Realistic Type IIB Supersymmetric Minkowski Flux Models without the Freed-Witten Anomaly
aa r X i v : . [ h e p - t h ] D ec ACT-07-08, MIFP-08-32
Realistic Type IIB Supersymmetric Minkowski Flux Modelswithout the Freed-Witten Anomaly
Ching-Ming Chen, Tianjun Li,
1, 2 and Dimitri V. Nanopoulos
1, 3, 4 George P. and Cynthia W. Mitchell Institute for Fundamental Physics,Texas A & M University, College Station, TX 77843, USA Institute of Theoretical Physics, ChineseAcademy of Sciences, Beijing 100080, P. R. China Astroparticle Physics Group, Houston Advanced Research Center (HARC),Mitchell Campus, Woodlands, TX 77381, USA Academy of Athens, Division of Natural Sciences,28 Panepistimiou Avenue, Athens 10679, Greece
Abstract
We show that there exist supersymmetric Minkowski vacua on Type IIB toroidal orientifold withgeneral flux compactifications where the RR tadpole cancellation conditions can be relaxed andthe Freed-Witten anomaly can be cancelled elegantly. We present a realistic Pati-Salam like fluxmodel without the Freed-Witten anomaly. At the string scale, we can break the gauge symmetrydown to the Standard Model (SM) gauge symmetry, achieve the gauge coupling unification nat-urally, and decouple all the extra chiral exotic particles. Thus, we have the supersymmetric SMswith/without SM singlet(s) below the string scale. Also, we can explain the SM fermion massesand mixings. In addition, the unified gauge coupling and the real parts of the dilaton and K¨ahlermoduli are functions of the four-dimensional dilaton. The complex structure moduli and one linearcombination of the imaginary parts of the K¨ahler moduli can be determined as functions of thefluxes and the dilaton.
PACS numbers: 11.10.Kk, 11.25.Mj, 11.25.-w, 12.60.Jv . INTRODUCTION The great challenge in string phenomenology is the construction of realistic string models,which do not have additional chiral exotic particles at low energy and can stabilize themoduli fields. Employing renormalization group equations, we may test such models at theupcoming Large Hadron Collider (LHC). In particular, the intersecting D-brane models onType II orientifolds [1], where the chiral fermions arise from the intersections of D-branes inthe internal space [2] and the T-dual description in terms of magnetized D-branes [3], havebeen very interesting during the last a few years [4].In the beginning [5], a lot of non-supersymmetric three-family Standard-like models andGrand Unified Theories (GUTs) were constructed on Type IIA orientifolds with intersect-ing D6-branes. However, these models generically have uncancelled Neveu-Schwarz-Neveu-Schwarz (NSNS) tadpoles and the gauge hierarchy problem. Later, semi-realistic supersym-metric Standard-like models and GUT models have been constructed in Type IIA theoryon the T / ( Z × Z ) orientifold [6, 7, 8] and other backgrounds [9]. We emphasize thatonly Pati-Salam like models can realize all the Yukawa couplings at the stringy tree level.Moreover, Pati-Salam like models have been constructed systematically in Type IIA the-ory on the T / ( Z × Z ) orientifold [7, 8]. Although the Standard Model (SM) fermionmasses and mixings can be generated in one of these models [10], we can not stabilizethe moduli fields and might not decouple all the chiral exotic particles. To stabilize themoduli via supergravity fluxes, the flux models on Type II orientifolds have