aa r X i v : . [ phy s i c s . g e n - ph ] A ug Non-Perturbative Unitary Mixing of Bosonic CausalSpinors
James Lindesay ∗ Computational Physics LaboratoryHoward University, Washington, D.C. 20059
Abstract
The fundamental fermion representations of causal spinor fields havepreviously been demonstrated to describe free Dirac fermions, as well as in-corporate only the observed degrees of freedom for local gauge invariance.In this paper, the first non-trivial boson representations of causal spinorswill be presented. The general unitary mixing of degenerate bosonic spinorswill be developed, and then applied to electro-weak bosons satisfying ap-propriate kinematic constraints. The resulting mass ratios are seen to beconsistent with reported measured values for these bosons.
Fundamental formulations of quantum dynamics must incorporate conserva-tion of momenta, positive-definite energies, and angular momenta, as well as themaintenance of various internal quantum numbers through unitarity and pairwisecreations/annihilations, when appropriate. Furthermore, correspondence of quan-tum interactions with classical formulations requires cluster decomposability, aswell as being able to establish causal relationships between interactions.Causality allows for space-like correlations in quantum systems (including‘spooky’ entanglements, etc.) without allowing space-like (faster-than-light) com-munications for separations satisfying ( x µ − y µ ) η µν ( x ν − y ν ) >
0, where twoevents have coordinates ~x and ~y . At the microscopic level, this requires thatfermion/boson fields must anti-commute/commute outside of the light-cone, i.e.,[ ˆ ψ (Γ) γ ( ~x ) , ˆ ψ (Γ) γ ( ~y )] ± = 0 for ~x − ~y space-like. This requirement connects the quantumstatistics to the spin, such that ( ± ) = − ( − J = − ( − .At their most primary level, causal spinor fields[1] necessarily involve two par-ticle types that pair to satisfy microscopic causality, while transforming underthe usual translations, rotations, and Lorentz boosts of the Poincare’ group. Thefundamental representation spinors are 4-dimensional, satisfying the usual Diracalgebra[2]. In addition to the generators of group transformations, there remain ∗ e-mail address, [email protected] xactly 12 additional hermitian generators of internal (non space-time) symme-tries that include a U(1), an SU(2), and for massive fermions an SU(3) that hasbeen shown to require mixing 3 generations of mass eigenstates[3]. The inter-actions generated by local gauge bosons associated with these generators mustbe incorporated into any geometric interaction via covariant curvilinear coordi-nate transformations from the flat space-time translations of the group. Further-more, the non-vanishing structure constants of the group directly constructs theMinkowski metric within the group parameter space defining group invariants.The spinor field equation requires the form ˆΓ µ ˆ P µ to be a Lorentz scalar oper-ation, satisfying Γ β · ~ i ∂∂x β ˆ ψ (Γ) γ ( ~x ) = − γmc ˆ ψ (Γ) γ ( ~x ) , (1.1)where m >
0, and Γ β are the finite dimensional matrix representations of the op-erators ˆΓ β . In (1.1), the particle mass squared m are eigenvalues of the Casimiroperator which labels the unitary representations of the (extended) Poincare groupalgebra. For massive particles, the label γ is the particular eigenvalue of the her-mitian matrix Γ , and for the Γ = (fermion) representation, the Dirac matricesare twice Γ β . In this paper, the Γ = 1 boson representation will be explored. Inparticular, since this 10-dimensioonal representation inherently combines scalarswith vectors, special attention will be given to an examination of electro-weakgauge bosons. In what follows, natural units with ~ = 1 , c = 1 will be utilized. Γ = 1 causal spinors
