- Email: [email protected]
Four-dimensional M-theory and supersymmetry breaking
13 ELSEVIER
Nuclear Physics B 507 (1997) 553-570
Four-dimensional M-theory and supersymmetry breaking Emilian Dudas a,b,l, Christophe Grojean a,c,2 a CERN-TIt, CH-1211 Geneva 23, Switzerland b lxzboratoire de Physique Th~orique et Hautes Energies, B~t. 211, Univ. Paris-Sud, F-91405 Orsay Cedex, France c CEA-SACLAY, Service de Physique Thdorique, F-91191 Gif-sur-Yvette Cedex, France
Received 2 May 1997; accepted 29 August 1997
Abstract We investigate compactifications of M-theory from 11 ---, 5 ~ 4 dimensions and discuss the geometrical properties of 4D moduli fields related to the structure of 5D theory. We study supersymmetry breaking by compactification of the fifth dimension and find that a universal superpotential is generated for the axion-dilaton superfield S. The resulting theory has a vacuum with (S) = 1, zero cosmological constant and a gravitino mass depending on the fifth radius as m3/2 ~ R 5 2 / M p l . We discuss the phenomenological aspects of this scenario, mainly string unification and the decompactification problem. (~) 1997 Elsevier Science B.V. PACS: 04.65+e; 11.25.Mj Keywords." Supergravity; Compactification; M-theory
1. Introduction In trying to describe four-dimensional physics from compactified string theories, it soon appeared [ 1 ] that the ten-dimensional string can hardly be weakly coupled, leading to far too large a Newton constant. So phenomenological attention must be paid to strongly coupled strings. Recently [2,3], a lot o f progress has been made in understanding this new physics: the strongly coupled regime is now viewed as the low energy limit I E-mail: [email protected]. 2 E-mail: [email protected]. 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0550-32 13 (97) 00590-7
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of M-theory and, in particular, the strongly coupled E8 × E8 heterotic string, traditionally considered as the most relevant one for phenomenology, can be described, in the low energy limit, by the eleven-dimensional supergravity with the two E8 gauge factors living each on a 10D boundary. The radius of the eleventh dimension is related to the 4 2/3 • So in the strongly coupled regime, Rll has to be large; in string coupling by RI1 "~ -st particular, it could be larger than the typical radius of the other six compact dimensions. Therefore, it appears in that case that the eleventh dimension has to be compactified after the Calabi-Yau internal manifold ( [ 4 - 6 ] ) . Describing four-dimensional physics from an E8 × E8 heterotic strongly coupled string should thus be equivalent to compactifing the eleven-dimensional supergravity on a Calabi-Yau manifold and then compactifing the fifth dimension on S 1/Z2. Our goal is to compactify the Lagrangian to 4D in a way compatible with N = 1 4D supersymmetry. As shown in Ref. [3], the presence of the boundaries and the interaction between the boundary fields and the bulk fields make this task difficult, as the 7D internal space is not really a direct product Q × S 1/Z2. However, we shall be mainly concerned with compactifying the (bulk) gravitational sector of the theory, where this difficulty does not appear and simply add the kinetic terms for the gauge fields on the boundaries. We ignore all the matter fields in 4D and their interactions. This is the point of view adopted here and we compare this pattern of compactification with the previous one, studied in Ref. [7], which corresponds to 11 ---+ 10---+4. Performing an explicit compactification on a CY manifold can be rather difficult, so we adopt an alternative way, by truncating with a symmetry of the compact space such as to maintain N = 1 supersymmetry in 4D. Of course, in this case we can describe only the analogue of untwisted fields of string theories, originating from 11D and 10D fields. In Section 2 of this paper, we identify, for different projections, the K~ihler structure for the moduli fields describing the shape of the internal manifold. We will observe very interesting geometrical properties, namely in all the cases the size of the compactified manifold is contained exclusively in the dilaton-axion superfield S, all the other moduli fields being invariant under dilatations of the 6D compactified space. Our main goal, to be studied in Section 3, is the N = 1 spontaneous supersymmetry breaking in 4D by compactification from 5D to 4D, by using the Scherk-Schwarz mechanism [8]. We argue that we obtain results that look non-perturbative from the perturbative heterotic string point of view, like a universal superpotential generation for S. The corresponding model spontaneously breaks supersymmetry with a zero cosmological constant, has the invariance S --~ 1/S and a minimum that is reached for S = 1. Section 4 shows that the use of the eleventh (or fifth, after the CY compactification) dimension to break supersymmetry offers a new perspective on the decompactification problem and the unification problem of perturbative string theories. The dependence of the gravitino mass on the fifth radius R5 (seen from 4D in Einstein units) is
m3/2 ~
R-2 5 h.4(4). ' " Pl
(1)
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By using this result, we argue that a value of the fifth radius of the order of 1012 GeV could solve both above-mentioned problems. We end with some conclusions and prospects.
2. Dimensional reduction The bosonic part of the strongly coupled E8 × E8 heterotic string Lagrangian is (we neglect for the moment higher-derivative terms) [ 3]
S'"'- 4,1__~f
_ ~__~GIJKLGIJKL)
MH
/ dl Ix el~...hzCt, t213Gt4...hG&...h,
3456K~1
MII
1 8~(4~-K~ 1)2/3
/ dl°x
v '~tr
FABFAB'
(2)
MlO
where I, J, K, L = 1 . . . . . 11 and GIJKL is related to the field strength of the three-form Cur by (in differential form language)
G=6dC +
K21{3~(Xll) dxll 2 X/~ -rr(4,n-) 2/3
to3v,
(3)
which is similar to the ten-dimensional relation H = d B
I
- ~w3r.
In (2), K~ is the l 1D
gravitational coupling that defines the l i D Newton constant K11 = M~I2/9, which is the M-theory scale. For the purposes of this section we set M~ = 1. The mass units will be discussed in detail later on. The above Lagrangian must be supplemented by the Horava-Witten Z2 projection, which projects out one gravitino and part of the other fields on the boundary. It acts in the tbllowing way: Xll ---'--Xll,
(4)
~/(--Xll) = Fll%(x11),
where g'1 is the l i D gravitino and F l l = F l . . . F I 0 is the 10D chirality matrix, while the three-form C is odd and the metric tensor is even. Throughout the paper, x 6 . . . . . x II will denote coordinates on the CY manifold Q, x 5 the extra eleventh dimension, and x 1. . . . . x 4 the ordinary 4D space-time. We introduce a complex structure on Q by defining
yi
X 2 i + 4 -Jr- ix 2i+5
,
yT_
x2i+4 _
ix2i+5
,
i=
1,2,3.
(5)
In the following,/z, v . . . . will refer to 4D and 5D Lorentz indices (to be distinguished whenever necessary) and i, j, k... to compact indices.
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We truncate the Lagrangian (2) in order to obtain N = 2 models 3 in 5D in the supergravity (Einstein) units, which we argue are the natural units in M-theory. Then we impose the Horava-Witten projection (acting by substituting Xll ~ x5 and Fll ~ F5 in ( 4 ) ) , which gives N = 1 models in 4D. A simple dimensional reduction can be performed to obtain an action in 5D. Our way of truncation (see below) is such that, in the metric tensor, there is no mixing between compact and non-compact indices. So, going into supergravity coordinates in 5D, we take g(lvl ) = G-I/3~(5) o/~, ,
~(11) ~;ij = gij,
(6)
where G is the determinant of the metric in the compact space g (to be distinguished from g(5) ). Similarly, the relevant components of the three-form are
C~,~p,
Ct,0 = A 0~) ,
Cijk = a•ijk,
(7)
Cok,
where a and Cok are complex scalars. Moreover, we shall take Cok = 0 in the following, these fields being irrelevant to our analysis. After algebraic manipulations, we obtain the desired 5D bosonic action (see Ref. [9] ), S(5) = - 1 2
-/
d5x g x / ~
./~(5) + - ~ t r ( g - O~g)tr(g-lO~g)
E
'
'
l
+ltr(g-l O.gg-Jc)~g) + -~GG~up~G ~u oP-
+186 I/3 ( gik~-[- g gi[) Fk3~ u(O)F
tzv(kD
+ 36 (detgiJ(ot, a)(aUa) + 2detgO(aua) (OUat) + detgTJ(Ouat)
(a"a*))]
f ( -36v~jdSxet~uPCrr~iC~zpP (a¢ra)(cgra')+ 16. 4 'km, 1[~a(i))F(ki)F(n'~)) ~ -up-~rr / 1
2~-(47-r) 2/3
fd4xv~G1/2trFABFaB"
(8)
In the following, we perform the rescalings (a, A ~0))~ ~ (a, A~J))/6. We now consider particular truncations and also compactify the fifth dimension to obtain a 4D action. For each case in our discussion, the gauge group on the observable boundary is broken by the usual embedding of the spin connection into the gauge group. To begin with, consider the simplest truncation, corresponding to a compactified space with just one radius e '~/2, the "breathing" mode. 4 Following Witten [ 10], we pick up an SU(3) subgroup of the SU(4) ~ SO(6) rotational group acting on x 6 . . . . . x 11. We define this SU(3) such that (yl,y2,y3) transforms like a representation 3 and (yf, y2, y3) like a representation 3. The 11D fields have to respect this SU(3) symmetry 3 We call N = 2 supersymmetry the smallest possible supersymmetry in 5D. 4 This simple truncation in the M-theory regime has recently also been performed in Ref. 16 ].
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and also the Z2 symmetry acting on x 5. The observable gauge group is E6. The massless spectrum in 5D is the universal hypermultiplet (Cusp, e 3o., a, a t) and the gravitational multiplet, which corresponds to a CY manifold with h(l,l) = 1 and h(2.1~ = 0. The resulting l i D tensor metric is written as
g(ll) 0
= e
o.
g(ll)
~i),
g(5) 55 = e2~',
iz ~
= e
-2o- (5) g~v,
g~5~ = e - ~,g~4~.
(9)
We have adopted here supergravity coordinates in which the 5D and 4D actions have a canonical supergravity expression. The scalar field e '~ is related to the radius of the CY manifold, while e ~' is related to the radius of the fifth dimension. Similarly, in 4D and after the Horava-Witten (Z~ w) projection, the three-form has only the massless components
C(iJ~ /zu5 =Cu,,s- '
C 50 (ll) =B60.
