Orbifolds with continuous Wilson lines and soft terms

FTUAM 96/41 KAIST-TH-96/18

arXiv:hep-ph/9612405v1 19 Dec 1996

Orbifolds with continuous Wilson lines and soft terms H.B. Kim Departamento de F´ısica Te´orica C–XI Universidad Aut´onoma de Madrid, Cantoblanco, 28049 Madrid, Spain [email protected]

C. Mu˜ noz∗ Department of Physics Korea Advanced Institute of Science and Technology, Taejon 305-701, Korea [email protected]

Abstract

Orbifold compactifications with continuous Wilson lines have very interesting characteristics and as a consequence they are candidates to obtain realistic models. We perform an analysis of the soft supersymmetry-breaking terms arising in this type of compactifications. We also compare these results with those of orbifolds without including continuous Wilson lines. Their phenomenological properties turn out to be similar.

FTUAM 96/41 KAIST-TH-96/18 December 1996

On leave of absence from Departamento de F´ısica Te´ orica C-XI, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain. ∗

The compactification of the E8 × E8 heterotic string on orbifolds [1] proved in the past to be a helpful way to construct semirealistic models. In particular, when quantized Wilson lines [2] were included in the analysis of orbifold compactifications, smaller gauge groups containing the standard model with appropriate matter representations were obtained [3]. Although the rank of these models is still 16 and the matter content too large, both can be reduced due to the effect of Fayet-Iliopoulos terms associated to anomalous U(1)′ s [4]. On the other hand, there is an alternative mechanism to produce the breaking of the gauge symmetry obtained from orbifold compactification. This can be carried out through the use of continuous Wilson line background fields [5]. They are untwisted moduli fields (in addition to the usual Ti and Ui moduli) whose vacuum expectation values (VEVs) break the gauge group lowering the rank. In addition they give mass to some matter fields. This is therefore a potential mechanism to obtain relevant models. Another interesting phenomenological property of continuous Wilson lines is the following. The discrepancy between the unification scale of gauge couplings and the string unification scale may be explained by the effect of string threshold corrections [6]. However, in order to achieve it, the VEVs of the T moduli must be quite large, Re T ≃ 16, far away from the self-dual point. The advantage of using continuous Wilson line moduli is that the same threshold effects may be obtained but with moderate values of the moduli [7]. As it is well known, in order to connect the above models with the low-energy phenomena, the computation of the soft supersymmetry (SUSY)-breaking terms is crucial [8]. This was carried out for the case of (0,2) symmetric Abelian orbifolds, just assuming that the seed of SUSY breaking is located in the dilaton/moduli sectors [9, 10, 11, 12], with interesting results. For example, non-universal scalar masses, even for particles with the same overall modular weight, are allowed and certain general sum-rules involving soft terms of different particles are obtained [12]. However, the case of orbifolds with continuous Wilson lines has not been addressed in the literature yet. Recently, due to the interesting phenomenological properties of the latter, there has been some activity in trying to obtain information about the structure of their low-energy effective Lagrangian [13, 14]. In particular, the explicit dependence of the K¨ahler potential on the Wilson line moduli has been obtained in [13] allowing the computation of the soft terms in this type of models. This is the aim of the present letter. We will ask what changes with respect to the previous analyses if one allows the presence of continuous Wilson lines. The work of [13] was mainly focus on the so-called factorizable ZN orbifolds for which the underlying six-dimensional torus lattice T6 can be decomposed into a direct sum of a four-dimensional and a two-dimensional sublattice T4 ⊕ T2 . Then, the following K¨ahler potentials for the untwisted moduli spaces associated with the torus T2 were obtained K = − log(T + T ∗ − AA∗ ) , K = − log[(T + T ∗ )(U + U ∗ ) − (B + C ∗ )(B ∗ + C)] ,

(1) (2)

where A, B, C are the complex Wilson line fields. The K¨ahler potential (2) is obtained in orbifold models where the complex plane associated with T2 has order two. In this 1

case T -, U-type moduli and two Wilson lines B and C are present and the K¨ahler potential possesses the required structure to solve the µ problem [15]. Let us concentrate first on case (1). Then, the K¨ahler potential associated with the three complex planes, including matter fields Ci , i = 1, 2, 3, is given by K = −

