On supersymmetry breaking in superstrings

4

PHYSICS

ON SUPERSYMMETRY

BREAKING IN SUPERSTRINGS

Volume 207 number

LETTERS

B

30 June 1988

*

I ANTONIADIS’ CERN, CH-1211 Geneva 23 Swrtzerlund

C BACHAS ‘, D LEWELLEN Stanford Linear Accelerator

Center, Stanford,

CA 94305, USA

and T TOMARAS Lahoratove Received

’

de Phyxque 29 March

Theonque,

Ecole Normale

Qpeneure,

F- 75231 Pam

France

1988

We prove that the existence of a shghtly mawve gravltmo or gaugmo m a class of gauwan string compactlficatlons, existence of an entrre towr of such states below M,,,.,k, slgnalmg the approach to a hmlt of decompactlficatlon

1. Introduction

so that the correspondmg momenta become quasicontinuous The fact that the scale of tree-level supersymmetry breaking is necessarily linked to the size of some mternal dimension 1s not a prlorz obvious This is, for instance, not the case if instead of supersymmetry one considers the breaking of gauge symmetries [ 1,6], neither 1s it true if instead of superstrmgs one conaders traditional Kaluza-Klein supergravities [ 7 ] It is thus tempting to consider this phemomenon as a characterlstlc stringy signature at low energy However, even though there is no direct experimental evldence against the existence of some extra dimension at say 100 TeV, such a scenario faces at least one serious difficulty #I couplings of order one at low energies should, by naive dlmenslonal analysis, become huge at the umticatlon scale, mvahdatmg the semlclassical description of the string and creating a new hierarchy problem For this reason, it IS more cautious to interpret our result as ruling out small tree-

Unless a completely different mechanism 1s responsible for protecting the gauge hierarchy, we expect that m a realistic string vacuum space-time supersymmetry 1s broken at a scale naer Mwcak, 1 e mfimteslmal m units of MPlanck The purpose of this letter is to prove that, at least m the restricted class of (gaussian) four-dlmeneonal models, constructed with free world-sheet fields in refs [ l-51, the exlstence of a gravitmo or gaugino of infinitesimal treelevel mass (M&MPlanCk < 1) implies the existence of an entire tower of such states, wlch masses equal to (2N+ 1 )MjiZ, (4N+ 1 )JW~,~, , where Nis some, not too large, integer This is a signal of decompactlficatlon, meaning that one or more internal radii become huge or, by duality, mfimteslmal m units of Mhinck, * Work supported by the Department of Energy, Contract DEAC03-76SF005 15 ’ Permanent address CPT, Ecole Polytechmque, F-9 1128 Palalseau, France ’ Permanent address Department of Physics, Umverslty of Crete, Irakhon, Greece

0370-2693/88/s ( North-Holland

03 50 0 Elsevler Science Publishers Physics Publishing Dlvlslon )

lmphes the

” There IS of course also the perenmal cosmological constant problem, which was studled m a similar context m ref [ 81

BV

441

Volume 207, number

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PHYSICS

level supersymmetry breakmg, at least m this restricted class of gausslan models Our interest m these Issues was spurred by attempts to construct string solutions with supersymmetry broken by a Scherk-Schwarz mechanism [ 7,91N2 Similar results to ours were derived by Dine and Selberg [ lo] and Banks and Dixon [ 111 Their arguments are not restricted to the gausslan compactlficatlons considered here, but then conclusions are weaker they show that supersymmetry cannot be broken contmuously by shdmg the vacuum expectation value of a scalar field at any analytic point and m a flat direction of its potential This does not, however, rule out a number of interesting posslblhtles for instance there could exist vacua with hlerarchltally suppressed supersymmetrlc breaking, which cannot be contmuously connected to supersymmetrlc ones Or the scale of supersymmetry breaking could be proportional to some internal radn, but with a constant of proportlonahty of the order, say, of lo-l6 St111another posslblhty 1s that broken supersymmetry could only characterize the nearly massless but not all of the massive string modes Last but not least, though plausible, it is by no means yet established that the only contmuous strmg parameters are vacuum expectation values of scalar fields, it 1s m fact noteworthy that m many interesting supergravity models [ 7,12 ] the scale of supersymmetry breakmg is not tuned by a Hlggs field In contrast to the proofs of refs [ 10, 1 1 ] , our proof would, If extended to arbitrary strmg compactlficatlons, exclude all of these posslblhtles, since we make no assumption about the mechanism producing a slightly massive gravitino or gaugino Finally, let us note that if supersymmetry is not broken at tree level, then non-renormalization theorems [ 131 guarantee that it will not break through radiative corrections A possible exception could occur if the gauge group contains anomalous U ( 1) factors [ 141, but as we will also show here the choral charge asymmetry can be bounded from below by a not too small number This makes it unlikely that supersymmetry will break, If at all, at a scale near Mweak by such a mechanism, leaving non-perturbatlve effects as the final alternatlve These dlfficultles m