(10)
Cg,,5 is a two-form in the 4D space-time while B is a 4D scalar field. The dimensional truncation of S (5) is easily performed and leads to
S ~4) = -
d4x ~
7"~(4) -k- ~ (O/zy) (cg/zT) -k- ~ (O#o-) (O/Zo -)
+le6o-GgvosGU~,p5 + 3e_2~,(OuB)(OUB) +
1
2zr(47r) 2/3
e3o.FupFU~" ~
J "
To be consistent with a true CY compactification, this 4D action has to derive from a Kahler potential; the supergravity coordinates are the appropriate ones for the identification of this K~ihler structure. The complex fields are indeed easily identified as
S = e 3°" + ial, T = e z' + iB,
(12)
and the corresponding K~ihler potential is/C = - I n ( S + S t ) - 3 l n ( T + T t). The imaginary part of S is obtained from a Hodge duality of G, V ~ e6o. Gtzl,p5 = e#upo.O°al .
( 13 )
Note, first of all, the well-known exchange of roles between S and T compared to string compactifications. Moreover, unlike the case of the direct compactification of the 10D string case, the definitions of T and S are completely decoupled, S being related to the volume of the 6D compactified space and T to the radius of the eleventh dimension, but seen from 5D. We call it the radius of the fifth dimension R5 in the following, in M-theory units (denoted by R~ M) in Section 4). In the last section, we shall see how, with another interpretation of this radius, we can reproduce the K~ihler structure of usual string compactifications.
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The SU(3) invariance requirement on the CY manifold allows one single field 0. in the metric. We can in fact mimic compactification on orbifolds by restricting to discrete subgroups of SU(3) that act non-trivially on the representation 3 and 3 in such a way that we maintain N = 2 in 5D, by using the methods employed in Ref. [ 11 ]. We shall consider three cases (the action is on the representation 3): (a) Z i2 symmetry acting like (ie 2i7rl3, iie2i~rl3, e2iTr/3) ; (b) Z3 symmetry acting like ( e 2izrl3, e 2i~r/3, e2i~rl3); (c) Z2 × Z~ symmetry acting like ( - 1 , 1 , - 1 ) × ( 1 , - 1 , - 1 ) . (a) Z12 symmetry. This model has a massless spectrum in 5D consisting of the universal hypermultiplet, two vector multiplets and the gravitational multiplet, corresponding to h(1,1) = 3, h(2,1) = 0. The observable gauge group is E6 × U(1) x U(1). In the supergravity coordinates, the metric tensor then takes the form (written directly from 11D ---+ 4D) g~lixl)
~--'y--2(tri+o'2+o'~)13O(4) ~
-
o i l , ix
55 g(ll)
=e2y-2(~l+~2+~)/3,
g~l 1) = eCri60 ,
(14)
where the shape of the orbifold is described by the three radii o'1,0"2 and 0-3. After the Z~ w projection, the massless modes of the three-form in 4D are Cur5,
C50 = Bi60.
(15)
It is easy to perform the dimensional reduction of S (5) and, as previously, to identify the K~Jhler structure. The 4D action now derives from /C=-In(S+S
t)-
~ ln(Tk+Tkt), l~k~3
(16)
and the complex fields are identified as S = e °'1+°'2+°'3 -I-
i
al,
Tk = ere -(~Z+~T2+~3)/3+<~k+ iBk,
(17)
where ai is still defined by a Hodge duality x/~e2<~"+<~2+
g(ll) = ~.-'rr-,-l/3,,(4) /it," u 6/i/..,
,
g(5~l) = e2"tG-l13,
g~7ll) = go"
(18)
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The shape of the orbifold is here described by the nine parameters appearing in the unrestricted go. The massless modes of the three-form in 4D are C~5, C50 = BO. The resulting compactified 4D action now derives from the following K~ihler potential: /C = - l n ( S + S t ) - lndet(T 0 + Tot),
(19)
and the complex fields are identified as
S = G 1/2 + ial, T0 = e~G-J/6g 0 + iB O,
(20)
where v/2GGu~p5 = eu~p~a"al. Note the relation dett 0 = e 3~' = R 3, which is again related to special geometry in 5D. (c) Z2 x Z~ symmetry. Here, in contrast with the previous cases, we obtain three additional hypermultiplets (in addition to the universal one) containing the bosonic f i e l d s (gii,Co~). The model is described by h~l,i) = 3, h~2.1) = 3 and the observable gauge group on the boundary is E6 x U( 1 ) x U(1). In the supergravity coordinates, the Z2 × Z~-invariant metric tensor now takes the form g~ll) #~
= e-y(Tr-1/3~(4) -o#v
and g(ll~ ij
_(11) =e27G-1/3
,
(21)
,g55
(g) g(2)
=
g(3)
(22) '
where g(i) are 2 x 2 symmetric matrices, Gi their determinant, and G the global determinant. The 4D massless modes of the three-form are Cu~5, C50 = Bi•i). The corresponding 4D K~ihler potential now is
)U=-ln(S+St)-
Z
ln(Tk + T k t ) -
Z
ln(Uk +Ukt)'
(23)
where
S = G t/2 + ial, Tk = e~G-J/6Glk/2 + iBk, (Gk) J/2 g(k) Uk = g~k~ + t~gk9) ' •
22
12
(24)
22
x/2 GG#~,p5 = e/z~,po-O°al . Note that all the models studied above are particular cases of no-scale models [ 12]. So the no-scale structure seems to be present in both the weak-coupling limit and the strong-coupling one of superstring compactifications [4,6]. with
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Recently, it was shown that the anomaly cancellation in M-theory asks for another term in the M-theory Lagrangian [3,13], which can be viewed as a term canceling fivebrane world-volume anomalies [ 14] or, by compactifying one dimension, as a one-loop term in the type IIA superstring [ 15]. In our conventions and by using differential form language, it reads W5 =
1
211/6(277.)10/3
/ C A Xs,
(25)
MII
where X8 -= - ½ trR 4 + 3~ (trR2)2" This term can be compactified to 4D in different compactification schemes by using the field definitions given above and the quantization rule ~r~.f tr R A R = m, where m is an integer and the integration is over a 4-cycle in the compactified space. Without entering into details and restricting to one overall radius, we find in 4D a higher-derivative term of the type TR 2 in superfield language (recently, similar 4D terms were studied [ 16] and claimed to be relevant for supersymmetry breaking). Armed with the above field definitions, we can now study our main concern, supersymmetry breaking from 5 --+ 4 dimensions by compactification.
3. The spontaneous breaking of N = 1 supersymmetry in four dimensions by compactification We are interested in the phenomenology of the M-theory compactifications, in particular we would like to break the N = 1 supersymmetry in 4D. To achieve this aim, we perform a mechanism proposed by Scherk and Schwarz [8] in a supergravity context and then generalized to superstrings in Ref. [ 17] (another possibility for breaking supersymmetry is the gaugino condensation of one of the E8 gauge group [ 18] ). As there is, for the moment, no quantum description of M-theory, we use its effective low energy description at the level of supergravity. The Scherk-Schwarz mechanism is a generalized dimensional reduction that allows for the fields a dependence in the compact coordinates. Nevertheless, to avoid ghost particles and to include ordinary dimensional reduction, this dependence must satisfy some properties: it has to be in a factorizable form and has to correspond to an Rsymmetry of the theory. The simplest example is the use of the compact SO(6) symmetry of the 6D compact manifold, which is readily applicable to the superstrings case. In this case, any component of a tensor with p compact indices and q non-compact indices takes the form A
P
i,',
Ti,...i,,u,...u,, ( x, y) = 1-I Ui,, ( y ) Ti,,...i;,u~...u,, ( x ) ,
(26)
n=l
where y denotes compact coordinates and x non-compact ones. This tensor decomposition is stable under product and exterior derivation.
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561
This extended dimensional reduction, when applied to the curvature term, generates a potential for the scalar fields corresponding to the metric in the compact space. The requirement for this scalar potential to be positive imposes further restrictions on the form of U. A solution was proposed by Scherk and Schwarz, by taking U = e M~'~,
(27)
where M is an antisymmetric matrix in the compact space with zeros in the row and the column corresponding to yl. Then the 4D scalar potential, in supergravity units, reads [8] V 0 = ~ - ~1l ( g , , l t r ( M 2 _ M g M g _ , ) _ ( g _ , M t g M g _ , ) l , , ) .
(28)
In (28) and the rest of this section, g is the metric in the 7D compact space and G its determinant. When applied to the kinetic term for the 11D spin -3 field, this extended dimensional reduction also generates, through the spin connection for compact indices, masses for the resulting 4D gravitinos. More important for our purposes are symmetries that mix fields of different parities, for reasons that will become transparent below. For the truncations we are discussing here, these must be subgroups of the Sp(8) symmetry present in the N = 8 supergravity in 5D (see the third reference in Ref. [8] ). We shall perform this mechanism in two cases: (a) with yl being the extra eleventh dimension, xS; (b) with yl being a Calabi-Yau or orbifoid coordinate. We use for this purpose the truncations derived in the preceding section. The projections P we use have to commute with the Scherk-Schwarz matrix U, which means explicitly [P, M] = 0
if
pyl=y~,
{P, M} = 0
if
pyl=_y~.
(29)
(a) An eleventh-dimension coordinate dependence. The characteristics of this case is the anticommutation relation
{z~ w, M} = O.
