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Centre de Physique Théorique∗ - CNRS - Luminy, Case 907 F-13288 Marseille Cedex 9 - France A DETAILED ACCOUNT OF ALAIN CONNES’ VERSION OF THE STANDARD MODEL IV Daniel KASTLER 1 5 and Thomas SCHU¨CKER 2 9 9 1 n a J 8 1 1 v 7 7 0 Abstract 1 0 5 We give a detailed account of the computation of the Yang-Mills action for the Connes-Lott 9 model with general coupling constant in the commutant of the K-cycle. This leads to tree- / h approximation results amazingly compatible with experiment, yielding a first indication on the t - Higgs mass. p e h : v i X r a PACS-92: 11.15 Gauge field theories MSC-91: 81E13 Yang-Mills and other gauge theories January 1995 CPT-94/P.3092 hep-th/9501077 anonymous ftp or gopher: cpt.univ-mrs.fr ∗ Unité Propre de Recherche 7061 1 and Université d’Aix-Marseille II 2 and Université de Provence The preceding papers I, II and III of this series presented computations based on the following Ansatz for the non-commutative Yang-Mills action: (1) YM = α (θ ,θ ) +α (θ ,θ ) (α +α = 1), l l l l q q q q l q with the scalar products (·,·)l and (·,·)q stemming as follows from the traces τDl and τDq: (2) (ω,ω′)l = ReτDl(ω⋆ω′) = ReTrω Dl−4πl(ω⋆ω′) , ω,ω′ ΩAn( ΩBn). (ω,ω′)q = ReτDq (ω⋆ω′) = ReTrωnDq−4πq(ω⋆ω′)o ∈ ∈ n o Thesescalar products are however not the most general and natural ones [3b]: with Γ , resp. l Γ (Γ′, resp. Γ′), positive elements of the respective commutants π (A),D ′, π (A),D ′ (qπ (Bl),D ′, qπ (B),D ′), the alternative Ansatz: { l l} { q q} l l q q { } { } (ω,ω′) = ReTr D−4π (ω⋆ω′)Γ (3a) l ω l l l , ω,ω′ ΩAn, (ω,ω′)q = ReTrωnDq−4πq(ω⋆ω′)Γqo ∈ n o (ω,ω′) = ReTr D−4π (ω⋆ω′)Γ′ (3b) l ω l l l , ω,ω′ ΩBn, (ω,ω′)q = ReTrωnDq−4πq(ω⋆ω′)Γ′qo ∈ n o yield indeed, in contrast to the previous Ansatz (2), a non-committed choice of (as required) gauge-invariant scalar products. According to the Poincaré-duality philosophy the new Yang- Millsactionwillbethesumofitselectroweak andchromodynamics partsrespectively stemming from the scalar products (3a) and (3b).3 Due to the product structure of the traces τD and l τD :4 q (4) τDl = τD ⊗Tr2 ⊗TrN , τDq = τD ⊗Tr2 ⊗TrN ⊗Tr3 togetherwiththe“fiberwise”natureofthecombinedspace-time-innerspacetheory,theproblem  of adapting our previous computations to the generalized Ansatz (3a), (3b) will de facto reduce to computations within the representations π andπ pertaining tothe inner space. The present l q treatment differs also from that of III in that the modular coalescence of the three U(1) groups – and their Lie algebras is performed after squaring the curvature and not before. The results reached through the new computations constitute a relatively mild modification of our previous results based on the former Ansatz (2), however leading to much more satisfying tree-approximation results, suppressing the former inconsistencies5 and amazingly compatible 3 The scalar products (3a), (3b) incorporate a generalized coupling constant “in the commutant of the K- cycle”. The leptonic and quark sectors appear independently since their intertwiners are trivial cf. (46), (47) below. 