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Gaugino masses from radiative corrections in superstring models
V o l u m e 180, n u m b e r 1,2
PHYSICS LETTERS B
6 N o v e m b e r 1986
GAUGINO MASSES FROM RADIATIVE CORRECTIONS IN SUPERSTRING MODELS John ELLIS, D.V. N A N O P O U L O S J, M. Q U I R 6 S 2 and F. Z W I R N E R 3,4 CERN, CH- 1211 Geneva 23, Switzerland R e c e i v e d 29 July 1986
We discuss contributions to the observable gaugino masses due to one-loop radiative corrections in the effective theory below the compactification and condensation scales of superstring models. We assume that some unspecifed dynamics determines the compactification radius, and that gaugino condensation in the hidden sector generates a gravitino mass smaller than the Planck mass. The largest contribution to the gaugino mass arises from radiative corrections to the effective scalar potential, which gen-
e r a t e mr/2 ~ m 5/3 3/2 m y2/3. Scalar m a s s e s a p p e a r in hi her o r d e r s o f p e r t u r b a t i o n theory. We c o m m e n t o n the i m p l i c a t i o n s for p h e n o m enological m o d e l c a l c u l a t i o n s .
One o f the key problems in superstring p h e n o m e nology :~ is to u n d e r s t a n d how s u p e r s y m m e t r y is broken to give a gravitino mass, and how this breaking of supersymmetry is subsequently c o m m u n i cated to the observable sector. A n o t h e r key p r o b l e m is to understand how the expectation values o f gaugesinglet " d i l a t o n " and " a x i o n " fields (S, T) [1] are determined. A promising suggestion for solving parts o f these problems is that hidden sector gauginos Z condense [3,4] after compactification, thereby determining dynamically the expectation value o f S [4] and generating a non-zero gravitino mass. However, this mechanism does not d e t e r m i n e the value o f the scalar field T which fixes the size o f the compaetified dimensions: Rc ~ T~/2, and also the mechanism for generating non-zero gaugino masses m j/2 and scalar masses mo in the observable sector remains obscure. In certain models for compactification [ 5 ], non-perturbative world-sheet instanton effects can generate a non-trivial effective potential for the T field [6], and might drive it towards a preferred ' P r e s e n t address: D e p a r t m e n t o f Physics, U n i v e r s i t y o f W i s c o n s i n , M a d i s o n , WI 5 3 7 0 6 , U S A . 2 O n leave f r o m I n s t i t u t o de E s t r u c t u r a de la M a t e r i a , M a d r i d , Spain. ~ Also at I n t e r n a t i o n a l School for A d v a n c e d Studies, 1-34100 Trieste, Italy. 4 Also at l s t i t u t o N a z i o n a l e di Fisica N u c l e a r e , !-35131 P a d u a , Italy. :~ F o r recent reviews, see ref. [ 1 ].
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
value. However, these models have theoretical and phenomenological defects [ 7 ] and we will not discuss them further here. We will just assume that some unspecified d y n a m i c s fixes ( 0 1 T L 0 ) = O ( 1 ) [8], so that the compactification scale is c o m p a r a b l e to the string excitation scale. This p a p e r is concerned with s u p e r s y m m e t r y breaking and its c o m m u n i c a t i o n to the observable sector. It has been argued previously [ 9] that hidden sector gauginos can only condense: ( 0 IZZ I0 ) # 0 if the hidden sector gauge group El is broken by the Hosotani m e c h a n i s m to some S U ( N ) subgroup. In this case, the scale Ac at which the hidden sector gauge interaction become strong is necessarily somewhat smaller than the compactification scale, and the gravitino mass m3/2~ ( 0 [ ) ~ ] 0 ) / m 2e ~ A z. 3c/. m . . .2 p ~ rap. If there is no inflation, or if this non-perturbative effect occurs after inflation, the universe contains S U ( N ) flux tubes which form cosmic strings that cause density perturbations 6 p i p ~ 102(A2/m~) and could seed galaxy f o r m a t i o n [ 10,11]. The resulting perturbations are acceptably small [11] if N~<4 so that Ac < 1016 G e V and m3/2 < 10 ~2 GeV. This is further m o t i v a t i o n for our assumption in the rest o f this p a p e r that 1TI3/2.~ rap. S u p e r s y m m e t r y breaking in the observable sector is absent at the tree level, but is p r e s u m a b l y generated from m3/2 by radiative corrections, and ml/2, mo could be much smaller than m3/2. One type o f one83
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PHYSICS LETTERS B
loop contribution [12] to m t/2 would be 0 ( (01NI O) × m3/2/mp), where N is some generic gauge non-singlet field. This is negligible if ( 01NI 0 ) is not much larger than mw, as we believe [7], and m3/2 4. mp as argued above. Subsequently, a possible two-loop contribution to mo has been identified [ 13 ]. We argue below that it yields mo~ m~/2/mp a n d is probably too small to be the d o m i n a n t source o f supersymmetry breaking in the observable sector. More recently, possible contributions to m~/2 which are o f order m3/2/m~ have been calculated [ 14]. We will c o m m e n t later on these contributions, which are also unlikely to be d o m i n a n t :2 The m a i n purpose o f this p a p e r is to present another contribution to ml/2 which is o f o r d e r m 5/31 3/2,rap2 / 3 , and hence likely to be the d o m i n a n t source of supersymmetry breaking in the observable sector. It arises because one-loop radiative corrections to the effective scalar potential shift the non-perturbatively d e t e r m i n e d value o f ( 0 1 S t 0 ) , thereby enabling the S field to contribute to m ~/2 which it d i d not do at the tree level. Previous phenomenological model calculations [ 17] o f low-energy spectra, which were m a d e assuming that observable sector s u p e r s y m m e t r y breaking was seeded by m j/2, are not substantially m o d i f i e d if m~/2 is in fact generated by the mechanism described here. This point is discussed at the end o f our paper, together with c o m m e n t s on the relations o f our calculations to previous ones, some cosmological remarks a n d some r e m i n d e r s o f open problems. At energies below the condensation scale Ac the tree-level potential for the m i n i m a l no-scale [18] supergravity theory o b t a i n e d from the superstring is given by
