The role of gauge singlets in three-generation models

Volume 240, number 3,4

PHYSICS LETTERS B

26 April 1990

T H E R O L E O F G A U G E S I N G L E T S IN T H R E E - G E N E R A T I O N M O D E L S F. DEL A G U I L A a, G.D. C O U G H L A N ..b.¢ and L.A.C.P. DA MOTA b • Departament de Fisica Tebrica, Universitat Autbnoma de Barcelona. E-08193 Bellaterra (Barcelona), Spain b Department of Theoretical Physics, 1 Keble Road, Oxford OXI 3NP, UK ¢ St. John's College. Oxford OXI 3JP, UK

Received 29 December 1989

The mass spectrum and gauge-couplingrenormalisation ofa phenomenologicallypromising three-generation model is analysed in detail. Gauge-singlet superfields play an important role in overcoming a major problem which is generic to a large class of models. Throughout a large region of parameter space, a naturally light Higgs doublet is compatible with perturbative gauge couplings, unification and the experimental value of sin20w.

1. Introduction

Since the emergence of the superstring [1] as a candidate quantum theory o f gravity which furthermore unifies all known forces, considerable effort has been directed towards elucidating its connection with the standard model [2,3]. Calabi-Yau manifolds have played an important role in this process. Moreover, there is a unique Calabi-Yau manifold [4] (up to diffeomorphism) which gives rise to precisely three generations of massless chiral fields [ 5 ]. One class of particle physics models emerging from this threegeneration manifold has already been studied in some detail [ 6 - 1 0 ] , and shows promise in providing a realistic description o f the low-energy physics. However, it has recently been pointed out [ l l ] that this class of models suffers from a generic problem. Under two "minimal" assumptions concerning the nature of the complete (non-renormalisable) superpotential, a detailed analysis of mass spectra [ I l ] has established that, w i t h o u t f u r t h e r input, the existence of a pair of naturally light Higgs doublets is incompatible with perturbative evolution of gauge couplings up to the compactification scale, Me, and their subsequent unification. These "minimal" assumptions were: (i) the discrete symmetries reliably generate the complete superpotential; and (ii) non-renormalisable terms are accompanied by a c o m m o n exponential suppression

factor, ~, arising from their non-perturbative origin [ 12 ]. O f course, it is possible that gauge couplings may not remain perturbative at very high mass scales and unification may take place non-perturbatively [13]. However, it is also possible (and desirable?) that perturbation theory remains valid even at very high scales, in which case a crucial aspect o f these models may have been overlooked, and may provide the "further input" necessary to evade the problem. This letter deals with a possible source of this "further input": the E 6 gauge-singlet superfields arising from the compactification [14]. These have an important bearing on the low-energy mass spectrum of this class, leading to acceptable perturbative unification and a pair o f light Higgs doublets. It is worth noting that singlets also have an important part to play in neutrino masses, supersymmetry breaking, gauge breaking, and cosmology. Hence, in any realistic analysis, they cannot be overlooked. On topological grounds one expects gauge-singlets associated with the complex structure, K~ihler and H ~(End T) modes to present in the theory. However, no method has been found of evaluating the structure of cohomology group H ~( E n d T ) for this manifold, and therefore the detailed properties of the associated singlets are unknown.

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Volume 240, number 3,4

PHYSICS LETTERS B

2. The role of singlets The

models

possess

the

gauge symmetry and the gauge non-singlet matter content at Mc is

SU(3)c×SU(3)L×SU(3)R(cE6)

2,~(1,3,3),

f.j ~ (1, 3, 3 ) e 3 × 2 7 + 6 ×

q,~(3,3,1),

Q,~(3,1,3)

#j~ (3, 3, 1),

O , ~ (3, 1, 3)

e3×27+4×

(27+27).

