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Physics Letters B

www.elsevier.com/locate/physletb

Realistic Type IIB supersymmetric Minkowski flux vacua

Ching-Ming Chena, Tianjun Lia'b'*, Yan Liub, Dimitri V. Nanopoulosa,c,d

a George P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843, USA b Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China

c Astroparticle Physics Group, Houston Advanced Research Center (HARC), Mitchell Campus, Woodlands, TX 77381, USA d Academy of Athens, Division of Natural Sciences, 28 Panepistimiou Avenue, Athens 10679, Greece

ARTICLE INFO ABSTRACT

We show that there exist supersymmetric Minkowski vacua on Type IIB toroidal orientifold with general flux compactifications where the RR tadpole cancellation conditions can be relaxed elegantly. Then we present a realistic Pati-Salam like model. At the string scale, the gauge symmetry can be broken down to the Standard Model (SM) gauge symmetry, the gauge coupling unification can be achieved naturally, and all the extra chiral exotic particles can be decoupled so that we have the supersymmetric SMs with/without SM singlet(s) below the string scale. The observed SM fermion masses and mixings can also be obtained. In addition, the unified gauge coupling, the dilaton, the complex structure moduli, the real parts of the Kahler moduli and the sum of the imaginary parts of the Kahler moduli can be determined as functions of the four-dimensional dilaton and fluxes, and can be estimated as well.

© 2008 Elsevier B.V. All rights reserved.

Article history■

Received 6 May 2008

Received in revised form 3 August 2008

Accepted 6 August 2008

Available online 9 August 2008

Editor: M. Cvetic

PACS: 11.10.Kk 11.25.Mj 11.25.-w 12.60.Jv

1. Introduction

One of the great challenging and essential problems in string phenomenology is the construction of the realistic string vacua, which can give us the low energy supersymmetric Standard Models (SMs) without exotic particles, and can stabilize the moduli fields. With renormalization group equation running, we can connect such constructions to the low energy realistic particle physics which will be tested at the upcoming Large Hadron Collider (LHC). During the last a few years, the intersecting D-brane models on Type II orientifolds [1], where the chiral fermions arise from the intersections of D-branes in the internal space [2] and the T-dual description in terms of magnetized D-branes [3], have been particularly interesting [4].

On Type IIA orientifolds with intersecting D6-branes, many non-supersymmetric three-family Standard-like models and Grand Unified Theories (GUTs) were constructed in the beginning [5]. However, there generically existed uncancelled Neveu-Schwarz-Neveu-Schwarz (NSNS) tadpoles and the gauge hierarchy problem. To solve these problems, semi-realistic supersymmetric Standardlike and GUT models have been constructed in Type IIA theory on the T6/(Z2 x Z2) orientifold [6,7] and other backgrounds [8]. Interestingly, only the Pati-Salam like models can give all the Yukawa couplings. Without the flux background, Pati-Salam like

* Corresponding author at: George P. and Cynthia W. Mitchell Institute for Fundamental Physics, Texas A&M University, College Station, TX 77843, USA. E-mail address: junlt@physics.tamu.edu (T. Li).

0370-2693/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2008.08.010

models have been constructed systematically in Type IIA theory on the T6/(Z2 x Z2) orientifold [7]. Although we may explain the SM fermion masses and mixings in one model [9], the moduli fields have not been stabilized, and it is very difficult to decouple the chiral exotic particles. To stabilize the moduli via supergravity fluxes, the flux models on Type II orientifolds have also been constructed [10-14]. Especially, for the supersymmetric AdS vacua on Type IIA orientifolds with flux compactifications, the RR tadpole cancellation conditions can be relaxed [13,14]. And then we can construct the flux models that can explain the SM fermion masses and mixings [14]. However, those models are in the AdS vacua and have quite a few chiral exotic particles that are difficult to be decoupled.

In this Letter, we consider the Type IIB toroidal orientifold with the Ramond-Ramond (RR), NSNS, non-geometric and S-dual flux compactifications [15]. We find that the RR tadpole cancellation conditions can be relaxed elegantly in the supersymmetric Minkowski vacua, and then we may construct the realistic Pati-Salam like models [16]. In this Letter, we present a concrete simple model which is very interesting from the phenomenological point of view and might describe Nature. We emphasize that we do not fix the four-dimensional dilaton via flux potential, and our model is a solution to the equations of motion for all the Type IIB fields.