also been con-structed [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. Especially, for the supersymmetric AdS vacuaon Type IIA orientifolds with flux compactifications, the Ramond-Ramond (RR) tadpolecancellation conditions can be relaxed [18, 20]. And then we can construct flux models thatcan explain the SM fermion masses and mixings [20]. However, these models are in the AdSvacua and have quite a few chiral exotic particles that are difficult to be decoupled. Re-cently, on the Type IIB toroidal orientifold with the RR, NSNS, non-geometric and S-dualflux compactifications [21, 22, 23], we showed that the RR tadpole cancellation conditionscan be relaxed elegantly in the supersymmetric Minkowski vacua [24]. Unfortunately, theFreed-Witten anomaly [25] can give strong constraints on model building [24], and the modelin Ref. [24] indeed has the Freed-Witten anomaly that might be cancelled by introducingadditional D-branes [11]. 2n this paper, we revisit the Type IIB toroidal orientifold with the RR, NSNS, non-geometric and S-dual flux compactifications [22]. We present supersymmetric Minkowskivacua where the RR tadpole cancellation conditions can be relaxed and the Freed-Wittenanomaly free conditions can be satisfied elegantly. We construct a realistic Pati-Salam likeflux model without the Freed-Witten anomaly. At the string scale, the gauge symmetry canbe broken down to the SM gauge symmetry, the gauge coupling unification can be achievednaturally, and all the extra chiral exotic particles can be decoupled so that we obtain thesupersymmetric SMs with/without SM singlet(s) below the string scale. The observed SMfermion masses and mixings can also be generated since all the SM fermions and Higgs fieldsarise from the intersections on the same two-torus. Moreover, the unified gauge couplingand the real parts of the dilaton and K¨ahler moduli are functions of the four-dimensionaldilaton. And the complex structure moduli and one linear combination of the imaginaryparts of the K¨ahler moduli can be determined as functions of the fluxes and the dilaton.The systematical model building and the detailed phenomenological discussions will be givenelsewhere [26].This paper is organized as follows: in Section II we review the Type IIB model buildingand study the supersymmetric Minkowski flux vacua. We construct a realistic Pati-Salamlike flux model and discuss its phenomenological consequences in Section III. Discussion andconclusions are presented in Section IV. II. TYPE IIB FLUX MODEL BUILDING
Let us consider the Type IIB string theory compactified on a T orientifold where T isa six-torus factorized as T = T × T × T whose complex coordinates are z i , i = 1 , , i th two-torus, respectively. The orientifold projection is implemented by gauging thesymmetry Ω R , where Ω is world-sheet parity, and R is given by R : ( z , z , z ) → ( − z , − z , − z ) . (1)Thus, the model contains 64 O3-planes. In order to cancel the negative RR charges fromthese 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. Con-cretely, for one stack of N a D-branes wrapped m ia times on the i th two-torus T , we turn on3 ia units of magnetic fluxes F ia for the center of mass U (1) a gauge factor on T , such that m ia π Z T i F ia = n ia , (2)where m ia can be half integer for tilted two-torus. Then, the D9-, D7-, D5- and D3-branescontain 0, 1, 2 and 3 vanishing