Fermion mass predictions from higher dimensions

Nuclear Physics B261 (1985) 461-490 © North-Holland Pubhshmg Company

FERMION MASS PREDICTIONS FROM HIGHER DIMENSIONS C WETTERICH CERN, Geneva, Swttzerland

Received 23 May 1985 Higher dimensional theories pre&ct ratios between Yukawa couphngs and gauge couphngs m the resulting four-&menslonal gauge theory After spontaneous symmetry breaking of weak mteracUons, these relattons are converted into relattons between particle masses We calculate exphcltly the Yukawa couphngs resulting from a slx-&menslonal SO(12) gauge theory We argue that the fermton masses follow the pattern mt ~, rob, mc, my ~- ms, m~ ~, rod, mu, me

1. Introduction A n e x p l a n a t i o n o f t h e o b s e r v e d m a s s e s o f q u a r k s a n d l e p t o n s as well as the m a t r i x d e s c r i b i n g m i x i n g angles a n d C P - v i o l a t i o n is o n e o f t h e o l d t h e o r e t i c a l p r o b l e m s in p a r t i c l e physics. I n f o u r - d i m e n s i o n a l unified g a u g e theories these q u a n t i t i e s are r e l a t e d to Y u k a w a c o u p h n g s b e t w e e n f e r m i o n s a n d scalars. U n f o r t u n a t e l y , even in g r a n d unified t h e o r i e s the m a t r i c e s o f Y u k a w a c o u p l i n g s c o n t a i n m a n y free p a r a meters A few r e l a t i o n s b e t w e e n f e r m i o n m a s s e s m a y be p r e d i c t e d [1, 2], b u t an u n d e r s t a n d i n g o f the full s p e c t r u m seems difficult in this context. H i g h e r d i m e n s i o n a l t h e o r i e s h a v e t y p i c a l l y o n l y a few free p a r a m e t e r s . The a c t i o n for s u p e r s t r l n g t h e o r i e s [3], for e x a m p l e , has no free d i m e n s i o n l e s s p a r a m e t e r s at all. I f such t h e o r i e s l e a d to a realistic f o u r - d i m e n s i o n a l m o d e l after d i m e n s i o n a l r e d u c t i o n , the s p e c t r u m o f f e r m i o n m a s s e s s h o u l d be h i g h l y p r e d i c t a b l e . F o u r d i m e n s i o n a l g a u g e fields a n d scalars often c o r r e s p o n d to different c o m p o n e n t s o f t h e s a m e h i g h e r d i m e n s i o n a l field. T h e e x i s t e n c e o f relations b e t w e e n Y u k a w a c o u p h n g s a n d g a u g e c o u p l i n g s s h o u l d t h e r e f o r e not surprise us. Even if h i g h e r d i m e n s i o n a l m o d e l s h a v e a few free p a r a m e t e r s , we e x p e c t r e l a t i o n s a m o n g f e r m i o n m a s s e s a n d a p r e d i c t i o n o n the relative scale o f f e r m i o n m a s s e s c o m p a r e d to the m a s s o f the W - b o s o n . In the n e x t s e c t i o n we o u t h n e the steps l e a d i n g to such p r e d i c t i o n s m a g e n e r a l f r a m e w o r k . T h e h i g h e r d i m e n s i o n a l t h e o r y m a y either be p u r e gravity c o u p l e d to s p m o r s o r i n c l u d e o t h e r fields like g a u g e fields in h i g h e r d i m e n s i o n s o r scalars. The d i s c u s s i o n a p p l i e s to the field t h e o r y limit o f a h i g h e r d i m e n s i o n a l string t h e o r y i n c l u d i n g p o s s i b l e s u p e r s y m m e t r i e s . F e r m i o n m a s s p r e d i c t i o n s can b e o b t a i n e d if the m o d e l a d m i t s a n a p p r o x i m a t e g r o u n d state with S U ( 3 ) c x SU(2)L X U ( 1 ) y symm e t r y l e a d i n g to a certain n u m b e r o f massless f o u r - d i m e n s i o n a l q u a r k s a n d l e p t o n s 461

462

C Wettench / Fermlon mass predtcnons

with the correct chlral quantum numbers. The requirement that all quarks and charged leptons acquire a non-vanishing mass from spontaneous breaking of SU(2)L x U(1)y imposes certain topological constraints on candidate ground states We argue that the problem of fermion masses and the gauge hierarchy problem are partly decoupled Similar to grand unified theories, fermion mass ratios may be predicted even though the explanation of a small symmetry breaking scale may remain unsatisfactory In sects, 3, 4 and 5 we discuss fermion mass ratios in the context of a sixdimensional SO(12) gauge theory [4, 5] We calculate all Yukawa couplings for the chiral quarks and leptons on a class of classical solutions were internal space is a sphere with gauge fields in a monopole configuration. Although the model IS not complete - six dimensional scalars have to be included - the spectrum of fermlon masses reproduces some features of the observed mass spectrum for certain solutions In the hmit of an unbroken U(1)R × U(1 )~ symmetry, three sorts of scalars contribute to fermion masses: Ha couples only to the top quark,/-/2 contributes to rob, mc and m, and h Induces masses for s and/z. These scalars have different quantum numbers so that their mixing may be small This suggests the pattern m t ~. rob, mc, m~ ~- ms, m~ ~" md, mu, me We give our conclusions in sect. 6 and two appendices display the conventions used in this paper for SO(12) generators in the splnor representations and for generalized spherical harmonics

2. Quark and lepton masses from higher dimensions Our starting point for a calculation of fermlon masses should be a model which leads to chiral fermIons in four dimensions There must be solutions of higher dimensional field equations with maximal four-dimensional symmetry [Polncar6 symmetry or (anti-) de Sitter symmetry in case of a small cosmological constant] and gauge symmetry SU(3)c x SU(2)L × U(1)v The gauge symmetry corresponds to isometrles of internal space or to special higher dimensional gauge transformations. The effective four-dimensional theory IS obtained by integrating over internal space. The harmonic expansion of spinors on this solution should lead to massless fermions with the chlral quantum numbers of quarks and leptons Chlrahty imphes that quarks and leptons can only get a mass of the order of the scale of spontaneous symmetry breaking of SU(2)L × U(1)v or smaller The number of chiral four-dimensional fermlons is given by an index [6, 9], the chlrality index No. This index measures the number of generations (For an example with SO(10) symmetry, N16 counts the number of unpaired left-handed fourdimensional fermions in the 16 representation.) A non-vanishing chirahty index with respect to SU(3)c xSU(2)LXU(1)v guarantees that the number of massless chlral fermion generations is stable against small changes of higher dimensional parameters or small deformations in the solution as long as SU(3)c × SU(2)L X U(1)v remains unbroken In higher dimensional theories, massless fermlons have now a

C Wettench / Fermlon mass predzctlons

463

similar status as massless gauge bosons: the zero mass is guaranteed by symmetry plus topology and any mass term involves the breaking of symmetry. In general, the chirality index is not a property of internal space alone but involves the geometry of full higher dimensional space-time [7]. In the case of a &feet product topology M 4 x c o m p a c t internal space, the chirality index is given by the character valued index on internal space alone Two general results are useful to remember. In higher dimensional riemannlan geometry without gauge fields, the properties of the Lorentz algebra imply [8] that chlral fermions are only possible for d = 2 m o d 4 Only d = 2 mod 8 can lead to a realistic spectrum without a doubling of fermions with the same mass. In riemannlan geometry with higher &mensional gauge fields, the same criteria imply d -- 2 mod 8, d = 6 mod 8 or d = 2 mod 2, depending whether the fermions are in a real, pseudoreal or complex representation of the gauge group. The second result implies [9, 10] that m riemannian geometry without gauge fields the chirality index vanishes for all possible symmetry groups and all representations if the ground state has topology M 4 × c o m p a c t internal space. Three different ways are known to obtain a non-vanishing chlrality index. The most popular starts with gauge fields in higher dimensions Several models leading to chiral fermions are known [ 11-13, 9, 4]. Among the anomaly-free models are the field theoretical limit of SO(32) or E8 × E8 superstrings [3] in 10 dimensions or the six-dimensional SO(12) model [4,5] which we will discuss later. The second possibility starts with higher dimensional rlemannlan gravity alone and abandons the requirement that the ground state should be M 4 × compact internal space Chlral fermions are found If internal space is non-compact [14, 7] or If geometry does not have the standard direct product form four-dimensional space ×internal space [7]. The simplest model starts with 18-dimensional nemannian gravity coupled to a Majorana-Weyl spinor This model has gravitational anomahes [15] In a first step of dimensional reduction, it leads to the anomaly-free chiral six-dimensional SO(12) model [4] The third alternative also remains m the context of gravity alone Chiral fermions are obtained from generalization [16, 14] of riemannian geometry There are several models leading to chlral fermions in standard generations of unification groups like SU(5), SO(10) or E6 or subgroups of them like SU(3)c x SU(2)L × U(1) a × U(1)a_L, etc This gives the required chiral quantum numbers for quarks and leptons At this stage, quarks and leptons are all massless and we now have to investigate the mechanism by which they get mass In a realistic model, colour and electric charge remain unbroken and all masses must be due to spontaneous breaking of SU(2)L×U(1)v Let us denote the scale of SU(2)L×U(1)y breaking by (4)). In a realistic model one needs (4))~ 174 GeV. This scale is very small compared to the Planck mass which is roughly the characteristic mass scale of all viable higher dimensional models. It is not clear how to find a natural explanation for such a small scale m higher dimensional theories and we will not tackle this problem - the gauge hierarchy problem - in this paper. We only note at this point that m higher dimensional models with a few free parameters one expects