X

i=1,2

log(Ti + Ti∗ − CiCi∗ ) − log(T3 + T3∗ − AA∗ − C3 C3∗ )

ˆ i , Ti∗ , A, A∗ ) + K ˜ i (Ti , Ti∗ , A, A∗ )Ci Ci∗ ≃ K(T

(3)

with ˜ i = (Ti + T ∗ )−1 , i = 1, 2 ; K ˜ 3 = (T3 + T ∗ − AA∗ )−1 , K i 3 X ˆ ˆ ˜ ˆ Ki , Ki = log Ki , K =

(4) (5)

i

where we associate the Wilson line A with (say) the third complex plane. If the moduli space contains more than one Wilson line modulus the above equations will be the same P with the modification AA∗ → k Ak A∗k . Notice that when continuous Wilson lines are turned off, the usual untwisted K¨ahler potential is recovered. Although the origin of SUSY breaking in strings is by itself an outstanding problem, we can get information about the SUSY-breaking sector of the theory parameterizing our ignorance by the VEVs of the dilaton and moduli auxiliary fields [10, 12]. Since the moduli metric is off-diagonal due to the T3 -A mixing (4,5), we will use for our calculation the parameterization introduced in [16] where the off-diagonal case was analyzed in order to compute Calabi-Yau soft terms. Then √ FS = 3m3/2 sin θ(S + S ∗ )e−iγS , √ Fα = 3m3/2 cos θ(P Θ)α , (6) where the angle θ and the complex parameters Θα just parameterize the direction of the Goldstino in the S, Ti , A field space. The analysis when more than one Wilson line modulus is present is straightforward and will be carried out below. P is a matrix ˆ = 1 where 1 stands for the canonically normalizing the moduli fields [16], i.e. P † KP P ∗ ˆ αβ ≡ ∂¯α ∂β K. ˆ Imposing α Θ Θα = 1 this parametrization has the unit matrix and K α virtue that, when we plug it in the general form of the supergravity scalar potential, the cosmological constant ˆ αβ Fβ − 3m23/2 V0 = (S + S ∗ )−2 |FS |2 + Fα† K

(7)

vanishes by construction. If the value of V0 is not assumed to be zero the modifications are straightforward [10, 12]. It is obvious, that the main difference with the previous computations of soft terms in orbifold compactifications is that now one must add in the VEV of the scalar potential a contribution due to the auxiliary field associated with the Wilson line modulus. Thus the soft terms are given by ˜ i (P Θ)β , m2i = m23/2 1 − 3 cos2 θ(P Θ)†α ∂¯α ∂β log K h





√ A123 = − 3m3/2 sin θe−iγS + cos θ(P Θ)α 2



  X 

j=1,2,3

i

˜ j − ∂α K ˆ ∂α log K

(8)   − ∂α log Y123 (9). 

The above scalar masses and trilinear scalar couplings correspond to matter fields which have already been canonically normalized. Here Y123 is a renormalizable Yukawa coupling involving three untwisted fields and A123 is its corresponding trilinear soft term. ˜i = Due to property (5) the soft masses can now easily be computed since ∂¯α ∂β log K † ˆ i )αβ . Recalling that P KP ˆ = 1 we obtain (K m21 = m23/2 (1 − 3 cos2 θ|Θ1 |2 ) ,

(10)

m22 = m23/2 (1 − 3 cos2 θ|Θ2 |2 ) ,

(11)

m23 = m23/2 (1 − 3 cos2 θ{|Θ3 |2 + |ΘA |2 }) .

(12)

Obviously, the masses of the matter fields associated with the first and second complex planes (10,11) are the same as in the multimoduli case of orbifolds without continuous Wilson lines [12] since the moduli metric has no T1,2 -A mixing. On the other hand, the soft mass (12) is similar to (10,11) with the only difference that the parameter ΘA associated to the Wilson line modulus A appears1 . Since its contribution is similar to those of the moduli Ti we expect no changes in the phenomenological properties obtained in [12]. In particular, non-universal scalar masses and tachyons are allowed. Besides, although there may be scalars with mass bigger than gauginos, on average the scalars are lighter than gauginos. For example, three particles linked via a renormalizable untwisted Yukawa coupling always fulfill the sum rule m21 + m22 + m23 = |M|2 .