(30)
We consider here two different possibilities. The first uses an SU(2) R-symmetry present in the 5D theory, which we argue to be the subgroup of Sp(8), which commutes with all the projections. The second uses the usual symmetry of the 6D internal manifold, which is relevant for the type IIA supergravity in 4D. (i) SU(2) R-symmetry. To keep things as easy as possible, consider for the moment the simplest truncation (9) corresponding to an SU(3) invariance in the compactified space. In this case, in 5D
E. Dudas. C. Grojean/Nuclear Physics B 507 (1997) 553-570
562
the only matter multiplet is the universal hypermultiplet, whose scalar fields parametrize su( 23 ) [ 19]. This structure can be simply viewed from 4D by a direct the coset su(z)×u(I) truncation (without the Z Hw projection). The truncated dependence on the universal hypermultiplet can be derived from the K~ihler potential [ 19] KS = - l n ( S + St - 2ata),
(31)
where the Hodge duality in 5D is x/2GGu~p~r = E~p~8(O~al + ia* 08 a) and S = e 3~r -t- a t a q- ial. The SU(2) symmetry acts linearly on the redefined fields 1 -
2a z2 = 1 + S '
S
zl = 1 + S '
(32)
which form a doublet (zl, z2). The Z~ w projection acts as z ~ W s = S, Z~Wa = - a , which translates on the SU(2) doublet in the obvious way,
(::)
(33)
The Scherk-Schwarz decomposition in this case reads explicitly (~,)=( ~2
cosmx5 - sin mx5
sinmxs~ ( z , ) z2
cosmx5 /
(34) '
corresponding to the matrix defined in (27) M = imo'2 and where m is a mass parameter. Note that, thanks to the anticommutation relation (30), which is clearly verified, the fields ~i have the same Z2Hw parities as the fields zi. The resulting scalar potential in 4D in Einstein metric is computed from the kinetic terms of the (el, f2) fields derived from (31 ). After making z2 = 0, corresponding to the projection Z2Hw, it is easily worked out and it is positive semi-definite. Expressed in terms of S and T, the result is 4m 2 I1 - 512 V = ( T + T t ) 3 S_q_St .
(35)
This can be seen as a superpotential generation for S. The 4D theory is completely described by 5 KS=-In(S+
St ) - 3 1 n ( T + T t ) ,
W = 2m( 1 + S).
(36)
Note that this superpotential for S should correspond to a non-perturbative effect from the heterotic string point of view. The minimum of the scalar potential corresponds to S = 1 and a spontaneously broken supergravity model with a zero cosmological constant. The order parameter for supersymmetry breaking is the gravitino mass m~/2 = e~:lwl 2 --
8m2/(T + Tt) 3. Note that the theory (36) is symmetric under the inversion S ~ 1/S, easily checked also on the scalar potential (35). This corresponds to a subgroup of the 5 This result is in agreement with general results on no-scale models (second reference in Ref. 117] ).
E. Dudas. C. Grojean/NuclearPhysicsB 507 (1997)553-570
563
SL(2, Z) acting on S as S --+ ( a S - i b ) / ( i c S + d), where a d - bc = 1, restricted here to a = d = 0, b = - c = 1. This already suggests a weak-coupling-strong-coupling symmetry of our resulting model, as in the S-duality models proposed some time ago [20]. We stress out that this 4D model presents some general features, independent of the details of the compactification. 6 The most general R-symmetry present in 5D in N = 8 supergravity is S p ( 8 ) . This symmetry is reduced once we impose truncations, and the truncation to N = 2 leaves an SU(2) symmetry appearing in the supersymmetry algebra. The scalar fields from the universal hypermultiplet, present in any compactification, transform as a doublet under SU(2). The Scherk-Schwarz mechanism uses only the antisymmetric part ( O ( 2 ) ) of this doublet, so we are led to (34). On the other hand, the 0 ( 2 ) part is exactly that required by the Horava-Witten Z~ w projection, which defines the E8 x E8 heterotic string. In more general CY models in 5D, the SU(2) symmetry acts also on other scalar fields than the universal hypermultiplet, but the piece (36) we computed should always exist and is universal. (ii) Symmetry of the compactified space. Since P ( S U ( 3 ) , Z3, ZI2 or Z2 × Z~) does not act on x 5, the commutation relation (29) simply reads [~M]
= 0.
(37)
For P = S U ( 3 ) , there is no antisymmetric matrix M respecting this condition, so we cannot perform the Scherk-Schwarz mechanism. For P = Z2 x Z~, the allowed M matrices involve three parameters and, in the basis (x 5, x 6 . . . . . x l l ) , read ¢'0
0 -ml M =
0
. . .
ml 0
:
0 -m2
(38)
m2 0 0 -m 3
m3 0
In this case, the Scherk-Schwarz matrix does not anticommute with Z~ w, so the mechanism is better adapted to type IIA supergravity in 4D. We are only interested here in a field spectrum common to type IIA and heterotic strings. As we will see in the next section, the qualitative features of this case concerning the decompactification problem are, however, similar to that o f the previous example. The scalar potential (28) reads, in this case, 3
vo=le-3rzm2
((
g(lil) - -
/ 2 4g(i ~)2) g(i)'~ 22" -'}-
'
i=1 6 The following arguments were developed in collaboration with R. Minasian.
(39)
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E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
which, expressed in terms of moduli fields (24), takes the simple expression 3
U/2_l
2 (40)
Vo - tl t2t3 i=l
This scalar potential has interesting consequences. In analogy with our previous example, the scalar potential is invariant under Ui --~ 1/Ui. The three U-fields develop a vev (Ui} = 1 (the self-dual points of duality transformations) and acquire masses m/2 m2 ui = (tlt2t3)"
(41)
At the same time, the single gravitino that remains after imposing the Z2 x Z~ x Z Hw symmetries also acquires a mass, which is computed to be mj + m2 + m3
m3/2=
(42)
2 V / ~ t2t3)
(b) A Calabi-Yau or orbifold coordinate dependence. We choose here yl to be x 6, a CY or orbifold coordinate. Then the commutation relation (29) reads [P, M x 6] = 0 ,
[ Z H W , M X 6] = 0 .
(43)
For P = SU(3), there is no antisymmetric matrix M respecting this condition, so we cannot perform the Scherk-Schwarz mechanism. For P = Z2 x Z~, the situation is different; since x 6 is now odd under the first Z2 and even under the second one and under Z Hw, the commutation relation simplifies again, {Z2, M} = 0 ,
[Z~,M] = 0 ,
[z~W,M] = 0 .
(44)
Thus we can always choose M to be of the off-diagonal following form, in the basis (X5,X 6 . . . . . X 11 ):
0
m 0 -m
0
0
mmt
0 mt
(45)
Then the scalar potential, written directly in terms of moduli fields, reads 1 (m2]t2U2 _ t3U312 -+- mt2[t2U3 _ 13U212 Vo = 4StlUl t2u2t3u3
,
2 ( ( m - m')2tzt3 - m m ' ( t 2 - t3) 2) (/-/2 - Ut2)(U3 - ut3)
)•
(46)
The minimum (for rn ~ m') of the scalar potential corresponds to the fiat directions
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
(t2) = (t3),
(u2) = (u3)
and
565
(Im/-/2) = (Im U3) = 0.
(47)
We do not write here the scalar mass matrix, but just notice that the leading matrix elements, which fix the physical masses, are of the type (consider m ~ m') 9
m°2 ~ m___~-,
(48)
Stl
where tj is the modulus field corresponding to the Scherk-Schwarz coordinate. This case is the analogue of the supersymmetry breaking by compactification in the weakly coupled superstrings [ 17]. Note that in the superstring case, the mass parameters m, m' in (45) are discrete, being related to automorphisms of the compactification lattice. All the masses we computed above are measured in M-theory units. In the following section, we translate these results in 4D supergravity units.
4. Phenomenological implications of M-theory compactifications We shall discuss in this section some phenomenological issues of M-theory compactifications focusing essentially on the 4D Newton constant and the gravitino mass. For simplicity, and without affecting our conclusions, we restrict ourselves to the simplest case of I i = e y. String compactifications do not easily allow [ 1,21 ] an adjustment of the string mass scale, the vev of the dilaton and the volume of the compact space to tune the three 4D observable quantities that are the Pianck scale, the GUT scale and the gauge coupling constant. For instance, in the weakly coupled regime of the E8 × E8 heterotic string, the well-known relation 4/3 ,I 2/3 O'GUT
(49)
GH ~ "-st M 2
GUT
disagrees with experimental values by a factor of order 20. Witten has shown [21] how in the strongly coupled regime the compatibility between string predictions land experimental values could be restored. It is interesting to worry about this issue in our patterns of compactification. In the l i D action, only the M-theory scale appears, which is the l i d Planck mass MI j, such that
877.(4,77.)2/3
' " 11' # v - xs_-0
(50) Now we compactify this action in the M-theory coordinates in which, as in the fivebrane units for the 10D string, there is no kinetic term for the "radius" of the extra dimension, that is, we adopt
E. Dudas, C. Grojean/NuclearPhysicsB 507 (1997)553-570
566
g(ll) izu = e_-2,,_(5) ,g,~u , g(5) 55 = e2~''
e~60 '
g(ll) 0 =
(51)
t,(5) ~(4) • o~, = 8tzu
It is transparent that these are the natural units of 5D supergravity. By performing the Hodge transformation (using the field definitions ( 1 2 ) ) S
cr
!
tG~z~p5 = etzvp,r3 a,
(52)
the 4D low energy effective action contains the terms
S(4' D @ f d4x v@ 4' x [tM~l (7p,(4) + l~s2(01zs )2 + ( 0 ~ a , ) 2 )
+ (41)5/3sf~F ~ ]- -
.
(53)
We can pass from the M-theory action (53) to the Einstein action (11) by the Weyl rescaling g(e4) = tg~ ~. Note the absence of a kinetic term for t in (53), the clear indication of a no-scale structure. This can therefore be associated with the M-theory (or 5D supergravity) units. The Lagrangian (53) allows us to identify the gauge coupling and the 4D reduced Planck mass to be 7 -
-
1
aGUT
r.J
-
-
1
(477")5/3s'
tM~j
=
M(p4~2.
(54)
These quantities are related to the radii of the Calabi-Yau space and the extra fifth dimension expressed in terms o f the M-theory units
e_,~/2 = s_l/6 _
MGUT MI1 '
t=
R~M)Mll.
(55)
However, the radius of the fifth dimension expressed in M-theory units is not convenient for physics in four dimensions, so we prefer to express it in terms of M(p4), the fourdimensional Planck scale, by defining hat(4) t -- ' " PI R- ~ • 5
(56)
Combining these equalities, we obtain the M-theory equivalent of (49), R - I 1/3 5 ¢-~GUT G.N" ~ ~(4) AA2 ' " P1 1"GUT
(57)
and, in order to be compatible with experimental constraints, the "radius" of the fifth dimension has an intermediate scale value, as already suggested by [4,5] 7 In the weakly coupled regime, the gauge coupling constant can have large threshold corrections. If this holds true in the strongly coupled regime, some of our considerations below will be modified.