464

C Wetterlch / Fermwn mass predzctzons

that the parameters can be adjusted to give a small value of (~b), similar to grand unified models Assuming a small scale (tb), it can be shown [6] that the masses of all quarks and leptons are of order (tb) or smaller. In higher dimensional models, spontaneous symmetry breaking corresponds to a continuous deformation of geometry and bosonic matter field configuration which reduces the symmetry of a given state Since all quarks and charged leptons are observed to be massive, we require that all chiral fermions (with the possible exception of neutrinos) must get a mass from such a continuous deformation breaking SU(2)L×U(1)y. This requirement is not trivial and places topological restrictions on candidate SU(3)c × SU(2)L × U(1).c symmetric states. The first restriction [6, 4] is that the Atlyah-Singer index [17] on this state must vanish. This index is a purely topological index counting the difference in the number of zero modes for the two inequivalent internal Weyl spinors (We always assume an even number of dimensions ) If this index is different from zero one always remains with the corresponding number of massless fermlons which cannot get mass by any continuous deformation. An equivalent formulation of this problem requires that the harmonic expansion of bosonic degrees of freedom on the SU(3)c × SU(2)L × U ( 1 ) y symmetric state must contain four-dimensional scalars with the appropriate quantum numbers to admit Yukawa couplings to the chiral quarks and leptons. A topological obstruction like a non-vanishing Atiyah-Singer index implies that certain fermions cannot have Yukawa couplings to any of the scalars in the harmonic expansion. (For a simple example In six-dimensional Einstein-Maxwell theory, see ref. [4] ) There can be additional topological obstructions if the higher dimensional model has more than one irreducible fermlon representation ~ . . • ~b,. • • ~b~ which are independent in the sense that the action has discrete symmetries of the sort F," Oj ~ t#j f o r j # t, ~b,--> -~,. (This is the case if there are more than one fermlon representation with minimal gravitational and gauge interactions.) We then have to require that the Atiyah-Singer index vanishes for each representation separately The total number of quarks and leptons are in a vector-like representation of SU(3)c × U(1)em. Nevertheless, this group admits complex representations In the above case we need l a vanishing chirahty index with respect to SU(3)c ×O(1)em for each fermion representation separately Otherwise SU(3)c × U(1)em symmetry forbids mass terms for some of the chiral fermions [5]. We note, however, that symmetries of the type F, may be broken by couplings to higher dimensional scalar fields. Fortunately, a vanishing Atiyah-Singer index does not exclude the chirahty index being different from zero A non-zero chirahty index is possible even on a trivial topology in riemannian gravity [7] or for a topologically trivial configuration of higher dimensional gauge fields [5] In fact, the chIrahty Index is not a property of topology alone. It is only defined with respect to a group of symmetries G and is a property of G-deformation classes [6] of geometries (Things are even more subtle if space-time does not have geometry M a x c o m p a c t internal space The chirahty Index may change [7] within a given G-deformation class ) A G-deformation class

C Wetterwh / Fermlon mass predwt~ons

465

consists of all states (geometries plus gauge field configurations, for example) which can be obtained from each other by continuous deformations, leaving G unbroken. The states with a given topology may contain different G-deformation classes and the chirality index may change from one to the other A good example [4] are SO(12) monopoles on S2 They are all topologically trivial, but they can be " u n w o u n d " only by using SU(3)c × SU(2)L X U(1)v non-singlet degrees of freedom. The different SU(3)c × SU(2)L × U(1)y deformation classes often lead to a different chirahty index. This points to a problem in models which combine ideas of grand unification and dimensional reduction. This combination was proposed early [18] because of two advantages the quarks and leptons form simple representations of unification groups like SU(5), SO(10) or E6 and the ratio between the gauge couplings of SU(3)c, SU(2)L and U(1)y fits well the experimentally observed values. However, the grand unified group must be broken to SU(3)c ×SU(2)L×U(1)v at a scale comparable to the typical scale of compactification Spontaneous symmetry breaking in the context of higher dimensional theories can lead to several unusual features [5]. For example, the number of chiral fermion generations may change in the course of the breaking SU(5) -->SU(3)c x SU(2)L × U(1)y if the broken and unbroken solutions correspond to different SU(3)c×SU(2)LXU(1)y deformation classes V Also, the Yuakawa couplings often do not obey the relations of grand unified symmetry [5, 19]. In general, one first has to find the SU(3)cXSU(2)L×U(1)V symmetric approximate ground state which differs from the true ground state only by corrections proportional to the scale (4,) of weak symmetry breaking. The contents of chlral fermions and their Yukawa couplings have to be calculated in this approximate ground state. Let us now assume that the S U ( 3 ) c × S U ( 2 ) L X U ( 1 ) v symmetric approximate ground state is known. The fermion mass problem can then be split Into several pieces First one has to identify the chiral quarks and leptons The discussion of their masses can neglect all heavy states in the harmonic expansion of splnors. All effects of mixing with heavy fermlons are suppressed [6] by a factor (~b)Lo compared to typical fermion masses of order (~b) Here Lo is the characteristic length scale of Internal space. Next one has to identify the weak Hlggs doublet ~b whose vacuum expectation value drives the breaking of SU(2)L ×U(1)y. In general this doublet is a linear combination of many four-&mensional scalar fields with the appropriate quantum numbers The third step involves the calculation of the Yukawa couplings of ~b to the chiral fermions. For a given approximate ground state they are uniquely determined in dependence of the parameters of the higher dimensional action Typically, Yukawa couplings are a function of the four-&menslonal gauge coupling g. Knowledge of Yukawa couplings allows predictions on fermlon mass ratios hke m c / m b , etc., at the scale Lo ~ The ratio between fermion masses and the mass Mw of the W-boson xs also pre&cted. Finally, one has to find the scale (~b) This last s t e p - t h e solution of the gauge hierarchy p r o b l e m - m a y be the most difficult

466

C Wettench / Fermmn mass predwtmns

Fortunately, it can be decoupled to a large extent from the problem of fermion mass predictions. In a realisUc model, it may not be easy to find the correct SU(3)c × SU(2) L × U(1)v symmetric approximate ground state. What can we learn from other known solutions which typically have a symmetry larger than SU(3)c ×SU(2)L ×U(1)v9 If such a solution is in a different SU(3)c ×SU(2)L×U(1)V deformation class from the approximate ground state, a discussion of fermlon mass relations for such a solution has nothing to do with the final fermion mass relations. The chiral fermions of such a solution are not related to the quarks and leptons of the approximate ground state, even if the number were the same. If a solution IS In the same deformation class as the approximate ground state, there is a continuous one-to-one connection between SU(3)c × SU(2)L X U(1)y representations in the harmonic expansion on this solution and on the approximate ground state Especially quarks and leptons on both states are continuously related. The same is true for possible candidates for the weak Hlggs doublet This permits at least a quahtatlve understanding of certain features of the fermion mass matrix by studying the representation content of fields for the known solution and evaluating their Yukawa couplings Two problems exist for quantitiatlve fermion mass predictions from such a solution, first, the harmonics for the chlral fermions may be different for this solution and the approximate ground state This corresponds to a mixing with heavy harmonics by going from one state to the other. The same is true for the Hlggs scalars. Even if all harmonics were approximately the same, the Yukawa couplings might change between the two states due to terms of the type ~$~b~b0. Here ~b0 IS a SU(3)c × SU(2)L × U(1)y slnglet scalar field whose vacuum expectation value differs for the two states. There is, however, an important special case where all these difficulties are minimized: this IS the case if the spontaneous symmetry breaking relating the known solution to the approximate ground state happens at a scale MI sufficiently small compared to Lo I All corrections to predictions on fermlon masses are then suppressed by factors M~Lo. In the remainder of this paper we will illustrate all these ideas m the context of a simple model.

3. Yukawa couplings from a six-dimensional S 0 ( 1 2 ) gauge theory As our specific example we will discuss the six-dimensional SO(12) model [4, 5]. This incorporates many characteristic features of higher dimensional gauge theories which are expected to be shared by more complicated models like those based on E8 x E8 in ten dimensions obtained from superstrings [3] It is possible that it can be obtained [5] from the E8 × E8 model in a first step of dimensional reduction. It may also be obtained from pure gravity in 18 dimensions [4]. Our model is simple enough to allow an explicit evaluation of the relevant Yukawa couplings. The Majorana-Weyl splnors $1 and $2 of this model have positive and negative six-dimensional hehclty respectively and they belong to the inequlvalent spinor

467

C Wettench / Fermmn mass predwtmns

representations 321 and 322 of SO(12). The model is free of all anomalies [15]. All Yukawa couplings are contained m the six-dimensional covariant kinetic term for the fermions: --

~

o~tp = l g l / 2 ~ 1 ~ l ' ~ ( V ~ - -

- A I 0 ))tPl+ l~A

~ - ' ~~ 2 |gl/21/72 ~P'(~7/2 - - t g- A

= & +&~.

))~2 (1)

Here V,z is the slx-dimensmnal covariant derivative, t~6= - d e t t~a~ (we use conventions of ref. [4]) and ~(1.2)!2an-r(1.2) - - 2-~'~ * A B

A, B = 1 • • • 12

,

(2)

with T(,,2) * a n the SO(12) generators m the spinor representations 321 and 322 (compare appendix A) We use Slx-dlmensmnal signature ~/~ = diag ( 1 , - 1 , - 1 , - 1 , - 1 , - 1 ) and

{9~, #/}=2n ~e, y^,a = T" ;

m =0, 1,2,3

~a= 5 F 1 =/Ts@,rl , ~S = "/5F2 = zys® r2,

y5= ly0yly2y3 .

