SO(10)-uni cation in noncommutative geometry revisited

hep-th/9804046

CPT-98/P.3631

-uni cation in noncommutative geometry revisited

SO(10)

Raimar Wulkenhaar Centre de Physique Theorique CNRS Luminy, Case 907 13288 Marseille Cedex 9, France e-mail: [email protected] April 6, 1998

Abstract

We investigate the SO(10)-uni cation model in a Lie algebraic formulation of noncommutative geometry. The SO(10)-symmetry is broken by a 45-Higgs and the Majorana mass term for the right neutrinos (126Higgs) to the standard model structure group. We study the case that the fermion masses are as general as possible, which leads to two 10-multiplets, four 120-multiplets and two additional 126-multiplets of Higgs elds. This Higgs structure diers considerably from the two Higgs multiplets 16 16 and 16c 16 used by Chamseddine and Frohlich. We nd the usual treelevel predictions of noncommutative geometry mW = 21 mt , sin2 W = 38 and g2 = g3 as well as mH  mt .

Keywords: grand uni cation, noncommutative geometry 1998 PACS: 11.15.Ex, 12.10.Dm, 02.20.Sv

1 Introduction Noncommutative geometry (NCG) has a long history in mathematics, but it were two discoveries by Alain Connes at the end of the last decade1 which made NCG so interesting for physicists: { Dirac operator and smooth functions, both acting on the spinor Hilbert space, contain all metric information of a spin manifold, { a nite dimensional version of that setting yields some sort of Higgs potential. In the following years, the idea that one should study the tensor product of ordinary dierential geometry and a matrix geometry has prospered. Numerous versions to construct gauge eld theories with spontaneous symmetry breaking (in particular the standard model) appeared, all more or less inspired by Connes'  supported by the German Academic Exchange Service (DAAD), grant no. D/97/20386. On leave of absence from Institut fur Theoretische Physik, Universitat Leipzig, Augustusplatz 10/11, 04109 Leipzig, Germany.

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R. Wulkenhaar

discovery, but dierent in details. These versions developed, competed with each other, died out { a perfect example of Darwin's theory of evolution. After all, one version survived2 , proposed by Alain Connes who { guided by a deep understanding of mathematics { completed a set of axioms3. These axioms have their origin in the classi cation of classical spin manifolds4. It is remarkable that the same axioms which are fruitful in the commutative case of spin manifolds provide the standard model as one of the simplest noncommutative examples. Sometimes it happens that branch lines of evolution have attractive features, which fascinate mankind long after the dead of the species (think of the dinosaurs). Such a species is grand uni cation. The axioms of NCG are compatible with the standard model but not with grand uni cation5. Nevertheless, grand uni cation is such an attractive idea that it will not die very soon, no matter how well the standard model is veri ed by experiment. Technically, the essential progress that NCG brings over the traditional formulation of gauge theories is that Yang-Mills and Higgs elds are understood as two complementary parts of one universal gauge potential. It is desirable to extend this unifying feature to grand uni ed theories (GUTs). The only chance to do so is to allow for a minor modi cation of the NCG-axioms so that GUTs are accessible as well. An inspiration how to modify the axioms comes from traditional gauge theories. They are formulated in terms of Lie groups or { on in nitesimal level { Lie algebras. Thus, replacing the associative algebra in NCG by a Lie algebra6, one can expect a formalism (section 2) that is able to produce grand uni ed models. In this paper we show that the SO(10)-GUT7;8 can indeed be formulated with this method. The SO(10)-model has the exceptional property that all known fermions and the supposed right-handed neutrino t into the irreducible 16representation of SO(10) (for each generation). Other GUTs such as SU(5), SU(5)  U(1) and SU(4)  SU(2)L  SU(2)R arise as intermediate steps of dierent SO(10) symmetry breaking chains. The treatment of the SO(10)-model by NCG-methods is not new. The rst approach9 by Chamseddine and Frohlich came in the early days of noncommutative geometry. Since then, NCG underwent the development sketched above that singled out the axioms incompatible with grand uni cation. Our construction diers in its conception and its results from the Chamseddine-Frohlich approach. The authors of ref. 9 start from the (associative) Cliord algebra of SO(10). The crucialPdierence of the two approaches lies in the Higgs sector. We denote by Y = i Yi M i the Yukawa operator. Its part Yi transforms the 16-representation into itself or into its charge conjugate 16c. The mass matrices M i mix the three fermion generations. Thus, for Yi we have the two possibilities 16 16 and 16c 16 , which are reducible representations under SO(10) or

SO(10)-uni cation in NCG revisited

3

so(10) and decompose into 16 16 = 1  45  210 ; 16c 16 = 10  120  126 : (1) The point is now that these two Yi-representations are not reducible under the Cliord algebra Cli(SO(10)). That is why the Higgs multiplets in the Chamseddine-Frohlich model are 16 16 and 16c 16, where only the latter occurs in the fermionic action (due to chirality). The consequence is that there is only one generation matrix M i in the fermionic action. This leads to the relation me : m : m = md : ms : mb = mu : mc : mt between the fermion masses, which is obviously not satis ed. Moreover, the analysis of the Higgs potential shows that also a Kobayashi-Maskawa matrix is not possible, but can be obtained by including an additional fermion in the trivial representation. In our version based upon the Lie algebra so(10) we do have the decomposition (1), and each irreducible representation occurring in (1) is tensorized by its own generation matrix. It is therefore no problem to get the fermion masses we want. For the SO(10)-symmetry breaking we have the 45 and 210-representations at disposal, which both do not occur in the fermionic action. We employ the 45representation to break SO(10) to SU(3)C  SU(2)L  SU(2)R  U(1)B L in the rst step (at about10 1016GeV). The corresponding (self-adjoint) generation matrix adds a freedom of 9 real parameters. The other symmetry breaking chain SO(10) ! SU(4)PS  SU(2)L  SU(2)R mediated by the 210 is possible as well, but we have to make a choice due to the length of the formulae. In the second step this intermediate symmetry is broken by the Majorana mass term for the right-handed neutrinos (126-generator) to the standard model symmetry group SU(3)C  SU(2)L  U(1)Y (at about10 109GeV). We then restrict ourselves to the case that the fermion masses are as general as possible, this implies the Higgs multiplets 10 (twice), 120 (four times) and 126 (twice). The surviving symmetry group is SU(3)C  U(1)EM . There is also a technical dierence to mention. The article ref. 9 was written in the pioneering epoch of noncommutative geometry where auxiliary elds emerged in the action. They eliminate themselves at the end via their equation of motion. Our version is based on a dierential calculus and we quotient out the ideal of auxiliary elds before building the action. We think this method is more transparent but the results are independent of the way of eliminating these unphysical degrees of freedom. Our paper is organized as follows: We review in section 2 our Lie algebraic approach to noncommutative geometry. In section 3 we write down our setting of the so(10)-model, where we acknowledge a lot of inspiration from ref. 9. Section 4 is devoted to the computation of the gauge potential and the Higgs part of the eld strength. This is the most cumbersome part, because the extremely rich Higgs structure leads to a big number of terms in the eld strength. It remains to calculate some traces to get the bosonic action (section 5) and to implement

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the various symmetries to get the fermionic action (section 6). We conclude with an outlook towards a minimal SO(10)-model.

2 The Lie algebraic formulation of noncommutative geometry The starting point is the Lie algebra g = C 1(M ) a acting via a representation  = id ^ on the Hilbert space H = L2 (M; S ) C F . Here, C 1(M ) is the

algebra of (real-valued) smooth functions on the (compact Euclidean) spacetime manifold M , L2 (M; S ) is the Hilbert space of square integrable bispinors and ^ a representation of the semisimple matrix Lie algebra a on C F . The treatment of Abelian factors is possible but more complicated. Moreover, we have the selfadjoint unbounded operator D = i@/ 1F + Y on H, with Y = 5 Y^ and Y^ 2 MF C . We also need a Z2-grading operator on H which commutes with (g) and anti-commutes with D. In many cases there will exist further discrete symmetries such as the charge conjugation J . A universal 1-form !1 2 1 has the structure

!1 =

X ;z

[fz az; [: : : [f1 a1 ; d(f0 a0 )] : : : ]] ;

with fi 2 C 1(M ) and ai 2 a. The commutators should be read as tensor products. The representation  of the universal calculus on H is obtained by taking  of fi ai and representing the universal d by the derivation [ iD; : ],

P

 = (!1) = ;z [fz ^ (az); [: : : [f1 ^ (a1); [ iD; f0 ^ (a0)]] : : : ]] P = ;z fz : : : f1@/ (f0 ) [^(az ); [: : : [^(a1); ^ (a0)] : : : ]] !A P + ;z 5 fz : : : f1f0 [^(az ); [: : : [^(a1 ); [ iY^ ; ^ (a0 )]] : : : ]] ! () The second and third line are independent for semisimple a. In the second line the commutators clearly yield an element of ^ (a) and f@/ f 0 a spacetime 1-form, together a Yang-Mills multiplet represented on L H. We decompose the nite F dimensional part of the Hilbert space into C = i ni C N , where ni are irreducible representations of a and NPis the number of fermion generations. Then, we have the decomposition Y^ = Y^ijr Mrij , where Y^ijr 2 ni nj is (for each r) a generator of an irreducible representation and Mrij 2 MN C a mass matrix. If we now evaluate the commutators in the third line above, the generators are P expanded to irreducible multiplets, and we obtain () = 5ijr Mrij , where ijr 2 C 1(M ) nij are function-valued irreducible representations, i.e. Higgs multiplets. The universal dierential of !1 is de ned as d! = 1

z XX ;z y=1

[fz az ; [: : : [d(fy ay); [: : : [f1 a1 ; d(f0 a0 )] : : : ]] : : : ]] :