(27+27),

(1)

The indices i and j label the different flavour copies of the nine ;t's, six ~s, seven q's and Q's, and four 4's and Q's. The different gauge components of these fields are listed in table i, together with their gauge quantum numbers and their conventional names. The 2, )Tmultiplets are identified with the lepton and Higgs superfields, the q, O multiplets with quark and 4, Q with antiquarks. To obtain the standard model at low energies, two stages of intermediate symmetry breaking are required at scales M~ and M2, with M¢>~M~>~M2.The first symmetry-breaking direction is taken to lie along the u~ component of one of the lepton superfields (i.e., the component ,%, "~ where the S U ( 3 ) a index a = 3 , and the SU(3)L index a = 3 ) . The second direction of symmetry breaking must then lie along the v4 component of a lepton superfield (i.e., 2.~, where a = 2 ,

26 April 1990

a = 3 ), as this is the only remaining standard-model singlet. Moreover, these directions are assumed to be D-flat, ( v s ) = ( ~ ) and ( v 4 ) = ( ~ ) , so that supersymmetry is preserved. So, intermediate-scale breaking generates mass terms for the above fields which are ordered in increasing powers of x = M~/Me and y = ,'~I2/M c. The E 6 singlets, which will be denoted generically by E, may play an important role in the low-energy theory in at least two possible ways. Firstly, they may be responsible for giving large masses to pairs of 27 and 27 multiplets, and secondly, as we shall explain, they may lead to term-dependent exponential suppression factors e, c', ... in the mass matrices. Each of these effects provides a possible way of overcoming the above problem, so that perturbation theory remains valid between Mz and Me, and one pair of Higgs doublets is naturally light. In what follows, both possibilities are investigated. But they are not unrelated. We shall argue that when singlets get large VEVs, both of these mechanisms can come into play. Furthermore, if singlets which transform non-trivially under the discrete symmetries obtain non-zero VEVs, then the associated symmetries are broken and new mass terms are generated from couplings such as 27 ~. (27.27)". 1"/M2¢. . . . , etc. Of course, the transformation properties of all the singlets under the discrete symmetries are not known; however, for deftniteness, this analysis assumes there are singlets E;

Table 1 Decomposition of the fundamental 27 representation of E6 giving hypercharges for the reduction Eo ~ SU ( 3 )c × SU ( 3 )L × SU ( 3 )a SU(3)cXSU(2)L×SU(2)axU(I)n_L~SU(3)c×SU(2)LXU(I)r. The indices attached to the fields q, Q and 2 correspond to SU (3)c X SU ( 3 )L × SU (3) R- In particular, A = 1, 2, 3 is an SU (3)c index, and a = I, 2 is the SU ( 2 )L part of an SU ( 3 )t. index. Fields

Usual notation

SU(3 ) c × S U ( 2 )L × SU (2)a

,/5 W'~ Y

/i "v'g Y/i' -

q..ga qA3 QA,

(u, d) d' u¢

(3, 2, I) (3, I, I ) "~

~ - ~ _~

1~ -~ _~

(2~'~

d"

j

(3, 1, 2)

~

Q.43 2~ ,,!.~ ~.~

d ": (v,e) (h'°, h ' - ) (h +, hO)

(3, l, 1 ) (1,2, 1)

~

"~ J

;.~

e°

"~

2]

v5

).]

390

v4

)

_~

( l, 2, 2)

-~ -t ~

-~ 0 0

(1, 1,2)

0

1

i

(1,1,1)

0

0

L

Volume 240, number 3,4

PHYSICS LETTERS B

Table 2 Discrete symmetry transformations. The transformations for the Q and O multiplets ( not shown ) follow from interchanging q ~ Q as well as # , ~ . The transformations for the gauge singlets, E, and ~, are assumed identical to those 2, and ~. Field

A = (BC)'*

B

C

DX Vd PX I/p

).1

0/2

0/

--*)'2

-').3

1

)-2 )-3 24 ).5+ 2s-

a2 0/ 0/ 1 1

1 a 2 I 1 1

'*). j

--*24

-'~4 --,). ~

~'3

"-*22

I I 1 1

27

1

1

-I

•"~s

l

0/

29

1

1 1

1 1

I

- ~',']. 9

---b2 9

0/2

--+Aa

--+2s

1 1

).-3

I 1 a

1 I a

I I -1

1 -I -,~

1 1 --1

•~-4

0/

0/

-- 1

--,~

-- 1

~5 ~6

(X2

0/2 0/2

-- 1 -- l

~ ~

- I - I

ql q2 q~ q4 q~ q6

Ot et2 I a 0/ 0/2

q7

q2 q--3 q--4

I 1 --.qs ~q4 ~q7

0~2

a c~2 I or: 1 0/ 1

~q6

--*Q4 ---'Q5

1

a

--*q2

m --,Q4

1 I 1

a z a 0/2

*q--~_ ~q4 .__,q~

--'Q_3_ ~Q2 ~Q,

I

~Q2 ~Qi --+Q3 --.Q6

G

q, ~ Q ,

-*Q7

t

q-,--,QZ

m

m

Masses

for 27-27

(2)

s'~2qr. 3r]

