aa r X i v : . [ h e p - ph ] J a n Standard Model gauge coupling unification
E.K. Loginov ∗ Ivanovo State University,Ermaka St. 39, Ivanovo, 153025, Russia
November 9, 2018
Abstract
We study the low energy evolution of coupling constants of thestandard model and show that gauge coupling unification can beachieved at the electroweak scale with a suitable normalization. Wechoose the grand unification group to be the semidirect product of
Spin (8) by S . In this case the three low energy gauge couplings andthe two scalar self-couplings are determined in terms of two indepen-dent parameters. In particular, it gives a precise prediction for themass of the Higgs boson. The standard model (SM) is a mathematically consistent renormalizable fieldtheory which predicts or is consistent with all experimental facts [1]. It suc-cessfully predicted the existence and form of the weak neutral current, theexistence and masses of the W and Z bosons, and the charm quark, as neces-sitated by the GIM mechanism. The charged current weak interactions, asdescribed by the generalized Fermi theory, were successfully incorporated, aswas quantum electrodynamics. The consistency between theory and experi-ment indirectly tested the radiative corrections and ideas of renormalizationand allowed the successful prediction of the top quark mass. Nevertheless, ∗ E-mail address: [email protected]
QCD ≪ M GUT that is basedon the logarithmic renormalization-group dependence of the gauge couplingconstant on the energy. Note, however, a similar analysis is not successful forthe electroweak interaction, whose coupling constants are small at the scale v ≈
246 GeV. It is unrelated to any dynamical scale and is introduced intothe theory as a free parameter. One immediate consequence of the grandunification hypothesis is a very simple explanation for the experimentallyobserved charge quantization. This is because the eigenvalues of the gener-ators of a simple non-Abelian group are discrete while those correspondingto the Abelian group are continuous. Unfortunately, by now LEP data haveshown [4, 5, 6] that simple non-SUSY grand unifications must be excluded,initially by the increased accuracy in the measurement of the Weinberg an-gle, and by early bounds on the proton lifetime [7]. In other words, gaugecoupling unification cannot be achieved in the SM if we choose the canonicalnormalization for the SM group, i.e., the Georgi–Glashow SU (5) normaliza-tion. Also, to avoid proton decay induced by dimension-6 operators via heavygauge boson exchanges, the gauge coupling unification scale is constrainedto be higher than about 5 × GeV.This latter restriction is not true for gauge-Higgs unification [8, 9, 10, 11],however. In gauge-Higgs models, the compactification scale may be of the or-der of the electroweak. Such unification is a very fascinating scenario beyondthe SM since the Higgs doublet is identified with the extra component of thehigher dimensional gauge field and its mass squared correction is predictedto be finite regardless of the non-renormalizable theory. This fact has openeda new possibility to solve the gauge hierarchy problem without, for exam-ple, supersymmetry. Obviously, the gauge-Higgs coupling unification can beachieved at the electroweak scale only with a suitable normalization. Note2lso that the unification group of the model does not necessarily be simple.For example, such group may be represented as a product of identical simplegroups (with the same coupling constants by some discrete symmetries) or itmay be obtained as an extension of a simple Lie group by means of a finitegroup of operators. The latter possibility will be considered in this paper. S ⋉ Spin (8) symmetry
