Intermediate scales in superstring models consistent with ten-dimensional symmetries

Volume 196, number 4 PHYSICS LETTERS B 15 October 1987

I N T E R M E D I A T E S C A L E S I N S U P E R S T R I N G M O D E L S C O N S I S T E N T W I T H T E N - D I M E N S I O N A L S Y M M E T R I E S

M. Q U I R T S Instituto de Estructura de la Materia, Serrano 119, E-28006 Madrid, Spain

Received 29 April 1987

We study intermediate-scale breaking along fiat directions (N) in four-dimensional theories consistent with classical symmetries of the ten-dimensional superstring. When supersymmetry is broken in the hidden sector by non-perturbative effects the flatness is lifted by non-renormalizable terms ( ~ m 2/2 I N I 4) in the tree level potential. One-loop gravitational corrections break supersymmetry in the observable sector providing a common mass to all scalars: If the effective theory obtained by integration of the heavy Kaluza-Klein modes is a small correction to the truncated theory, then: (i) Flat directions are necessary to stabilize the 0-fields (corresponding to non-fiat directions) at their origin; (ii) The tree level (plus one-loop gravitational corrections) minimum is at (q~) = 0, (N) ~> n ~nm 4/4M 3/4 ~ 10 ~ 5 GeV, where mo N O ( 1 ) TeV is the scale of supersymmetry breaking in the observable sector and n-~ O (102) the total number of scalar superfields; (iii) At this minimum the soft-breaking scalar masses in the observable sector are m~ ~ (N) 2Mp-2 mN2 << mN;2 (iv) Non-gravitational radiative corections will communicate the supersymmetry breaking m~ from the N-sector to the 0-sector, providing an effective breaking ( m o ) a f = m o 2 - - z ~ (aotrr/n)2 m2 . The critical temperature associated with the N-phase transition is Tc ~ (n/aGv~c) mo ~ 104 GeV. The softening of cosmological problems and the possibility &low-energy baryogenesis at temperatures below Tc are briefly outlined.

The ten-d imens ional heterot ic string [ 1] with gauge group E8 × E~ is a good candida te to (consistent ly) unify gravity with the strong and electroweak interactions. When the theory is compact i f ied on M4 X K, K being a compact s ix-manifold with a Kgh- ler structure, S U ( 3 ) ho lonomy and spin connect ion embedded into the gauge group [2] , there is an unbroken N = 1 supersymmetry in four dimensions. Massless mat te r superfields are neutral with respect to e; and t ransform like ng 2 7 + ~ ( 2 7 + 2 7 ) with

respect to E6, where ng= [b l ,2-b l . l [ and f i = m i n (bl,l, bl,2) are de te rmined by the topology o f K [2] . More- over i f tel (K) S 0 E6 can be broken down by the flux breaking mechanism [3,4] to a smaller gauge group H which contains S U ( 3 ) × S U ( 2 ) × U ( 1 ) . Light mat te r fields t ransform under H as [ 4 ] ng complete 27 generat ions plus some self-conjugate parts o f ~ ( 2 7 + 2 7 ) .

Inc luded in__the charged sector there are two fields in each 27 (27) which are s tandard model singlet (a conjugate sneutr ino and an SO (10) singlet) . We will denote them generically by N ( N ) . The potent ia l is flat along the di rect ion N = lq* i f ~N > O. In that case

the field N that corresponds to the fiat d i rect ion can acquire a vacuum expectat ion value at some intermediate scale [ 5 ] ( N ) = MI >> Mw provided that: (i) The flatness o f the potent ia l is l i f ted by non-renor- mal izable terms, a n d ( i i ) a negative square-mass (soft-breaking) te rm for the N field is deve loped at the origin.