m ia s, respectively. Introducing for the i th two-torus the evenhomology classes [ i ] and [ T i ] for the point and two-torus, respectively, the vectors of theRR charges of the a th stack of D-branes and its image are[Π a ] = Y i =1 ( n ia [ i ] + m ia [ T i ]) , [Π ′ a ] = Y i =1 ( n ia [ i ] − m ia [ T i ]) , (3)respectively. The “intersection numbers” in Type IIA language, which determine the chiralmassless spectrum, are I ab = [Π a ] · [Π b ] = Y i =1 ( n ia m ib − n ib m ia ) . (4)Moreover, for a stack of N D(2n+3)-branes whose homology class on T is (not) invariant un-der Ω R , we obtain a ( U ( N )) U Sp (2 N ) gauge symmetry with three (adjoint) anti-symmetricchiral superfields due to the orbifold projection. The physical spectrum is presented in TableI. The flux models on Type IIB orientifolds with four-dimensional N = 1 supersymmetry areprimarily constrained by the RR tadpole cancellation conditions that will be given later, thefour-dimensional N = 1 supersymmetric D-brane configurations, and the K-theory anomalyfree conditions. For D-branes with world-volume magnetic field F ia = n ia / ( m ia χ i ) where χ i is the area of the i th two-torus T in string units, the condition for the four-dimensional N = 1 supersymmetric D-brane configurations is X i (cid:0) tan − ( F ia ) − + θ ( n ia ) π (cid:1) = 0 mod 2 π , (5)where θ ( n ia ) = 1 for n ia < θ ( n ia ) = 0 for n ia ≥
0. The K-theory anomaly free conditionsare [14] X a N a m a m a m a = X a N a m a n a n a = X a N a n a m a n a = X a N a n a n a m a = 0 mod 2 . (6)4 ABLE I: General spectrum for magnetized D-branes on the Type IIB T orientifold. Sector Representation aa U ( N a ) vector multiplet3 adjoint multiplets ab + ba I ab ( N a , N b ) multiplets ab ′ + b ′ a I ab ′ ( N a , N b ) multiplets aa ′ + a ′ a ( I aa ′ − I aO ) symmetric multiplets ( I aa ′ + I aO ) anti-symmetric multiplets We turn on the NSNS fluxes h and a i , the RR fluxes e i and q i , the non-geometric fluxes b ii , and the S-dual fluxes f i [21, 22, 23, 26]. To avoid subtleties, these fluxes should be evenintegers due to the Dirac quantization. For simplicity, we assume a i ≡ a , e i ≡ e , q i ≡ q , b ii ≡ β i . (7)We can show that the constraints on fluxes from the Bianchi indentities are satisfied. Theconstraints on fluxes from the SL (2 , Z ) S-duality invariance give aβ i = qf i . (8)The RR tadpole cancellation conditions are X a N a n a n a n a = 16 − aq , X a N a n ia m ja m ka = 12 qβ i ,N NS7 i = 0 , N I7 i = 0 , (9)where i = j = k = i , and the N NS7 i and N I7 i denote the NS 7-brane charge and the other 7-brane charge, respectively [22, 26]. Thus, if aq < qβ i <
0, the RR tadpole cancellationconditions are relaxed elegantly because − aq/ − qβ i / s , three5¨ahler moduli t i , and three complex structure moduli u i . With the above fluxes, we canassume u = u = u ≡ u . (10)Then the superpotential becomes W = 3 ieu − qu + s ( ih − au ) − β i t i u − f i st i . (11)In addition, the holomorphic gauge kinetic function for a generic stack of D(2n+3)-branesis given by [26, 27, 28] f a = 1 κ a (cid:0) n a n a n a s − n a m a m a t − n a m a m a t − n a m a m a t (cid:1) , (12)where κ a is equal to 1 and 2 for U ( n ) and U Sp (2 n ), respectively. And the K¨ahler potentialfor these moduli is of the usual no-scale form [29] K = − ln( s + ¯ s ) − X i =1 ln( t i + ¯ t i ) − X i =1 ln( u i + ¯ u i ) . (13)For the supersymmetric Minkowski vacua, we have W = ∂ s W = ∂ t i W = ∂ u W = 0 . (14)From ∂ s W = ∂ t i W = 0, we obtain f i t i = ih − au , s = − qa u , (15)then the superpotential turns out to be W = (cid:18) e − qh a (cid:19) iu . (16)Therefore, to satisfy W = ∂ u W = 0, we obtain3 ea = qh . (17)Because Re s >