( i = 1..... 9) and Es ( j = 1..... 6) with the same transformations as the leptons ;t,~ 27 and ).s e 27 (see table 2) ~'

2.1.

large masses, while the electroweak Higgs remain light. However, even in the most favourable case there are three vector-like singlet quarks lighter than M~, in a d d i t i o n to the three pairs o f vector-like S U ( 2 ) L doublets, which are c a n d i d a t e electroweak Higgs pairs. In other words, three complete 27's remain light because o f chirality. This can also be seen from the explicit form o f the mass matrices. Recall that the (21 × 24) lepton/Higgs doublet mass matrix takes the form [ 11 ]

-1

1

0/2

26 April 1990

pairs

Singlet self-couplings o f the form 1" / M ' ~ - 3 ( H ~ 2 ) may lead to very large VEVs, ( E ) = O ( M c ) [151, leading in turn to the pairing up o f 27 and 27 fields through mass terms o f the form 27.27.1 n~Men - ' (n>~ 1 ). Thus many vector-like fields m a y be given *~ These exhaust all combinations of the symmetries A and B.

It" \).~k/

where i, j, k, p, q, r are flavour indices; a, b are SU (2) L indices; and the subscripts 1, 2, 3 are S U ( 3 ) R indices. The 27-27.1" contributions with which we are dealing only contribute to the three 9)< 6 submatrices B, which have a m a x i m u m rank o f six. Hence there are at most 3)< 6 = 18 massive pairs o f doublets. This leaves three vector-like pairs plus the three chiral doublets without any contribution from terms of this type. So, there are only three pairs o f vector-like doublets lighter than Me which are candidate electroweak Higgs pairs ~2. A similar argument applies to the d-type singlet quarks: their 1 5 × 1 8 mass matrix is given by

H

,

qfe,,'

LIfe/,

," M

s

N

(3)

IT_J\Q2k/

where the superscript 3 is an S U ( 3 ) L index, and the subscripts 2, 3 are SU (3) a indices (colour indices are suppressed). Thc submatrix I has a m a x i m u m rank o f four and, therefore, at most 3 ) < 4 = 12 pairs o f quarks are massive, with three vector-like and three chiral quarks light. Moreover, the 6 × 9 mass matrix for the charged lepton singlets, e c, has the same form as B T in eq. ( 2 ) , leaving only the three chiral lepton singlets light. Similarly, the 4)< 7 mass matrices for both the quark doublets (u, d ) and u-type anti~2 The three chiral families originate from 27's of E6, and the only (effective) couplings which efficiently give them mass are cubic, 273. Hence, the relevant Higgsshould come (largely) from a 27; otherwise, mixing introduce several powers of x and y. reducing the couplings unacceptably. 391

Volume 2 4 0 , n u m b e r 3,4

PHYSICS

LETTERS

quarks, u c, have the same form as I T in eq. (3), and only the three chiral fields are light. However, this most favourable case cannot be realised with just one singlet getting a VEV, as is evident from the first column of table 3. Although the quark sub-matrix I can be of maximal rank (i.e., four), the lepton/Higgs sub-matrix B cannot, leaving a large number o f vector-like doublets and singlets massless to this order. This results in the same problems that were uncovered in ref. [ 11 ], viz., naturally light Higgs doublets are incompatible with perturba-

B

26 A p r i l 1 9 9 0

rive unification. This is illustrated in table 3 for intermediate-scale symmetry breaking along the favoured directions ( ( I / x/2 ) (2 s + 2 9 ) ) = ( 21 ) = Mi and ((I/x//2)(2,-X2)>=(~-5)=M2, which are precisely those analysed in ref. [ 1 1 ]. The table shows the mass spectra for M, =M2 and just one singlet VEV, ( E ) = M~. The only singlet directions that leave a pair of Higgs doublets relatively light (mH ~ ~ x 3 M c ) are Es+, E7, E, and E2. But the large numbers of fields with masses of ~x2Mcdestroy the perturbative evolution o f gauge couplings. It is interesting to note that

