Loop Fayet-Iliopoulos terms in T 2 / Z 2 models: instability and moduli stabilization
Hiroyuki Abe, Tatsuo Kobayashi, Shohei Uemura, Junji Yamamoto
EEPHOU-20-003WU-HEP-20-03KUNS-2788
Loop Fayet-Iliopoulos terms in T /Z models:instability and moduli stabilization Hiroyuki Abe , Tatsuo Kobayashi , Shohei Uemura and Junji Yamamoto Department of Physics, Waseda University, Tokyo 169-8555, Japan Department of Physics, Hokkaido University, Sapporo 060-0810, Japan CORE of STEM, Nara Women’s University, Nara 630-8506, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan
Abstract
We study Fayet-Iliopoulos (FI) terms of six-dimensional supersymmetric Abelian gaugetheory compactified on a T /Z orbifold. Such orbifold compactifications can lead tolocalized FI-terms and instability of bulk zero modes. We study 1-loop correctionto FI-terms in more general geometry than the previous works. We find inducedFI-terms depend on the complex structure of the compact space. We also find thecomplex structure of the torus can be stabilized at a specific value corresponding to aself-consistent supersymmetric minimum of the potential by such 1-loop corrections,which is applicable to the modulus stabilization. a r X i v : . [ h e p - t h ] A ug Introduction
Effective theory of superstring includes various dimensional objects, i.e., branes. Branesare important components for particle phenomenology. Branes can break the supersym-metry (SUSY) and realize the chiral spectrum [1–4]. They can be a source of generationsof matter fields, and flavor structure [5, 6]. Anti-branes can induce the positive cosmo-logical constant [7]. Such a brane mode behaves as a localized mode in effective theory.Therefore it is important to investigate interactions between bulk fields and localizedoperators [8, 9].The Fayet-Iliopoulos term (FI-term) in supersymmetric Abelian gauge theory wasintroduced as a source of spontaneous SUSY breaking at first [10]. Later it was shownthat FI-term is not only a source of the SUSY breaking, but has vast implications fortheoretical particle physics. The FI-term is prohibited by local SUSY unless the gaugegroup is related to U (1) R [11–13] or associated with non-linear terms [14]. Especially inhigher dimensional supersymmetric theory, it is related to anomaly [15], and introducesinstability of bulk superfields [16, 17].(See also [18].)Even in the higher dimensional theory, the bulk FI-term is prohibited by local SUSY,but the FI-term localized at special points, i.e., orbifold fixed points can appear [19].Such a FI-term is called localized FI-term. The localized FI-term is induced by quantumcorrections in orbifold compactification even if the FI-term is set to zero at the treelevel [20]. This is formally calculated by infinite sum of all KK-modes of fields which havecharges of the corresponding U (1). In the trivial background without the localized FI-term, mode expansion of bulk fields is given by plane waves. Their infinite sum convergesto the Dirac delta function. Hence the localized FI-term is induced. Since it is localized,the FI-term induces a local potential for bulk fields. To cancel the FI-term, the vacuumexpectation values (VEVs) of auxiliary fields must also be localized. It affects the wavefunction profiles of the bulk fields. For the model of five-dimensional Abelian gaugetheory compactified on S /Z , the localized FI-term induces localization of bulk zeromodes at the fixed points, and rejects wave functions of all the massive modes from thefixed points [16, 17]. Similar results are obtained also for six-dimensional SUSY theorycompactified on T /Z orbifold [21]. Thus it is a quite general consequence for higherdimensional SUSY theory compactified on orbifolds.If the value of the localized FI-term is not zero, VEVs of the auxiliary fields are shifted.The massive modes can not penetrate to the fixed points in this 1-loop corrected vacuum.Hence 1-loop corrections to the FI-terms are only due to the zero mode. The zero modeis localized at the fixed points, and reproduces the localized FI-term, but it is not thesame as that of the infinite sum of the plane waves. Bulk contribution is not canceled bybrane mode contributions in general, and the FI-term receives further corrections. Thusthis background is unstable. In our previous work we investigated this instability for the S /Z compactification model [22]. In the present paper we investigate instability for T /Z compactification. Toroidal compactification is a more realistic compactification forphenomenology; it has a concrete stringy origin [1]. It also can realize the chiral spectrumof the Standard Model (SM). (See e.g. Refs. [23, 24].) The localized FI-term on toroidal1rbifold may affect the flavor structure of the SM [25]. As well as S /Z compactification,loop correction of the FI-term can lead to the instability of 1-loop corrected vacuum. Wefind that the instability is related to the complex structure of the torus. There are someapplications for moduli stabilization and extra dimensional models.This paper is organized as follows. In section 2, we examine the localized FI-termand zero mode of bulk scalar field in six-dimensional SUSY gauge theory compactifiedon T /Z orbifold, whose geometry is described by an arbitrary value of the complexstructure modulus τ ( ∈ C ). The localized FI-term is induced by quantum corrections,and it leads to nonzero VEVs of auxiliary fields. It affects equations of motion for bulkfields and their wave function profiles. In Section 3, we focus on an untilted torus, i.e., atorus whose complex structure is pure imaginary, and recalculate the 1-loop correctionsto the FI-term in the SUSY vacuum which has nonzero VEV of the auxiliary field. We seethat 1-loop corrections can cause the instability of the SUSY vacuum. In Section 4, weextend the consequences in section 3 to the torus that has an arbitrary value of τ . We findthese quantum corrections depend on τ . We show the complex structure modulus musttake a specific value for the cancellation between loop corrections from bulk and branemodes. In other words, modulus stabilization of the complex structure is realized. Section5 is devoted to our conclusion. In Appendix A, we study the validity of our evaluation ofthe FI terms. We also confirm the localization of the wave function of bulk zero mode byuse of an explicit regularization of the Dirac delta function. In Appendix B, we show themodular transformation of elliptic theta functions. T /Z model In this