0, Re t i > u i >
0, we require f i Re t i a < , qa < . (18)6n short, in our constructions, we have fixed a linear combination of the K¨ahler moduli t i and the complex structure moduli u as follows from Eq. (15) f i Re t i = 3 a q Re s , Re u = − aq Re s ,f i Im t i = h + 3 a q Im s , Im u = − aq Im s . (19)In general, this kind of D-brane models might have the Freed-Witten anomaly [11, 18,25]. In the world-volume of a generic stack of D-branes we have a U (1) gauge field whosescalar partner parametrizes the D-brane position in compact space. These U (1)’s usuallyobtain St¨uckelberg masses by swallowing RR scalar fields and then decouple from the low-energy spectra. In the mean time these scalars participate in the cancellation of U (1) gaugeanomalies through a generalized Green-Schwarz mechanism [30]. For the generic a stack ofD-branes, the U (1) a gauge field couples to the RR fields in four dimensions as follows F a ∧ N a X I =0 c aI C (2) I , (20)where I = 0 , , ,
3, and c a ≡ m a m a m a ; c a ≡ m a n a n a ; c a ≡ n a m a n a ; c a ≡ n a n a m a , (21)where C (2)0 and C (2) i are two-form fields that are Poincare duals to the Im s and Im t i fieldsin four dimensions, respectively. In terms of them the couplings have a Higgs-like form A aµ ∂ µ ( c a Im s − c a Im t − c a Im t − c a Im t ) . (22)Thus, certain linear combinations of the imaginary parts of the s and t i fields obtain massesby combining with open string vector bosons living on the branes. In addition, these linearcombinations of Im s and Im t i fields will transform with a shift under U (1) a gauge transfor-mations, like Goldstone bosons do. If the shift symmetry for any D-brane stack is violatedby the flux potential, we shall have the Freed-Witten anomaly [25]. From the superpotentialin Eq. (11), we obtain that the flux potential may fix Im s and one linear combination ofIm t i . Thus, we derive the Freed-Witten anomaly free conditions c a = 0 , f c a + f c a + f c a = 0 . (23)Or equivalently, we have c a = 0 , β c a + β c a + β c a = 0 . (24)7 II. A REALISTIC MODEL
In this Section, we shall present a realistic model. We choose the following fluxes a = 8 , q = − , β = 2 , β = 6 , β = 6 ,f = − , f = − , f = − , h = − e , (25)where the flux e is not fixed. We present the D-brane configurations and intersection numbersin Table II, and the resulting spectrum in Tables III and IV. One can easily check that ourmodel satisfies the Freed-Witten anomaly free conditions in Eq. (23) or Eq. (24). Stack N ( n , l ) ( n , l ) ( n , l ) A S b c c ′ d d ′ e e ′ f ga b c d e f g χ = χ = χ TABLE II: D-brane configurations and intersection numbers where l i ≡ m i . The complete gaugesymmetry is [ U (4) C × U (2) L × U (2) R ] Observable × [ U (2) × U Sp (2) ] Hidden , and the SM fermionsand Higgs fields arise from the intersections on the first two-torus.