Table 3 M a s s s p e c t r a f o r v a r i o u s s i n g l e t V E V s < E > = M c f o r M~ = M 2 a n d a c o m m o n t e r m s . N~/H, N ~ a n d N a give the n u m b e r s o f l e p t o n / H i g g s

e x p o n e n t i a l s u p p r e s s i o n f a c t o r , t, f o r t h e n o n - r e n o r m a l i s a b l e

d o u b l e t s , c h a r g e d l e p t o n s i n g l e t s a n d d - t y p e q u a r k s i n g l e t s that get m a s s at a

g i v e n o r d e r . Nq g i v e s both the number o f ( u , d ) q u a r k d o u b l e t s a n d u ¢ s i n g l e t s . Singlet with VEV El, E2, E3 o r E4

Es+ o r E ,

E7

Es_

E3, E,. Es or E 6

Es o r E9

1 N~/. N~

6 2

11 ")

cr 2

~x 3

~x 4

4

-

-

-

4

-

-

-

-

-

Na

12

3

N~

4

.

.

.

ex 6

.

A,'~/.

3

12

3

I ")

1

1

A,~

1

-

2

-

2

l

Nd

-

I1

3

-

1

Nq

-

-

3

-

1

Nt/H

3

I2

4

1 a)

1

N~,

1

-

2

-

3

Na

-

11

3

-

1

5,~

-

-

3

-

1

N~/H

3

5

-

-

N~

1

-

3

-

2

Nd N,~

-

11 -

3 3

-

1 1

N~/H

3

12

3

I ")

2

N~,

1

-

2

-

2

A,~

-

11

3

-

1

NQ

-

-

3

-

I

N~/,

3

N,~

1

13 ~)

l 3 a) -

5

-

3

-

Nd

12

-

-

3

Na

4

-

-

-

N~/H

6

N,,

2

-

4

Na

3

9

3

N,

l

-

3

") T h e l i g h t e s t H i g g s p a i r h a s a m a s s o f t h i s o r d e r .

392

x

1 1 *'

4

Volume 240, number 3,4

PHYSICS LETTERS B

these singlet directions correspond to the breaking of at most the C or D symmetry. If two singlets get VEVs, the situation is little improved, with either no light Higgs and/or too many light fields. Moreover, smaller values of M2 do not help (at least in this particular case).

2.2. Term-dependent suppression factors All the analysis so far has assumed that non-renormalisable terms in the superpotential are accompanied by an exponential suppression factor which is the same for all terms. However, relaxing this assumption can have a drastic effect on the mass spectra of this class of models. As an example, which will be more fully dealt with below, one might consider different exponential suppression factors ~ and e' associated with terms 2 7 3 . ( 2 7 . 2 7 ) " / M 2'' and (2727)~/M 2~-3. There is an obvious way (but it need not be the only one) in which such term-dependent suppression factors might arise, namely, via the singlets. If the singlets themselves obtain large masses, ME=O(M~), then at scales below ME they may be integrated out of the theory to yield effective interactions of the form (27.27)L In particular, renormalisable couplings of the form 27.27.1 can generate effective non-renormalisable couplings of the form (27-27) ~. This suggests the possibility that these terms may not be suppressed by the same exponential factor. To illustrate the importance of this effect, again consider the favoured model with ( ( I / x / ~ ) (;% + ,~9)>=(2~=Mt and ( ( l / x / / 2 ) ( 2 ~ - , ; t 2 ) ~ = ( ~ ) =M2, and no singlet VEVs. Let us take the 273. (27.2~) ~ and 273.(27.2~) ~ suppression factor ~ to be smaller than the ( 27.27 )" factor, which is taken to be ~' = 1. Then, the mass spectrum is quite different from that given in ref. [ 11 ]. The most drastic change is that the Higgs mass is given by m n = h max{~x 3, y ~2/x} × Me, where x = M~/M~, y = M2/M~ and h is a coupling less than one. Moreover, this is the only mass eigenvalue in which ~ appears. This suggests that very small values of~ (i.e. ~
26 April 1990

model is at best marginal. The requirement of a light Higgs, mH < 1 TeV, translates into an upper bound on the compactification scale of

Me <...x/hyiZ× 10 3 GeV~< 3h-11015 G e V ,

(4)

where the best possible case, sin20w=0.224, was used to obtain the last inequality. For believed small values of h, say 0.1, this value of M¢ is probably still too small. However, the situation can be improved by combining these term-dependent suppression factors with non-zero VEVs for the singlets.