section, we evaluate the localized FI-term induced by quantum corrections in the T /Z orbifold. We take the following strategy. First we consider 1-loop FI-term inducedby tree level wave functions. Then we investigate mode expansion of the bulk fields in the1-loop corrected background including a singular configuration of the gauge field. Finallywe recalculate the quantum correction of the FI-term induced by the 1-loop correctedwave functions, and search a consistent configuration.Before describing the multiplets that are contained in T /Z models, we describe thetorus T and orbifold action of Z . We define the orthogonal coordinates of T as x , x ,and we denote the two-dimensional metric by g ij : g ij = (cid:18) (cid:19) ( i, j = 5 , . (2.1)The coordinates ( x , x ) satisfy the following periodic boundary conditions: (cid:40) ( x , x ) ∼ ( x + 2 πR, x ) , ( x , x ) ∼ ( x + 2 πR Re τ, x + 2 πR Im τ ) . x (cid:48) = x − Re τ Im τ x Rz = x (cid:48) + τ x (cid:48) x (cid:48) = τ x R ¯ z = x (cid:48) + ¯ τ x (cid:48) boundary x (cid:48) ∼ x (cid:48) + 2 πR z ∼ z + 2 π conditions x (cid:48) ∼ x (cid:48) + 2 πR z ∼ z + 2 πτ metric g i (cid:48) j (cid:48) = (cid:18) τ Re τ | τ | (cid:19) g mn = R (cid:18) (cid:19) Table 1: Coordinates and metrics on the toruswhere we introduced a complex structure τ , which takes an arbitrary value in the upperhalf plane H . We define the Z orbifold action as Z : ( x , x ) → ( − x , − x ) , (2.2)and there are four fixed points: (0 , , ( πR, , ( πR Re τ, πR Im τ ) and ( πR (1+Re τ ) , πR Im τ ).Hereafter these fixed points are denoted by z , z , z and z , respectively.We introduce non-orthogonal coordinates ( x (cid:48) , x (cid:48) ) which are along the lattice vectorsof the torus. In these non-orthogonal coordinates, the two periodic boundary conditionscan be represented as (cid:40) ( x (cid:48) , x (cid:48) ) ∼ ( x (cid:48) + 2 πR, x (cid:48) ) , ( x (cid:48) , x (cid:48) ) ∼ ( x (cid:48) , x (cid:48) + 2 πR ) . (2.3)We also define complex coordinates ( z, ¯ z ) as Rz ≡ x (cid:48) + τ x (cid:48) and R ¯ z ≡ x (cid:48) + ¯ τ x (cid:48) . Fromnow on, we use the notation of indices as M, N ∈ { , , , , , } , µ, ν ∈ { , , , } , and i, j, m, n ∈ { , } . We also use the indices with prime, M (cid:48) , N (cid:48) , i (cid:48) , j (cid:48) to represent the non-orthogonal coordinates ( x (cid:48) , x (cid:48) ). We summarize the relations of the coordinates and themetrics in Table 1.We consider six-dimensional SUSY Abelian gauge theory defined below the cutoff scaleΛ. Such a theory is described by four-dimensional N = 2 supermultiplets: Abelian vectormultiplet and hypermultiplets. In addition to the N = 2 multiplets, we can introducebrane modes at the fixed points. The brane modes preserve N = 1 SUSY and we assumethat they consist of only chiral multiplets; there are no extra gauge fields at the fixedpoints. We introduce brane mode Φ I = ( φ I , ψ I ) at each fixed point z I . The multipletsare summarized as follows: • bulk mode: N = 2 Abelian vector multiplet= { gauge field A M , gaugino Ω , auxiliary field (cid:126)D } hypermultiplet = { real scalars A i , hyperino ζ }• brane mode: chiral multiplet = { complex scalar φ I , Weyl fermion ψ I } . We should pay attention to the auxiliary fields in N = 2 Abelian vector multiplet. It isdecomposed into an N = 1 vector multiplet and a single chiral multiplet. The auxiliary3eld D of the N = 1 vector multiplet is given by a linear combination of a part of theauxiliary field (cid:126)D and the field strength F . We choose D = − D + F in this paper. The Z orbifold action is defined to preserve this four-dimensional N = 1 structure, e.g., theparity assignment to D is even and those to other two auxiliary fields D and D are odd.We also introduce two complex scalar fields φ + and φ − , which are linear combinations ofthe real scalars of the hypermultiplet. φ + is parity even and φ − is parity odd. The bosonic Lagrangian is written as follows: L = − F MN F MN + i ¯ΩΓ M ∂ M Ω + 12 (cid:126)D + (cid:88) ± ( D M φ †± D M φ ± ∓ gφ †± qφ ± D ) + · · · + (cid:88) I =1 δ ( x − x I ) δ ( x − x I ) (cid:104) D µ φ † I D µ φ I + gφ † I q I φ I ( − D + F ) + · · · (cid:105) , (2.4)where D M φ ± = ∂ M φ ± ± igqφ ± A M . The quantities q and q I are charges of the hypermultiplet and the brane modes respectively. g is the gauge coupling constant. Four-dimensional effective potential is represented asfollows: V d = (cid:90) dx dx (cid:104) g | φ T + qφ − | + (cid:88) ± ( D φ ± + i D φ ± ) † ( D φ ± + i D φ ± )+ 12 (cid:0) F − ξ − g ( φ † + qφ + − φ †− qφ − ) − g (cid:88) I φ † I q I φ I δ (2) ( x − x I , x − x I ) (cid:1) − (cid:0) D − ξ − g ( φ † + qφ + − φ †− qφ − ) − g (cid:88) I φ † I q I φ I δ (2) ( x − x I , x − x I ) (cid:1) −
12 ( D + gφ T + qφ − + gφ †− qφ ∗ + ) −
12 ( D + igφ T + qφ − − igφ †− qφ ∗ + ) (cid:105) , (2.5)where we include the contributions of FI-term L FI = ξ ( − D + F ).From (2.5), the SUSY conditions are written by D = F = ξ + g ( φ † + qφ + − φ †− qφ − ) + g (cid:88) I φ † I q I φ I δ (2) ( x − x I , x − x I ) , (2.6) φ T + qφ − = 0 , D φ ± + i D φ ± = 0 . (2.7)We study the situation where the U (1) is unbroken, i.e., (cid:104) φ ± (cid:105) = (cid:104) φ I (cid:105) = 0. The SUSYsolution is as follows: (cid:104) F (cid:105) = ξ ( x , x ) . (2.8) For precise calculation, see [21].
4e also obtain the equation of motion (EOM) for the scalar fields φ ± in terms of thecompact directions:(zero mode) : ( D + i D ) φ ± = 0 , (2.9)(massive mode) : ( −D + i D )( D + i D ) φ ± = λφ ± . (2.10)The SUSY solution and the zero mode equation in the non-orthogonal coordinates arerepresented simply as follows:(SUSY condition) : (cid:104) F (cid:48) (cid:48) (cid:105) = | Im τ | ξ ( x (cid:48) , x (cid:48) ) , (2.11)(zero mode EOM) : ( τ D (cid:48) − D (cid:48) ) φ ± ( x (cid:48) , x (cid:48) ) = 0 . (2.12)By evaluating (2.11) and (2.12), we will confirm that, if the localized FI-term has anonzero value, the zero mode of the bulk field is localized at the fixed points z = z I , thatis similar to [21]. ξ = 0 We calculate the FI-term induced by 1-loop corrections of the scalar fields φ ± . As thefirst step, we use the mode expansions in the SUSY vacuum with ξ = 0. In the SUSYvacuum with ξ = 0, the EOMs (2.9) and (2.10) become ∂ ¯ ∂φ ± ( z, ¯ z ) = R λφ ± ( z, ¯ z ) , (2.13)where we represent them in the complex coordinates ( z, ¯ z ). The general solutions ofEOMs are given by φ ± ( z, ¯ z ) = A e cz − c (cid:48) ¯ z , (2.14)where