana fermions. Any information about CP violation in theleptonic sector with Majorana neutrinos are contained in the neutrino mass matrix andtheir charged current interactions. Without loss of generality we can work in a basis,in which the charged lepton mass matrix is diagonal, so that the complex phases in theneutrino masses and mixing matrix will determine the CP violation in any model.Some of the complex phases in the neutrino mass and mixing matrices can be removedby the rephasing of the neutrinos, so it is a general practice to construct rephasinginvariant measures to study the CP violation. In this section we shall demonstrate thatthese CP violating rephasing invariant measures are the Bargmann invariants with theMajorana neutrinos and are related to the geometric phases. In particular, we shallemphasize on the lepton number violating CP violating measures, which are the newBargmann invariants with the Majorana fermions,We begin with the charged current interactions of the neutrinos with the chargedleptons and the neutrino mass matrix for the Majorana neutrinos: L CC = − g √ X α = e,µ,τ ¯ ν αL γ ρ l − αL W + ρ , L mass = ν TiL C − ν iL = m i ν iLc ν iL , (25)where ν iL , i = 1 , , m i and l αL , α = e, µ, τ are the7eak charged lepton eigenstates, which are the states with diagonal charged lepton massmatrix. The Majorana neutrinos in this basis ν αL are related to the physical neutrinos ν iL by a unitary transformation [5] given by ν iL = X α = e,µ,τ ( U ∗ αi ν αL + λ i U αi ν cRα ) . (26) U is the neutrino mixing matrix. Since the right-handed fermions are blind to the SU (2) L interactions, we shall be working with only the left-handed Majorana neutrions.So we shall drop the index L .The mixing matrix U αi that relates the weak neutrino eigenstates ν α to the physicalneutrino eigenstates ν i : | ν i i = X α = e,µ,τ ( U ∗ αi | ν α i + λ i U αi | ν cα i ) . (27)appears in the charged current interactions of the physical neutrinos ν i with the physicalcharged leptons l α : L CC = − g √ X α = e,µ,τ ¯ ν i ( U † ) iα γ ρ l − α W + ρ + H.c. and it relates the neutrino mass matrix in this weak interaction basis to the physicalneutrino mass matrix (diagonal) by( U T ) iα M αβ U βj = λ ∗ i M diag ij . (28) λ i are the Majorana phases defined by equation 3, so that the physical Majorana neutrinos ν i also satisfy the Majorana condition ν ci = λ ∗ i ν i . Any complex phases in the mass matrix M αβ may be tansferred to the mixing matrix U αi by the rephasing of the physical neutrinos and the weak basis states of neutrinos, but theMajorana phases may not be removed independently. Rephasing the gauge basis and themass basis of the neutrinos ν i → e − iδ i ν i and ν α → e − iη α ν α , (29)would then imply rephasing of the mixing matrix and the Majorana phase matrix as: U αi → e − i ( η α − δ i ) U αi , λ i → e − iδ i λ i and ˜ λ i → e iδ i ˜ λ i . (30)where we defined λ ∗ = ˜ λ .Both Dirac and Majorana neutrinos can have CP violation coming from the phasesin the mixing matrix U. The simplest rephasing invariant combination with the mixingmatrix can be defined as [15] T Dαiβj = U αi U βj U ∗ αj U ∗ βi (31)While the rephasing invariant measure T Dαiβj contains all the CP violating phases in themixing matrix, it does not include the Majorana phases, and hence, any CP violation in8 lepton number violating interaction will not have any contribution from this measure T Dαiβj .The simplest rephasing invariant measure containing the Majorana phases consists oftwo mixing matrix and the Majorana phase matrices [15] s Mαij = U αi U ∗ αj ˜ λ ∗ i ˜ λ j . (32)Although this rephasing invariant measure contains the Majorana phases, it may not ap-pear in the probability or cross-section of any lepton number violating physical processes.The rephasing invariant measure that contains the Majorana phase and also enter in thephysical processes may be defined as [17] T Mαiβj = U αi U ∗ βj U ∗ αj U βi λ i λ ∗ j (33)We shall now demonstrate that the measures T Dαiβj and T Mαiβj may be defined as Diracand Majorana type Bargmann invariants and can be viewed as geometric phases.We start with the state vectors | ν i i , | ν α i and | ν cα i in the Hilbert space H . The statevector | ν i i satisfy the Majorana condition of equation 3, that is, ν ci = λ i ν i and can beexpressed in terms of the state vectors | ν α i and | ν cα i as given by equation 27. We canthen utilize the orthogonality conditions of the state vectors | ν i i , | ν α i and | ν cα ih ν a | ν b i = ( ν a , ν b ) = δ ab where ν a,b ≡ ν α,β , ν i,j or ν cα,β , and using equation 27, express the inner products of thenon-orthogonal state vectors as ( ν i , ν α ) = U αi ( ν i , ν cα ) = λ ∗ i U ∗ αi (34)We now construct Bargmann invariants without Majorana phases and with only themixing matrix, which is∆ D ( ν i , ν α , ν j , ν β ) = ( ν i , ν α )( ν α , ν j )( ν j , ν β )( ν β , ν i )= Tr h ρ D ( ν i ) ρ D ( ν α ) ρ D ( ν j ) ρ D ( ν β ) i = Tr h ( ν i ν † i )( ν α ν † α )( ν j ν † j )( ν β ν † β ) i = U αi U ∗ αj U βj U ∗ βi = T Dαiβj (35)Similarly we can construct the Bargmann invariants with Majorana fermions, and hence,Majorana phases λ i . It should contain both ν α and ν cα and the simplest one is given by∆ M ( ν i , ν α , ν j , ν cβ ) = ( ν i , ν α )( ν α , ν j )( ν j , ν cβ )( ν cβ , ν i )= Tr h ρ D ( ν i ) ρ D ( ν α ) ρ D ( ν j ) ρ M ( ν cβ ) i = Tr h ( ν α ν † α )( ν i ν † i )( ν j ν † j )( ν cβ ν cβ † ) i = U αi U ∗ αj λ ∗ j U ∗ βj λ i U βi = T Mαiβj (36)This is the rephasing invariant measure of CP violation with Majorana fermions. It isclear from the form of s αij that it can enter in any Bargmann invariants, but it cannot be a9argmann invariant because it is not closed. The Bargmann invariants we constructed ∆ D and ∆ M , are the conventional rephasing invariant measures T Dαiβj and T Mαiβj , respectively.Moreover, the phases in s αij are related to these BIs T D and T M T Dαiβj = s αij s ∗ βij and T Mαiβj = s αij s βij (37)The Bargmann variables, ∆ D and ∆ M , are defined on the ray space R and the points ρ ( ν i ) , ρ ( ν α ) , ρ ( ν cα ) on R are non-orthogonal and pair wise linearly independent. Thesepoints are connected by geodesics, which form a closed loop in R , representing a cyclicevolution in the state space. This establishes that the Bargmann invariants, and hence,the rephasing invariant measures, gives us the geometric phases φ g = − arg(∆ )The Majorana nature of the neutrinos implies lepton number violation. So, ∆ M is therephasing invariant measure of CP violation [17] that enters in the lepton number violatingCP violating interactions like the neutrinoless double beta decays or W − W − → e − e − . Ifwe extend this analysis to include the right-handed neutrinos, then a similar CP violatingmeasure with the right-handed neutrinos would appear in the lepton number violatingCP asymmetry as in the models of leptogenesis [18]. In the previous section we constructed the Bargmann invariants with Majorana neutrinosand demonstrated that the CP violation coming from the Majorana phase λ i and also theunitary mixing matrix U iα can be presented in the form of rephasing invariant measuresor equivalently as BIs. This establishes that the CP violating phases are geometric phasesin the leptonic sector. In this section we shall extend this analysis to study the unitarymixing matrices in the framework of recursive parametrization and demonstrate how toinclude the Majorana phases in this formalism.We shall first define the recursive parametrization of any unitary matrix, keeping inmind that we shall be applying this formalism to discuss the neutrino mixing matrixdefined by equations 27 and 28. Then we shall generalize the formalism to incorporatethe Majorana phases, defined by equation 28. For the unitary matrix without includingthe Majorana phases, we shall use the notation and conventions of reference [16].Any n × n matrix A n ∈ U ( n ) can be uniquely decomposed [4] into n block matricesgiven by A n = A n ( ζ ) A n − ( η ) A n − ( ξ ) ...A ( γ ) A ( β ) A ( α ) A ( χ ) (38)The elements a jk of A n ( ζ ) matrix can be constructed in the following way a jn = ζ j ; j = 1 , , ...na j j − = ρ j − ρ j ; j = 2 , ...na jk = − ζ j ζ ∗ k +1 ρ k ρ k +1 ; j ≤ k ≤ n − a jk = 0; ∀ j ≥ k + 2 (39)10 is the unit vector forming the basis of the n × n space implying