PHYSICAL REVIEW D 102, 065004 (2020)

Implications of the weak gravity conjecture in anomalous quiver gauge theories

† Yoshihiko Abe,1,* Tetsutaro Higaki ,2, and Rei Takahashi2 1Department of , Kyoto University, Kyoto 606-8502, Japan 2Department of Physics, Keio University, Yokohama 223-8533, Japan

(Received 12 June 2020; accepted 12 August 2020; published 9 September 2020)

We argue a smallness of gauge couplings in Abelian quiver gauge theories, taking the cancellation condition into account. In theories of our interest there exist chiral fermions leading to chiral gauge anomalies, and an anomaly-free gauge coupling tends to be small, and hence can give a nontrivial condition of the weak gravity conjecture. As concrete examples, we consider Uð1Þk gauge theories with a discrete symmetry associated with cyclic permutations between the gauge groups, and identify anomaly- free Uð1Þ gauge symmetries and the corresponding gauge couplings. Owing to this discrete symmetry, we can systematically study the models and we find that the models would be examples of the weak coupling conjecture. It is conjectured that a certain class of chiral gauge theories with too many Uð1Þ symmetries may be in the swampland. We also numerically study constraints on the couplings from the scalar weak gravity conjecture in a concrete model. These constraints may have a phenomenological implication to model building of a chiral hidden sector as well as the visible sector.

DOI: 10.1103/PhysRevD.102.065004

I. INTRODUCTION The latter extension is called the scalar weak gravity conjecture (SWGC). These conjectures also have been Swampland conjectures attract much attention recently checked in several aspects [14,15], and indicate that in various aspects [1–7]. The conjectures are expected to repulsive forces of gauge interactions among the same constrain effective field theories to be consistent with species of particles are stronger than attractive forces of quantum gravity, and give us new insights into not only gravity and Yukawa interactions among them [16]. the theory as a candidate of quantum gravity but also The situation may not be so simple in chiral gauge physics beyond the Standard Model (SM). theories. IR symmetries are often obtained through the Among them, the weak gravity conjecture (WGC) breaking of UV symmetries, and an IR gauge coupling is requires theories consistent with quantum gravity to include given by a linear combination of UV gauge couplings as a charged state with a charge q and a mass m satisfying the in the SM. The linear combinations are determined by weak gravity bound [3], the Stückelberg couplings among the gauge bosons and m would-be Nambu-Goldstone bosons (or axions) associated eq ≥ pffiffiffi ; ð1Þ with the symmetry breaking. This is applicable not only 2 MPl to anomaly-free gauge theories but also to consistent theories possessing anomalous Uð1Þ gauge groups. In so that an extremal black holes can have a decay channel. theories with an anomalous Uð1Þ, an axion field plays The WGC briefly states that the gravity is the weakest an important role to cancel the gauge anomalies: the gauge force. Here, e is an anomaly-free gauge coupling and MPl is invariance is (nonlinearly) restored owing to the axion the reduced Planck mass. The WGC can be extended to coupling to topological terms of the gauge fields on theories with multiple Uð1Þ groups [8] and also to a scalar top of the Stückelberg couplings.1 As in the ordinary exchange force such as a Yukawa interaction [9–13]. spontaneous symmetry breaking, these Stückelberg cou- plings lead to the gauge boson mass and determine *[email protected] the eigenstate of massless gauge boson. Thus the gauge † [email protected] boson of anomalous Uð1Þ symmetry is decoupled in the low energy limit.2 In the , this anomaly Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to 1See also a recent work [17]. the author(s) and the published article’s title, journal citation, 2Some of gauge bosons in the anomaly-free gauge groups can and DOI. Funded by SCOAP3. also become massive through the Stückelberg couplings.

2470-0010=2020=102(6)=065004(20) 065004-1 Published by the American Physical Society ABE, HIGAKI, and TAKAHASHI PHYS. REV. D 102, 065004 (2020)