The ten-dimensional representations of the Γ β matrices are presented in theappendix (A.1). In what follows, motions will be confined to the z-axis, sinceeigenspinors with general momenta can be obtained through a simple rotation.The eigenspinors satisfy Γ β p β Ψ (1) γ ( ~p, J, s z ) = − γm Ψ (1) γ ( ~p, J, s z ) , (2.2)where γ and J are integers no greater than Γ = 1. Microscopic causality requires[4][5]that the space-time dependent boson fields combine a spinor labeled by γ withone labeled by − γ , (i.e., an appropriate combining of particle with anti-particle).Relevant hermitian normalized eigenspinors for z-moving systems are presentedin Equations (A.4) and (A.5) in the appendix.The ten standard state ( ~p → { m, , , } at rest) spinors include a scalar with J = 0 , γ = 0, as well as vector triplets with J = 1 , s z = +1 , , − γ = +1 , , −
1. It is worth mentioning that this also corresponds withthe number of charged and neutral weak bosons, including the Higgs. In the nextsection, the unitary mixing of the spinors will be examined. .2 Unitary mixing of spinors
A noteworthy property of (1.1) that is not true of the Dirac representation isthat fields with γ = 0 are all degenerate regardless of mass, including masslessfields. This implies that unitary mixing of degenerate fields can result in newfields with differing masses that continue to satisfy the field equation. Further-more, for massless particles m →
0, fields given by ˆ A (Γ) µγ ( ~x ) ≡ Γ µ ˆ ψ (Γ) γ ( ~x ) definecontravariant field components that inherently satisfy the Lorentz gauge condition ∂∂x µ ˆ A (Γ) µγ ( ~x ) = 0.It is thus of interest to examine the mixing of Γ = 1 vector (J=1) and scalar(J=0) bosons, especially in relation to the mixing of electro-weak bosons. Nomixing that preserves normalization between spinors of differing masses with γ = 0can be found. However, orthogonal mixing of the degenerate γ = 0 spinors in (A.4) can be developed. Using the identifications m B → iQ B , with p B → ∓ Q B m W q m W + p W , q m B + p B → ± Q B m W p W , (2.3)a time-like W and orthogonal space-like B are described. In this expression, the ± sign corresponds with the sign of p W . Thus, the four γ = 0 degenerate W and B spinors can mix to generate new spinor forms Z and A according tocos φ W B Ψ (1)0 ( ~p W , J, s z ) + sin φ W B Ψ (1)0 ( ~p B , J, s z ) = Ψ (1)0 ( ~p Z , J, s z ) , − sin φ W B Ψ (1)0 ( ~p W , J, s z ) + cos φ W B Ψ (1)0 ( ~p B , J, s z ) = Ψ (1)0 ( ~p A , J, s z ) , (2.4)where for z-moving systems, the parameter s z labels the helicity of the particle.In this development, it is sufficient to examine only z-moving momenta p , sincegeneral motions can be defined by a simple rotation into the direction of motion.It is convenient to define the dimensionless parameter ζ m as follows: ζ m ≡ sin − p m p m + 2 p m ! , p m = m sin ζ m p cos(2 ζ m ) . (2.5)Using this definition, a typical γ = 0 normalized spinor demonstrated in appendixEq. (A.4) takes the form Ψ (1)0 ( ~p m , ,
1) = sin ζ m √ ζ m sin ζ m √ . (2.6)or massive particles, − π < ζ m < π , while massless particles have ζ = ± π , andspace-like 4-momenta satisfy π < | ζ iQ | < π . Expressed using this parameteriza-tion, the mixing satisfies sin φ W B = sin( ζ W − ζ Z ) , (2.7)where ζ W is defined using the superposition of the γ = 0 spinors in (2.4). Thereare alternative expressions of this mixing angle in terms of ζ B and ζ A (with | ζ W − ζ B | = π for orthogonality), but the above form is most convenient for thekinematic identifications to be discussed in what follows.Under (active) Lorentz boosts γ LT ≡ √ − β LT , the parameters ζ m transformaccording to ζ m = ⇒ β LT ˜ ζ m = sin − sin ζ m + β LT cos ζ m p β LT sin(2 ζ m ) + β LT ! . (2.8)Since this expression is completely independent of mass m , this means that any twodifferent masses m A = m B with equal ζ values ζ A = ζ B are necessarily co-moving ,but displaying differing momenta. In particular, the Lorentz transformation thatboosts a mass with finite ζ m to rest is given by β restLT = − tan ζ m .General W − B spinor mixing kinematics can be expressed in terms of Lorentzinvariants by defining ( P µW + P µB ) η µν ( P µW + P µB ) ≡ − M using ζ W MQ B = sin − q − m W Q B + p M + 2 M ( Q B − m W ) + ( Q B + m W ) √ M − m W ) + 2( M + m W ) Q B + Q B ) (2.9)This defines the spinor mixing in (2.4) completely in terms of the invariant overallrest energy M and the masses m W and m B = iQ B . W + B Mixing into Electro-Weak Bosons
