The one-loop effective potential and supersymmetry breaking in superstring models

= 4M3/2 7a 2 Since M is generated non-perturbatxvely, it may be much smaller than My, allowmg for small values o f c Furthermore, in certain compacnficatlons M and a may be determined by the complex structure which, m turn, is determined by the VEVs of gauge-singlet scalar fields In this case, c will not be determmed on compacufication, but only after mimmlsing the effectwe potentml. We shall &scuss in detail each of these posslbd~ties for c We now turn to the specific form o f the low-energy lagranglan. In what follows, we ignore the gauge-slnglets, N, except where reference to c is made. In parncular, as a first approximation, the contnbuUon to V(~ eft~ coming from the smglets xs neglected, it should, however, be included and we leave the analys~s of this to a future pubhcatlon. The effective d = 4 supergravlty theory is defined by [ 4,16 ] f.p = S6~p, G=-ln(S+S)-31n(T+T-21~{

2)

- l n l P ( @ ) + W(s) l 2, P(@) =d.b~¢~¢vo~, W(s) = c + h exp ( - 3S/2bo)

(1)

Here. P ( ~ ) is the superpotentlal for N chlral superfields ~ , and W(s) is the effective superpotentxal for S reduced by the supersymmetry breaking The corresponding scalar potential is

~2 We use the same notation, c, for the combined effect of a vacuum expectatmn value of P ( N ) and/or of the ant~symmetnc tensor field

Volume 198, number 4

PHYSICS LETTERS B V(I ) ~

1

~rr

V (°) = ( S + S ) ( T + T - 2 1 q ~ I 2 ) 3 -v

D-terms, (2)

and is mlmmlsed by (@) =0,

(3a)

(S)=~bocoS.t.(l+co)exp(-Ico)=-c/h

(3b)

Note that h is a constant of order the compacnficanon scale For definiteness, we take h = - 1 so that c may be expressed m terms of co alone. h=-l,

c = (1 +co) e x p ( - 1 co).

(3c)

The value o f ( T ) ~s undetermined at tree level and hence so is the gravxtmo mass. m312=

1

(S+S) (TWT) 3

col---~ -

+AV">,

AV") =V(m,,Ac)+[V(fit,,AG)-V(fit,,Ac)]

× )P(@) + c + h [ 1 +3(S+S)/2bo] exp( -3S/2bo)12 + 6 ( S + S ) (T+7"-2[~]2) 2

V (0)

3 December 1987

(4)

In order to fix both ( T ) and m3/2, It lS necessary to compute the one-loop effective potentml The m m a l attempts computed v- - e(~) f f by integrating the loop mom e n t u m up to the condensate scale A~, above whmh the gaugmo condensate d~ssolves They found, as already alluded to, that r/(~> - - e f t decreases without bound as T - , 0 +, suggesting that the perturbat~ve analysis breaks down Recently, however, Bxn6truy et a l [ 1 1 ] have discovered a possible perturbat~ve solunon to the problem, arguing that the contribution to ,vO) err from the m o m e n t u m region between Ac and the compacnficatmn (or G U T scale), AG, stabdlses the potennal as T - 0 + and gives a global m l m m u m for a fimte gravm n o mass For p2>A~ the g' gauge couphng becomes weak and the gaugmo condensate d~ssolves, leawng (H/,,,~) and/or ( P ( N ) ) as the only supersymmetry-breaking effects (We discuss this in more detail shortly ) The mass spectrum of the theory ~s now changed dramancally, as can be seen by setting h = 0 , c ~ 0 m eq (2) (see table 2) Most importantly, the gauge-non-sanglet scalars ¢~ become massive and, being bosons, make a postttve stablhsmg c o n t n b u t m n to Ve(1f t ~ The full expressmn for the oneloop potential ~s therefore

(5)