6 cancellation is realized by the Green-Schwarz mechanism case. Similar conditions for theories with a discrete Zk [18] involving string theoretic axions. 4D string models (gauge) symmetry are also discussed in Refs. [28,29]. with anomalous Uð1Þ’s have been well discussed for Equation (4) could be regarded as an example of the weak realizing the SM [19–24]. coupling conjecture [29]. In this paper, we will focus on models with multiple A notion of quiver is often used for theories Uð1Þ symmetries and chiral fermions. For models with in the presence of multiple gauge groups and bifundamental Uð1Þk, an anomaly-free Uð1Þ is given by a linear combi- chiral fermions, and matches model building involving nation of the original symmetries: D- well [30–40]. Instead of concrete string models, in this paper we will consider quiver gauge theories with Xk Uð1Þk gauge groups and focus on the anomaly-free gauge ð1Þ ¼ ð1Þ ð Þ U anomaly-free ciU i; 2 groups and the (S)WGC in a bottom-up approach, suppos- i¼1 ing that the remaining anomalies are canceled and then the where k is the number of Uð1Þ symmetries, c (i ¼ 1; anomalous gauge bosons get massive. In general, compu- i tation of anomalies depends on the content in 2; …;k) is a model-dependent Oð1Þ coefficient and Uð1Þ i models. In order to check anomaly-free Uð1Þ’s systemati- is the ith gauge group. We will discuss some examples in cally and study concretely the (S)WGC constraints on the the following section. Then, the corresponding anomaly- gauge couplings, we restrict ourselves to several types of free gauge coupling e is given by models controlled by discrete symmetries. However, a 1 Xk 2 behavior of the anomaly-free gauge coupling in Eq. (3) ¼ ci ð Þ Uð1Þ’ 2 2 ; 3 does not change in general models with anomalous s. e i¼1 gi The (S)WGC can constrain range of free parameters in low energy theories and show what parameter values are ð1Þ where gi is the gauge coupling of the U i symmetry. The favored by UV theory in the view point of IR physics. gauge coupling e will become necessarily very weak In some quiver gauge theories of our interest, there exist a and smaller than the original coupling gi as the number discrete symmetry associated with cyclic permutations 3 of Uð1Þ gauge groups increases in the large k limit. Thus, between the gauge groups in certain quiver gauge theories, the WGC condition in Eq. (1) looks hard to be satisfied and the symmetry can generally be broken in anomaly-free with an assumption that chiral anomalies can be canceled. Uð1Þ theories by a linear combination of Uð1Þ’sasin In other words, the repulsive force among particles will Eq. (3). Some of quiver gauge theories remind us of then become very weak. It is conjectured that a certain class deconstructed extra dimension [41,42], which could relate of chiral gauge theories with too many Uð1Þ symmetries our approach to the weak coupling conjecture in hologra- 4 can be in the swampland. It is noted that the gauge groups phy [29]. Also new insights can be given to chiral Abelian in 10D superstring theories are restricted [26], while those gauge theories which may be a candidate of hidden sectors in 4D models seem less-constrained in the view point of dark matter models in [43]. 5 of tadpole condition. When magnitude of all the gauge This paper is organized as follows. In Sec. II,wegive couplings is comparable to each other, the Eq. (3) is a brief review of the (S)WGC and anomalous Uð1Þ rewritten as symmetries. In Sec. III, we will discuss concrete quiver gauge theories with Uð1Þk, then identify the anomaly-free g˜ m eq ∼ pffiffiffi q ≳ ; ð4Þ Uð1Þ symmetries. In Sec. IV, we numerically show the k MPl SWGC constraint on the gauge couplings and Yukawa couplings in a Uð1Þ4 quiver gauge theory. We discuss also a ˜ ∼ ∀ where g gi for i is the average of the gauge couplings. toy model from 5D compactification similarly. ∼ −1=2 We find that the gauge coupling is scaling as e k for a Section V is devoted to summary and conclusion. In this ≲ ð ˜ MPlÞ2 large k and there exists an upper bound on k, k qg m , paper, we will discuss the above arguments with the tree if the WGC is correct and the ratio of m=MPl remains level parameters. fixed in the large k limit. This upper bound on k is similar to the species bound [27],butk is not the number of II. BRIEF REVIEWS OF THE (S)WGC ð1Þ species but the number of U gauge groups in our AND ANOMALOUS Uð1Þ’S A. The WGC and the SWGC 3The WGC with a similar gauge coupling is discussed in Ref. [25]. In this subsection, we give a brief review of the WGC 4 This will generally be applicableQ to theories with a semi- and the SWGC in four dimension. The WGC claims that ¼ k simple gauge group of G i¼1 Gi in the large k limit, when G is spontaneously broken to a simple group. Here Gi is a simple 6 group. If we have too large ci’s, the theory would be in the 5In the heterotic string, the rank of the gauge group is sixteen. swampland owing to the appearance of very weak coupling.