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570 R -1 5
1012-13 G e V .
567
(58)
The fifth radius also appears in the gravitino mass generated by the Scherk-Schwarz mechanism, so we have to take care that R~-I be compatible with the phenomenologically preferred value for the gravitino mass. In the expressions for the gravitino mass from (36) and (42), the mass parameters that appear in the Scherk-Schwarz mechanism using the fifth dimension are naturally of the order of the M-theory scale, so we obtain 8 Mll m3/2 "~ V / ~
_ R52
(59)
""/1"4(4)p1'
where to arrive at the last identity we have used the identification of the moduli fields (12) and the definitions (54) and (56). This expression is similar to a usual supergravity form in which an auxiliary field would have developed a vev of the order of R 5-2 " Phenomenologically, the gravitino mass should be of the order of 1 TeV. 9 This leads to a value for the extra radius quite compatible with the one deduced from Newton's constant, R 5- l
101H2 GeV.
(60)
Consequently, the presence of the extra dimension could offer a solution to the decompactification puzzle usually encountered in string compactifications. Indeed, generically, the Scherk-Schwarz mechanism in strings leads to [17] 1 m.~/9 ~
" -
--
Rcy
~ MGUT.
(61 )
This is actually similar to the result we obtained in the last section in the case where a CY coordinate was used for the Scherk-Schwarz mechanism. To check it, rewrite the masses (48) in 4D supergravity units as m 2 ~ M ( p 4 ) 2 / ( s t 2 ) ~ g 2/R2/5, where g is the gauge coupling. This relation is phenomenologically hard to accommodate. Two issues have been proposed to solve this problem: (i) Compactify six dimensions on an asymmetric Calabi-Yau space with two or more different radii [22]. However, to agree with phenomenological values tbr the gravitino mass, these two (or more) radii must have hierarchical values within about fifteen orders of magnitude. / p n + l /i//(4 ~n ~ (ii) Construct models of strings such that the gravitino mass is m3/2 ~ 1 / ~ , , C y ' " PI , (4) n (MGuT/MpI) MCUT. However, phenomenology asks for n = 4,5, and it is difficult to explicitly construct such models. XA similar formula was conjectured by Antoniadis and Quiros 151, but replacing R5 with the eleventh radius. In our expression, R5 is the radius of the fifth dimension measured in 4D Planck units ( 5 6 ) . 9 This mass will be transmitted to the observable sector if part of the gauge group originates from 5D vector multiplets or if the moduli contributing to supersymmetry breaking give threshold corrections to gauge coupling. If not, the transmission occurs through gravitational interactions [24], in which case we need m3/2 ~ 1012 GeV so R 5- I ~ 1015 GeV.
568
E. D u d a s , C G r o j e a n / N u c l e a r P h y s i c s B 5 0 7 ( 1 9 9 7 ) 5 5 3 - 5 7 0
Table 1 Identification of the real parts of the moduli fields in string compactification in various units Units
Five-brane
String
Supergravity e (b/6
A
I
e 24'/3
s
e 3~r
e3~r- 2dp
e3tr - (a/2
t
e ~r+2¢b/3
e ~r
e ~r+~/2
So the extra dimension brings a new and more satisfying alternative to the decompactification problem of compactified strings. To conclude this phenomenological study, we would like to compare our pattern of compactification (11 --* 5 --~ 4) with another one, already studied [7], in which the extra dimension was compactified first (11 --~ 10 ~ 4). From a strongly coupled 10D point of view, the radius of the eleventh dimension x 5 is R~j ~ g~l) ~ e2:,_2% and this radius is related to the string dilaton by [21] Rlt '-~ e 2'/'/3.
(62)
Combining those relations, the expressions of the real part of T becomes t ,-~ e24'/3e °, which is the expression obtained by Binrtruy [23] by compactifying the 10D heterotic string in the five-brane units that are supposed to be appropriate to the strongly coupled regime. Actually we can, in the same way, reproduce the moduli's identification obtained by string compactification down to 4D in the various units. Suppose that we are working in units characterized, in 10D, by the tensor metric g~lO). These units are related to the five-brane ones by a Weyl rescaling g(10)
-
(10)B
~,~ = a g ~
(63)
So the typical radius of the Calabi-Yau space, measured in five-brane units, is now Rc¥ "~ ( A - l e '~) 1/2 =_ ( e , ~ ) l / a ; the radius of the extra dimension, viewed from 10D in five-brane units, is Rll ~ e y - ' ~ , which is still related to the string dilaton by (62). The real parts of the fields are now identified as S = e 3°'n = , ~ . - 3 e 3 ° ,
t = e r = A - l e '~+24)/3.
(64)
For instance, string (S) and supergravity (E) units are related to five-brane ( B ) units by g(JO)B -24,/3.(lo) s -~b/6.(IO) E u~ = ~ ~;)~ = ~ ~
(65)
Eq. (64) reproduces the results of string compactification summarized in Table 1. However, from a 5D point of view, i.e. after the compactification of the CY manifold, the radius of the eleventh dimension should be R11 "~ e z', so the identification of the moduli fields is now different, leading to the previous conclusions.
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570 S
569
1/3 s 11~10
Weakly coupled regime
~4
1/2
I
11~
5 ~4
Strongly coupled regime Fig. 1.
From an 11D point of view in M-theory coordinates (51), the 11 -+ 5 --~ 4 pattern of compactification is consistent if R5 > Rcy, which reads t > s ~/2. From a 10D point of view in five-brane coordinates (63), in the 11 ~ 10 --, 4 pattern of compactification, the radii of the Calabi-Yau space and of the extra dimension are now Rcy"-" e 'r/2 and R11 "-' e 24'/3, while the consistency condition of compactification is Rcy > Rxl, which, in term of moduli fields (Table 1), now becomes t < s t/2. In both cases, the 10D string strongly coupled regime condition e 4' > 1 simply reads t > s 1/3. So we can draw, on the (t) line, the different consistent patterns of compactification (we assume that s > 1), see Fig. 1. We have seen that phenomenology asks for large values of t, so our pattern of compactification is self-consistent.
5. Conclusions We studied truncations of M-theory from 11 --~ 5 --, 4 dimensions, applying our results to the supersymmetry-breaking mechanism by compactification from 5D to 4D. The geometric properties of the moduli fields, related to the special geometry of the 5D theory, are peculiar and important for phenomenological purposes. Re S is the volume of 6D internal manifold, the various hl,i moduli fields T are related to the radius of the fifth dimension and are invariant under dilatations of the 6D manifold and the h2.1 moduli U characterize, as usual, the complex structure of the 6D manifold. Using different symmetries of the 5D theory, we are able to break supersymmetry by compactification in various ways. The most interesting one is by using an SU(2) symmetry related to the universal hypermultiplet of the 5D theory. In this case, we obtain (for the simplest possible truncation) a model described by /C=-In(S+
St ) - 3 1 n ( T + T t ) ,
W = 2 m ( l + S).
(66)
This universal superpotential for S should correspond to a non-perturbative string effect. The minimization gives S = 1 and a spontaneously broken supergravity model with a zero cosmological constant. We hope that this result will shed some light on problems such as dilaton stabilization and supersymmetry breaking in effective strings. Another example, using symmetries of the compactified 6D space, gives rise to a potential for the U moduli, the corresponding vacuum is Ui = 1 with a zero cosmological constant.
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
570
D e f i n i n g 4 D scales a n d c o u p l i n g s , w e find that the r e s u l t i n g g r a v i t i n o m a s s in su-
pergravity (R5 ~
~'-2/A,~(4) u n i t s is m3/2 ~ "'5 / ' " P 1 , w h i c h b e t t e r s t h e usual d e c o m p a c t i f i c a t i o n l i m i t
0 ) p r o b l e m a n d c o r r e l a t e s it w i t h t h e u n i f i c a t i o n p r o b l e m in a way that l o o k s
phenomenologically promising.
Acknowledgements It is a p l e a s u r e to t h a n k E B i n & r u y , C. K o u n n a s , R. M i n a s i a n , J. M o u r a d , B. P i o l i n e a n d C. S a v o y for h e l p f u l d i s c u s s i o n s a n d c o m m e n t s .
References 111 M. Dine and N. Seiberg, Phys. Rev. Lett. 55 (1985) 366; V.S. Kaplunovsky, Phys. Rev. Lett. 55 (1985) 1036; M. Dine and N. Seiberg, Phys. Lett. B 162 (1985) 299. 12] C.M. Hull and P.K. Townsend, Nucl. Phys. B 438 (1995) 409; E. Witten, Nucl. Phys. B 443 (1995) 85; For recent reviews, see for instance, P.K Townsend, hep-th/9612121. [3] P. Ho~ava and E. Witten, Nucl. Phys. B 460 (1996) 506; B 475 (1996) 94. [41 T. Banks and M. Dine, Nucl. Phys. B 479 (1996) 173, hep-th/9609046; E. Caceres, V.S. Kaplunovsky and I.M. Mandelberg, hep-th/9606036; T. Li, J.L. Lopez and D.V. Nanopoulos, hep-ph/9702237. 151 1. Antoniadis and M. Quiros, Phys. Lett. B 392 (1997) 61. 16] T. Li, J.L. Lopez and D.V. Nanopoulos, hep-th/9704247. 17] E. Dudas and J. Mourad, hep-th/9701048. [ 8] J. Scherk and J.H. Schwarz, Nucl. Phys. B 153 (1979) 61; Phys. Lett. B 82 (1979) 60; E. Cremmer, J. Scherk and J.H. Schwarz, Phys. Lett. B 84 (1979) 83; P. Fayet, Phys. Lett. B 159 (1985) 121; Nucl. Phys. B 263 (1986) 649. 191 M. Giinaydin, G. Sierra and P.K. Townsend, Nucl. Phys. B 242 (1984) 244; B 253 (1985) 573; A.C. Cadavid, A. Ceresole, R. D'Auria and S. Ferrara, Phys. Lett. B 357 (1995) 76. I10] E. Witten, Phys. Lett. B 155 (1985) 151. 1111 S. Ferrara, C. Kounnas and M. Porrati, Phys. Lett. B 181 (1986) 263. 112] E. Cremmer, S. Ferrara, C. Kounnas and D.V. Nanopoulos, Phys. Lett. B 133 (1983) 61; J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B 241 (1984) 406; B 247 (1984) 373. [ 13] S.P. de Alwis, Phys. Lett. B 388 (1996) 291. [14] M.J. Duff, J.T. Liu and R. Minasian, Nucl. Phys. B 452 (1995) 261. [15] C. Vafa and E. Witten, Nucl. Phys. B 447 (1995) 261. 1161 A. Hindawi, B. Ovrut and D. Waldram, hep-th/9602075; R. Ledu, Ph.D. thesis, CPT Marseille, 1997. 117t C. Kounnas and M. Porrati, Nucl. Phys. B 310 (1988) 355; S. Ferrara, C. Kounnas, M. Porrati and E Zwirner, Nucl. Phys. B 318 (1989) 75. [ 181 P. Hofava, Phys. Rev. D 54 (1996) 7561. 1191 S. Ferrara and S. Sabharwal, Class. Quantum Grav. 6 (1989) L77; Nucl. Phys. B 332 (1990) 317. 1201 A. Font, L. Ib~ifiez, D. Liist and F. Quevedo, Phys. Lett. B 249 (1990) 35; S.J. Rey, Phys. Rev. D 43 (1991) 526; Z. Lalak, A. Niemeyer and H.P. Nilles, Phys. Lett. B 349 (1995) 99; P. Bin6truy and M.K. Gaillard, Phys. Lett. B 365 (1996) 87. [211 E. Witten, Nucl. Phys. B 460 (1996) 506. [221 1. Antoniadis, C. Mufioz and M. Quiros, Nucl. Phys. B 397 (1993) 515; 1. Antoniadis and K. Benakli, Phys. Lett. B 326 (1994) 69. [23] P. Bin6truy, Phys. Lett. B 315 (1993) 80. 124] I. Antoniadis and M. Quiros, preprint CERN-TH/97-90.