The usefulness o f in te rmedia te mass scales is two- fold: ( i ) Included in the 27 of E6 there are self-conjugate color tr iplets (weak singlets) quarks, D and D e , that can media te pro ton decay by dimension-f ive operators . D-quarks get a mass f rom the superpotent ial coupling N D D c so a large enough value o f Mi can al leviate the pro ton stabil i ty problem in superstr ing models. ( i i ) In models with several self- conjugate generat ions the gauge coupling constants can blow up very fast and a Landau singulari ty can appear at scales below Mp. However , all the fields coupled to N in the superpotent ia l will get decoupled at scales below MI and (depending on the value o f MI) the Landau singulari ty can disappear .

In termediate-scale models present some difficul- t ies [ 6 -9 ] which have been recently po in ted out. On

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d P h y s i c s Publ ishing Div i s ion)

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the one hand the analysis o f baryon decay via dimension-five operators [10] strongly suggests that m I ~ 1016 GeV [ 9 ]. On the other hand the radiative scenario for the intermediate-scale breaking [ 11 ] is in trouble [9] for such a high value of MI because the N-field has no renormalizable interactions below that scale and the logarithmic range between Mp and MI is so small that the needed Yukawa coupling might be inconsistent with the perturbative expansion and the very renormalization group resummation may not be appropriate. Finally, the proposed mechanism to lift flatness of the potential has some inher- ent problems. First o f all the presence o f non- renormalizable terms in the superpotential (non- renormalizable supersymmetric terms) as (27 27)", n >i 2, would trigger an intermediate scale at [ 5,12 ] Ml ~ m l / ( 2 n - 2 ) M ~ , 2 n ' 3 ) / ( 2 n - 2 ) , where mo < O(1 ) TeV

is the scale o f supersyrnmetry breaking in the observable sector, so for n = 2, M~ ~ 10 ~L GeV is not large enough to solve the proton stability problem. Actually the condition Mx > 1016 GeV requires n >~ 4 and very specific discrete symmetries o f the internal manifold would be required to prevent smaller values o f n ~ The second problem concerns the possible origin o f non-renormalizable corrections to the superpotential. Non-renormalization theorems [ 13 ] for F-terms obeyed by string loop and sigma model corrections to all finite orders of perturbation theory prevent any correction to the superpotential among massless superfields that would be derived from ten-dimensional field theory [ 14] (i.e. 273) coming from integration of massive string states #2. On the other hand an explicit calculation of the most general superpotential consistent with the symmetries of the ten- dimensional field theory has proved [ 16 ] non-renormalization of the superpotential from integration o f heavy Kaluza-Klein modes.

A first step towards the solution o f these problems was given in refs. [ 17,18 ]. There we have built the

most general four-dimensional effective action consistent with classical symmetries of the Es×E~ superstring [ 16,19] and compared the result with the theory obtained by performing a truncation [ 14,16,20 ] of D = 10 supergravity in which all heavy Kaluza-Klein modes are discarded. In the particular case [ 14] b1.1 = 1, b~,2=O (one generation) the truncated theory is exact because the only (1,1)-form is the covariantly constant form defining the Kghler structure of the internal manifold [ 2 ] and the heavy Kaluza-Klein modes do not couple linearly to the massless modes. In the general case bl , l> 1 and/or bl,2> 0 integration of heavy Kaluza-Klein modes will modify the truncated theory [ 16,21] and we have found that many features o f the latter one were just artifacts o f the truncation approximation [17]. In particular, flat directions N = lq* are lifted by non- renormalizable terms in the potential which vanish in the truncated theory [17] and behave like #3 m 2/2 I NI 4. Therefore these terms can trigger an intermediate scale breaking if supersymmetry is broken in the observable sector.

In this paper we will analyse the possibility o f intermediate scale breaking along a flat direction in the general class of four-dimensional theories (stud- ied in ref. [ 17 ]) consistent with the classical symmetries o f the ten-dimensional superstring. The charged scalar sector will be denoted by ~i= {N, ~a), where N is the field corresponding to the flat direction ~4 N = N * and ~a are the rest o f fields, corresponding to non-flat directions.