In our model, the anomalies from the global U (1)s of the U (4) C , U (2) L , U (2) R , U (2) d and U (2) e gauge symmetries are cancelled by the generalized Green-Schwarz mechanism, andthe gauge fields of the corresponding anomalous U (1)s obtain masses via the linear B ∧ F couplings. In addition, we can break the global U (1) a , U (1) L , U (1) R , U (1) d and U (1) e gaugesymmetries respectively of U (4) C , U (2) L , U (2) R , U (2) d and U (2) e by giving the string-scalevacuum expectation values (VEVs) to S a , S a , S iL , S iL , S iR , S iR , X de , T d , T d , S id , S id , T e , T e , S ie , and S ie . Without loss of generality, we can assume that their VEVs satisfy the D-flatnessconditions for the global U (1) a , U (1) L , U (1) R , U (1) d and U (1) e gauge symmetries. Thus,the effective gauge symmetry in the observable sector is indeed SU (4) C × SU (2) L × SU (2) R .In order to break the gauge symmetry down to SU (3) C × SU (2) L × U (1) I R × U (1) B − L , on thefirst two-torus, we split the a stack of D-branes into a and a stacks with 3 and 1 D-branes,respectively, and split the c stack of D-branes into c and c stacks with 1 D-brane for each8 uantum Number Q Q L Q R Field ab × (4 , ¯2 , , , , ,
1) 1 -1 0 F L ( Q L , L L ) ac × (¯4 , , , , , ,
1) -1 0 1 F R ( Q R , L R ) bc × (1 , , ¯2 , , , ,
1) 0 1 -1 Φ i ( H iu , H id ) ac ′ × (4 , , , , , ,
1) 1 0 1 X iac ′ × (¯4 , , ¯2 , , , ,
1) -1 0 -1 X iac ′ bc ′ × (1 , , , , , ,
1) 0 1 1 Φ ′ ( H ′ u , H ′ d )1 × (1 , , , , , ,
1) 0 -1 -1 Φ ′ aa ′ × (1 , , , , , ,
1) 2 0 0 S a × (¯1 , , , , , ,
1) -2 0 0 S a ad × (¯4 , , , , , ,
1) -1 0 0 X iad ad ′ × (4 , , , , , ,
1) 1 0 0 X ad ′ ae × (4 , , , , ¯2 , ,
1) 1 0 0 X iae ae ′ × (¯4 , , , , ¯2 , ,
1) -1 0 0 X ae ′ af × (4 , , , , , ,
1) 1 0 0 X iaf ag × (¯4 , , , , , ,
2) -1 0 0 X iag bb ′ × (1 , , , , , ,
1) 0 2 0 S iL × (1 , ¯1 , , , , ,
1) 0 -2 0 S iL bd × (1 , , , ¯2 , , ,
1) 0 1 0 X ibd bd ′ × (1 , ¯2 , , ¯2 , , ,
1) 0 -1 0 X bd ′ bg × (1 , , , , , ,
2) 0 1 0 X ibg cc ′ × (1 , , , , , ,
1) 0 0 2 S iR × (1 , , ¯1 , , , ,
1) 0 0 -2 S iR ce × (1 , , ¯2 , , , ,
1) 0 0 -1 X ice ce ′ × (1 , , , , , ,
1) 0 0 1 X ce ′ cf × (1 , , ¯2 , , , ,
1) 0 0 -1 X icf TABLE III: The chiral and vector-like superfields in the observable sector, and their quantumnumbers under the gauge symmetry U (4) C × U (2) L × U (2) R × U (2) × U Sp (2) . one. We can break the U (1) I R × U (1) B − L gauge symmetry further down to the U (1) Y gauge symmetry by giving the string-scale VEVs to the vector-like particles with quantumnumbers ( , , / , − ) and ( , , − / , ) under SU (3) C × SU (2) L × U (1) I R × U (1) B − L from a c ′ D-brane intersections. Similar to the discussions in Ref. [10], we can explain theSM fermion masses and mixings via the Higgs fields H iu , H ′ u , H id and H ′ d because all the SMfermions and Higgs fields arise from the intersections on the first two-torus. Note that wegive the string-scale VEVs to the fields S iL , S iL , S iR , S iR , X de , T d , T d , S id , S id , T e , T e , S ie , and9 uantum Number Q Q L Q R Field dd ′ × (1 , , , , , ,
1) 0 0 0 T d × (1 , , , ¯3 , , ,
1) 0 0 0 T d × (1 , , , , , ,
1) 0 0 0 S id × (1 , , , ¯1 , , ,
1) 0 0 S id de × (1 , , , ¯2 , , ,
1) 0 0 0 X de df × (1 , , , ¯2 , , ,
1) 0 0 0 X idf ee ′ × (1 , , , , , ,
1) 0 0 0 T e × (1 , , , , ¯3 , ,
1) 0 0 0 T e × (1 , , , , , ,
1) 0 0 0 S ie × (1 , , , , ¯1 , ,
1) 0 0 S ie eg × (1 , , , , , ,
2) 0 0 0 X ieg f g × (1 , , , , , ,
2) 0 0 0 X ifg × (1 , , , , , ,
