Three Dimensional Rotations of the Electroweak Interaction

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by  Mark L. Raphaelian

Three Dimensional Rotations of the Electroweak Interaction
Mark L. A. Raphaelian
National Security Technologies, LLC., Livermore CA 94550 ∗
Anextensiontothestandardelectroweakmodelispresentedthatisdifferentfrompreviousmodels.
Thisextensioninvolvesathreedimensionalrotationofaplanedefinedbytwocharge-neutralcurrent
axes. InthismodeltheplanealsocontainsthethreealgebraicallycoupledSU(2)groupsquantization
axes that comprise the SU(3) group. This extension differentiates leptonic currents from baryonic
currents through hypercolor-charge channels and predicts additional symmetries and currents due
7 to different rotational orientations of the interaction projecting onto the plane defining the SU(3)
1 symmetry group and a U(1) symmetry axis.
0 PACS numbers: 02.20.-a, 11.10.Nx, 11.40.-q, 12.10.Dm, 12.60.Cn, 12.90.+b, 14.70.-e, 95.30.Cq
2

n
The unification between electromagnetic and weak positive cyclic ordering (12:45:67) are:
a
J gauge theories producing the standard electroweak the-
8 ory occurs via a rotation within a plane defined by two (λ1,λ2,λ12), (λ5,λ4,λ45), and (λ6,λ7,λ67). (2)
1 orthogonal gauge fields. These fields are rotated about 0 0 0
a perpendicular axis such that their admixtures pro-
Each of the three groups follows the cyclic intragroup
] duce the observedU(1)symmetry (electromagnetic)and
h (same axis) SU(2) algebra [λ ,λ] = +2iλ ǫ and
k l m klm
SU(2) symmetry (weak) gauge fields. The mixing angle
p therefore the three “0-component” matrices represent
- is called the Weinberg angle [1]. Another way to view
three SU(2) symmetry quantization axes. These three
n
the above rotation is that of the rotationof a plane con-
e SU(2) symmetry quantization axes lie on the plane de-
tainingthequantizationaxesoftheU(1)andSU(2)sym-
g fined by the two diagonal SU(3) group matrices λ3 and
s. metry groups. In this perspective, the plane containing λ8.
the two perpendicular axes is rotated about an orthogo-
c In addition to the cyclic intragroup algebra of the
i nal “line of nodes” axis in a three dimensional space by
s above three SU(2) groups, each of these groups couple
theWeinbergangle. Theprojectionofthe rotatedSU(2)
y
to one another through an intergroup(different axis) al-
h quantizationaxisontothenon-rotatedU(1)quantization
gebraic relationships. These algebraic relationships fol-
p axisproducestheequivalencebetweentheweakcoupling
[ charge and electromagnetic coupling charge. low the form of: [λa0,λl1] = ±iλl2 where the l1 and
This work proposes an extension to the standard the- l2 subscriptsrepresenttwodifferent non-zero-component
1 members from the same the SU(2) group and λa is a
v ory involving a two-dimensional rotation with one angle 0
“0-component” member from a different SU(2) group.
8 on a plane to a three-dimensionalrotation involving two
The importance of intergroup coupling is that each of
9 angles within a three dimensional space. In this rota-
the SU(2) groups is not an isolated SU(2) group but in-
1 tion,threegaugefieldsarerotatedinspacesuchthatad-
3 mixtures of them produce the observed electromagnetic steadforma SU(3)groupstructure whencoupledinthis
0 manner.
gaugefield andtwo orthogonalneutralgaugefields. The
.
2 two neutral gauge fields axes define a plane perpendicu- Another relationship between the three SU(2) groups
0 lar to the U(1) quantization axis and contain the charge is λ λ = 0 for the zero-componentmatrices:
×
7
quantization axes found in the SU(3) group[2]. Because
:1 this extension contains U(1), SU(2), and SU(3) group λ102,λ405 =03, λ405,λ607 =03, λ607,λ102 =03.(3)
v structures, this extension has the properties of a unified (cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
Xi field theory. The importance of the above commutations is that si-
TheSU(3)groupisdefinedbytheGell-Mannmatrices multaneous eigenvalues along each of the three SU(2)
r
a λi where the subscript runs from 1 to 8. Two of the ma- groups “0-component” quantization axes are possible.
trices, λ3 and λ8, are diagonal and from them we define Theeigenvaluemeasurementalonganaxisisdistinctand
the following three “0-component” matrices: independentfromtheeigenvaluemeasurementsalongthe
differentaxes. Theseeigenvalueslieonatwo-dimensional
λ102 = λ3, λ405 = −21 +λ3 + √3λ8 , plane defined by the three SU(2) quantization axes.