The one-loop c o n t n b u t m n comes m two parts. The first, V(m,,Ac), ~s from performing the loop integration from p ' - = 0 to p2=A2, w~th the mass spectrum given by m,. The second contribution (shown m brackets) comes from integrating between p2= A;9 a n d p 2 = A 8"~ w~th the modified mass spectrum fit, The general expressmn for V(M, A) is

V( M, A) -- (1/64~z 2) Str[ A2 M 2 + A 4 ln(1 + M2 /A 2) - M 4 In(1 q-A2/M2)]

(6)

When eq (5) ~s implemented, the potential becomes pos~uve defimte (for Nlarge enough) and the problem o f refs [ 6 - 8] is c~rcumvented However, the global m i m m u m now lies at T - , m , correspondlng to a supersymmetrlc (m3/2 = 0), uncompactlfied ( A G = 0 ) ground state Nevertheless, an acceptable mmxmum for a finite value o f T was obtained by B1n6truy et al [ 1 1 ] through the introduction o f varmus parameters whach, it was argued, reflect uncertainties m the theory Such a parameter p>~0 (10) was needed m ref [ 1 1 ] It gave rise to a term m V{~I,) of the form -Str M 2 In p whmh produced the m i n i m u m The origin o f this term was argued to be the prescription-dependent fimte parts in the renormallsanon We note, however, that since AG and A~ are physmal (fimte) scales, there are no lnfinites in - (~) calculating v~f r Furthermore, the a d d m o n o f thxs term actually leads to a potential unbounded from below m the small-T reglme where m,, fit, >>Ao, Ac In the next section, we reanalyse the suggestmn of Bm6truy et al [ 11 ] to show that eq (6) needs to be modified shghtly to take account of mass thresholds, and that this mod~ficatmn leads naturally to a bounded potennal and a global m l m m u m . Furthermore, this m l m m u m gives rise to an acceptable hmrarchy

3 Mmlmtsanon of the one-loop effective potential Let us first estabhsh some n o t a n o n A convenient variable to define is Z=\

(m3/e )2=~ coe e-°~/(T+ f') 2 AG /

(7)

469

Volume 198, number 4

PHYSICS LETTERS B

In the last equahty, we have used the tree-level mlnl m i s a t m n c o n d m o n s eq (3) We note that there are three variables u p o n which V(l --eft) d e p e n d s S, T a n d c ~3 Eq (3b) reduced these to two independent ones, which m a y be taken to be Z and o)=3(S)/bo All q u a n t m e s m a y now be expressed m terms o f these variables In particular, we have

m~/2 = (12/bo) (1/oa 2) exp( ½09)

Z 3/2 ,

A ~ - (Re S Re T) - i = ( 1 2 / b o ) (1/0) 2) exp(½ o~) Z t'~,

T + T = ½~o exp( - ½o9) Z-~/2, A~ = e x p ( - ~ o~) A~.

(8)

In table 1, the mass spectrum in the two regimes o f the loop m o m e n t u m ~s hsted m units o f the appropriate g r a v m n o mass The raUo o f the mass o f the g r a v m n o in the two regimes is (1~l~/2/m3/2) 2 = (l "{'-1/09) 2

(9)

Evaluatmg eqs ( 5 ) a n d (6) with these spectra leads to an expression for A V ~ that a p p r o a c h e s zero as Z ~ 0 . and, as Z ~ , "~ The scales A(, and ,1~ are not independent quantmes the) actually depend on ( T ) and ( S ) Furthermore, h is a constant which we have taken to be h = - 1 Table 1 Mass-squared elgenvalues m the two regimes of the loop momentum in umts of the relevant gravltlno mass N a n d N¢, correspond to the number of gauge-non-smglet scalars and gauglnos respectively

scalars S T NXO"

p2
A~
(1 + 6a)-'m~,2 0,0 0, 0

2ff1~ 2, 10t~/~,2 0,0

2th~z2, 0

fermlons

Zs

o92m~,2

thai2

Zt

4m~/2

Zo

0

9rh3,2 0

0

rh~,~

m~,

r~ 2

gaugmos NoX2 gravltlno

~',,

470

t mlK

3 December 1987

AVt~ ~ c o n s t × Z l n Z

×[(N-2N~-4)-(Ac/A~)4(N-2NG)]

