A generic problem for a class of three-generation models

Volume 227, number 1

PHYSICS LETTERS B

17 August 1989

A G E N E R I C P R O B L E M F O R A C L A S S O F T H R E E - G E N E R A T I O N M O D E L S "~ F. D E L A G U I L A a, G . D . C O U G H L A N

a,b,c

and M. M A S I P a

a Departament de Fisica Te6rica, Universitat AutOnoma de Barcelona, E-08193 Bellaterra (Barcelona), Spain b Department of Theoretical Physics, 1 Keble Road, O.~ford OXI 3NP, UK St. John's College, Oxford OX1 3JP, UK

Received 12 May 1989

A promising example from a class of three-generation superstring models is analysed. Under broad assumptions, the existence of a H iggs doublet light enough to allow for the standard electroweak breaking is found to be incompatible with (i) the perturbative evolution of gauge couplings up to the compactification scale, (ii) gauge unification, and (iii) a correct prediction of the electroweak mixing angle. This seems to indicate that discrete symmetries and order-of-magnitude mass ratios are not enough to reproduce the standard model.

The superstring [1] at present offers a unique framework for a consistent q u a n t u m theory o f gravity and for the unification o f all known forces. Although definite string-based predictions for low-energy p h e n o m e n o l o g y are p r o b a b l y still some way off, some a t t e m p t s have been m a d e to chart a passage from the superstring down to the s t a n d a r d model [2,3 ]. In particular, a class o f models based on the three-generation C a l a b i - Y a u m a n i f o l d [4] has been studied in some detail [ 5 - 8 ] , firstly for its potential phenomenological relevance, and secondly, for its role as a playground in which the m o r e general features o f superstring models m a y be explored. In this letter, a specific example from this class o f models is analysed. It was chosen because it a p p e a r e d to be one o f the most p r o m i s i n g candidates for a viable m o d e l [ 5 - 7 ] , as it incorporates a m a t t e r parity s y m m e t r y which forbids dangerous renormalisable couplings m e d i a t i n g baryon- and lepton-number-violating processes [ 9 ]. Moreover, the necessary interm e d i a t e gauge s y m m e t r y breaking takes place along directions in flavour space which a study o f flat directions [6,10] has revealed as favoured, and which naturally lead to large i n t e r m e d i a t e scales [ 11 ].

Through a detailed analysis o f the mass spectrum, it is shown that the requirement o f a light standardmodel Higgs pair is i n c o m p a t i b l e with the perturbative evolution o f gauge coupling constants up to the compactification scale, Me, their subsequent unification and a correct prediction o f the electroweak mixing angle. This analysis is based on the known properties of these models together with a m i n i m a l set o f assumptions: ( i ) the discrete symmetries reliably generate the complete superpotential, and (ii) nonrenormalisable terms in the superpotential are acc o m p a n i e d by a ( c o m m o n ) exponential suppression factor, ~, arising from their non-perturbative origin [ 12 ]. The distinguishing feature o f this analysis is the consideration o f the complete set o f non-renormalisable terms allowed by the discrete symmetries. Allowance is m a d e for the known zeroes in the 273 and 273 couplings [7], but other possible zeroes, coming from "accidental" cancellations, singular points in the m o d u l i space, etc., are assumed absent. Note that the possible role o f gauge singlet superfields ~j and the possibility o f t e r m - d e p e n d e n t suppression factors are not considered. There is considerable uncertainty associated with these effects, and they are hence not in the spirit o f " m i n i m a l assumptions". In a longer c o m p a n i o n p a p e r [13], an exhaustive case-by-case

~" Work partially supported by research project CICYT.

~ This possibility is dealt with in a forthcoming paper.