4 Tr2 denotes the R L 2 2 matrix trace with entries matrices given by the weak isotopic spin (overall − × 4 4 matrices for quarks, cf. (16), (17), (18) below; and 3 3 matrices for leptons, cf. (19), (20), (21) below; × × whilst TrN, Tr3, and τD respectively standfor the fermionfamily N N matrix trace,the colour 3 3 matrix × × trace, and the Dirac-operator trace τ . D 5 e.g. the fact that, for x=0, the values of g3/g2 =1 and sin2θ =3/8 were of the “grand unification type” whilst the mass-ratio m /m =2 was near the experimental value [5b]. t W 1 with the experimental evidence. For a specific choice of the “coupling constant” within the commutant, one computes the tree-approximation values of the ratio between strong and electroweak coupling constant, the weak angle, and the ratios between the top and the W, and the Higgs and the top masses. One can fit the three first items with the known experimental values: in fact g /g and sin2θ turn out to be mutually uncorrelated, and uncorrelated with 3 2 W the ratios m /m and m /m which determine each other, thus yielding, since the top has t W H W been found, a “prediction” of the Higgs mass: the latter equals 1.5698 m for m /m = 2, the t t W value (near experiment !) fixed by the (canonical ?) choice of the “coupling constant” in the center of the K-cycle, a choice without incidence on g /g and sin2θ . 3 2 W Of course these results are “classical”. Reliable results await a renormalized field quantiza- tionwhich should beeffected with due consideration ofthe (still to befound) esoteric symmetry brought about by the Higgs boson as a fifth gauge boson. This paper is the companion paper to [5a] with whom it shares its subject matter with a different style and emphasis. Whilst [5a], destined to a physical audience, adopts a notational setting congenial to the habits of elementary particle physicists, and insists on the global strategy [6] without giving all computational details, we here address mathematical physicists in the notation of our former reports [4b], giving a line-by-line account of computations. These computations have been performed independently in the two papers in different notation, thus affording a mutual check. For the convenience of our reader, we begin by recalling the definitions of the inner space structure with its Poincaré-dual A B -K-cycles (H ,D ,χ ) and (H ,D ,χ ). We begin ew⊗ chrom l l l q q q with the uncolored leptonic and quarkonic A -K-cycles. ew [0] Reminder (the inner space). a b The algebra A . With H = ;a,b C , we have: ew ( b a! ∈ ) − p 0 (5) A = C H = (p,q);p C,q H = ,q ;p C,q H ew ⊕ { ∈ ∈ } ( 0 p! ! ∈ ∈ ) (6) (p,q)⋆ = (p,q⋆), p C, q H, ∈ ∈ (7) G = u = (u,v) A ;u⋆u = uu⋆ = 1,v⋆v = vv⋆ = 1 = U(1) SU(2) ew ew { ∈ } × The leptonic and quarkonic A -K-cycles (H ,D ,χ ) and (H ,D ,χ ). ew l l l q q q Leptonic K-cycle: Hilbert space: H = (C1 C2) CN, (8) l R ⊕ L ⊗ e ν e R L L 2 Operators (endomorphisms of H as 3 3 matrices with entries in M (C)): l N × e ν e R L L 1 0 0 e N R (9) χ = 0 1 0 ν . l  N  L − 0 0 1 e N L  −    e ν e R L L p1 0 0 e N R p 0 a b (10) π ((p,q)) = 0 a1 b1 ν , p = ,q = A , l  N N  L ew 0 p! b a!! ∈ 0 b1 a1 e N N L −  −    e ν e R L L 0 0 M⋆ e e R (11) D = 0 0 0 ν . l   L M 0 0 e e L     Quarkonic K-cycle: Hilbert space: H = (C2 C2) CN, (12) q R ⊕ L ⊗ u d u d R R L L Operators (endomorphisms of H as 4 4 matrices with entries in M (C)): q N × u d u d R R L L 1 0 0 0 u N R 0 1 0 0 d (13) χ = N  R . q 0 0 1 0 u N L  −   0 0 0 1 d  − N  L   u d u d R R L L p1 