2. Type IIB flux compactifications

We consider the Type IIB string theory compactified on a T6 orientifold where T6 is a six-torus factorized as T6 = T2 x T2 x T2 whose complex coordinates are z,, i = 1, 2, 3 for the ith two-torus,

Table 1

General spectrum for magnetized D-branes on the Type IIB T6 orientifold

Sector Representation

aa U (Na) vector multiplet

3 adjoint multiplets

ab + ba Iab (Na, Nb ) multiplets

ab' + b'a ¡ab' (Na, Nb) multiplets

aa' + a'a 2(Iaa' — Iao3) symmetric multiplets

2(Iaa' + IaO3) anti-symmetric multiplets

respectively. The orientifold projection is implemented by gauging the symmetry QR, where Q is world-sheet parity, and R is given by

R : (zi, Z2, Z3) ^ (-zi, -Z2, -Z3).

Thus, the model contains 64 O3-planes. In order to cancel the negative RR charges from these O3-planes, we introduce the magnetized D(3 + 2n)-branes which are filling up the four-dimensional Minkowski space-time and wrapping 2n-cycles on the compact manifold. Concretely, for one stack of Na D-branes wrapped m'a times on the 'th two-torus T2, we turn on n'a units of magnetic

fluxes Fa for the center of mass U(1)a gauge factor on T2, such that '

-L ZV:

a 2n J Fa

where m'a can be half integer for tilted two-torus. Then, the D9-, D7-, D5- and D3-branes contain 0, 1, 2 and 3 vanishing m'as, respectively. Introducing for the 'th two-torus the even homology classes [0' ] and [T2] for the point and two-torus, respectively, the vectors of the RR charges of the ath stack of D-branes and its image are

ina] = Y\ n [0i ]+ma[ T?]), i=1

[K] = U n [0 ]-ma [T?]),

respectively. The "intersection numbers" in Type IIA language, which determine the chiral massless spectrum, are

lab = [na] ■ [nb] = n(n<am<b - n'bm'a)

Moreover, for a stack of N D(2n + 3)-branes whose homology classes on T6 is (not) invariant under QR, we obtain a USp(2N) (U(N)) gauge symmetry with three anti-symmetric (adjoint) chiral superfields due to the orbifold projection. The physical spectrum is presented in Table 1.

The flux models on Type IIB orientifolds with four-dimensional N = 1 supersymmetry are primarily constrained by the RR tadpole cancellation conditions that will be given later, the four-dimensional N = 1 supersymmetric D-brane configurations, and the K-theory anomaly free conditions. For the D-branes with world-volume magnetic field Fa = n'a/(m'aX') where X' is the area of T'2 in string units, the condition for the four-dimensional N = 1 supersymmetric D-brane configurations is

tan-1(F'a) + e(n'a)n) = 0 mod 2n,

where 0(n'a) = 1 for n'a < 0 and 0(n'a) = 0 for n'a > 0. The K-theory anomaly free conditions are [17]

Y Namam?ma = Y Namlnln3 = Y Nanlm2n3

= E Nany?m3a = 0 mod 2.

And the holomorphic gauge kinetic function for a generic stack of D(2n + 3)-branes is given by [16,18,19]

fa = -1 (n^3s - nam2m3ti Ka

- n?mamaf2 - n3mam2f^,

where Ka is equal to 1 and 2 for U(n) and USp(2n), respectively.

We turn on the NSNS flux h0, RR flux e', non-geometric fluxes b'' and b'', and the S-dual fluxes f', g'j and g" [15]. To avoid the subtleties, these fluxes should be even integers due to the Dirac quantization. For simplicity, we assume

e' = e, b'' = p, b'' = ft, f' = f, g'j = -g'' = g, (8)

where ' = j. Then the constraint on fluxes from Bianchi identities is

f P = gp. (9)