SO(10)-uni cation in NCG revisited

5

Its representation on H reads after elementary calculation z XX

(d!1) =

;z y=1

[fz ^ (az ); [: : : f[ iD; fy ^ (ay)]; [: : : [ iD; f0 ^ (a0 )] : : : ]g : : : ]]

P

= f iD; g + ;z [fz ^ (az); [: : : [f1 ^ (a1); [D2; f0 ^ (a0)]] : : : ]]  f iD; (!1)g + (!1) (2) P 1 2 0 0 z z 1 = f@/ ; Ag + ;z [f ^ (a ); [: : : [f ^ (a ); [ @/ 1F ; f ^(a )]] : : : ]] + f@/ ; ()g + f iY; Ag + f iY; ()g + ^ () ; with

P ^ () = ;z fz : : : f1f0 [^(az ); [: : : [^(a1); [Y^ 2 ; ^ (a0 )]] : : : ]] : After a lengthy calculation6 one nds f@/ ; Ag + P;z [fz ^ (az ); [: : : [f1 ^ (a1 ); [ @/ 2 1F ; f0 ^ (a0)]] : : : ]] = dA + C 1(M ) f^ (a); ^ (a)g ; where d is the exterior dierential (which anti-commutes with 5) and the f^ (a); ^ (a)g-part is independent of A and (). Moreover, we have f@/ ; ()g = d(). We now decompose Y^ 2 into generators of irreducible representations, Y^ 2 = Y^k2 + Y^?2 + ^ (1) : Here, ^ (1) contains trivial representations which commute with ^ (a). Those generators which also occur in Y^ , denoted Y^k2 , generate obviously the corresponding representations which occur already in (). The other (non-trivial) generators, denoted Y^?2, generate representations independent of () and A. It is now crucial to notice that the same  can be written in many ways as (!1) so that the de nition of a dierential \d = (d!1)" is ambiguous. The usual way out is to consider equivalence classes modulo the ideal J 2 = (d(ker  \ 1 )). We have just shown that if (!1) = 0 then there remain only the f^ (a); ^ (a)g-part and the representations generated by Y^?2 , which gives (3) J 2 = C 1(M ) P(f^ (a); ^ (a)g + [^(a); [: : : [^(a); Y^?2] : : : ]]) : The nal formula for the dierential of  = A + () is therefore d = d + f iY; g + ^ () mod J 2 : (4) In the same way one represents the space n of universal forms of degree n on H and determines the corresponding ideal J n = (d(ker  \ n 1)). The generalization of (2) is d((!n) + J n) = [ iD; (!n)]] + (!n) + J n+1 ; !n 2 n ; (5)

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R. Wulkenhaar

where [ : ; : ] is the graded commutator, i.e. the anti-commutator if both entries are odd under Z2 and the commutator else. This yields the graded dierential Lie algebra D = ( )=J . We propose to de ne the connection r as a generalization of the dierential d and the covariant derivative D as a generalization of the operator D. This means that D is a linear unbounded selfadjoint operator on H and odd under Z2, and r : nD ! nD+1 is linear. Both D and r are related via the same formula (5):

r((!n) + J n) = [ iD; (!n)]] + (!n) + J n+1 ; for !n 2 n and any degree n. The general solution is

D = D + i ; r = d + [ ; : ] ; [ ; ( n )]]  ( n+1 ) ; [ ; J n]  J n+1 : One obvious solution is  = A + () 2 ( 1 ). But there are further solutions

possible, depending on the setting. These additional solutions allow us to formulate gauge theories with u(1)-factors such as the standard model. Demanding that  commutes with functions, we have the decomposition

0 = 1 r0 + 0 5 r1 of the additional solutions, where the matrices ri 2 MF C commute with ^ (a). The essential step is to check f0; ( 1 )g  ( 2 ), which yields several conditions for r0 and r1 . Finally, one has to verify the compatibility with J . The curvature is now r2 = [F ; : ], where one nds the usual formula

F = d + 21 f; g for the eld strength. The dierential and (anti)commutator are de ned via the (graded) Leibniz rule and Jacobi identity; for it one has to enlarge the ideal J by the graded centralizer C of ( ). Then, the general formula (4) continues to work so that for  = A + () one has

F = (dA + A2j ) + (d() + fA; (() iY )g)
+ (())2 + f iY; ()g + ^() mod J 2 + C 2 : 2

(6)

Here, A2 j2 is the restriction of A2 to the spacetime 2-form part. The bosonic action is de ned via the Dixmier trace and can be rewritten as

Z Z 2 SB = g F dx tr(F?) = dx (L2 + L1 + L0) M ZM  1

2

= g21F

dx tr((dA + A2j2 )2) + tr((d() + fA;(() iY )g)2) M
+tr( (())2 + f iY; ()g + ^ () 2?) :

(7)

SO(10)-uni cation in NCG revisited

7

Here, g is a coupling constant and F the dimension of the matrix part. The trace includes the trace over gamma matrices and F? is the component of F orthogonal to J 2. The bosonic action consists of three parts, the Yang-Mills Lagrangian L2, the covariant derivative L1 of the Higgs elds and the Higgs potential L0. The fermionic action is

SF =

1 2s

Z

M

dx

D

=

1 2s

Z

M

dx i  (@/ 1F + A + () iY ) ;

(8)

where s is the number of discrete symmetries of the setting, and 2 H is invariant under these symmetries (possibly only after passing to Minkowski space). The fermionic action contains the minimal coupling to the Yang-Mills elds A and the Yukawa coupling to the Higgs elds ().
Let us study the part  = (())2 + f iY; ()g +^() ? of the eld strength, whose square gives the Higgs potential. Since the covariant derivative D can be written as D = i@/ 1F + iA + i(() iY ), it is natural to consider (~) = (() iY ) as the analogue of a classical Higgs multiplet. This is because (~) transforms as (~) 7! u(~)u under gauge transformations D 7! uDu, with u 2 C 1(M ) exp(a + r0 ). In terms of ~ we have  = ((~)2 + ^ () + Y 2 )? = ((~)2 + ^ () + (1) + Y?2 + Yk2)? = ((~)2 + ^ (~) + (1))? ; (9) because Y?2 has by de nition no component orthogonal to J 2 and P if (~) = iY + ;z [(az ); [: : : [(a1); [ iY; (a0)]] : : : ]] P then ^ (~) = Yk2 + ;z [(az ); [: : : [(a1); [Yk2 ; (a0)]] : : : ]] ; with ai 2 g. Thus,  can be expressed completely in terms of ~ so that gauge invariance of the Higgs potential V = tr(2 ) is obvious. The point is now that we know a priori the Higgs vacuum: it is h(~)i0 = iY . At this con guration we have  = 0 and V = 0. On the other hand V is non-negative so that iY is a global and local minimum. In the vicinity of iY we have V = tr(( i(Y () + ()Y ) + ^ ())2?), i.e. something bilinear in the physical Higgs elds. The coecients are the Higgs masses, after diagonalization and rescaling. There are however massless modes, the Goldstone bosons. They are of the form () = [(a); iY ] with a 2 g. In this case we have ( i(Y () + ()Y ) + ^ ())? = ( [(a); Y 2 ] + [(a); Yk2])? = ( [(a); Y?2 ])? = 0 : The masses of the Yang-Mills elds come from the part tr(fA; iY g)2 = tr([(A); Y ][(A ); Y ]) of the Lagrangian L1 . This is a form of the GoldstoneHiggs theorem: The massive Yang-Mills elds are those which do not commute

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R. Wulkenhaar

with Y and to each of them there corresponds a massless Goldstone boson. The Higgs mechanism consists in removing the Goldstone bosons by those gauge transformations which do not commute with Y , and which are xed in this way. The remaining unconstrained gauge degrees of freedom are those which commute with Y , and to each of them there corresponds a massless Yang-Mills eld.