Fo = e a [ G ' ( G '' - ' ) { G j - 3 1 + ½D'~D c~ ,
(l)
where G is the K~ihler potential [ 2 ]: G = - l n ( S + S * ) - 3 In( T + T* - 2¢/0~) + l n l f ( O ) + W ( S ) 12 ,
(2)
with f(q~)--f,..c¢om¢~" the superpotential for m a t t e r fields and [ 4 ] :2 We have nothing to add to previous comments [ 15 ] on the interpretation of Wess-Zumino terms, which some authors [ 16] had thought might also contribute to rn~/2and mo. 84
6 November 1986
W(S)=c+hexp(-2S),
~.= 8 7 ~ 2 / c 2 ( G ' ) ,
(3)
is the effective superpotential i n d u c e d by non-perturbative effects in the hidden sector. The second term in W ( S ) , eq. ( 3 ) , results from G ' gaugino condensation, where G' is the biggest simple c o m p o n e n t o f the subgroup into which E~ is b r o k e n during compactification. The constant term in (3) m a y result from a v a c u u m expectation value o f the a n t i s y m m e tric tensor field H~p. The vacuum expectation values o f the matter fields are d e t e r m i n e d at the tree level by
Ga(G"-l)~Gb=3,
a,b=T,c/J,
(4)
which leads to ( 0 1 ¢ 1 0 ) = 0 but leaves ( 0 [ T I 0 ) undetermined. As m e n t i o n e d earlier, we assume that TR is fixed by some other m e c h a n i s m as yet unspecified. Using ( 4 ) , the tree-level potential can be written as
Vo =eaGS( G " -~);Gs,
(5)
which is positive semi-definite. Its v a c u u m state is therefore given by G~ = 0, which occurs when cos ).Sl = -- 1 ,
(6a)
(1 +2Zo) e---° =c/h,
(6b)
where z - 2 S R and we have d e n o t e d S = SR + iS~. The vanishing o f (3, translates into a vanishing gaugino mass, since
ml/2 = ( 1 / 2 S R ) eG/ZGs(G " -J)s .
(7)
F o r this reason, although local s u p e r s y m m e t r y is spontaneously broken in the hidden sector, at the tree level this breaking does not c o m m u n i c a t e to the observable sector. Moreover, other sources o f supers y m m e t r y breaking including scalar masses and trilinear couplings are known to vanish at the tree [ 18 ] and one-loop levels [ 19]. However, when radiative corrections to the effective scalar potential are considered, the tree level v a c u u m expectation value ( 6 ) will be corrected by a very small a m o u n t A z = O ( h ) 4 ` 1. This smallness is in agreement with the consistency o f the perturbation expansion, which leads us to expect small shifts in the tree-level v a c u u m state unless the tree-level potential is fiat. Although small, the one-loop correction Az=2ASR has d r a m a t i c consequences. The
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shifted value of Gs no longer vanishes, and is a source of supersymmetry breaking in the observable sector at the one-loop level :3 The one-loop correction to the effective scalar potential, which arises from supersymmetry breaking below the condensation scale Ac, is in general given by [ 13,20] V~ = (1/64n 2) {A~ Str M 2 + Str [ M 4 In (M 2/A~) ]
+Ac4 Str in(1 + m 2 / A 2 c ) - S t r [ M
4 ln(1 + M 2 / A 2 ) ] } , (8)
where the condensation scale is Ac=()./TRZ) ~/2 exp( - ~z) .
(9)
All masses in (8) are to be taken at the tree level, and so are proportional to the gravitino mass m3/2 = e ~;/2 =( 2116 T3RZ) I/2( c-- h e --) ,
(11 )
The general expression for the supertrace in (11) using the vacuum (4) for the matter fields, is [ 19,21 ] StrM2=2m~/2(-1-kGS(G
''
~)~G~) ,
(12)
where k is a constant which in principle depends on the number of chiral fields and on the number of gauginos. Our final result does not depend on the value of k. Thus the one-loop effective scalar potential for z can be cast as (13)
Vcr~(z) = Vo(z) + V , ( z ) ,
where Vo(z)=(2/16T3R)(l/z)[c-h(l+2z)
e---] 2
(14)
is the tree-level potential, and V~(z)=-(1/16n2)A2(z)[mZ/2(z)+kVo(z)]
The m i n i m u m of Veff(z), eq. (13), can easily be obtained by expanding Z = Z o + A Z around the treelevel vacuum. A straightforward calculation gives in leading order 2 2Zo+3 e (2/3)zo 24n2 TR (2Zo_ 1)2 ,
dz=
(15)
is the one-loop correction, with the functions A~(z) and m 3/2(z) defined in eqs. (9) and (10) respectively. :3 This effect was considered in ref. [ 12], but discarded because there the dynamical hypothesis was made that radiative corrections to the effective potential were O(mw). 4
(16)
from which we see that indeed dz,~ Zo as required by our approximation. The function Gs corresponding to the vacuum (16) does not vanish, but takes the value 22 2Zo+3 1 G~= 24n2TR 2Zo-- 1 4z~ e -(2/3)zo
(17)
This gives a c o m m o n value before renormalization for all the gaugino masses (7), which is hierarchically smaller than m3/2 and is given by
(10)
where we put & equal to its vacuum expectation value (6a). Therefore, if the tree-level vacuum expectation value (6b) is at Zo> 1 (i.e. m3/2 ~. 1 ), then A~ (9) is also ~ 1, while (m3/2/Ac)2,~ 1 and soM2/A2~,~ 1 in eq. (8). In this limit we can approximate (8) by [20] V~ =(1/32n2)A:c S t r M 2 .