The K~ihler potential consistent with the ten- dimensional symmetries is [16,17] (in units of (87r) - 1/2 Mr)

G = - l n ( S + S * ) - 3 I n ( T + T*) +h(u) + I n IPI 2, (1)

where S and T are E6-singlets [ 13,14] and #s

u = 2 1 N I 2 + l ~ " l 2. (2)

~ A model with discrete symmetries of the internal manifold preventing n = 2 and with MI ~ 10 TM GeV has been presented in ref. [12].

~2 Of course non-perturbative effects can violate the non-renormalization theorems. The case of world-sheet instantons for certain compactifications is an example and was explicitly exhibited in ref. [ 15 ]. We will not consider the effect of world- sheet instantons throughout this paper.

#3 Notice that in the supersymmetric limit m3/2---~0 these terms vanish, the direction N=/q* remains fiat and the non-renormalization theorems [ 13 ] are not violated.

oq We are assuming for simplicity that only one direction is available for the breaking of H. This happens, in particular, if X is an Y-space: 8 =b1.1 = 1. In that case H has to be rank-six and there is an extra U(1 ) at low energies. The generalization to several flat directions is straightforward.

~ We have rescaled the matter fields as t- 1/20i~ q~t, with t--- T+ T*.

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The superpotential in (1) is P=g2(S) +t3/2W(~) i) where t2(S) = c + r exp( - 3S/2bo) is the superpotential generated by gluino condensat ion in the hidden sector and a non-vanishing torsion for the internal manifold [22], and W=dijgq)i#~ k is the superpotential o f the observable sector. The scalar potential corresponding to (1) is given by [ 17 ]

Vtree = Vl ~- V2 -~- V3 ~- VD, (4)

where

VI - exp [h (u ) ] I P - ( S + S * ) P s I 2 ( S + S , ) t 3 , (5)

exp [h (u ) ] g 2 - (S+S*)h ' (u ) IW'12' (6)

exp [h (u ) ] 2(u) V 3 - (S+S*) t 3 a(u) f l (u )h ' (u )

× l uh'(u)(2+ t3/2a(U) W 12, (7)

[h ' (u ) ]2 d~d ~, (8) VD= S + S *

where

2 = ( h ' ) 2 - 3 h ", a = 3 + u h ' , f l=h '+uh" ,

d ~ = 0~(T'~)j~ j. (9)

The function h is constrained by the positivity of the matr ix G}, which determines the kinetic lagran- gian of the chiral superfields, and by the semi-positive definiteness of the scalar potential (4) to satisfy the conditions [17] p ( u ) > 0 , h ' ( u ) > 0 and 2(u) /a(u) >>. O. The flatness of the potential along the N field is lifted by V3, eq. (7). We see that there is no sof t -supersymmetry breaking in the observable sector ~ at tree level.

The one-loop contr ibution to the effective potential (including gravitational radiative corrections) is given by [ 17 ]

=--am3/2m3/z(u ) [ 1 - -p (u ) u] V l - l o o p 2/3 2

- a(gGuw,/2) rn~: 3 m3/2 (u) exp [ h (u) /2 ]

× ( 1 - ( 3 n + 1 0 ) 2 ( u ) ~ h ' ( u ) a ( u ) B ( u ) ] [ W+h.c . ] +0(m4/2) ,

(10)

where

p(u) = 2 ( n + 1) 2(u)/o~(u)fl(u), (11)

n is the number of chiral superfields, n ~ O ( 1 0 2 ) , a = (gGux/4g) 2 (bogcvT/6r) 2/3, and the gravitino mass is given by

exp[h (u ) ] 1£2(S)12, (12) m~/2(u) - ( S + S . ) t 3

with m3/2-m3/2(0). From eq. (10) we see that one- loop corrections to the effective potential provide a c o m m o n soft-breaking mass to all scalars in the observable sector (first term in (10)) given by - m2u, where

m2=am8~3h'(O) (1 - r/), (13)

with

q = 2 ( n + 1 ) 2 (0) /3 [h ' (0 ) ] 2, (14)

and a soft-breaking trilinear coupling (second te rm in (10)) A ~ m ~ 3 M ~ 2/3. The soft-breaking induced by the trilinear coupling is negligible as compared to that induced by scalar masses, A~m(m3/2/Mr,) 1/3 ~ m(m/Me) 1/4 << m, and we can consistently neglect the second te rm of eq. (10).