3 The so(10)-setting Let I 2 M32 C , I = 0; : : : ; 9, be so(10)-gamma matrices represented in terms of

tensor products of ve sets of Pauli matrices9 i ! 12 12 12 12 i ; i = 1 3 i ;  i ! 12 12 12  i 12 ; i+3 = 1 1 i ;  i ! 12 12  i 12 12 ; (10) i+6 = 1 2 i ; i ! 12 i 12 12 12 ; 0 = 2 ; i ! i 12 12 12 12 ; 11 = i 0 1


9 = 3 ; where i = 1; 2; 3. The tensor products are interpreted such that i is 2  2 and i is 32  32. The matrix Lie algebra is a = so(10) represented as so(10) 3 a = aIJ IJ , with IJ a = aJI 2 R , and where I1I2:::In = (1=n!) [I1 I2


In] is the completely anti-symmetrized product of -matrices. Summation over equal so(10) indices I; J; : : : from 0 to 9 and over equal spacetime indices ; ; : : : from 0 to 3 is understood. We introduce the projection operators P = 21 (1  11 ) 13 ; P = diag(14 P+ ; 14 P+ ; 14 P ; 14 P ) ; and the so(10)-conjugation matrix B = 1 3 4 6 8 13 = B = B T = B y ; B 2 = 196 ; B ( I m)B = TI m 8m 2 M3 C ; BPB = P ; and de ne the Z2-grading operator = diag( 5 P+ ; 5 P+ ; 5 P ; 5 P ) : Then, the Hilbert space is (11) H = P (L2 (M; S ) C 32 C 3 C 4 ) = L2 (M; S ) C 192 : The representation  of g = C 1(M ) a 3 f a on H is de ned by 1 0 P (a 1 )P 0 0 0 + 3 + CC : B 0 P+(a 13)P+ 0 0 (f a) = f 14 B A @ 0 0 P (a 13)P 0 0 0 0 P (a 13)P (12)

SO(10)-uni cation in NCG revisited

9

Note that P commutes with a 13 and that commutes with (a). The selfadjoint Yukawa operator anti-commuting with is 1 0 0

5 P+MP+ 5 P+N P 0 B 5 P+MP+ 0 0

5  P+N P C C Y =B 5  y 5 @ P N P+ 0 0

 P B MBP A : 0 0

5 P N yP+ 5 P B MBP (13) We distinguish explicitly 5 and 5 , which are equal in Euclidean space, in Minkowski space however we have 5 = 5  . This allows for a parallel development of our model in both Euclidean and Minkowski space. The matrices M and N are obtained by tensorizing the generators given in ref. 9 by independent generation matrices: M = i( 45 + 78 + 69 ) M1 ; N = i 0 Ms + 3 Mp + 120 Ma0 i 123 Ma (14) 0 + ( 450 + 780 + 690 ) Mb i( 453 + 783 + 693 ) Mb i( 01245 + 01278 + 01269 ) Mc ( 31245 + 31278 + 31269 ) Mf 1 8 i( 1 i 2 ) 3 ( 4 i 5 )( 6 i 9 )( 7 i 8 ) M2 : Here, Ms; Mp; Mc; Mf ; M2 are symmetric and Ma ; Ma0 ; Mb ; Mb0 anti-symmetric 3  3-matrices and M1 = M1y. That implies B N B = N T and M = My. We stress that here lies the essential dierence between our Lie formalism and the algebraic version9 : There, the matrices Ms; Mp; Mc; Mf ; M2 are all proportional to each other, the same is true for the matrices Ma ; Ma0 ; Mb; Mb0 . This is dictated by the fact that 16c 16 is irreducible under the Cliord algebra of SO(10). The above setting is chosen in such a way that it has two symmetries J and S . First, the charge conjugation is given by 0 0 0 C B 0 1 B 0 0 0 C B CC P  c:c ; J =PB @ C B 0 0 0 A 0 C B 0 0 where C is the spacetime conjugation matrix, C  C = , and c:c stands for complex conjugation. We use the following convention for Euclidean gamma matrices 1 0  0 1   0 ia 2 5 0 1 2 2 0 a

= 1 0 ; = ia 0 ; = = 02 1 ; C = 0 2 : 2 2 Our Minkowskian gamma matrices are 1 0  0 1   0 a 2 5 0 1 2 3 a 0

= 1 0 ; = a 0 ; = i = 02 1 ; C = 2 : 2 2

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R. Wulkenhaar

Observe that J 2 = P in Minkowski space but J 2 = P in Euclidean space. Second, we have an exchange symmetry 0 0 i1 0 0 1 00 1 0 01 384 384 C B B i 1 0 0 0 C 1 0 0 0 384 C C B P ; S = P SE = P B M A @ @ 384 0 0 0 i1384 A P ; 0 0 0 1384 0 0 i1384 0 0 0 1384 0 where SE is realized in Euclidean space and SM in Minkowski space. This yields in both Euclidean and Minkowski spaces [J; D] = [J; (a)] = [S ; D] = [S ; (a)] = [J; S ] = 0 ; where D = P (i@/ 132 13 14)P + Y .

4 The gauge potential and its eld strength The gauge potential  2 ( 1 ) is composed of two parts, of a a-valued spacetime

1-form A =  (A) and (up to 5) a a- representation valued spacetime 0-form (),  = A + (). The second part has the general structure X () = [(az ); [: : : ; [(a1); [ iY; (a0)]] : : : ]] ; ;z

where ai 2 g. Products of -matrices are generators of irreducible representations, but as some of them occur more than once in Y , we must check that they are linear independent. For instance, 1 1 1 4 ad 01  ad 01 ( 0 )  4 [ 01 ; [ 01 ; 0 ]] = 0 ; 4 ad 01  ad 01 ( 3 ) = 0 ; which establishes the independence of the two 10-dimensional representations generated by 0 Ms and 3 Mp. Next, application of 41 ad 01  ad 01 to the four 120-dimensional representations generated by IJK establishes the independence of 123 and ( 450 + 780 + 690) from 120 and ( 453 + 783 + 693 ). Application of 81 ad 16  ad 64  ad 41 leads to independence of all these four 120-dimensional representations. Finally, application of 41 ad 69  ad 69 and 41 ad 01  ad 01 to the three 126-dimensional representations generated by IJKLM shows that they are independent. In conclusion, and using the identity B (a 13 )B = a 13 , the general form of () is 1 0 0 5 M 5 N 0 B 5  M 0 0

5  N C C (15)  ( ) = iP B y @ 5 N 0 0

5 BMB A P ; 0 0

5 Ny 5 BM B M = i M1 N = i1 Ms + 2 Mp + 1 Ma0 i2 Ma + 3 Mb0 i4 Mb i 1 Mc 2 Mf i 3 M2 ;

SO(10)-uni cation in NCG revisited

11

where  2 C 1(M ) 45, i 2 C 1(M ) 10 and i 2 C 1(M ) 120, all of them being real representations, and i 2 C 1(M ) 126 (complex representation). The next step is to compute the ideal J 2 and the part ^ () of the curvature. Both are related to Y 2, decomposed into irreducible representations. Those which occur both in Y and Y 2 contribute to ^ (), the other ones give rise to the ideal. Using 5 5 = 14 , with  = 1 in Euclidean space and  = 1 in Minkowski space, we have

0 P+Y(1) P+ 0 0 B 0 P Y P P Y + (2) P (1) + Y 2 = 14 B y P P B Y BP B@ 0 P+Y(2) (1) + y P Y(2) P+ 0 0 Y1 = M2 + NN y ; Y2 = MN + N B MB :

Y(2) P 0 0

P B Y(1) BP

1 CC CA ;

(16)

In detail, we nd

Y(1) = 132 (3M1M1y + MsMsy + MpMpy + Ma0 Ma0 y + Ma May

+ 3Mb0 Mb0 y + 3MbMby + 3McMcy + 3Mf Mfy + 41 M2 M2y) i( 45 + 78 + 69) (MsMb0 y + Mb0 Msy + MpMby + MbMpy + Ma0 Mcy + McMa0 y + Ma Mfy + Mf May + 2Mb0 Mfy + 2Mf Mb0 y + 2MbMcy + 2McMby 41 M2M2y ) + i 03 Z1 i 12 Z2 ( 4578 + 4569 + 7869 ) Z3 + ( 0345 + 0378 + 0369 ) Z4 ( 1245 + 1278 + 1269 ) Z5 + 0123 Z6 + 81 ( 2567 1467 + 1568 + 2468 1489 2479 + 2589 1579 ) Z7 + 81 ( 1567 + 2467 2568 + 1468 + 2489 1479 + 1589 + 2579 ) Z8 : Y(2) = i 0 (3M1Mb0 3Mb0 M 1 ) + 3 (3M1Mb 3MbM 1) + 120 (3M1 Mc 3McM 1 ) i 123 (3M1 Mf 3Mf M 1) i( 453 + 783 + 693 ) (M1 Mp iMpM 1 + 2M1 Mc 2McM 1 ) + ( 450 + 780 + 690 ) (M1 Ms MsM 1 + 2M1Mf 2Mf M1 ) i( 01245 + 01278 + 01269 ) (M1 Ma0 Ma0 M 1 + 2M1Mb 2MbM 1) ( 31245 + 31278 + 31269 ) (M1Ma Ma M 1 + 2M1Mb0 2Mb0 M 1) 1 i( 1 i 2 ) 3 ( 4 i 5 )( 6 i 9 )( 7 i 8 ) ( 3M1 M2 3M2 M 1 ) ; 8

where (h:c denotes the Hermitian conjugate of the preceding term)

Z1 = (MsMpy + Ma0 May + 3Mb0 Mby + 3McMfy + 81 M2 M2y) + h:c ; Z2 = (MsMa0 y + MpMay + 3Mb0 Mcy + 3MbMfy 81 M2 M2y) + h:c ;