I. Introduction

0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

55

Volume 227, number 1

PHYSICS LETTERS B

study of all possible intermediate symmetry breaking directions (not excluded by fast proton decay) reveals that this problem is a generic one: under these minimal assumptions, all models in this class share the above incompatibilities. The class of models under consideration is that arising from Yau's construction [4] based on C p 3 x C P 3 and with a group of discrete symmetries of order 72 [ 5 ]. The models possess the gauge symmetry S U ( 3 ) c × S U ( 3 ) t × S U ( 3 ) R ( ~ E 6 ) and the gauge non-singlet matter content at M~ is 2 , ~ (1, 3, 3),

/ ~ ~ (1, 3, 3)

~3×27+6× (27+27), q,~ (3, 3, 1),

Q , ~ (3, 1, 3) t

q,~(3,3,1),

(~,

e3×27+4×

(3,1,3)

(27+27).

(1)

Table I Decomposition of the fundamental 27 representation of E 6 giving hypercharges for the reduction E 6 = S U ( 3 ) c × S U ( 3 ) L × SU(3)R ~ S U ( 3 ) ¢ x S U ( 2 ) L x S U ( 2 ) R × U ( I ) ~ _ , . = S U ( 3 ) c × SU (2)L × U (1)> The indices attached to the fields q, Q and correspond to S U ( 3 ) c × S U ( 3 ) L × S U ( 3 ) R . In particular, A = 1, 2, 3 is an SU(3)c index, and a = l , 2 is the SU(2)L part of an SU(3)L index. /Y Y SU(3)c v'5 × SU (2) L ×SU(2)R

Usual notation

q,,

(u,d)

qJ3 Q"

d' u"

Q,2

dc

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

Q,3 2~ 2~

d,C (v,e) (h'°,h '-)

"(

.4~

(h+,h °)

f

~_ 23

56

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

e~

;

v~

j

v~

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

~ -~ - 3~ ~ ~ -½ -½

\ //% a y *~-t.

-g

1

_~ _

2. The m o d e l -¼ 0

½

o

1

¼

0

a

_~

o

(1, 1, 1)

quantum numbers and their conventional names• The 4, £multiplets are identified with the lepton and Higgs superfields, as there are more copies of these and hence they can be expected to obtain non-zero VEVs first. Furthermore, the smaller number of q, Q, c], Q multiplets means that, below Me, the corresponding gauge coupling will not fall as fast as the others, and can, therefore, be identified with the strong coupling of QCD. To obtain the standard model at low energies, two stages of intermediate symmetry breaking are required at scales MI and M2, with Mc > M~ > M2. The first symmetry-breaking direction is taken to lie along the u5 component of one of the lepton superfields [i.e., the component 2~, where the SU(3)R index c~= 3, and the SU (3) L index a = 3 ]. The second direction of symmetry breaking must then lie along the v4 component (t.e., 2,, where a = 2, a = 3 ), as this is the only remaining standard-model singlet. Moreover, these directions are assumed to be D-flat, ( v5 ) = ( ~ 5 ) and ( v4 ) = ( v4 ), so that supersymmetry is preserved. This analysis assumes the most general (non-renormalisable) superpotential consistent with the discrete symmetries (see table 2), except for the known zeroes in the 273 and 273 couplings [7]. Hence, after the choice of directions in flavour space along which the gauge symmetry is broken, the different fields get mass contributions from an infinite set of terms ordered by increasing powers of the (small) ratios x = M ~ / M c , y = ] P I 2 / M ~ . Then the thrust of the argument is rather simple. In general, all Higgs fields receive masses to some order in x and/or y, and the requirement that the lightest Higgs be lighter than O ( 1 TeV) then constrains M1 and M> This, in turn, constrains the mass spectrum and hence the renormalisation-group evolution of the gauge couplings. The final result is that the different gauge coupling constants all explode below the compactification scale, with no unification and no prediction of sin20w. •

The indices i and j label the different flavour copies of the nine 2's, six 2s, seven q's and Q's, and four q+s and Q's. The different gauge components of these fields are listed in table 1, together with their gauge

Fields

17 August 1989

0

For the specific model of interest, the first stage of symmetry breaking lies along the v5 component of (28+29) and 2~, and the second along the v4 component of ( 2 ~ - 2 2 ) and ~ . Hence, ( ( 1 / x / 2 ) 0 t s + 29))=(21 )=MI, and ((1/x~)(2,-42))= ( 2 s ) = M2. Given the superpotential and this choice

Volume 227, number 1

PHYSICS

LETTERS

Table 2

Non-singlet matter content and discrete symmetry transformations. The transformations for the Q and Q multiplets ( n o t shown ) follow from interchanging q'~--~Qas well as ~/~-~O.