We begin by discussing the following simple construction. This constructionarises in connection with the following question: is it possible to embed anarbitrary group G in some group e G with the property that every automor-phism of G is the restriction of some inner automorphism of e G ? Let Φ be asubgroup of Aut G . Then for e G one may take the set of ordered pair φg withmultiplication defined by the rule φg · φ ′ g ′ = φφ ′ g φ ′ g ′ , (1)there φ ∈ Φ and g ∈ G . (We are writing pairs without their customarycomma and brackets.) The group axioms are straightforward to verify. Fromthe rule for multiplication (1) it is immediate that φ − gφ = g φ . Hence theproblem is solved. The group e G is called the extension of the group G bymeans of the automorphisms in Φ and denoted as Φ ⋉ G . Alternatively onesays that e G is a semidirect product of G by Φ.Now let Φ be a subgroup of the outer automorphisms group of G . Suppose V is a representation space of G . Then the representation of G in V inducesa representation of e G in the direct sum e V = V ⊕ · · · ⊕ V n , where each directsummand V i is isomorphic to V and n = | Φ | . For the space V i one may takethe ordered pair φ i V , where φ i ∈ Φ. Then the action of e G on e V can bewritten as φg · ⊕ i φ i V = ⊕ i φφ i g φ i V. (2)Let G be a set of elements of G such that g φ = g for all φ ∈ Φ. Clearly,it is a Φ-invariant subgroup of G . Using the formula (2) one may define arepresentation of G on e V . It is easy to prove that G acts equivalently oneach direct summand of e V .Let G be a simple gauge group. If the normalization of the generatorsof G are fixed, then the gauge couplings will be the same for both G and G . Suppose that for the energy scale µ > M an G gauge theory possesses3oth discrete and gauge symmetries, whereas for µ = M the symmetriesbreaking e G → G to take place. Then the representation space of G reducesto V and hence the normalization of the generators of G must be changed.Since the definition of coupling constants depends on the normalization ofthe generators it follows that the gauge coupling of G should be also change,namely g → g | Φ | as e V → V .Now we suppose G = Spin (8). This group has the outer automorphismsgroup S = h ρ, σ i , where ρ = σ = ( ρσ ) = 1, and two Majorana–Weylreal eght-dimensional representations that related to the eght-dimensionalreal vector representation by the action of S . The group Spin (8) cannotcontain the SM group as a subgroup, but it contains disjoint subgroups G and G × G that are isomorphic to SU (3) and SU (2) × U (1), respectively.Moreover, we always can choose these subgroups in the following manner:(i) g φ = g for 1 = φ ∈ S and g ∈ G ,(ii) g φ = g = g ρσ for 1 = φ = ρσ and g ∈ G ,(iii) g φ = g for all φ ∈ S and g ∈ G .Hence, if for the energy scale µ = M the discrete and gauge symmetriesbreaking of S ⋉ Spin (8) to take place, then the gauge couplings of G , G ,and G should be satisfy g = √ g = √ g (3)as µ = M .Just as for Spin (8), the group S ⋉ Spin (8) cannot contain the SM groupas a subgroup. Nevertheless, in the next section we shall show that there isa way to break the symmetry by S ⋉ Spin (8) SU (2) −−−→ SU (3) × U (1) . (4)Moreover, the symmetry breaking is such that the gauge couplings of theSM group also satisfy (3). This possibility of the breaking is based on thefollowing mathematical construction [12].As was remarked before, the group G admits the outer automorphisms ρ and σ . Let G σ = { g ∈ G | g σ = g } (5)(i.e. G σ is the centralizer of σ in G ). We denote by S the factor space G/G σ .Our nearest aim is to define a binary composition on the cosets of G σ . Define L = { g − g ρσ | g ∈ G } . (6)4or each left coset gG σ , there exists exactly one element L a in L such that L ∩ gG σ = { L a } . This defines a permutation representation of G on S ; if gL a G σ = L b G σ for g ∈ G let gL a = L b . Using this permutation representa-tion, we may define a binary composition in S . If L a L b G σ = L c G σ , define ab = c . This binary composition make S into a nonassociative loop isomor-phic to S , which is defined in Appendix A. In particular, this defines thepermutation representation of G on S . Extending this action by linearityon O , we obtain the eight-dimensional spinor representation of G .Apart from the left action of G on S , there exists the trivial right actionof G on S with G σ acting on itself by multiplication on the right. Suppose G ⊂ G σ and ( G × G ) ∩ G σ = 1. Then the (left ant right) actions of G on S induce the left action of G × G on S and the right action of G on G σ .Obviously, these group actions are independent of each other. In this section we briefly discuss the gauge-Higgs model based on the group S ⋉ Spin (8). Consider the