A is a complex constant, and c, c (cid:48) are also complex constants satisfying cc (cid:48) = − R λ. (2.15)By imposing the boundary conditions φ ± ( z + 2 π ) = φ ± ( z ) and φ ± ( z + 2 πτ ) = φ ± ( z ) thecomplex constants c, c (cid:48) are quantized:2 π ( c − c (cid:48) ) = 2 πin ( n ∈ Z ) , (2.16)2 π ( cτ − c (cid:48) ¯ τ ) = 2 πi(cid:96) ( (cid:96) ∈ Z ) . (2.17)Thus the solutions that satisfy the boundary conditions are represented as follows: φ ± ,n(cid:96) ( z, ¯ z ) = A n(cid:96) e τ (cid:0) n ( τ ¯ z − ¯ τz )+ (cid:96) ( z − ¯ z ) (cid:1) , (2.18) λ = − R (Im τ ) (cid:110) ( n Re τ − (cid:96) ) + ( n Im τ ) (cid:111) . (2.19)5n the coordinates ( x (cid:48) , x (cid:48) ), these can be more simple form as φ ± ,n(cid:96) ( x (cid:48) , x (cid:48) ) = A n(cid:96) e i ( nR x (cid:48) + (cid:96)R x (cid:48) ) . (2.20)Since ( n, (cid:96) ) and ( − n, − (cid:96) ) correspond to the same eigenvalue, φ ± ,n(cid:96) ( x (cid:48) , x (cid:48) ) = A n(cid:96) e i ( nR x (cid:48) + (cid:96)R x (cid:48) ) + B n(cid:96) e − i ( nR x (cid:48) + (cid:96)R x (cid:48) ) , (2.21)where n runs from 0 to + ∞ and (cid:96) runs from −∞ to + ∞ . Under the action of Z , thewave functions behave as φ + ( − x (cid:48) , − x (cid:48) ) = φ + ( x (cid:48) , x (cid:48) ) , (2.22) φ − ( − x (cid:48) , − x (cid:48) ) = − φ − ( x (cid:48) , x (cid:48) ) . (2.23)We obtain mode expansions of the bulk scalars: φ + ,n(cid:96) ( x (cid:48) , x (cid:48) ) = A λ cos (cid:18) nR x (cid:48) + (cid:96)R x (cid:48) (cid:19) , (2.24) φ − ,n(cid:96) ( x (cid:48) , x (cid:48) ) = A λ sin (cid:18) nR x (cid:48) + (cid:96)R x (cid:48) (cid:19) , (2.25)where the normalization factor A λ is 1 /πR √ Im τ for λ (cid:54) = 0 up to phases, which are notrelevant to the following discussions. Zero modes are constant solutions. They are givenby φ + , = A ( A = 1 / πR √ Im τ ) , (2.26) φ − , = 0 , (2.27)up to a phase, which is not relevant to the following discussions. 1-loop diagrams con-tributing to the FI-term are written as Figure 1 in the case of S /Z . We can evaluatethe divergent part of the FI-term that is induced by 1-loop diagrams of bulk scalars: ξ bulk ( x (cid:48) , x (cid:48) ) = g tr( q ) (cid:16) Λ π + 14 ln Λ π g i (cid:48) j (cid:48) ∂ i (cid:48) ∂ j (cid:48) (cid:17) ∞ (cid:88) n =0 ∞ (cid:88) l = −∞ {| φ + ,nl | − | φ − ,nl | } = g tr( q ) (cid:16) Λ π + 14 ln Λ π g i (cid:48) j (cid:48) ∂ i (cid:48) ∂ j (cid:48) (cid:17) | Im τ | (cid:88) I =1 ,..., δ ( x (cid:48) − x (cid:48) I ) δ ( x (cid:48) − x (cid:48) I ) , (2.28)where the second derivative g i (cid:48) j (cid:48) ∂ i (cid:48) ∂ j (cid:48) = R ∂ ¯ ∂ is originated from the log divergent term+ λ ln Λ by use of the EOM. In the second row, we use the Fourier expansion of theDirac delta function: δ ( y ) = 1 πR + 2 πR ∞ (cid:88) n> cos (cid:16) nyR (cid:17) ( − πR < y < πR ) . (2.29) The loop diagram around which the scalars φ ± run induces only the linear term of D . The samecontribution to the linear term of F (cid:48) (cid:48) arise from fermion’s loop as same as the ∂ y Σ in the S /Z modelunless the SUSY is broken. / | Im τ | is multiplied, which comes from (cid:112) det g i (cid:48) j (cid:48) = | Im τ | when wenormalize the wave function. Considering the contributions from the brane modes, weobtain the 1-loop induced FI-term: ξ ( x (cid:48) , x (cid:48) ) = ξ bulk + ξ brane = 1 | Im τ | (cid:88) I =1 ,..., ( ξ I + ξ (cid:48)(cid:48) g i (cid:48) j (cid:48) ∂ i (cid:48) ∂ j (cid:48) ) δ ( x (cid:48) − x (cid:48) I ) δ ( x (cid:48) − x (cid:48) I ) , (2.30) ξ I = g Λ π (cid:16)
14 tr( q ) + tr( q I ) (cid:17) , ξ (cid:48)(cid:48) = g π
14 tr( q ) . (2.31)The FI-term is localized at the fixed points of the orbifold. Thus we obtain a localizedFI-term. ξ (cid:54) = 0 On the untilted torus, i.e., Re τ = 0, the zero mode of scalar field is localized at thefixed points by the localized FI-term [21]. Here, we show that the FI-term localizes thezero mode of scalar field similarly at the fixed points in the general T /Z orbifold witharbitrary τ .From (2.11) and (2.12), the SUSY conditions and the EOM of the zero mode for thebulk scalar are represented by (cid:104) F (cid:48) (cid:48) (cid:105) = | Im τ | ξ ( x (cid:48) , x (cid:48) ) , (2.32)( τ D (cid:48) − D (cid:48) ) φ ± , ( x (cid:48) , x (cid:48) ) = 0 . (2.33)We concentrate on the parity even mode. We write explicitly them by the derivatives ∂ (cid:48) , ∂ (cid:48) and gauge fields A (cid:48) , A (cid:48) : ∂ (cid:48) (cid:104) A (cid:48) (cid:105) − ∂ (cid:48) (cid:104) A (cid:48) (cid:105) = | Im τ | ξ ( x (cid:48) , x (cid:48) ) , (2.34) (cid:110) ( τ ∂ (cid:48) − ∂ (cid:48) ) + igq ( τ (cid:104) A (cid:48) (cid:105) − (cid:104) A (cid:48) (cid:105) ) (cid:111) φ + , ( x (cid:48) , x (cid:48) ) = 0 . (2.35) Obviously the parity odd modes have no zero mode. (cid:40) A (cid:48) = (Im τ ) − (cid:0) Re τ ∂ (cid:48) − ∂ (cid:48) (cid:1) W,A (cid:48) = (Im τ ) − (cid:0) | τ | ∂ (cid:48) − Re τ ∂ (cid:48) (cid:1) W. (2.36)In this gauge, the SUSY condition and EOM become1Im τ (cid:0) | τ | ∂ (cid:48) − τ ∂ (cid:48) ∂ (cid:48) + ∂ (cid:48) (cid:1) (cid:104) W (cid:105) = | Im τ | ξ ( x (cid:48) , x (cid:48) ) , (2.37) (cid:110) ( τ ∂ (cid:48) − ∂ (cid:48) ) − gq ( τ ∂ (cid:48) − ∂ (cid:48) ) (cid:104) W (cid:105) (cid:111) φ + , ( x (cid:48) , x (cid:48) ) = 0 . (2.38)In the complex coordinates Rz = x (cid:48) + τ x (cid:48) and R ¯ z = x (cid:48) + ¯ τ x (cid:48) , the derivatives ∂ z , ∂ ¯ z aregiven by (cid:18) ∂ z ∂ ¯ z (cid:19) = − Rτ − ¯ τ (cid:18) ¯ τ − − τ (cid:19) (cid:18) ∂ (cid:48) ∂ (cid:48) (cid:19) . (2.39)Eqs. (2.37) and (2.38) are written as follows: ∂ ¯ ∂ (cid:104) W (cid:105) = R ξ, (2.40) (cid:8) ¯ ∂ − gq ( ¯ ∂ (cid:104) W (cid:105) ) (cid:9) φ + , ( z, ¯ z ) = 0 , (2.41)where the 1-loop FI-terms (2.30) and (2.31) are represented in the complex coordinate as ξ ( z, ¯ z ) = 2 R (cid:88) I =1 ,..., ( ξ I + ξ (cid:48)(cid:48) R ∂ ¯ ∂ ) δ (2) ( z − z I ) , (2.42) ξ I = g Λ16 π ( 14 tr( q ) + tr( q I )) , ξ (cid:48)(cid:48) = g π