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R. Wulkenhaar

Z3 = (M1 M1y + 81 M2 M2y + MsMfy + MpMcy + Ma0 Mby

+ Ma Mb0 y + Mb0 Mb0 y + Mb Mby + Mc Mcy + Mf Mfy) + h:c ; Z4 = (MsMby +MpMb0 y+Ma0 Mfy+Ma Mcy+2Mb0 Mcy+2MbMfy 81 M2 M2y) + h:c ; Z5 = (MsMc y+MpMfy+Ma0 Mb0 y+Ma Mby +2Mb0 Mby+2McMfy+ 81 M2 M2y) + h:c ; Z6 = (MsMay + MpMa0 y + 3Mb0 Mfy + 3MbMcy 81 M2 M2y) + h:c ; Z7 = i(((Ms + Mp + Ma0 + Ma + 3Mb0 + 3Mb + 3Mc + 3Mf )M2y) h:c) ; Z8 = ((Ms + Mp + Ma0 + Ma + 3Mb0 + 3Mb + 3Mc + 3Mf )M2y) + h:c : This gives

0 ^((1)) 0  BB 0 ^((1)) ^(0(2)) ^(0(2) ) ^ () = P 14 @ 0 ^ ( )y B ^ ( )B 0 (1)

(2)

1 CC P ; A

^ ((2) )y 0 0 B ^ ((1) )B 0 y 0 y ^ ((1) ) = i (MsMb + Mb Ms + MpMby + Mb Mpy + Ma0 Mcy + McMa0 y +Ma Mfy+Mf May+2Mb0 Mfy +2Mf Mb0 y+2Mb Mcy+2McMby 41 M2 M2y) ^ ( ) = i (3M M 0 3M 0 M ) +  (3M M 3M M ) (2)

and

1

1

b

b

1

2

1

b

b

1

+ 1 (3M1 Mc 3McM 1 ) i2 (3M1 Mf 3Mf M 1) + 3 (M1 Ms MsM 1 + 2M1 Mf 2Mf M 1) i4 (M1 Mp MpM 1 + 2M1 Mc 2McM 1 ) i 1 (M1 Ma0 Ma0 M 1 + 2M1 Mb 2Mb M 1) 2 (M1 Ma Ma M 1 + 2M1 Mb0 2Mb0 M 1) i 3 ( 3M1 M2 3M2 M 1 )

J 2 = P([(g); [: : : ; [(g); Y?2]] : : : ] + f(g); (g)g) 1 0P J P 0 0 0 + (1) + CC ; B 0 P+J(1)P+ 0 0 =B A @ 0 0 0 P B J(1) BP 0 0 0 P B J(1) BP 1 1 J(1) = C (M ) 1 C 13 + C (M ) i45 (C Z1 + C Z2 ) + C 1(M ) 210 (C Z3 + C Z4 + C Z5 + C Z6 + C Z7 + C Z8 + C 13 ) : The eld strength F of the gauge potential  = A + () is given in (6). Let  be the spacetime 0-form component of F? orthogonal to J 2, as given in (9).

SO(10)-uni cation in NCG revisited

13

Introducing ~ =  + ( 45 + 78 + 69 ) ; ~ 1 = 1 + 0 ; ~ 2 = 2 + 3 ; ~ 1 = 1 + 120 ; ~ 2 = 2 + 123 ; ~ 3 = 3 + ( 453 + 783 + 693 ) ; ~ 4 = 4 + ( 450 + 780 + 690 ) ; ~ 1 = 1 + ( 01245 + 01278 + 01269 ) ; ~ 2 = 2 + ( 31245 + 31278 + 31269 ) ; ~ 3 = 3 + 81 ( 1 i 2) 3 ( 4 i 5)( 6 i 9)( 7 i 8) ; we nd after straightforward but apparently lengthy calculation

0P  P 0 0 P+(2) P BB + 0(1) + P+(1) P+ P+(2) P 0  = 14 B y P (2) P+ P B(1) BP 0 @ 0y P (2) P+ 0 0 P B(1) BP P P P k (Q210 )? (1) = i 1i (Q1i )? + j 45j (Q45j )? + k 210 k

1 CC CA ;

~ 21 ) M~ f11g = 21 (3 132 (i) + 21 (132 (~ 21)1 ) M~ fssg + 21 (132 (~ 22 )1) M~ fppg + 12 (132 (i~ 1)21) M~ fa a g + 21 (132 (i~ 2)21 ) M~ faag + 21 (3 132 (i~ 3)21 ) M~ fb b g + 21 (3 132 (i~ 4)21) M~ fbbg + 21 (3 132 ( ~ 1 ~ y1)1) M~ fccg + 21 (3 132 ( ~ 2 ~ y2)1) M~ fff g + 21 (16 132 ( ~ 3 ~ y3)1) M~ f22g (~ 1 ~ 2)1 M~ [sp] + (~ 1 ~ 2)1 M~ [a a] + (~ 1 ~ 3 )1 M~ fa b g + (~ 1 ~ 4 )1 M~ [a b] (~ 2 ~ 3 )1 M~ [ab ] + (~ 2 ~ 4 )1 M~ fabg + (~ 3 ~ 4 )1 M~ [b b] 1 ~ ~y 1 ~ ~y ~ ~y ~ ~ ~y ~ 2 i( 1 2 2 1 )1 Mfcf g 2 ( 1 2 + 2 1 )1 M[cf ] + 21 i( ~ 1 ~ y3 ~ 3 ~ y1)1 M~ [c2] 12 ( ~ 1 ~ y3 + ~ 3 ~ y1)1 M~ fc2g + 21 i( ~ 2 ~ y3 ~ 3 ~ y2)1 M~ ff 2g + 21 ( ~ 2 ~ y3 + ~ 3 ~ y2)1 M~ [f 2] i(~ 1~ 2)45 Mfspg + i(~ 1~ 1 )45 Mfsa g + i(~ 1~ 2 )45 M[sa] + i(~ 1~ 3 ~ )45 Mfsb g + i(~ 1~ 4 )45 M[sb] i(~ 2~ 1 )45 M[pa ] + i(~ 2~ 2 )45 Mfpag ~ 45 Mfpbg i(~ 2~ 3 )45 M[pb ] + i(~ 2~ 4 ) + i(~ 1~ 2 )45 Mfa ag i(~ 1~ 3 )45 M[a b ] + i(~ 1~ 4)45 Mfa bg 1 ~ ~y 1 ~ ~y ~ ~ ~ ~ ~ 2 i(1 1 + 1 1 + 2)45 Mfa cg 2 (1 1 1 1 )45 M[a c] 1 i( ~ 1 ~ y2 + ~ 2~ 1 )45 M[a f ] + 21 (~ 1 ~ y2 ~ 2~ 1 )45 Mfa f g 2 1 ~ ~y 1 ~ ~y ~ ~ ~ ~ 2 i(1 3 + 3 1 )45 Mfa 2g 2 (1 3 3 1 )45 M[a 2] 0