Nuclear Physics B 507 (1997) 553-570
Four-dimensional M-theory and supersymmetry breaking Emilian Dudas a,b,l, Christophe Grojean a,c,2 a CERN-TIt, CH-1211 Geneva 23, Switzerland b lxzboratoire de Physique Th~orique et Hautes Energies, B~t. 211, Univ. Paris-Sud, F-91405 Orsay Cedex, France c CEA-SACLAY, Service de Physique Thdorique, F-91191 Gif-sur-Yvette Cedex, France
Received 2 May 1997; accepted 29 August 1997
Abstract We investigate compactifications of M-theory from 11 ---, 5 ~ 4 dimensions and discuss the geometrical properties of 4D moduli fields related to the structure of 5D theory. We study supersymmetry breaking by compactification of the fifth dimension and find that a universal superpotential is generated for the axion-dilaton superfield S. The resulting theory has a vacuum with (S) = 1, zero cosmological constant and a gravitino mass depending on the fifth radius as m3/2 ~ R 5 2 / M p l . We discuss the phenomenological aspects of this scenario, mainly string unification and the decompactification problem. (~) 1997 Elsevier Science B.V. PACS: 04.65+e; 11.25.Mj Keywords." Supergravity; Compactification; M-theory
1. Introduction In trying to describe four-dimensional physics from compactified string theories, it soon appeared [ 1 ] that the ten-dimensional string can hardly be weakly coupled, leading to far too large a Newton constant. So phenomenological attention must be paid to strongly coupled strings. Recently [2,3], a lot o f progress has been made in understanding this new physics: the strongly coupled regime is now viewed as the low energy limit I E-mail: [email protected]. 2 E-mail: [email protected]. 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PII S0550-32 13 (97) 00590-7
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
554
of M-theory and, in particular, the strongly coupled E8 × E8 heterotic string, traditionally considered as the most relevant one for phenomenology, can be described, in the low energy limit, by the eleven-dimensional supergravity with the two E8 gauge factors living each on a 10D boundary. The radius of the eleventh dimension is related to the 4 2/3 • So in the strongly coupled regime, Rll has to be large; in string coupling by RI1 "~ -st particular, it could be larger than the typical radius of the other six compact dimensions. Therefore, it appears in that case that the eleventh dimension has to be compactified after the Calabi-Yau internal manifold ( [ 4 - 6 ] ) . Describing four-dimensional physics from an E8 × E8 heterotic strongly coupled string should thus be equivalent to compactifing the eleven-dimensional supergravity on a Calabi-Yau manifold and then compactifing the fifth dimension on S 1/Z2. Our goal is to compactify the Lagrangian to 4D in a way compatible with N = 1 4D supersymmetry. As shown in Ref. [3], the presence of the boundaries and the interaction between the boundary fields and the bulk fields make this task difficult, as the 7D internal space is not really a direct product Q × S 1/Z2. However, we shall be mainly concerned with compactifying the (bulk) gravitational sector of the theory, where this difficulty does not appear and simply add the kinetic terms for the gauge fields on the boundaries. We ignore all the matter fields in 4D and their interactions. This is the point of view adopted here and we compare this pattern of compactification with the previous one, studied in Ref. [7], which corresponds to 11 ---+ 10---+4. Performing an explicit compactification on a CY manifold can be rather difficult, so we adopt an alternative way, by truncating with a symmetry of the compact space such as to maintain N = 1 supersymmetry in 4D. Of course, in this case we can describe only the analogue of untwisted fields of string theories, originating from 11D and 10D fields. In Section 2 of this paper, we identify, for different projections, the K~ihler structure for the moduli fields describing the shape of the internal manifold. We will observe very interesting geometrical properties, namely in all the cases the size of the compactified manifold is contained exclusively in the dilaton-axion superfield S, all the other moduli fields being invariant under dilatations of the 6D compactified space. Our main goal, to be studied in Section 3, is the N = 1 spontaneous supersymmetry breaking in 4D by compactification from 5D to 4D, by using the Scherk-Schwarz mechanism [8]. We argue that we obtain results that look non-perturbative from the perturbative heterotic string point of view, like a universal superpotential generation for S. The corresponding model spontaneously breaks supersymmetry with a zero cosmological constant, has the invariance S --~ 1/S and a minimum that is reached for S = 1. Section 4 shows that the use of the eleventh (or fifth, after the CY compactification) dimension to break supersymmetry offers a new perspective on the decompactification problem and the unification problem of perturbative string theories. The dependence of the gravitino mass on the fifth radius R5 (seen from 4D in Einstein units) is
m3/2 ~
R-2 5 h.4(4). ' " Pl
(1)
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
555
By using this result, we argue that a value of the fifth radius of the order of 1012 GeV could solve both above-mentioned problems. We end with some conclusions and prospects.
2. Dimensional reduction The bosonic part of the strongly coupled E8 × E8 heterotic string Lagrangian is (we neglect for the moment higher-derivative terms) [ 3]
S'"'- 4,1__~f
_ ~__~GIJKLGIJKL)
MH
/ dl Ix el~...hzCt, t213Gt4...hG&...h,
3456K~1
MII
1 8~(4~-K~ 1)2/3
/ dl°x
v '~tr
FABFAB'
(2)
MlO
where I, J, K, L = 1 . . . . . 11 and GIJKL is related to the field strength of the three-form Cur by (in differential form language)
G=6dC +
K21{3~(Xll) dxll 2 X/~ -rr(4,n-) 2/3
to3v,
(3)
which is similar to the ten-dimensional relation H = d B
I
- ~w3r.
In (2), K~ is the l 1D
gravitational coupling that defines the l i D Newton constant K11 = M~I2/9, which is the M-theory scale. For the purposes of this section we set M~ = 1. The mass units will be discussed in detail later on. The above Lagrangian must be supplemented by the Horava-Witten Z2 projection, which projects out one gravitino and part of the other fields on the boundary. It acts in the tbllowing way: Xll ---'--Xll,
(4)
~/(--Xll) = Fll%(x11),
where g'1 is the l i D gravitino and F l l = F l . . . F I 0 is the 10D chirality matrix, while the three-form C is odd and the metric tensor is even. Throughout the paper, x 6 . . . . . x II will denote coordinates on the CY manifold Q, x 5 the extra eleventh dimension, and x 1. . . . . x 4 the ordinary 4D space-time. We introduce a complex structure on Q by defining
yi
X 2 i + 4 -Jr- ix 2i+5
,
yT_
x2i+4 _
ix2i+5
,
i=
1,2,3.
(5)
In the following,/z, v . . . . will refer to 4D and 5D Lorentz indices (to be distinguished whenever necessary) and i, j, k... to compact indices.
E. Dudas, C Grojean/NuclearPhysicsB 507 (1997) 553-570
556
We truncate the Lagrangian (2) in order to obtain N = 2 models 3 in 5D in the supergravity (Einstein) units, which we argue are the natural units in M-theory. Then we impose the Horava-Witten projection (acting by substituting Xll ~ x5 and Fll ~ F5 in ( 4 ) ) , which gives N = 1 models in 4D. A simple dimensional reduction can be performed to obtain an action in 5D. Our way of truncation (see below) is such that, in the metric tensor, there is no mixing between compact and non-compact indices. So, going into supergravity coordinates in 5D, we take g(lvl ) = G-I/3~(5) o/~, ,
~(11) ~;ij = gij,
(6)
where G is the determinant of the metric in the compact space g (to be distinguished from g(5) ). Similarly, the relevant components of the three-form are
C~,~p,
Ct,0 = A 0~) ,
Cijk = a•ijk,
(7)
Cok,
where a and Cok are complex scalars. Moreover, we shall take Cok = 0 in the following, these fields being irrelevant to our analysis. After algebraic manipulations, we obtain the desired 5D bosonic action (see Ref. [9] ), S(5) = - 1 2
-/
d5x g x / ~
./~(5) + - ~ t r ( g - O~g)tr(g-lO~g)
E
'
'
l
+ltr(g-l O.gg-Jc)~g) + -~GG~up~G ~u oP-
+186 I/3 ( gik~-[- g gi[) Fk3~ u(O)F
tzv(kD
+ 36 (detgiJ(ot, a)(aUa) + 2detgO(aua) (OUat) + detgTJ(Ouat)
(a"a*))]
f ( -36v~jdSxet~uPCrr~iC~zpP (a¢ra)(cgra')+ 16. 4 'km, 1[~a(i))F(ki)F(n'~)) ~ -up-~rr / 1
2~-(47-r) 2/3
fd4xv~G1/2trFABFaB"
(8)
In the following, we perform the rescalings (a, A ~0))~ ~ (a, A~J))/6. We now consider particular truncations and also compactify the fifth dimension to obtain a 4D action. For each case in our discussion, the gauge group on the observable boundary is broken by the usual embedding of the spin connection into the gauge group. To begin with, consider the simplest truncation, corresponding to a compactified space with just one radius e '~/2, the "breathing" mode. 4 Following Witten [ 10], we pick up an SU(3) subgroup of the SU(4) ~ SO(6) rotational group acting on x 6 . . . . . x 11. We define this SU(3) such that (yl,y2,y3) transforms like a representation 3 and (yf, y2, y3) like a representation 3. The 11D fields have to respect this SU(3) symmetry 3 We call N = 2 supersymmetry the smallest possible supersymmetry in 5D. 4 This simple truncation in the M-theory regime has recently also been performed in Ref. 16 ].