6 November 1986
I/3 O~GUT • -~- 6 n a o u x { 2 n o ~ G u v ' ~
m,,2= 6 ~ - ~ ~ \
"
h~
"J
m~ 3
-m2/3 - ,
(18) 2 where we have used [2,4] a o u x __ = gGux/4n = ( 4 n S °) ~. We observe that the value obtained for m~/2, eq. (18), depends on TR only implicitly through m3/2. It does not have any explicit dependence on TR. Numerically, using h and g ~ u v = O ( 1 ), a value of m ~/2~ 102-103 GeV corresponds to a gravitino mass m3/2~_(1.O-4.1)22/S×lO l° GeV. This result combined with eq. (10) leads to 2 - 20 which yields
m3/2 -~(0.34-1.3) × 10 ~ G e V ,
(19a)
and hence Ac~['l'~
-Y~
2
2 //~]~ 1/3 1/3 2/3 OLGUTtr~/~] m3/2mp
_~ ( 2 . 7 - 4 . 2 ) × 10 ~5 G e V .
(19b)
This value of Acis naturally obtained if G ' is SU(4). The implications of these results are discussed later, after some comments on other estimates of supersymmetry breaking effects in the observable sector. Other possible one-loop sources ofgaugino masses have recently been studied in ref. [ 14]. This paper analyzes: (1) gravitino and gravitino/gauge boson loops; (2) analogous diagrams with the gravitino replaced by an g fermion; and (3) S boson and S boson/gaugino loops. The third class of diagrams is logarithmically divergent in four dimensions, with a 85
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physical cut-off being provided by the momentum IPI ~Ac above which the gaugino condensate dissolves and the theory becomes supersymmetric. In the limit of large z, the leading contribution of these diagrams is m~I /32~~_( m 33/2/nm~) 16zo3 In (Aflm 3/2)
- ( 32z~/37rm 2) m3/2,
( 20 )
which is much smaller than our contribution (18) to mjn. It is not possible to calculate reliably the nonlogarithmic finite part to be added to (20) unless one knows how the condensate dissolves at IPI >A~. This means that one cannot reliably calculate corrections to (20) which involve lower powers of z. Since z = O ( a - J ) , such terms would in any case be of higher order in a perturbative expansion in a and should be compared to diagrams containing additional gauge loops that might not be universal. Next we note that all loop calculations in the low-energy effective theory are done in a consistent regularization scheme. We believe that the physics of the superstring in general and the melting of the gaugino condensate in particular favour the momentum space regularization used above to evaluate eq. (20). If this scheme is also used to evaluate the diagrams of classes (1) and (2) above, their net contributions to ml/2 vanish +.4.We conclude that our contribution (18) to ml/2 is larger than all the contributions studied in ref. [ 14]. Chiral gauge non-singlet matter superfields ¢ in the observable sector do not feel supersymmetry breaking at the one-loop level [19]. One particular twoloop diagram which can contribute to the scalar masses has been mentioned in ref. [ 14]. It involves the superpotential f ( ~ ) =f,,,,,~/#"O" and a scalar singlet S propagator with a supersymmetry breaking (mass) z insertion ocm 2/2. However, the four-scalar f/,,,,,¢#"q)"S vertex is O(m3/2), and we estimate that this particular diagram induces a scalar mass 2 with a typical suppression factor f / 4 n 2 m o ~ m3/2, coming from the two loop integrations and the Yukawa coupling f This source of supersymmetry breaking is also negligible compared to the one computed in this paper. We plan to return in the future to a more complete discussion [22] of higher loop :4 Even if dimensional regularization were used, their contributions would be of lower order in z, and hence of higher order in a, than eq. (20). 86
6 November t986
contributions to scalar masses. We now comment on the phenomenological implications of our main result (18) for model calculations. In some recent papers [17], detailed phenomenological studies of radiative symmetry breaking and of the resulting particle spectrum were presented, in the framework of the minimal superstring model that could be obtained from E 6 after symmetry breaking by Wilson loops. These studies were made assuming that the gravitino mass m3/2 and the condensation scale Ac were both of order of the Planck mass. The result (18) of the present paper suggests, however, that m3/2 (19a) and Ac (19b) could be significantly smaller than rap. Nevertheless, we believe that the dominant source of supersymmetry breaking in the observable sector is still a universal gaugino mass rn~/2, as was assumed in ref. [17], with smaller trilinear couplings and with scalar masses generated by higher order radiative corrections. Apart from this, the only other ingredient used in the old approach [17] was a consistency condition requiring that the vacuum expectation values of the Higgs doublets H and IZIand of a neutral singlet N (which transformed non-trivially under the additional U (1)E) all be non-zero. However, this requirement must be imposed in any case, if one wants a phenomenologically acceptable spectrum of fermion and gauge boson masses. Thus the only practical difference between the old approach and that exposed here will be the scale at which the initial boundary conditions are imposed on the soft supersymmetry breaking parameters in the renormalization group equations. These only apply at renormalization scales below Ac, since above Ac the gaugino condensate dissolves and there is no supersymmetry breaking. In the old approach [17] we were taking Ac~Mx~ 1017-1018 GeV, whereas now we should use (19b). This difference has little effect on the previous discussion of radiative symmetry breaking triggered by the scalar squared mass m ~ being driven negative, and details of the resulting particle spectra will be discussed in ref. [22]. The slight reduction in the range of Q2 available for evolution of the scalar masses can be compensated by an increase in the Yukawa couplings governing their evolution. Since the operative Yukawa couplings were those giving masses to the vectorlike charge - 1 / 3 quarks, and to a lesser extent to the Higgsinos and