(10)

This is p o s m v e as long as co>3/21n[(N-2N~)/ ( N - 2N6 - 4) ], and N is large enough, showing that the b a d behavxour as T--,0 + m a y be avoided. We are now in a positron where some mterpretan o n is necessary before "v ell ~1 is b h n d l y calculated It should be noted, first of all, that although ( ~ ) = 0 as soon as the m o m e n t u m c a m e d by the gaugmos exceeds Ac, the mass spectrum does not change abruptly Consider, for example, the c o n t n b u U o n s to the mass o f the gauge-non-stagier scalars ~" xn fig 1 At tree level, m~ = 0 due to the cancellauon of contrlbuUons from ( ; ~ ) and c - regardless o f p 2~ Even i f the m o m e n t u m carried by the m a t t e r fields p2 >> A~, the fracUon of that m o m e n t u m carried by the gaugmos is zero at tree level and ( ~ ) ¢ 0 However, higher loops r e n o r m a h s e these c o n t n b u u o n s differently (fig l b ) a n d the cancellation ~s spoilt as/7 gets large One expects, therefore, that the scalar masses switch on slowly (logartthmlcally), at p2 ~A~, and it is possible that even at p2=A~, the masses m a y only be fractions of the values shown on the right-hand sxde o f table 1 The above &scusslon is m e a n t to m o t i v a t e a parameter K, which we introduce to account for the slow change m the mass spectrum above p2=A~ In thxs regime, all gauge-non-smglet scalar masses ~4 are scaled relaUve t o m~/2 by a n u m b e r 0 < K < 1, ~ e , n~=2K(l+l/09)

2m3/2

(P2>A~)

(11)

The precise value of K we shall treat as a p a r a m e t e r for now, but it ~s, in principle, calculable from graphs such as fig l b It ~s worth noting that the b e h a v m u r o f "v t.ff ~1 at large Z ( Z ~ o o ) is i n d e p e n d e n t o f K So K only modxfies the s m a l l - Z b e h a v i o u r o f the potential, and, as we shall s e e , produces a m i m m u m But, we need more than just a m m t m u m , we need a hierarchy between the Planck scale a n d the effecUve scale o f s u p e r s y m m e t r y breaking m the low-energy theory. To generate such a hierarchy, the value o f ( Z ) must be reasonably small (1 e., << 1) Eq (7) suggests that this might be o b t a i n e d through .4 It m~ght be more reahsUc to interpolate between m 2 and rh 2 for all fields, except gauglnos However, such an interpolation did not change the results slgmficantly

Volume 198, number 4

PHYSICS LETTERS B (X ?,)

3 December 1987

~c q~a

I

q~ . . . . . . . . P

0 ~ + 4)< . . . . P

I. . . . . . I i



+

.... P

- ~I - . . . .

~°= 0

I I

~c

(a)
--

p

c -~o

+

&_~_~2 _ t ..... P

+ •

~0

,I



~c

(b) Fig 1 (a) Tree-level c o n t n b u t m n s to gauge-non-smglet scalar masses from ( ~ ) and c These contrlbuUons cancel at the tree-level m i n i m u m (b) At the loop level, they are renormahsed differently and the tree-level cancellanon is spoilt at high m o m e n t a v = 0 h, N or S

rather (t) a large value o f ( T ) , or ( n ) a large value o f o) Each o f these possibilities can be reahsed dep e n d i n g on the values o f the parameter, see fig 2 We have also found it ~mportant to introduce a second p a r a m e t e r r / t o allow for the p o s s l b l h t y that the condensate scale A~, whtch we define in terms o f the condensate

(7~>=-A2,

(12)

ts different from the m o m e n t u m scale Peru, at which

V(1) eff

i=

(w)