B

9

9

9

9 (1) 3

A=

(BC)

4

B

C

D × Va

2

16 16 12 12 20 2 0

• 2 18 18 10 2 2 12 10 10

9

9

8

8

4

4

12

~

9

9

8

8

4

4

12 12

17 17 9 21 9 9

3

13 • (1) 1

_424

5 21 21 13 (~) • 13 13 9

9

.

. . . .

12 12

• [~]2525

•

12

. . . . 20 20

•

6

6

• 18 18 10 2 2 12 10 10

181810 1818

2

•

.

181818

222214

14 •

-

6

1212121 2 l S [6] 1010 2

4 18 0

18 18

-

I010 4

2 18 6

18 -

18

16 10 8

8

_4 12 • 2

2

( i ) 25

. . . .

11113

3

•

• 191919

16 16 8

8 24 12 • 24 24

25 ( i )

. . . .

11119

9

-

,

12 12 4

4

-

1

12 12 4

4

20 20 12 12

1

23

o~

a2

~2a

~2~

24

o~

1

"-'23

--'22

1

25+

1

1

1

1

1

25

l

1

1

1

27

1

1

-I 1

•

•

17 17 ( 1 )

7 7 2 3 23 15 5

• 15 15

.

- 20 20 20

•

.

17 17 5 ( i )

7 7 23 23 15 5

• 15 15

6

4

4

(1) ~

20 20 12 12

.

.

6

4

4

23 23 15 15

•

•

7

7

23 23 15 15

•

- [7] 7

.

19 19 11 11 5

15 [-] 3

19 19 11 11 5

15 •

9

1515 7

7 2 3 11

23 23

• , ~(!)9 9

1515 7

7 23 11

. 23 23

7

•

.

2

6

10 10

1016 8

8

• (2) _2

7

-

•

2

8

10 10

161618] 8 24112] • 24 24

2

2

. . . .

3

6

8

. . . .

12 12 4 (4) •

. ll

11

10 10

. . . .

2 0 2 0 12 12 •

6

19 19 11 23

. 11 11

10 10

. . . .

202012

[6] 4

1

OL

~)-9

~29

1

1

o~2

" ---,.%

--,28

1

k[3] 3

2~

1

1

1

1

1

1

23

o~

o~

-1

,'2~

--1

25

a 2

o~2

-- 1

--~A 4

-- l

~66

~z

o~2

- 1

~2~

- 1

3

121214] 4

O~

OL2

~q5

-~Q6

a o~2

1 a

--*q4 --~q7

-~ Qv ~O4

q7

0:'2

1

-'q6

~Q5

1

o~

~

-'a4

]

11

a~ O~

--'q~ ~q~4

~Q3 ---,Q~

I

1

a ~-

--*q~

-~O,

20

20 2 0 2 0 4 (4) 4

(~)

16 16

8

8

24 ( 1 2 )

12 12

( i i

20 20 12

12

20 20 12

qs q6

12 •

[-] 2 0 2 0

e

1

q4

,

(v)

1

~

_4 12

19 19 11 23

28

3

9

3

29

--*Q~

191919

20 29 20

3

1

6

4

1

1

6

• 18 18

4

-~2,

1

2

10102

_2 24 20 20 4

---,23

off

4

_2 24 20 20 4

~2~

1

2

3

~2_~

a2

14

1 13 25 3 ( ! )

1

q,

2

14 [21 4

9

a

q3

2

9 (1) 13 25 1

a ~'

-I

•

, 10 2

17 17 9

a:

1

16 16 12 12 20 20

17 17 9

22

1

17 5 17 17 17

( ! ) 17 17 9 21 9

1717

P × Vt,

2~

-1

~ (~) 17 5 17 17 17

17 17 9 5

Field

17 A u g u s t 1 9 8 9

-

•

20

(,) (6)

4

eel

Q3

(b)

q,-'Q,

--q,-*Qi

of symmetry breaking directions, it is straightforward to calculate the mass spectrum as a function of the two intermediate scales. Note that all mass terms involving a power of M~ and/or M2 greater than unity have a suppression factor, e ~ O ( 1 ), which is due to the non-perturbative origin o f the non-renormalisable terms in the superpotential [12]. The mass matrices for the leptons/Higgs fields are given in fig. 1, and for the quarks in fig. 2, for the case of 342 = M~/M~, so that y may be expressed as y = x 2. Each entry gives the power, k, of M~ in the lowest-order (i.e., the biggest) contribution to a particular mass term, i.e., k denotes a contribution of the form ex k 21y~M~=~xkM~ (0~<2/~
Fig. 1. Lepton mass matrices M2 = M ~ / M , : (a) lepton/Higgs doublets, and (b) charged lepton singlets. Each entry corresponds to the smallest power o f x contributing to a given mass term. Entries in parentheses denote the major contribution to a particular eigenvalue (mixing between entries of the same magnitude must also be taken into account ). The 2 × 2 and 3 × 3 submatrices indicated by those entries within brackets must be diagonalised more carefully. Underlined entries are independent of

m2. l o r YMc=M2. The absence o f an entry indicates that the coefficient of the corresponding mass term is zero to all orders; underlined entries are in-

xMc=M

dependent of 342. Note that the mass matrix for the charged lepton singlets, e c, is just the lower right submatrix o f the lepton-doublet mass matrix in fig. la. Similarly, the mass matrices for the singlet up quarks, u c, and the quark doublets, (u, d), are given by the lower right submatrix of the singlet-down-quark mass matrix in fig. 2a. the mass eigenvalues can be obtained through a straightforward (perturbative) diagonalisation of the four matrices. The orders of magnitude of these

57

Volume 227, number 1 5 (1) i 951(i) 9 9 9 5~9 9 5 5

222

5

3(1)

8

8

8

5 (!)9517 1 ] (i) 953199 1 95(i) 3 9 9

8 6

8 4

8 6

4

6

4

'17 3

~;

PHYSICS LETTERS B

8 8 8 8

8 4

395

1917

5

428 428

8 8

64 4 6

428 428

8 8

6 4

8 4

8 4

i 8 8 4 6

(1) /1) 3

4 6

5 1111 3 1111 5 1111

6

8

6

8

6

8

6

8

6

3 1175

3 1111

8

6

8

• 1606120 6

62

(2) 2 1 0 1 0 2 10 lO 6 6 2 lO 1 0 6 6 2 1 0 6 6 2 4 2 106 6 4 2

311 7 311 7 3117 311 7

,5 ~ - ( 1

3 1! ? 3 1175 3 1173

106 106

•

3 5 3 5

51111 31111 51111 31111

8(4) 2 886 84(_2)884 8 4 2 8 8

4 6 6 (4)

~184 ~, 8 s(~6

d 'c ,

d¢,

(a)

-(i4

-u~i'( 9 d

i

4 _88(

4 (2) 8 8 (4) _ 8 8

4

4 6

-¢, ( Zu

17 A u g u s t 1989

slightly more careful diagonalisation must be performed. For example, for the 3 X 3 submatrix, the eigenvalues are found to be xZMc, e2xSMc, ext°M,, for e > x 2, a n d x2M,., (:XTMc,e2xSMc for e < x 2. From table 3, the evolution of gauge couplings between the electroweak scale, Mz, and the compactification scale, M~., may be calculated as a function of x and e (and M,.). However, there is a physical constraint on the values of x and e, viz., that, whatever the spectrum may be, the mass of the lightest standard-model Higgs must be below 1 TeV, in order to allow for electroweak breaking. Table 3 tells us that the lightest pair of SU (2)L doublets have a mass of either ex ~°Mc o r e 2 X 8 M o depending on the values of x and e: but this is not the relevant Higgs pair. The electroweak Higgs giving mass to the chiral up-type quarks must have the quantum numbers of (h +, h °) in table 1, and originate (mainly) from the 27 representation of E6 ~2. Hence the relevant eigenvalue, which in this case is independentof M2, is ex3M~, and the constraint may be written as mHigg s = ( _ m l x - . . ~ 1 TeV.