Spin (8)-invariant gauge theory defined on themanifold M = M , × S , where M , is the Minkowski spacetime and S isthe seven-dimensional sphere. (We assume that this sphere is equipped withthe octonionic multiplication.) Suppose that this theory possess a symmetryunder a discrete group K of inner automorphisms. Further, let A ( x, y ) and C ( x, y ) be all gauge fields in the theory and only the fields C ( x, y ) ∈ so (7) v ,where so (7) v is the Lie algebra of G σ . Following [13], we declare that onlyfield configurations invariant under the action K : ( A ( x, y ) → M ( k ) A ( x, k − [ y ]) C ( x, y ) → N ( k ) C ( x, k − [ y ]) , (7)are physical. Here M ( k ) and N ( k ) are matrix representations of K and k [ y ]is the image of the point y ∈ S under the operation of k ∈ K . But unlikethe standard orbifolding conditions, we suppose that M ( K ) = N ( K ).In order to determine automorphisms that are responsible for the sym-metry breaking (4), we must first select the group K . Suppose K = Z × Z .Further, suppose that the subgroup H in Spin (8) is generated by the opera-tors I = R and J = L − L L , which are defined in Appendix B. Obviously, IJ = J I and I = J = 1. It follows from this that H = H I × H J ,5here the subgroups H I and H J are generated by I and J , respectively,i.e., H ≃ Z × Z . Define the action of K on S and the representations K → M ( K ) and K → N ( K ) as follows. Let f , f I , and f J be homomor-phisms of K onto H , H I , and H J respectively. Then M ( k ) A ( x, y ) = f ( k ) − A ( x, y ) f ( k ) , (8) N ( k ) C ( x, y ) = f I ( k ) − C ( x, y ) f I ( k ) , (9) k − [ y ] = f J ( k ) − y. (10)We now focus our attention on the symmetry breaking at the fixed points.Since the factor group K/K J acts on S trivially, it follows that the Spin (8)gauge symmetry is reduced (under the given action) to the centralizer of I in Spin (8), i.e., to SU (4) s × U (1) (see also Appendix C). Therefore thefields A ( x, y ) and C ( x, y ) take its values in the Lie algebra su (4) s ⊕ u (1).In particular, C ( x, y ) takes its values in su (3) s (which is the intersection of su (4) s and so (7) v ). Now consider the SU (4) s × U (1) symmetry breaking by K J . It follows from (4) that the SU (3) s symmetry must be preserved underthe action of K J on S . This is possible only if the components of C ( x, y )are independent of y , i.e., if C M ( x, y ) = ig s C pµ ( x ) λ p , (11)where λ p are the usual Gell-Mann matrices for SU (3) (cf. the formulas inAppendix C).On the contrary, the unbroken symmetries of A ( x, y ) must belong to thecentralizer of J in SU (4) s × U (1). Using (B.2) and the explicit form of thegenerators of SU (4) s × U (1), we prove that the fields A ( x, y ) take its valuesin the Lie algebra su (2) s ⊕ u (1). Hence we can define these fields by A M ( x ) = igA kµ ( x ) σ k , (12) B M ( x, y ) = { ig ′ B µ ( x ) , g ′ φ a ( x ) } , (13)where σ k are the standard Pauli matrices and φ a are the components of φ ( x ) ∈ O . Strictly speaking, we would have to write φ ( x ) ∈ S . However, inthis case the field φ ( x ) will be unobservable. The point is that for the Kaluza–Klein type theories to be able to describe the observed four-dimensional worldit is necessary for the extra spatial dimensions to be compactified down to a6ize which we do not probe in particle physics experiments (e.g. the Plancklength). Therefore we suppose that the field φ ( x ) has quantum fluctuationssuch that φ ( x ) ∈ O but not S . In this case exactly one component of φ ( x )(scalar field) will be observed.We now consider the action of K J on S . It is easily shown that thecondition J y = y is equivalent to ( e , e , y ) = 0. Hence y belong to anassociative subalgebra of O generated by e and e . Obviously, this is thealgebra of quaternions with the basis 1 , e , e , e . The complexification of O transform this subalgebra into a two-dimensional unitary space. (Denoteit by the symbol Φ .) We have proved, in fact, that all nonzero fields φ ( x )in (13) must belong to Φ . Thus, if we identify φ ( x ) as the massless Higgsdoublet, then we obtain a complete set of boson fields of the SM.Finally, consider the SM Lagrangian for the field φ ( x ) L H = ( D µ φ ) † ( D µ φ ) + µ φ − λφ , (14)and suppose that the fields B µ ( x ) and φ a ( x ) in (14) are components of B ( x, y ). Then we have λ = g ′ . (15)As usual the Higgs vacuum expectation value breaks the SU (2) × U (1) sym-metry down to U (1). As a result we have the symmetry breaking (4). The conditions (3) are valid in the S ⋉ Spin (8) limit. Now we need to studythe regime µ < M . The evolution of the SM gauge coupling constants in theone-loop approximation is controlled by the renormalization group equation dα − n ( µ ) d ln µ = b n π , (16)where b = − n , b = 22 − n , b = 33 − n , and α n = g n / π . (We haveignored the contribution coming from the Higgs scalar and higher-order ef-fects.) It follows from (3) that the generators of SM group in the fundamentalrepresentation should be normalize by the condition 6 n = 2 n = n = N f ,where N f is the number of quark flavors. Expressing the low-energy cou-plings in terms of more familiar parameters, we can represent the solutions7f Eq. (16) as α − s ( µ ) = α − ( M ) − b π ln M µ , (17) α − ( µ ) sin θ µ = α − ( M ) − b π ln M µ , (18)35 α − ( µ ) cos θ µ = α − ( M ) − b π ln M µ . (19)Taking the linear combination [12 × Eq. (17) − × Eq. (18) + 7 × Eq. (19)]and using the relations (3), we havesin θ µ = 737 + 60111 α ( µ ) α s ( µ ) . (20)Obviously, Eq. (20) implies a non-trivial consistency condition among thegauge couplings. Taking the linear combination [ − × Eq. (17)+6 × Eq. (18)+10 × Eq. (19)] and again using the relations (3), we haveln M µ = 6 π (cid:2) α − ( µ ) − α − s ( µ ) (cid:3) . (21)This determines the unification scale M . Also, combining Eqs. (20) and(21), we obtain sin θ µ = 313 − α ( µ )39 π ln M µ . (22)Finally, it follows easily from Eqs. (17)–(19) that the running electroweakand strong gauge coupling constants satisfy α − ( µ ) = α − ( M ) − − N f π ln M µ , (23) α − s ( µ ) = α − s ( M ) − − N f π ln M µ , (24)where the gauge couplings are connected by the relation α s ( M ) = 13 α ( M ).Note that the choice of normalization of the generators essentially influenceson the behavior of the gauge couplings by changing its values in fixed points.So for example Eqs. (23) and (24) will differ from that obtained in the SM.Therefore we must have a rule which permits to compare the gauge coupling8onstants in our (non-canonical) and the canonical normalizations. We willextract this rule from the renormalization on-shell scheme.The on-shell scheme [14, 15, 16, 17, 18, 19] (see also Ref. [20]) promotes thetree-level formula sin θ W = 1 − M W /M Z to a definition of the renormalizedsin θ W to all orders in perturbation theory, i.e.,sin θ W = παv M W (1 − ∆ r ) , (25)where ∆ r summarizes the higher order terms. Here α is the fine structureconstant, M W is the mass of the charged gauge boson, and v = ( √ G F ) − / is the vacuum expectation value. One finds ∆ r = ∆ r − ∆ r ′ , where ∆ r =1 − α/α ( M Z ) is due to the running of α and ∆ r ′ represents the top quark mass m t and the Higgs boson mass M H dependence. Using the formal expansion(1 + ∆ r ′ ) − = 1 − ∆ r ′ + . . . , we can rewrite the formula (25) in the formsin θ W = πα ( M Z ) v M W (cid:18) − α ( M Z ) α ∆ r ′ + . . . (cid:19) . (26)In the on-shell scheme the value of sin θ W is independent of the normalizationof the generators. We suppose that the value of α ( µ ) in the fixed point µ = M Z is the same for both the canonical and non-canonical normalizations.Using this condition, we can now compare values of the running couplingconstant α ( µ ) in the two normalizations.Following Ref. [21, 22, 23], we remove the ( m t , M H ) dependent term from∆ r and write the renormalized sin θ µ (in the non-canonical normalizations)as sin θ µ = πα ( µ ′ ) v M W (27)for M Z ≤ µ ≤ µ ′ ≤ v . Further, we suppose that the unification scalecoincides with the electroweak scale (i.e., M = v ) and that the left-handside of Eqs. (22) and (27) are equal as µ = M . In this case M W M = r πα , (28)where α = α ( M ). It follows from (20)–(24) and (28) that the three param-eters α ( M Z ), α s ( M Z ), and sin θ W of the SM are now determined in terms ofone independent parameters α . We show the gauge coupling unification inFig. 1. Thus, there are two predictions.9 (M)= α M = 246.2204 GeV (M) (M)= µ Figure 1: One-loop gauge coupling unification for the SM with the non-canonical normalization.In conclusion, we show that the values of the coupling constants andthe masses of the gauge bosons which are deduced from the SM are com-patible with these predictions. Using α − = 127 .
