Z2

Nuclear Physics B 564 Ž2000. 441–468 www.elsevier.nlrlocaternpe

Scale, gauge couplings, soft terms and toy compactification in M-theory on S 1rZ 2 Tianjun Li Department of Physics, UniÕersity of Wisconsin, Madison, WI 53706, USA Received 11 August 1999; accepted 15 September 1999

Abstract In M-theory on S 1rZ 2 , we point out that to be consistent, we should keep the scale, gauge couplings and soft terms at next order, and obtain the soft term relations M1r2 s yA, < M0rM1r2 < ( 1r '3 in the standard embedding, and M1r2 s yA in the non-standard embedding with five branes and K 5,n s 0. We construct a toy compactification model which includes higher order terms in the four-dimensional Lagrangian in standard embedding, and discuss its scale, gauge couplings, soft terms, and show that the higher order terms do affect the scale, gauge couplings and especially the soft terms if the next order correction was not small. We also construct a toy compactification model in non-standard embedding with five branes and discuss its phenomenology. We argue that one might not push the physical Calabi–Yau manifold’s volume to zero at any point along the eleventh dimension. q 2000 Elsevier Science B.V. All rights reserved. PACS: 11.25.Mj; 04.65.qe; 11.30.Pb; 12.60.Jv Keywords: M-theory; Compactification; Supersymmetry; Scale; Coupling

1. Introduction M-theory on S 1rZ 2 suggested by Horava and Witten w1x is an eleven-dimensional supergravity theory with two boundaries on which the two E8 Yang–Mills fields live. Many studies of compactification Žstandard embedding and non-standard embedding. and phenomenology have been carried out w2–53x, and, as we know, the key which connects the M-theory on S 1rZ 2 to the low energy phenomenology is the SUSY breaking soft terms, because if we knew the soft terms, by running RGE, we can obtain the low energy SUSY particle spectra and discuss their search in future colliders. On the other hand, if we knew the low energy SUSY particle spectra from future colliders, we might derive the soft terms by solving RGE w54x. Therefore, we need to understand the SUSY breaking soft terms as well as possible. The original eleven-dimensional Lagrangian obtained by Horava and Witten w1x is at order of k 2r3. Witten’s solution to the compactification of M-theory on S 1rZ 2 , which 0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 0 0 . 0 0 6 0 2 - 1

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has N s 1 supersymmetry in four dimensions just considered the next order expansion of the metric w2x. Moreover, after compactifying the theory on the deformed Calabi–Yau manifold, Lukas, Ovrut and Waldram obtained the four-dimensional Lagrangian which is at order of k 4r3 w27x. The explicit expansion parameter, defined in Section 2, is x. However, the soft terms M1r2 , M02 , A, which were obtained previously in standard embedding for the simplest compactification w30x, include the higher order correction, i.e. when we expand the soft terms M1r2 , M02 , A as polynomials in x for < x < - 1, we have the terms which are proportional to x n where n ) 1. This is inconsistent with the original calculation, because we did not include higher order corrections to the previous Lagrangian, or the Kahler potential, gauge kinetic function and superpotential. In short, ¨ the correct soft terms should be at order of x, and this is a good approximation only when x is small. In this paper, we calculate the soft terms M1r2 , M02 , A at order of x in standard embedding, and compare with previous calculations. We find that if x - 0.2, we have good approximation; we will have an observable deviation if x is large, i.e. x ) 0.5. Therefore, the phenomenology discussions, like low energy SUSY particle spectra and collider search, should stay in the small x region. In addition, we also discuss the scale and gauge couplings at order x, and the detailed discussions are similar to those in Refs. w20,21,51x with small x. Furthermore, we obtain the following soft term relations in standard embedding or non-standard embedding without five brane for the simplest compactification: M1r2 s yA ,

M0 M1r2

(

1

'3

.

Ž 1.

Because the original eleven-dimensional Lagrangian is at order of k 2r3, and Witten’s solution is at next order, it is very difficult to do the calculation which includes higher order correction unless one makes some simplification. In order to discuss higher order correction and compare the differences of the soft terms, we use the following ansatz in our toy model: Ž1. we just consider the Lagrangian which was obtained by Horava and Witten w1x; Ž2. we do not consider the higher order expansion of the deformed metric, and Ž3. we do not consider the higher order correction to the bulk fields. Because the massive modes’ contributions are suppressed by the factor Vp1r6 pr p which is about the order of 0.1 and very small when we consider the intermediate unification w20,51x, we just consider the massless modes. In standard embedding, with this ansatz and considering the Calabi–Yau manifold with Hodge–Betti number hŽ1,1. s 1,hŽ2,1. s 0, we calculate the higher order correction to the Lagrangian, or the scale, gauge couplings, Kahler potential, gauge kinetic function ¨ and superpotential. In the x n s 0 limit for n ) 1, our results are the same as those obtained previously w22–24,27x. In this toy model compactification, we find that ŽI. In order to keep g 11,11 positive, we can not push the physical Calabi–Yau manifold’s volume in the hidden sector to zero w2x. ŽII. x s 1 is not the M-theory limit. x s 3 or y s 1 Žwhere y is one-third of x . is the limit which pushes the physical Calabi–Yau manifold’s volume in the hidden sector to zero, but, in order to keep g 11,11 positive, we require that x - 3r2 or y - 1r2. x s 3r2 or y s 1r2 is considered as the M-theory limit in this paper. In other words, we define the M-theory limit as the limit which keeps the signature of the metric invariant, i.e. the

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total high order corrections to the metric are less than the zeroth order metric in each component of the metric. ŽIII. We have strong constraints on the gauge coupling a H and GUT scale MH in the hidden sector: 1 27 1

'3

a GU T - a H - 27a GUT , MGU T - MH - '3 MGUT .

Ž 2. Ž 3.

ŽIV. We notice that, if y is small Ž y - 0.2., in large parameter space, the magnitude of the gaugino mass is larger than that of the scalar mass, and if y is large Ž y ) 0.3., in most of the parameter space, the magnitude of the gaugino mass is smaller than that of the scalar mass. However, in the previous soft term analysis w30x, the magnitude of the gaugino mass is often larger than that of the scalar mass in the standard embedding. In addition, we compare the soft term deviations with the two scenarios discussed in the second paragraph. Non-standard embedding in M-theory on S 1rZ 2 is also an interesting subject w35,38–40x. We can embed the spin connection into the two E8 gauge fields, so, after compactification, the gauge groups in the observable sector and hidden sector will be G O and G H , which are subgroups of E8 . Also we can include the five branes, states which are essentially non-perturbative in heterotic string. The presence of the five-branes is very important to the three generation GUT model building because they introduce more freedom in the anomaly cancellation condition that consistent vacua need to satisfy w39,40,48,49x. Furthermore, introducing five branes will affect the gauge kinetic function, Kahler potential and non-perturbative superpotential by gaugino condensation in ¨ the next order because one introduces more moduli whose real parts are the positions of the five-branes. Without five-brane, the non-standard embedding’s results are similar to those in the standard embedding. With five-brane correction, we discuss the scale, gauge couplings and soft terms to the next order, and we find that if there is no Kahler ¨ potential for the five-brane moduli Ž K 5 ., we obtain the soft term relation M1r2 s yA .