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top quarks, these particles could be somewhat heavier. Next we c o m m e n t on astrophysical consequences of our results. It was already m e n t i o n e d in the introduction to this paper that if E~ is broken down to some SU(N) subgroup which condenses after inflation, hidden sector flux tubes will create cosmic strings which cause density perturbations 8p/p~ 102(A2/m2e) [ 10,1 1 ]. These perturbations are too large if N > 4, and are uninterestingly small if N < 4. However, they have just the right magnitude to seed galaxies if N = 4, which is the case favoured by our relation [21 ] between ml/2 and m3/2. If, on the other hand, inflation occurs after compactification during [23] the formation of the gaugino condensate, it will generate scale-free perturbations of m a g n i t u d e S p / p ~ 202 3/2Ac/mp. 3 3 These are of the right magnitude if N = 4 or 5. Thus, galaxy formation and the magnitude of the gauge hierarchy may be intimately linked. We will return to cosmology in ref. [22]. However, all is not a bed of roses in the scenario outlined in this paper, and we would like to close by recalling three important open problems. One is that we have no dynamical mechanism to fix TR and hence the scale of the compactified dimensions. No way has been found to fix TR using radiative corrections [13,20], and here we have only discussed their impact on the d e t e r m i n a t i o n of SR. In certain circumstances, non-perturbative dynamics [6 ] can give a nontrivial potential for TR, but the interpretation of this is unsatisfactory [7]. A second problem is that in order to fix m3/2 .~. mp, and assuming T R= O (1) as is usually argued [8], eq. (10) tells us that we need Zo,> 1, and hence (according to eq. ( 6 b ) ) c ~ h in the superpotential (3). Since m w ~ m l / 2 ~ m3/2/m 5/3 P 2/3, it • is actually this c/h hierarchy which guarantees the gauge hierarchy: m w ~ (c/h)5/3mp. The superpotential term c is normally assumed [4] to be a relic of a v a c u u m expectation value of the antisymmetric tensor strength H~,~p in the six compactified dimensions: ( 0 I H I j A q 0 ) =C(-1JK, though any other effect which gave a non-trivial superpotential to beat against the h e ~s term (3) due to gaugino condensation could do as well. However, it is known [ 6 ] that non-perturbative world-sheet i n s t a n t o n effects do not give any interactions to the S field. Thus the origin of the hierarchy c/h~. 1, and hence of m3/2,~m P and mw,~.mp,
6 November 1986
is still open. Finally, we note that the small value of the cosmological constant is also not yet understood. Radiative corrections of the form (8) or (15) make a c o n t r i b u t i o n to the v a c u u m energy which is much larger than the m2w which had been assumed previously [ 12], and which is in turn much larger than the experimental upper limit on the cosmological constant. One of us (F.Z.) acknowledges partial financial support from the F o n d a z i o n e Ing. A. Gini.
References [ 1] J. Ellis, CERN preprint TH.4439/86 (1986); H.-P. Nilles, CERN preprint TH.4444/86 (1986); L.E. Ib~ifiez,CERN preprint TH.4459/86 (1986 ). [2] E. Witten, Phys. Len. B 155 (1985) 151. [3] J.-P. Derendinger, L.E. Ib~ifiezand H.-P. Nilles, Phys. Lett. B 155 (1985) 65. [4] M. Dine, R. Rohm, N. Seibergand E. Witten, Phys. Lett. B 156 (1985) 55. [5] E. Witten, Nucl. Phys. B 268 (1986) 79; C.M. Hull, Phys. Lett. B 167 (1986) 51; A. Strominger, Superstrings with torsion, Santa Barbara preprint (1985). [6] X.-G. Wen and E. Witten, Phys. Len. B 166 (1986) 397, J. Ellis, C. G6mez, D.V. Nanopoulos and M. Quir6s, Phys. Len. B 173(1986) 59; M. Dine, N. Seiberg, X.-G. Wen and E. Witten, Princeton preprint ( 1986). [7] J. Ellis, K. Enqvist, D.V. Nanopoulos, K. Olive, M. Quir6s and F. Zwirner, Phys. Lett. B 176 (1986) 403. [8] M. Dine and N. Seiberg, Phys. Rev. Len. 55 (1985) 366; Phys. Lett. B 162 (1985) 299; V. Kaplunovsky,Phys. Rev. Lett. 55 (1985) 1036. [9] E. Cohen, J. Ellis, C. G6mez and D.V. Nonopoulos, Phys. Left. B 160 (1985) 62. [ 10] E. Witten, Phys. Lett. B 153 (1985) 243. [ 11] B.A. Campbell, J. Ellis, D.V. Nanopoulos, K. Olive and S. Rudaz, CERN preprint TH.4479/86 (1986). [ 12] E. Cohen, J. Ellis, K. Enqvist and D.V. Nanopoulos, Phys. Lett. B 161 (1985) 85. [ 13] Y.J. Ahn and J.D. Breit, Nucl. Phys. B 273 (1986) 75. [ 14] P. Bin6truy,S. Dawson and I. Hinchliffe,Phys. Lett. B 179 (1986) 262. [ 15] J. Ellis, C. G6mez and D.V. Nanopoulos, Phys. Lett. B 168 (1985) 215. [ 16] J.-P. Derendinger,L.E. Ib~ifiezand H.-P. Nilles,Nucl. Phys. B 267 (1986) 365; L.E. lb~ifiezand H.-P. Nilles, Phys. Len. B 169 (1986) 354. [ 17] E. Cohen, J. Ellis, K. Enqvist and D.V. Nanopoulos, Phys. Lett. B 165 (1985) 76; J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Nucl. Phys. B 276 (1986) 14; Mod. Phys. Lett. A 1 (1986) 57. 87
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[ 18] J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B 247 (1984) 373. [ 19 ] J.D. Breit, B.A. Ovrul and G. Segr~, Phys. Lett. B 162 (1985) 303; P. BinOtruy and M.K. Gaillard, Phys. Lett. B 186 (1986) 347. [20] M. Quir6s, Phys. Left. B 173 (1986) 265.