(3)

Here y " are the usual four-dimensional Dirac matrices The Pauli matrices r~ and r2 act on internal spinor indices. (Two-dimensional Dwac spmors have two components.) Let us now assume the existence of a classical solution with Poincar6 invanance and SU(3)c x SU(2)L × U(1)v gauge symmetry. This solution is considered to be the approximate ground state of the theory up to small effects of spontaneous SU(2)L x U(1)v breaking. Pomcar6 invariance implies, for the vlelbein and the gauge field in the ground state [20, 21]

.,~ (crl/2(y)6~

ea =

0 ) ~(y) ,

.~=(

0 ) .~(y)

(4)

Known classical solutions of th~s type for the six-dimensional SO(12) EinstemYang-Mills theory include monopole solutions [11, 13, 4, 5] on an internal sphere S 2 or generahzation to non-compact internal space [21]. All these solutions lead to symmetries larger than SU(3)c x SU(2)L X U(1)y, but additional classical solutions are expected once six-dimensional scalar fields are included in the action [4, 5]. The spinor representations of SO(12) contain fields with the usual quantum numbers of quarks and leptons as well as mirror particles (With respect to the SO(10) subgroup, both 321 and 322 transform as 16+ 16.) For all known classical solutions one finds an asymmetry between standard fermions and mwror fermions m the harmonic expansion, provided a certain monopole number n [4, 5] is different from

468

C Wettertch / Fermton mass predictions

zero The model accounts for n generations of chlral quarks and leptons In the resulting effective four-dimensional theory Except for the massless chlral modes, all other infinitely many fermlonlc modes in the harmonic expansion have masses 91017 GeV. We concentrate on Yukawa couplings of the chiral fermlons and truncate the harmonic expansion ~b(y, x) = ~b° = ~boj(y)~b°J(x).

(5)

H e r e j is a multi-index denoting the different quantum numbers of the chlral splnors. The chlral splnors form a complex representation of the subgroup of SO(12) left mvarlant by the ground state. The only possible Yukawa couplings Involve scalars in non-trivial representations of this subgroup They are contained in the internal components ,4~ of the SO(12) gauge fields and belong to the 66-dimensional adjolnt representation of SO(12). We denote /~ot

~--- e°a. ( A ,A~

- f~,~) ,

,4± = x/~ (-41T 'fi'2) •

(6)

The harmonic expansion for the internal corr/ponents of SO(12) gauge fields reads [13, 5] .4±(y, x ) = b ± , ( y ) ~ b ± ' ( x )

(7)

The index t denotes the various quantum numbers of the four-dimensional scalar fields ~b±'(x) Retaining only those scalars, the fermlonlc action (1) for the chlral fermions reduces to ~/2~°2{,,~-'/2r"o,,

+ l G ~T5(.4+~-+ + .4_~-_)}~°.2.

(8)

Here we have defined ~± = ½('r, :1: '~'2) •

(9)

The first term in eq (8) leads to the standard kinetic term whereas the second contains all Yukawa couphngs. There is no mixing between 61 and qs2 and we can discuss the Yukawa couplings separately for both spinors. We define internal and four-dimensional Weyl splnors ~b~(y) = 1(1 + ~'3)g%(Y), I/#0LJ,R(X) = 1(1 + T')qs°J(x).

(10)

Using the six-dimensional Weyl constraints (~,~®~3)~, = ~ , ,

(Ts® ~-3)62= -~b2,

(11)

and integrating over the coordinates y of internal space, one finds for the four-

C Wettench / Fermmn mass predlctmns

469

dimensional Yukawa couplings of ~1 ~L~'(l)Vuk----~°~(x)'ySh~,dP-'(x)O~(x) + ~b~(x)tyShf~,gb+'(x)~b°r'(x) ,

(12)

with Yukawa couplings h~, = x/2 ~ f d2y gl/20"2(Oo~(y))*b_,(y)T_~(y), hj-~,= x/2 ~ f dEy g2~/2tr2(0~(y))*b+, (y) ~'+~bor,(y)

(13)

For the spinor 02 the subscripts L and R designing four-dimensional hehcity are interchanged. The spinors 01 and ~b2 are subject to a six-dimensional Majorana constraint

B41B21B~2~O~ 2 = ~bl.2 .

(14)

The SO(12) charge conjugation matrix B12 acts on SO(12) indices and fulfils (see appendix A)

B*2B12 = - 1 .

(15)

The four- and two-dimensional charge conjugation matrices are defined [22] (ym), = _B4ymB~l,

B'B4 = 1,

(Fa) * = B2FaB; 1 ,

B'B2 = - 1 .

(16)

Internal charge conjugation maps every complex representation in the harmonics of ~b+(y) into a complex conjugate representation in ff-(y) and vice versa [8] + _ c~21/Jos(y) =

-1

--1

+

,

B2 B12 (00)(Y)) = •o](Y).

(17)

The Majorana constraint (14) therefore identifies (~4~/0J(X) ~

B2l($°J(x))*= O°S(x),

(18)

and one has

d~°~(x) = o°k(x)=-- ( oOk(x) )T B4y °

(19)

This allows us to bring the Yukawa couplings of ~ and ~2 into a manifestly symmetric form .=.~(1)-

"Ok

5 (1)

-,

Oj

~(2)-

"Ok

5 (2)

+l

Oj

Wk--@L(X)iY hk~,~b ( X ) $ L ( X ) + h c . ,

w k - - ~ L (X)ly hkj, C~ (X)ffL(X)+hc.,

t l k j , L g( 1 ) --4 2-

f

,,~(2)--kj, -- --x/2 ~

d

I

2

1/2

Yg2

2

+

T

+

tr (If/0k) B12b_Aboj

=

/,(1) - "jk,

--

(20)

h~, ,

11(2) + d 2Yg21/2 o"2 (OOk)T B~2b+,$~-- ,,jk, = hr~, .

(21)

C Wetterlch / Fermlon mass pre&cUons

470

Here we have used the fact that the two terms in (12) are hermitlan conjugate to each other We also have taken an explicit representation B2 = / . 2 = tr 2 In the above form it becomes apparent that the factor ty 5 in (20) can be absorbed by a suitable chlral phase rotation of ~b~(x) and we may drop it in the future We also note that the harmonics b+, and b_, are related by hermitlan conjugation (b+,) t = b-r.

(22)

This identifies the four-dimensional fields (~b+'(x)) * = t b - ; ( x ) .

(23)

For an evaluation of the Yukawa couplings hkj, we have to specify the normalization of g,~(y) and b±,(y). It is chosen so that the four-dimensional fields @~,R(X) and cb±'(x) have the standard normalization of their kinetic terms For the fermionic harmonics this implies (compare (8)) f rl 2, ~1/2~3/2[,l,+ "it,l,+ u. .Y ,~2 t., \WOk] tl.POj

f,..12, ~.L y

=

.1/2 3/2(.,--~*,,,52 , 1 k WO/~)

is: tl*O] -- 2t*k 3•

(24)

The factor ½ is due to the identification of c¢4~0°J and q,0i due to the Majorana constraint [8]. With this convention, the four-dlmensmnal kinetic term only contains independent left-handed fields

~ . . = ,qT~(x) y," o,¢,~(x).

(25)

Inserting the ansatz (6) and (7) into the six-dimensional kinetic term for the S0(12) gauge bosons integrated over internal coordinates ~

=-

d2y g~,/2o'2 Tr G ~ G

,

(26)

should give the standard kinetic term for the independent four-dimensional scalars 4~+'(x):

~,~,, = O"(~b+'(x))*OMb+'(x)

(27)

This gives the normalization f d2y gl/2o" Tr (b+,)*b+~ = 280 .

(28)

Here and in eq. (26) the trace is taken in the vector representation of SO(12) (for conventions see ref [5].) To proceed further, we will choose an explicit basis for the SO(12) spInor representations This is described in appendix A. In table 1 we give in this basis

C Wettench / Fermmn mass predlctmns

471

TABLE 1 Abehan quantum numbers for SO(12) spmor representation 32~* 321"

Tl2

7"34

T56

T78

T9,1o

TH 12"

ISL

ISP.

Ya-t.