(cid:0) (cid:1) (1)
and λ607 = −12 +λ3 − √3λ8 . thrUeseinignttrhinesdicefiannitgiuolnarτ m=oλm/e2n,ttuhmeab(oorveisgorsopuipns)pgrrooduupcse.
(cid:0) (cid:1)
From these definitions, the three SU(2) groups in a Each of these intrinsic angular momentum groups has a
“spin-up” (α) and a “spin-down” (β) basis state defined
along one of the quantization axes (12), (45), and (67).
Inasphericalcoordinatesystem,thethreeintrinsicangu-
∗ Aemlsaoil: [email protected]:/e/.gwowvw.linkedin.com/in/markraphaelian; laanrdm(oτm67e,nτt6u7m,τ6g7r)ou[3p]s. aTrhe:e(aτl+1g2e,bτr−1a2i,cτp01r2o),p(eτr+t4i5e,sτ−o45f,tτh04e5s)e,
+ − 0
2
intrinsic angular momentum groups are as follows: and an associated charge-raising and charge-lowering
gauge field,
τa,τa = 03, τa,τb = 03,
0 0 0 0
(cid:2) (cid:3) (cid:2) (cid:3) G±38µ(x)=∓√12(G3µ(x) ±iG8µ(x)), (8)
τa,τa = τa , τa,τb = 1τb ,
h 0 ±1i ± ±1 h 0 ±1i ∓2 ±1 such that,
(4)
τa ,τa = τa, τa ,τb = 03, Jµ+(x)G− (x) + Jµ−(x)G+ (x)
h +1 −1i − 0 h +1 −1i 38 38µ 38 38µ
τa ,τa = 03, τa ,τb = 1 τc , = J3µ(x)G3µ(x) + J8µ(x)G8µ(x). (9)
h ±1 ±1i h ±1 ±1i −√2 ∓1 Notonlydoestheequivalenceshowsthatthetwoneutral
where “a,” “b,” and “c” are each one of the three cyclic currents can also be viewed as a form of charge-raising
ordered (12:45:67)SU(2) groups contained within SU(3) and charge-loweringcurrents, but also due to the nature
and a=b=c [4]. oftheSU(3)groupstructureconstants,thecurrentalong
6 6
Aswiththestandardelectroweaktheorywithitsthree the one neutral current axis intrinsically has three times
currents, the above intrinsic angular momentum groups the strength of the current along the other axis.
allow us to define eight coordinate style currents of the Inthe proposedextensionto the standardelectroweak
formJiµ(x) = ψγµτiψ. Oftheeightcurrents,sixareas- rotation, the total interaction current has a coupling
sociatedwiththethreecoplanarSU(2)quantizationaxes strength g and projects onto a plane defined by the
s
and two with the two SU(3) perpendicular axes defining SU(3) group’s neutral current axes and an orthogonal
the plane. Traditionally the six coplanar SU(2) currents axistothe planethatdefinesthe U(1)quantizationaxis.
are reconfigured into spherical coordinate style currents Using g and g as the coupling constants on the group
p 0
andcoupledtoagaugefieldtoproducethecharge-raising planeandU(1)axisrespectively,andαasthe anglethat
and charge-loweringinteraction forms: the projection on the plane makes relative to a planar
axis, it follows that the interaction current can be writ-
g Jµ(x)G (x)
− p i i iµ ten as:
P
= −gp(J1µ2(x)G†12µ(x)+J1µ2†(x)G12µ(x)) −gs Pi Jiµ(x)Giµ(x)
−gp(J4µ5(x)G†45µ(x)+J4µ5†(x)G45µ(x)) (5) = −gpJpµ(x)Gpµ(x) − g0J0µ(x)G0µ(x), (10)
where g is a c−ougppl(inJg6µ7c(oxn)sGta†6n7µt(oxf)t+heJ6iµn7†t(exr)aGct6io7µn(xin))t,he = −g3J3µ(x)G−3gµ0(xJ)0µ(−x)gG80Jµ8µ(x(x),)G8µ(x)
p
SU(3) groupplane and G (G ) is the gauge field associ- where g and g are the coupling constants associated
a †a 3 8
ated with the respective current axis. These six interac- with the projections onto the two SU(3) group’s neutral
tionterms representthe chargechangingcurrentswithin current axes. The coupling constants are defined by the
the SU(3) symmetry group. These terms also represent relations:
the SU(3) interaction current projecting onto a SU(2)
g =g cosα, g =g sinα, tanα=g /g . (11)
symmetry quantization axes. 3 p 8 p 8 3
In addition to the six charge changing current terms,
Unlike the standard two-dimensional rotation that
there are two charge-neutral current terms associated
allows for a mixture of two gauge fields, the above
with the two orthogonal SU(3) axes defining the plane.
