Family unification in SO16

Volume 149B, number 6 PHYSICS LETTERS 27 December 1984

FAMILY UNIFICATION IN SO16 ~

Peter ARNOLD Institute for Theoretical Physics and Department of Physics, Stanford University, Stanford, CA 94305, USA

Received 20 August 1984

We present a symmetry-breaking pattern for SO 16 × (U 1 )PQ which predicts only four super-light neutrinos. Thus, un- like other models of orthogonal family unification, our model satisfies cosmological bounds on neutrino species.

Introduction. Orthogonal groups are natural candi- dates for family unification since a spinor representation of SO10+N decomposes into multiple families with similar particle content under SO10+N --> SO10 X SO N. Half of these families have V-A weak interactions and half have V + A ,1

SO15 is the smallest orthogonal group that could account for the three observed V-A families. Experi- ment has not yet found any V+A families. As we shall see later, the only mechanism we know that explains their absence will not work for SO15. We turn next to SO16 whose two spinor representations are real. To guarantee that fermions do not obtain GUT-scale masses, one usually considers only complex representations; so we would cast aside SO16 and move on to a 16 family SO18 model. Alternatively, we may intro- duce a global Peccei-Quinn symmetry for mass protection. In this paper, we consider an SO16 X (U1)pQ theory which has four V-A and four V+A families.

We use the Bagger-Dimopoulos mechanism [2] to make the V+A families heavier than the V - A . The SO18 model o f Bagger and Dimopoulos has eight light neutrinos. A primary virtue of our model is that it has but four. This is consistent with the constraints of big- bang nucleosynthesis [3]. Finally, our model has eight families and so may be consistent with experimental limits on proton decay [4].

Warm-up models. Under SO16 ---r SO10 × SO6, the

Supported in part by National Science Foundation, con- tract NSF-PHY-83-10654.

,1 For a review of orthogonal unification, see ref. [1 ].

0370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

spinor representation decomposes as 12816 (1610 , 46) + (1--'610, 46)- (Subscripts will always indi- cate the orthogonal group.) If the four 1610 families are said to have V-A weak interactions, then the four 1610 wilt have V+A.

We wish to consider symmetry breakings SO16 X (U1)pQ ~ SO10 X SO 6 X (U1)pQ ~ . . .~ (SU 3 × SU 2 × U 1) × G. The nature of G _C SO 6 × (U 1 )po will deter- mine the number of light families and very-light neutrinos. For G = U1, we found two interesting candi- dates.

Before showing the results, let us fix some notation. The spinors of SO2N will be written as [e I e 2 ...e N) where e! = +1. The two spinor representations are dis- tinguished by IIej = +1 and IIej = - 1 . Under SO16 SO10 × SO 6, we write

[ el e2 "'" e8)16 -+ [ el e2 e3)6 [ e4 e5 e6 e7 e8 )10"

Now consider the U 1 's generated by the following charges:

X 1 = 4 e 1 +2e 2 + e 3 - q , (1)

X 2 = 61 + e 2 + e 3 + q, (2)

where q is the PQ charge normalized to be +1 on the fermions. The charges of the families are listed in table 1.

Suppose G were given by (1). What happens if SO16 X (U1)pQ -+(SU 3 X SU 2 X U1) X (U1)xI some- where near the GUT scale? The V-A and V+A families transform as conjugate representations under SU 3 × SU 2 × U1, and so will pair up to form masses if they

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Table 1 The charges X l and X 2 of the eight families.

V-A families X1 X 2 V+A families X1 X2

I+++) 6 4 I - - - ) - 8 - 2 I + - - ) 0 0 I -++ ) - 2 2 I - + - ) - 4 0 I + - + ) 2 2 I - - + ) - 6 0 I++- ) 4 2

have opposite X 1 charges. The X 1 = +-4 families therefore become super-heavy leaving six light families. Also, each remaining family contains a singlet under SU 3 X SU 2 × U 1 . Such neutrinos can pair up to gain masses if they have opposite X 1 charges. The X 1 = +-6 neutrinos pair up, as do the X 1 = +-4 neutrinos, and the X 1 = 0 neutrino pairs with itself. We therefore have five GUT- scale neutrinos in the light families. By the standard see-saw mechanism, this leads to five very-light neutrinos when the families get mass at the weak scale.