uI u2 u3 v

-1/2 -1/2 -1/2 -1/2

1/2 1/2 1/2 1/2

-1/2 1/2 1/2 -1/2

1/2 -1/2 1/2 -1/2

1/2 1/2 -1/2 -1/2

1/2 1/2 1/2 1/2

1/2 1/2 1/2 1/2

0 0 0 0

1/3 1/3 1/3 -1

n/2+p/2 n/2+p/2 n/2+p/2 n/2-3p/2

dt d2 d3 e

1/2 1/2 1/2 1/2

-1/2 -1/2 -1/2 -1/2

-1/2 1/2 1/2 -1/2

1/2 -1/2 1/2 -1/2

1/2 1/2 -1/2 -1/2

1/2 1/2 1/2 1/2

-1/2 -1/2 -1/2 -1/2

0 0 0 0

1/3 1/3 1/2 -1

n/2+p/2 n/2+p/2 n/2+p/2 n/2-3p/2

u~ u~ u~ vc

1/2 1/2 1/2 1/2

1/2 1/2 1/2 1/2

1/2 -1/2 -1/2 1/2

-1/2 1/2 -1/2 1/2

-1/2 -1/2 1/2 1/2

1/2 1/2 1/2 1/2

0 0 0 0

-1/2 -1/2 -1/2 -1/2

-1/3 -1/3 -1/3 1

n/2-p/2+m n/2-p/2+m n/2-p/2+m n/2+3p/2+ m

d~ d~ d~ ec

-1/2 -1/2 -1/2 -1/2

-1/2 -1/2 -1/2 -1/2

1/2 -1/2 -1/2 1/2

-1/2 1/2 -1/2 1/2

-1/2 -1/2 1/2 1/2

1/2 1/2 1/2 1/2

0 0 0 0

1/2 1/2 1/2 1/2

-1/3' -1/3 -1/3 1

n/2-p/2-m n/2-p/2-m n/2-p/2-m n/2+3p/2-m

~ ~ ~ pc

-1/2 -1/2 -1/2 -1/2

-1/2 -1/2 -1/2 -1/2

-1/2 1/2 1/2 -1/2

1/2 -1/2 1/2 -1/2

1/2 1/2 -1/2 -1/2

-1/2 -1/2 -1/2 -1/2

0 0 0 0

1/2 1/2 1/2 1/2

1/3 1/3 1/3 -1

-n/2+p/2-m -n/2+p/2-m -n/2+p/2-m -n/2-3p/2- m

d~ d~ ~¢ e~

1/2 1/2 1/2 1/2

1/2 1/2 1/2 1/2

-1/2 1/2 1/2 -1/2

1/2 -1/2 1/2 -1/2

1/2 1/2 -1/2 -1/2

-1/2 -1/2 -1/2 -1/2

0 0 0 0

-1/2 -1/2 -1/2 -1/2

1/3 1/3 1/3 -1

-n/2+p/2+m -n/2+p/2+m -n/2+p/2+m -n/2-3p/2+m

fil ~2 da

1/2 1/2 1/2 1/2

-1/2 -1/2 -1/2 -1/2

1/2 -1/2 -1/2 1/2

-1/2 1/2 -1/2 1/2

-1/2 -1/2 1/2 1/2

-1/2 -1/2 -1/2 -1/2

-1/2 -1/2 -1/2 -1/2

0 0 0 0

-1/3 -1/3 -1/3 1

-n/2-p/2 -n/2-p/2 -n/2-p/2 -n/2+3p/2

d~ d2 ds

-1/2 -1/2 -1/2 -1/2

1/2 1/2 1/2 1/2

1/2 -1/2 -1/2 1/2

-1/2 1/2 -1/2 1/2

-1/2 -1/2 1/2 1/2

-1/2 -1/2 -1/2 -1/2

1/2 1/2 1/2 1/2

0 0 0 0

-1/3 -1/3 -1/3 1

-n/2-p/2 -n/a-p/2 -n/2-p/2 -n/2+3p/2

N*

* For the representation 322 reverse sign of all eigenvalues of T~t32 and of n m the column for N

the a b e h a n q u a n t u m n u m b e r s for the f e r m l o n s c o n t a i n e d in 32~. F o r the f e r m i o n s m 322 the sign o f the e i g e n v a l u e o f T~1.12 has to be reversed. I n s t e a d o f the g e n e r a l n o t a t i o n ~Jo,(Y) for the chlral s p l n o r h a r m o n i c s we often w d l use the m o r e intuitive n o t a t i o n T(y) for the h a r m o n i c s c o r r e s p o n d i n g to the z - l e p t o n , etc. M o r e generally, we use ej(y) for the h a r m o n i c s o f c h a r g e d l e p t o n s in the 16 o f SO(10), etc. W e are i n t e r e s t e d m t h e Y u k a w a c o u p l i n g s o f q u a r k s a n d l e p t o n s to c o l o u r singlet, SU(2)L d o u b l e t s c a l a r fields. T h e a d j o m t r e p r e s e n t a t i o n o f SO(12) c o n t a i n s a

472

C Wetterwh / Fermton mass predwtmns TABLE 2 SO(12) Quantum numbers for neutral components of Hlggs doublets I3L

I3R

q

Ya-L

SU(4)c × SU(2)L ×SU(2)R rep

SU(5) rep

HI /-/2

-½ 3

3 -3

1 1

0 0

(1, 2, 2) (1,2,2)

5 5

10 10

h~26 h~26

12 12

_12 _ !2

0 0

0 0

(15, 2, 2) (15, 2, 2)

45 5

126 126

h~2O 212o /~2o "120 ( h"120 i ) $ =h2

_ !2 ~1 -3 3

-~ 2 - ~1 3

0 0 0 0

0 0 0 0

(15,2,2) (15,2,2) (1, 2, 2) (1,2,2)

5,45 5,45 5, 45 5, 45

120 120 120 120

(h 12o 1 ) .= h

--3

SO(10) rep

complex 10 plet of SO(10) It has charge q = + 1 with respect to the abehan symmetry m SO(12) which commutes with SO(10). This 10 plet contains two candtdate Hlggs doublets. It is sufficient to calculate the Yukawa couphngs of the electrically neutral components of these Higgs fields which we denote by H1 and H2. Abellan quantum numbers of H1 and /-/2 are given m table 2 The scalars H~+, HE+ with q = 1 and positive two-dimensional hehctty (and their complex conjugates) couple to ordinary quarks and leptons. The scalars Hi_, H2_ with q = 1 and negative heliclty couple to the mtrror fermxons [5]. In our basis, the harmontcs for Hi+ and H2+ are (compare appendix A) b+nl, = ½t( T3,11 + IT3A2 - IT4,H + T 4 A 2 ) D I , ( y ) = IYI1Dl,(y), b+/42, = -½t( Ta,H + tT3,~2+ zTa,H - T4,12)D2,(y) =/4202,(y) •

(29)

Here D,(y) are complex scalar functions of y and t now labels the harmonic expansion for these parttcular fields. The normalization (28) xmphes f

d 2y g21/2 t r D l (:g2 ) ~ D l ( 2 ) s

=

So.

(30)

Using the exphcit form of B12 one finds in our basis B12/~1(2 ) = ~Zo@ 1 To® {To® T1® ~'3® `/-3-- T0® T1® T0® T0 "1- '1"3® '/'1 ® TO® 7"3:q7 T3 ® r l ® 3"3® T0}"

(31)

Inserting this expression mto (21) reduces the Yukawa couplmgs to integrals over scalar functions Constder first the Yukawa couplmgs of the chiral left-handed quarks and leptons from the splnor 02- As reqmred by the abehan quantum numbers [5], H1 couples only to u and v whereas /-/2 couples only to d and e One finds the following

C Wettench / Fermton mass predwnons

473

non-vamshlng Yukawa couplings h(2) c

2

f dEygl/Eo'2UkU~-Dh,

d2ys2 o ,-k~-j ,--~,,

h(~E)~,Hh=X/2~

h(2) o1/2 v 2 ~r /k- d~3c - zJ2t, y-~ dgd~HEt _-- .//'0 v ~. g f d : y S2

(2) 2 g f d 2Yg21/2 tr 2 e k- e jc- D2, hekeTnE,='f2

(32)

The quarks and leptons from ~b, couple to the hermitian conjugate of H,+ and HE+ [5]. Observing b _ - f f y r = ( b + n h ) * = H,D,,'* * , b--ASr = (b+H2,)* = H2D2, "* * ,

(33)

"* = ¼to® ro®{~'o® ~',® z3® z3 + ~'0® r , ® Zo® ~'0 BI2HI(2) :z[z,t-3~) ,/-, ~) ,/-0(~ ,/-3::~ T3 ~) -/-1~) ,/-3(~ '/'0} ,

(34)

one finds the Yukawa couplings

L(,)c It UkU2H2 r = ~ g f d 2 y g ~ / 2

+ J~+,-.. O"2 tikU LI2t

~,k~,~n2r=x/~ o~ f d2y ~s2 k vj¢+m, "'2,, h(,) . , / 2 o"2z,+ h(1) akdj n l

r = x/2 g f

he~e~,~,~=,/2 ~ f

d2y °1/2~2t/+'4C-~ ? ~ ts* vt 1 k5utj2 xJ , , ,

""12"u Y82~1/2~2~+~C+ l r ~~.kt~j * t ~ ' a.J,I

(35)

Among the four-dimensional fields ~b+'(x) we have discussed the electrically neutral components of weak doublet colour smglet fields H ' t ( x ) and H ~ ( x ) . Let us now assume that these fields acqmre vacuum expectatmn values (H',), (H~) breaking the weak SU(2)L X U(1)y group. This reduces a mass term for the quarks and leptons from (20)" ~(U)

-c]

k

M = UL (Mu).~kgL -b b.c. =

a~(Mu)jkUkL+h.c..

(36)

474

C Wettench

/

Fermlon mass predwtlons

and similar for d, v and e The mass matrices are given by

t (MD)jk = 2{~

h~!d~llg(H'l)* --~~ dCldkH2,\..2/i ,

(Me),k = 2

h~7~ffir(H,) + 2 h e ~ e k H 2 , ( H 2 ) !

(37)

.

Here the sum over j and k runs over all independent chiral spinors contained m ~b+(y) and ~ - ( y ) for the spinors ff~ and ~2, respectwely. The factor 2 arises since there are equal contributions from h~%~m, and h ~ m , , etc Vacuum expectation values (VEV) of Higgs doublets also give mass to the W and Z bosons. In general, the neutral component of the low energy scalar doublet 4, will be a mixture of different H'j, H ; and possibly even other doublets contained in a more complete model. We may denote (H'I) = ~/,(4,),

(H~) = y-~(~b),

(38)

,2+ 21~,1 21~;1=~1 i !

(39)

with

Equality holds if no other doublets are present In a realistic model one needs (6) = 174 GeV,

(40)

M~v= ½g2I(q~)[ 2 .