Fzg 2 The form o f the one-loop effective potentml in scheme I and s c h e m e II for c o n s t a n t S = ( S )

the new mass spectrum is relevant. Thus we write Per,, =~TAc,

(13)

where we expect r/=O(1 ), but not exactly equal to unity In fact, a rough calculation revolving graphs o f the form o f fig l b ymlds 10-2~<~/~< 1 In practice, this means that A¢ should be replaced by qAc in eqs ( 5 ) and (10) and m table l, but nowhere else This effectively Increases the interval for which the m o d i f i e d mass spectrum (r~,) ~s relevant, a n d increases the hierarchy, 1 e., small r / l e a d s to a large ( T ) and hence smaller ( Z ) ( 0 S c h e m e I, c f i x e d W i t h c fixed during compactlficaUon and o) fixed at tree level, V~rt) is a functmn o f Z only (equivalently, it m a y be v~ewed as a function o f T only) Now, V(e~r~ ) m a y be m i n l m l s e d with respect to Z, with q adjusted to generate the required hierarchy and K fine-tuned to g~ve a zero cosmological constant It is i m p o r t a n t to reahse that fine-tuning K to cancel the cosmological constant simultaneously finetunes the hierarchy. In parttcular, if K ts n o t finetuned the cosmological constant ~s typmally large and negative, a n d Z = O ( 1 ) despite q being small Hence, the cosmological constant p r o b l e m a n d the hwrarchy p r o b l e m have been connected m such a way that o n e fine t u n m g fixes both 471

V o l u m e 198, n u m b e r 4

PHYSICS LETTERS B

3 D e c e m b e r 1987

Table 2 C a s e A S c h e m e I, w i t h c = O ( I ) [ a n d h e n c e f r o m eq ( 3 b ) to = O ( 1 ) ] , ~ m u s t be small to g e n e r a t e a n a c c e p t a b l e h i e r a r c h y N = 7, N~ = l C a s e B S c h e m e I, w~th q = 1 a n d c<< 1 (~ e , ~o >> 1 ) N = 7, N G = 1 C a s e C S c h e m e I, w~th q small a n d o) large, b u t ( S ) a n d ( T ) o f order umty N=7, No,= 1 Case

17

A

1 10 - ° 5 10 - I 10 - ~ 5 10--"

B

C

(aJ)

(Z)

,+'bo m312

3 3 3 3 3

51×10 40×10 52×10 71X10 92X10

-3 -> -v -9 -tL

47X10 12×10 47×10 19X10 73X10

1 1 1 1 1 1

3 12 24 36 48 60

51X10 74X10 49X10 47X10 53X10 63X10

-s -s -7 -~ L~ -~a

4 7 X 1 0 -2 46X10 -3 1 1 X I 0 -3 44X10 -4 23X10-4 13XI0 -4

10 t 10 - ~ 10 - 2

18 24 27

4 4 X 1 0 -8 2 6 X 1 0 -~° 9 8 X 1 0 -~-"

53X10 -5 38X10 -6 6 1 X 1 0 -7

The results are t a b u l a t e d for a few values o f q in table 2 where we have taken, for the p u r p o s e s o f Illustration, N = 7 and N~ = 1 If we insists that c should be of o r d e r unity, then r / m u s t be chosen to be quite small ( t / ~ 0 03) to generate an acceptable hierarchy However, ff c can be small, then a perfectly acceptable hierarchy can be o b t a i n e d through a large o) (co >> 1 ) a n d r/= 1 N o t e that, although co is large, S need not be since S = -~boo), and bo can be quite small xf E~ ~s b r o k e n to a small enough h i d d e n sector group g'. The value o f ( Re T ) , on the other hand, is exceptionally small. This can be seen in table 2, case B Finally, we note that an acceptable hierarchy can be o b t a i n e d with both ( S ) a n d ( T ) o f o r d e r umty, as shown in table 2, case C. In view o f the remarks m a d e in ref. [ 5 ], this seems the most likely scenario In each case, the value o f K was fine-tuned to y~eld a vanishing cosmological constant, but, In all cases, its numerical value lies m the range 0 42 < K < 0.49 G a u g m o masses in this scheme are generated via the m e c h a m s m o f Elhs et al. [ 19], 1.e, the value o f co is shifted by a very small a m o u n t from the treelevel of eq ( 3 b ) , go)<<~o Leading to a non-zero value o f G ~. Scalar masses are then r e d u c e d at two loops [20] roll2