Physical W , Z , and H bosons undergo processes that obey kinematic con-servation conditions. The strategy will be to examine combinations of variousparameters of mixing associated with a co-moving H , since co-moving spinorshave identical components, independent of mass. W + H elastic scattering As previously mentioned, meaningful relationships amongst the mixing param-eters are expected to imply on-shell conservation satisfying p µ i + p µ i = p µ f + p µ f .Consider the scattering W + H → W + H . In the center-of-momentum frame, H = − p W . This can be expressed using (2.5) as a relationship between ζ H and ζ W given by ζ H ( ζ W ) = − sin − m W sin ζ W p m H cos(2 ζ W ) + 2 m W sin ζ W ! . (3.10)Appropriate forms for ζ W and ζ Z in (2.7) should be ascertained.A particular W − B orthogonal mixing, having overall invariant mass M = m Z with m B → iQ B = im H , involves all electro-weak mass parameters, and canbe constructed from (2.9), resulting in the form ζ W m Z m H = sin − q − m H m W + p ( m H + m W ) + 2( m H − m W ) m Z + m Z √ m H + m W ) + 2( m H − m W ) m Z + m Z ] . (3.11)Direct substitution of (3.11) into (3.10) results in the expression ζ H ( ζ W m Z m H ) = − sin − √ s m W (cid:16) − m H m W + √ ( m H + m W ) +2( m H − m W ) m Z + m Z (cid:17) m H + m W (cid:16) − m H m W + √ ( m H + m W ) +2( m H − m W ) m Z + m Z (cid:17) ! . (3.12)For this particular value of ζ H , the momenta indeed satisfy p H ( ζ H ) = − p W ( ζ W m Z m H ),providing kinematically consistent values of opposing momenta with energy con-servation for elastic scattering of H + H , W + W , and W + H .Finally, the mixing angle in (2.7) must be determined. The values for ζ W and ζ Z will be chosen so that the mixing parameters are co-moving with the H ’saccording to ζ W = − ζ H ( ζ W m Z m H ) = − ζ Z , as demonstrated in Figure 1. HHW B W B -P H P W -P W P H z W, -z H z W m Z m H z , Z z H Figure 1: W − B mixings ζ W m Z m H (into invariant mass m Z ) that are kinematicallyconsistent for H elastic scattering with the W , satisfying | P W | = | P H | . Momentaare represented using solid arrows. The H ’s simultaneously co-move with themixing according to ζ W = − ζ H = − ζ Z , as represented by the dashed arrows.he resulting mixing angle satisfiescos φ ( W H ) W B ⇒ m H m H − m H m W + m W p ( m H + m W ) + 2( m H − m W ) m Z + m Z = 2 µ HZ µ HZ − µ HZ µ W Z + µ W Z p µ HZ + (1 − µ W Z ) + 2 µ HZ (1 + µ W Z ) , (3.13)where the mass ratios are defined by µ W Z ≡ m W m Z and µ HZ ≡ m H m Z . Substitutionof recent values of the electro-weak boson masses from the Particle Data Group (PDG)[6] gives a mixing angle satisfying cos φ W B ≃ . ∆ cos φ WB µ WZ ≃ − . × − , well within the experimentaluncertainty. W + W − , Z Z , H H scattering re-arrangements
Next, consider scatterings involving boson pairs Z + ¯ Z , W + ¯ W , and H + ¯ H .For a given H momentum p H , the kinematics constrains p Z , defining ζ Z ( ζ H ) givenby ζ Z ( ζ H ) = sin − √ s m H + ( m H − m Z ) cos(2 ζ H ) m H + ( m H − m Z ) cos(2 ζ H ) ! . (3.14)Thus, kinematic systems co-moving with the H ’s will be used to specify the pa-rameter ζ Z in (2.7).As was done in the previous section, the kinematic system should be chosen tomix parameters involving W , Z , and H in a meaningful manner such that energy-momentum is conserved. Consider the momentum p W ∗ for W + ¯ W scattering inthe rest frame of the H ’s (with energies ǫ W ∗ = m H ). This defines ζ W ∗ using anequation analogous to (3.14) given by ζ W ∗ = sin − √ s m H − m W m H − m W ! , (3.15)as demonstrated in Figure 2. Next, consider a system involving Z + ¯ Z and H + ¯ H . z W * H HW +* W -* H H W +* W -* Figure 2: Kinematic diagram for W ¯ W creation/annihilation in H ’s rest frame.f the H in this system co-moves with an aforementioned W ∗ , i.e. ζ H = ζ W ∗ , thensubstituting (3.15) into (3.14) results in the expression ζ Z ( ζ H ( ζ W ∗ )) = sin − s − m H m H − m W m Z . (3.16)This relationship indeed satisfies the kinematic requirement ǫ Z = ǫ H → m H m W in thecenter-of-momentum system. These parameters are demonstrated in Figure 3. ZZ HH P Z P H z Z W B -z W m W m W z H z , W * -z H -z , W * e =e H Z