Ž 4.

This may be interesting in the low energy phenomenology analysis. Furthermore, we consider the toy model compactification and calculate its Kahler potential, gauge kinetic ¨ function, superpotential, and discuss its scale, gauge couplings and soft terms. Our results are as follows: ŽI. In order to keep g 11,11 positive, we can not push the physical Calabi–Yau manifold’s volume in the hidden sector to zero w2x. ŽII. The M-theory limit is y5b s 12 or y5b s y1, where y5b is defined in Section 2. In order to keep g 11,11 positive, we require that y5b - 21 , and in order to keep the signature of metric gmn invariant, we require that y5b ) y1. In short, we obtain y1 - y5b - 12 . ŽIII. We have strong constraints on the gauge coupling a H and GUT scale MH in the hidden sector: 1 64

a GU T - a H - 64a GUT ,

Ž 5.

T. Li r Nuclear Physics B 564 (2000) 441–468

444 1 2

MGU T - MH - 2 MGUT .

Ž 6.

Of course, we have more freedom in the phenomenology discussion if we include five-branes, for which we introduce the new parameters b i where i s 1, N. Therefore, we do not do the numerical analysis of the soft terms because we just have three soft term parameters: M1r2 , M02 , and A, and we can make them into free parameters by varying b i and e , which is defined in Section 2. Finally, we argue that, in general, we might not push the physical Calabi–Yau manifold’s volume to zero at any point along the eleventh dimension. Our conventions are the following. We denote the eleven-dimensional coordinates by x 0 , . . . , x 9 , x 11 and the corresponding indices by I, J, K, . . . s 0, . . . ,9,11. The orbifold S 1rZ 2 is chosen in the x 11 direction, so we assume that x 11 g wypr ,pr x with the endpoints identified as x 11 ; x 11 q 2pr . The Z 2 symmetry acts as x 11 yx 11 . Then, there exist two ten-dimensional hyperplanes, Mi10 with i s 1,2, locally specified by the conditions x 11 s 0 and x 11 s pr , which are fixed under the action of the Z 2 symmetry. When we compactify the theory on a Calabi–Yau three-fold, we will use the indices A, B,C, . . . s 4, . . . ,9 for the Calabi–Yau coordinates, and the indices m , n . . . s 0, . . . ,3 for the coordinates of the remaining, uncompactified, four-dimensional space. Holomorphic and antiholomorphic coordinates on the Calabi–Yau space will be labeled by a,b,c, . . . and a,b,c, . . . . In addition, we denote the hyperplane M110 as the observable sector, and the hyperplane M210 as the hidden sector.

™

2. Scale, gauge couplings, soft terms and toy compactification in standard embedding

2.1. Scale, gauge couplings, Kahler function and soft terms reÕisited ¨ First, let us review the gauge couplings, gravitational coupling and the physical eleventh dimension radius in the M-theory w21x. The relevant eleven-dimensional Lagrangian is given by w1x LB s y

1 2k

2

HM

d 11 x g R y

'

11

1

Ý is1,2

2 2r3

2p Ž 4pk .

HM

10 i

d 10 x g 14 FIaJ F a I J .

'

Ž 7.

In the eleven-dimensional metric 1 , the gauge couplings and gravitational coupling in four dimensions are w2,17,21x

k2 8p GNŽ4. s

a GU T s

1

2pr pVp

,

1 2Vp Ž 1 q x .

Ž 8. Ž 4pk 2 .

2r3

,

Ž 9.

Because we think that eleven-dimensional metric is more fundamental than the string metric and Einstein frame, our discussion in this paper uses te eleven-dimensional metric.

T. Li r Nuclear Physics B 564 (2000) 441–468

w aH xW s

1 2Vp Ž 1 y x .

2r3

Ž 4pk 2 .

,

445

Ž 10 .

where x is defined by xsp 2

rp

2r3

k

ž /

HX

Vp2r3 4p

vn

tr F n F y 12 tr R n R 8p 2

,

Ž 11 .

where r p , Vp are the physical eleventh dimension radius and Calabi–Yau manifold’s volume Žwhich is defined by the middle point Calabi–Yau manifold’s volume between the observable sector and the hidden sector., respectively. From the above formula, one obtains xs

a H ay1 GUT y 1

.

a H ay1 GUT q 1

Ž 12 .

The GUT scale MGU T and the hidden sector GUT scale MH when the Calabi–Yau manifold is compactified are y6 MGU T s Vp Ž 1 q x . ,

Ž 13 .

My6 H s Vp Ž 1 y x . ,

Ž 14 .

or we can express the MH as MH s

1r6

aH

ž / a GU T

MGUT s

ž

1qx 1yx

1r6

/

MGUT .

Ž 15 .

Noticing that M11 s ky2 r9 , we have M11 s 2 Ž 4p .

y2r3

y1 r6

a GUT

MGUT .

Ž 16 .

The physical scale of the eleventh dimension in the eleven-dimensional metric is

w pr p x y1 s

8p 1qx

Ž 2 a GUT .

y3r2

3 MGUT

M Pl2

,

Ž 17 .

where M p l s 2.4 = 10 18 GeV. From the constraints that MGUT and MH are smaller than the scale of M11 , one obtains

a GU T (

Ž 4p .

2r3

2

,

aH (

Ž 4p . 2

2r3

,

Ž 18 .

or

a GU T ( 2.7; a H ( 2.7 .

Ž 19 .

For the standard embedding, the upper bound on x is 0.97 Ž x - 0.97., for a GU T s 251 . Second, let us review the Kahler potential, gauge kinetic function, superpotential and ¨ soft terms in the simplest compactification of M-theory on S 1rZ 2 . The Kahler potential, ¨ gauge kinetic function and superpotential are w22–24,27x K s Kˆ q K˜ < C < 2 ,

Ž 20 .

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446

Kˆ s yln w S q S x y 3ln w T q T x , K˜ s

ž

a

3 q TqT

SqS

/

Ž 21 .


Ž 22 .

O Re fab s Re Ž S q a T . da b ,

Ž 23 .

H Re fab s Re Ž S y a T . da b ,

Ž 24 .

W s dx yzC xC yC z ,

Ž 25 .

where S, T and C are dilaton, moduli and matter fields respectively. a is the next order correction constant which is related to the Calabi–Yau manifold and reads

as

xŽ SqS. TqT

.

Ž 26 .

The non-perturbative superpotential due to the gaugino condensation is w29,55x

ž

Wnp s hexp y

8p 2 C2 Ž G H .