557
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
and also the Z2 symmetry acting on x 5. The observable gauge group is E6. The massless spectrum in 5D is the universal hypermultiplet (Cusp, e 3o., a, a t) and the gravitational multiplet, which corresponds to a CY manifold with h(l,l) = 1 and h(2.1~ = 0. The resulting l i D tensor metric is written as
g(ll) 0
= e
o.
g(ll)
~i),
g(5) 55 = e2~',
iz ~
= e
-2o- (5) g~v,
g~5~ = e - ~,g~4~.
(9)
We have adopted here supergravity coordinates in which the 5D and 4D actions have a canonical supergravity expression. The scalar field e '~ is related to the radius of the CY manifold, while e ~' is related to the radius of the fifth dimension. Similarly, in 4D and after the Horava-Witten (Z~ w) projection, the three-form has only the massless components
C(iJ~ /zu5 =Cu,,s- '
C 50 (ll) =B60.
(10)
Cg,,5 is a two-form in the 4D space-time while B is a 4D scalar field. The dimensional truncation of S (5) is easily performed and leads to
S ~4) = -
d4x ~
7"~(4) -k- ~ (O/zy) (cg/zT) -k- ~ (O#o-) (O/Zo -)
+le6o-GgvosGU~,p5 + 3e_2~,(OuB)(OUB) +
1
2zr(47r) 2/3
e3o.FupFU~" ~
J "
To be consistent with a true CY compactification, this 4D action has to derive from a Kahler potential; the supergravity coordinates are the appropriate ones for the identification of this K~ihler structure. The complex fields are indeed easily identified as
S = e 3°" + ial, T = e z' + iB,
(12)
and the corresponding K~ihler potential is/C = - I n ( S + S t ) - 3 l n ( T + T t). The imaginary part of S is obtained from a Hodge duality of G, V ~ e6o. Gtzl,p5 = e#upo.O°al .
( 13 )
Note, first of all, the well-known exchange of roles between S and T compared to string compactifications. Moreover, unlike the case of the direct compactification of the 10D string case, the definitions of T and S are completely decoupled, S being related to the volume of the 6D compactified space and T to the radius of the eleventh dimension, but seen from 5D. We call it the radius of the fifth dimension R5 in the following, in M-theory units (denoted by R~ M) in Section 4). In the last section, we shall see how, with another interpretation of this radius, we can reproduce the K~ihler structure of usual string compactifications.
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E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
The SU(3) invariance requirement on the CY manifold allows one single field 0. in the metric. We can in fact mimic compactification on orbifolds by restricting to discrete subgroups of SU(3) that act non-trivially on the representation 3 and 3 in such a way that we maintain N = 2 in 5D, by using the methods employed in Ref. [ 11 ]. We shall consider three cases (the action is on the representation 3): (a) Z i2 symmetry acting like (ie 2i7rl3, iie2i~rl3, e2iTr/3) ; (b) Z3 symmetry acting like ( e 2izrl3, e 2i~r/3, e2i~rl3); (c) Z2 × Z~ symmetry acting like ( - 1 , 1 , - 1 ) × ( 1 , - 1 , - 1 ) . (a) Z12 symmetry. This model has a massless spectrum in 5D consisting of the universal hypermultiplet, two vector multiplets and the gravitational multiplet, corresponding to h(1,1) = 3, h(2,1) = 0. The observable gauge group is E6 × U(1) x U(1). In the supergravity coordinates, the metric tensor then takes the form (written directly from 11D ---+ 4D) g~lixl)
~--'y--2(tri+o'2+o'~)13O(4) ~
-
o i l , ix
55 g(ll)
=e2y-2(~l+~2+~)/3,
g~l 1) = eCri60 ,
(14)
where the shape of the orbifold is described by the three radii o'1,0"2 and 0-3. After the Z~ w projection, the massless modes of the three-form in 4D are Cur5,
C50 = Bi60.
(15)
It is easy to perform the dimensional reduction of S (5) and, as previously, to identify the K~Jhler structure. The 4D action now derives from /C=-In(S+S
t)-
~ ln(Tk+Tkt), l~k~3
(16)
and the complex fields are identified as S = e °'1+°'2+°'3 -I-
i
al,
Tk = ere -(~Z+~T2+~3)/3+<~k+ iBk,
(17)
where ai is still defined by a Hodge duality x/~e2<~"+<~2+
g(ll) = ~.-'rr-,-l/3,,(4) /it," u 6/i/..,
,
g(5~l) = e2"tG-l13,
g~7ll) = go"
(18)
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
559
The shape of the orbifold is here described by the nine parameters appearing in the unrestricted go. The massless modes of the three-form in 4D are C~5, C50 = BO. The resulting compactified 4D action now derives from the following K~ihler potential: /C = - l n ( S + S t ) - lndet(T 0 + Tot),
(19)
and the complex fields are identified as
S = G 1/2 + ial, T0 = e~G-J/6g 0 + iB O,
(20)
where v/2GGu~p5 = eu~p~a"al. Note the relation dett 0 = e 3~' = R 3, which is again related to special geometry in 5D. (c) Z2 x Z~ symmetry. Here, in contrast with the previous cases, we obtain three additional hypermultiplets (in addition to the universal one) containing the bosonic f i e l d s (gii,Co~). The model is described by h~l,i) = 3, h~2.1) = 3 and the observable gauge group on the boundary is E6 x U( 1 ) x U(1). In the supergravity coordinates, the Z2 × Z~-invariant metric tensor now takes the form g~ll) #~
= e-y(Tr-1/3~(4) -o#v
and g(ll~ ij
_(11) =e27G-1/3
,
(21)
,g55
(g) g(2)
=
g(3)
(22) '
where g(i) are 2 x 2 symmetric matrices, Gi their determinant, and G the global determinant. The 4D massless modes of the three-form are Cu~5, C50 = Bi•i). The corresponding 4D K~ihler potential now is
)U=-ln(S+St)-
Z
ln(Tk + T k t ) -
Z
ln(Uk +Ukt)'
(23)
where
S = G t/2 + ial, Tk = e~G-J/6Glk/2 + iBk, (Gk) J/2 g(k) Uk = g~k~ + t~gk9) ' •
22
12
(24)
22
x/2 GG#~,p5 = e/z~,po-O°al . Note that all the models studied above are particular cases of no-scale models [ 12]. So the no-scale structure seems to be present in both the weak-coupling limit and the strong-coupling one of superstring compactifications [4,6]. with
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
560
Recently, it was shown that the anomaly cancellation in M-theory asks for another term in the M-theory Lagrangian [3,13], which can be viewed as a term canceling fivebrane world-volume anomalies [ 14] or, by compactifying one dimension, as a one-loop term in the type IIA superstring [ 15]. In our conventions and by using differential form language, it reads W5 =
1
211/6(277.)10/3
/ C A Xs,
(25)
MII
where X8 -= - ½ trR 4 + 3~ (trR2)2" This term can be compactified to 4D in different compactification schemes by using the field definitions given above and the quantization rule ~r~.f tr R A R = m, where m is an integer and the integration is over a 4-cycle in the compactified space. Without entering into details and restricting to one overall radius, we find in 4D a higher-derivative term of the type TR 2 in superfield language (recently, similar 4D terms were studied [ 16] and claimed to be relevant for supersymmetry breaking). Armed with the above field definitions, we can now study our main concern, supersymmetry breaking from 5 --+ 4 dimensions by compactification.
3. The spontaneous breaking of N = 1 supersymmetry in four dimensions by compactification We are interested in the phenomenology of the M-theory compactifications, in particular we would like to break the N = 1 supersymmetry in 4D. To achieve this aim, we perform a mechanism proposed by Scherk and Schwarz [8] in a supergravity context and then generalized to superstrings in Ref. [ 17] (another possibility for breaking supersymmetry is the gaugino condensation of one of the E8 gauge group [ 18] ). As there is, for the moment, no quantum description of M-theory, we use its effective low energy description at the level of supergravity. The Scherk-Schwarz mechanism is a generalized dimensional reduction that allows for the fields a dependence in the compact coordinates. Nevertheless, to avoid ghost particles and to include ordinary dimensional reduction, this dependence must satisfy some properties: it has to be in a factorizable form and has to correspond to an Rsymmetry of the theory. The simplest example is the use of the compact SO(6) symmetry of the 6D compact manifold, which is readily applicable to the superstrings case. In this case, any component of a tensor with p compact indices and q non-compact indices takes the form A
P
i,',
Ti,...i,,u,...u,, ( x, y) = 1-I Ui,, ( y ) Ti,,...i;,u~...u,, ( x ) ,
(26)
n=l
where y denotes compact coordinates and x non-compact ones. This tensor decomposition is stable under product and exterior derivation.
E. Dudas. C. Grojean/NuclearPhysicsB 507 (1997) 553-570
561
This extended dimensional reduction, when applied to the curvature term, generates a potential for the scalar fields corresponding to the metric in the compact space. The requirement for this scalar potential to be positive imposes further restrictions on the form of U. A solution was proposed by Scherk and Schwarz, by taking U = e M~'~,
(27)
where M is an antisymmetric matrix in the compact space with zeros in the row and the column corresponding to yl. Then the 4D scalar potential, in supergravity units, reads [8] V 0 = ~ - ~1l ( g , , l t r ( M 2 _ M g M g _ , ) _ ( g _ , M t g M g _ , ) l , , ) .