(13)

This result is obvious from the property α Θ∗α Θα = 1. Let us remark√ that the tree-level gaugino masses for any compactification scheme are given by Ma = 3m3/2 sin θe−iγS . P ˜ j − ∂α K ˆ = 0 due to property (5) and the unFurthermore, since j=1,2,3 ∂α log K twisted Yukawa couplings do not depend on the moduli, from (9) one also obtains P

A123 = −M ,

(14)

as in the case of orbifolds without continuous Wilson lines. If the moduli space contains more than one Wilson line modulus, as mentioned below (5), we just have to define new parameters ΘAk and the modification introduced in the soft mass (12) is straightforwardly computed using again property (5)2 , |ΘA |2 → P 2 ahler potential (2). k |ΘAk | . A similar situation occurs in the case of the K¨ In this case the most general K¨ahler potential including matter fields can be written as K3 = − log[(T3 + T3∗ )(U3 + U3∗ ) − (B + C ∗ )(B ∗ + C) − (C3 + C3′∗ )(C3∗ + C3′ )] ˆ3 + K ˜ 3 (C3 + C3′∗ )(C3∗ + C3′ ) ≃K (15) 1

2

3

˜ p = (T1 +T ∗ )np (T2 +T ∗)np (T3 +T ∗ −|A|2 )np , This is also true for twisted matter Cp . In this case K 3 2 1 2 ˆ ˜ p = − P ni K where nip are the modular weights, and therefore log K i p i . Using (8) we obtain mp = 2 2 1 2 2 2 3 2 2 m3/2 [1 + 3 cos θ{np |Θ1 | + np |Θ2 | + np (|Θ3 | + |ΘA | )}]. For particles belonging to the untwisted sector one has nip = −δpi and (10,11,12) are recovered. 2 Since twisted moduli appear in the K¨ahler potential (to lowest order) in the same way as Wilson line moduli [17], the same result holds for this case. 1

3

with ˜ 3 = [(T3 + T3∗ )(U3 + U3∗ ) − (B + C ∗ )(B ∗ + C)]−1 , K ˆ 3 = log K ˜3 , K

(16) (17)

Due to property (17) we can use again the same argument than above to obtain 2 2 2 2 2 2 m23 = m′2 3 = m3/2 (1 − 3 cos θ{|Θ3 | + |ΘU3 | + |ΘB | + |ΘC | }) .

(18)

Therefore the comments below (12) still hold. ˜ 3 C3 C ′ + h.c. in (15) possesses the required structure On the other hand, the term K 3 to solve the µ problem [15]. The µ and B parameters are given by   √ ˜3 µ = m3/2 1 − 3 cos θ(P Θ)†α∂¯α log K (19) n o h √ ˜ 3 + (P Θ)† ∂¯α log K ˜3 Bµ = m23/2 2 − 3 cos θ (P Θ)α ∂α log K α ) # ( ¯α ∂β K ˜3 ∂ ˜ 3 )(∂β log K ˜ 3) − (P Θ)β (20) +3 cos2 θ(P Θ)†α 2(∂¯α log K ˜3 K Using e.g. (8) one can easily check that 2 m23 + |µ|2 = m′2 3 + |µ| = Bµ .

(21)

This is precisely the same result obtained in the case of orbifold compactifications without including continuous Wilson lines [12]3 . In summary, we have performed an analysis of the soft SUSY-breaking terms arising in orbifold compactifications taking into account the whole untwisted moduli space. The latter includes not only the moduli Ti which determine the size and shape of the compact space but also the continuous Wilson line moduli which are generically present in any orbifold compactification. The final formulae for soft terms are different in general from those of previous computations where only Ti moduli were considered, due to the extra contributions of the Wilson line auxiliary fields. However, the phenomenological properties turn out to be similar and, in this sense, previous analyses are more general than expected.

Acknowledgments We thank D. L¨ ust for drawing our attention to the issue studied in this paper and for extremely useful discussions. HBK is supported by a Ministerio de Educaci´on y Ciencia grant. CM is a Brain-Pool Fellow. 3

The phenomenological consequences of this relation in connection with the electroweak symmetry breaking were studied in [18].