0

0 0

0

0 0

0

0

0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

14

R. Wulkenhaar

i(~ 2~ 3 )45 Mfab g i(~ 2~ 4 )45 M[ab] + i(~ 3~ 4 )45 Mfb bg + 21 i(~ 2 ~ y1 + ~ 1 ~ 2 )45 M[ac] 12 (~ 2 ~ y1 ~ 1~ 2 )45 Mfacg 1 ~ ~y 1 ~ ~y ~ ~ ~ ~ ~ 2 i(2 2 + 2 2 + 2)45 Mfaf g 2 (2 2 2 2 )45 M[af ] + 21 i(~ 2 ~ y3 + ~ 3 ~ 2 )45 M[a2] 12 (~ 2 ~ y3 ~ 3 ~ 2 )45 Mfa2g 1 ~ ~y 1 ~ ~y ~ ~ ~ ~ 2 i(3 1 + 1 3 )45 Mfb cg 2 (3 1 1 3 )45 M[b c] 1 i( ~ 3 ~ y2 + ~ 2~ 3 )45 M[b f ] + 21 (~ 3 ~ y2 ~ 2 ~ 3 4i) ~ 45 Mfb f g 2 1 ~ ~y 1 ~ ~y ~ ~ ~ ~ 2 i(3 3 + 3 3 )45 Mfb 2g 2 (3 3 3 3 )45 M[b 2] ~ 45 Mfbcg + 21 i(~ 4 ~ y1 + ~ 1 ~ 4 )45 M[bc] 12 (~ 4 ~ y1 ~ 1 ~ 4 + 4i) 1 i( ~ 4 ~ y2 + ~ 2~ 4 )45 Mfbf g 21 (~ 4 ~ y2 ~ 2~ 4 )45 M[bf ] 2 + 21 i(~ 4 ~ y3 + ~ 3 ~ 4 )45 M[b2] 12 (~ 4 ~ y3 ~ 3~ 4 )45 Mfb2g 1 ( 1 ~ ~y 1 ~ ~y ~ ~ ~y 2 ( 1 1 )45 Mfccg 2 2 2 )45 Mfff g 2 (( 3 3 )45 16i) Mf22g 1 ~ ~y 1 ( ~ ~y ~ ~y ~ ~y 2 i( 1 2 2 1 )45 Mfcf g 2 1 2 + 2 1 )45 M[cf ] + 21 i( ~ 1 ~ y3 ~ 3 ~ y1)45 M[c2] 21 ( ~ 1 ~ y3 + ~ 3 ~ y1)45 Mfc2g + 12 i( ~ 2 ~ y3 ~ 3 ~ y2)45 Mff 2g + 21 ( ~ 2 ~ y3 + ~ 3 ~ y2 )45 M[f 2] 1 (i) ~ 2210 M^ f11g 2 + (~ 1~ 1 )210 M^ [sa ] (~ 1~ 2 )210 M^ fsag + (~ 1~ 3 )210 M^ [sb ] (~ 1 ~ 4)210 M^ fsbg + (~ 2~ 1 )210 M^ fpa g + (~ 2~ 2 )210 M^ [pa] + (~ 2~ 3 )210 M^ fpb g + (~ 2~ 4 )210 M^ [pb] + 21 (~ 1 ~ y1 + ~ 1~ 1)210 M^ fscg 21 i(~ 1 ~ y1 ~ 1~ 1)210 M^ [sc] + 21 (~ 1 ~ y2 + ~ 2~ 1)210 M^ [sf ] 12 i(~ 1 ~ y2 ~ 2~ 1 )210 M^ fsf g + 21 (~ 1 ~ y3 + ~ 3~ 1)210 M^ fs2g 21 i(~ 1 ~ y3 ~ 3 ~ 1)210 M^ [s2] 1 i( 1 ~ ~y ~ ~ ^ ~ 2 ~ y1 ~ 1~ 2)210 M^ fpcg 2 (2 1 + 1 2 )210 M[pc] 2 + 21 (~ 2 ~ y2 + ~ 2~ 2)210 M^ fpf g 12 i(~ 2 ~ y2 ~ 2 ~ 2)210 M^ [pf ] 1 i( 1 ~ ~y ~ ~ ^ ~ 2 ~ y3 ~ 3~ 2)210 M^ fp2g 2 (2 3 + 3 2 )210 M[p2] 2 + 21 (~ 21 )210 M^ fa a g + 12 (~ 22)210 M^ faag + 21 (~ 23 )210 M^ fb b g + 12 (~ 24 )210 M^ fbbg + (~ 1 ~ 2)210 M^ [a a] + (i~ 1~ 2 11 )210 M^ fa ag + (~ 1 ~ 3 )210 M^ fa b g (i~ 1~ 3 11 )210 M^ [a b ] + (~ 1 ~ 4 )210 M^ [a b] + (i~ 1~ 4 11 )210 M^ fa bg (~ 2 ~ 3 )210 M^ [ab ] (i~ 2~ 3 11 )210 M^ fab g + (~ 2 ~ 4)210 M^ fabg (i~ 2~ 4 11 )210 M^ [ab] + (~ 3 ~ 4 )210 M^ [b b] + (i~ 3~ 4 11 )210 M^ fb bg 1 ~ ~y 1 ~ ~y ~ ~ ^ ~ ~ ^ 2 (1 1 1 1 )210 M[a c] 2 i(1 1 + 1 1 )210 Mfa cg 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0 0

0 0

0

0

0

0

0

0

0

0

SO(10)-uni cation in NCG revisited

15

+ 21 (~ 1 ~ y2 ~ 2~ 1 )210 M^ fa f g 12 i(~ 1 ~ y2 + ~ 2 ~ 1)210 M^ [a f ] 1 ~ ~y 1 ~ ~y ~ ~ ^ ~ ~ ^ 2 (1 3 3 1 )210 M[a 2] 2 i(1 3 + 3 1 )210 Mfa 2g 1 ~ ~y 1 ~ ~y ~ ~ ^ ~ ~ ^ 2 (2 1 1 2 )210 Mfacg + 2 i(2 1 + 1 2 )210 M[ac] 1 ~ ~y 1 ~ ~y ~ ~ ^ ~ ~ ^ 2 (2 2 2 2 )210 M[af ] 2 i(2 2 + 2 2 )210 Mfaf g 1 ~ ~y 1 ~ ~y ~ ~ ~ ~ ^ ^ 2 (2 3 3 2 )210 Mfa2g + 2 i(2 3 + 3 2 )210 M[a2] 1 ~ ~y 1 ( ~ ~y ~ ~ ^ ~ ~ ^ 2 3 1 1 3 )210 M[b c] 2 i(3 1 + 1 3 )210 Mfb cg + 12 (~ 3 ~ y2 ~ 2~ 3 )210 M^ fb f g 21 i(~ 3 ~ y2 + ~ 2~ 3 )210 M^ [b f ] 1 ( 1 ~ ~y ~ ~y ~ ~ ^ ~ ~ ^ 2 3 3 3 3 )210 M[b 2] 2 i(3 3 + 3 3 )210 Mfb 2g 1 ~ ~y 1 ( ~ ~y ~ ~ ^ ~ ~ ^ 2 4 1 1 4 )210 Mfbcg + 2 i(4 1 + 1 4 )210 M[bc] 1 i( 1 ( ~ ~y ~ ~ ^ ~ 4 ~ y2 + ~ 2~ 4 )210 M^ fbf g 2 4 2 2 4 )210 M[bf ] 2 1 ~ ~y 1 ( ~ ~y ~ ~ ^ ~ ~ ^ 2 4 3 3 4 )210 Mfb2g + 2 i(4 3 + 3 4 )210 M[b2] ( ~ 1 ~ y1)210 M^ fccg ( ~ 2 ~ y2)210 M^ fff g ( ~ 3 ~ y3 )210 M^ f22g 1 ~ ~y 1 i( ~ ~y ^ ~ 1 ~ y2 ~ 2 ~ y1)210 M^ fcf g 2 ( 1 2 + 2 1 )210 M[cf ] 2 1 ~ ~y 1 ~ ~y ~ ~y ^ ~ ~y ^ 2 ( 1 3 + 3 1 )210 Mfc2g + 2 i( 1 3 3 1 )210 M[c2] + 21 ( ~ 2 ~ y3 + ~ 3 ~ y2)210 M^ [f 2] + 21 i( ~ 2 ~ y3 ~ 3 ~ y2 )210 M^ ff 2g ; 0

0

0

0

0

0

0

0

0

P

P

0

P

j Q120 + k 126 (2) = i 10i Q10i + j 120 j k 126 Qk = (~ ~ 1)10 Mf1sg + i(~ ~ 2)10 Mf1pg + (~ ~ 1 )10 M[1a ] i(~ ~ 2 )10 M[1a] + ((~ ~ 3 )10 + 3~ 1) M[1b ] i((~ ~ 4 )10 + 3~ 2) M[1b] + ((i~ ~ 1)120 i~ 3) M[1s] i((i~ ~ 2 )120 i~ 4) M[1p] + (i~ ~ 1 )120 Mf1a g i(i~ ~ 2 )120 Mf1ag + (i~ ~ 3 )120 Mf1b g i(i~ ~ 4 )120 Mf1bg ((i~ ~ 1)120 + 3i~ 1 + 2~ 4 ) M[1c] + i((i~ ~ 2)120 + 3i~ 2 2~ 3 ) M[1f ] (i~ ~ 3 )120 M[12] + ((~ ~ 1 )126 ~ 1) M[1a ] i((~ ~ 2)126 ~ 2) M[1a] + ((~ ~ 3 )126 + 2i ~ 2) M[1b ] i((~ ~ 4)126 2i ~ 1) M[1b] + (~ ~ 1)126 Mf1cg i(~ ~ 2)126 Mf1f g + ((~ ~ 3)126 + 3i ~ 3) Mf12g ; 0

0

0

0

0

0

with M~ f g = (M M y + M My )(?) ; Mf g = (M M y + M My )? ; M^ f g = (M M y + M My )? ; Mf1g = (M1 M + M M 1) ;

M~ [ ] = (iMM y M[ ] = (iMM y M^ [ ] = (iMM y M[1] = (iM1M

iM My )(?) ; iM My )? ; (=1) =1) iM My )? ; iMM 1 ) ; ( 6= 1)

16

R. Wulkenhaar

(Q1i )(?) = Qi 31 tr(Qi)13 ; 45 P2 45 (Q45 j )? = Qj a;b=1 tr(Qj Za )Tab Zb ; ? 210 P9 210 (Q210 j ) = Qj a;b=3 tr(Qj Za )Tab Zb ;