LETTERS

B

30 June 1988

breaking space-time supersymmetry should, among other thmgs, make us rethink about the necessity of having it as an approximate low energy symmetry m string theories

2. Fermiomc proof The Idea of our proof 1s that world-sheet supersymmetry severely restricts the form of massless, or mfinlteslmally massive space-time spmors Together with modular invariance this places strong constraints on the allowed superstrmg spectra Consider the fermlomc construction [ 2,4,5 ] of four-dlmenaonal heterotlc string theories In addition to the four spacetime coordmates P and their left-moving superpartners vP, one has an extra 18 left-moving and 44 nghtmoving fermlons x” and qA, respectively The worldsheet supercharge 1s [ 15 ] TF = vbq

+ IfahrXaXhXr,

(1)

with& the structure constants of some semi-simple Lie group G Since G has dimension 18, it 1s either SU(2)6,0rSU(3)x0(5)orfinallySU(2)XSU(4) A space-time spmor must belong to a Hllbert-space sector H n’, m which the fermlomc fields have the following boundary condltlons when transported around the string w“+V> x”+ ljl’GaDxD, VA_*J&&t

(2a) AB

rl

B (‘&cl

where dRlght is an orthogonal matrix, while &G must m addition belong to the group of automorphlsms [5] of G &&cAut (G) This last requirement 1s due to the fact that the supercurrent must be periodic We may always choose some, generally complex basis f( ’ ), , fcK) for the fermlons that dlagonahzes the boundary condltlons (2 ) f(‘)+-exp(i7ra(‘))f(‘)

(3)

We denote by cy= ( 1, aG, aRlght) the correspondmg vector of phases, restricted so that - 1
DzKounnas and Porratr [ 91 have m this context obtamed slmllar results to ours

442

t(l+a(‘))+mteger

(4)

Volume 207, number

PHYSICS

4

The mass of a state 1s determined Vlrasoro gauge condltlons M’=

by the zeroth-order

C (frequencies)-++i+$cr, 1

‘4

(frequencies)

(5a)

- 1 + i&!Rlght %+ht

(5b)

of the two In eq (5a) the factor Q1s the contrlbutlon real, or one complex, transverse fermlons v” which are periodic, since we are consldermg a Ramond sector Next, we examme the phase-vector length QI~ a!o as & ranges over Aut( G) This 1s mmlmlzed for some special automorphlsms J& with the following two crucial properties (1) & cy&=i dim G=3, which 1s precisely the value necessary to make the left-moving vacuum m the sector H <,massless, and (11) N a% =O (mod 2) for some small integer N (,V< 6 for the groups that mterest us), which means that ZZ!~1s some small root of the identity These properties were established m ref [ 5 1, where all the minima &E were classified exhaustively Here we will restrict ourselves to mner automorphlsms, and give an alternative elegant proof which we owe to Peis a group ter Goddard [ 161 An inner automorphlsm element in the adjoint representation, up to a conjugatlon it can be written as &=exp(inf3

H),

(6)

with iY, the mutually commutmg Cartan generator The elgenvalues of this matrix are 1 for each of the rank (G) commutmg generators, and exp (m&p) for every generator correspondmg to the root vector p Thus we find o!<, ~l~=frank(G)+

1 (f+~-l)~ + “
(7)

Let us normalize the length of the long roots to 2 Then 1 +rcpPIpl= ha” where h IS the (integer) dual Coxeter number (h=nforSU(n),andh=n-2forO(n) with n > 5 ) Defining f30=

B

30 June 1988

we may rewrite eq ( 7 ) as follows LYE~1~ =i dim G+h(8-f&)*

LyG

L =

LETTERS

(10)

It follows immediately that the mmimum phase-vector length is idim G, as advertlzed, and it 1s obtained for the special automorphlsm ti!