Figure 3: Kinematic diagrams involving Z, ¯ Z, H , and ¯ H . The on-shell H isco-moving with a kinematically-consistent W ( ζ H = ζ W ∗ ), which coincides withenergy-momentum conservation ǫ H = ǫ Z in the center-of-momentum system.Finally, an appropriate parameter ζ W MQ B for the W + B will be identified with ζ W in (2.7). Since all of the kinematically-consistent mixing of the bosons havebeen incorporated in ζ Z , the mixing in (2.9) should involve orthogonal W mixinginto W , with M = m W , Q B = m W . This gives the only (finite mass) mixing thatresults in a mass-independent pure number: ζ W m W m W = sin − s − √ ! . (3.17)Furthermore, this is the only mixing form ζ W MQ B (constructed from available massparameters) indirectly generated using the forms (2.5) for ζ W and ζ Z (substitutedinto (2.7)) with ǫ Z = ǫ H = m H m W (and consistent momenta), that directly results ina value for cos φ W B consistent with m W m Z . Using (3.16) and (3.17) to calculate themixing angle in (2.7) gives the resultcos φ W B = ⇒ Z ¯ Z → H ¯ H p
50 + 20 √ m H + p − √ p m H − m W m Z p m H − m W m Z = p √ µ HZ + p − √ p µ HZ − µ W Z p µ HZ − µ W Z . (3.18)n this case, substitution of mass values from the PDG[6] results in a mixing angleof cos φ W B ≃ . ∆ cos φ WB µ WZ ≃ . × − , also well within the experimental uncertainty. As can be seen from the form of the equations (3.13) and (3.18), the mixingangle and mass ratios are completely independent of the overall mass scale. TheH-Z mass ratio can be expressed in the closed form µ HZ = 1 √ µ W Z h − √ − √ µ W Z + 10 µ W Z + µ W Z p − µ W Z i − µ W Z + 5 µ W Z = ⇒ cos φ WB → mWmZ p cos φ W B " √
52 cos(2 φ W B ) − sin(2 φ W B ) ! . (3.19)Although a closed form expression for µ W Z has yet to be obtained, this purenumber is a direct prediction from consistency of (3.13) and (3.18). Choosing m Z as the most precisely measured mass, the predicted mass values are given by m W m Z = 0 . ... m H m Z = 1 . ...φ W B = 0 . ...m Z = 91 . ± . GeVm W = 80 . ± . GeVm H = 125 . ± . GeV. (3.20)These values fall well within the present uncertainties in measurements of themass ratios reported by the Particle Data Group[6]: m W m Z = 0 . ± . ,θ W Z = 0 . ∓ . ,m Z = 91 . ± . GeV,m W = 80 . ± . GeV,m H = 125 . ± . GeV. (3.21)Since only on-shell mass ratios are predicted, if the formulation is indeed physi-cally meaningful, any deviations can only come from incorrect associations in thekinematic processes presented. Other meaningful processes must necessarily beredundant.
Discussion and Conclusions
Causal spinor fields have fundamental Γ = fermion representations with thecorrect number and types of gauge-field generated interactions, while requiringany geometric interaction that is covariant under curvilinear coordinate trans-formations must incorporate those interactions via the principle of equivalence.Unitary mixing of degenerate Γ = 1 , γ = 0 boson spinors consistent with on-shellkinematics has been utilized to determine the mixing angle in terms of mass ratios.This allows all boson masses to be determined by a single particle mass scale.The formulation has been applied using the experimental masses of the electro-weak bosons. By requiring a co-moving H spinor associated with each mixing, itsvector ( J = 1) components can be associated with the identical spinor componentsof the associated massive gauge boson. This leaves only the scalar ( J = 0)component to associate with the Higgs boson. The predicted mass ratios havebeen shown to be consistent with measured values.The formulation also suggests the possibility of a Ψ (1)0 ( J = 0) electroweakboson as a dark matter candidate. Promising expressions for the ”effective” mixingangle and its energy-scale dependence, as well as additional relations amongstparticle couplings, are actively being explored as on-going research. Acknowledgement
The author is grateful for the unwavering support of Eileen Johnston throughoutmuch of this extensive effort.