Now consider G given by (2). The reasoning is the same, but we get eight light families and again five very- light neutrinos ,2. We will now consider problems with the two models.

The axion must be invisible. X 1 and X 2 generate global continuous symmetries which are anomalous since they include anomalous PQ transformations. When G is broken, it will then leave behind a pseudo- Goldstone boson called the axion [5]. To sufficiently decouple the axion from ordinary matter so that it satisfies experimental and cosmological bounds, G must be broken above "109 GeV [6].

In general, any family group G which survives to the weak scale must be discrete. Any U 1 C G with a PQ component must be broken above "~109 GeV. Any other U 1 C G will be gauged and so must be broken above ~105 GeV to satisfy experimental bounds on flavor-changing neutral currents mediated by the gauge bosons. In the case at hand, we might be able to break (U1)x1 or (U 1)X2 to Z n subgroups which will still implement the necessary mass protection.

,2 It is interesting to consider extending these symmetries in a natural way to SO18. We get X 1 = 8e 1 + 4e 2 + 2e 3 + e 4 - q andX2 = el + e2 + e3 + e4 + q . The second is exactly the U1 considered by Senjanovik et al. [7] (they break it to a discrete Z 6 subgroup). The first yields an alternative six family model with five very-light neutrinos.

These are too many light neutrinos. We introduced this article touting the cosmological bound on the number of neutrinos. Each of the symmetries presented yields five, but the bound is four [3].

How do we break SO 6 X (U1)pQ -~ G? So far, we have made no mention of the Higgs structure needed for the desired breaking. The Higgs sector might well be so complicated as to make even a hardened veteran feel faint. We don't know what to do with X1, but searching for a way to implement X 2 led us to a simple and natural breaking scheme which will solve the pre- vious problems as well. This model is the subject of the next section.

The mode l If we could arrange a discrete family symmetry that mapped (left-handed) Weyl fermions as

X V _ A "+ X V _ A , XV+ A -~ iXv+A ,

then we would protect against ( V - A ) / ( V + A ) masses and allow GUT-scale neutrino masses for only the V - A families.

The discrete family symmetry in our model will be similar but more complicated. The complexity of the symmetry group will be offset by the natural way in which it arises. The breaking pattern is shown in fig. 1.

Breaking o f SO 6 X (U1)pQ. Recall that SO 6 is local- ly isomorphic to SU 4 . Now consider a VEV (@) in the 106 , and write (@) as a 4 X 4 symmetric matrix. We as- sume only that the norms of its eigenvalues are all distinct. What is the unbroken subgroup of SO 6 X (U 1)PQ

IO is S0=6 X (U l )pg

S010 x SO 6 X (UL)po <xy. I0 ~ - SU 3XSU z x U(

GeV I - .X{Z2 )~'

i0 ~

10 2 SU3x Utx Z z x Z~,

Fig. 1. Symmetry breaking pattern for SO16 X (U1)pQ.

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if this Higgs couples directly to the fermions? t ransforms as

q~ -+ (e- iOgt) T q~(e-i°gt)

under SO 6 X (U 1 )pQ where g is an e lement of SU 4 a n d 0 is the PQ rota t ion of the fermions. For the unb roken group elements , ~b -+ q~, so

g*~b = e i2° q~g.

Now choose a basis where ~ is diagonal , a . Writing q~i/= aiSii (no sum), we get

aig = ei2°5.g 0 (no sum).