P2 T tr(Z Z ) =  ; Pa=19 aTa tr(Za Zb ) =ab ; 0

0

a=3 a a 0

a b

ab 0

with Z9 = 13. We have the following 8 constraints due to the Zi of our ideal: 0 = Mfspg + Mfa ag + 3Mfb bg + 3Mfcf g + 8Mf22g ; 0 = Mfsa g + Mfpag + 3Mfb cg + 3Mfbf g 8Mf22g ; 0 = M^ f11g + 8M^ f22g + M^ fsf g + M^ fpcg + M^ fa bg + M^ fab g + M^ fb b g + M^ fbbg + M^ fccg + M^ fff g ; 0 = M^ fsbg + M^ fpb g + M^ fa f g + M^ facg + 2M^ fb cg + 2M^ fbf g 8M^ f22g ; (17) 0 = M^ fscg + M^ fpf g + M^ fa b g + M^ fabg + 2M^ fb bg + 2M^ fcf g + 8M^ f22g ; 0 = M^ fsag + M^ fpa g + 3M^ fb f g + 3M^ fbcg 8M^ f22g ; 0 = M^ [s2] + M^ [p2] + M^ [a 2] + M^ [a2] + 3M^ [b 2] + 3M^ [b2] + 3M^ [c2] + 3M^ [f 2] ; 0 = M^ fs2g + M^ fp2g + M^ fa 2g + M^ fa2g + 3M^ fb 2g + 3M^ fb2g + 3M^ fc2g + 3M^ ff 2g : 0

0

0

0

0

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

According to the general theory we have to investigate whether or not the connection form  receives an extra contribution 0 = 1 r0 + 0 5 r1. Due to the symmetries of our setting and the requirement that the ri commute with ^ (so(10)), the matrices ri have the general form

0 iP (1 M )P 0 + 32 0 + B 0 iP+(132 M0 )P+ r0 = B @ 0 0 iP

0 0

0 0 0

(132 M0 )P 0 0 0 iP (132 M0 )P 0 0 P+(132 M00 )P+ 0 0 0 B P ( 1

M ) P 0 0 0 + 32 0 + r1 = B @ 0 0 0 P (132 M00 )P 0 0 P (132 M00 )P 0

1 CC ; A

1 CC ; A

with M0 = M0y and M00 = M00 y. The condition [r0; ^ ()] 2 ^ ( 1(a)) yields from the 45-sector i[M0; M1 ] 2 R M1 , which xes M0 up to three parameters. The 10-sector yields i(M0Ms + MsM0) 2 R Ms + iR Mp and i(M0Mp + MpM0 ) 2 R Mp + iR Ms . The r.h.s. are two-dimensional so that both i(M0 Ms + Ms M0 ) and i(M0Mp + MpM0 ) are orthogonal to 7-dimensional spaces. It is clear that there exists no solution for M0 in general. The condition fr1; ^ ()g 2 ^ ( 2 (a))+ f^(a); ^ (a)g derived from f0 ; (!1)g 2 ( 2 ) gives from the 45-sector no condition at all, because i (M00 M1 +M1M00 ) is contained in (1) for any M00 . From the 10-sector we get 2 times (9 6)

SO(10)-uni cation in NCG revisited

17

conditions, from the 120-sector 4 times (9 9) conditions and from the 10-sector 3 times (9 7) conditions. This means that there will not exist a solution for M00 in general. There are no extra contributions to the gauge potential possible. In the same way one shows that the graded centralizer C 2 is trivial, C 2 = C 1(M )P  f(g); (g)g. There is also no extra contribution to the ideal J 2 .

5 The bosonic action The bosonic action (7) is now given by 

SB = 192g2

Z

M

tr(F dx = 2)

Z M

(L2 + L1 + V ) dx ;

where g is the so(10)-coupling constant. In this formula, V = Higgs potential, which in more detail is given by

V = 48g2 + +

X

X k;k

X

tr(P+1i 1i ) tr((Q1i )(?)(Q1i )(?)) + 0

i;i

0

tr(P+

k

k

Q

210 )? ( 210 )? ) +

Qk

210 210 ) tr(( k 0

0

10 y tr(P+10i (10i )y) tr(Q10 i (Qi ) ) + 0

X

0

2 ) is the

45 tr(P+45j 45j ) tr((Q45 j )?(Qj )?) 0

0

Xj;j

0

k (k )y) tr(Q126 (Q126 )y ) tr(P+126 126 k k 0

0

0

0

i;i

X

0



192g2 tr(

k;k

0 0

0

j;j




j (j )y) tr(Q120 (Q120 )y ) : tr(P+120 120 j j

0

The other parts of the Lagrangian are

L2 = 192g tr((dA + A2j )2 ) ; L1 = 192g tr((d() + fA; () iY g)2) : As there are 23 dierent 1-terms, 48 2 = 46 dierent 45-terms, 70 6 = 64 dierent 210-terms, 6 dierent 10-terms, 9 dierent 120-terms and 7 dierent 10-terms in , there occur 21 (23
24 + 46
47 + 64
65 + 6
7 + 9
10 + 7
8) = 2

2

2

3531 dierent gauge invariant terms in the Higgs potential. All of them are compatible with the con guration M and N speci ed by the Yukawa operator Y as the vacuum. This means that the most general gauge invariant Higgs potential that leads to the desired spontaneous symmetry breaking depends on 3531 parameters. Of course, not all these terms are really necessary, and in the classical formulation one puts most of the coecients equal to zero. But there is no justi cation for doing so, the Higgs potential in the classical formulation does contain 3531 parameters. Our theory reduces this huge number to 9, namely the parameters of the unknown matrix M1 . All other matrices occur in the fermionic action and can be measured, in principle. Thus, although our SO(10)model has more independent parameters than the standard model, the ratio of the xed parameters to the classical parameters is much better than in corresponding treatments of the standard model.

18

R. Wulkenhaar

We now study the Yang-Mills part L2 192g2 tr(fA; Y g2) of the Lagrangian. For that purpose we introduce chiral -matrices

f

 = p1 ( 1 2  = p1 ( 4

2

 i 5) ; 0  i 3) ; 4

p; q g = 2 pq1 ; i j ij 32

 = p1 ( 2 2  = p1 ( 5

2

 i 9) ; 1  i 2) ; 6

ijpq = ij pq

 = p1 3

2( 7

pq ij

 i 8) ;

= 21 [ pi; qj ] = ( pijq)y ;

where p; q 2 f+; g and i; j = 1; : : : ; 5. The bar in q changes the sign. The Yang-Mills eld A can now be decomposed as

A = P ( 4p1 2 gAijpq;  pqij 13 14) = P (  A  13 14) ; pq 1 Aij Aijpq; = Ajiqp; = Aijpq; 2 C 1(M ) ; A  = 4p 2 pq; ij ;

(18)

where 13 acts on the generation space and 14 is the 4  4-matrix structure in (12). De ning  = 21 (    ) we have

dA = P ( 8p1 2 g(@Aijpq; @ Aijpq;)  pqij 13 14 ) ; A2 = P ( 321 g2Aijpq;Aklrs;   pqij rskl 13 14 ) 1 g 2 Aij Akl (   +   ) ( pq rs + rs pq ) 1 1 ) = P ( 128 3 4 pq; pq; kl ij ij kl pq pq 2 ij kl     rs rs 1 + P ( 128 g Apq;Apq; ( ) ( ij kl kl ij ) 13 14) : This gives with mn;pqrs tu ; ( pqij rskl rskl pqij ) = 2 ftu;ijkl mn mn;pqrs q r  m;p n;s p r  m;q n;s q s  m;p n;r n;r ; ftu;ijkl = jk t;i u;l ik t;j u;l jl t;i u;k + ilpst;jm;q u;k i;q =  i  q , the nal results where p;j j p mn;pqrsAij Akl  tu 1 1 ) ; A2 j2 = P ( 321 g2ftu;ijkl 3 4 pq; rs; mn mn  tu 2 1 dA + A j2 = P ( 8p2 gFtu; mn 13 14 ) ; mn = @ Amn @ Amn + p1 gf mn;pqrsAij Akl : Ftu;  tu;  tu; 2 2 tu;ijkl pq; rs;

Using tr(  )=4(gg gg) and tr( at

pq rs )=32( ps qr ij kl il jk

ikprjlqs), we arrive

ij F kl
tr(   )
tr((P +P ) ij kl )
6 =  F ij F pq; : L2 = 192g
1281 g2Fpq; + rs; pq rs 8 pq; ij It is now convenient to identify the su(3)  su(2)L  su(2)R  u(1)B L gauge 2

SO(10)-uni cation in NCG revisited

19

elds (Gk ; V i; V~ i; G0), where from now on i; j = 1; 2; 3 and k = 1; : : : ; 8: A12+ ; = p12 i(G6 iG7) ; A23+ ; = p12 (G4 +iG5) ; A31+ ; = p12 (G1 iG2) ; 11 3 11 22 8 p12 (A33 p16 (A33 + ; A+ ; ) = iG ; + ; +A+ ; 2A+ ; ) = iG ; 22 33 0 p1 (A11 3 + ; +A+ ; +A+ ; ) = iG ; 55 3 p1 (A44 A45+; = p12 (V1 iV2) =: V+ = (V )y ; 2 + ; A+ ; ) = iV ; 55 ~3 p1 (A44 A45++; = p12 (V~1 iV~2) =: V~+ = (V~ )y ; 2 + ; +A+ ; ) = iV : in this notation, the Yang-Mills Lagrangian L2 takes the form
L2 = 4 Gk Gk + @[ G0]@ [ G0] + Vi Vi + V~i V~i (19) ; ] + @ Ai4 @ [ A+ ; ] 4 [ + 2 ( @[ Aij++;]@ [ Aij ;] + @[ Ai++ [ +; ] i4 ; ]@ Ai4 ; ] + ; ] i 5 [  i 5 [  +@[ A++;]@ Ai5 + @[ A +;]@ Ai5 ) + I:T ; where I:T stands for interaction terms between 3 or 4 Yang-Mills elds and Gk = @[ Gk] gfkk k Gk Gk ; Vi = @[ Vi] gijj Vj Vj ; V~i = @[ V~i] gijj V~j V~j ; 0