gauge fields allow for admixtures from three gauge fields
Thesecurrenttermsfollowtheformofthetwogenerators
through a three-dimensional rotation of the form:
defining the SU(3) group:
J3µ(x) = √12 ψ1γµψ1−ψ2γµψ2 , GG38µµ((xx))  = R(α,β,γ) XX83µµ((xx)) , (12)
(cid:0) (cid:1) (6) G0 (x) X0 (x)
J8µ(x) = √16 ψ1γµψ1+ψ2γµψ2−2ψ3γµψ3 ,  µ   µ 
(cid:0) (cid:1) whereR(α,β,γ) is a standardrotationmatrix. The pro-
and couple to fields defined along their respective quan- posedstrongomagneticextensionreducestothestandard
tization axis. electroweak rotation in a left handed coordinate system
As with the charge-changing currents, these two under the rotation matrix R(Z ,Y ,Z )=R(0,β,0).
1 2 3
charge-neutral currents can also be written in another Followingthestandardelectroweakunificationscheme,
more symmetrical manner by defining a charge-raising we define a strongomagnetic hypercharge X that cou-
sm
neutral current and charge-loweringneutral current, ples to the U(1) field. For the extended theory, the
sameunificationmethodologyresultsintherelationQ =
J3µ8±(x)=∓21(J3µ(x) ±i√13J8µ(x)), (7) I3 + Xsm,whereQisthe electricchargeassociatedwith
3
In the electroweak unification, the quantization axis
TABLE I. This table shows the relationship between the
of the SU(2) group is rotated about the “line of nodes”.
quantum numbers associated with the electric, isospin, and
Theprojectionofthisaxisontothenon-rotatedU(1)axis
strongomagnetic charges for first generation particles.
allows for the equivalence between the SU(2) coupling
constant and the U(1) coupling constant. We can do
Electric Weak Strongomagnetic the same for this rotation by defining the electric charge
Charge Isospin Hypercharge
in terms of the projected coupling coefficients, one from
Particle Symbol Q I X
3 sm each neutral current axis.
anti-down d +1/3 +1/2 −1/6 g cosβ =g cosγsinβ+g sinγsinβ =e (17)
anti-up u −2/3 −1/2 −1/6 0 3 8
up u +2/3 +1/2 +1/6 The above equivalence produces two solutions:
down d −1/3 −1/2 +1/6
(g /e)=(g /e)tanβ for γ = +α;
positron e+ +1 +1/2 +1/2 0 p
anti-neutrino ν 0 −1/2 +1/2 =(g /e)tanβcos2γ for γ = α. (18)
p −
neutrino ν 0 +1/2 −1/2
electron e− −1 −1/2 −1/2 Using the γ = +α constraint and identifying X0 (x)
µ
withA (x),theinteractionneutralcurrentsfoundwithin
µ
the U(1) and SU(3) groups are:
tchiaeteUd(1w)itshymthmeeStUry(2g)rosyump,mIe3triys tghroeuwpe,aakndisoXspsminisastshoe- − gpJpµ(x)Gpµ(x) − g0JXµ0(x)G0µ(x) =
sstyrmonmgeotmryaggnroeutipc.hTypheerrcehlaartgioenaasslsoociraetperdeswenitthstthhee SSUU((32)) − gp/cosβ(cid:18)(cos2(sγin−2cβocsoβssγi)ns2µγ()xJ)3µ/(ex) (cid:19)X3µ(x)
−
and SU(3) groups’ quantum number projections onto a (sin2γ cosβcos2γ)Jµ(x)
quantization axis having U(1) symmetry. In terms of + gp/cosβ(cid:18) (sin−2βsinγ)sµ(x)8/e (cid:19)X8µ(x)
currents, the relation is equivalent to: −
sµ(x)A (x). (19)
Jµ(x) = Jµ (x) = sµ(x)/e Iµ(x). (13) − µ
0 Xsm − 3
For the α = γ constraint, the interaction neutral
Thestrongomagnetichyperchargeandthecorrespond- −
currents found within the U(1) and SU(3) groups are:
ing electric charge and isospin state for first generation
fsetrrmonigoonms aisgnperteiscenhtyepdericnhatrhgeetmabalgeniIt.uIdtesihsoiwnsdetpheantdtehnet − gpJpµ(x)Gpµ(x) − g0JXµ0(x)G0µ(x) =
(cos2γ+cosβsin2γ)Jµ(x)
owfhtehtheeirsotshpeinfesrtmaitoenoifstahebaferrymonioonralnepdtodne.peTnhdespoonllayriotny − gp/cosβ(cid:18) (sin2βcos2γcosγ)s3µ(x)/e (cid:19)X3µ(x)
−
of the hypercharge depends on whether the fermion is a (sin2γ+cosβcos2γ)Jµ(x)
particle or anti-particle. + gp/cosβ(cid:18) (sin2βcos2γsinγ)s8µ(x)/e (cid:19)X8µ(x)