Taking the n o r m of bo th sides, lail = la/I ifgi/4= O. Since lail 4= la/I f o r i :~/" by a s sumpt ion ,g must also be diagonal. For the diagonal e lements , the last equat ion gives gil = ei2°gii. Also the uni ta r i ty of a diagonal ma-

trix implies Igiil = 1, so

gii = -+ e-j0 (no sum).

By applying the condi t ion det g = 1 to restrict 0,

we may now consider all o f the possibilities for g. These are shown in table 2 by listing the diagonal elements . The V - A families t ransform as a 4 under ei°g. The V+A families t ransform as a 4 under ei°gt .

The name of this group is Z 4 × ( Z 2 ) 3 , 4 . Case 2 cor-

responds to the XV-A -+ XV-A and XV+A -+ iXV+A

#a We are assured that the complex VEV (¢) is diagonalizeable because we have assumed that its eigenvalues are distinct.

,4 The name of the group is unimportant but can be deduced as follows. The group is f'mite and abelian and so can be written as the direct sum of Z n subgroups. The order of the group is 32, some elements have order 4, and none have higher order. It must then be Z 4 X (Z2) 3 or (Z4) 2 × Z 2. Counting the number of elements of order 4 pins it down to Z4 X (Z2) a.

Table 2 The discrete family group given by entries of 4 X 4 diagonal matrices. Take all permutations of the entities. _+ and _+' are distinct.

g V-A : eiOg V+A : eiOg'f

Case 1 -+e-i0(1,1,1,1) -+(1,1,1,1) _+'(1,1,1,I) Case 2 +-e-i0 (1,1,1,-1) _+(1,1,1,-1) _+'(i,i,i,-i) Case 3 e-i°(1,1,-1 , -1) (1,1 , - 1 , - 1 ) +_(1,1,-1,-1)

symmet ry discussed earlier. These symmetries give the desired mass protect ion. Note also that each member of case 2 is equivalent to the Z 4 subgroup of the X 2 symmet ry of our warm-up models.

Now that we've found a VEV which gives the desired symmet ry breaking, we must check that it can couple

to the fermions in the SO16 theory. [It must do so in order that it t ransforms under (U 1)PQ, the breaking of which is essential to our argument.]

To directly couple to the fermions, an SO16 Higgs must lie in

(12816 X 12816)s = [0] 16 + [4] 16 + [8] 16"

Our VEV is in the 1"-'06 = [3] 6 or SO 6 . This may be ob-

tained from a ( [1]10, [316) C_C_ [4116 or ([5110, [316) _c [8] 16- Since the breaking is nea rMGUT, we must preserve SU 3 X SU 2 × U 1 . The [1]10 does no t conta in a direct ion which does so, bu t the [5] 10 does , s . So we

take our Higgs ~b to be in ([5110, [316)"

Breaking at the weak scale. The model predicts four

V - A and four V+A families, bu t why don ' t we see the V+A families? Fol lowing Bagger and Dimopoulos [2], we will let the weak Higgs couple directly only to the

V+A families, giving them masses of o r d e r M w "" 300 GeV as in fig. 2. The V - A families obta in masses radiatively and so are lighter.

In SO 6 ~ SU4,

4 X 4 = 6 + 1 0 , 7l X 4 = 6 + 1--'0,

H w must belong to the 106 i f it is to couple directly to only the V+A families. The V - A families obta in masses, through graphs like fig. 3, o f order aMw(¢)2 / ( c o ) 2 .

, s This direction also preserves SU s . The [5] lO has no direction which breaks SU s but preserves SU 3 × SU2 × U 1 , so we must break SUlo -+ SU 3 X SU 2 X U1 and SO s × (Ut)pQ

Z 4 × (Z2) 3 separately.

V-t-A I V + A I I I I

r//~//)

(Nw) Fig. 2. The V+A families obtain masses directly from the weak Higgs.