(28)
In (28) and the rest of this section, g is the metric in the 7D compact space and G its determinant. When applied to the kinetic term for the 11D spin -3 field, this extended dimensional reduction also generates, through the spin connection for compact indices, masses for the resulting 4D gravitinos. More important for our purposes are symmetries that mix fields of different parities, for reasons that will become transparent below. For the truncations we are discussing here, these must be subgroups of the Sp(8) symmetry present in the N = 8 supergravity in 5D (see the third reference in Ref. [8] ). We shall perform this mechanism in two cases: (a) with yl being the extra eleventh dimension, xS; (b) with yl being a Calabi-Yau or orbifoid coordinate. We use for this purpose the truncations derived in the preceding section. The projections P we use have to commute with the Scherk-Schwarz matrix U, which means explicitly [P, M] = 0
if
pyl=y~,
{P, M} = 0
if
pyl=_y~.
(29)
(a) An eleventh-dimension coordinate dependence. The characteristics of this case is the anticommutation relation
{z~ w, M} = O.
(30)
We consider here two different possibilities. The first uses an SU(2) R-symmetry present in the 5D theory, which we argue to be the subgroup of Sp(8), which commutes with all the projections. The second uses the usual symmetry of the 6D internal manifold, which is relevant for the type IIA supergravity in 4D. (i) SU(2) R-symmetry. To keep things as easy as possible, consider for the moment the simplest truncation (9) corresponding to an SU(3) invariance in the compactified space. In this case, in 5D
E. Dudas. C. Grojean/Nuclear Physics B 507 (1997) 553-570
562
the only matter multiplet is the universal hypermultiplet, whose scalar fields parametrize su( 23 ) [ 19]. This structure can be simply viewed from 4D by a direct the coset su(z)×u(I) truncation (without the Z Hw projection). The truncated dependence on the universal hypermultiplet can be derived from the K~ihler potential [ 19] KS = - l n ( S + St - 2ata),
(31)
where the Hodge duality in 5D is x/2GGu~p~r = E~p~8(O~al + ia* 08 a) and S = e 3~r -t- a t a q- ial. The SU(2) symmetry acts linearly on the redefined fields 1 -
2a z2 = 1 + S '
S
zl = 1 + S '
(32)
which form a doublet (zl, z2). The Z~ w projection acts as z ~ W s = S, Z~Wa = - a , which translates on the SU(2) doublet in the obvious way,
(::)
(33)
The Scherk-Schwarz decomposition in this case reads explicitly (~,)=( ~2
cosmx5 - sin mx5
sinmxs~ ( z , ) z2
cosmx5 /
(34) '
corresponding to the matrix defined in (27) M = imo'2 and where m is a mass parameter. Note that, thanks to the anticommutation relation (30), which is clearly verified, the fields ~i have the same Z2Hw parities as the fields zi. The resulting scalar potential in 4D in Einstein metric is computed from the kinetic terms of the (el, f2) fields derived from (31 ). After making z2 = 0, corresponding to the projection Z2Hw, it is easily worked out and it is positive semi-definite. Expressed in terms of S and T, the result is 4m 2 I1 - 512 V = ( T + T t ) 3 S_q_St .
(35)
This can be seen as a superpotential generation for S. The 4D theory is completely described by 5 KS=-In(S+
St ) - 3 1 n ( T + T t ) ,
W = 2m( 1 + S).
(36)
Note that this superpotential for S should correspond to a non-perturbative effect from the heterotic string point of view. The minimum of the scalar potential corresponds to S = 1 and a spontaneously broken supergravity model with a zero cosmological constant. The order parameter for supersymmetry breaking is the gravitino mass m~/2 = e~:lwl 2 --
8m2/(T + Tt) 3. Note that the theory (36) is symmetric under the inversion S ~ 1/S, easily checked also on the scalar potential (35). This corresponds to a subgroup of the 5 This result is in agreement with general results on no-scale models (second reference in Ref. 117] ).
E. Dudas. C. Grojean/NuclearPhysicsB 507 (1997)553-570
563
SL(2, Z) acting on S as S --+ ( a S - i b ) / ( i c S + d), where a d - bc = 1, restricted here to a = d = 0, b = - c = 1. This already suggests a weak-coupling-strong-coupling symmetry of our resulting model, as in the S-duality models proposed some time ago [20]. We stress out that this 4D model presents some general features, independent of the details of the compactification. 6 The most general R-symmetry present in 5D in N = 8 supergravity is S p ( 8 ) . This symmetry is reduced once we impose truncations, and the truncation to N = 2 leaves an SU(2) symmetry appearing in the supersymmetry algebra. The scalar fields from the universal hypermultiplet, present in any compactification, transform as a doublet under SU(2). The Scherk-Schwarz mechanism uses only the antisymmetric part ( O ( 2 ) ) of this doublet, so we are led to (34). On the other hand, the 0 ( 2 ) part is exactly that required by the Horava-Witten Z~ w projection, which defines the E8 x E8 heterotic string. In more general CY models in 5D, the SU(2) symmetry acts also on other scalar fields than the universal hypermultiplet, but the piece (36) we computed should always exist and is universal. (ii) Symmetry of the compactified space. Since P ( S U ( 3 ) , Z3, ZI2 or Z2 × Z~) does not act on x 5, the commutation relation (29) simply reads [~M]
= 0.
(37)
For P = S U ( 3 ) , there is no antisymmetric matrix M respecting this condition, so we cannot perform the Scherk-Schwarz mechanism. For P = Z2 x Z~, the allowed M matrices involve three parameters and, in the basis (x 5, x 6 . . . . . x l l ) , read ¢'0
0 -ml M =
0
. . .
ml 0
:
0 -m2
(38)
m2 0 0 -m 3
m3 0
In this case, the Scherk-Schwarz matrix does not anticommute with Z~ w, so the mechanism is better adapted to type IIA supergravity in 4D. We are only interested here in a field spectrum common to type IIA and heterotic strings. As we will see in the next section, the qualitative features of this case concerning the decompactification problem are, however, similar to that o f the previous example. The scalar potential (28) reads, in this case, 3
vo=le-3rzm2
((
g(lil) - -
/ 2 4g(i ~)2) g(i)'~ 22" -'}-
'
i=1 6 The following arguments were developed in collaboration with R. Minasian.
(39)
564
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
which, expressed in terms of moduli fields (24), takes the simple expression 3
U/2_l
2 (40)
Vo - tl t2t3 i=l
This scalar potential has interesting consequences. In analogy with our previous example, the scalar potential is invariant under Ui --~ 1/Ui. The three U-fields develop a vev (Ui} = 1 (the self-dual points of duality transformations) and acquire masses m/2 m2 ui = (tlt2t3)"
(41)
At the same time, the single gravitino that remains after imposing the Z2 x Z~ x Z Hw symmetries also acquires a mass, which is computed to be mj + m2 + m3
m3/2=
(42)
2 V / ~ t2t3)
(b) A Calabi-Yau or orbifold coordinate dependence. We choose here yl to be x 6, a CY or orbifold coordinate. Then the commutation relation (29) reads [P, M x 6] = 0 ,
[ Z H W , M X 6] = 0 .
(43)
For P = SU(3), there is no antisymmetric matrix M respecting this condition, so we cannot perform the Scherk-Schwarz mechanism. For P = Z2 x Z~, the situation is different; since x 6 is now odd under the first Z2 and even under the second one and under Z Hw, the commutation relation simplifies again, {Z2, M} = 0 ,
[Z~,M] = 0 ,
[z~W,M] = 0 .
(44)
Thus we can always choose M to be of the off-diagonal following form, in the basis (X5,X 6 . . . . . X 11 ):
0
m 0 -m
0
0
mmt
0 mt
(45)
Then the scalar potential, written directly in terms of moduli fields, reads 1 (m2]t2U2 _ t3U312 -+- mt2[t2U3 _ 13U212 Vo = 4StlUl t2u2t3u3
,
2 ( ( m - m')2tzt3 - m m ' ( t 2 - t3) 2) (/-/2 - Ut2)(U3 - ut3)
)•
(46)
The minimum (for rn ~ m') of the scalar potential corresponds to the fiat directions
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
(t2) = (t3),
(u2) = (u3)
and
565
(Im/-/2) = (Im U3) = 0.
(47)
We do not write here the scalar mass matrix, but just notice that the leading matrix elements, which fix the physical masses, are of the type (consider m ~ m') 9
m°2 ~ m___~-,
(48)
Stl
where tj is the modulus field corresponding to the Scherk-Schwarz coordinate. This case is the analogue of the supersymmetry breaking by compactification in the weakly coupled superstrings [ 17]. Note that in the superstring case, the mass parameters m, m' in (45) are discrete, being related to automorphisms of the compactification lattice. All the masses we computed above are measured in M-theory units. In the following section, we translate these results in 4D supergravity units.
4. Phenomenological implications of M-theory compactifications We shall discuss in this section some phenomenological issues of M-theory compactifications focusing essentially on the 4D Newton constant and the gravitino mass. For simplicity, and without affecting our conclusions, we restrict ourselves to the simplest case of I i = e y. String compactifications do not easily allow [ 1,21 ] an adjustment of the string mass scale, the vev of the dilaton and the volume of the compact space to tune the three 4D observable quantities that are the Pianck scale, the GUT scale and the gauge coupling constant. For instance, in the weakly coupled regime of the E8 × E8 heterotic string, the well-known relation 4/3 ,I 2/3 O'GUT
(49)
GH ~ "-st M 2
GUT
disagrees with experimental values by a factor of order 20. Witten has shown [21] how in the strongly coupled regime the compatibility between string predictions land experimental values could be restored. It is interesting to worry about this issue in our patterns of compactification. In the l i D action, only the M-theory scale appears, which is the l i d Planck mass MI j, such that
877.(4,77.)2/3
' " 11' # v - xs_-0
(50) Now we compactify this action in the M-theory coordinates in which, as in the fivebrane units for the 10D string, there is no kinetic term for the "radius" of the extra dimension, that is, we adopt
E. Dudas, C. Grojean/NuclearPhysicsB 507 (1997)553-570
566
g(ll) izu = e_-2,,_(5) ,g,~u , g(5) 55 = e2~''
e~60 '
g(ll) 0 =
(51)
t,(5) ~(4) • o~, = 8tzu
It is transparent that these are the natural units of 5D supergravity. By performing the Hodge transformation (using the field definitions ( 1 2 ) ) S
cr
!
tG~z~p5 = etzvp,r3 a,
(52)
the 4D low energy effective action contains the terms
S(4' D @ f d4x v@ 4' x [tM~l (7p,(4) + l~s2(01zs )2 + ( 0 ~ a , ) 2 )
+ (41)5/3sf~F ~ ]- -
.