The requirement of not having isospin couple to the −
sµ(x)A (x). (20)
hyperchargeproduces an isospincurrentcomprisedfrom − µ
currents along both neutral current axes in the SU(3)
The above SU(3) neutral current terms involve only
groupplane. ThisisospincouplingtotheU(1)symmetry
two angles: β, the amount of angular rotationabout the
axis vanishes at rotational angles β = 0, π and α =
± “line of nodes”, and γ, the amount of angular rotation
2π/3.
± perpendicular to the “line of nodes.” Even when we set
g Iµ(x) = g (cosαJµ(x)+sinαJµ(x))tanβ. (14) γ = 0, meaning that the gauge field defined along the
0 3 p 3 8 ˆe8 direction lies on and parallel to the “line of nodes”
When we use this relation to eliminate the remaining positivedirectionality,theabovetotalinteractionreduces
weak isospin current terms that couple to the neutral to:
currentaxes,I3µ(x)X3µ(x)andI3µ(x)X8µ(x),weproduce
additional crossterms between neutral currents defined − gpJpµ(x)Gpµ(x) − g0JXµ0(x)G0µ(x) =
along one axis and gauge fields defined along the other
axis. To remove these crossterms, the following angular − (gp/cosβ)(J3µ(x)−sin2βsµ(x)/e)X3µ(x)
relationshipmust holdforthe coupling coefficients along − gpJ8µ(x)X8µ(x) − sµ(x)Aµ(x), (21)
the two axes defining the plane:
and has one more term over what is found in the sin-
tan2γ = tan2α, gle neutral current axis case of the standardelectroweak
(15)
tanγ = ± tanα = ±g8/g3, theory. In this rotation there is an extra current lying
along the “line of nodes” that is absent in the stan-
or γ = α, from which we can choose the constraints:
± dard electroweak theory. This extra current is orthog-
cosγ = +cosα and sinγ = sinα. (16) onal to both the U(1) symmetry and electroweak axes
±
4
and therefore representsa non-electromagnetic and non- straint γ =+α is given by:
electroweak current. For this rotation, only the gauge
field defined along the ˆe8 axis does not couple to the − gs Jiµ(x)Giµ(x) = −sµ(x)Aµ(x)
electromagnetic current. Xi
tioAntliγes=on+πbu/2t,atnhtei-pnaeruatlrlaell ctourtrheent“alilnoengofthneodˆee3s”diproecs-- − gp/cosβ(cid:18)(cos2(sγin−2cβocsoβssγi)ns2µγ()xJ)3µ/(ex) (cid:19)X3µ(x)
−
itive directionality. Here the above interaction reduces (sin2γ cosβcos2γ)Jµ(x)
to: + gp/cosβ(cid:18) (sin−2βsinγ)sµ(x)8/e (cid:19)X8µ(x)
−
−++g(pggJpppµ/J(c3xµo)(sxGβ)pX)µ(3(Jµx8µ()(xx)−)−−g0ssJiµnXµ(2x0β()xsA)µµG((x0x)µ)/(.ex))X=8µ(x) (22) −−gp−g(pgJ(p1µJ2(4µ(5Jx(6µ)x7G)(xG†1)2†4µG5(µ†6x(7)xµ+)(x+J)1µ+J2†4µ(5J†x6(µ)7x†G)(xG1)24µG5(µ6x(7)xµ))(x))). (25)
There are many similarities between the interaction
As with the γ = 0 rotation, there is an additional term currentfoundinthestandardelectroweaktheoryandthe
overwhatis found in the single neutralcurrentaxiscase interaction current found in the strongomagnetic exten-
and is representative of a non-electromagnetic and non- sion to the theory. To achieve the standard electroweak
electroweakneutral currentinteraction. In this rotation, theory limit, R(0,β,0), the additional neutral current
only the gauge field defined along the ˆe3 axis does not interaction term must cancel out the additional charge-
couple to the electromagnetic current. raising and charge-loweringinteraction terms:
IntableIthestrongomagnetichyperchargemagnitude J8µ(x)X8µ(x) =
equals 1/2 for leptons and 1/6 for baryons. These quan-
tum numbers are equivalent to the normalization con- (J4µ5(x)G†45µ(x)+J4µ5†(x)G45µ(x)) (26)
sJtµa(nxt)saanssdocJiµat(exd)cwuirthretnhtse.gSeinnecreattihnegsemtawtorixcudrerfienntinsgintdhie- +(J6µ7(x)G†67µ(x)+J6µ7†(x)G67µ(x)).