0 0 0 u N R 0 p1 0 0 d p 0 a b (14) π ((p,q)) =  N  R , p = ,q = A , q 0 0 a1N b1N uL 0 p! b a!! ∈ ew   −  0 0 b1 a1 d  − N N  L   u d u d R R L L 0 0 M⋆ 0 u u R 0 0 0 M⋆ d (15) D = d  R , q M 0 0 0 u u L    0 M 0 0 d  d  L   Remark: One passes from the matrices (5), (6), (7) to the matrices (8), (9), (10) through the changes M 0, M M followed by restriction to the right-lower corner 3 3 matrix. This u d e → → × procedure applied to a 3 3 depending upon M and M is called leptonic reduction. u d × We will in fact also use the following 3 Two-by-two matrix versions. Quark sector: version with 2 2 matrices with entries in × M (C) M (C), corresponding to the decomposition: 2 N ⊗ R L p 1 0 R (16) π ((p,q)) = ⊗ N , q 0 q 1 L N ! ⊗ R L 0 M⋆ R (17) D = , q M 0 L ! R L 1 1 0 R (18) χ = ⊗ N . q 0 1 1 L N ! − ⊗ Lepton sector: version with 2 2 matrices with entries × M M (C) M(C2,C) M (C) 1 N N ⊗ ⊗ : M(C,C2) MN(C) M2(C) MN(C) ! ⊗ ⊗ R L p 1 0 R a b (19) p ((p,q)) = ⊗ N , p C, q = A , l 0 q 1N !L ∈ b a! ∈ ew! ⊗ − R L 0 (0 M⋆) e R (20) D = 0 . l  0 L M e (cid:18) (cid:19)   R L 1 1 0 R (21) χ = ⊗ . l 0 1 1 L N ! − ⊗ Coloured Poincaré-dual A B -K-cycles H ,D ,χ and H ,D ,χ . ew ⊗ chrom l l l q q q The algebra B . (cid:16) (cid:17) (cid:16) (cid:17) chrom (22) B = C M (C) = (p′,M);p C, m M (C) chrom 3 3 ⊕ { ∈ ∈ } (23) (p′,m)⋆ = (p′,m⋆), p′ C, m M (C), 3 ∈ ∈ (24) G = u′ = (u′,v) A ;u⋆u = uu⋆ = 1,v⋆v = vv⋆ = 1 = U(1) U(3). chrom ew { ∈ } × The leptonic and quarkonic A B -K-cycles (H ,D ,χ) and (H ,D ,χ ). ew ⊗ chrom l l l q q q Leptonic K-cycle (H ,D ,χ): l l l H = H C , χ = χ 1 l l ⊗ chrom l l ⊗ chrom D = D 1 (25) πll(p,q)l=⊗πl(cphr,oqm)⊗1chrom . ′ ′ ′ π (p,m) = 1 p = p  l l ⊗ 4 Quarkonic K-cycle (H ,D ,χ ): q q q H = H C3 , χ = χ 1 q q ⊗ chrom q q ⊗ chrom D = D 1 (26) πqq(p,q)q=⊗πq(cphr,oqm)⊗1chrom ′ π (p,m) = 1 m ′ q q ⊗′ (note the relation [D ,π (p,m)] = [D ,π (p,m)] = 0 implying the algebraic Poincaré duality l l q q condition). We recall the formulae (concerning the quark sector): M 0 M = u = E M +F M u d  0 Md! ⊗ ⊗ (27) with E = 1 0 , and F = 0 0 . 0 0! 0 1!  COMMUTANTS OF THE INNER SPACE ELECTROWEAK, RESP. CHROMO- DYNAMICS K-CYCLE. We now analyze the commutants of the inner space electroweak, resp.chromodynamics K- cycle, calling so the respective subalgebras of End (H H ) consisting of the elements com- l ⊕ q muting with D D and with all π (a) π (a), a A , resp. all π (b) π (b), b B . l ⊕ q l ⊕ q ∈ ew l ⊕ q ∈ chrom We begin with a remark relative to a notation which we shall use in order to spare writing: [1] Remark. With a and b linear operators of the respective complex vector spaces H and K, we write Int(a,b) for the set of linear maps: H K intertwining a and b: → (28) Int(a,b) = S End (H,K);Sa = bS) . { ∈ } We then have that: (i): With a,H and b,K as above, and using a 2 2 matrix notation for the endomorphisms × of H K, we have that: ⊕ Int(a,a) Int(a,b) (29) Int(a b,a b) = . ⊕ ⊕ Int(b,a) Int(b,a)! (ii): With a,H and b,K as above, H and K finite-dimensional, and a and b self-adjoint with non-intersecting respective sets of eigenvalues, we have Int(a,b) = 0 . { } 5 Proof: (i): follows from: a b S 0 aS Sa bT Sb (30) , = − − . " c d! 0 T !