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<+> (+>

\ / \ /

V - A V ÷ A i v + A v - A I I I

( H w )

(o)

(+> <+>

x / \ /

/ \ / \

[ \

V - A V+A [ V+A V-A I I I

< Hw> (b)

Fig. 3. The V-A families obtain masses radiatively.

Any VEV of the 10 6 will break Z 4 X (Z2)3 to at least (Z2)4, corresponding to cases 1 and 3 of table 1. The full (Z2) 4 would forbid any mixing between families and so is disallowed by experiment . Almost any random VEV, however, will break Z 4 X (Z2)3 -~ Z 2 X Z 2 . (Specifically, (Hto) and the GUT-scale VEV <~b) should share no eigenvectors in SU 4 .) This Z 2 X Z 2 takes

XV_A "+ +Xv_A , XV+ A ~ +fXv+A

and forbids mixing between V - A and V+A families. Finally, let us again check that the appropriate VEV

is contained in S016. We may use either the ([1 ] 10, [3] 6) C [4116 or ([g] 10, [3]6) C_ [8116. Only the [1 ] 10 contains both of the Weinberg-Salam doublets needed: (1,2)y_- 1 and (1,2)y=_ 1 under SU 3 X SU 2 X U 1. So we take H w to be in ([1]10, [316 ).

In passing, we can now see the problem with a model based on SO15 X (U1)pQ. Both V - A and V+A families transform under the same representation of the SO 5 family group. We cannot arrange a Higgs to couple to only the V+A families.

Concluding remarks. If the measurement of the Z 0 width indicates that there are three or four light neutrino species but the V+A families are still found near the weak scale, then models such as this are viable. There are a few problems, however, which until now I have swept under the rug.

The introduction of the anomalous PQ symmetry leads to well-known domain wall problems in the early universe [8]. There is, however, an aesthetic problem with introducing the PQ symmetry in the first place. Most symmetries are introduced a priori, but the PQ symmetry is not exact. Why shouldn' t we be allowed to write down terms in the lagrangian which break PQ if PQ is not to be a true symmetry of the theory?

Some authors [9] have tried to arrange the gauge group and particle representations so that the require- ments of gauge-invariance and renormalizability forbid PQ-violating terms. Unfortunately, no such solution can work in SO16. All the representations of SO16 are real. So, no matter what representation a Higgs field

is in, we can construct a gaugeqnvariant quadratic term ~bM ~b which violates PQ.

I would like to thank Savas Dimopoulos for his guidance and suggestions.

References

[1] F. Wilezek and A. Zee, Phys. Rev. D25 (1982) 553. [2] J. Bagger and S. Dimopoulos, Nucl. Phys. B244 (1984)

247. [3] J. Yang, D. Sehramm, G. Steigman and R. Rood, Astro-

phys. J. 227 (1979) 697; K. Olive, D. Schramm, G. Steigman, M. Turner and J. Yang, Astrophys. J. 246 (1981) 557; J. Yang, M. Turner, G. Steigman and K. Olive, Enrico Fermi Institute preprint 83-43 (1983).

[4] J. Bagger, S. Dimopoulos and E. Masso, Phys. Lett. 145 (1984) 211.

[5] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223; F. Wilczek, Phys. Rev. Lett. 40 (1978) 279.

[6] M. Dine, W. Fisehler and M. Srednieki, Phys. Lett. 104B (1981) 199.

[7] G. Senjanovik, F. Wilczek and A. Zee, University of Cali- fornia, Santa Barbara preprint (1984).

[8] P. Sikivie, Phys. Rev. Lett. 48 (1982) 1156. [9] H. Georgi, L. Hall and M.B. Wise, Nucl. Phys. B192 (1981)

409; J.E. Kim, Phys. Rev. D24 (1981) 3007; D26 (1982) 3221; S. De Alwis and P. Frampton, Phys. Rev. D24 (1981) 3345; P. Frampton, Phys. Rev. D25 (1982) 294.

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Family unification in SO16