3. A working model The results of the previous sub-sections, along with table 3, suggest that combining term-dependent suppression factors with large a VEV for either Es÷, E7, E7 o r E2 should yield an acceptable model with M~ =Mz. In the first two directions, all the discrete symmetries remain unbroken; in the latter two, C or D is broken. Consider first a VEV for E7, which breaks the D symmetry. The mass spectrum is given in table 4. It is identical to that in table 3, except for the following changes. The Higgs mass is given by

mH =h max{~x3, xtt} ×Mc ,

(5)

and, again, this is the only eigenvalue in which c appears. Finally, in the spirit of this order-of-magnitude analysis, let us introduce a number, k, of order unity which multiplies the masses of the d-type singlet quarks. This number, k, allows for the fact that the computed eigenvalues are only approximate and provides a means of adjusting mass thresholds. Of Table 4 The order of magnitude of the mass eigenvalues, in units of the compactification scale for ( E7 ) = Mc, M~ = Mz, and term-dependent suppression factors ~' = I, ~< 1.

Nl/il

Nt.,

-~

Nq

3×M¢ 12×xM¢ 4×xZM¢ I xx4Mc

l×Mc 2xx2Mc 3 xx4M¢

11XxMc 3xx2M¢ 1×X4Mc

3 X x2M¢ 1 xx4M¢

1X max{6r ~, x ' i } M ~ "~ al The lightest Higgs pair has a mass of this order.

393

Volume 240, number 3,4

PHYSICS LETTERS B

course, different such factors could be introduced for all fields; however, this gains little. The essence of threshold effects is captured in the single parameter, k. The analysis of this model then proceeds in two parts, depending on the value ofe. In both cases, five constraints must be satisfied: mH ~< 1 T e V , 1

(6a) 1

- - > 0 , at (Me)

~ > 0 , aR (,~/'c)

1

- - > 0 , at(Me)

aL(:~,-) = a R ( M ¢ ) =ac(M~) •

27~

26 April 1990

- -

= 6 9 . 0 6 2 + 18 l n x + 15 In k,

(7)

2n - -

=804.248 sinZOw + 7.158+62 l n x ,

(8)

at(Me) aL(Mc)

2n - _ - 804.248 sin20w + 588.867 aR(Mc)

~'2.0~ +1391nx+ (7.5)Ink'

(9)

(6b,c,d) (6e)

These are shown in fig. 1. Furthermore, the model must predict a value ofsin20w consistent with the experimental value of 0.228 + 0.004, with the only adjustable parameters being h and k. In the favoured regime of small c (i.e., ~
where the upper (lower) number in eq. (9) corresponds to k > l ( k < l ) . Imposing the unification constraint then yields Inx=-2.775+

1361

'

0009

sin20w=0.229+ (0.01 l J I n k ,

(11)

and the upper bound on the compactification scale, eq. (4), becomes M~ ~
(12)

[Oglo x -30

-10

-20

""

\ ta o t~ o

-10

i¢i]

. . : M~ s)n

,

...

I

I

~..9=0228

....

Fig. 1. The constraints on e and x for ,t42 = M~, sin20w = 0.228 and h = 0. I. The Higgs constraint corresponds to the half-plane to the left of the dashed line labelled by H; those of finite aL.R.C tO the half-planes to the right of the lines labelled by L,R,C. The shaded region is consistent with all four constraints, and is virtually unchanged as sin20w varies from 0.224 to 0.232. Finally, the dotted line gives the unification constraint, which for ~ < x s, is independent of~.