# cS Tc dT Td! − − (ii): With S = (Si), a = (λ δi), b = (µ δi), we have k i k i k (31) (Sa bS)i = Σ Siλ δh µ δiSh = (λ µ )Si = 0. − k h h h k − i h k k − i k (cid:16) (cid:17) In what follows we shall comply to the common usage of choosing our fermion mass- matrices such that M and M are diagonal, positive matrices, whilst M = C M , with e u d d | | C (the Kobayashi-Maskawa matrix) unitary and M strictly positive. Furthermore d | | we assume that all fermion masses are different (the eigenvalues of M , M and M e u d | | consists of positive numbers (the masses of leptons and quarks) all different from one another – experiment!). We further assume that no eigenstate of M is an eigenstate of d | | C (experiment!). We use the shorthands: µ = MM⋆, µ = M⋆M, µ = M2 (32) µue = Meu2 e µ = M M⋆, µ = M⋆M ,  d d d d d d we then have: e (33) µ = M 2 = Cµ C⋆, d d d | | e (33a) µ = M 2 = CµC⋆, | | where C = id C in the second line. ⊕ e [2] Lemma. We have that: ′ (i): The most general self-adjoint element Γ of Int(D ,D ) is as follows: one has in 3 3 l l l × matrix notation: h(µ ) 0 k(µ ) e e ′ (34) Γ =  0 δ 0 , l k(µ ) 0 h(µ )  e e  where h and k are arbitrary real functions, and δ is anyself-adjoint element of M (C). N 6 ′ (ii): The most general self-adjoint element Γ of Int(D ,D ) is as follows: one has in 4 4 q q q × matrix notation: f(µ ) 0 l(µ ) 0 u u ′  0 g(µd) 0 m(µd)C⋆  (35) Γ = , q  l(µ ) 0 f(µ ) 0   u e u e     0 Cm(µ ) 0 Cg(µ )C⋆  d d    where f,g,l and m are arbitrary real functions. Thus the most general self-adjoint element e e Γ′ of Int(D ,D ) is of the form Γ′ S with Γ′ as in (35) and S M (C) self-adjoint. q q q q ⊗ q ∈ N (iii): The self-adjoint elements of Int(D ,D ) or of Int(D ,D ) vanish. The same holds for l q q l self-adjoint elements of Int(D ,D ) or of Int(D ,D ). l q q l Proof: α β µ ′ (i): With Γ = β δ ν, in 3 3 matrix notation, equating: l × µ ν σ     0 0 M α β µ M µ M ν M σ e e e e ′ (36) DlΓl =  0 0 0 β δ ν =  0 0 0  M 0 0 µ ν σ M α M β M µ  e    e e e       and α β µ 0 0 M µM 0 αM e e e ′ (37) ΓlDl = β δ ν 0 0 0  = νMe 0 βMe, µ ν σM 0 0  σM 0 µM    e   e e      yields β = ν = 0; further M µ = µM and M µ = µM , whence M2µ = M µM = µM2, e e e e e e e e ′ ′ ′ whence µ = k(M ), µ = k (M ) for some functions k,k , with in addition k = k since e e ′ the relation M µ = µM now reads M k(M ) k (M ) = 0; finally M α = σM and e e e e e e e − M σ = αM , whence M2α = M σM =(cid:16)αM2, whence α(cid:17)= h(M2) which also equals σ e e e e e e e because the relation M α = σM now reads αM = σM . e e e e a b (ii): Wehave, using the2 2 matrixnotation,withΓ′ = , a,b,c,d M (C) M (C): × q c d! ∈ 2 ⊗ N a b 0 M⋆ bM M⋆c aM⋆ M⋆d ′ (38) Γ ,D = , = − − , q q " c d! M 0 !# dM Ma cM⋆ Mb ! h i − − we must thus have: (39a) dM Ma = 0, − (39b) aM⋆ M⋆d = 0, − (39c) bM M⋆c = 0, − (39d) cM⋆ Mb = 0, − 7 We compute a and d: (39a) and (39b) entail: (40) aM⋆M = M⋆Ma, dMM⋆ = MM⋆d. whence the existence of a real function F with a = F (µ).