The whole theory is governed by the parameter r/. I f r/> 1, m 2 < 0 and the origin along the N and ~a directions is stable. In that case a non-vanishing ( N ) at some intermediate scale can only be generated by non-gravitat ional radiative corrections between Mp and MI. This scenario was analysed in ref. [ 17 ] and

~lzr .c ~ 1/4 ;I,4-3/4 1014 leads to ~,11 ~,,~ o ~vl v ~ GeV, with mo < O (1) TeV, whilst the above ment ioned difficulties associated with the radiative breaking in such a small logari thmic interval [ 9 ] still remain. On the other hand if r/< 1 ~6 the origin along N and ¢~ is unstable. However we will prove that there is a stable mini- m u m at ( N ) ~0 , ( ~ a ) =0 . The reason being that a non-vanishing ( N ) generates in (7) a positive square-mass te rm for the ¢~-fields which exactly can- cels the negative contribution f rom (10). Notice that i f r/< 1 the existence of fiat directions is a necessary requirement for the stability o f the theory.

~6 Notice that r/is equal to zero in the truncated theory [ 14]. On the other hand integration of heavy Kaluza-Klein modes gives rise to terms which are suppressed by inverse powers of the heavy mass. Therefore it is thinkable (at least for some compactifications) the effective theory to be just a small perturbation of the truncated one, so q < 1 might be a plausible case.

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We will now compute (N> at ~"=0. The only non- vanishing contribution to the tree level potential comes from V3 and can be written as

Vtree (N,q} a = 0 ) = 4 ~ ( 0 ) m2/2 ] N [ 4 [ 1 + O ( INI2)],

(15)

while the one-loop contribution is

1 Vl_loop(N,¢ a = 0 ) = h ' (0) (1-~/) m2

-2m2INIZ[ I+2AINI2+O(INI4 ) ] , (16)

where A is the constant

A= ½ h"(O)-p ' (O)-p(O)h'(O) h ' ( O ) - p ( O )

(17)

The minimum of the total potential

V = Vtree -~- Vl_loop ( 1 8 )

is at

N 2 Ng = 3 m 2 - - (19 ) IS12= 1 - 4 A N 2 ' 42(0) m2/2"

We will now prove that the point ~a = 0 and N given by (19) is a minimum of the total potential (18). Concerning Vl_Joop, only the first term in (10) will contribute to the soft-breaking square-masses in the observable sector because W(0~=0)=0 . A straightforward calculation gives, at Oa = 0

dVl-l°°p m2 4m2(A+~) INle[ 1 +O(INI2)], d[O. 12 -

(20)

where A is given by (17) and

1 h ' ( 0 ) 2 27= 2 h ' (0) - p ( 0 ) " (21)

Notice that the negative sign in (20) seems to point towards a destabilization of the potential at the origin. However the important feature of the tree level potential (7) is that the same mechanism that lifts the flatness along the N direction and triggers intermediate-scale breaking will produce positive soft- breaking square-masses in the non-flat sector. They

can be readily computed, at ~ a 0, as ~7

dVt~e 4 2(0) m~/2 INI 2 d l&[2 =~

× [ 1 +9A INI z + O ( / N I 4)], (22)

where A is the constant

1 &'(O) 1 h"(O) 2 A= 3 2(0) 3 h'(O~ + ~h ' (O) . (23)

Using now eq. (19), eqs. (20) and (22) can be cast as

d VI _loop m 2 d[+:12 - - 4 m 2 ( A + X ) N~+O(mZN4),

d Vtree ~l-(-~i ~ =m e + mZ( 4A+ 9A) N~ +O(mZU4) ,

(24)

from where the soft-breaking mass in the non-flat sector can be written as

m~o = ( 9 A - 4X) NZm 2 +O(mZN4) , (25)

and the stability condition in the observable sector is

9 A - 4 X > 0 . (26)

Up to now we have proved that the field configu- ration Ca = 0, N given by (19) is a local minimum of the total potential (18), modulo the stability condition (26). To prove it is the global minimum we should prove it is either the only minimum or the deepest one. Of course we cannot exclude in general the existence of other minima in the field configu- ration space at N = 0 and some non-fiat fields Ca ~s 0. However the presence of renormalizable terms for non-fiat directions would trigger a vacuum expec- tation value (~a) ~ m and the depth of these minima would be ( ~ m 4) much smaller that the depth of the minimum (19) (~m2Ng) .