=exp(in/&*H),

(11)

which, among inner automorphlsms, 1s unique modulo the choice of the subset of positive roots Furthermore, it 1s straightforward to check that JZ& 1s an Nth root of the identity, with N= h for simply laced group, while N= 2h for 0 (5 ), N 1s therefore a small integer The special automorphlsms ( 11 ), have been studied extensively by Coxeter, and also arise m some other amusing context [ 17 ] For our purposes what makes them special is that they can yield massless space-time gravltmos #3 axl;10>,, wlthS= eq (5), periodic lators, it a slightly

(12) (1, &, (YR,.&= 0) Using the mass-formula, and the fact that near S there are no almost fermlons with mfimteamal-frequency osal1s now easy to see that the only candldate for massive gravltmo is of the form

axr10>,+&s

>

(13)

where 6s= (0, sac,, &!R,ght) -=K1, and the mass is @,,

= i (&% )2= 8 tsaR,ght

(14)

1’

Here the absence of a linear term 1s due to the fact that (YE 6a, =O since &. mmlmlzed vector length Suppose now that this state belongs to the Hllbert space of a four-dimensional fermlomc string model This means that S+&S 1s in the group Z of allowed boundary condltlons, and that the state (13) survives the generalized GSO-type projections exp(ij3 F)=

-c

1

s+6s [

P

=1

lf&j=l,

=+1

d&=-l,

(15)

for all /?EE We are here usmg the notation of ref [ 5 ] operators, the dot product is lorentzlan, left mmus right,, c[ p”] IS the

F ISthe vector of fermlon-number

1; + UP

and using the Freudenthal-de

Vrles strange formula ” As a result they can be used to generate N= 2 superconformal

f30*80=(1/3h)

dim G

(9)

models, as well as Gfunctlon

Identitles

443

Volume 207, number

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PHYSICS

coefficient with which the correspondmg spm-structure contributes to the one-loop amplitudes, and S,= 1 or - 1 according asp 1s a Neveu-Schwarz or Ramond boundary condltlon Eq (15) follows from the fact that on the state ( 13 ), exp (i/3 F) acts, respectively, as the identity or as a chlrahty operator Now E 1s a group under addition of phase-vectors modulo 2 Furthermore, 2N S= 0 mod 2, from which we conclude that S+ (2N+ 1)&Y, S+ (4N+ 1 )&S, , etc are also in 3 We therefore have an entire tower of candidate low-mass gravltmos a??; IO) S+(2N+1)65,

,

ax:~O>.+,,,+,,,,,

with mass-differences equal to 2N M3,2 These states satisfy left-right level matching due to the absence of a hnear term in eq ( 14) We must still show that they survive all GSO-type proJectlons, I e , that the coefficients c S+ (2NS 1)6S p [

S+ 1[

(4N+ 1)6S P

>c

1 ] >,

also satisfy eq ( 15) To prove this one must use the duality and factorrzatlon condltlons [ 2,4,5 =exp(fina!j3)c

[ P1 _-(y

= integer

and more general, but we are unable to obtain a very tight bound on N Recall that to a given state m H /, with fermlon numbers F, we assign a momentumwinding number vector [ 21 p=$a+F

(18)

This has 10 left and 22 right components Allowed momenta belong to a shlfted lattice AST where the shift vector 1sd = ( 1, 0, ,0 ), the entry 1 correspondmg to the bosomzed transverse @’ Modular mvanante dictates that p’=p: -pi be an odd integer, and that the lattice r be integer and self-dual, this means that elements of rhave integer dot products with each other, and that any q such that (q-A) has integer dot products with Tls an allowed momentum [ 21 These constraints generalize the usual condltlons of an even, integer, self-dual lorentzlan lattice [ 11, with the extra complications arising because one treats the bosomzed world-sheet fermlons on the same footing as the internal-space coordmates Now the most general Lorentz-Invariant supercharge IS of the form [ 18 1,

+ C b(l)

(17)