[ I ~4/3 5/3 t o~,g) m312,

mo ~,~ (oL/g)lf'mll2 •

(14)

F o r mj,,_ = 0 ( 1O- 7 ), an acceptable hierarchy, with 472

-~ -3 -5 -° -8

~ xi bo A o 065 019 66×10 -z 2 2 X 10--" 7 6 X 1 0 -3 065 054 15 65 32 17X10 z 025 023 019

,/7v bo p~,,

(Re T)

040 37×10-'4 0 × 1 0 -3 43X10 4 46X10 -s

23 26 23×102 20X103 17X104

040 73X10 -z 2 8 X 1 0 -2 16X10 -z 11xl0-'7 6 X 1 0 -3

23 086 53X10 20XI0 62X10 18X10

1 2 X 1 0 -3 14X10 -4 22X10 -5

26 23 30

-~ -3 -s -6

m o = O ( 1 TeV), results F r o m table 2, we see that this IS p o s s i b l e for reasonable choices o f the parameters. ( u ) S c h e m e II, c varzable C o n s i d e r now the case where c is due to (H~,,n) a n d / o r ( P ( N ) ) , such that it is not fixed during compactlficatlon T h e n we can interpret eq. ( 3 b ) as d e t e r m i n i n g the value o f c, i e , H a n d / o r N will adjust themselves so that the treelevel potenUal V ~°) is mxmmlsed In this case eq. (3b) is an identity giving V~°~-0, but leaving ( S ) und e t e r m i n e d at tree level, So, one m u s t m l m m i s e V eff ~ = A V ~t~ with respect to Z a n d o) (or, equavalently S and T) However, one might object that eq. (10) implies -eft v ~tl is u n b o u n d e d from below as Z ~ o e for small values o f o3. To see that this does not, recall that we have m a d e the replacement Ac--,qAc a n d by choosing q small enough Verr ~) Is positive semi-definite for all (positive) o~ and Z We have m l n l m l s e d v• err t~ for various values o f K and t / ( s e e table 3) Note that the hierarchy is again driven by a large value o f ( R e T ) , b u t is only obt a i n e d through the fine-tumng o f K (which simultaneously adjusts the cosmological constant to zero) In all cases, we find that K is roughly the same as m scheme I, i.e., 0 42~
3 December 1987

PHYSICS LETTERS B

Volume 198, number 4 Table 3 Scheme II for various values oft/

q

(09)

(Z)

x/b~mv2

vboAc,

.,, bop~Nt

(ReT)

063 10 -°~ 10 )

17 I9 20

31×10 6 5 9 × 1 0 -~ 1 9×10 -9

2 3×10-4 1 1×10 -5 82)<10 -7

1 3×10-~ 4 5 × 1 0 -2 1 9)<10 -2

b 2×10-2 1 0XI0 -2 1 4)<10 3

i 0)<10 z 7 6 × 1 0 ~" 42)<103

e v e n larger h i e r a r c h y t h a n m s c h e m e I

3 ml/2 ~ m3/2,

mo ~ (odn) ~P-m~/2

References

(15)

This nine, rn3/2=O(10 -4) generates an acceptable hierarchy with mo=O(1 TeV) From table 3, we see thzs ~s easily a c h i e v e d It is w o r t h n o t i n g that ~o ~s always o f o r d e r M p