We have seen that a non-vanishing ( N ) will produce soft-breaking scalar masses in the N-sector, m ~ m 2, and in the ~a-sector, ~2 Ay2m2~lZr--2

which is much smaller than m 2. On the other hand

~7 Notice that we have not included in (22) the supersymmetric mass of self-conjugate fields in the 27, Xi, X7, that can be coupled to N in the superpotential, XiXTN, and get a mass pro- porfional to ( N ) . These fields are decoupled below MI.

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supersymmetry breaking will be communicated from the N to the ~a-sector through (two-loop) non-gravitational radiative corrections ~7, providing an effective mass

m2o ~ (o~6uT/n)2m 2 <O(1 ) TeV 2, (27)

where the factor (OZGuT/n) 2 in (27) comes from the two-loop calculation. For No < 10 2, rn~ << mE and (27) will be the main source of supersymmetry breaking in the scalar masses of the observable sector.

Notice that a non-vanishing ( N ) will also produce soft-breaking trilinear couplings from (7). However it is easy to see that A~;t(0)m3/2 × N g ~ m2/m3/2 ~ m ( m/Mp)1/4, which is of the same order of magnitude as the trilinear soft breaking induced by the second term of the one-loop effective potential (10) and can therefore be consistently neglected as compared to (27).

We will now estimate the order of magnitude of the different scales appearing in the theory. The effective supersymmetry breaking in the (non-flat) observable sector mo has to satisfy the condition (27) to protect the gauge hierarchy from high-energy destabilization. Using now in (13) ~8 bo = 3 C2 (H')/16n2, with C2(H' )<4 , and ~9 r=O(1) in the superpotential £2(S), the factor a in (13) takes the value

0"~ (gGUT/47~2) 2 " O ( 1 ), (28)

the breaking mass in the N-sector

m ~ (gGUT/47~ 2) m4~32Mp ~/3, (29)

and the effective scale of supersymmetry breaking in the observable sector

mo ~ [g3UT/(47C2)2 ] 'rt3/2sva~4/3 ;1/--1/3p , (30)

where we have made use of (13) and (17)-(29) . Taking now gGux~O(1) and mo bounded by (27) we obtain

m3/2 <35< 109 GeV, (31)

which is consistent with the bound C2(H') < 4 used in (28), and

m ~ ( 4 z c 2 / g ~ u T ) mo ~ 10 T e V . (32)

~8 H' is the biggest subgroup into which E~ is broken down by the flux breaking mechanism and responsible for gluino conden- sation in the hidden sector.

~9 This bound is obtained from m3/2 ~ exp [ - 8~rZ/g~UT C2(H')] Mr,, mo~m3/zMr,4/3 -1/3, see eq. (13), and the bound (27).

Notice that the (radiatively induced) scalar mass mo, eq. (27), is the main source of supersymmetry breaking in the observable sector. The trilinear soft breaking is

A/mo ~ (47~ 2) ( m / M p ) TM ~ 10 -2, (33)

while the gaugino mass induced by the one-loop gravitational effective potential is [23]

ml/2 ~ (OZcuT/4Zt2)4/3m~3My2/3 < 10 GeV, (34)

according to the bound (31 ). Finally, using (19), (27) and (30) the intermediate scale can be written as

No 1/2 2 1/2/~-1/2(0 ) ml/4M3p/4, ~ (3g~m-/16zc) (35)

and numerically, using (14) with n -O(102) , we obtain

No ~t / -m.1015 GeV, (36)

where q< 1. We will conclude this paper with two comments.