394

As k is varied from 0.65 to 1.4, acceptable values of sin20w of 0.224-0.232 are predicted, with x = M ~ / Mc~ 0.06 and the compactification scale ranging from 4h - ) to lh - ' × 10 '6 GeV. This is quite acceptable for h = 0.1. Furthermore, by introducing more threshold parameters, M,. can be raised higher still. This analysis was also repeated for ( E s + ) , ( E , ) , and (E~) ~ 0, with very similar results. Of particular interest are the E5÷ and E, directions, for which no discrete symmetries are broken. The numerical results in this case are very close to those above for E7, with a compactification scale which is again of order h - I x 1016 GeV.

4. Conclusion It can be concluded from the above analysis, that the E6 gauge singlet superfields have an important bearing on the low-energy physics predicted by this class of models. By giving masses to 27-27 pairs and by selectively enhancing some types of mass terms relative to others, they can lead to acceptable perturbative unification which is consistent with a pair of

Volume 240, number 3,4

PHYSICS LETTERS B

light Higgs d o u b l e t s . T h e b e s t c a n d i d a t e s for t h i s role are s i n g l e t s w h i c h t r a n s f o r m t r i v i a l l y u n d e r t h e disc r e t e s y m m e t r i e s , o r at m o s t b r e a k C o r D.

Acknowledgement The authors acknowledge discussions with G.G. R o s s a n d M. M a s i p . O n e o f us, L . A . C . P . M . , w o u l d like to t h a n k C N P q ( B r a z i l i a n G o v e r n m e n t ) for financial support.

References [ 1 ] P. Ramond, Phys. Rev. D 3 ( 1971 ) 2415; A. Neveu and J.H. Schwarz, Nucl. Phys. B 31 ( 1971 ) 86; M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117; D. Gross, J. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 55 (1985) 502; Nucl. Phys. B 256 (1985) 253. [2] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46. [31E. Witten, Phys. Lett. B 155 (1985) 151. [4] S.T. Yau, Proc. Natl. Acad. Sci. 74 (1987) 1. [5] P. Aspinwall, B. Greene, K. Kirklin and P. Miron, Nucl. Phys. B 298 (1987) 193;

26 April 1990

P. Candelas, A. Dale, C. Liitken and R. Schimmrigk, Nucl. Phys. B 298 (1988) 493; B. 306 ( 1988 ) I 13. [6] B.R. Greene, K.H. Kirklin, P. Miron and G.G. Ross, Phys. Lett. B 180 (1986) 69; Nucl. Phys. B 278 (1986) 667; B 292 (1987) 606. [7] G.G. Ross, lectures Trieste Summer Workshop ( 1987); lectures 1988 Banff Summer Institute, CERN preprint CERN-TH-5109/88 (1988). [8 ] B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, Phys. Lett. B192(1987) l l l ; J. Distler, B.G. Greene, K.H. Kirklin and P.J. Miron, Phys. Lett. B 195 (1987) 41. [91S. Kalara and R.N. Mohapatra, Phys. Rev. D 35 (1987) 3143; D 36 (1987) 3474; R. Arnowitt and P. Nath, Phys. Rev. Left. 60 (1988) 1817; 62 (1989) 1437, 2225; Phys. Rev. D 40 (1989) 191. [ I 0 ] F. del Aguila and G.D. Coughlan, Phys. Lett. B 215 ( 1988 ) 93. [ 11 ] F. del Aguila, G.D. Coughlan and M. Masip, Phys. Lett. B 227 ( 1989 ) 55; preprint OUTP-89-07P, UAB-FT-206/89. [ 12] M. Dine, N. Seiberg, X.G. Wen and E. Witten, Nucl. Phys. 278 (1986) 769; B 289 (1987) 319; J. Ellis, C. G6mez, D.V. Nanopoulos and M. Quir6s, Phys. Lett. B 173 (1986) 59; M. Cvetic, Phys. Rev. Lett. 59 ( 1987 ) 1795. [ 13] L. Maiani and R. Petronzio, Phys. Lett. B 176 (1986) 120. [ 14] E. Witten, Nucl. Phys. B 268 (1986) 79. [ 15 ] G.D. Coughlan, G. Germ~in, G.G. Ross and G. Segr6, Phys. Lett. B 198 (1989) 467; G.D. Coughlan, Intern. J. Mod. Phys. A 4 (1989) 41 I.

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