726 and N f = 6 yield α − ( M Z ) = 127 . α s ( M Z ) = 0 . θ W = 0 . α s ( M Z ) = 0 . α s ( M Z )may also be chosen the same for both the canonical and non-canonical nor-malizations. Using M = 246 . M W = 80 . M Z = 91 . The condition (15) gives a precise prediction about the Higgs mass m φ . Herewe follow the presentation of Coleman and Weinberg [28] (see also [29]). Theone-loop effective potential of SU (2) × U (1) gauge theory is given by V ( φ ) = − µ φ + λφ + Cφ ln φ M , (29)10here M is an arbitrary mass parameter and C = 116 πv X b m b + m φ − X f m f ! . (30)Here the indices b and f run over the vector bosons and fermions (the topquark contribution is excluded), and the mass of the Higgs scalar is taken tozeroth order, i.e., m φ = 2 µ = 2 λv . (31)With (29), we can obtain the mass of the Higgs particle. It is given by m φ = 2 v (cid:20) λ + C (cid:18) ln v M + 32 (cid:19)(cid:21) . (32)We now use the condition (15). It follows from the formulas (3) and (28)that λ = 310 g = 26 πα . (33)Substituting this expression into (31), one finds the mass of the Higgs scalarto zeroth order ( m φ = 124 .
53 GeV). Knowing the masses of the vector bosonsand fermions, one may calculate C . (We obtain C = 0 . M = M Z , we have the Higgs boson mass m φ =126 .
15 GeV. This is agrees quite well with the experimental results that wererecently obtained in [30, 31].
In this paper, we have shown that the gauge-Higgs model based on the group S ⋉ Spin (8) can be considered as a candidate for the real physical theory.It do not contradict SM (at least at the bosonic sector) and gives precisepredictions for the masses of the gauge bosons and the Higgs scalar. Herewe make two general remarks.The group
Spin (8) occupies a special position among the simple Liegroups since only it has outer automorphism group S . Namely this prop-erty of Spin (8) permits to define the non-canonical normalization in a naturalway and to get the gauge coupling constants unification. To describe the S -symmetry breaking, we embedded the Spin (8)-gauge theory in the model11ith larger global symmetry groups. The motivation for this is that what-ever the high energy physics producing the spontaneous breaking of the gaugegroup, it is likely to possess a larger global symmetry than the gauged one.We risk to suppose that the existence of discrete S -symmetry is related to aduplication of the fermionic structure in the SM. We suppose that S ⋉ K isthe discrete flavor symmetry group. But of course this is only a hypothesis.In order to describe the gauge-Higgs unification, we have used the re-lation between Spin (8) and the algebra of octonions. Generally speaking,this approach is not new. Properties of octonions was used earlier to de-scribe various mechanisms of compactification of d = 11 supergravity downto d = 4 [32] (see also the review [33]) and to find solutions of the low-energyheterotic string theory [34]. There have been many other attempts over theyears to incorporate this algebra into physics. The present paper is a nextstep in this direction. A Octonions
We recall that the algebra of octonions O is a real linear algebra with thecanonical basis e = 1 , e , . . . , e such that e i e j = − δ ij + c ijk e k , (A.1)where the structure constants c ijk are completely antisymmetric and nonzeroand equal to unity for the seven combinations (or cycles)( ijk ) = (123) , (154) , (167) , (264) , (275) , (347) , (365) . (A.2)The algebra of octonions is not associative but alternative, i.e. the associator( x, y, z ) = ( xy ) z − x ( yz ) (A.3)is totally antisymmetric in x, y, z . Consequently, any two elements of O generate an associative subalgebra. The algebra O permits the involution(anti-automorphism of period two) x → ¯ x such that the elements t ( x ) = x + ¯ x, n ( x ) = ¯ xx (A.4)are in R . In the canonical basis, this involution is defined by ¯ e i = − e i . Itfollows that the bilinear form( x, y ) = 12 (¯ xy + ¯ yx ) (A.5)12s positive definite and defines an inner product on O . It is easy to provethat the quadratic form n ( x ) permits the composition n ( xy ) = n ( x ) n ( y ) . (A.6)It follows from this that the seven-dimensional sphere S = { x ∈ O | n ( x ) = 1 } (A.7)is closed relative to the multiplication in O . Finally, since the quadratic form n ( x ) is positive definite, it follows that O is a division algebra. B Triality