Appendix
Matrix Representation of
Γ = 1
Systems
The Γ = 1 representations of the various group generators consist of 10 × (cid:2) Γ , Γ k (cid:3) = i K k , [Γ , J k ] = 0 , [Γ , K k ] = − i Γ k , (cid:2) Γ j , Γ k (cid:3) = − i ǫ jkm J m , [Γ j , J k ] = i ǫ jkm Γ m , [Γ j , K k ] = − i δ jk Γ . Choosing the order of quantum num-bers in the columns of the matrices as γJs z → , , , − , , , − , − , − , − − : = − − − Γ x = − − − − − − − − − − − − Γ y = i − − − − − − − − − − −
10 0 − − − − − Γ z = √ − − − −
10 0 0 0 − − (A.1)he representation transforms a scalar ( J = 0 , γ = 0) component, as well asvector ( J = 1) and “iso-vector” ( γ = 1 , , −
1) components. For present purposes,the 10-spinors will be Hermitian normalized (rather than “Dirac” normalized)such that for states labeled by γ (the eigenvalue of Γ ), Ψ (1) † γ ′ ( ~p, J ′ , s ′ z ) Ψ (1) γ ( ~p, J, s z ) = δ γ ′ γ δ J ′ J δ s ′ z s z . (A.2)The spinors satisfy the eigenvalue equation Γ µ ˆ P µ Ψ (1) γ ( ~p, J, s z ) = − γmc Ψ (1) γ ( ~p, J, s z ) , (A.3)with m > always . For pairwise kinematics, it is sufficient to consider only z-moving particles, since general states can be developed via rotation of the z-axis.The normalized degenerate ( γ =0) eigenstates take the form { Ψ (1)0 ( ~p, , , Ψ (1)0 ( ~p, , , Ψ (1)0 ( ~p, , , Ψ (1)0 ( ~p, , − } ⇒ √ m + p z √ m +2 p z − p z √ √ m +2 p z p z √ √ m +2 p z , p z √ √ m +2 p z √ m + p z √ m +2 p z p z √ √ m +2 p z , , − p z √ √ m +2 p z √ m + p z √ m +2 p z − p z √ √ m +2 p z (A.4)Degenerate eigenstates labeled with differing m all satisfy (A.3) with its right-hand side set to zero, and thus they can freely mix. For completeness, spinors { Ψ (1)1 ( ~p, , , Ψ (1)1 ( ~p, , , Ψ (1) − ( ~p, , } are displayed: (cid:18) m √ m + p z (cid:19) p z √ √ m + p z (cid:18) − m √ m + p z (cid:19) , − p z √ √ m + p z (cid:18) m √ m + p z (cid:19) − (cid:18) − m √ m + p z (cid:19) , p z √ √ m + p z − (cid:18) − m √ m + p z (cid:19) (cid:18) m √ m + p z (cid:19) . (A.5) eferences [1] J. Lindesay, “Linear Spinor Fields in Relativistic Dynamics”,arXiv:1312.0541 [hep-th] (2013) 13 pages.[2] P.A.M. Dirac, Proc. Roy. Soc. (London), A117, 610 (1928); ibid, A118, 351(1928).[3] J. Lindesay, “An Alternative Approach to Unification of Gauge andGeometric Interactions”, arXiv:submit/ 1768076 [hep-th] (3 Jan 2017),arXiv:1701.01332v2 [physics.gen-ph] 10 pages. See also, J. Lindesay,“AModel of Unified Gauge Interactions”, arXiv:submit/149541 [hep-th] (6 Mar2016), arXif:1604.01375 [physics.gen-ph] 12 pages.[4] J. Lindesay, Foundations of Quantum Gravity , Cambridge University Press(2013), ISBN 978-1-107-00840-3. See chapter 4.[5] S. Weinberg,
The Quantum Theory of Fields , Cambridge University Press,Cambridge (1995).[6] M. Tanabashi, et al. (Particle Data Group), Phys. Rev. D98