/

Ž SyaT . ,

Ž 27 .

where the group in the hidden sector is G H and C2 Ž G H . is the quadratic Casimir of G H . In this case, G H is E8 and C2 Ž G H . s 30. With the standard formulae w56–58x, one can easily obtain the following soft terms w19,30x: M1r2 s

'3 C0 M3r2 1qx

2 M02 s V0 q M3r2 y

ž

sin u eyi u S q

2 3C0 M3r2

Ž3qx .

2

x

'3

cos u eyi u T ,

/

Ž x Ž 6 q x . sin2u q Ž 3 q 2 x . cos 2u

y2'3 xcos Ž u S y u T . sin u cos u . , Asy

'3 C0 M3r2 Ž3qx .

Ž 28 .

Ž Ž 3 y 2 x . sin u eyi u

S

q '3 xcos u eyi u T . ,

Ž 29 . Ž 30 .

where M3r2 is the gravitino mass, F S s '3 M3r2 C0 Ž S q S . sin u eyi u S ,

Ž 31 .

F T s M3r2 C0 Ž T q T . cos u eyi u T ,

Ž 32 .

C02 s 1 q

V0 2 3 M3r2

,

Ž 33 .

and V0 for the tree level vacuum density. Generically, the dynamics of the hidden sector may give rise to both ² F S : and ² F T :, but one type of F-term often dominates. Therefore, we concentrate on the two

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limiting cases: dilaton dominant SUSY breaking Ž F T s 0. and moduli dominant SUSY breaking Ž F S s 0. w56–58x 2 : ŽI. Dilaton dominant SUSY breaking scenario Ž F T s 0.. In this case, the soft terms become M1r2 s

'3 M3r2 1qx

2 M02 s M3r2 y

Asy

,

Ž 34 .

2 3 M3r2

Ž3qx .

'3 M3r2 3qx

2

x Ž6qx . ,

Ž 35 .

Ž3y2 x . .

Ž 36 .

ŽII. Moduli dominant SUSY breaking scenario Ž F S s 0.. In this case, the soft terms become x M1r2 s M , Ž 37 . 1 q x 3r2 x M0 s M , Ž 38 . 3 q x 3r2 Asy

3x 3qx

M3r2 .

Ž 39 .

Therefore, we have M0rA s y1r3,

3 0 M1r2rM0 0 2 .

Ž 40 .

However, the previous calculation is expanded in x Ž x is considered to be small. and the Lagrangian, Kahler potential and gauge kinetic function are of order x. Therefore, to ¨ be consistent, we should keep the scale, gauge couplings and soft terms of order x, because there may exist other high order corrections Ž x 2 and higher. to the Lagrangian, or Kahler potential and gauge kinetic function. Considering x n s 0 for n ) 1, the ¨ following previous scale and gauge coupling equations need to be changed:

a GU T s

1yx 2Vp

w aH xW s

Ž 4pk 2 .

1qx 2Vp

2r3

Ž 4pk 2 .

,

2r3

Ž 41 . ,

Ž 42 .

MH s Ž 1 q xr3 . MGUT ,

w pr p x y1 s 8p Ž 1 y x . Ž 2 a GUT . y3r2

2

Ž 43 . 3 MGUT

M Pl2

,

Ž 44 .

In the discussion of the dilaton dominant SUSY breaking scenario and moduli dominant SUSY breaking scenario, in order to obtain the simple soft term relations, we set V0 s 0, C0 s1 and u S s u T s 0.

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448

and the next order soft terms are M1r2 s '3 C0 M3r2 Ž 1 y x . sin u eyi u S q

ž

2 M02 s V0 q M3r2 y

2 C02 M3r2

3

x

'3

cos u eyi u T ,

/

Ž 45 .

Ž 6 xsin2u q 3cos 2u

y2'3 xcos Ž u S y u T . sin u cos u . , A s y'3 C0 M3r2 Ž 1 y x . sin u eyi u S q

ž

Ž 46 . x

'3

cos u eyi u T .

/

Ž 47 .

Obviously, we have M1r2 s yA. We obtain the following soft terms in the two limiting cases: ŽI. Dilaton dominant SUSY breaking scenario Ž F T s 0.: M1r2 s '3 M3r2 Ž 1 y x . ,

Ž 48 .

M0 s Ž 1 y x . M3r2 ,

Ž 49 .

A s y'3 M3r2 Ž 1 y x . .

Ž 50 .

The soft term relations are the same as those obtained in the weakly coupled string with dilaton dominant SUSY breaking, M1r2 s yA s '3 M0 . ŽII. Moduli dominant SUSY breaking scenario Ž F S s 0.: M1r2 s xM3r2 ,

Ž 51 .

M0 s 0 ,

Ž 52 .

A s yxM3r2 .

Ž 53 .

The soft term relations M1r2 s yA, M0 s 0 are similar to the no-scale case w59–63x, M1r2 / 0 and M0 s A s 0, for A is not very important in the RGE running. Because in general we have M1r2 s yA, we might need to know the relations between the magnitude of M1r2 and that of M0 . Taking V0 s 0,C0 s 1, we can express < M1r2 < 2 as follows 2 < M1r2 < 2 s 3 M02 q M2r3 x 2 Ž 3sin2u q cos 2u y 2'3 cos Ž u S y u T . sin u cos u . .

Ž 54 .

We obtain M1r2 s yA ,

M0 M1r2

(

1

'3

,

Ž 55 .

for the last term in Eq. Ž51. is obviously positive, and only when u S y u T s 0 Žp . and u s p 6 or u s 7p 6 Ž u s 5p 6 or u s 11p 6., < M1r2 < s '3 < M0 <. In Fig. 1, choosing

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449

Fig. 1. M0 r M1r2 versus u . Solid line, dotted, dot-dashed and dashed line represent x s 0.05,0.1,0.2,0.3, respectively.

u S s u T s 0, we draw M0 M1r2 versus u by taking x s 0.05, 0.1, 0.2, 0.3 which are represented by the solid, dots, dot-dashed and dashed lines, respectively. We can see that the deviation from " 1 '3 is large when u is close to 0 or p , or x is large. The soft terms are different from those obtained in the weakly coupled string where one has w56,57x M0

M1r2 s yA ,

1 s

M1r2

'3

.

Ž 56 .

2.2. Toy model compactification and its phenomenology If we did not consider the higher order correction to the bulk fields, one can easily write down the most general parametrized Kahler potential, gauge kinetic function and ¨ non-perturbative superpotential. For the simplest compactification, they are w20x K s Kˆ q K˜ < C < 2 ,

Ž 57 .

Kˆ s yln w S q S x y 3ln w T q T x ,

Ž 58 .

`

ž

K˜ s 1 q

Ý ci

a ŽTqT . SqS

is1 `

O fab sS 1q

Ý is1

di

aT

ž / S

i

/ž

3 TqT

/


Ž 59 .

i

da b ,

Ž 60 .