(39a) then reads dC M = | | C M F (µ) = CF (µ) M whence dC = CF (µ), d = CF (µ)C⋆. | | | | We now compute b and c: (39c) and (39d) entail M⋆Mbe= M⋆cM⋆ = bMM⋆, µb = e e e e bµ = bCµC⋆, µbC = bCµ, whence the existence of a real function G with b = G(µ)C⋆. For c = b⋆, G is real: indeed (39d) then reads Mb = C M b = b⋆M⋆ = b⋆ M C⋆e, i.e. | | | | M bC =eC⋆b⋆eM , whenece, since M bC = bC M , bC = C⋆b⋆. e | | | | | | | | We showed the existence of real functions F and G for which: F (µ) G(µ)C⋆ ′ (41) Γ = , q CG(µ) CF (µ)C⋆! e e which comes to (35) in 4 4 matrix notation. × e e α β γ δ ′ ′ ′ ′ (iii): Let S =  α β γ δ  Int(Dq,Dl): equating ∈ ′′ ′′ ′′ ′′ α β γ δ      ′′ ′′ ′′ ′′ 0 0 M α β γ δ M α M β M γ M δ e e e e e ′ ′ ′ ′ (42)  0 0 0  α β γ δ  =  0 0 0 0  ′′ ′′ ′′ ′′ M 0 0 α β γ δ   M α M β M γ M δ   e    e e e e  and      0 0 M 0 α β γ δ u γM δM αM βM⋆  0 0 0 M⋆ u d u d (43) α′ β′ γ′ δ′  d =  γ′Mu δ′Md α′Mu β′Md⋆  M 0 0 0  α′′ β′′ γ′′ δ′′ 0u M 0 0  γ′′Mu δ′′Md α′′Mu β′′Md⋆   d      yields α′ = β′ = γ′ = δ′ = 0; further M α′′ = γM and M γ = α′′M whence M2γ = e u e u e M α′′M = γM2 implying γ = 0 by [1](ii), and thus α′′ = 0 – the changes α′′ γ′′ and γ e α uthen yieulding α = γ′′ = 0; analogously M β′′ = δM and M δ = β′′M→⋆ whence M→2β′′ = M δM = β′′M⋆M implying β′′ = 0 = δ;ethe changdes β′′ eδ′′ and δ d β then yieelding β =e δ′′d= 0. Wedprodved that S = 0. For S Int(D ,D ), w→riting H =→H C3 ∈ q l q q ⊗ as a direct sum, the restrictions of S to all summands vanish. [3] Lemma. We have that: (i): The commutant of the K-cycle (H ,D ,χ ) of A 6 coincides with 1 1 M (C).7 q q q ew 2 N 3 ⊗ ⊗ 6 We recall that we call commutant of the K-cycle (H,D,χ) of an algebra A the set of operators of H commuting with π(A) and D. 7 12 denotes here the unit of the 2 2 matrix algebrawith entries in M2(C). Note that in factthe colorless × K-cycle (Hq,Dq,χq) is irreducible in the sense that the only operatorsof Hq commuting with πq(Aew) and Dq are the scalars. Equivalently πq(Aew) and Dq generate B(Hq). 8 (ii): A self-adjoint Γ acting on H belongs to the commutant of the K-cycle (H ,D ,χ ) of l l l l l A iff it is of the type 1 Γ , with Γ a real function of M .8 ew 2 N N e ⊗ (iii): The commutant of the K-cycle (H ,D ,χ ) of B coincides with Int(D ,D ) 1 . q q q chrom q q ⊗ 3 (iv): The commutant of the K-cycle (H ,D ,χ ) of B coincides with Int(D ,D ). l l l chrom l l Proof: (i): If S commutes with D and π (A ), it is of the form S T, T M (C), S commuting q q ew ⊗ ∈ N a b with D and π (A ). S = , a,b,c,d M (C) M (C) is then both of the q q ew 2 N c d! ∈ ⊗ form (35) and fulfills for all p H , q H: diag ∈ ∈ (44a) [a,p 1 ] = 0, N ⊗ (44b) [d,q 1 ] = 0, N ⊗ (44c) b(q 1 ) (p 1 )b = 0, N N ⊗ − ⊗ (44d) c(p 1 ) (q 1 )c = 0, N N ⊗ − ⊗ Now (44c), resp. (44d), with p = 0 yield b(q 1 ) = 0, resp. (q 1 )c = 0 whence N N ⊗ ⊗ b = c = 0 since q 1 is invertible. Since H generates M (C) linearly, (44b) further N 2 ⊗ implies that d = 1 d, with d M (C). Together these imply that l = m = 0, N ⊗ ∈ f(µ ) = C (µ )C⋆ being a multiple of the identity: this then also holds for S. u g d (ii): If S commutes with D and π (A ), it is of the form (34) and is contained in e l l ew 1 X 0 (45) π (A )′ = 1 ⊗ ;X,Y M (C) , l ew N ( 0 12 Y ! ∈ ) ⊗ (cf. [1](i)). Both facts together imply k(µ ) = 0, and h(µ ) = δ = X = Y, whence the e e claim. [4] Proposition. We have that: (i): The positive elements of the commutant of the K-cycle (H H ,D D ,χ χ ) of q ⊕ l q ⊕ l q ⊕ l A are the operators of the form ew Γ 0 q (46) Γ = 0 Γ ! l a b 812denotesheretheunit2 2matrixofthetypeA= ,a M1 MN(C),b M(C2,C) MN(C), × c d ∈ ⊗ ∈ ⊗ c ∈ M(C,C2)⊗MN(C), d ∈ M2(C)⊗MN(C). Not(cid:18)e that (cid:19)A belongs to the commutant of (Hl,Dl,χl) iff one has b=c=0 and a=1 Γ , d=1 Γ with Γ an arbitrary function of M . N N N e ⊗ ⊗ 9