Let x be any element of O . The left multiplications L x and right multiplica-tions R x of O which are determined by x are defined by L x y = xy, R x y = yx (B.1)for all y in O . Clearly L x and R x are linear operators on O . We choose thecanonical basis and denote by L i and R i the operators L e i and R e i respec-tively. Then from (A.1) and the fully antisymmetry of the associator (A.3),we get L i L j + L j L i = − δ ij I, (B.2)where I is the identity 8 × L , . . . , L are generators of the Cliffordalgebra Cl , ( R ), and therefore they generate the Lie algebra so (8). This isthe Lie multiplication algebra of O .In this algebra, we separate the subspaces L spanned by the operators L i and the subalgebra so (7) s spanned by the operators S i = L i + 2 R i , (B.3) D ij = L [ e i ,e j ] − R [ e i ,e j ] − L i , R j ] . (B.4)(The latter linearly generate the 14-dimensional exceptional simple Lie al-gebra g .) This imply that the algebra so (8) decomposes into the directsum so (8) = so (7) s ⊕ L. (B.5)13he algebra so (8) admits the outer automorphisms ρ and σ of orders 3 and2 respectively. We may define them by L ρi = R i , R ρi = − L i − R i ,L σi = − R i , R σi = − L i . ) (B.6)Obviously, the automorphisms ρσ , σρ , and σ fixe all elements of so (7) s , so (7) c = so (7) ρs , and so (7) v = so (7) ρ s , respectively. The elements of intersec-tion of the subalgebras, i.e. the elements of g , is fixed by ρ .Just as for so (8), the group Spin (8) also admits the outer automorphisms ρ and σ . According to (B.6), they are defined by L ρa = R a , R ρa = L − a R − a ,L σa = R − a , R σa = L − a , ) (B.7)where a ∈ S . The automorphisms ρσ , σρ , and σ fixe the elements of SO (7) s , SO (7) c = SO (7) ρs , and SO (7) v = SO (7) ρ s respectively. The intersection ofthe group, i.e. G , is fixed by ρ . Here SO (7) v is generated by the elements R − a L a = L a R − a , (B.8) L − ab L a L b = R ab R − a R − b . (B.9)Note also that these automorphisms permute inequivalent irreducible repre-sentations s , c , and v of the Spin (8) group having the same dimension-ality.
C Complexification
Suppose C is a subalgebra of O spanned by the elements 1 and i = e . Wemay consider O as a four dimensional complex (or rather unitary) spacerelative to the multiplication ax , where a ∈ C and x ∈ O . This space isinvariant under the unitary group, SU (4) s × U (1), the Lie algebra of whichdecomposes into the direct sum su (4) s ⊕ u (1) = su (3) s ⊕ su (2) s ⊕ u (1) ⊕ V s (C.1)of the subspaces (but not the Lie subalgebras). We write down the generatorsof SU (4) s × U (1) in the explicit form.14) su ( ) s : D − D = 6( e − e ) (C.2) D − D = 6( e − e ) (C.3) D − D = 6( e − e ) (C.4) D − D = 6 i ( e + e ) (C.5) D − D = 6 i ( e + e ) (C.6) D − D = 6 i ( e + e ) (C.7) D − D = 6 i ( e − e ) (C.8) D = 2 i ( e + e − e ) (C.9)2) su ( ) s ⊕ u ( ): L + L L = 2( e − e ) (C.10) L + L L = 2 i ( e + e ) (C.11) L + L L = 2 i ( e − e ) (C.12) R = i ( e + e + e + e ) (C.13)3) V s : L + L L = 2( e − e ) (C.14) L + L L = 2( e − e ) (C.15) L + L L = 2 i ( e + e ) (C.16) L + L L = 2 i ( e + e ) (C.17)Here e ij is the 4 × i, j )th entry 1, and all other entries 0.Note that the automorphisms ρ and σ fixe the elements of su (3) s and theautomorphism ρσ fixes the elements of su (2) s and V s , while R ∈ u (1) isnot invariant under any element of S . Note also that SU (4) s × U (1) is thecentralizer of R in Spin (8).
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