T. Li r Nuclear Physics B 564 (2000) 441–468

450 ` H fab sS

1q

Ý di is1

Wnp s hexp y

ya T

ž

S

8p 2

i

/

da b ,

Ž 61 .

`

C2 Ž G H .

ž

S 1q

Ý is1

di

ž

ya T S

i

//

.

Ž 62 .

Using standard methods w56–58x, one can easily calculate the soft terms. But here one introduces many parameters, which is useless to the low energy phenomenology analysis. Therefore, we construct a toy simple model as an example which contains high order corrections. In fact, in order to consider high order corrections to the scale, gauge couplings and soft terms in detail, we need to consider the higher order Lagrangian. However, the original Lagrangian is of order k 2r3 w1x, and Witten’s solution to the compactification of M-theory on S 1rZ 2 , which has N s 1 supersymmetry in four dimensions, just considered the next order expansion of the metric w2x. Therefore, it is really very tough to consider the higher order terms realistically. The ansatz of our toy model is that we just consider the Lagrangian which was obtained by Horava and Witten; we do not consider the higher order expansion of the metric and higher order correction to the bulk fields. With this ansatz, we can discuss the higher order corrections to the scale, gauge couplings, Kahler potential, gauge kinetic function and superpotential. ¨ The bosonic part of the eleven-dimensional supergravity Lagrangian is given by w1x LB s

1

k2

HM

11

'2 y 3456

ž

d 11 x g y 12 R y 481 GI JK L G I JK L

'

e I1 I 2 . . . I11 CI1 I 2 I3GI4 . . . I 7 GI8 . . . I11 1

y

Ý is1,2

2p Ž 4pk 2 .

2r3

HM

10 i

/

d 10 x g 14 FIaJ F a I J ,

'

Ž 63 .

where

k2

G 11 I JK s Ž E 11CI JK " 23 permutations. q

l2 s 2p Ž 4pk 2 .

2r3

11 '2 l2 d Ž x . v I JK ,

.

In this paper, we consider the compactification on a Calabi–Yau manifold with Hodge–Betti numbers hŽ1,1. s 1 and hŽ2,1. s 0. For the zeroth-order metric, there are only one dilaton and one modulus controlling the overall size of the Calabi–Yau space and the length of the orbifold interval. We write 2

A B 2c 11 Ž0. ds 2 s gmn dx m dx n q e 2 a g Ž0. A B dx dx q e Ž dx . ,

Ž 64 .

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451

so that the physical Calabi–Yau volume Vp is e 6 a V and the physical length of the orbifold interval pr p is e cpr . For the next order correction, because the massive modes are suppressed by the factor Vp1r6 pr p which is about the order of 0.1 and very small, when we consider the intermediate unification we just consider the massless mode. The deformed metric isw27x 2

ds s 1 q b e

cy4 a

ž

2 x 11

y1

pr

/

Ž0. gmn dx m dx n q

A B cy4 a =e 2 a g Ž0. A B dx dx q 1 y 2 b e

2 x 11

ž

pr

1ybe

y1

/

cy4 a

ž

2 x 11

pr

y1

/

2

e 2 c Ž dx 11 . ,

Ž 65 .

where 1 3

bs p

r

2

tr F n F y 12 tr R n R

2r3

k

ž /

vn . Ž 66 . V 2r3 4p 8p 2 X The M-theory limit is b e cy 4 a s " 12 . In order to keep g 11,11 positive, we require that y 12 - b e cy 4 a - 12 , which is enough to keep the physical Calabi–Yau manifold’s volume non-zero at any point along the eleventh dimension. Defining CA B11 s 16 xv A B , Cmn 11 s 16 Bmn , Ž 67 .

H

and

E w m Bnr x s 13 ey1 2 aemnr d Ed s ,

cˆ s c q 2 a ,

Ž 68 .

one can obtain the Lagrangian in four dimensions to the zeroth order w17,27x, pr V S01 s 2 y g yR y 18 Em aE m a y 32 Em cˆE m cˆ k M4

H'

y3ey2 cEˆm xE mx y ey12 a Em sE ms , S02 s

pr V k

2

HM 'y g 4

3k 2 y 4

y3eycˆDm CD m C y

p q<2 ˆ a< d ey2 cy6 y pqrC C

y 14 e 6 a FmnH F H mn y S03 s

pr V k2

HM 'y g 4

3k 2 y 8

Ž 69 .

'2 s 4

3 y2 cˆ 8

e

3k 2 32

&

O Omn Fmn F y

2

ž C D CD m

m

3i

'2

ey2 cˆ CDm C y CDm C E mx

ž

/

2

ˆ a CT i C y 1 e 6 a F O F Omn ey2 cy6 Ž . mn 4

'2 s 4

&

FmnH F H mn ,

C q C 2 Dm CD m C y 2 < C < 2 Dm CD m C

p q r <2 ˆ a< d ey3 cy6 , pqrC C C

Ž 70 .

/ Ž 71 .

where k s 4 2 r p Ž4prk .1r3 and for the gauge fields and gauge matter fields, we rescale them as follows w17,27x

'

Cp

™ p'2 r ž 4kp /

1r3

Cp,

Am

™ p'2 r ž 4kp /

1r3

Am .

Ž 72 .

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In the above equations, S01 is from the bulk Lagrangian, S02 from the boundary Lagrangian and S03 from the boundary Lagrangian with an additional d Ž0. factor which is not well defined. Considering the higher order correction from the deformed metric to the boundary terms, we obtain S2 s

pr V k2

HM 'y g 4

3i y

ˆ 2 L D CD m C y3eycL q y m

Lq Ly ey2 cˆ CDm C y CDm C E mx

ž

'2 3k 2

/

2 y2 cy6 Lq Ly e ˆ a< dpqrC pC q < 2 y

y 4

3k 2

2 y2 cy6 Lq Ly e ˆ a Ž CT i C .

32

2

3 6a O 3 6a H y 14 Lq e Fmn F Omn y 14 Ly e Fmn F H mn

'2 s y 4 S3 s

pr V k2

&

3 O Omn Lq Fmn F y

HM 'y g 4

3 8

'2 s 4

&

3 Ly FmnH F H mn ,

Ž 73 .

Lq Ly ey2 cˆ C 2 Dm CD m C q C 2 Dm CD m C

ž

y2 < C < 2 Dm CD m C y

/

3k 2 8

2 y3 cy6 Ly e ˆ a < dp qr C pC qC r < 2

1

Ž 1 q 2 b e cy4 a .

1r2

,

Ž 74 . where

Lqs 1 q b e cy 4 a ,

Lys 1 y b e cy 4 a .

Ž 75 .

Because S 3 is proportional to d Ž0. in the original Lagrangian and is of order k 4r3, we use S 2 to obtain the Kahler potential, gauge kinetic function and the superpotential in ¨ this toy compactification: K s Kˆ q K˜ < C < 2 ,

Ž 76 .

Kˆ s yln w S q S x y 3ln w T q T x ,

Ž 77 .