4 Conclusions T h e f o u r - & m e n s l o n a l " n o - s c a l e " field t h e o r y that arises f r o m c o m p a c t ~ f i c a U o n o f t h e d = 10 superstring can g e n e r a t e a large h i e r a r c h y a n d g~ve acc e p t a b l e l o w - e n e r g y p h e n o m e n o l o g y F o l l o w i n g the o b s e r v a t i o n [ 11 ] that the s p e c t r u m o f p a r n c l e masses changes drastically at high m o m e n t a , A~
[ I ] P Ramond, Phys Rev D3(1971)2415, A NeveuandJ Schwarz, Nucl Phys B31 (1971)86 M B Green and J H Schwarz, Phys Lett B 149 (1984) 117, D Gross, J Harvey, E Martmec and R Rohm, Phys Rev Lett 55 (1985) 502, Nucl Phys B 258 (1985) 253 [2]A Chamseddme, Nucl Phys B 185 (198l)403, G Chaphne and N Manton, Phys Lett B 120 (1983) 105 [3] P Candelas, G Horowltz, A Strommger and E Wltten, Nucl Phys B 258 (1985) 46 [4]E W~tten, Phys Lett B 155 (1985)151 [5] V Kaplunovsky, Phys Rev Lett 55 (1985) 1033, M Dine and N Se~berg, Phys Rev Lett 55 (1985)366, Phys Lett B 162 (1985) 299 [6] J Elhs C Gomez and D V Nanopoulos. Phys Lett B 171 (1986) 203 [7] C P Burgess, A Font and F Quevedo, Nucl Phys B 272 (1986) 661, J -P Derendmger and L E Ibfifiez, Nucl Phys B 267 (1986) 365 [8] J Brelt, B Ovrut and G Segr~, Phys Lett B 162 (1985) 303 P Bm6truy and M K Galliard, Phys Lett B 168 (1986) 347, M Mangano, Z Phys C 28 (1985)613 [9] Y J Ahn andJ D Bren, Nucl Phys B 273 (1986) 75 /10] M Qmros, Phvs Lett B 173 (1986) 256 [ 11 ] P Bm6truy, S Dawson, M K Gafllard and I Hlnchllffe, preprmt LBL-22339 (1986) [ 12] E Cremmer, S Ferrara, C Kounnas and D V Nanopoulos, Phys Let/ B 133 (1983) 61, J Elhs, C Kounnas and D V Nanopoulos, Nucl Ph~s B 241 (1984) 406, B 247 (1984) 373, Phys Lett B 143 (1984) 410, J Elhs, K Enqwst and D V Nanopoulos, Phys Lett B 147 (1984) 99, J Elhs, A B Lahanas, D V Nanopoulos and K Tamvakls, Phys Lett B 134 (lq84) 429 [ 13] T Hubsch, Umverslty of Maryland preprmt (1986), P M~ron, D Phil Thesis, Oxford (1987), K Karkhn and P J Mlron, to be pubhshed [14] E Wltten, Nucl Phys B268 (1986)79, M Dine N Selberg, X-G Wen and E W~tten, Princeton preprmt (1987) [ 15] S Ferrara, L Glrardello and H P Ndles, Phys Lett B 125 (1983) 457, 473

Volume 198 number 4

PHYSICS LETTERS B

E Cohen, J Elhs, C Gomez and D V Nanopoulos, Phys Lett B 160 (1985) 62 [ 16] M Dine, R Rohm, N Selberg and E W~tten, Phys Lett B 156 (1985) 55 [ 17 ] J -P Derendmger, L E Ibaf~ez and H P Nllles, Phys Lett B 155 (1985) 65 [18] R R o h m a n d E Wltten, Ann Phys 170 (1986) 454

474

3 December 1987

[ 19] J Elhs, D V Nanopoulos, M Qmros and F Zwlrner, Phys Lett B 180 (1986) 83 [20] J Elhs, A B Lahanas, D V Nanopoulos. M Qmr6s and F Zwlrner, Phys Lett B 188 (1987) 408 [21 ] P Bmetruy, S Dawson and I Hmchhffe, Phys Lett B 179 (1986) 262, Phys Rev D 35 (1987) 2215