(53)
We can pass from the M-theory action (53) to the Einstein action (11) by the Weyl rescaling g(e4) = tg~ ~. Note the absence of a kinetic term for t in (53), the clear indication of a no-scale structure. This can therefore be associated with the M-theory (or 5D supergravity) units. The Lagrangian (53) allows us to identify the gauge coupling and the 4D reduced Planck mass to be 7 -
-
1
aGUT
r.J
-
-
1
(477")5/3s'
tM~j
=
M(p4~2.
(54)
These quantities are related to the radii of the Calabi-Yau space and the extra fifth dimension expressed in terms o f the M-theory units
e_,~/2 = s_l/6 _
MGUT MI1 '
t=
R~M)Mll.
(55)
However, the radius of the fifth dimension expressed in M-theory units is not convenient for physics in four dimensions, so we prefer to express it in terms of M(p4), the fourdimensional Planck scale, by defining hat(4) t -- ' " PI R- ~ • 5
(56)
Combining these equalities, we obtain the M-theory equivalent of (49), R - I 1/3 5 ¢-~GUT G.N" ~ ~(4) AA2 ' " P1 1"GUT
(57)
and, in order to be compatible with experimental constraints, the "radius" of the fifth dimension has an intermediate scale value, as already suggested by [4,5] 7 In the weakly coupled regime, the gauge coupling constant can have large threshold corrections. If this holds true in the strongly coupled regime, some of our considerations below will be modified.
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570 R -1 5
1012-13 G e V .
567
(58)
The fifth radius also appears in the gravitino mass generated by the Scherk-Schwarz mechanism, so we have to take care that R~-I be compatible with the phenomenologically preferred value for the gravitino mass. In the expressions for the gravitino mass from (36) and (42), the mass parameters that appear in the Scherk-Schwarz mechanism using the fifth dimension are naturally of the order of the M-theory scale, so we obtain 8 Mll m3/2 "~ V / ~
_ R52
(59)
""/1"4(4)p1'
where to arrive at the last identity we have used the identification of the moduli fields (12) and the definitions (54) and (56). This expression is similar to a usual supergravity form in which an auxiliary field would have developed a vev of the order of R 5-2 " Phenomenologically, the gravitino mass should be of the order of 1 TeV. 9 This leads to a value for the extra radius quite compatible with the one deduced from Newton's constant, R 5- l
101H2 GeV.
(60)
Consequently, the presence of the extra dimension could offer a solution to the decompactification puzzle usually encountered in string compactifications. Indeed, generically, the Scherk-Schwarz mechanism in strings leads to [17] 1 m.~/9 ~
" -
--
Rcy
~ MGUT.
(61 )
This is actually similar to the result we obtained in the last section in the case where a CY coordinate was used for the Scherk-Schwarz mechanism. To check it, rewrite the masses (48) in 4D supergravity units as m 2 ~ M ( p 4 ) 2 / ( s t 2 ) ~ g 2/R2/5, where g is the gauge coupling. This relation is phenomenologically hard to accommodate. Two issues have been proposed to solve this problem: (i) Compactify six dimensions on an asymmetric Calabi-Yau space with two or more different radii [22]. However, to agree with phenomenological values tbr the gravitino mass, these two (or more) radii must have hierarchical values within about fifteen orders of magnitude. / p n + l /i//(4 ~n ~ (ii) Construct models of strings such that the gravitino mass is m3/2 ~ 1 / ~ , , C y ' " PI , (4) n (MGuT/MpI) MCUT. However, phenomenology asks for n = 4,5, and it is difficult to explicitly construct such models. XA similar formula was conjectured by Antoniadis and Quiros 151, but replacing R5 with the eleventh radius. In our expression, R5 is the radius of the fifth dimension measured in 4D Planck units ( 5 6 ) . 9 This mass will be transmitted to the observable sector if part of the gauge group originates from 5D vector multiplets or if the moduli contributing to supersymmetry breaking give threshold corrections to gauge coupling. If not, the transmission occurs through gravitational interactions [24], in which case we need m3/2 ~ 1012 GeV so R 5- I ~ 1015 GeV.
568
E. D u d a s , C G r o j e a n / N u c l e a r P h y s i c s B 5 0 7 ( 1 9 9 7 ) 5 5 3 - 5 7 0
Table 1 Identification of the real parts of the moduli fields in string compactification in various units Units
Five-brane
String
Supergravity e (b/6
A
I
e 24'/3
s
e 3~r
e3~r- 2dp
e3tr - (a/2
t
e ~r+2¢b/3
e ~r
e ~r+~/2
So the extra dimension brings a new and more satisfying alternative to the decompactification problem of compactified strings. To conclude this phenomenological study, we would like to compare our pattern of compactification (11 --* 5 --~ 4) with another one, already studied [7], in which the extra dimension was compactified first (11 --~ 10 ~ 4). From a strongly coupled 10D point of view, the radius of the eleventh dimension x 5 is R~j ~ g~l) ~ e2:,_2% and this radius is related to the string dilaton by [21] Rlt '-~ e 2'/'/3.
(62)
Combining those relations, the expressions of the real part of T becomes t ,-~ e24'/3e °, which is the expression obtained by Binrtruy [23] by compactifying the 10D heterotic string in the five-brane units that are supposed to be appropriate to the strongly coupled regime. Actually we can, in the same way, reproduce the moduli's identification obtained by string compactification down to 4D in the various units. Suppose that we are working in units characterized, in 10D, by the tensor metric g~lO). These units are related to the five-brane ones by a Weyl rescaling g(10)
-
(10)B
~,~ = a g ~
(63)
So the typical radius of the Calabi-Yau space, measured in five-brane units, is now Rc¥ "~ ( A - l e '~) 1/2 =_ ( e , ~ ) l / a ; the radius of the extra dimension, viewed from 10D in five-brane units, is Rll ~ e y - ' ~ , which is still related to the string dilaton by (62). The real parts of the fields are now identified as S = e 3°'n = , ~ . - 3 e 3 ° ,
t = e r = A - l e '~+24)/3.
(64)
For instance, string (S) and supergravity (E) units are related to five-brane ( B ) units by g(JO)B -24,/3.(lo) s -~b/6.(IO) E u~ = ~ ~;)~ = ~ ~
(65)
Eq. (64) reproduces the results of string compactification summarized in Table 1. However, from a 5D point of view, i.e. after the compactification of the CY manifold, the radius of the eleventh dimension should be R11 "~ e z', so the identification of the moduli fields is now different, leading to the previous conclusions.
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570 S
569
1/3 s 11~10
Weakly coupled regime
~4
1/2
I
11~
5 ~4
Strongly coupled regime Fig. 1.
From an 11D point of view in M-theory coordinates (51), the 11 -+ 5 --~ 4 pattern of compactification is consistent if R5 > Rcy, which reads t > s ~/2. From a 10D point of view in five-brane coordinates (63), in the 11 ~ 10 --, 4 pattern of compactification, the radii of the Calabi-Yau space and of the extra dimension are now Rcy"-" e 'r/2 and R11 "-' e 24'/3, while the consistency condition of compactification is Rcy > Rxl, which, in term of moduli fields (Table 1), now becomes t < s t/2. In both cases, the 10D string strongly coupled regime condition e 4' > 1 simply reads t > s 1/3. So we can draw, on the (t) line, the different consistent patterns of compactification (we assume that s > 1), see Fig. 1. We have seen that phenomenology asks for large values of t, so our pattern of compactification is self-consistent.
5. Conclusions We studied truncations of M-theory from 11 --~ 5 --, 4 dimensions, applying our results to the supersymmetry-breaking mechanism by compactification from 5D to 4D. The geometric properties of the moduli fields, related to the special geometry of the 5D theory, are peculiar and important for phenomenological purposes. Re S is the volume of 6D internal manifold, the various hl,i moduli fields T are related to the radius of the fifth dimension and are invariant under dilatations of the 6D manifold and the h2.1 moduli U characterize, as usual, the complex structure of the 6D manifold. Using different symmetries of the 5D theory, we are able to break supersymmetry by compactification in various ways. The most interesting one is by using an SU(2) symmetry related to the universal hypermultiplet of the 5D theory. In this case, we obtain (for the simplest possible truncation) a model described by /C=-In(S+
St ) - 3 1 n ( T + T t ) ,
W = 2 m ( l + S).
(66)
This universal superpotential for S should correspond to a non-perturbative string effect. The minimization gives S = 1 and a spontaneously broken supergravity model with a zero cosmological constant. We hope that this result will shed some light on problems such as dilaton stabilization and supersymmetry breaking in effective strings. Another example, using symmetries of the compactified 6D space, gives rise to a potential for the U moduli, the corresponding vacuum is Ui = 1 with a zero cosmological constant.
E. Dudas, C. Grojean/Nuclear Physics B 507 (1997) 553-570
570
D e f i n i n g 4 D scales a n d c o u p l i n g s , w e find that the r e s u l t i n g g r a v i t i n o m a s s in su-
pergravity (R5 ~
~'-2/A,~(4) u n i t s is m3/2 ~ "'5 / ' " P 1 , w h i c h b e t t e r s t h e usual d e c o m p a c t i f i c a t i o n l i m i t
0 ) p r o b l e m a n d c o r r e l a t e s it w i t h t h e u n i f i c a t i o n p r o b l e m in a way that l o o k s
phenomenologically promising.
Acknowledgements It is a p l e a s u r e to t h a n k E B i n & r u y , C. K o u n n a s , R. M i n a s i a n , J. M o u r a d , B. P i o l i n e a n d C. S a v o y for h e l p f u l d i s c u s s i o n s a n d c o m m e n t s .
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