(b)

Fig. 2. Quark mass matrices as for fig. 1: ( a ) singlet down-quarks, and (b) quark doublets and up-antiquarks.

eigenvalues are listed in table 3. The origin of each eigenvalue is indicated in fig. 1 or fig. 2 by parentheses around the appropriate entry in the corresponding matrix. A minor complication arises in the 2 × 2 and 3 × 3 submatrices in fig. 1a formed by the entries within brackets. The eigenvalues of these subrnatrices cannot be read directly from the entries: a

(2)

In the renormalisation-group analysis, all eigenvalues in table 3 smaller than mHigg~ were set equal to mHiggs. This can be attributed to other fields (e.g. singlets) obtaining VEVs at this scale. Irrespective of whether this is realistic or not, it is a conservative asE6, and the only (effective) couplings which efficiently give them mass are cubic, 273. Hence, the relevant Higgs should come (largely) from a 27; otherwise, mixing introduces several powers o f x, reducing the coupling unacceptably.

~2 The three chiral families originate from 27's of

Table 3 The order of magnitude of the mass eigenvalues, in units of the compactification scale. The eigenvalues of the lightest two lepton/Higgs doublets depend on whether e < x 2 ( g i v e n first ) or e > x 2 ( g i v e n second). Lepton/Higgs doublets

e c lepton singlets

d-type quark singlets

Quark doublets + u c antiquarks

13 x x M , . al

1 X6x2M,. ~1 3 X ex%~/,.

10 × x M c a) 1Xx2~/c 1 X ex2M~ a) 3 ×ex%~/~

1 Xex2M~ ~ 3 X ex4~[c

1 Xx2Mc 1 Xex2Mc ~) 1 Xex3Mc ") 3 X ex4M~ l X tx7Mc

o r e2xS~'v[c

1 X e2xSMc o r ~x'°Mc "~ These eigenvalues are independent of M2.

58

1 X(.x6M~. 1 XexleM~

Volume 227, number 1

PHYSICS LETTERS B

sumption without which the renormalisation constraints would be much stronger. Moreover, the absence of observable proton decay demands (conservatively) [14] that M~ > 10 ~5 GeV, and, so eq. (2) gives ~x2~<10 -~2,

(3)

implying that x a n d / o r e must be small. If e is small, then the renormalisation of gauge couplings is such that they explode below M,; if x is small, they explode between M, and M,, ( if not at a lower scale). An appreciation of the extent to which the above constraints are incompatible may be gained from fig. 3. The Higgs constraint, eq. (2), corresponds to the half-plane to the left of the dashed line labelled by H. The solid lines labelled by L, R, and C correspond to the values o f x and e at which the compactificationscale gauge couplings, c~L,a.c,(M~), explode. The halfplane to the right of each line corresponds to values o f x and e for which the appropriate gauge coupling remains perturbative between Mz and Me. These are the perturbation-theory constraints:

f

F

,

,

i

l

l

l

l

T

l

l

,

W2=M~2/Mc

20

o ..

--

L=R

0

--

_

"~

20

~Jllkl -10

i

I

5

-5

10

L o g io x

Fig. 3. The constraints on e and x for m 2 =M~/M~. The Higgs constraint corresponds to the half-plane to the left of the dashed line labelled by H; those of finite O~LR.Cto the half-planes to the right of the lines labelled by L, R, C. The shaded region is consistent with all four constraints, but is far from the "physical" region, x, e< 1. Finally, the dotted lines, L=C and L=R, together give the unification constraint, indicated by a square (or by "' +" and "'X" for sin20,,= 0.22, 0.24 respectively).