the components ζ i toobey n P i =1 ζ i = 1 and ρ j = s j P i =1 | ζ i | . Any A m ∈ U ( m ) ( m < n ) is an unitary matrix withdiagonal elements equal to 1 and all other elements equals to zero for trivial rows andcolumns. This method can also be used to construct SU ( n ) matrices by multiplying thefirst column of the obtained U ( n ) matrix with ( − n − ζ ∗ ζ assuming ζ = 0.We shall now proceed to construct the rephasing invariants in this formalism. Oncewe have the required U ( n ) matrix, we consider the freedom to rephase the various statesby multiplying U ( n ) matrix with the diagonal phase matrices U ′ = D ( θ ) U D ( θ ′ )where D ( θ ) = diag(e i θ , e i θ , e i θ , ..., e i θ n ) . (40)For the appropriate choice of the phases θ and θ ′ , one can get explicit form of the rephasinginvariants in this formalism. Before we consider explicit construction of such invariants,we shall demonstrate how these constructions differ for the Dirac and Majorana cases, sothat we can demonstrate the Majorana phases for the groups SU ( n ) for n = 2 , , H with only the Dirac neutrinos. Themass term for a Dirac neutrino can be written as L Dmass = m D ¯ ν L M Dν ν R which can be diagonalized by a bi-unitary transformation U † L M Dν U R = M diagν . (41)where U L and U R diagonalizes the matrices M Dν M Dν † and M Dν † M Dν , respectively.If we make any phase transformation to the left-handed and the right-handed neutri-nos by the matrices D ( θ ) and D ( θ ′′ ) then M diagν = U † L M Dν U R = U † L D † ( θ ) M Dν D ( θ ′′ ) U R = D † ( θ ′ ) U † L D † ( θ ) M Dν D ( θ ′′ ) U R D ( θ ′ )= U ′ L † M Dν U ′ R . (42)Thus the phase transformation of the left-handed neutrinos can be represented by thetransformation of U L as U L → U ′ L = D ( θ ) U L D ( θ ′ ) . (43)Thus for the Dirac fermions, we have the freedom to make two sets of rephasing with theparameters θ and θ ′ .Since the right handed neutrinos ν R do not enter the SU (2) L × U (1) Y charged currentinteractions, their transformation (rephasing of the matrix U R by D ) will not affect ouranalysis.In case of Majorana neutrinos, the mass term may be written as L Mmass = m M ν Lc M Mν ν L . (44)11his mass matrix M Mν is symmetric and may be diagonalized by only one matrix U , andhence, the same unitary matrix U will diagonalize the neutrino mass matrix, and hence, λ i M diagν = U T M Mν U = U T D ( θ ) M Mν D ( θ ) U (45)Thus rephasing of the Majorana fermions may be represented by U → U ′ = D ( θ ) U . (46)Given the prescription for the rephasing of Dirac neutrinos by equation 43 and for therephasing of the Majorana neutrinos by equation 46, we can now explicitly construct therephasing invariants in this recursive parametrization formalism for both the Dirac andMajorana fermions, as demonstrated below for the groups SU ( n ), n = 2 , , SU (2) The SU (2) matrix obtained in the formulation of recursive parametrization is given by U = A ( α ) = (cid:18) α ∗ α − α ∗ α (cid:19) Choosing the diagonal phase matrix as D ( θ ) = diag( e i ( θ ) , e i ( − θ ) ) and rephasing accordingto equation 43, for the Dirac neutrinos, it is clear that the matrix remains unchanged i.e. U ′ = U . Whereas, rephasing for Majorana neutrinos according to equation 46 gives U ′ = D ( θ ) U = (cid:18) α ∗ e iθ α e iθ − α ∗ e − iθ α e − iθ (cid:19) . The action of rephasing changes the elements of the matrix as α → α ′ = α e iθ ,α → α ′ = α e − iθ ;which can be written in general as ζ j → ζ ′ j = ζ j e in θ and can be represented in a tabularform in table 1. ζ j → ζ ′ j ζ j e in θ n α → α ′ +1 α → α ′ − SU (2)The number of independent elements of the unit vector (in this case ~α ) decreases byone after rephasing due to the constraint α ′ = α ′ ∗ , leaving only one rephasing invariantquantity given by ( α α ). It is important to note that no such quantities are found forDirac type rephasing which reflect the fact that only two generations of quark cannotproduce CP violating pure Dirac phase whereas it is sufficient to produce CP violatingmajorana phase in the lepton sector with two generations of Majorana neutrinos.12 .2 Recursive parametrization of SU (3) The recursive parametrization