K˜ s

3 TqT

ž

O fab sS 1q

ž ž

H fab sS 1y

1q

bT S

bT S

b ŽTqT . SqS

2

/ž

1y

b ŽTqT . SqS

/

,

Ž 78 .

3

/ /

da b ,

Ž 79 .

da b ,

Ž 80 .

3

T. Li r Nuclear Physics B 564 (2000) 441–468

ž

Ws 1q

3r2

bT

/ ž

S

Wnp s hexp y

1y

8p 2 C2 Ž G H .

bT S

453

3r2

kd x y z C x C y C z ,

/

ž

S 1y

bT S

Ž 81 .

3

/

,

Ž 82 .

where S s e 6 a q i'2 s , T s e cˆ q i'2 x . Ž 83 . One can easily prove that, when b is very small, i.e. taking b 2 s 0, one obtains the same results as those obtained previously w22–24,27x. Now, we will consider the phenomenology in the toy compactification. First, let us discuss the scale and gauge couplings,

k2 8p GNŽ4. s

a GU T s aH s

,

2pr pVp d 1

2Vp Ž 1 q y . 1

2Vp Ž 1 y y .

3

3

Ž 84 . Ž 4pk 2 .

Ž 4pk 2 .

2r3

2r3

,

Ž 85 .

,

Ž 86 .

where ys

b ŽTqT .

s

SqS 1

ds

a H1r3ay1r3 GUT y 1 a H1r3ay1r3 GUT q 1

Hy1'1 y 2 yx Ž 1 q yx . 1

2

,

Ž 87 .

3 Ž 1 y yx . dx

.

Hy1'1 y 2 yx dx

Fig. 2. d versus y.

Ž 88 .

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454

Fig. 3. M1r 2 versus x in the unit of gravitino mass. Solid, dotted and dashed lines represent scenarios O, S and T, respectively.

By the way, comparing with the compactification in the last subsection, we have y s x 3, b s a 3. The GUT scale MGU T and the hidden sector GUT scale MH when the Calabi–Yau manifold is compactified are 3

Ž 89 .

My6 H s Vp Ž 1 y y . ,

Ž 90 .

y6 MGU T s Vp Ž 1 q y . , 3

Fig. 4. M0 versus x in the unit of gravitino mass. Solid, dotted and dashed lines represent scenarios O, S and T, respectively.

T. Li r Nuclear Physics B 564 (2000) 441–468

455

Fig. 5. A versus x in the unit of gravitino mass. Solid, dotted and dashed lines represent scenarios O, S and T, respectively.

or we can express the MH as MH s

1qy

1r2

ž / 1yy

M11 s 2 Ž 4p .

MGUT ,

y2r3

Ž 91 . y1 r6

a GUT

MGUT .

Ž 92 .

And the physical scale of the eleventh dimension is

w pr p x y1 s

8pd

Ž1qy.

2 a GUT . 3 Ž

y3r2

3 MGUT

M Pl2

.

Fig. 6. Soft terms versus angle u with y s 0.15 in units of gravitino mass.

Ž 93 .

T. Li r Nuclear Physics B 564 (2000) 441–468

456

Fig. 7. Soft terms versus angle u with y s 0.2 in units of gravitino mass.

In order to keep g 11,11 positive, we require that y 12 - y - 12 . In addition, in Fig. 2, we plot the d versus y, and we find that d is about 1. So, d will not change the previous scale picture w20,21,51x. The bounds on MH and a H are 1 27 1

'3

a GU T - a H - 27a GUT ,

Ž 94 .

MGU T - MH - '3 MGUT .

Ž 95 .

In addition, because y 12 - y - 12 , the discussions of the scale and gauge couplings are similar to those in Refs. w20,21,51x with small x. So, we will not redo the discussion here. Second, let us discuss the soft terms from above Kahler potential and gauge kinetic ¨ function. Using standard tree level formulae w56–58x, we obtain the following soft terms: M1r2 s

'3 C0 M3r2 1qy

2 M02 s M3r2 q V0 y

sin u Ž 1 y 2 y . eyi u S q '3 ycos u eyi u T , 2 M3r2 C02

Ž1 y y 2 .

2

Ž 3 y Ž2 y 9 y q 3 y 3 . sin2u q Ž1 y 5 y 2 q 2 y 3 y 2 y 4 .

=cos 2u q 2'3 ysin u cos u cos Ž u S y u T . Ž y1 q 6 y y y 2 . . , Asy

'3 1yy2 3y

y

1yy2

Ž 96 .

Ž 97 .

M3r2 C0 Ž 1 y 3 y q 8 y 2 . sin u eyi u S M3r2 C0 Ž 1 y 3 y . cos u eyi u T ,

Ž 98 .

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457

Fig. 8. Soft terms versus angle u with y s 0.3 in units of gravitino mass.

where F S s '3 M3r2 C0 Ž S q S . sin u eyi u S , F T s M3r2 C0 Ž T q T . cos u eyi u T , C02 s 1 q

V0 2 3 M3r2

,

Ž 99 . Ž 100. Ž 101.

and V0 for the tree level vacuum density. Now, we discuss the numerical results for the soft terms by taking V0 s 0,C0 s 1, and u S s u T s 0. First, we compare the soft terms. We denote the original soft terms ŽEqs.

Fig. 9. Soft terms versus angle u with y s 0.4 in units of gravitino mass.

458

T. Li r Nuclear Physics B 564 (2000) 441–468

Fig. 10. Soft terms versus angle u with y s 0.5 in units of gravitino mass.

Ž28. – Ž30.. as scenario O, the soft terms ŽEqs. Ž45. – Ž47.. at next to the leading order, which is simple, as scenario S, and the soft terms ŽEqs. Ž96. – Ž98.. in the above toy compactification as scenario T. Choosing u s p 4, we draw the soft terms: M1r2 , M0 , A versus x in Fig. 3, Fig. 4, and Fig. 5 respectively. We notice that, when x - 0.2, the three scenarios agree very well; when x ) 0.5, we can see obvious differences among them. In addition, we can compare the magnitudes of the soft terms in these three S T O scenarios: M1r2 - M1r2 - M1r2 , M0S - M0O - M0T, < A < S - < A < O - < A < T. Therefore, if we discuss M-theory phenomenology from the soft terms, we should stay in the small x region in order to be consistent, because we do not know the higher order correction, although in a realistic model, x might be large. Second, we analyze the soft terms in the toy compactification, and draw the soft terms versus u by choosing y s 0.15,

Fig. 11. Soft terms versus angle u with y sy0.3 in units of gravitino mass.

T. Li r Nuclear Physics B 564 (2000) 441–468

459

0.2,0.3,0.4,0.5 in Figs. 6–10, respectively. We notice that, if y is small Ž y - 0.2., in large parameter space, the magnitude of the gaugino mass is larger than that of the scalar mass; if y is large Ž y ) 0.3., in most of the parameter space, the magnitude of the gaugino mass is smaller than that of the scalar mass. However, in the previous soft term analysis in standard embedding, the magnitude of the gaugino mass is often larger than that of the scalar mass w30x. We also discuss the case that y - 0, where we just use y s y0.3 as an example in Fig. 11.