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[21 ] M.T. Grisaru, M. Ro6ek and A. Karlhede, Phys. Lett. B 120 (1983) 110. [22] J. Ellis, A.B. Lahanas, D.V. Nanopoulos, M. Quir6s and F. Zwirner, CERN preprint, in preparation. [ 23] J. Ellis, K. Enqvist, D.V. Nanopoulos and M. Quir6s, N ucl. Phys. B 277 (1986) 231.
PHYSICS LETTERS B
6 N o v e m b e r 1986
GAUGINO MASSES FROM RADIATIVE CORRECTIONS IN SUPERSTRING MODELS John ELLIS, D.V. N A N O P O U L O S J, M. Q U I R 6 S 2 and F. Z W I R N E R 3,4 CERN, CH- 1211 Geneva 23, Switzerland R e c e i v e d 29 July 1986
We discuss contributions to the observable gaugino masses due to one-loop radiative corrections in the effective theory below the compactification and condensation scales of superstring models. We assume that some unspecifed dynamics determines the compactification radius, and that gaugino condensation in the hidden sector generates a gravitino mass smaller than the Planck mass. The largest contribution to the gaugino mass arises from radiative corrections to the effective scalar potential, which gen-
e r a t e mr/2 ~ m 5/3 3/2 m y2/3. Scalar m a s s e s a p p e a r in hi her o r d e r s o f p e r t u r b a t i o n theory. We c o m m e n t o n the i m p l i c a t i o n s for p h e n o m enological m o d e l c a l c u l a t i o n s .
One o f the key problems in superstring p h e n o m e nology :~ is to u n d e r s t a n d how s u p e r s y m m e t r y is broken to give a gravitino mass, and how this breaking of supersymmetry is subsequently c o m m u n i cated to the observable sector. A n o t h e r key p r o b l e m is to understand how the expectation values o f gaugesinglet " d i l a t o n " and " a x i o n " fields (S, T) [1] are determined. A promising suggestion for solving parts o f these problems is that hidden sector gauginos Z condense [3,4] after compactification, thereby determining dynamically the expectation value o f S [4] and generating a non-zero gravitino mass. However, this mechanism does not d e t e r m i n e the value o f the scalar field T which fixes the size o f the compaetified dimensions: Rc ~ T~/2, and also the mechanism for generating non-zero gaugino masses m j/2 and scalar masses mo in the observable sector remains obscure. In certain models for compactification [ 5 ], non-perturbative world-sheet instanton effects can generate a non-trivial effective potential for the T field [6], and might drive it towards a preferred ' P r e s e n t address: D e p a r t m e n t o f Physics, U n i v e r s i t y o f W i s c o n s i n , M a d i s o n , WI 5 3 7 0 6 , U S A . 2 O n leave f r o m I n s t i t u t o de E s t r u c t u r a de la M a t e r i a , M a d r i d , Spain. ~ Also at I n t e r n a t i o n a l School for A d v a n c e d Studies, 1-34100 Trieste, Italy. 4 Also at l s t i t u t o N a z i o n a l e di Fisica N u c l e a r e , !-35131 P a d u a , Italy. :~ F o r recent reviews, see ref. [ 1 ].