3 8
vidually couple to the electromagnetic current at inter- Inconclusionwehavepresentedastrongomagneticex-
action rotation angles γ = 0, π, and γ = π/2 respec- tensiontotheelectroweaktheoryinwhichathreedimen-
± ±
tively, then the generators’ normalization constants are sionalrotationiscapableofproducingaSU(3)groupuni-
representative of a scaling factor associated with the in- fication with the weak isospin and U(1) groups. In this
teractioncurrentswhenprojectedontheU(1)symmetry extensiontherearethreeorthogonalneutralcurrentaxes,
axis. The J3µ(x) interaction current for leptonic inter- two of which form a two-dimensional plane containing
actions and the J8µ(x) interaction current for baryonic weak isospin, SU(2), and SU(3) algebraic group struc-
interactions. tures. Unification occurs through neutral current pro-
jections onto a U(1) symmetry axis whose isospin inde-
Other rotational angles produce interaction current
pendent magnitude, the SU(3) hypercharge, depends on
configurations of interest. For β = 0 and γ =
thenormalizationconstantassociatedwiththegenerator
0, π/2, π, only decoupled neutral-charge SU(3) and
± ± defining the quantization axis. Manifestations of non-
electromagnetic interaction currents remain:
electromagnetic and non-electroweak interaction neutral
currents occur at specific rotational angles. The exten-
− gpJpµ(x)Gpµ(x) − g0JXµ0(x)G0µ(x) = shiaonndreedducoceosrdtointahteesstyasntedmardunedleecrtarotwheraeke-tdhimeoernysiinonaallerfot--
± gp(J3µ(x)X3µ(x)+J8µ(x)X8µ(x)) tationofR(0,β,0)andaneutralcurrentinteractionterm
sµ(x)A (x). (23) canceling out two associated charge-raising and charge-
− µ
lowering interaction terms.
At β = 0 and γ = π/4, the only neutral-charge inter-
±
action current that remains is electromagnetic: ACKNOWLEDGMENTS
− gpJpµ(x)Gpµ(x) − g0JXµ0(x)G0µ(x) = SecTuhriistymTaencuhsncorilpotgiehsa,sLbLeCen, uanudtheorreCdonbtyratchteNNoa.tioDnEa-l
sµ(x)A (x). (24) AC52-06NA25946 with the U.S. Department of Energy.
− µ
The United States Government and the publisher, by
accepting the article for publication, acknowledge that
The full strongomagnetic-electroweak interaction the- the United States Government retains a nonexclusive,
oryhas twocharge-neutralcurrents,three charge-raising paid-up, irrevocable, world-wide license to publish or
currents,threecharge-loweringcurrents,andforthecon- reproduce the published form of this manuscript, or
5
allow others to do so, for United States Government search in accordance with the DOE Public Access Plan
purposes. The U.S. Department of Energy will provide (http://energy.gov/downloads/doe-public-access-plan),
public access to these results of federally sponsored re- DOE/NV/25946–2648.
[1] F.MandlandG.Shaw,QuantumFieldTheory.NewYork: [3] Pushpa,P.S.Bisht,T.Li,andO.P.S.Negi,Int.J.Theor.
Wiley, 1986, p271. Phys., 51, 1866 (2012).
[2] F. Halzen and A. D. Martin, Leptons and Quarks, John [4] M. L. Raphaelian, Lecture Notes - Concepts of Rotation
Wiley & Sons, New York,NY (1984). Within Quantum Physics, unpublished (2016).