The RR tadpole cancellation conditions are

Y, Nan>2n3 = 16, Y, Naniam>amka = -- eß,

NnsIî = 0, Ni7i = 0,

where ' = j = k = ', and the NNS7i and Ni7' denote the NS7 brane charge and the other 7-brane charge, respectively [15]. Thus, if ep < 0, the RR tadpole cancellation conditions are relaxed elegantly because —epi/2 only needs to be even integer. Moreover, we have 7 moduli fields in the supergravity theory basis, the dilaton s, three Kahler moduli t', and three complex structure moduli U'. With the above fluxes, we can assume

t = ti + t2 + t3, Ui = U2 = U3 = u. Then the superpotential becomes W = 3ieu + ih0s - t{ßu - ißu2) - st(f - igu). The Kähler potential for these moduli is

K = - ln(s + s) -Y ln(ti + ti) -Y ln(ui + ui). i=i i=i

In addition, the supergravity scalar potential is

V = eK(KiiDiWDjW - 3|W|2),

where Kij is the inverse metric of j = di djK, Di W = (diK)W, and di = dfr where & can be s, ti, and ui. Thus, supersymmetric Minkowski vacua, we have

W = ds W = dt W = du W = 0.

From dsW = dtW = 0, we obtain

t = ■

s = -ß- u, f ,

f — 'gu

then the superpotential turns out

vw U hoP V

W = I 3e--— \iu.

Therefore, to satisfy W = du W = 0, we obtain 3ef = ph0.

di W + for the

Because Re s > 0, Re ti > 0 and Re ui > 0, we require

— < 0, g

в < 0

In general, this kind of D-brane models might have the Freed-Witten anomalies [10,20]. Interestingly, the Freed-Witten anomalies can be cancelled by introducing additional D-branes [10]. In particular, the additional D-branes will not affect the main properties of the D-brane models, for example, the four-dimensional N = 1 supersymmetry and the chiral spectra, etc. [10]. Therefore, we can construct this kind of D-brane models by ignoring the subtlety of the Freed-Witten anomalies.

3. Model

Choosing eft = -12, we present the D-brane configurations and intersection numbers in Table 2, and the resulting spectrum in Table 3. The anomalies from three global U(1)s of U(4)C, U (2)L and U (2)r are cancelled by the Green-Schwarz mechanism, and the gauge fields of these U(1)s obtain masses via the linear B A F couplings. So, the effective gauge symmetry is SU(4)C x SU(2)L x SU(2)r. In order to break the gauge symmetry, on the first two-torus, we split the a stack of D-branes into a1 and a2 stacks with 3 and 1 D-branes, respectively, and split the c stack of D-branes into ci and c2 stacks with 1 D-brane for each one. Then, the gauge symmetry is further broken down to SU(3)c x SU(2)l x U(1)i3R x U(1)b-l. We can break the U(1)i3R x U (1)b-l gauge symmetry down to the U (1)y gauge symmetry by giving vacuum expectation values (VEVs) to the vector-like particles with quantum numbers (1,1, 1/2, —1) and (1,1,-1/2,1) under SU(3)C x SU(2)L x U(1)i3R x U(1)B-L from a2c1 D-brane intersections. Similar to the discussions in Ref. [9], we can explain the SM fermion masses and mixings via the Higgs fields H'u, H'u, Hd and Hd because all the SM fermions and Higgs fields arise from the intersections on the first torus. To decouple the chiral exotic particles, we assume that the T Ri and SiR obtain VEVs at about the string scale, and their VEVs satisfy the D-flatness U (1)R. The chiral exotic particles can obtain masses via the following superpotential

W D iSiRSjTkRTlR + TiRXjxk, (20)

where MSt is the string scale, and we neglect the O(1) coefficients in this Letter. In addition, the vector-like particles S'L and S'l in the anti-symmetric representation of SU(2)L can obtain the VEVs close to the string scale while keeping the D-flatness U(1)L. Thus, we can decouple all the Higgs bidoublets close to the string scale except one pair of the linear combinations of the Higgs doublets for the electroweak symmetry breaking at the low energy by fine-tuning the following superpotential

W D $ (SL& + SR$') + $ (TR $' + SJL$')

+wSS Фк ф'+SiTRФк Ф

+ s [tr ф 'Ф' + slsR Ф'Ф').

In short, below the string scale, we have the supersymmetric SMs which may have zero, one or a few SM singlets from S'L, S[, and/or SR. And then the low bound on the lightest CP-even Higgs boson

mass in the minimal supersymmetric SM can be relaxed if we have the SM singlet(s) at low energy [21].