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
value. However, these models have theoretical and phenomenological defects [ 7 ] and we will not discuss them further here. We will just assume that some unspecified d y n a m i c s fixes ( 0 1 T L 0 ) = O ( 1 ) [8], so that the compactification scale is c o m p a r a b l e to the string excitation scale. This p a p e r is concerned with s u p e r s y m m e t r y breaking and its c o m m u n i c a t i o n to the observable sector. It has been argued previously [ 9] that hidden sector gauginos can only condense: ( 0 IZZ I0 ) # 0 if the hidden sector gauge group El is broken by the Hosotani m e c h a n i s m to some S U ( N ) subgroup. In this case, the scale Ac at which the hidden sector gauge interaction become strong is necessarily somewhat smaller than the compactification scale, and the gravitino mass m3/2~ ( 0 [ ) ~ ] 0 ) / m 2e ~ A z. 3c/. m . . .2 p ~ rap. If there is no inflation, or if this non-perturbative effect occurs after inflation, the universe contains S U ( N ) flux tubes which form cosmic strings that cause density perturbations 6 p i p ~ 102(A2/m~) and could seed galaxy f o r m a t i o n [ 10,11]. The resulting perturbations are acceptably small [11] if N~<4 so that Ac < 1016 G e V and m3/2 < 10 ~2 GeV. This is further m o t i v a t i o n for our assumption in the rest o f this p a p e r that 1TI3/2.~ rap. S u p e r s y m m e t r y breaking in the observable sector is absent at the tree level, but is p r e s u m a b l y generated from m3/2 by radiative corrections, and ml/2, mo could be much smaller than m3/2. One type o f one83
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loop contribution [12] to m t/2 would be 0 ( (01NI O) × m3/2/mp), where N is some generic gauge non-singlet field. This is negligible if ( 01NI 0 ) is not much larger than mw, as we believe [7], and m3/2 4. mp as argued above. Subsequently, a possible two-loop contribution to mo has been identified [ 13 ]. We argue below that it yields mo~ m~/2/mp a n d is probably too small to be the d o m i n a n t source o f supersymmetry breaking in the observable sector. More recently, possible contributions to m~/2 which are o f order m3/2/m~ have been calculated [ 14]. We will c o m m e n t later on these contributions, which are also unlikely to be d o m i n a n t :2 The m a i n purpose o f this p a p e r is to present another contribution to ml/2 which is o f o r d e r m 5/31 3/2,rap2 / 3 , and hence likely to be the d o m i n a n t source of supersymmetry breaking in the observable sector. It arises because one-loop radiative corrections to the effective scalar potential shift the non-perturbatively d e t e r m i n e d value o f ( 0 1 S t 0 ) , thereby enabling the S field to contribute to m ~/2 which it d i d not do at the tree level. Previous phenomenological model calculations [ 17] o f low-energy spectra, which were m a d e assuming that observable sector s u p e r s y m m e t r y breaking was seeded by m j/2, are not substantially m o d i f i e d if m~/2 is in fact generated by the mechanism described here. This point is discussed at the end o f our paper, together with c o m m e n t s on the relations o f our calculations to previous ones, some cosmological remarks a n d some r e m i n d e r s o f open problems. At energies below the condensation scale Ac the tree-level potential for the m i n i m a l no-scale [18] supergravity theory o b t a i n e d from the superstring is given by
Fo = e a [ G ' ( G '' - ' ) { G j - 3 1 + ½D'~D c~ ,
(l)
where G is the K~ihler potential [ 2 ]: G = - l n ( S + S * ) - 3 In( T + T* - 2¢/0~) + l n l f ( O ) + W ( S ) 12 ,
(2)
with f(q~)--f,..c¢om¢~" the superpotential for m a t t e r fields and [ 4 ] :2 We have nothing to add to previous comments [ 15 ] on the interpretation of Wess-Zumino terms, which some authors [ 16] had thought might also contribute to rn~/2and mo. 84
6 November 1986
W(S)=c+hexp(-2S),
~.= 8 7 ~ 2 / c 2 ( G ' ) ,
(3)
is the effective superpotential i n d u c e d by non-perturbative effects in the hidden sector. The second term in W ( S ) , eq. ( 3 ) , results from G ' gaugino condensation, where G' is the biggest simple c o m p o n e n t o f the subgroup into which E~ is b r o k e n during compactification. The constant term in (3) m a y result from a v a c u u m expectation value o f the a n t i s y m m e tric tensor field H~p. The vacuum expectation values o f the matter fields are d e t e r m i n e d at the tree level by
Ga(G"-l)~Gb=3,
a,b=T,c/J,
(4)
which leads to ( 0 1 ¢ 1 0 ) = 0 but leaves ( 0 [ T I 0 ) undetermined. As m e n t i o n e d earlier, we assume that TR is fixed by some other m e c h a n i s m as yet unspecified. Using ( 4 ) , the tree-level potential can be written as
Vo =eaGS( G " -~);Gs,
(5)
which is positive semi-definite. Its v a c u u m state is therefore given by G~ = 0, which occurs when cos ).Sl = -- 1 ,
(6a)
(1 +2Zo) e---° =c/h,
(6b)
where z - 2 S R and we have d e n o t e d S = SR + iS~. The vanishing o f (3, translates into a vanishing gaugino mass, since
ml/2 = ( 1 / 2 S R ) eG/ZGs(G " -J)s .
(7)
F o r this reason, although local s u p e r s y m m e t r y is spontaneously broken in the hidden sector, at the tree level this breaking does not c o m m u n i c a t e to the observable sector. Moreover, other sources o f supers y m m e t r y breaking including scalar masses and trilinear couplings are known to vanish at the tree [ 18 ] and one-loop levels [ 19]. However, when radiative corrections to the effective scalar potential are considered, the tree level v a c u u m expectation value ( 6 ) will be corrected by a very small a m o u n t A z = O ( h ) 4 ` 1. This smallness is in agreement with the consistency o f the perturbation expansion, which leads us to expect small shifts in the tree-level v a c u u m state unless the tree-level potential is fiat. Although small, the one-loop correction Az=2ASR has d r a m a t i c consequences. The
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shifted value of Gs no longer vanishes, and is a source of supersymmetry breaking in the observable sector at the one-loop level :3 The one-loop correction to the effective scalar potential, which arises from supersymmetry breaking below the condensation scale Ac, is in general given by [ 13,20] V~ = (1/64n 2) {A~ Str M 2 + Str [ M 4 In (M 2/A~) ]
+Ac4 Str in(1 + m 2 / A 2 c ) - S t r [ M
4 ln(1 + M 2 / A 2 ) ] } , (8)
where the condensation scale is Ac=()./TRZ) ~/2 exp( - ~z) .