The first comment concerns the feasibility of the physical picture drawn here. We have stated that the whole low-energy theory depends on the parameter r/in eq. (14), which in turn is induced by integration of the heavy Kaluza-Klein modes. Actually in the truncated theory (where heavy modes are put equal to zero) q=0, i.e. [14] h o ( u ) = - 3 1 n ( 1 - 2 u ) . Although an explicit calculation of the effective theory via integration of the infinite number of heavy modes is not available, we are tempted to speculate about a possible example where the factor of 3 in ho(u) gets corrected by a tiny amount, i.e. hcff(u) = - [3/(1 - e ) ] In ( 1 - 2 u ) with 1 >> E>0. In this case 2elf(u)=Eh'Zeff(U) and it satisfies all the required positivity conditions. The parameter r / in (14) is related to e by t / = 2 ( n + 1)d3 and the stability condition (26) translates into the mild condition r/< 1/3.

The second comment concerns cosmology. The potential V(N), eqs. (15) and (16), is very flat (because mN/(N) << 1 ) and there is a late first-order phase transition [6,7] at T¢~ mN with a huge entropy release that will dilute the baryon-to-entropy ratio and so low energy baryogenesis is required. A first observation is that the critical temperature is now T c ~ 104 GeV, higher than in the usual scenario of intermediate-scale breaking [6,7], and some cosmological problems may now be alleviated. In par-

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t icular p r imord ia l nucleosynthesis puts the severe l imit [ 7 ] MI < 1021~m ~2, where ~ is the pa ramete r in the decay rate o f the N-f ie ld F N ~~emN/M~3 e and can be O(1 ) i f there is a large number o f contr ib- uting intermediate heavy states and decay modes [ 6] for N. Using the value (32) for mN and the absolute bound ~ < O ( 1 ) we obta in the harmless bound MI < Mp. A second observat ion is that the high value of Tc would seem to favor mechanisms for baryogenesis at the electroweak scale. In par t icular the Aff leck-Dine mechanism [24] for producing baryon asymmet ry f rom decays of coherent oscil lat ions of squarks and sleptons, ¢, in conjunct ion with the fiat direct ion N has been recently analysed [ 9 ] in the usual in termediate-scale breaking scenario. As in ref. [ 9 ] the most interest ing case arises when the reheat- ing tempera ture after inflat ion is TR < (m M~) 1/2 (i.e. TR< 109 GeV for M I = i014 GeV, TR< 10m), as use to happen in supergravi ty inf la t ionary models, and the N-symmet ry is not res tored after inflation. A straightforward general izat ion of the results o f ref. [9 ] shows that the generated baryon-to-entropy ratio is

nB/s ~ ~O(b 2 mo ( M p / m ) 1/2 M~- 1M~2, (37)

where 0 is the CP-violat ing angle [24] , ~o the ini t ial ampli tude of the ~ f i e ld after inflation, given by [25 ] q)2o ~HMp, H being the Hubble constant during the inf lat ionary epoch, and M x the grand unif icat ion scale. Taking for H the absolute upper b o u n d [26] dictated by energy density fluctuations, H < 10-5Mp, for Mx the conservat ive lower bound M x ~> 1017 GeV and for the parameters ~ and O, {, 0 < O(1 ), the condi t ion nB/s>lO -1° t ranslates into M~<Me. Thi s means that producing baryon asymmet ry requires rather special condi t ions that will be model depend- ent. Actual ly i f the parameters ¢ and /o r 0 turn out to be much smaller than one, or the Hubble constant much smaller than its upper bound, the baryon- to- entropy rat io for values o f M~ solving the pro ton sta- bi l i ty p rob lem would be too small for p r imord ia l baryogenesis.

I wish to thank John Ellis for useful in format ion concerning refs. [9,25 ]. This work was par t ly sup- por ted by CAICYT under contract AE-85-0023-3.

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Intermediate scales in superstring models consistent with ten-dimensional symmetries