14 tr( q ) , (2.43)where the factors come from the coordinate transformation. From (2.40) and (2.42), wecan split the SUSY solution into two parts: (cid:104) W (cid:105) = (cid:104) W (cid:48) (cid:105) / (cid:104) W (cid:48)(cid:48) (cid:105) , (2.44) ∂ ¯ ∂ (cid:104) W (cid:48) (cid:105) = (cid:88) I =1 ,..., ξ I δ (2) ( z − z I ) , (cid:104) W (cid:48)(cid:48) (cid:105) = 2 R (cid:88) I =1 ,..., ξ (cid:48)(cid:48) δ (2) ( z − z I ) . (2.45)The equation for (cid:104) W (cid:48) (cid:105) is the Poisson equation with the source ξ I at the fixed points. Thesolution is obtained as (cid:104) W (cid:48) (cid:105) = 12 π (cid:88) I ξ I (cid:20) ln (cid:12)(cid:12)(cid:12) ϑ (cid:16) z − z I π (cid:12)(cid:12)(cid:12) τ (cid:17)(cid:12)(cid:12)(cid:12) − π Im τ { Im ( z − z I ) } (cid:21) . (2.46) Considering Re τ = 0 and the differences of scale between x and x (cid:48) , we see that this gauge (2.36)intrinsically corresponds to the gauge in [21]. ϑ ( z | τ ) is the elliptic theta function, and our convention is given by ϑ ab ( z, τ ) = ∞ (cid:88) n = −∞ e πi ( n + a/ τ +2 πi ( n + a/ z + b/ , (2.47) ϑ ( z | τ ) ≡ − ϑ ( z, τ ) , ϑ ( z | τ ) ≡ ϑ ( z, τ ) , ϑ ( z | τ ) ≡ ϑ ( z, τ ) , ϑ ( z | τ ) ≡ ϑ ( z, τ ) . (2.48)With this gauge background, the solution of the EOM (2.41) can be formally representedby φ + , ( z, ¯ z ) = f ( z ) e gq (cid:104) W (cid:105) . (2.49)The holomorphic function f ( z ) must be constant because it is a periodic holomorphicfunction. The zero mode of φ + is represented as follows: φ + , ( z, ¯ z ) = f (cid:89) I =1 ... (cid:12)(cid:12)(cid:12) ϑ (cid:16) z − z I π (cid:12)(cid:12)(cid:12) τ (cid:17)(cid:12)(cid:12)(cid:12) gqξ I / π exp (cid:110) − gqξ I π Im τ { Im ( z − z I ) } + 2 gqξ (cid:48)(cid:48) R δ (2) ( z − z I ) (cid:111) . (2.50)Since this wave function includes the Dirac delta function in the argument of exponential,it is not well defined. The Dirac delta function implies that this wave function hasserious divergences at the fixed points, while the fixed points are the zero points for thetheta function. Integral of the wave function on any small region including a fixed pointseems to be divergent. Whereas wave functions must be canonically normalized. Thisdivergence must be canceled by the normalization factor f . As a result, normalized wavefunction would be a localized mode at the fixed points such as the Dirac delta function.Such a localized mode appears in an explicit regularization scheme for the case of S /Z compactification [16, 22]. It is also true for toroidal orbifolds. We can show it by use ofan explicit regularization of the delta function. ξ (cid:54) = 0 Calculation of the 1-loop FI-term is affected by the zero mode localization. It impliesthe instability of the supersymmetric vacuum for the S /Z model [22]. Such a vacuuminstability may happen in the present T /Z model. Thus we should reevaluate the 1-loop FI-term again with the background given by (2.44), (2.45) and (2.46), and we shouldexamine how stable configurations for the brane mode are.In our evaluation, we make the following two assumptions: Assumption 1 : The massive mode profiles of the bulk scalar are excluded at thefixed points.
Assumption 2 : Corrections to the FI-term can be evaluated by the square values ofwave functions near the fixed points only. See Appendix A. S /Z model [22]. The second assumptionmeans that the ratio of the 1-loop FI-term at each fixed point z = z I is equal to the ratioof | φ + , ( z I + (cid:15), ¯ z I + ¯ (cid:15) ) | of (2.50). From (2.50), the zero mode near the fixed point z = z I is written as below: φ + , ( z I + (cid:15), ¯ z I + ¯ (cid:15) ) = f exp (cid:110) gqξ (cid:48)(cid:48) R δ (2) ρ ( (cid:15) ) (cid:111) | ϑ (cid:16) (cid:15) π (cid:12)(cid:12)(cid:12) τ (cid:17) | gqξ I / π × (cid:89) J (cid:54) = I (cid:12)(cid:12)(cid:12) ϑ (cid:16) z I + (cid:15) − z J π (cid:12)(cid:12)(cid:12) τ (cid:17)(cid:12)(cid:12)(cid:12) gqξ J / π exp (cid:110) − gqξ J π Im τ { Im ( z I + (cid:15) − z J ) } (cid:111) , (2.51)where we introduce δ (2) ρ ( z ), which is a regularization of delta function; δ (2) ρ ( z ) is finiteand δ (2) ρ ( z ) → δ (2) ( z ) as ρ → +0. We introduce ξ min , which denotes the minimum of ξ I . ϑ ( (cid:15)/ π | τ ) is approximated by η ( τ ) (cid:15) near the origin [21], where η ( τ ) is the Dedekind etafunction. We find ϑ ( (cid:15)/ π | τ ) → (cid:15) →
0. We redefine the normalizationfactor by f (cid:48) ≡ f exp (cid:110) gqξ (cid:48)(cid:48) R δ (2) ρ ( (cid:15) ) (cid:111) (cid:12)(cid:12)(cid:12) ϑ (cid:16) (cid:15) π (cid:12)(cid:12)(cid:12) τ (cid:17)(cid:12)(cid:12)(cid:12) gqξ min / π . (2.52) f (cid:48) is a finite constant. The zero mode near the fixed point is represented as φ + , ( z I + (cid:15), ¯ z I + ¯ (cid:15) ) = f (cid:48) (cid:12)(cid:12)(cid:12) ϑ (cid:16) (cid:15) π (cid:12)(cid:12)(cid:12) τ (cid:17)(cid:12)(cid:12)(cid:12) gq ( ξI − ξ min)2 π (cid:89) J (cid:54) = I (cid:12)(cid:12)(cid:12) ϑ (cid:16) z I + (cid:15) − z J π (cid:12)(cid:12)(cid:12) τ (cid:17)(cid:12)(cid:12)(cid:12) gqξ J / π × exp (cid:110) − gqξ J π Im τ { Im ( z I + (cid:15) − z J ) } (cid:111) . If ξ I is not equal to ξ min , because of the suppression of | ϑ ( (cid:15)/ π | τ ) | , the wave functionmust vanish near the fixed point z = z I . Thus the part | ϑ ( (cid:15)/ π ) | gq ( ξ I − ξ min ) / π determinesthe point where the zero mode is localized. For instance, if ξ I ∗ is the only minimum and ξ J (cid:54) = I ∗ > ξ I ∗ , the wave function is localized only at z ∗ I . Thus it is represented as φ + , = (cid:112) δ ( z − z I ∗ ) , (2.53)where the square root of the delta function denotes that the wave function is localized atthe fixed point z I ∗ and canonically normalized. If several ξ I are the minimum simultane-ously, the zero mode is localized at the several fixed points z I where ξ I = ξ min . The massive mode of the bulk scalar field is evaluated in [21]. The evaluation was performed exceptthe small regions that contain the fixed points, and the analysis near the fixed points are difficult. Since the zero mode wave function is localized at the fixed points, this description is not exactly true.We provide a more rigorous treatment and justify the second assumption in Appendix A. For a concrete example, see Appendix A. Im ( z I − z J ) } J = 1 J = 2 J = 3 J = 4 I = 1 0 0 π (Im τ ) π (Im τ ) I = 2 0 0 π (Im τ ) π (Im τ ) I = 3 π (Im τ ) π (Im τ ) I = 4 π (Im τ ) π (Im τ ) { Im ( z I − z J ) } .The ratio of the zero mode of bulk scalar fields at the fixed points can be practicallyevaluated by r I ≡ (cid:89) J (cid:54) = I (cid:12)(cid:12)(cid:12) ϑ (cid:16) z I − z J π (cid:17)(cid:12)(cid:12)(cid:12) gqξ J / π exp (cid:110) − gqξ J π Im τ { Im ( z I − z J ) } (cid:111) . (2.54)In the complex coordinates ( z, ¯ z ), the fixed points are z I = { , π, πτ, π (1 + τ ) } . (2.55)The explicit forms of { Im ( z I − z J ) } are summarized in Table 2. We define T I as T I ≡ (cid:89) J (cid:54) = I (cid:12)(cid:12)(cid:12) ϑ (cid:16) z I − z J π (cid:17)(cid:12)(cid:12)(cid:12) gqξ J / π , (2.56)which is the elliptic theta function part of r I . From (2.55), we find T I = { × | ϑ ( − | τ ) | ξ × | ϑ ( − τ | τ ) | ξ × | ϑ ( − τ | τ ) | ξ } gq/ π { | ϑ ( | τ ) | ξ × × | ϑ ( − τ | τ ) | ξ × | ϑ ( − τ | τ ) | ξ } gq/ π { | ϑ ( τ | τ ) | ξ × | ϑ ( − − τ | τ ) | ξ × × | ϑ ( − | τ ) | ξ } gq/ π { | ϑ ( τ | τ ) | ξ × | ϑ ( τ | τ ) | ξ × | ϑ ( | τ ) | ξ × } gq/ π , where the first, second, third and fourth rows correspond to T , T , T and T respectively.The elliptic theta function ϑ satisfies the following relations: ϑ ( v + 1 | τ ) = − ϑ ( v | τ ) , (2.57) ϑ ( v + τ | τ ) = − e − iπ (2 v + τ ) ϑ ( v | τ ) , (2.58)and the elliptic theta functions ϑ i ( i = 2 , ,