(9)
All masses in (8) are to be taken at the tree level, and so are proportional to the gravitino mass m3/2 = e ~;/2 =( 2116 T3RZ) I/2( c-- h e --) ,
(11 )
The general expression for the supertrace in (11) using the vacuum (4) for the matter fields, is [ 19,21 ] StrM2=2m~/2(-1-kGS(G
''
~)~G~) ,
(12)
where k is a constant which in principle depends on the number of chiral fields and on the number of gauginos. Our final result does not depend on the value of k. Thus the one-loop effective scalar potential for z can be cast as (13)
Vcr~(z) = Vo(z) + V , ( z ) ,
where Vo(z)=(2/16T3R)(l/z)[c-h(l+2z)
e---] 2
(14)
is the tree-level potential, and V~(z)=-(1/16n2)A2(z)[mZ/2(z)+kVo(z)]
The m i n i m u m of Veff(z), eq. (13), can easily be obtained by expanding Z = Z o + A Z around the treelevel vacuum. A straightforward calculation gives in leading order 2 2Zo+3 e (2/3)zo 24n2 TR (2Zo_ 1)2 ,
dz=
(15)
is the one-loop correction, with the functions A~(z) and m 3/2(z) defined in eqs. (9) and (10) respectively. :3 This effect was considered in ref. [ 12], but discarded because there the dynamical hypothesis was made that radiative corrections to the effective potential were O(mw). 4
(16)
from which we see that indeed dz,~ Zo as required by our approximation. The function Gs corresponding to the vacuum (16) does not vanish, but takes the value 22 2Zo+3 1 G~= 24n2TR 2Zo-- 1 4z~ e -(2/3)zo
(17)
This gives a c o m m o n value before renormalization for all the gaugino masses (7), which is hierarchically smaller than m3/2 and is given by
(10)
where we put & equal to its vacuum expectation value (6a). Therefore, if the tree-level vacuum expectation value (6b) is at Zo> 1 (i.e. m3/2 ~. 1 ), then A~ (9) is also ~ 1, while (m3/2/Ac)2,~ 1 and soM2/A2~,~ 1 in eq. (8). In this limit we can approximate (8) by [20] V~ =(1/32n2)A:c S t r M 2 .
6 November 1986
I/3 O~GUT • -~- 6 n a o u x { 2 n o ~ G u v ' ~
m,,2= 6 ~ - ~ ~ \
"
h~
"J
m~ 3
-m2/3 - ,
(18) 2 where we have used [2,4] a o u x __ = gGux/4n = ( 4 n S °) ~. We observe that the value obtained for m~/2, eq. (18), depends on TR only implicitly through m3/2. It does not have any explicit dependence on TR. Numerically, using h and g ~ u v = O ( 1 ), a value of m ~/2~ 102-103 GeV corresponds to a gravitino mass m3/2~_(1.O-4.1)22/S×lO l° GeV. This result combined with eq. (10) leads to 2 - 20 which yields
m3/2 -~(0.34-1.3) × 10 ~ G e V ,
(19a)
and hence Ac~['l'~
-Y~
2
2 //~]~ 1/3 1/3 2/3 OLGUTtr~/~] m3/2mp
_~ ( 2 . 7 - 4 . 2 ) × 10 ~5 G e V .
(19b)
This value of Acis naturally obtained if G ' is SU(4). The implications of these results are discussed later, after some comments on other estimates of supersymmetry breaking effects in the observable sector. Other possible one-loop sources ofgaugino masses have recently been studied in ref. [ 14]. This paper analyzes: (1) gravitino and gravitino/gauge boson loops; (2) analogous diagrams with the gravitino replaced by an g fermion; and (3) S boson and S boson/gaugino loops. The third class of diagrams is logarithmically divergent in four dimensions, with a 85
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physical cut-off being provided by the momentum IPI ~Ac above which the gaugino condensate dissolves and the theory becomes supersymmetric. In the limit of large z, the leading contribution of these diagrams is m~I /32~~_( m 33/2/nm~) 16zo3 In (Aflm 3/2)
- ( 32z~/37rm 2) m3/2,
( 20 )
which is much smaller than our contribution (18) to mjn. It is not possible to calculate reliably the nonlogarithmic finite part to be added to (20) unless one knows how the condensate dissolves at IPI >A~. This means that one cannot reliably calculate corrections to (20) which involve lower powers of z. Since z = O ( a - J ) , such terms would in any case be of higher order in a perturbative expansion in a and should be compared to diagrams containing additional gauge loops that might not be universal. Next we note that all loop calculations in the low-energy effective theory are done in a consistent regularization scheme. We believe that the physics of the superstring in general and the melting of the gaugino condensate in particular favour the momentum space regularization used above to evaluate eq. (20). If this scheme is also used to evaluate the diagrams of classes (1) and (2) above, their net contributions to ml/2 vanish +.4.We conclude that our contribution (18) to ml/2 is larger than all the contributions studied in ref. [ 14]. Chiral gauge non-singlet matter superfields ¢ in the observable sector do not feel supersymmetry breaking at the one-loop level [19]. One particular twoloop diagram which can contribute to the scalar masses has been mentioned in ref. [ 14]. It involves the superpotential f ( ~ ) =f,,,,,~/#"O" and a scalar singlet S propagator with a supersymmetry breaking (mass) z insertion ocm 2/2. However, the four-scalar f/,,,,,¢#"q)"S vertex is O(m3/2), and we estimate that this particular diagram induces a scalar mass 2 with a typical suppression factor f / 4 n 2 m o ~ m3/2, coming from the two loop integrations and the Yukawa coupling f This source of supersymmetry breaking is also negligible compared to the one computed in this paper. We plan to return in the future to a more complete discussion [22] of higher loop :4 Even if dimensional regularization were used, their contributions would be of lower order in z, and hence of higher order in a, than eq. (20). 86
6 November t986