3. Scale, gauge couplings, soft terms and toy compactification in non-standard embedding

3.1. Scale, gauge couplings, Kahler function and soft terms reÕisit ¨ The non-standard embedding without 5-branes is similar to the standard embedding in Subsection 2.1. In general, non-standard embedding includes 5-brane correction, which introduces additional moduli fields Zn , whose real parts are the five-brane positions along the eleventh dimension. The Kahler potential, gauge kinetic function and ¨ superpotential are w39,40x K s Kˆ q K˜ < C < 2 ,

Ž 102.

Kˆ s yln w S q S x y 3ln w T q T x q K 5 ,

Ž 103.

K˜ s

ž

3 ez

3 q TqT

SqS

/


Ž 104.

N

ž ž

Ý Ž 1 y Zn . 2 b n

O fab s S q 3e T b 0 q

ns1

/

da b ,

Ž 105.

N

H fab s S q 3 e T b Nq1 q

Ý Ž Zn . 2 b n ns1

/

da b ,

Ž 106.

W s dx yzC xC yC z ,

½

Wnp s hexp y

Ž 107.

8p 2 C2 Ž G H .

N

ž

S q 3 e T b Nq1 q

Ý Ž Zn . 2 b n ns1

/5

,

Ž 108.

where

k es

2r3

ž p/

2p 2r V 2r3

4

,

N

zsb 0q

Ý Ž 1 y 12 Ž Zn q Zn . . ns1

Ž 109. 2

bn ,

Ž 110.

T. Li r Nuclear Physics B 564 (2000) 441–468

460

and K 5 is the Kahler potential for the moduli fields Zn . In addition, b 0 , b Nq 1 are the ¨ instanton number in the observable sector and hidden sector, b n where n s 1, . . . , N is the magnetic charge on each five-brane. The cohomology constraint on b i is Nq1

b i s 0, ,

Ý

Ž 111.

ns0

which means that the net charge should be zero. With the assumption ² Zn : s ²Re Zn :, ² S : s ² S :, ²T : s ²T :, we discuss the scale and gauge couplings first w51x:

k2 8p

GNŽ4. s

a GU T s aH s

2pr pVp d 5b 1

2Vp Ž 1 q e . 1

,

Ž 112.

Ž 4pk 2 . Ž 4pk 2 .

2Vp Ž 1 q e h .

2r3

2r3

,

Ž 113.

,

Ž 114.

where pr

d 5b s

H0

Ž 1 y 2 y5b . dx 11

pr

H0 es

.

3 ez Ž T q T .

eh s

Ž 115.

Ž 1 y y5b . dx 11

SqS 3e Ž T q T . SqS

,

Ž 116. N

ž

b Nq 1 q

Ý Ž 12 Ž Zn q Zn . .

2

ns1

/

bn ,

Ž 117.

and where y5b s

e ŽTqT . SqS

n

2

Ý

Nq1

b m Ž
ms0

Ý Ž Ž Re Zm . 2 y 2Re Zm . b m

,

ms0

Ž 118. in the interval Re Zn (
x 11

,

Re Zn s

x 11 n

, Ž 119. pr pr where n s 1, . . . , N. y5b is a function of x, and in general d 5b will not be 1 in non-standard embedding with five-branes. However, d 5b may be close to 1 for we consider the small next order correction, so, it will not change the previous scale picture w20,21,51x. The GUT scale MGU T and the hidden sector GUT scale MH when the Calabi–Yau manifold is compactified are y6 MGU Ž 120. T s Vp Ž 1 q e . , My6 H s Vp Ž 1 q e h . ,

Ž 121.

T. Li r Nuclear Physics B 564 (2000) 441–468

461

or we can express the MH as MH s

1r6

1qe

ž

/

1 q eh

M11 s 2 Ž 4p .

MGUT ,

y2r3

Ž 122. y1 r6

a GUT

MGUT .

Ž 123.

The physical scale of the eleventh dimension is

w pr p x y1 s

8pd 5b 1qe

Ž 2 a GUT .

y3r2

3 MGUT

M Pl2

.

Ž 124.

Second, we can also obtain the following soft terms w50,51x: 1

M1r2 s

ž F˜ q eF˜ S

1qe

2 M02 s V0 q M3r2 y

T

q eF˜ nzn ,

Ž 125.

/

1

Ž3 qe.

2

e 6 q e . < F˜ S < 2 q 3 Ž 3 q 2 e . < F˜ T < 2 y 6 eReF˜ S F˜ T

žŽ

q Ž ez Ž 3 q e . zn m y e 2zn zm . F˜ n F˜ m y 6 ezn Re F˜ S F˜ n q 6 ezn Re F˜ T F˜ n ,

/

Ž 126. Asy

1 3qe

ž Ž 3 y 2 e . F˜

S

q 3eF˜ T q Ž 3ezn y Ž 3 q e . z K 5, n . F˜ n ,

/

Ž 127.

where

V0 s

2

FS

q3

SqS

2

FT TqT

2 q F n F m K 5, n m y 3 M3r2

Ž 128.

and F˜ n s

F˜ S s

Fn

z

,

FS SqS

Ž 129.

,

F˜ T s

FT TqT

,

Ž 130.

and K 5, n s E K 5rE Zn , K 5, n m s E 2 K 5rŽ E Zn E Zm ., zn s EzrE Zn , zn m s E 2zrŽ E Zn E Zm .. The original calculation is at order of e . Therefore, to be consistent, we should keep the scale, gauge couplings, and soft terms at order of e , because there may exist other high order correction Ž e 2 and higher. to the Lagrangian, or the Kahler potential and ¨

T. Li r Nuclear Physics B 564 (2000) 441–468

462

gauge kinetic function. Considering e n s 0 for n ) 1, the following scale and gauge coupling equations need to be changed: 1ye 2r3 a GU T s Ž 4pk 2 . , Ž 131. 2Vp

aH s

1 y eh 2Vp

Ž 4pk 2 .

2r3

,

Ž 132.

MH s Ž 1 q er6 y e hr6 . MGUT ,

w pr p x y1 s 8pd5b Ž 1 y e . Ž

Ž 133.

3 MGUT y3r2 2 a GUT M Pl2

,

.

Ž 134.

and the soft terms at order e are M1r2 s Ž 1 y e . F˜ S q eF˜ T q eF˜ nzn , 2 M02 s V0 q M3r2 y

ž

2 3

Ž 135.

e < F˜ S < 2 q < F˜ T < 2

y 23 eRe F˜ S F˜ T q 13 ezzn m F˜ n F˜ m y 23 ezn Re F˜ S F˜ n q 23 ezn Re F˜ T F˜ n ,

/

A s y Ž 1 y e . F˜ S q eF˜ T q Ž ezn y z K 5, n . F˜ n .