If&=-l,

which helps to get rid of phases Eq ( 17) follows from the fact that ol& +po + 1, or cr: +/3o are the phasevectors of an automorphlsm, if pG 1s an automorphism (S,= - 1) or antlautomorphlsm (SD= 1 ), respectively, and that Nol: =O mod Z, and finally that a!: mmimlzes vector length over the group of automorphlsms This then completes our proof, which goes through slmllarly for outer automorphlsms Note that the integer N governing the mass differences between the tower of gravltmos 1s at most equal to 6

+ C a(r) exp(lr

9)

It is instructive to translate this proof in the bosonic language The proof is, as we will see simpler

&ylcxp(lt

P),

(19)

where the vectors r, t and b entermg m the sums have nonvamshmg components only along the nine mternal bosomzed left-movmg coordmates T, IS a good weight $ conformal field provides r2 = 3, t2 = 1 and b(t) t=O Further condltlons must be imposed m order that the antlcommutator of two supercharges yield the energy-momentum tensor, but the solutions to these complicated equations have not yet been completely classified Consider next the transformation properties of TF as one moves around the string parametrized by O
3. Bosonlc proof

444

30 June 1988

TF =yPd,X,,

and the fact that if S, = 1,

B

,

(16)

NS p= even integer

LETTERS

07(0+2x)

=ew(2w

1 Ip)

p+w2)

w[lq

v(fl) 1 Ip)

(20)

Since TI. must be either antlperlodlc (NeveuSchwarz) or perlodlc (Ramond), we conclude that

Volume 207. number

Ap=rp=tp=O =$

PHYSICS

4

modZ

(NS),

modZ

(R),

(22)

such that 2NqETwhenever qcrE To see why, choose among all r, t and d some linearly independent basis , e, spanning Ed Then for any momentum p we 6, have 2Ic’q (P-d) =2N,$,

(q eI)(g-‘),J[eJ

(~~11,

(23)

where glJ=el e,EZ Now q e, and (p-d) e, are halfintegers or integers, while the components ofg-’ are ratlonals with a common denominator that divides det (g) Choosmg N to be twice this denominator we see that 2Nq (p-A) IS always integer, proving that 2Nq belongs to the self-dual lattice r The bound (22) follows from the fact that det (g) 1s the square volume of the paralleleplped (e,, , ed) which is less than 39, since e: = 1 or 3 for all I The next, crucial observation 1s that a would-be massless gravltmo must have a momentum p. lying entirely in Ed, and hence belonging to the lattice r, Indeed recall the mass formula hf’=$pt-f+ =ip$-11

1 (freqs ) L C (freqs ) K

B

30

June 1988

(21)

for all Y, t entering m expression ( 19)) and for any momentum p It follows by “self-duality” that all the vectors r and t are themselves allowed momenta of the theory Let now E” be the euchdean space, of dimension d< 10, spanned by r, t and d, and let r, be the lattice of vectors m Ed which have half-integer or integer dot products with all r, t and d For any given theory the elements of r, do not belong m general to the lattice r of shifted momenta, but there exists some integer N bounded from above by Iv-<~~x~,

LETTERS

(24)

Since to create a gravitmo we must act on the right with a aF{ oscillator, p. can have only left non-vanishing components, p0 R= 0 On the other hand, it follows from the super-Virasoro algebra alone that m a Ramond sector

where TF(0) 1s the zeroth moment of the supercurrent This means that for a massless fermlomc state p. L must minimize vector length, subject to the constraints (2 1) of having half-integer dot products with r, t and d Thus it cannot have a component normal to all these vectors simultaneously, and must hence he m Ed We are now almost ready Indeed, consider a theory contammg a slightly massive gravltmo Its momentum 1s necessarily of the form p=po+Gp with pOErE and Spa 1 But from our previous dlscusslon we conclude that if p IS an allowed momentum, then so are po+(2Nk+1)6p