(41)

and the mass of the W boson is

The weak gauge coupling g can be calculated at the compactification scale Mc from the six-dimensional kinetic term for the gauge fields (26)

g-2(M¢) = ~-2 f d2yg~/2= ~-2V2"

(42)

Once the Yuka~ra couplings (32) and (35) are calculated for a specific ground state, we can establish relations among fermion masses. If in addition some coefficients Y'~(2) are known, relations between the fermlon masses and the W boson mass can be established If the low energy doublet 4) is uniquely determined (all 7 coefficients known), the total contribution of six-dimensional SO(12) gauge fields to the mass matrices of quarks and leptons is calculable In the next section we will calculate

C Wettench / Fermlon mass predtcttons

475

the Yukawa couplings for some simple "trial ground states" and compare the results with the observed spectrum of fermlon masses.

4. Fermion m a s s relations

So far our discussion did not specify the properties of the SU(3)c × SU(2)L × U( 1)v symmetric approximate ground state except the feature that it should lead to chiral quarks and leptons. We only have used algebraic properties related to sixdimensional Lorentz symmetry and SO(12) gauge lnvarlance. Actual calculation of the Yukawa couplings h involves knowledge of the ground state g e o m e t r y - t h e functions tr(y) and g ~ / 2 ( y ) _ and of the harmonics for chiral fermions and twodimensional v e c t o r s / ~ . As an example we will evaluate the Yukawa couphngs on a geometry M4× S 2 with gauge fields in a monopole configuration on S 2 The most general SU(3)c × S U ( 2 ) L × U ( 1 ) v symmetric solution of this type is characterized [5] by three integers n, m and p with n + p even The number of chlral fermlon generations is given by n. These solutions have at least a symmetry SU(3)c x SU(2)L × U(1)R ×U(1)B-L xU(1)q ×SU(2)c. The abehan symmetries U(1)R and U(1)B-L are subgroups of SO(10) with generators I3R and B - L. The group U(1)q is the subgroup of SO(12) commuting with SO(10) and SU(2)c Is related to lsometries acting on S 2. Both U(1)q and SU(2)c differentiate between fermions of different generations and we may call them generation symmetries in this sense This does not imply that all fermions within a standard generation have the same SU(2)G × U(1)q transformation properties. Indeed this is not the case whenever m or p differ from zero. These m o n o p o l e solutions have not all required properties of the SU(3)c × SU(2)L × U(1)y symmetric approximate ground state since their symmetry is larger than SU(3)c × SU(2) ×U(1)v. In a more realistic model, six-dimensional scalar fields have to be introduced [4] in order to break U ( 1 ) R × U ( 1 ) B - L × U ( 1 ) q Z S U ( 2 ) G to U(1)v Nevertheless, some of the qualitative features of the structure of the fermion mass matrices can be understood by evaluating Yukawa couplings on these solutions We also will discuss the modifications expected for a more reahstic approximate ground state For the m o n o p o l e solution on S 2 we use polar coordinates 0 and ~b with (r(0, q~) = 1,

(43)

g '/2( O, qb ) = L2o sin 0 V2 = 47rL2o,

(44)

g2 = g2/47.rL2o,

(45)

where the radius Lo of S 2 can be calculated from the parameters of the sixdimensional action [5] The generalized speherical harmonics in presence of the monopole field are characterized by a hellcity A One has

u~-( O, ~b ) = ,/~ D ~ u )( O, ~b) ,

(46)

476

C Wettench / Fermzon mass pre&ctrons 3.~ =½-½N~,

(47)

and similar for other quarks and leptons Properties of the harmonics DI~ ~ are displayed in appendix B In table 1 we have listed the values of N for the different fields contained in ~b1. The values of N for ~b2 are obtained by reversing the sign of n. For the representation 321, chiral harmonics u+(0, ~b) etc., exist [5] whenever N > 0 for

/=13.1, k = - l , - l + 1,

,l

(48)

Similarly, the "hehcity" 3. for harmonics u-(O, oh) is 3.v-

t49)

21 ½ N u

The chlral harmomcs u-(0, ~b) in 322 are obtained for N < 0, 1 = IAI The generalized spherical harmonics for the Hlggs scalars are D I , ( 0 , t~) = D~l)(0, t~), D2,(O, ~b) = DI~2)(O, 0 ) ,

(50)

Xl = 1-½n +½m, 3.2 = 1 --~n 1 --~m 1

(51)

Using the integral over three-spherical harmonics in appendix B, we can express the Yukawa couplings in terms of the four-dimensional gauge coupling g and the Wlgner 3-./symbols: t,(2) ~

"UkUjHll

h(l)

, ~ . , . 2c - -

= g(½(2Au + 1)(2Au~+ 1)(_2kl + 1))1/z

(::

A,,~ -,3. l "~(3.u 3.uc

3.1 ] k l k

3.u¢ 13

I1

'

- ( - 1)"-a2g(l(-23.. + 1)(-23..c + 1)(-23.2+ 1)) '/2

~ --

x

3..

3..~

..

-3.2}\ Ik

/j

-3.2) -I,

(52)

Here I Is the third component of SU(2)c angular m o m e n t u m and the requirement Ik + Ij + I, = 0 (Ik + / j --/, = 0) for non-vanishing h t2) (h m) reflects conservation of L The scalars H1 (/-/2) have non-vanishing Yukawa couphngs to quarks and leptons whenever 3.1 <~0 (3.2 ~<0) In this case conservation of hehclty (h~ + h.¢+ A, = 0 for h ~2), h. + huo-3.2 = 0 for h (')) is automatically fulfilled for the couphngs (32) and (35). Conservation of total angular m o m e n t u m reqmres that only the lowest modes in the harmonic expansion of H1 (HE) with l =-3.1 (l =-3.2) have non-vanishing Yukawa couphngs to the chiral quarks and leptons [5]

C

477

Wettench / Fermmn mass predzcnons

Let us investigate the SU(5) symmetric solution characterized b y m = 1 The SU(2)c

×U(1)q

quantum

numbers

of the chlral fermions

n -- 3 , p = 1

and

are [5]

uL) 21/2+1_1/2 , dE

-

UL. 2 1 / 2 + _1-1/2 d~"

3_-1/2,

c /'PL

_41/2

=c. //L

11/2

e~. _21/2+ 1_1/2 . (We have indicated The first column ~b2.) I n t a b l e various

the dimension comes

from

of the SU(2)G representation

the harmonic

3 we give the quantum

fermion

bilinears.

(53)

expansion

number

with q as a subscript.

o f ~bl a n d t h e s e c o n d

I f o r t h e f i e l d s t', c ' . . .

(The prime indicates

from

as well as

that these states are not necessarily

TABLE 3 SU(2)G x U(1 )q q u a n t u m numbers for chlral fermlons (n = 3, p = m = 1)

21+1 I q

21+1 I q

21+1 I q

t'

t ~'

c'

c c'

1 0 _½

1 0 _1

2 1 !

2 1 !

2

f' lc'

C' Cc'

1 o -1

3 -1 1

b , b ~,

s , s ~,

3 1 --1

4,2 ~ 0

¢%.c, 21+1 I q

2

3 1 -1

/z,p c, 4,2 ½ 0

u'

u ~'

b'

b ¢'

2 !2

1 0

3 1

2 ½ !

s'

s c'

d'

d ~'

r'

z c'

/z'

/x ~'

2 ½

3 0

2 -½

3 -1

3 1

1 0 -~

3 0 -~

2 ~ ~

-

2

Ut U c'

3 1 1

t~ cc '

tt uc '

uP tct

c' uc'

U' Cct

2 ½ 0

3,1 o 1

3,1 o 1

d , b ~,

s , d ~,

d , s c,

3 -1 --1

4,2 ½ 0

4,2 -½ 0

4,2 -½ 0

/.L,~.c,

~.,ec,

e%.C,

pt,e c,

e,pC,

3 0 -1

4,2 ½ 0

3 -1 -1

4,2 -½ 0

4,2 -½ 0

2

2

-½

0

0

2 ~ 0

d , d ~,

b , s ~,

s , b ¢,

b , d ¢,

4 -3 0

3 0 --1

e,e c,

.r,p c, 4 3 0

4 3 0

e ~'

3 2 -1 -½ ½

c' tC'

-½

4 _3 0

e'

C

478

Wetterlch / Fermmn

mass

pred~cnons

TABLE 4 SU(2)G x U(1)q

quantum

numbers

wxth Yukawa couphngs

for scalars

(n=3,p=m=l)

21+1 I q

l+½q

H~

H~

H°

H;

1 0 -1

3 1 1

3 0 1

3 -1 1

-½

I h *1(2,~)

2/+1

t

q

_½

h2(2,½)

2

2

2

0

0

0

½

-~

h *1(4,~) t

h *1(4,--)),

h *1(4,--~) 3

4

~1

-½

h2(2,-1)

2

_½

~

½

h*I (4,~.)3 21+1

~

h *1(2,-~)1

_½

1+½q

3

0

4

4

4

t

-5

-½

½

3

q

0

0

0

0

h.2(4,_~)

I+½q

_3

_½

½

3

h2(4 3)

h2(4,21)

h2(4 _½)

2/+1 I q

4 ~ 0

4 ~3 0

4 _1 0

4 _3_2 0

l+½q

~

½

-½

-5

eigenstates o f the mass matrix ) The s a m e quantum numbers for Hlggs scalars can be found in table 4. In this basis, the fermlon m a s s matrices n o r m a l i z e d to the mass o f the W b o s o n is found tt

Ct

Ur

0

0

Mu =

4~(T~-)*

44~(T°) *

Mw

4x/~ (y°) * 4~/~-(T2-)*] uc'

2!1

b'

-~--.---2T ° 2T~

s'

c °'

(54)

d'

o o

I lc'

9

o / d ~'

(55)

C Wenerwh / Fermlon mass pre&cttons ~"

I~'

ME ~ 2 0 ~ 2 - - 2 Y ° Mw0

\o

0

479

e'

2 y ~ -ct 0 /p. ~'

(56)

0 / e c'

The three fermlon mass matrices are easily diagonalized. For generic Yl, Y2-, 3,o and y~ the unitary transformations ( t , c , u ) = ( t ' , c ' , u ' ) U ~ and ( b , s , d ) = (b', s' d')Uro are different and a non-zero Cabibbo angle between the first two generations is induced We note that two charged leptons and two charge - 1 quarks always remain massless. This is due [5] to the discrete symmetry F : ~bl --> ~b~,02 -->-~b2 Let us suppose that the low-energy Higgs doublet is essentially given by H~ with a small admixture of H2. In this limit '~1 ~---"1 ,

To=

Y2+ = 0 ,

(57)

we have several relations for the fermion masses (at the scale Lo t) mt= 2 M w ,

(58)

mb = my = a M w ,

(59)

me = w/~ rob,

(60)

ms = m~ = mu = md = me = O.