4) are related to ϑ as ϑ (cid:16) (cid:12)(cid:12)(cid:12) τ (cid:17) = ϑ (0 | τ ) , (2.59) ϑ (cid:16) τ (cid:12)(cid:12)(cid:12) τ (cid:17) = ie − iπτ/ ϑ (0 | τ ) , (2.60) ϑ (cid:16) τ (cid:12)(cid:12)(cid:12) τ (cid:17) = ϑ (cid:16) τ (cid:12)(cid:12)(cid:12) τ (cid:17) = e − iπτ/ ϑ (0 | τ ) . (2.61)11herefore, by using ϑ i (0 | τ ) ( i = 2 , , T I is simply rewritten as T I = { | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × e π Im τ ( ξ + ξ ) } gq/ π { | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × e π Im τ ( ξ + ξ ) } gq/ π { | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × e π Im τ ( ξ + ξ ) } gq/ π { | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × e π Im τ ( ξ + ξ ) } gq/ π . (2.62)The ratio of the absolute value of the wave functions at the fixed points is evaluated as r I = { | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ } gq/ π { | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ } gq/ π { | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ } gq/ π { | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ × | ϑ (0 | τ ) | ξ } gq/ π . (2.63)Since the zero mode is localized at the fixed points, the normalized wave function of thezero mode is given by | φ + . | = (cid:115) (cid:80) ξ I = ξ min r I δ ( z − z I ) (cid:80) ξ I = ξ min r I . (2.64) r I are transformed each other by the modular symmetry. The modular symmetry isgenerated by two elements, S and T , and these generators transform the modulus τ as S : τ → − τ , T : τ → τ + 1 . (2.65)The elliptic theta functions are transformed each other by S and T , and transformationbehavior is shown in Appendix B. The S transforms zero mode values at z and z ,and z and z , i.e., φ + , ( z , ¯ z ) ←→ φ + , ( z , ¯ z ) and φ + , ( z , ¯ z ) ←→ φ + , ( z , ¯ z ). Onthe other hand, the T transforms zero mode values at z and z , and z and z , i.e., φ + , ( z , ¯ z ) ←→ φ + , ( z , ¯ z ) and φ + , ( z , ¯ z ) ←→ φ + , ( z , ¯ z ). When ξ = ξ = ξ = ξ ,the above zero mode profile is invariant under the modular symmetry. In the previous section, we have finished the preparations to calculate the localized FI-term in the new SUSY background, where the VEV of F (cid:48) (cid:48) has nonzero value. In stableconfiguration, the bulk mode contribution cancels the brane mode contributions. Thus weexamine configurations where the cancellation occurs. Under the second assumption, the1-loop FI-term that is induced by the bulk mode can be evaluated by r I . In the configura-tions where the cancellation cannot occur, the 1-loop FI-term changes the supersymmetricvacuum further, which leads to the instability of the SUSY vacuum.In this section, we investigate the stability of the SUSY vacuum in the untilted torus,i.e., Re τ = 0. In the untilted torus, except for the differences from the scale of x and x (cid:48) ,the zero mode profile φ + , and gauge field W coincide with the results in [21].12 .1 Completely symmetric configuration First we consider the completely symmetric configuration of the brane charges, i.e., q = q = q = q . We assume the sum of U (1) charges is set to zero, which means that thebulk charge is four times as big as that of the localized charge: q = − q . Furthermore,we assume the tree level Lagrangian has no FI-term and (cid:104) F (cid:48) (cid:48) (cid:105) = 0. From (2.30) and(2.31), we obtain the 1-loop induced FI-term: ξ = ξ bulk + ξ brane = 2 R (cid:88) I =1 ,..., ( ξ I + ξ (cid:48)(cid:48) R ∂ ¯ ∂ ) δ (2) ( z − z I ) , (3.1) ξ = ξ = ξ = ξ = 0 , ξ (cid:48)(cid:48) = gq
16 ln Λ π . (3.2)Solving the D-flat condition (2.32) in the gauge (2.36), we obtain the corrected SUSYbackground solution: (cid:104) W (cid:105) = 2 R (cid:88) I =1 ,..., ξ (cid:48)(cid:48) δ (2) ( z − z I ) . (3.3)In this new SUSY background, we recompute the zero mode of φ + . The zero mode canbe evaluated from (2.63): φ + , ( z, ¯ z ) = √ R (cid:115) (cid:88) I =1 ,..., δ (2) ( z − z I ) , (3.4)where the square root of the delta function denotes that the wave function is localized atthe fixed points and canonically normalized as mentioned before.Substituting (3.4) into the KK expansion of the bulk fields in (2.28), we obtain the1-loop FI-term again. From the assumption 1 in section 2.3, the massive modes do notcontribute to the 1-loop FI-term. We can evaluate the contribution of the bulk fields: ξ bulk = gq Λ π
14 2 R (cid:88) I =1 ,..., δ (2) ( z − z I ) . (3.5)The contribution of the brane fields is unchanged. It is written as ξ brane = g Λ π R (cid:88) I =1 ,..., q I δ (2) ( z − z I ) . (3.6)As a result, we obtain the quantum correction to the FI-term in the new SUSY back-ground, ξ ( z, ¯ z ) = ξ bulk + ξ brane = 0 . (3.7)The quantum correction vanishes. The bulk zero mode shields the brane charges com-pletely. Thus the SUSY vacuum does not shift further, i.e., it is a stable vacuum.13 .2 Partially symmetric configuration Next, we consider a partially symmetric configuration where the U (1) charges of the branefields are given by q = 0 and q = q = q . We assume the sum of U (1) charges is set tozero, which means that the bulk charge is three times as big as that of the localized charge: q = − q . Furthermore, we assume the tree level Lagrangian has vanishing FI-term and (cid:104) F (cid:48) (cid:48) (cid:105) = 0. The 1-loop induced FI-term is calculated as ξ = ξ bulk + ξ brane = 2 R (cid:88) I =1 ,..., ( ξ I + ξ (cid:48)(cid:48) R ∂ ¯ ∂ ) δ (2) ( z − z I ) , (3.8) ξ = κ, ξ = ξ = ξ = − κ/ (cid:18) κ ≡ gq Λ π (cid:19) , (3.9) ξ (cid:48)(cid:48) = gq