k=l,2,3,

,

giving an entire tower of spin-; states with masses much less than MPlanck A second consequence of our discussion 1s that, if there 1s a choral charge-asymmetry, it is bounded from below by l/a, and hence cannot be made arbitrarily small The reason is that if p and 3 stand for the momenta of two choral, massless fermlons, then pt =j: = f and pi =a; = 1 or 0, and from our previous arguments 2Np, fiL~IZ But since p d=p, oL--pR & IS an integer, we conclude that so 1s N(p,k&)’ Hence the charge asymmetry 1pR 2 jR 1, where the sign depends on the relative hellclty, cannot be less than l/,/‘?? The induced oneloop Fayet-Ihopoulos D-term can thus be made arbitrarily small only if the string coupling constant 1s infinitesimal Let us conclude with some remarks Firstly, the proof of the existence of a tower of states below M,,,anck goes through identically if, instead of a slightly massive gravltmo we assume the existence of a slightly massive gaugmo correspondmg to the U ( 1) ** maxlma1 abehan subgroup of the gauge group Secondly, the proof m the bosomc language encompasses a wider class of supercurrents [ 18 1, but the bound (22 ) 1s for all the examples we know, overestlmatmg N by three orders of magnitude Without a complete classlficatlon of supercharges of type ( 19) we do not however know how to slgmficantly lower this bound Finally, we believe the arguments presented here can be extended with some modlficatlons to generic orblfold constructions [ 191, we do not have however an ex445

Volume 207, number

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PHYSICS

haustlve proof, because the complete rules of the game are not yet known exphcltly

Acknowledgement We acknowledge Banks, C Kounnas,

useful conversations with K S Naram and M Porratl

T

References [l]KS Naram,Phys Lett B 169 (1986)41, KS Naram, M H Sarmadl and E Wltten, Nucl Phys B 279(1987)369 H Kawal, DC Lewellen and S H H Tye, Phys Rev Lett 57(1986) 1832,Nucl Phys B288(1987)1 W Lerche, D Lust and A N Schellekens, Nucl Phys B 287 (1987) 477 1I Antomadis, C Bachas and C Kounnas, Nucl Phys B 289 (1987) 87 I Antomadls and C Bachas, Nucl Phys B 298 ( 1988) 586 [ 61 M Mueller and E Witten, Phys Lett B 182 ( 1986) 28, S Ferrara, C Kounnas and M Porratl, preprint CERN-TH 4800/87 (corrected version), I Antomadq C Bachas and C Kounnas, Phys Lett B 200 (1988) 297 [7] J Scherk and J H Schwarz, Phys Lett B 82 (1979) 60, Nucl Phys B 153 (1979) 61

446

LETTERS

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[ 81 H Itoyama

30 June 1988 and T Taylor, Phys

Lett B 186 ( 1987) 129

[9] R Rohm, Nucl Phys B 237 (1984) 553, C Kounnas and M Porratl, preprmt UCB-PTH

88/l

(1988), D Chang and A Kumar, Northwestern preprint 87112 [lo] M Dme and N Selberg, preprint IASSNS/HEP-87/50 [ 111 T Banks and L Dixon, Princeton and Umverslty of Cahforma at Santa Cruz preprint (1988) [ 121 E Cremmer, S Ferrara, C Kounnas and D V Nanopoulos, Phys Lett B 133 (1983) 61, J Elhs, C Kounnas and D V Nanopoulos, Nucl Phys B 241 (1984)406 [13]E Martmec,Phys Lett B 171 (1986) 189, M DmeandN Selberg, Phys Rev Lett 57 (1986) 2625 [ 141 M Dine, N Selberg and E Wltten, Nucl Phys B 289 (1987) 589, J J AttIck, G Moore and A Sen, SLAC and IASSNS preprint (December 1987) [ 151 I Antomadls, C Bachas, C Kounnas and P Wmdey, Phys Lett B 171 (1986) 51 [ 161 P Goddard, private commumcatlon [17]HSM Coxeter,DukeMath J 18(1951)765, B Kostant, Am J of Math 81 ( 1959) 973, C Itzykson, Intern J Mod Phys A 1 (1986) 65 [ 18 ] H Kawai, D Lewellen and S H H Tye, Intern J Mod Phys A3 (1988) 279, W Lerche, B E W Nllsson and A N Schellekens, Nucl Phys B294(1987)136 [ 191 L Dixon, J A Harvey, C Vafa and E Witten, Nucl Phys B 261 (1985) 678, B 274 (1986) 285, K S Naram, M H Sarmadl and C Vafa, Nucl Phys B 288 (1987) 551