(61)

Also all mixing angles vanish and there is no CP-vlolation For a ~ ~0 the quarks and leptons divide into three groups: only the top quark has a mass of order Mw The next group consists of b, c and r with masses of a few GeV. The third group consists of s, tz, u, d and e which are massless at this level. These qualitative features seem to reflect the structure of the observed fermion mass matrices The quantitative prediction mb m~ is in satisfactory agreement with experiment However, the charm quark mass me = x/~ mb is tOO high by a factor of three We should remember, however, that the monopole solutions on S2 are not a satisfactory approximate ground state. In a more realistic approximate ground state, the rotation symmetry on the sphere will be broken. Unless protected by some symmetry, the quantitative mass relations are expected to change. This is certainly the case for relations (58) and (60). In contrast, corrections to m b my will be proportional to the scale where both SU(5) and SU(4)c are spontaneously broken and may be suppressed [2]. What are the conditions for the qualitative features like mt >>rob, me, m, >>ms, m~,, rod, mu, me to persist? This brings us to the question why certain fermlon masses are much smaller than others, the topic of the next section. =

=

480

C Wettench / Fermlon mass pre&ctmns 5. The hierarchy of fermion masses

The ratio between the electron mass and the mass of the tau-lepton is about 3 x 10-4 Similar small ratios are observed for mu/mt or md/mb Ratios like ms~rob or m . / m ~ are of order ~ - ~ We call this the hierarchy of fermion masses How are such small numbers as 10 -4 to be understood in a higher dimensional framework9 In four-dimensional grand unified theories, Yukawa couplings are free parameters and we may take them as small as we want In higher dimensional theories, however, the four-&mensional Yukawa couplings become calculable. As we have seen in the preceding section, they all tend to be of the same order as the gauge coupling g unless they vamsh due to some symmetries If both electron and tau-lepton have non-vanishing Yukawa couplings to the same Higgs scalars, there seems to be little chance to obtain me<< m,. (Here we work In a basis where the Hlggs scalars are classified according to some symmetries which will be spontaneously broken like SU(5) × U ( 1 ) x S U ( 2 ) × U ( 1 ) in the preceding section The low energy weak Higgs doublet will be a hnear combination of such scalars ) Things are different if the Higgs scalar, whose VEV gives the leading contrabution to m , does not couple to muons or electrons. Muon and electron must then get their masses from other Higgs scalars The admixture of these scalars to the low energy doublet may be small (y, << 1) and a fermion mass hierarchy could be explained At first sight this seems only to shift the problem from explaining small Yukawa couplings to an explanation of small mlxmgs y,. The second problem, however, is easier to treat. If two scalars have different quantum numbers with respect to some spontaneously broken symmetry G, their mixing must be proportional to some power of Mc, the scale where G is spontaneously broken. Typical mixing coefficients 3', are then of order (MGLo) ~ with /~/some model-dependent positive integer and Lo I the scale of spontaneous compactlfiCatlon. Even a moderate "fine structure" in the scales relevant for compactificatlon M c ~ - ~ L o ~, say) may lead to rather small T, and therefore rather small fermion mass ratios. In four-dimensional grand unified theories the fermion mass hierarchy is usually related to the generation problem: the first generation has the lightest masses, the masses of the second generation are in an intermediate range and the third generation consists of the heavy quarks and leptons This pattern is underlined by the smallness of the mixing angles between different generations. Looking a bit closer, however, the hierarchy of masses does not follow exactly the generation pattern. The fermlon masses rather divide into four groups ?nt , /'~b~ m c , /'n~., ms, mp.,

rod, mu, me

(62)

There is roughly a factor ~o between the overall mass scales of the different groups

C Wettench / Fermzon mass pred~ctzons

481

This suggests that a s m a l l q u a n t i t y MGLo a p p e a r s with &fferent p o w e r s / ( / i n the m i x i n g coefficients %. As we have seen in the p r e v i o u s section, there is no r e a s o n w h y the f e r m l o n mass h i e r a r c h y s h o u l d f o l l o w e x a c t l y the g e n e r a t i o n pattern. T h e structure o f the f e r m i o n m a s s matrices IS r a t h e r d i c t a t e d b y the SU(2)G×U(1)q q u a n t u m n u m b e r s . In g e n e r a l , the v a r i o u s q u a r k s a n d l e p t o n s within a g e n e r a t i o n have different t r a n s f o r m a t i o n p r o p e r t i e s with r e s p e c t to the g e n e r a t i o n g r o u p SU(2)G x U(1)q Let us n o w c o m e b a c k to o u r m o d e l . T h e r e are three t y p e s o f s y m m e t r i e s w h i c h can p r e v e n t m i x l n g s b e t w e e n different Hlggs scalars in the limit w h e r e t h e y r e m a i n u n b r o k e n . T h e first is the discrete s y m m e t r y F t r a n s f o r m i n g ~,tI "> ~gl, []/2-'> --[]/2 This tells us t h a t the scalars c o n t a i n e d in the h a r m o n i c e x p a n s i o n o f the s i x - d i m e n s i o n a l g a u g e b o s o n s can o n l y i n d u c e f e r m i o n m a s s e s c o n s i s t e n t with u n b r o k e n S U ( 3 ) c × U(1)em b y p a i r i n g a q u a r k u w i t h i n ~b~ with an a n t i q u a r k u e also w i t h i n q'l- A l t h o u g h for b o t h s p l n o r s ~bl a n d ~2 t o g e t h e r the n u m b e r o f u equals the n u m b e r o f u e, there m a y be an a s y m m e t r y b e t w e e n q u a r k s a n d a n t l q u a r k s within ~bl o r ~02 a l o n e T h u s F s y m m e t r y can f o r b i d m a s s terms, a n d in the a b o v e e x a m p l e with n = 3, m = p -- 1, this is the case for s,/~, d a n d e. The s e c o n d s y m m e t r y is SU(2)G ×U(1)q g e n e r a t i o n s y m m e t r y o r a suitable s u b g r o u p o f it F o r e x a m p l e , in the limit o f u n b r o k e n S U ( 2 ) c the scalars H ~ a n d H1 c a n n o t mix F i n a l l y , there is the s y m m e t r y U(1)R w h i c h p r e v e n t s m i x i n g b e t w e e n H~ a n d /-/2 while a l l o w i n g mixlngs b e t w e e n H * a n d //2 Before g o i n g on with a d i s c u s s i o n h o w these different s y m m e t r i e s m a y be b r o k e n b y the i n t r o d u c t i o n o f a s i x - d i m e n s i o n a l s c a l a r field [4], let us briefly investigate the structure o f the m a s s m a t r i c e s for o t h e r values o f n, p a n d m. This will illustrate t h a t a s e m i - r e a l i s t i c m a s s p a t t e r n as in the p r e v i o u s section (mt >> rob, me, mr >> ms, m~,, mo, mu, me) Is far f r o m trivial W e start with three g e n e r a t i o n m o d e l s (n = 3 ) . using the g e n e r a l results o f ref. [5] for the S U ( 2 ) ~ × U(1)q t r a n s f o r m a t i o n p r o p e r t i e s o f chlral f e r m i o n s for v a r i o u s n, p a n d m, one finds that for p/> 3 the s y m m e t r y F f o r b i d s m a s s terms for all charge - I q u a r k s or for all charge 2 q u a r k s d e p e n d i n g on m i> 0 or m <~ 0 I f we w a n t that quarks o t h e r t h a n t o p a n d c h a r m a c q u i r e a mass c o n s i s t e n t with F s y m m e t r y , we r e m a i n with n = 3, p = 1 N o w we find that for m/> 2 the field HI has no Y u k a w a c o u p l i n g s to chiral quarks. Both mt a n d mb are i n d u c e d b y the s a m e field a n d t h e r e f o r e m the s a m e o r d e r o f m a g n i t u d e . F o r m ~< 0 the s y m m e t r y F f o r b i d s m a s s e s for all c h a r g e d l e p t o n s a n d r e l a t i o n s h k e m b = m r are e x c l u d e d . T h e o n l y r e m a i n i n g case with t h r e e g e n e r a t i o n s is the e x a m p l e o f the p r e w o u s section with n -- 3, p = m = 1. F o r f o u r - g e n e r a t i o n m o d e l s (n = 4), the case p I> 4 l e a d s a g a i n to a s i t u a t i o n w h e r e either all u p - t y p e q u a r k s o r all d o w n - t y p e q u a r k s r e m a i n massless d u e to F s y m m e t r y F o r p = 0, F s y m m e t r y implies m c o m p l e t e massless g e n e r a t i o n s H o w e v e r , t h e r e IS an a d d i t i o n a l discrete s y m m e t r y [5] for p = 0 which Implies that the m a s s e s o f charge ~ a n d charge - ~1 quarks are e q u a l E v e n if this s y m m e t r y is s p o n t a n e o u s l y b r o k e n b y s i x - d i m e n s i o n a l s c a l a r fields, a h i e r a r c h y mt >> mb IS difficult to u n d e r s t a n d T h e r e m a y be an e x c e p t i o n for n = 4, p = 0, m = 3 w h e r e all q u a r k s a n d l e p t o n s o f the first three g e n e r a t i o n s m u s t