16 ln Λ π . (3.10)Solving the D-flat condition (2.32) in the gauge (2.36), we obtain the SUSY backgroundsolution corrected by 1-loop effects as (cid:104) W (cid:105) = 14 π (cid:88) I =1 ,..., ξ I (cid:20) ln (cid:12)(cid:12)(cid:12) ϑ (cid:16) z − z I π (cid:12)(cid:12)(cid:12) τ (cid:17)(cid:12)(cid:12)(cid:12) − π Im τ { Im ( z − z I ) } (cid:21) + 2 R (cid:88) I : f.p. ξ (cid:48)(cid:48) δ (2) ( z − z I ) . (3.11)The ratio of the zero mode at the fixed points in this new background can be evaluatedfrom (2.63): r I = { | ϑ (0 | τ ) | κ × | ϑ (0 | τ ) | − κ/ × | ϑ (0 | τ ) | − κ/ } gq/ π { | ϑ (0 | τ ) | − κ/ × | ϑ (0 | τ ) | − κ/ × | ϑ (0 | τ ) | κ } gq/ π { | ϑ (0 | τ ) | − κ/ × | ϑ (0 | τ ) | κ × | ϑ (0 | τ ) | − κ/ } gq/ π . Note that the wave function of the zero mode vanishes at z since ξ is bigger than ξ min = − κ/
3. The zero mode is given by φ + , ( z, ¯ z ) = √ R (cid:115) | ϑ (0 | τ ) | gqκ π δ (2) ( z − z ) + | ϑ (0 | τ ) | gqκ π δ (2) ( z − z ) + | ϑ (0 | τ ) | gqκ π δ (2) ( z − z ) | ϑ (0 | τ ) | gqκ/ π + | ϑ (0 | τ ) | gqκ/ π + | ϑ (0 | τ ) | gqκ/ π . (3.12)Substituting (3.12) into the KK expansion of the bulk fields in (2.28), we obtain the1-loop FI-term again. The contribution of the bulk field is given by ξ bulk = gq Λ π R × | ϑ (0 | τ ) | gqκ π δ (2) ( z − z ) + | ϑ (0 | τ ) | gqκ π δ (2) ( z − z ) + | ϑ (0 | τ ) | gqκ π δ (2) ( z − z ) | ϑ (0 | τ ) | gqκ/ π + | ϑ (0 | τ ) | gqκ/ π + | ϑ (0 | τ ) | gqκ/ π . (3.13)14he charges of brane modes stability of the vacuum q = q = q = q stable q = q = q (cid:54) = q unstable q = q (cid:54) = q = q stable { q = q (cid:54) = q , q } and { q (cid:54) = q } unstable q I (cid:54) = q J ( I (cid:54) = J ) unstableTable 3: Stable and unstable configurations of brane modes.The contribution of the brane fields is unchanged, and is written as ξ brane = g Λ π R (cid:88) I =1 ,..., q I δ (2) ( z − z I ) . (3.14)As a result, we obtain the quantum correction to the FI-term in the new SUSY back-ground, ξ ( z, ¯ z ) = ξ bulk + ξ brane (cid:54) = 0 . (3.15)The quantum correction does not vanish. Therefore, the SUSY vacuum shifts furtherby the 1-loop FI-term, i.e., it is an unstable vacuum. Unless we introduce a fine-tunedFI-term at tree level, the vacuum is unstable in the partially symmetric configuration. We have examined the stability of the SUSY vacuum in the two configurations: completelysymmetric one and partially symmetric one. The former has the supersymmetric stablevacuum, but the latter does not.We summarize stability of various configurations in Table 3. In all of these examples,we assume that the bulk mode has a charge q which cancels the charges of the brane modes,i.e., q + (cid:80) I q I = 0. The first and second rows correspond to the results in the section3.1 and 3.2, respectively. In the table, “stable” means that the FI-term is not inducedin the new SUSY vacuum. On the other hand “unstable” means that the FI-term isinduced in the new SUSY vacuum. It is always possible to introduce a localized FI-termat tree level which makes the zero mode wave function of the bulk field shield the branecharges completely. If such a fine-tuned FI-term is available, unstable configurations canbe stabilized. To add the tree level FI-term, we should pay attention for flux quantization.The localized FI-term corresponds to localized magnetic flux [26–28]. The Wilson looparound the fixed points in the SUSY background of (2.32) is non-trivial, W I = exp (cid:18) − iq (cid:73) C I A (cid:19) = exp (cid:18) − iq (cid:90) D I ξ (cid:19) , (3.16)where C I is a circle around z I and D I is the disc including z I , and we use EOM of thegauge field (2.8). Thus ξ can be interpreted as a localized flux. Since W I must be ± ξ I which make the localization of the zero mode shield the brane chargescompletely, satisfying the quantization condition. It might be interesting to investigateit. Vacuum (in)stability will be also related to the anomaly on the compact space. Weobserve that the stable configurations are anomaly free since the charge of the bulk zeromodes is canceled by that of the brane modes everywhere. On the other hands, anomalyis not canceled in the unstable configurations locally. This may imply inconsistencyof the model. The local anomaly requires additional fields, e.g., antisymmetric fields,which cancel the anomaly via Green-Schwarz mechanism, or other local operators. Theseadditional terms may change the localized FI-term and vacuum structure. For instance,the loop diagrams including antisymmetric fields would contribute to the localized FI-term, and shift it. It may be interesting to investigate stability of the bulk mode includingsuch additional effects. We would study it elsewhere. We examine the stability of the SUSY vacuum in the tilted torus T /Z , i.e., Re τ (cid:54) = 0.Basically, the results are the same as those of the untilted torus. The difference comesonly from the profiles of the zero modes, which generally depend on the backgroundgeometry. Taking into account general τ , we find a part of unstable vacuum can bestabilized. Especially, the partially symmetric configuration leads to different results. We are interested in the partially symmetric configuration, i.e., the charges of three branemodes are the same, and the charge of the other one is zero. Similar to section 3.2, weconcentrate on the configuration that the charges of the brane modes in the fixed points z = z , z , z are the same for concreteness. The charge of the bulk mode is three timesas big as that of the localized charge, which is required for (cid:80) I ξ I = 0. (See Figure 2.)In the SUSY vacuum with (cid:104) F (cid:48) (cid:48) (cid:105) = 0, the 1-loop induced FI-term is written by ξ = κ, ξ = ξ = ξ = − κ/ , ξ (cid:48)(cid:48) (cid:54) = 0 , (4.1)where κ ≡ gq Λ π .The FI-term corrects the SUSY vacuum as (cid:104) F (cid:48) (cid:48) (cid:105) = | Im τ | ξ ( x (cid:48) , x (cid:48) ). Again we evalu-ate the 1-loop FI-term in the new SUSY vacuum. Since (4.1) satisfies ξ min = ξ = ξ = ξ ,the φ + , ( z I , ¯ z I ) is already given by (3.12). The ratio of zero mode profiles at fixed pointsis given by | φ + , ( z ) | : | φ + , ( z ) | : | φ + , ( z ) | : | φ + , ( z ) | = 0 : | ϑ (0 | τ ) | gqκ π : | ϑ (0 | τ ) | gqκ π : | ϑ (0 | τ ) | gqκ π . The 1-loop FI-term induced by the bulk field in the new vacuum is induced as this ratioat the fixed points. In order not to generate the 1-loop FI-term in the new vacuum, the16igure 2: The configuration of brane modesbulk contribution must cancel that from the brane modes. Since the charges of the branemodes are the same at the three fixed points of z , z , z , we obtain the following stabilitycondition: | ϑ (0 | τ ) | = | ϑ (0 | τ ) | = | ϑ (0 | τ ) | . (4.2)These conditions cannot be satisfied if Re τ = 0. This is the reason why we insisted thatthis configuration is unstable in the untilted torus in section 3.3. Whereas, in the tiltedtorus, the condition (4.2) can be satisfiedBy use of modular transformation behavior of the elliptic theta functions as shownin Appendix B, we find that the complex structure, e.g., τ = e iπ/ , satisfies the abovecondition (4.2). The point τ = e iπ/ is on the boundary of the fundamental domain ofthe modular group. Thus, in the torus which has the complex structure τ = e iπ/ , the1-loop induced FI-term in the new SUSY vacuum vanishes. Accordingly the configurationof three brane modes has a stable vacuum.The 1-loop FI-term generates a D-term potential: V D ∝ (cid:90) dx (cid:48) dx (cid:48) (cid:112) det g i (cid:48) j (cid:48) ( ξ + · · · ) . (4.3) ξ contains the divergent term of cutoff Λ if ξ is not zero. The D-term potential wouldbe dominant. Thus, we consider that τ would be stabilized in the value that cancels the1-loop FI-term in the new SUSY vacuum. In the configuration of three brane modes, we insist that the complex structure is stabilizeddynamically at τ = e iπ/ by the potential V D . For other combinations of three fixed points where the three brane modes are located, the equivalentconditions appear.
17e show the stable configuration in Figure 3. In this configuration, there are the braneFigure 3: Torus of τ = e iπ/ modes in the fixed points except the origin, and the bulk mode is localized at the fixedpoints except the origin, too. Figure 3 shows when the vacuum is stable, the positionalrelations of fixed points where the branes are located are equidistant each other. Weexpect that the complex structure is stabilized in such a way that the fixed points wherethe branes are located have symmetric positional relations. Otherwise there are no stableSUSY vacuum, and SUSY or gauge symmetry would be broken.Four-dimensional CP can be embedded into proper Lorentz transformation in higherdimensional theory, where extra dimensions are also reflected [29–34]. For example, in sixdimensional theory, four-dimensional CP is combined with the reflection, z → − ¯ z, (4.4)so as to be embedded into six-dimensional proper Lorentz transformation. Under theabove reflection, the modulus transforms τ → − ¯ τ . (4.5)Thus, when Re τ = 0, CP is conserved. For other values of Re τ , CP can be broken.Hence, the value τ = e iπ/ has implication in CP violation physics. We have investigated the quantum corrections to the localized FI-terms in six-dimensionalSUSY Abelian gauge theory compactified on the T /Z orbifold. If theory has modular symmetry, the transformation (4.5) is meaningful up to the modular sym-metry.(See e.g. [35–37].) That implies that CP is conserved at the values of τ at the boundary of thefundamental domain including τ = e iπ/ .
18n the S /Z orbifold, the localization of bulk zero mode causes the instability ofthe vacuum. Similarly, the bulk zero mode is localized in the untilted T /Z model,too [21]. We find that the new supersymmetric vacuum which is changed by 1-loopFI-term can be unstable in untilted compactification. The instability is related to theconfiguration of brane modes and their U (1) charges. We have shown that the 1-loopcorrection vanishes for the completely symmetric configurations, but it is not true for theasymmetric configurations. It is because the zero mode profile and brane charges canceleach other for the former case, but it does not happen for the latter case. Therefore, inthe asymmetric configurations the vacuum receives further corrections and is unstable. Ifwe put a fine-tuned FI-term in the tree level Lagrangian, we can realize a stable vacuumeven for asymmetric configuration. In such a stable vacuum, zero mode profile shields thebrane charges completely, and their corrections are canceled each other. This result is thesame as the one derived on the S /Z orbifold [22].As opposed to the S /Z orbifold, the complex structure exists in the T /Z orb-ifolds. The 1-loop FI-term depends on the complex structure, i.e., the complex structureassociates with the instability of the vacuum. Especially, we can stabilize the complexstructure τ by using the cancellation of 1-loop FI-term that is induced in a new supersym-metric vacuum. We have considered the configuration with three brane modes that arelocated at each of three fixed points and have the same charge. We have found that thecomplex structure τ is stabilized at the value of e iπ/ , which makes the three fixed pointsequidistant each other. We expect that the stabilization mechanism which is caused bythe cancellation of 1-loop FI-term occurs in more general orbifolds, and the stabilizedcomplex structures make the positions of fixed points symmetric. It contrasts with thetraditional moduli stabilization mechanism by three form flux [38–40]. We have focusedon 1-loop corrections and mainly investigated stable configurations in the present paper.For unstable vacuum, SUSY or gauge symmetry would be broken, and higher loop correc-tion might play important role. It is interesting to consider these effects. We will studyit elsewhere.Magnetic flux also affects the profiles of the wave function of the bulk fields, andincrease the number of the chiral zero modes [23, 41–43]. It is interesting to extend ouranalysis to the T /Z orbifolds with magnetic fluxes. Its flavor structure would be differentfrom that of magnetized orbifold models without FI-terms [24, 44, 45]. In magnetizedorbifold models, zero modes transform each other under the modular symmetry [46–48].In addition, our FI-term has already non-trivial behavior under the modular symmetry.Thus, it is interesting to study localized FI-terms from the viewpoint of modular flavormodels [49] and their modulus stabilization [37, 50]. Toroidal orbifolds have K¨ahler moduli in general. The effective potential of our model does notinclude the K¨ahler moduli, and its stabilization by the bulk instability is not realized. We need anothermoduli stabilization mechanism such as non-perturbative effects for the K¨ahler moduli [7]. δ (2) ( x, y ) Acknowledgments
H. A. was supported in part by Waseda University Grant for Special Research Projects(Project number: 2019Q-027) and also supported by Institute for Advanced Theoreticaland Experimental Physics, Waseda University. T. K. was supported in part by MEXTKAKENHI Grant Number JP19H04605.