Next, we consider the gauge coupling unification and moduli stabilization. The real parts of the dilaton and Kahler moduli in our model are [16]

Re s =

-Ле—ф4 4n '

Re t1 =

л/6в-ф4

Table 2

D-brane configurations and intersection numbers

U (4)c x U (2)i x U(2)r x USp(10)

N (nl, m') ns Па b b' c c' O3

a 4 (1,0) x (1, —1/2) x (1,1) 0 0 3 0(3) —3 0(3) 0(1)

b 2 (1, —3) x (1,1/2) x (1, 0) 0 0(6) - - 0(6) 0(1) 0(3)

c 2 (1,3) x (1,1/2) x (0, —1) —6 6 - - - - 3

O3 5 (1,0) x (2, 0) x (1, 0) - - 6X1 = X2 = 2X3 =2

Table 3

The chiral and vector-like superfields, and their quantum numbers under the gauge symmetry SU(4)c x SU(2)l x SU(2)r x USp(10)

Quantum number Q 4 Q2i Q 2R Field

ab 3x (4, 2,1,1) 1 —1 0 Fi (Qi, li)

ac 3x (4, 1, 2, 1) —1 0 1 Fr (Qr , iR )

cs 6x (1,1,3,1,1) 0 0 —2 T' TR

CA 6x (1,1,1,1,1) 0 0 2 S' SR

cO3 3x (1,1, 2,10) 0 0 1 X'

ac' 3x (4,1, 2,1) 1 0 1

3x (4, 1,2,1) —1 0 —1

bc 6x (1, 2, 2, 1) 0 1 —1 Ф'(H'u, Hd)

6x (1, 2, 2, 1) 0 —1 1 Ф'

bc' 1x (1, 2, 2, 1) 0 1 1 Ф'№, Hd )

1x (1,2,2,1) 0 —1 —1 Ф'

bb' 6x (1,1,1,1) 0 2 0 S' Si

6x (1,1,1,1) 0 —2 0 S ' Si

Re t2 =

46г-ф4 12п ,

Re t3 =

46г-ф4 6n '

where 04 is the four-dimensional dilaton. From Eq. (7), we obtain that the SM gauge couplings are unified at the string scale as follows

—2 _ —2 _ 3 —2 _

gSU(3)c = gSU(2)i = 5 gU (1)y =

л/6г-ф4

Using the unified gauge coupling g2 ~ 0.513 in supersymmetric SMs, we get

~ —1.61.

For moduli stabilization, we first obtain t from Eqs. (16) and (22)

Re t =

З46е—ф4

Thus, we have

1 в Im s = — Im t + в,

Im t = ±

3ßho 27e—

Re u = —

л/6 fe—ф 4nß

ff Im u = — Im t--.

Let us present a set of possible solutions to the fluxes

h0 = —18^, e = 6n, P = fi = -2n, f = — 2n', g = 2n,

where n = ±1 and n' = ±1. Choosing 04 = —1.61, n = n' = 1, we obtain the numerical values for the moduli fields

Re s = Re u = 0.975, Re t1 = 1.95, Ret3 = 0.650,

Re t2 = 0.325,

^Im ti = ±4.30, Im s = Im u = ^1.43 + 1.

4. Conclusions

We showed that the RR tadpole cancellation conditions can be relaxed elegantly in the supersymmetric Minkowski vacua on the Type IIB toroidal orientifold with general flux compactifications. And we presented a realistic Pati-Salam like model in details. In this model, we can break the gauge symmetry down to the SM gauge symmetry, realize the gauge coupling unification, and decouple all the extra chiral exotic particles around the string scale. We can also generate the observed SM fermion masses and mixings. Furthermore, the unified gauge coupling, the dilaton, the complex structure moduli, the real parts of the Kahler moduli and the sum of the imaginary parts of the Kahler moduli can be determined as functions of the four-dimensional dilaton and fluxes, and can also be estimated.

Acknowledgements

This research was supported in part by the Mitchell-Heep Chair in High Energy Physics (C.M.C.), by the Cambridge-Mitchell Collaboration in Theoretical Cosmology (T.L.), and by the DOE grant DE-FG03-95-Er-40917 (D.V.N.).

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