0

00

00

0

0

0

0

and fkk k and ijj are su(3) and su(2) structure constants. The lesson is that the choice 1921g2 for the global normalization constant was correct, where g is the coupling constant of su(3) and the two su(2) Lie subalgebras. It remains to compute the mass terms  1921g2 tr(fA; Y g2) =  1921g2
4
tr(  5   5 (P+[A  ; M][A  ; M] + P+[A  ; N ][A  ; N y])) = 121g2
( 4pg 2 )2
4
16
4 A ; ) tr(2M 2 + M 0 M 0 y + M M y + 3M 0 M 0 y + 3M M y 4 (Ai4+;A+i4 ; + Ai++ a a b b ; i4 1 a a b b +3McMcy + 3Mf Mfy + MpMpy + MsMsy) 5 A ; ) tr(M 2 + M 0 M 0 y + M M y + M 0 M 0 y + 8 (Ai5+;A+i5 ; + Ai++ a a ; i5 1 a a b b y y +Mb Mb + 2McMc + 2Mf Mfy) 5 A ; ) tr(M M y ) + 2 (Ai+4 ;Ai4+; + Ai++ 2 2 ; i5 + ; + ; 4 A i4 0 0 y Ma M y + M 0 M 0 y Mb M y + 4 (Ai++ ; i4 + A ; Ai4 ) tr(Ma Ma a b b b y +McMc Mf Mfy + Ms Msy MpMpy) 4 A +; + Ai4 A+ ; ) tr(M 0 M y M M 0 y + M 0 M y M M 0 y + 4 ( Ai++ a a b b ; i4 ; i4 a a b b y y y +McMf Mf Mc + MsMp MpMsy ) 5 A ; Ai5 A +; ) tr(M M y + M M y + M 0 M 0 y + M 0 M 0 y 8 (Ai++ b a a b ; i5 + ; i5 a b b a +2McMfy + 2Mf Mcy ) 0

00

0

20

R. Wulkenhaar

+ 32 Aij++;Aij ; tr(M12 + Mb0 Mb0 y + Mb Mby + Mc Mcy + Mf Mfy + 161 M2M2y ) ; jj y ; y y y i + 8iijj Ajj++;A++ i5 tr(M2 Mf +M2 Mc ) 8ijj A ; Ai5 tr(Mf M2 +Mc M2 ) + 2 (2V+V  + 2V~+V~  + (V3 V~3)(V3 V~3 )) tr(Ma0 Ma0 y + Ma May + 3Mb0 Mb0 y +3MbMby + 3McMcy + 3Mf Mfy + MsMsy + MpMpy ) 4(V+V~  + V V~+ ) tr( Ma0 Ma0 y + Ma May + 3Mb0 Mb0 y 3Mb Mby 3Mc Mcy + 3Mf Mfy + MsMsy MpMpy ) 4(V+V~  V V~+ ) tr(Ma Ma0 y Ma0 May + 3Mb0 Mby 3Mb Mb0 y +3Mf Mcy 3McMfy + MsMpy MpMsy) q q
+ 2(V~+V~  + ( 32 G0 +V~3 )( 32 G0 +V~3 )) tr(M2 M2y) ; 0

0

0

0

where we used tr(   ) = 4g and 5 5  = 14. The factor 4 comes from the anti-symmetry Aijpq = Ajiqp and the 16 from the trace over P+ times -matrices. We anticipate (section 6) the relation between fermion masses and the matrices Ms;p;a ;a;b ;b;c;f , which reads (tp = transpose of the preceding term) 0

0

Ms = 161 (Mn+3Mu+Me+3Md) + tp ; Mp = 161 (Mn+3Mu Me 3Md ) + tp ; Mc = 161 (Mn Mu Me+Md ) + tp ; Mf = 161 (Mn Mu +Me Md ) + tp ;

Ma = 161 (Mn +3Mu+Me+3Md) tp ; Ma0 = 161 (Mn +3Mu Me 3Md ) tp ; Mb = 161 (Mn Mu Me+Md ) tp ; Mb0 = 161 (Mn Mu +Me Md ) tp : (20)

This gives

Ma0 Ma0 y+Ma May+3Mb0 Mb0 y+3MbMby +3McMcy +3Mf Mfy+MsMsy +MpMpy) = 121 tr(Mn Mny + MeMey + 3MuMuy + 3Md Mdy) =  ; y y 0 0y y 0 0y y y y 4 3 tr( Ma Ma +Ma Ma +3Mb Mb 3Mb Mb 3Mc Mc +3Mf Mf +Ms Ms Mp Mp ) = 61 tr(MnMey + MeMny + 3MuMdy + 3Md Muy) = 2~ ; 4 tr(M M 0 y M 0 M y +3M 0 M y 3M M 0 y +3M M y 3M M y +M M y M M y ) a a b b f c c f s p p s a a b b 3 y y y y 1 = 6 tr(MeMn MnMe +3MdMu 3MuMd ) = 2i^ : 2 3 tr(

The numbers ; ~; ^ are not determined by the experimental data because the Dirac mass matrix for the neutrinos Mn is unknown. Let us assume that the largest eigenvalue of Mn is smaller than the mass mt of the top quark. Then, we have in leading approximation  = 14 m2t and ~ = 14 mt mb . We now diagonalize the V -q V~ -G0 sector. The photon is the massless linear comq bination P = 12 G0 38 V3 38 V~3, which is perpendicular to the plane spanned q2 3 q3 0 1 3 3 ~ p by 2 (V V ) and 5 V~ + 5 G . Now, abbreviating M 0 2= 13 tr(M2 M2y), the

SO(10)-uni cation in NCG revisited

21

mass terms of the V -V~ -G0 sector are

 2 + M 0 2 2M 02  0 3 3 3 3 1 ~ ~ p2 (V V ); p8 (V +V )+ 2 G 2M0 2 4M 0 2    + ~ +  V ~ i^

 1 1

2

p3

+ V ; V

~ + i^  + M 0 2

:

V~

!

~3 3 2 (V Vp) p18 (V~3 +V3 )+ 23 G0 p1

After a unitary-orthogonal transformation !        p1 (V3 +V~3 ) i sin V  W Z = cos  sin  cos  e 2   p sin  cos  p18 (V~3 V3 )+ 23 G0 ; W~  = sin  ei cos  V~ ; Z~ 0

0

 , the physical particles obtain the following masses: where ei = p~^+i^ 2 +~ 2 0

m2W m2W~ m2Z m2Z~

q

 + 21 M 0 2 q 14 M 0 4+^2+~2   + (^2+~2)=M 02  + 21 M 0 2 + q 14 M 0 4 +^2+~2  M 0 2 +   + 52 M 0 2 q 254 M 0 4 3M 0 2 + 2  58  2516M22  + 52 M 0 2 + 254 M 0 4 3M 02 + 2  5M 0 2 + 52  0

This means

mW  21 mt ;

sin2 W  38

mZ = mW = cos W ;

m2t

80 M 0 2

:

(21)

We recall that the mass prediction for mW depends crucially on the assumption that the Dirac mass for the neutrinos can be neglected. But as mW cannot be much larger than 21 mt , this assumption seems topbe correct. For the rotation angles we get cot 2 = 43 2M 2 and tan 2 = 2 ~2 + ^2 =M 02. Hence, there is a violation of the standard model of the order m2t =M 02 , for instance a coupling of the W  bosons to the right-handed fermions, and the Weinberg angle is not universal any more. We neglect the mixing of the order kMa;a ;b;b ;c;f;s;pk=kM1;2k between the very massive leptoquarks. Denoting M 2 = 34 tr(M12 ), the masses of the leptoquark are: 0

0

0

4 5 Ai+4 Ai++ Ai+5 Ai++ Aij++ p 2 02 p 2 02 p 2 02 M +M M M M +M 4M +M

The renormalization group analysis10 suggests M  1016 GeV. Below that energy, the original gauge group SO(10) is broken to the intermediate symmetry group SU(3)C SU(2)LSU(2)RU(1)B L. At the scale M 0  109GeV, the subgroups SU(2)R and U(1)B L are broken to the standard model symmetry