ž

Ž 136. Ž 137.

/

Obviously, if K 5, n s 0, we obtain M1r2 s yA in this case, too. For the scale and gauge couplings, because we consider small e or the small next order correction, the discussions are similar to those in Refs. w20,21,51x with small x. And because of too many parameters in the soft term expressions, we will not do the numerical analysis here, for we can just let M1r2 , M0 , A be free parameters. 3.2. Toy compactification and its phenomenology In this section, we will consider the toy compactification in non-standard embedding. First, we consider the case without five-brane, which is similar to the case in Subsection 2.2. Therefore, we just write down the Kahler potential, gauge kinetic function and ¨ superpotential, K s Kˆ q K˜ O < CO < 2 q K˜ H < CH < 2 ,

Ž 138. Ž 139.

Kˆ s yln w S q S x y 3ln w T q T x , K˜ s

3

O

K˜ s

TqT 3

H

TqT

O fab sS

Re

ž

1q

H fab sS

ž

ž ž

eb 0 Ž T q T .

1q

SqS

eb 0 Ž T q T .

1q

SqS

eb 0 T S

1y

2

/ž /ž

1y

1y

eb 0 Ž T q T .

/

SqS

eb 0 Ž T q T . SqS

,

Ž 140.

,

Ž 141.

2

/

3

/

da b ,

eb 0 T S

Ž 142.

3

/

da b ,

Ž 143.

T. Li r Nuclear Physics B 564 (2000) 441–468

ž ž

WO s 1 q WH s 1 q

3r2

eb 0 T

/ ž / ž

S

1y

3r2

eb 0 T S

1y

8p 2

Wnp s hexp y

eb 0 T

C2 Ž G H .

S

eb 0 T S

ž

S 1y

3r2

/ /

bT S

463

kd x y z COx COy COz ,

Ž 144.

kd x y z CHx CHy CHz ,

Ž 145.

.

Ž 146.

3r2

3

/

The soft terms in the observable sector are similar to those in Subsection 2.2. The only difference between here and Subsection 2.2 is that in Subsection 2.2 the gauge group in the hidden sector is E8 , and here the gauge group in the hidden sector is the subgroup of E8 . Now we consider the case with five-brane. The next order metric is the following w39,40x: 2

A B 2c 11 Ž0. ds 2 s Ž 1 q y5b . gmn dx m dx n q Ž 1 y y5b . e 2 a g Ž0. A B dx dx q Ž 1 y 2 y5b . e Ž dx . .

Ž 147. The M-theory limit is y5b s 12 or y5b s y1. In order to keep g 11,11 positive, we require that y5b - 12 , and in order to keep the signature of metric gmn invariant, we require that y5b ) y1. In short, we obtain y1 - y5b - 12 . The physical Calabi–Yau manifold’s volume is obviously non-zero at any point along the eleventh dimension, and if one defined Vpmin and Vpmax as the minimum and maximum physical Calabi–Yau manifold’s volume along the eleventh dimension respectively, one obtains Vpmax - 64Vpmin .

Ž 148.

Using the same ansatz as in Subsection 2.2, we obtain the Kahler potential, gauge ¨ kinetic function and the superpotential K s Kˆ q K˜ O < CO < 2 q K˜ H < CH < 2 ,

Ž 149.

Kˆ s yln w S q S x y 3ln w T q T x q K 5 ,

Ž 150.

K˜ O s

3 TqT

ž

ž

= 1y

K˜ s

3

H

TqT

ž

ž

= 1y

ž

O fab sS 1q

1q

e ŽTqT .

Nq1

SqS

ms0

e ŽTqT .

Nq1

SqS

ms0

1q

e ŽTqT .

Nq1

SqS

ms0

Nq1

SqS

ms0

S

Ý Ž 1 y Ž Zm q Zm . .

Ý Ž 1 y 12 Ž Zm q Zm . .

e ŽTqT .

eT

2 1 2

1 2

Ý Ž Ž Zm q Zm . .

Ý Ž 12 Ž Zm q Zm . .

Ý

2

b

/ /

bm ,

2

/

b

bm ,

m

m

/ Ž 151.

2

Ž 152.

3

Nq1 ms0

2

2

2 Ž 1 y Zm . b m da b ,

/

Ž 153.

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464

H fab sS

ž

1q

eT

Zm2

Ý

S

Wnp s hexp y

3

Nq1 m

b

ms0

8p 2 C2 Ž G H .

/

da b , eT

ž

S 1q

Ž 154. Zm2

Ý

S

3

Nq1

b

m

ms0

/

.

Ž 155.

With the assumption ² Zn : s ² Zn :, ² S : s ² S :, and ²T : s ²T :, we discuss the scale and gauge couplings first,

k2 8p GNŽ4. s

a GU T s

1 T 3

Ž 156. Ž 4pk 2 .

2r3

,

Ž 157.

2Vp Ž 1 q e . 1

aH s

,

T 2pr pVp d 5b

Ž 4pk 2 .

3 2Vp 1 q e Th

Ž

.

e ŽTqT .

Nq1

SqS

ms0

e ŽTqT .

Nq1

SqS

ms0

2r3

,

Ž 158.

where eT s

e Th s

Ý Ž 1 y 12 Ž Zm q Zm . . Ý Ž 12 Ž Zm q Zm . .

pr T d 5b s

H0 (1 y 2 y

2

2

bm ,

Ž 159.

bm ,

Ž 160.

2 3 Ž 1 q y5b . Ž 1 y y5b . dx 11

5b

.

pr

H0 (1 y 2 y

Ž 161.

11 5b dx

The GUT scale MGU T and the hidden sector GUT scale MH are, when the Calabi–Yau manifold is compactified, 3

T My6 GU T s Vp Ž 1 q e . ,

Ž 162.

3

T My6 H s Vp Ž 1 q e h . ,

Ž 163.

i.e. we can express the MH as MH s

1 q eT

1r2

ž / 1 q e Th

M11 s 2 Ž 4p .

y2r3

MGUT ,

Ž 164.

y1 r6

a GUT

MGUT .

Ž 165.

The physical scale of the eleventh dimension is

w pr p x y1 s

T 8pd 5b

Ž 1 q eT .

2 a GUT . 3 Ž

y3r2

3 MGUT

M Pl2

.

Ž 166.

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465

In order to keep g 11,11 and g 1,1 positive, we require that y1 - y5b - 12 . So, we obtain y 12 - e T - 1 ,

y 12 - e Th - 1

Ž 167.

and the bounds on MH and a H are 1 64 1 2

a GU T - a H - 64a GUT ,

Ž 168.

MGU T - MH - 2 MGUT .

Ž 169.