2) 0 0 X ifg TABLE IV: The chiral and vector-like superfields in the hidden sector, and their quantum numbersunder the gauge symmetry U (4) C × U (2) L × U (2) R × U (2) × U Sp (2) . S ie , the chiral exotic particles can obtain masses around the string scale via the followingsuperpotential from three-point and four-point functions W ⊃ ( T d + S id ) X jad X ad ′ + ( T e + S ie ) X jae X ae ′ + X de ( X iad X jae + X ad ′ X ae ′ )+( X ifg + X ifg ) X jaf X kag + ( T d + S id ) X jbd X bd ′ + S iL X jbg X kbg +( T e + S ie ) X jce X ce ′ + S iR X jcf X kcf + ( T d + S id ) X jdf X kdf +( T e + S ie ) X jeg X keg + 1 M St (cid:16) S iL ( T d + S jd ) X kbd X lbd + S iR ( T e + S je ) X kce X lce + ( T d + S id )( T e + S je ) X de X de (cid:17) , (26)where M St is the string scale, and we neglect the O (1) coefficients in this paper. In addition,we can decouple all the Higgs bidoublets close to the string scale except one pair of thelinear combinations of the Higgs doublets for the electroweak symmetry breaking at the lowenergy by fine-tuning the following superpotential W ⊃ Φ i ( S jL Φ ′ + S jR Φ ′ ) + 1 M St (cid:16) S iL S jR Φ k Φ l + S iL S jR Φ ′ Φ ′ + S iL S jR Φ ′ Φ ′ (cid:17) . (27)In short, below the string scale, we have the supersymmetric SMs which may have zero,one or a few SM singlets from S iL , S iL , S iR , and S iR . Then the upper bound on the lightest10P-even Higgs boson mass in the minimal supersymmetric SM can be relaxed if we havethe SM singlet(s) at low energy [31].Next, we consider the gauge coupling unification and moduli stabilization. Note that3 χ = χ = χ as given in Table II are derived from the supersymmetric D-brane configura-tions, we define χ ≡ χ , χ = χ ≡ χ . (28)Thus, the real parts of the dilaton and K¨ahler moduli in our model are [26]Re s = e − φ πχ √ χ , Re t = 3 √ χe − φ π , Re t = √ χe − φ π , Re t = √ χe − φ π , (29)where φ is the four-dimensional dilaton. From Eq. (12), we obtain that the SM gaugecouplings are unified at the string scale as follows g − SU (3) C = g − SU (2) L = 35 g − U (1) Y = e − φ π (cid:18) χ √ χ + 3 √ χ (cid:19) . (30)From the real part of the first equation in Eq. (15) or the first equation in Eq. (19), weobtain χ = 23 . (31)Therefore, χ i are determined by the supersymmetric D-brane configurations and the condi-tions for the supersymmetric Minkowski vacua. Using the unified gauge coupling g ≃ . φ ≃ − . . (32)From Eq. (19), we obtainRe u = 4Re s , Im u = 4Im s , t + 24Im t + 24Im t = − h + 96Im s . (33) IV. DISCUSSION AND CONCLUSIONS
We showed that the RR tadpole cancellation conditions can be relaxed and the Freed-Witten anomaly can be cancelled elegantly in the supersymmetric Minkowski vacua on11he Type IIB toroidal orientifold with general flux compactifications. And we presenteda realistic Pati-Salam like flux model in details. In this model, we can break the gaugesymmetry down to the SM gauge symmetry, realize the gauge coupling unification, anddecouple all the extra chiral exotic particles around the string scale. We can also generatethe observed SM fermion masses and mixings. Futhermore, the unified gauge coupling andthe real parts of the dilaton and K¨ahler moduli are functions of the four-dimensional dilaton.And the complex structure moduli and one linear combination of the imaginary parts of theK¨ahler moduli can be determined as functions of the fluxes and the dilaton.
Acknowledgments
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