C Wettench / Fermlon mass predlctmns

482

acquire their masses from effects violating F symmetry What remains is the case n = 4, p = 2: for rn <~ 1, no mass term is allowed for charged leptons. For m t> 3, only /42 has Yukawa couplings to chiral fermlons. Only three up-type quarks, one down-type quark and between two and four charged leptons can acquire a mass consistent with F symmetry. There is no reason to expect relations like mb = m,. We are left with the SU(5) symmetric solution n = 4 , m = p = 2 This leads to a similar structure of the mass matrix as the three-generation example in sect. 4. However, mass relations now mainly involve fermlons of the unobserved fourth generation. We conclude that the qualitative structure of the mass matrix is a very restrictive criterion for model building, leaving only very few possibilities in the present model. In the remainder of this section we discuss some qualitative features of the breaking of F symmetry and S U ( 2 ) G X U ( 1 ) q symmetry for the solution n = 3 , p = m = 1 We find that In addition to the mass splitting (58)-(61), there is a mechanism for a mass split between ms, m+, and rod, mu, me. T h i s gives us some hope to obtain the mass pattern (62). The best candidates for a breaking of F symmetry are six-dimensional scalar fields with Yukawa couplings to the splnors ~bl and q'2. Possible representations are the vector, third and fifth rank antisymmetrIc tensor representations of SO(12) with dimension 12, 220 and 792, respectively. At most, three additional independent Yukawa couplings are possible in our sixdimensional model. The components of these fields with possible Yukawa couplings to the four-dimensional chiral quarks and leptons (not mirrors) have the followIngl SO(10) × U( 1)q transformation properties:

{12} = (10,0), {220} D (120, 0 ) + (10, 0), {792} ~ (126+ 126, 0) + (120, 0).

(63):

All three fields have U(1)q charge q = 0 Comparison with table 3 shows that Yukawa' couplings are only possible to the blllnears t'c c', c't c', t'u ~', u't ~', s's ~', d'd ~', s'b ~', d'b ~', s'd ~', d's ~', ~'/z ~', e'e ~', r'/z c', z'e ~',/x'e c' and e'tz ¢' These additional terms in the mass matrices can now induce masses for all quarks and leptons. Let us restrict our discussion to the scalar in the 792-dimensional representation This is the minimal setting where the gauge group can be broken to SU(3)c × SU(2)L × U(1)v for the approximate ground state. We add a scalar piece to the six-dimensional action [4]: t"

S~

J d6~ ~ / 2 { _ ½D~ ~D~ ~ + ( f ~ , ~ 2 + h c ) + V(~b)},

(64)

with 4) In the 792-dlmenslonal representation, the Yukawa coupling f a new free parameter and the potential V(¢~) so far unspecified. With respect to SO(10) × U(1)q,

C Wettench / Fermton mass pre&ctmns

483

{792}-> (126+ 126, 0) + (210, ± 1) + (120, 0).

(65)

792 decomposes into

The 210 contains three SU(3)c x SU(2)L × U ( 1 ) y singlets transforming as (1, 1, 1), (15, 1, 1) and (15, 1, 3) under SU(4)c xSU(2)L xSU(2)R or 1, 24, 75 under SU(5). The SU(3)c x S U ( 2 ) L X U ( 1 ) y slnglet in 126 transforms as (10, 1, 3) under SU(4)c x SU(2)L X SU(2)R and as a sInglet under SU(5). These are the fields whose vacuum expectation values can induce the spontaneous symmetry breaking SU(5) x 1](1) x SU(2)G × U(1)q") SU(3)c x SU(2)L x U(1)y

(66)

We assume that the approximate ground state is in the same SU(3)c x SU(2)L X U(1)y deformation class as the monopole solution with n - - 3 , r n - - p = 1. We can then use harmonic expansion on the monopole solution on S2 to gain some insight into the qualitative structure of contributions to the mass matrices induced by the Yukawa couphng f. The 126 contains two colour singlet SU(2)L doublets belonging to (15, 2, 2) of SU(4)c x SU(2)L x SU(2)R whose electrically neutral components are denoted by hi126 and h 126. The 120 contains a (15, 2, 2) with electrically neutral i,120 and h212°= (hl~2°)* and a (1,2,2) with neutral fields /~2o and colour singlets ,,1 /~2o= (h~12o), We give various quantum numbers of these fields in table 2. The hehcity for scalars m a monopole configuration is A = -½N

(67)

The harmonic expansion for all these fields contains SU(2)G angular momentum l = ½, 3, I" • " • Comparing with the quantum numbers of fermion bllinears in table 3 we find that only the doublets and quartets can have non-vanishing Yukawa couplings with the chiral quarks and leptons The SU(2)G X U(1)q quantum numbers of all scalars with Yukawa couplings are listed in table 4 Comparison with table 3 together with I3R conservation tells us the allowed Yukawa couplings for the various fields. At this stage, predictions on the fermion mass matrices appear to be rather hopeless unless we have some information on the mixing coefficients 3,. These coefficients are determined to a large extent by the vacuum expectation values of the SU(3)c × S U ( 2 ) L × U ( 1 ) y singlet fields In 210 and 126 Let us assume that the symmetry breaking (66) takes place in two steps. The first step, mediated by a vacuum expectation value of 210 with definite SU(2)G spin I, breaks SU(5) × l~l(1) × SU(2)G × U(1)q ~ SU(3)c × SU(E)L × U(1)R × U(1) ~-L × U(1)~.

(68)

The symmetry U(1)RXU(1)~ will then restrict possible mixings of scalars. The S U ( 3 ) c X S U ( 2 ) L × U ( 1 ) y singlet fields in 210 have all q = ± l , I3R=0, YB_L=0, N = ±3, A = =t=3. SU(2)G angular momentum I is therefore half integer and starts with 1=~-. Assume the vacuum expectation value responsible for (68) has q - - 1 ,

484

C Wetterlch/ Fermtonmasspredlcnons

I =--½. The charge 4 = I +½q,

(69)

is then conserved. The symmetry F transforms ~b -* -~b and is broken by this vacuum expectation value. The monopole solutions also have a conserved CP symmetry [5]. This is broken by (210)*# (210) and CP-violatlng phases are expected in the mass matrices The symmetries U(1)R × U(1)fi imply that the only Higgs fields which can mix at this stage with H~ must have I3R =--½, I + ½ q = --½ These fields are H~-, h2 ( I = - ½ ) and h~ ( I =½)* In this approximation the mass matrices have the form

a32(h)i) a23(oh)az2(H2)o

/a33(H1) Mu= /

MD =

b22(h)* 0

ME =

c22(h)*

\ c13(h)*

'

(70)

,

(71 )

(72)

0

Here h is an appropriate combination of fields h~ and h* Once the approximate ground state and the corresponding harmonics for chiral fermions and scalar fields are known, the coefficients a33 etc, can be calculated The scale of the coefficients in front of (h) is given by the six-dimensional Yukawa coupling f and is therefore a free parameter In the limit of unbroken U(1)R ×U(1)~ symmetry, one finds md = rnu = me ----0.

(73)

In addition to the masses discussed in the previous section, we now have also non-vanishing ms and m~, It seems not to be difficult to arrange for a scale of a few hundred MeV for these masses using the free parameter f. We also note the appearance of non-vamshing mixing angles from the dlagonalization of ME and MD. Finally, the symmetry U(1)RXU(1)B_L×U(1)~ must be broken to U(1)v This can be done by fields in 210 with q = 1, I # -½ and by the SU(5) slnglet in 126 The scale of this breaking should not be too low, otherwise the masses mu, md and me would come out too small and masses for left-handed neutrinos [23] would be too large.

6. Conclusions

We have discussed the general method how to calculate quark and lepton masses in higher dimensional theories. This method was illustrated in a six-dimensional model with gauge group SO(12). In this model we have calculated the fermion

C Wetterwh/ Ferm~onmass predwtzons

485

mass matrices for all SU(3)c × SU(2)L × U(1)v symmetric monopole solutions on an internal sphere S 2. We expect the qualitative features of these mass matrices to be valid for a more realistic SU(3)c × SU(2)L × U ( l ) v symmetric approximate ground state Already at this qualitative level this model is rather predictive. Most of the monopole solutions lead to a qualitatively unacceptable fermlon mass spectrum and should be discarded. Only a few solutions remain in agreement with the structure of fermion mass matrices. This shows how the observed hierarchy of fermlon masses becomes an important restriction for reahstic higher dimensional models. We have proposed a semi-realistic six-dimensional SO(12) model with scalars in the fifth rank antisymmetric tensor representation of SO(12). It includes fields with appropriate quantum numbers for a realistic spontaneous symmetry breaking. Interesting fermlon mass matrices are obtained for solutions in the same S U ( 3 ) c × SU(2)L × U(1)v-deformation class as the SU(5) symmetric monopole solution with n = 3, m = p = 1. These solutions lead to three generations of quarks and leptons In the hmlt of unbroken U(1)R X U(1)~ symmetry, three types of scalar fields couple to the chlral fermlons The scalar H1 only couples to the top quark, H f contributes to rob, mc and m~ whereas h gives masses to s and ~ and induces non-zero mixing angles In this limit the up and down quark and the electron remain massless. The first generation will finally get a mass from effects breaking U(1)R )>(H~-) >>(h) can be obtained, leading to the mass pattern mt >>rob, me, m~ >>ms, m~ >>rod, mu, me. In conclusion, we propose that dltierences in symmetry breaking scales around the compactification scale lead to a suppression of mixings between different scalar doublets. This in turn implies small ratios between fermlon masses In other words, the fine structure of scales for spontaneous compactificatlon is responsible for the observed structure in the fermion mass matrices. If this program succeeds, the particular chemistry of our world, which crucially depends on the masses mu, rnd and me, would finally be traced back to some fine structure of scales around 1017GeV The author would like to thank M. Gell-Mann and the Theory Group at C A L T E C H , where part of this work was performed, for their hospitality.