A Localization of the zero mode
Here we show the zero mode of the bulk scalar in 1-loop corrected background is localizedat the fixed points. Since the wave function includes the exponential of the delta function,this function is not well defined. Here we evaluate it by use of an explicit regularizationof the delta function. We regularize the delta function as follows (see Figure 4): δ (2) ρ ( x, y ) = (cid:40) πρ (1 − (cid:112) x + y /ρ ) ( (cid:112) x + y ≤ ρ ) , (cid:112) x + y > ρ ) . (A.1)20e can check (cid:82) dxdy δ (2) ρ ( x, y ) = 1 immediately. (cid:90) dxdy δ (2) ρ ( x, y ) = (cid:90) dr (cid:90) dθ rδ (2) ρ ( r, θ )= (cid:90) ρ dr (cid:90) dθ r πρ (1 − r/ρ )= 2 π (cid:90) ρ dr πρ ( r − r /ρ )= 2 π πρ × ρ . (A.2)The wave function of the zero mode is given by (2.50). Substituting the regularization(A.1) into the wave function, we obtain | φ + , ( z, ¯ z ) | ∼ (cid:40) | f | (cid:81) I =1 ,..., | ψ I ( z, ¯ z ) | exp (cid:110) kπρ (1 − | z − z I | /ρ ) (cid:111) ( | z − z I | ≤ ρ ) | f | (cid:81) I =1 ,..., | ψ I ( z, ¯ z ) | ( | z − z I | > ρ ) (A.3)where k = gqξ (cid:48)(cid:48) R and ψ I ( z, ¯ z ) is given by | ψ I ( z, ¯ z ) | ≡ (cid:12)(cid:12)(cid:12) ϑ (cid:16) z − z I π (cid:12)(cid:12)(cid:12) τ (cid:17)(cid:12)(cid:12)(cid:12) gqξ I / π exp (cid:110) − gqξ I π Im τ { Im ( z − z I ) } (cid:111) . (A.4)We define D I as the disc with radius ρ around the fixed points z I . Since ψ I ( z, ¯ z ) is finiteexcept for the vicinities of the fixed points, we can evaluate the norm of the wave functionby the sum of integrals on D I : (cid:90) T dzd ¯ z | φ + , | = (cid:88) I =1 ,..., (cid:90) D I dzd ¯ z | φ + , | + C, (A.5)where C is a finite constant, which is almost independent of ρ . (More precisely ρ depen-dence is sub-leading.) C is ignorable in the limit of ρ to zero. In the vicinity of the fixedpints, ϑ ( z − z I ) is singular. It is approximated as ϑ (cid:16) z − z I π (cid:12)(cid:12)(cid:12) τ (cid:17) ∼ η ( τ ) ( z − z I ) , (A.6)where η ( τ ) is the Dedekind eta function. Thus we can evaluate the wave function aroundthe fixed point z I by | φ + , ( z, ¯ z ) | ∼ | f | (cid:32)(cid:89) J (cid:54) = I | ψ J ( z I , ¯ z I ) | (cid:33) (cid:12)(cid:12) η ( τ ) ( z − z I ) (cid:12)(cid:12) gqξ I /π exp (cid:20) kπρ (1 − | z − z I | /ρ ) (cid:21) . (A.7)21ntegral on D I is calculated as N I ≡ (cid:90) ρ rdr (cid:90) π dθ | η ( τ ) r | gqξ I /π exp (cid:20) kπρ (1 − r/ρ ) (cid:21) = 2 π | η ( τ ) | gqξ I /π e kπρ (cid:18) πρ k (cid:19) gqξ I /π (cid:90) kπρ dr (cid:48) r (cid:48) gqξ I /π e − r (cid:48) ∼ π | η ( τ ) | gqξ I /π e kπρ (cid:18) πρ k (cid:19) gqξ I /π Γ (cid:18) gqξ I π (cid:19) , (A.8)where Γ( z ) is the gamma function, and we have approximated the integration range by R + . Γ (cid:0) gqξ I π (cid:1) is not zero since gqξ I /π is positive definite. N I diverges in the limit of ρ → +0. To normalize the zero mode, we obtain f = (cid:32) (cid:88) I =1 ,..., N I (cid:89) J (cid:54) = I | ψ J ( z I , ¯ z I ) | (cid:33) − / → . Except for D I , we find | φ + , ( z ) | = f (cid:81) | ψ I ( z ) | → ρ → +0. Thus the zeromode wave function is localized at the fixed points. It behaves as a linear combination ofthe delta functions δ ( z − z I ): | φ + , ( z, ¯ z ) | = (cid:88) I =1 ,..., C I δ (2) ( z − z I ) . (A.9)The coefficients C I are calculated by the surface integrals of | φ + , ( z, ¯ z ) | on small disc D I : C I = (cid:90) D I d z | φ + , ( z, ¯ z ) | ∼ (cid:90) D I | f | (cid:32)(cid:89) J (cid:54) = I | ψ J ( z I , ¯ z I ) | (cid:33) (cid:12)(cid:12) η ( τ ) ( z − z I ) (cid:12)(cid:12) gqξ I /π exp (cid:20) kπρ (1 − | z − z I | /ρ ) (cid:21) = N I (cid:81) J (cid:54) = I | ψ J ( z I , ¯ z I ) | (cid:80) I =1 ,..., N I (cid:81) J (cid:54) = I | ψ J ( z I , ¯ z I ) | . (A.10)Extracting ρ dependence of N I , we obtain N I ∝ e kπρ ρ gqξIπ ) , (A.11)while ρ dependence of the denominator of (A.10) is evaluated as (cid:88) I =1 ,..., N I (cid:89) J (cid:54) = I | ψ J ( z I , ¯ z I ) | ∼ e − kπρ ρ − gqξminπ ) , (A.12)22here ξ min is the minimum of ξ , ..., ξ . If ξ I is bigger than ξ min , C I vanishes in the limitof ρ to zero. We obtain C I C I = (cid:81) J (cid:54) = I | ψ J ( z I , ¯ z I ) | ( (cid:80) ξI = ξmin (cid:81) J (cid:54) = I | ψ J ( z I , ¯ z I ) | ) , ( ξ I = ξ min )0 . ( ξ I > ξ min )This is nothing but (2.54). Thus we can evaluate C I by the absolute value of the wavefunction near the fixed point. B Modular symmetry of elliptic theta functions
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