contributions to scalar masses. We now comment on the phenomenological implications of our main result (18) for model calculations. In some recent papers [17], detailed phenomenological studies of radiative symmetry breaking and of the resulting particle spectrum were presented, in the framework of the minimal superstring model that could be obtained from E 6 after symmetry breaking by Wilson loops. These studies were made assuming that the gravitino mass m3/2 and the condensation scale Ac were both of order of the Planck mass. The result (18) of the present paper suggests, however, that m3/2 (19a) and Ac (19b) could be significantly smaller than rap. Nevertheless, we believe that the dominant source of supersymmetry breaking in the observable sector is still a universal gaugino mass rn~/2, as was assumed in ref. [17], with smaller trilinear couplings and with scalar masses generated by higher order radiative corrections. Apart from this, the only other ingredient used in the old approach [17] was a consistency condition requiring that the vacuum expectation values of the Higgs doublets H and IZIand of a neutral singlet N (which transformed non-trivially under the additional U (1)E) all be non-zero. However, this requirement must be imposed in any case, if one wants a phenomenologically acceptable spectrum of fermion and gauge boson masses. Thus the only practical difference between the old approach and that exposed here will be the scale at which the initial boundary conditions are imposed on the soft supersymmetry breaking parameters in the renormalization group equations. These only apply at renormalization scales below Ac, since above Ac the gaugino condensate dissolves and there is no supersymmetry breaking. In the old approach [17] we were taking Ac~Mx~ 1017-1018 GeV, whereas now we should use (19b). This difference has little effect on the previous discussion of radiative symmetry breaking triggered by the scalar squared mass m ~ being driven negative, and details of the resulting particle spectra will be discussed in ref. [22]. The slight reduction in the range of Q2 available for evolution of the scalar masses can be compensated by an increase in the Yukawa couplings governing their evolution. Since the operative Yukawa couplings were those giving masses to the vectorlike charge - 1 / 3 quarks, and to a lesser extent to the Higgsinos and
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top quarks, these particles could be somewhat heavier. Next we c o m m e n t on astrophysical consequences of our results. It was already m e n t i o n e d in the introduction to this paper that if E~ is broken down to some SU(N) subgroup which condenses after inflation, hidden sector flux tubes will create cosmic strings which cause density perturbations 8p/p~ 102(A2/m2e) [ 10,1 1 ]. These perturbations are too large if N > 4, and are uninterestingly small if N < 4. However, they have just the right magnitude to seed galaxies if N = 4, which is the case favoured by our relation [21 ] between ml/2 and m3/2. If, on the other hand, inflation occurs after compactification during [23] the formation of the gaugino condensate, it will generate scale-free perturbations of m a g n i t u d e S p / p ~ 202 3/2Ac/mp. 3 3 These are of the right magnitude if N = 4 or 5. Thus, galaxy formation and the magnitude of the gauge hierarchy may be intimately linked. We will return to cosmology in ref. [22]. However, all is not a bed of roses in the scenario outlined in this paper, and we would like to close by recalling three important open problems. One is that we have no dynamical mechanism to fix TR and hence the scale of the compactified dimensions. No way has been found to fix TR using radiative corrections [13,20], and here we have only discussed their impact on the d e t e r m i n a t i o n of SR. In certain circumstances, non-perturbative dynamics [6 ] can give a nontrivial potential for TR, but the interpretation of this is unsatisfactory [7]. A second problem is that in order to fix m3/2 .~. mp, and assuming T R= O (1) as is usually argued [8], eq. (10) tells us that we need Zo,> 1, and hence (according to eq. ( 6 b ) ) c ~ h in the superpotential (3). Since m w ~ m l / 2 ~ m3/2/m 5/3 P 2/3, it • is actually this c/h hierarchy which guarantees the gauge hierarchy: m w ~ (c/h)5/3mp. The superpotential term c is normally assumed [4] to be a relic of a v a c u u m expectation value of the antisymmetric tensor strength H~,~p in the six compactified dimensions: ( 0 I H I j A q 0 ) =C(-1JK, though any other effect which gave a non-trivial superpotential to beat against the h e ~s term (3) due to gaugino condensation could do as well. However, it is known [ 6 ] that non-perturbative world-sheet i n s t a n t o n effects do not give any interactions to the S field. Thus the origin of the hierarchy c/h~. 1, and hence of m3/2,~m P and mw,~.mp,
6 November 1986
is still open. Finally, we note that the small value of the cosmological constant is also not yet understood. Radiative corrections of the form (8) or (15) make a c o n t r i b u t i o n to the v a c u u m energy which is much larger than the m2w which had been assumed previously [ 12], and which is in turn much larger than the experimental upper limit on the cosmological constant. One of us (F.Z.) acknowledges partial financial support from the F o n d a z i o n e Ing. A. Gini.
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[ 18] J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B 247 (1984) 373. [ 19 ] J.D. Breit, B.A. Ovrul and G. Segr~, Phys. Lett. B 162 (1985) 303; P. BinOtruy and M.K. Gaillard, Phys. Lett. B 186 (1986) 347. [20] M. Quir6s, Phys. Left. B 173 (1986) 265.
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[21 ] M.T. Grisaru, M. Ro6ek and A. Karlhede, Phys. Lett. B 120 (1983) 110. [22] J. Ellis, A.B. Lahanas, D.V. Nanopoulos, M. Quir6s and F. Zwirner, CERN preprint, in preparation. [ 23] J. Ellis, K. Enqvist, D.V. Nanopoulos and M. Quir6s, N ucl. Phys. B 277 (1986) 231.