In addition, the discussions of the scale and gauge couplings are similar to those in Refs. T w20,21,51x with small x if we defined the physical eleventh dimension length as pr p d 5b . T So, we will not redo the discussions here. Furthermore, d 5b might be around about 1 because of the constraints on b i and the charge distributions. In addition, we obtain the following normalized soft terms: 1

M1r2 s

e ŽTqT . V

ž

FS 1y2

SqSqe ŽTqT . V

SqS

/

q3 e F TV y 3 e F n Ž T q T . V n ,

2 M02 s M3r2 q V02 y < F S < 2

yFn

y2

ž

ž

K˜ mOn

y

K˜ O

K˜ SOn K˜ O

O K˜ SS

ž

K˜ O

K˜ mO K˜ nO K˜ O 2

K˜ SO K˜ nO

y

K˜ O 2

/

/

y

Ž 170.

K˜ SO K˜SO K˜ O 2

Fm y2

ž

/

y< FT <2

O K˜ ST

y

K˜ O

Re Ž F n F S . y 2

ž

K˜TOT

ž

K˜ O

K˜ SO K˜TO

/

K˜ O 2

K˜TOn K˜ O

y

y

K˜TO K˜TO K˜ O 2

/

Re Ž F T F S .

K˜TO K˜ nO K˜ O 2

/

Re Ž F n F T . ,

Ž 171.

ž

AsFS y

1 y SqS

3 K˜ SO K˜ O

/ ž

qFT y

3 y TqT

3 K˜TO K˜ O

/ ž

q F n K 5, n y

3 K˜ nO K˜ O

/

,

Ž 172. where

˜

K SO s 3

O K˜ SS s6

O K˜ ST s6

 

eV

y

ž

Ž SqS.

2

q2

eV

Ž SqS.

3

y3

3

q3

Ž SqS.

Ž SqS. 4

e 3ŽTqT . V 3

Ž SqS.

4

q3

3

e 2 ŽTqT . V 2

Ž SqS.

e 2V 2

2

e 2 ŽTqT . V 2

e 3ŽTqT . V 3

Ž SqS.

4

0

,

Ž 173.

2

y6

e 3ŽTqT . V 3

Ž SqS.

/

,

5

0

,

Ž 174.

Ž 175.

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466



˜

K SOn s 3

ž ž ž

eV n

K˜TO s 3 y

 

˜

ŽTqT .

2

y

Ž SqS. e 3V 3

3

e 2VV n

Ž SqS. eV n SqS

2

y

Ž SqS. q3

q2

y

e 3ŽTqT . V 3

Ž SqS.

Ž SqS.

2

/

3

,

,

Ž 176. Ž 177.

2

,

Ž 179. 2

q3

e 3 Ž T q T . V 2V n

Ž SqS.

e 2 Ž T q T . Vb n

e 2 Ž T q T . Vn Vl

/

0

Ž 178.

e 2 Ž T q T . VV n

Ž SqS.

3

4

,

e 3 Ž T q T . V 2V n

Ž SqS.

2Ž S q S .



/

3

e 3 Ž T q T . V 2V n

Ž SqS.

y2

Ž SqS.

eb n

q6 y

2

y9

3

e 2V 2

1

ŽTqT .

K˜ nO s 3 y

K nOl s 3

Ž SqS.

1

K˜TOT s 6 K˜TOn s 6

y4

2

Ž SqS.

2

e 2 Ž T q T . VV n

2

3

2

3 2

y

e 3 Ž T q T . V 2b n

Ž SqS.

3

0 0

,

Ž 180.

dn l

2

y3

e 3 Ž T q T . VV n V l

Ž SqS.

3

0

,

Ž 181.

and Nq1

Vs

Ý Ž 1 y 12 Ž Zm q Zm . .

2

bm ,

Ž 182.

ms0

V n s Ž 1 y 12 Ž Zn q Zn . . b n .

Ž 183. i

Because we introduce the new parameters b , where i s 1, . . . , N, if we included-five branes, we have a lot of freedom in phenomenology discussions. We do not do the numerical analysis of the soft terms because we just have three soft term parameters M1r2 , M02 , and A, and we can make them free parameters by varying b i and e . 4. The physical Calabi–Yau manifold’s volume In the above toy compactifications, we notice that the physical Calabi–Yau manifold’s volume at any point along the eleventh dimension can not be zero from the metric directly. We can argue that this may be the general result. The next order metric can be written as w2,27x

ž

ds 2 s 1 q

ž

'2 6

q 1y

/

Ž0. B gmn dx m dx n q

'2

/

ž

1y

'2 3

/

B g AŽ B0 . q '2 B A B dx A dx B

2

B Ž dx 11 . ,

3 1 where E w I1 B I 2 . . . I 7 x s 7 )GI1 . . . I 7 , and B a b is defined as follows w27x: Bmnrs a b s emnrs B a b ,

Ž 184. Ž 185.

T. Li r Nuclear Physics B 564 (2000) 441–468

467

and B s g ( 0 )AB BAB . Because the massive modes are suppressed, B A B can be written as a '2 B A B have the same linear function of the massless modes w27x. So, '2 B g Ž0. A B and 3

' sign. In addition, the magnitude of '2 B g Ž0. A B is larger than that of 2 B A B , for example, 3

in the toy compactification, the magnitude of '2 B A B is half of that of '2 B g Ž0. A B which 3

can be easily proved at next order in general. Therefore, in order to keep g 11,11 positive, we may not push the physical Calabi–Yau manifold’s volume to zero at any point along the eleventh dimension. 5. Conclusion In M-theory on S 1rZ 2 , we point out that to be consistent, we should keep the scale, gauge couplings and soft terms at next order in the standard embedding and non-standard embedding, and obtain the soft term relations M1r2 s yA, < M0rM1r2 < ( 1r '3 in the standard embedding and the soft term relation M1r2 s yA in non-standard embedding with five-branes and K 5, n s 0. Furthermore, we construct a toy compactification model which includes higher order terms in the four-dimensional Lagrangian in standard embedding, and discuss its scale, gauge couplings, soft terms, and explicitly show that the higher order terms do affect the scale, gauge couplings and especially the soft terms if the next order correction was not small. We also construct a toy compactification model in non-standard embedding with five-branes, calculate its Kahler potential, gauge ¨ kinetic function, and discuss the scale, gauge couplings and soft terms. The M-theory limit gives strong constraints on the gauge coupling Ž a H . and GUT scale Ž MH . in the hidden sector in these toy models. Finally, we argue that, in general, we might not push the physical Calabi–Yau manifold’s volume to zero at any point along the eleventh dimension. The phenomenological consequences of the soft terms obtained in this paper and the multi-moduli toy compactification are under investigation. Acknowledgements This research was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-95ER40896 and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. References w1x w2x w3x w4x w5x w6x w7x w8x w9x

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468 w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x w34x w35x w36x w37x w38x w39x w40x w41x w42x w43x w44x w45x w46x w47x w48x w49x w50x w51x w52x w53x w54x w55x w56x w57x w58x w59x w60x w61x w62x w63x

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