22

R. Wulkenhaar

q

q

group SU(3)C SU(2)L U(1)Y , with the hyper-charge given by 35 V~3 25 G0 . Finally, the fermion masses break the standard model symmetry at the scale mt  102GeV to the remaining symmetry SU(3)C U(1)EM . Hence, the only massless Yang-Mills elds are the photon P and the eight gluons Ga . The Higgs mechanism consists in using the other 45 9 = 36 SO(10)-gauge parameters to eliminate the 36 Goldstone bosons [(a); iY ], which in turn breaks the symmetry from SO(10) to the fermion symmetry SU(3)C U(1)EM . Thus, there are 45+2
10+4
120+3
2
126 36 = 1301 36 = 1265 independent Higgs components. We compute now the upper limit for the mass of the standard model Higgs eld. It is obtained by evaluating the Higgs potential V at the con guration ~ = ( 45 + 78 + 69 ) ; ~ 1 = (1 + ) 0 ; ~ 2 = (1 + ) 3 ; ~ 1 = (1 + ) 120 ; ~ 2 = (1 + ) 123 ; ~ 3 = (1 + )( 453 + 783 + 693 ) ; ~ 4 = (1 + )( 450 + 780 + 690 ) ; ~ 1 = (1 + )( 01245 + 01278 + 01269 ) ; ~ 2 = (1 + )( 31245 + 31278 + 31269 ) ; (22) ~ 3 = 81 ( 1 i 2) 3( 4 i 5)( 6 i 9)( 7 i 8) : It is important that  is a real eld, because the con guration corresponding to a imaginary part is the Goldstone boson given by the commutator with +44 . One easily nds (1) = 14 132 (M~ fssg + M~ fppg + M~ fa a g + M~ faag + 3M~ fb b g + 3M~ fbbg +3M~ fccg + 3M~ fff g ) + 214 i( 45+ 78 + 69 ) (Mfsb g+Mfpbg +Mfa cg+Mfaf g +2Mfbcg+2Mfb f g ) ; (2) = 0 ; 0

0

0

0 0

0

0

up to the Higgs self-interaction 2 . The other 45 and all 210-contributions are cancelled due to (17). We insert (20) and arrive at (1) = 161 14 132 ( M~ fnng + M~ feeg + 3M~ fuug + 3M~ fddg +M~ fntntg + M~ fetetg + 3M~ fututg + 3M~ fdt dtg) + 161 14 i( 45 + 78 + 69 ) ( Mfnng + Mfeeg Mfuug Mfddg Mfntntg Mfetetg + Mfututg + Mfdt dtg ) : where M~ tt := (2MT MT y)?. We neglect again Mn and choose Mu = diag(mu; mc; mt ), where the entries are the masses of the u; c; t-quarks. There is a neglectable contribution of 45 to the Higgs potential, and we have in leading approximation

(1) =

1 4

m2t 14 132 diag( 1; 1; 2) :

SO(10)-uni cation in NCG revisited

23

This leads to

V = 48g2 161 2m4t
4
16
6 =  2g12 m4t 2 : Inserting the con guration (22) into the part L1 of the Lagrangian we get L1 = 192g2
4
tr(P+(  @ 5N )(  @ 5 N )) = 121g2 tr(P+@ N y @  N ) = 121g2 @  @  
16
tr(Ma0 Ma0 y + Ma May + 3Mb0 Mb0 y + 3MbMby + 3McMcy + 3Mf Mfy + MsMsy + MpMpy) = g22  @  @   : This means that the physical Higgs boson is obtained by rescaling ' = 2pg   = g=mt, and it receives the mass m' = mt : (23) This value is an upper bound for the Higgs mass, because we have only calculated the diagonal matrix element of the whole mass matrix. Due to the o-diagonal matrix elements, the masses = eigenvalues are dierent from the diagonal matrix elements, and the smallest eigenvalue is smaller than the smallest diagonal matrix element. It is plausible that this smallest diagonal matrix element is just m2' , because any other Higgs con guration obtains a mass contribution from (2) of the order tr((M1Mi  MiM 1)2 )=tr(Mi2 ). The computation of the masses of the remaining 1264 Higgs bosons is analogous.

6 The fermionic action

To write down the fermionic action (in Minkowski space!), recall that our setting has two symmetries J and S . It is therefore natural to demand that the fermionic con guration space has the same symmetries, H = f 2 L2 (M; S ) C 384 : P = J = S = g : As usual we impose a Weyl condition in Minkowskian case. This means to look for a chirality operator that commutes with both J and S . The unique choice up to the sign is  =  ;  = P diag( 5 196 ; 5 196 ; 5 196 ; 5 196)P : Thus, elements of H are of the form 0 ( 1 (1 5 ) P ) ~ 1 BB ( 2i 12 (414 5) P++) ~ CC ~ 2 L2(M; S ) C 96 : = B ( 1 (1 + 5) 2 P B ) ~ C ; @ 2 4 A (i 12 (14+ 5 ) 2 P B ) ~

24

R. Wulkenhaar

In order to eliminate the projection operators we introduce 0 b B = b 0 ; b = 2 22 2 13 2 C 48 ; A 0   1 A  13 = 0 bAT b ; A 2 C (M ) so(10) 13 ;  0 H~  P+(N + N )P = 0 0 ; 0 = ~ 0 = 12 ; a = ~ a ; a = 1; 2; 3 ;
( 21 (14 5) P+) ~ = 0 ; L ; 0 ; 0 t ; L 2 L2 (M )C 2 C 16 C 3 : Now, the fermionic action (8) is

SF = = =

Z

ZM Z

M M

dx 41 y 0(D + i) dx

1 2





y

L;

 i~(@ + A)

T 2 L b

dx i Ly ~  (@ + A) L +

H~ y

n1

yH ~(

2 L



H~

i(@ bAT b)

o

2b L ) + h:c

L

2b :

 L

(24)

To get the last line one has to take into account that fermions i are Grassmannvalued, which means 1T X T 2 = 2yX 1 , for any matrix X . The correct fermionic parameterization dictated by the electric charge is 1 2 3 1 2 3 L = nL ; uL ; uL; uL; eL ; dL ; dL; dL ;
2 d3R; 2 d2R; 2 d1R; 2eR ; 2 u3R; 2u2R ; 2u1R; 2 R t ; 2 b L = nR ; u1R ; u2R; u3R ; eR ; d1R; d2R ; d3R;
2 d3L; 2d2L; 2 d1L; 2 eL; 2 u3L; 2u2L; 2u1L; 2 L t; where t does not transpose the entries of the vector. It it now easy to verify that y 1 y N (  2 b ) + h:c =  uy (M 1 )u L 3 R dL (Md 13 )dR L u 2 L eyL MeeR Ly M R RT 2M2 R + h:c ; with the mass matrices Mu;d;;e given implicitly in (20). The right neutrinos receive a large Majorana mass of the order kM2 k and the see-saw mechanism produces very small masses for the left-handed neutrinos of the order km2nk=kM2k.

7 Outlook What we have presented here is a maximal SO(10)-model which allows the fermion masses to be as general as possible. This is in contrast to the original idea of grand uni cation, namely, to reduce the number of free parameters of the standard model. The number of Higgs multiplets can be reduced

SO(10)-uni cation in NCG revisited

25

by imposing appropriate relations between the fermion masses. For instance, the minimal SO(10)-model containing one complex 10, one complex 126 and the 45 (or 210) is obtained by putting Ms = 1Mp, M2 = 2Mc = 3Mf and Ma = Ma0 = Mb = Mb0 = 0, with real parameters i. This model is very predictive in the fermion sector and one can calculate the neutrino masses10 . However, in our formulation the ideal J 2 becomes so large that the only surviving terms in the Higgs potential are 1 and 10. This is not sucient. There seems to be a strong evidence that a 120 representation must be included. This next-tominimal SO(10)-model will be studied elsewhere.

Acknowledgment I am grateful to Bruno Iochum, Daniel Kastler, Thomas Krajewski, Serge Lazzarini and Thomas Schucker for very useful discussions and for the hospitality at the CPT Luminy. 1. A. Connes, Geometrie non commutative, InterEditions (Paris 1990). 2. A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995) 6194{ 6231. 3. A. Connes, Gravity coupled with matter and the foundation of noncommutative geometry, Commun. Math. Phys. 182 (1996) 155{176. 4. D. Kastler, Noncommutative geometry and fundamental physical interactions (Historical sketch and outline of the present situation), Monsaraz summer school (1997). 5. F. Lizzi, G. Mangano, G. Miele and G. Sparano, Constraints on uni ed gauge theories from noncommutative geometry, Mod. Phys. Lett. A 11 (1996) 2561{2572. 6. R. Wulkenhaar, Noncommutative geometry with graded dierential Lie algebras, J. Math. Phys. 38 (1997) 3358{3390. 7. H. Fritzsch and P. Minkowski, Uni ed interactions of leptons and hadrons, Annals Phys. 93 (1975) 193{266. 8. H. Georgi, The state of the art{gauge theories, in: Proc. 1974 Williamsburg AIP conf., ed by C. E. Carlton, AIP (New York 1975) 575{582. 9. A. H. Chamseddine and J. Frohlich, SO(10) uni cation in noncommutative geometry, Phys. Rev. D 50 (1994) 2893{2907. 10. R. N. Mohapatra, Minimal SO(10) grand uni cation: Predictions for proton decay and neutrino masses and mixings, Warsaw Elem. Part. Phys. (1993) 158{172, and hep-ph/9310265.