scheme 39 can be easily extended to obtain SU (3) matricesin the form given by U = A ( β ) A ( α )= − β ∗ α ∗ + β ∗ β α σ − β ∗ α + β ∗ β α σ β β ∗ α ∗ + β ∗ β α ∗ σ β ∗ α − β ∗ β α σ β − σ α ∗ σ α β , σ = q | β | + | β | . Choosing the diagonal phase matrix as D ( θ ) = diag( e i ( θ + θ ) , e i ( − θ + θ ) , e i ( − θ ) ) andafter rephasing by Dirac type i.e. U ′ → U = D ( θ ) U D ( θ ′ ), we get the changed componentsof unit vectors given in the table 2 ζ j → ζ ′ j = ζ j e in θ + in θ + in ′ θ ′ + in ′ θ ′ n n n ′ n ′ α → α ′ = 0 +2 − − α → α ′ = 0 − − β → β ′ = +1 +1 0 − β → β ′ = − − β → β ′ = 0 − − SU (3)It is interesting to note that the only quantity which is linear in every componentof unit vectors (in this case ~α and ~β ) and remains invariant after rephasing is given by α α ∗ β ∗ β ∗ β . This fact reflects that only one pure Dirac phase can occur in the mixingmatrix for CP violation with three generations.However, using Majorana type rephasing for the mixing matrix (i.e U ′ → U = D ( θ ) U )leads to more invariant quantities. Also the number of independent components of unitvectors reduces to three after rephasing due to the constraint α ′ = α ′ ∗ = β ′ . The twosmallest forms of rephasing invariants,each of which is linear in each component of unitvectors in this case are ( α α ) and ( β β β ). SU (4) The procedure can be extended easily to four generations of leptons to get SU (4) mixingmatrix in the form of U = A ( γ ) A ( β ) A ( α ). We choose the diagonal phase matrix forDirac type rephasing to be D ( θ ) = diag( e i ( θ + θ + θ ) ,e i ( − θ + θ + θ ) , e i ( − θ + θ ) , e i ( − θ ) ). The set of changes in the components of the unit vectorsafter rephasing in the form U → U ′ = D ( θ ) U D ( θ ′ ) is given in table 3.13 j → ζ ′ j = ζ j e in θ + in θ + in θ + in ′ θ ′ + in ′ θ ′ + in ′ θ ′ n n n n ′ n ′ n ′ α → α ′ = 0 0 +3 − − − α → α ′ = 0 0 − − β → β ′ = 0 +2 +2 0 − − β → β ′ = 0 − − β → β ′ = 0 0 − − γ → γ ′ = +1 +1 +1 0 0 − γ → γ ′ = − − γ → γ ′ = 0 − − γ → γ ′ = 0 0 − − SU (4)There are three pure Dirac phases appearing in the mixing matrix for four generationsand are related to the invariant quantities which are given as ( α α ∗ β ∗ β ∗ β ), ( β β ∗ γ ∗ γ )and ( β β ∗ γ ∗ γ ∗ γ ).On the other hand, Majorana type rephasing constraints the number of independentcomponents of unit vectors by α ′ ∗ = α ′ = β ′ = γ ′ and β ′ = γ ′ . The three invariantquantities of smallest forms are given by ( α α ), ( β β β ) and ( γ γ γ γ ).It is quite evident that rephasing invariant quantities are structurally different due tothe additional constraints appearing only in Majorana type rephasing. All the invariantquantities are of lower order and linear in every component of unit vectors required toconstruct the unitary matrix recursively. Higher order invaraints can be constructedout of these lower order invariants. For n generations of leptons [15], the number of CPviolating pure Dirac phases in the mixing matrix is ( n − n − /
2, whereas the number ofpure Majorana phases is ( n −
1) with a total number of n ( n − / ~ζ (in equation 38) vanishes. In such acase [19], a different recursive approach produces different form of unitary matrix whichupon rephasing changes the components of the unit vector in a different way, however,the number of rephasing invariants and the forms remain unchanged. Bargmann invariants have been shown to connect the rephasing invariant quantities withthe geometric phase for the Dirac fermions. We extend this analysis to include theMajorana fermions and show how to relate the Majorana phases with the Bargmann in-variants, by defining proper quantum mechanical ray and Hilbert spaces for the Majoranafermions. As an explicit example, we included Majorana neutrinos in the leptonic sectorand constructed the Bargmann invariants for both the Dirac and the Majorana neutrinos.14his allows us to interpret the CP violating phases of Dirac and Majorana fermions to ageometric phase. We then explained how to incorporate Majorana phases in a recursiveparametrization of unitary matrices with explicit examples of SU(n), n = 2,3,4.
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