Appendix A SO(12) SPINOR RESPRESENTATIONS The Clifford algebra in twelve dimensions is obtained from the twelve anticommuting Dlrac matrices FA. {FA, FB} =

2"OAB

=

--28AB,

A, B = 1 • "" 12.

(A 1)

486

C Wettertch / Fermzon mass predtctzons

The FA are 64 × 64 matrices and their commutators form the geneators ZAB of SO(12) m the reduoble Dirac representation .,Y,Aa =

-

¼[FA, Fa]

(A.2)

We will use hermitlan generators (A.3)

TAB = -- I~AB ,

which obey the usual SO(12) commutation relations [ TAB , TCD ] = I S A c T B D -- I S A D T B c + lt~BDTAc -- lt~BcTAD.

(A.4)

The two lnequivalent irreducible pseudo-real spmor representations of SO(12) have d~menslon 32 and are obtained from the Dirac representation ~0, using the standard Weyl projection operators. qJl = ~b(320:½(1 +/~,2)qJ, q/2 = ~b(322) = ½(1 -/~,2) q/, F12 = - F I F 2

• " " -F12,

F2z = 1,

{F,2, FA} = 0

(A.5)

(A.6)

In our conventions, the Dirac matrices are anti-hermitian and r12 is hermitian F*A= - F a ,

/~'2

=/~,2-

(A.7)

We next gwe an explicit representation for the matrices FA m a basis which is convement for the descriptions of quarks and leptons. With respect to the subgroup SO(10) × U(1)q, the SO(12) spinor representations decompose 321 --) 161/2+ 16_1/2, 322"-) 16-1/2+ 1---61/2

(A.8)

The subscript indicates the abehan charge q. We write the 64-component Dwac spmor ~0 in the form ~0(161/2)t (~(321)~ = ~O= \q/(32~)]

(A9)

~b(i"6-1/2)

~b(16_~/2) ' q/(161/2

with c

c

c

c

c

c

c

~(16) T= (ul, u2, u3, v, dl, d2, d3, e, ul, u2, u3, v , d~, d2, d3, eC),

~ / ( ~ ) T (1~, 1~, /~, ~c, d~, dE, d~, ~c, al ' a2 ' a3 ' ~, Jl, J2, J3, ~)-

(A. 10)

The fields in 1-6 have the quantum numbers of mirror particles. In the basis (A 9)

487

C Wettertch / Fermzon mass pre&ctmns

and (A.10) the Dlrac matrices are given in terms of direct products of Pauh matrices z°=(10

~)'

rl=(~

10)'

z2=(0/

O1)'

r3=(10

?1)'

(All)

as

E 1= fro® 7-o®7.z® r3® 7h®'rl F,,= rro® "to® 'rl ® "to® 'rl ® "rl F 3 = iTo®ro®r3®ro®,rl®rl

~ro®ro® To®ro® r2® ~ it2® r l ® r2®'r2® rl®'r~

/'6= lrl®r2®~2®r~®rl®zl

F.,= It3 ®

T2 ® 'F2® T1 ® 'rl ® "/'1

_G= iro®'r2® rE® r2® ~h® rl G= Clo

1"/'2® "/'0® "/'2® "/'1® "/'1® "/'1 l'r2 ® "r3® "/'2® ~'2® "/'1® "/"1

F1~ = ZZo® ro®'ro® ro® r3® zl

r~2 = ~ro@~'o@to@ ~'o@to@ z2

(A 12)

The six generators of the Cartan sub-algebra of SO(12) read T12 = -- ½to® t o ® r3 ® ~'3® t o ® To, T34 --- ITo ® -/-0® '/'3® '/'0® T3 ® To ,

7"56= - ½z3®Zs® ~'o®~'s® to® Zo, T78 = ½r3® To® t o ® r 3 ® 7"0® 7-0, T9,10 --- l'l'o (~) "/'3® 'to® 'r3® ~'0® Zo, 1 Tl 1,12 : i"/'o® '/'0® To® '/'0® "/'3® "/'3.

(A. 13)

In our basis the standard assignment of left-handed and right-handed weak isospin and of the B - L charge is given by I3L = _1( 712 -- r34), I3R

=

--

½(T12+ T34),

Yn-L =2(7"56+ TTS+ T9,1o) •

(A.14)

The abehan quantum numbers of the representation 321 are shown in table 1 For the representation 322, the sign of q = T~,~2 has to be reversed. Our basis is adapted

488

C Wettench / Fermton mass predtcttons

to the s u b g r o u p SO(6) x S O ( 4 ) = S U ( 4 ) c X S U ( 2 ) L × S U ( 2 ) R o f SO(10). The generators o f SO(10) are given by TmN, M, N = 1 • • • 10, the SU(2)L X SU(2)R generators are TMN; M, N = 1 • • • 4 and SU(4)c is obtained for TMN, M, N = 5 • • • 10. The projection operators for representations o f SU(2)L x SU(2)R, SU(4)c, SO(10) and U(1)G respectwely are given slmdar to (A.5) by

F4 = - F , F2F3F4 = -~'0® to® to® r3® r3® to, F 6 = - t F s F 6 • • • Flo = - - r o ® ~ ' o ® % ® r 3 ® t o ® ro,

Flo = F4F6= to® ~0® ~0® to® r3® ~0, F2 = zFnF~2= ro®ro®zo®Zo®r3®z3, (A 15)

/~12 = /~10/~2 = '/'0® "TO® "/'0® "/'0® 7"0® ~'3"

We finally define the SO(12) charge conjugatxon m a m x

B12 by

( r~)* = -B,2FABT~ , B*12B~2= - 1 .

(A.16)

In our convention we also have ( r a ) T = B,2rAB?~,

( A 17)

and = -B,~.

(A.18)

= fro® to® Zo® zl® ~2® r3.

(A.19)

B,= = B*~ = - B ; ~ = - ~ h

One finds

Bl2 = F1F4FsF8FlOF12

Appendix B

GENERALIZED SPHERICAL HARMONICS In this a p p e n d i x we collect the formulae for generahzed spherical h a r m o m c s used m this paper. We have a d a p t e d our conventions to the b o o k o f E d m o n d s [24]. The h a r m o n i c expansion o f a field with helicity A on a sphere is 0~A)( 0, & ) = E a,kD~)( O, &),

B.1)

lk

with "--'tkr~A)defined by

-,O6D~ ) = {

-

sln~

rn2

ffiD~ ) = (k - A )D~ ) ,

1-c°s0)

(B.2)

"A'

a°(sm °~°) + s~--~--0+zA(x + r~) ---sin 2 0 J~D'tk'=l(l+l)D~ )

(B.3)

C Wettench / Fermlon mass predwtmns

489

The quantum numbers l, r~ and k correspond to total angular momentum, the third component of orbital angular momentum and the third component of total angular momentum. They have the range /--IAI, IAI+I ....

,

k = A + r~ = -l, - l + l,

.,l

(B.4)

We normalize the functiuons D ~ ) according to ~1/2[ F)(A)~$ F')(A) I r12" Y ~ 2 I,L"lk ] L"l'k'-- ~ll,~kk,

(B.5)

with d2y gl/2 = L2d(cos 0) d~b the volume element on the sphere with radius Lo. The harmonics D ~ ) are related to the reduced matrix elements of rotations [24]

[21+1~ 1/2 D ~ )( O, qb) = \ 4--~o] d tk,( O) exp (,ffz~b).

(B.6)

The real functions dtk~(O) have the symmetries

d~kA(O) = d~-a,_k(O) = (--1)k-ad~k(O) = (--1)k-Xdt_k_x(O),

(B 7)

and are normalized (without summation over k and h)

I~ d(cosO)d~k~d~-21+l2 6 u '

(B.8)

•

For h = 0 o n e recovers the Legendre functions

d l o ( O ) = [ ( l - k ) ! ~ '/2 \(l+k),,I pk(o)' and

d°o(O) =

(B.9)

1 We also note that the symmetries (B.7) imply

(D~)) * = (--l)k-aD~,Z~,)

(B 10)

The integral over three generalized spherical harmonics can be expressed in terms of the WIgner 3-j symbols"

f d 2, -'/2D('X,)D(a2)D(a3)-((21a+l)(212+l)(213+l))l/2 Y ~2 Ilkl 12k2 13k3-4~L 2

\A1

A2 As

kl

k2

ks

"

This Integral vanishes unless AI + A 2 + A3 = 0 ,

kl + k2+ k3 = 0,

(B.12)

490

C Wettench / Fermmn mass predictions

and total angular momenta obey the restrictions

111-121<~13<~111+121

(B 13)

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