1

o~L,R,,:(Mc )

17 August 1989

>0,

(4)

where sin20w=0.23 and the one-loop beta functions have been used (see ref. [ 13 ] for details). However, the renormalisation-group equations really only make sense for x,~ ~< 1 and for aL,R,C finite and positive. The shaded region in fig. 3 is the region consistent with the Higgs constraint, eq. (3), and all the perturbation-theory constraints, eq. (4). It clearly lies a long way from the "physical" region, x,e ~< 1, and one must conclude that these constraints are incompatible. Finally, imposing O~c(Mc):O~L(mc) and o~R(Mc)= C~L(M,.) (given by the dotted lines), one arrives at the unification constraint O~n(me) =aR(Mc) = O!c(me),

(5)

which is marked by a square in fig. 3 (the " + " and " X " correspond to sin20w=0.22 and 0.24 respectively, with little change in the shaded region [ 15 ] ). However, this point is not physical: it lies outside the physical region and the coupling constants are furthermore negative, again illustrating the incompatibility of the various constraints. Very similar results have also been obtained for other choices o f M2. In fact, in this model, the specific value of M2 is largely irrelevant. It should be emphasised that the foregoing analysis is extremely robust, in the sense that the results are stable with respect to perturbations in mass thresholds, gauge couplings and the value of M2. Hence, for the above choice of VEV directions, one may conclude that discrete symmetries and order-ofmagnitude mass estimates cannot, without further input, yield acceptable phenomenology for this particular model. It may well be the case that the very general considerations presented herein will be relevant for many superstring models. Indeed, under the above assumptions, this is so for the entire class of models to which our example belongs [ 13 ].

Acknowledgement The authors acknowledge discussions with G.G. Ross.

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Volume 227, number 1

PHYSICS LETTERS B

References [ l ] P. Ramond, Phys. Rev. D 3 ( 1971 ) 2415; A. Neveu and J.H. Schwarz, Nucl. Phys. B 3 l ( 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. [3] E. Witten, Phys. Lett. B 155 (1985) 151. [4] S.T. Yau, Proc. Natl. Acad, Sci. 74 (1987) 1. [5 ] B.R. Greene, K.H. Kirklin, P. Miron and G.G. Ross, Phys. Lett. B 180 (1986) 69; Nucl. Phys. B 278 (1986) 667; B292 (1987) 606. [6]G.G. Ross, lectures given Trieste Summer Workshop (1987). [ 7 ] B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, Phys. Lett. B 192 (1987) 111; J. Distler, B.G. Greene, K.H. Kirklin and P.J. Miron, Phys. Lett. B 195 (1987) 41.

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17 August 1989

[8] S. Kalara and R.N. Mohapatra, Phys. Rev. D 35 (1987) 3143;D 36 (1987) 3474; R. Arnowitt and P. Nath, Phys. Rev. Lett. 60 (1988) 1817; 62 ( 1989 ) 1437; preprint CTP-TAMU-74/88, NUB#2960 ( 1988 ); preprint CTP-TAMU-82/88, NUB~2962 ( 1988 ). [9] M.C. Bento, L. Hall and G.G. Ross, Nucl. Phys. B 292 ( 1987 ) 400. [10] G.G. Ross, lectures given 1988 Banff Summer Institute, preprint CERN-TH-5109/88 ( 1988 ). [ 11 ] F. del Aguila and G.D. Coughlan, Phys. Lett. B 215 ( 1988 ) 93. [ 12 ] M. Dine, N. Seiberg, X.G. Wen and E. Witten, Nucl. Phys. B 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 ] F. del Aguila, G.D. Coughlan and M. Masip, preprint OUTP° 89-07P, UAB-FT-206/89 (1989). [ 14] J. Ellis, K. Enqvist, D.V. Nanopoulos and K.A. Olive, Phys. Lett. B 188 (1987)415. [ 15 ] L. Maiani and R. Petronzio, Phys. Lett. B 176 (1986) 120.