4

References [1] L.J. Dixon, J. Harvey, C. Vafa and E. Witten, Nucl. Phys. B261 (1985) 678, B274 (1986) 285. [2] L.E. Ib´an ˜ ez, H.P. Nilles and F. Quevedo, Phys. Lett. B187 (1987) 25. [3] L.E. Ib´an ˜ ez, J.E. Kim, H.P. Nilles and F. Quevedo, Phys. Lett. B191 (1987) 282. [4] J.A. Casas, E.K. Katehou and C. Mu˜ noz, Nucl. Phys. B317 (1989) 171; J.A. Casas and C. Mu˜ noz, Phys. Lett. B209 (1988) 214, B214 1988 157; A. Font, L.E. Iba˜ nez, H.P. Nilles and F. Quevedo, Phys. Lett. B210 (1988) 101. [5] L.E. Ib´an ˜ ez, H.P. Nilles and F. Quevedo, Phys. Lett. B192 (1987) 332; L.E. Ib´an ˜ ez, J. Mas, H.P. Nilles and F. Quevedo, Nucl. Phys. B301 (1988) 157; T. Mohaupt, Int. J. Mod. Phys. A9 (1994) 4637. [6] K. Choi, Phys. Rev. D37 (1988) 1564; L.E. Ib´an ˜ ez, D. L¨ ust and G.G. Ross, Phys. Lett. B272 (1991) 251; L.E. Ib´an ˜ ez and D. L¨ ust, Nucl. Phys. B382 (1992) 305. [7] H.P. Nilles and S. Stieberger, Phys. Lett. B367 (1996) 126. [8] For recent reviews, see: C. Mu˜ noz, “Soft supersymmetry-breaking terms and the µ problem”, hep-th/9507108; “Soft terms from Strings”, Nucl. Phys. B (Proc. Suppl.) 49 (1996) 96, hep-th/9601325, and references therein. [9] L.E. Ib´an ˜ ez and D. L¨ ust, Nucl. Phys. B382 (1992) 305. [10] A. Brignole, L.E. Ib´an ˜ ez and C. Mu˜ noz, Nucl. Phys. B422 (1994) 125 [Erratum: B436 (1995) 747]. [11] S. Ferrara, C. Kounnas and F. Zwirner, Nucl. Phys. B429 (1994) 589 [Erratum: B433 (1995) 255]; T. Kobayashi, D. Suematsu, K. Yamada and Y. Yamagishi, Phys. Lett. B348 (1995) 402; P. Brax and M. Chemtob, Phys.Rev. D51 (1995) 6550. [12] A. Brignole, L.E. Ib´an ˜ ez, C. Mu˜ noz and C. Scheich, FTUAM 95/26, hepph/9508258, to appear in Z. Phys.C. [13] G. Lopes Cardoso, D. L¨ ust and T. Mohaupt, Nucl. Phys. B432 (1994) 68; P. Mayr and S. Stieberger, hep-th/9412196; M. Cvetic, B. Ovrut and W.A. Sabra, Phys. Lett. B351 (1995) 173. [14] D. Bailin, A. Love, W.A. Sabra and S. Thomas, Nucl. Phys. B447 (1995) 85; G. Lopes Cardoso, D. L¨ ust and T. Mohaupt, Nucl. Phys. B450 (1995) 115; P. Mayr and S. Stieberger, Phys. Lett. B355 (1995) 107.

5

[15] G.F. Giudice and A. Masiero, Phys. Lett. B206 (1988) 480; J.A. Casas and C. Mu˜ noz, Phys. Lett. B306 (1993) 288; V.S. Kaplunovsky and J. Louis, Phys. Lett. B306 (1993) 269; G. Lopes-Cardoso, D. L¨ ust and T. Mohaupt, in [13]; I. Antoniadis, E. Gava, K.S. Narain and T.R. Taylor, Nucl. Phys. B432 (1994) 187. [16] H.B. Kim and C. Mu˜ noz, FTUAM 96/29, hep-th/9608214, to appear in Z. Phys. C. [17] W.A. Sabra, S. Thomas and N. Vanegas, Mod. Phys. Lett. A11 (1996) 1307. [18] A. Brignole, L.E. Ib´an ˜ ez and C. Mu˜ noz, Phys. Lett. B387 (1996) 769.

6