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Z X there exists a state with a charge q and a mass m satisfying pffiffiffiffiffiffi S ⊃ d4x −g y ðφ¯ Þφaψψ¯ ; ð10Þ the inequality matter a a m eq ≥ pffiffiffi ð5Þ ∂m 2 y ðφ¯ Þ ≔ ðφ¯ Þ¼∂ mðφ¯ Þ: ð11Þ MPl a ∂φa a in a theory consistent with quantum gravity [3]. The factor pffiffiffi Then the SWGC for ψ is given by of 1= 2 comes from the relative normalization of the Newton force against the Coulomb one, and a generaliza- X m2 X tion to an arbitrary dimension is straightforward [44]. This fijq q ≥ þ Kaby y ; ð12Þ i j 2M2 a b conjecture makes (super)extremal black holes decay into i;j Pl a;b lighter ones. ij ab The WGC can be extended to theories including a scalar where f and K are the inverse matrix of the fij and Kab exchange force such as a Yukawa interaction. This is called respectively. This inequality can be interpreted as the total the SWGC [9,10,14]. Let us consider a theory with multiple gauge repulsive force is stronger than the sum of the Uð1Þ gauge groups: attractive forces of the gravity and the total Yukawa Z  interactions when we focus on forces acting between the pffiffiffiffiffiffi 2 X 1 ψ j ⃗ j ≥ j ⃗ jþj⃗ j 4 MPl a μ b test particle : FCoulomb Fgravity FYukawa . The abso- S ¼ d x −g R − K ∂μϕ ∂ ϕ EM 2 2 ab lute value of long-range force mediated by massless fields a;b 1 X in four dimension is expressed as ðiÞ ðjÞμν − fijðϕÞFμν F ð6Þ 4 A i;j jF⃗j¼ ; ð13Þ 4πr2 ðiÞ where R is a Ricci scalar, ϕa is a real scalar field, Fμν is a where a numerator A is the factor corresponding to each field strength of Uð1Þ , K is a scalar kinetic matrix, f is i ab ij force: a gauge kinetic function, and i, j (¼ 1; 2; …;k) and a, b denote the labels of Uð1Þ gauge groups and those of scalar 2 X m fields respectively. The diagonal parts of f give the gauge ¼ ij ¼ ij ACoulomb f qiqj;Agravity 2 2 ; ð1Þ ’ i;j MPl couplings of U i s and the off-diagonal components are X kinetic mixings. The matter part action is given by ¼ ab ð Þ AYukawa K yayb 14 Z  pffiffiffiffiffiffi 1 X a;b 4 a μ b S ¼ d x −g − K ∂μΦ ∂ Φ matter 2 ab φa a;b If the scalars are heavy, Yukawa interactions are short-  X   range forces and neglected. Then the SWGC gets back to þ ψ¯ γμ ∇ þ ðjÞ ψ − ðΦÞψψ¯ ð Þ i μ i qjAμ m 7 the WGC. j B. Anomalous Uð1Þ symmetries where ψ is a Dirac spinor of a test particle for the SWGC ð1Þ In this subsection, we review cancellation of chiral Uð1Þ and has a charge qi under the gauge group U i and a mass mðΦÞ, and Φa is a real scalar field which may be different gauge anomalies by axion fields. In 4D effective field a from ϕ in general. Here, the covariant derivative ∇μ theories, gauge transformation of the axions can cancel the includes the spin connection. The Φa is decomposed as chiral anomalies produced by light chiral fermions in the presence of topological terms of the gauge fields Φa ¼ φ¯ a þ φa; ð8Þ and the Stückelberg couplings. In field theories with an anomaly-free Uð1Þ gauge symmetry, such axions are where φ¯ a is the background configuration of Φa and φa would-be Nambu-Goldstone bosons associated with the denotes a fluctuation around the background. With these, spontaneous breaking of the Uð1Þ symmetry. After inte- the mass mðΦÞ is rewritten as grating out heavy fermions with chiral Uð1Þ charges, we can obtain anomalous Uð1Þ in the low energy limit [17,45]. ∂m In 4D string models, anomalies can be canceled by the mðΦÞ¼mðφ¯ Þþ φa þ: ð9Þ ∂φa Green-Schwarz mechanism involving string theoretic axi- ons that originate from tensor fields, when tadpoles of mðφ¯ Þ is the mass of the ψ in the background φ¯ a, and the brane charges are canceled [46,47]. higher order terms of φa give the interaction terms between We shall consider the 4D action involving axions in φa’s and ψ. Thus the Yukawa coupling reads: addition to chiral fermions leading to chiral anomalies:

065004-3 ABE, HIGAKI, and TAKAHASHI PHYS. REV. D 102, 065004 (2020) Z    X 1 2 X 2 III. QUIVER GAUGE THEORIES ¼ 4 − mi ∂ θ þ ðiÞ Saxion d x 2 2 BiI μ I Aμ AND THE WGC ∈ ð1Þ gi ∈ i U anomaly I axions X  In this section, we discuss quiver theories with Uð1Þk CiIθI μνρσ ðiÞ ðiÞ þ ϵ Fμν Fρσ : ð Þ gauge symmetry and identify anomaly-free gauge groups. 32π2 15 I∈axions In general, computation of anomalies depends on the matter content in models. To check anomaly-free Uð1Þ’s system- θ Here, I is an axion, BiI and CiI are constants, mi is the atically and identify the gauge couplings concretely, we gauge boson mass. For the anomalous Uð1Þ symmetries, focus on several types of models controlled by discrete the fields transform as symmetries. However, an anomaly-free gauge coupling will be given by Eq. (3) in general cases. As for a quiver ðiÞ ðiÞ θI → θI − DIiΛi;Aμ → Aμ þ ∂μΛi; ð16Þ diagram in this paper, each node implies a gauge group whereas each arrow among two nodes shows a left-handed Λ where i is the transformationP parameter, and we assume chiral fermion charged under two gauge groups. The ¼ δ that DIi satisfies I BiIDIj ij. The theory is invariant number of arrows shows that of and a direction in the presence of chiral anomalies produced by gauge of an arrow is corresponding to the representation against transformations against chiral fermions: two gauge groups. An arrowhead corresponds to anti- Z   fundamental representation while its opposite side means X X ð1Þ 4 CiIDIi μνρσ ðiÞ ðiÞ fundamental one. For theories only with multiple U S ¼ d x Λ ϵ Fμν Fρσ ; anomaly i 32π2 groups, (anti-)fundamental representation is supposed to ∈ ð1Þ I∈axions i U anomaly have a charg e þ1 (−1). A solid line shows a chiral (left- ð17Þ handed) fermion whereas a dashed line shows a complex scalar. δ ¼ þ δ ¼ 0 such that ΛStotal Sanomaly ΛSaxion . Thus, in terms At first, we shall focus on nonsupersymmetric gauge of axions the anomaly-free Uð1Þ’s are determined such theories with bifundamental chiral fermions of ðN1; N2Þ 7 ð 1Þ ð 2Þ that the coefficients of CiI ’s are vanishing. 4D effective representation under U N × U N × gauge group, action from 5D theory is also discussed, for instance, which is inspired by D-brane models. Although there in Refs. [48,49]. The anomalous gauge bosons become exist many types of quiver diagrams corresponding to massive as gauge theories, for simplicity we focus on theories includ- ing only Uð1Þ groups in the diagrams such as Fig. 1. Since 1 2 1 2 there exist chiral fermions, chiral gauge anomalies can gen- mi ðiÞ 2 mi ˜ ðiÞ 2 − ðB ∂μθ þ Aμ Þ ≕ − ðAμ Þ ; ð18Þ 2 iI I 2 g2 erally be produced as a consequence. We study cancellation i i condition of chiral anomalies to identify anomaly-free θ’ gauge couplings at the tree level, and apply the couplings after s are eaten by them as in spontaneous gauge sym- 3 metry breaking. Further, for some nonanomalous gauge to the WGC. Anomaly-free conditions for UðNÞ and 4 bosons, there can exist Stückelberg couplings UðNÞ are discussed in Appendix A. For instance, in ð Þk Z  SU N theories with a general N, non-Abelian gauge X 1 m2 anomaly cancellations require that the number of incoming S ¼ d4x − i axion 2 g2 arrows is equal to that of outgoing ones at each node. In i∈Uð1Þ i  nonanomalous   Uð1Þk theories we will simply mimic SUðNÞk cases X 2 0 0ðiÞ because in D-brane models a gauge group can be given × B ∂μθ þ Aμ : ð19Þ iI I by UðNÞ¼Uð1Þ × SUðNÞ rather than just SUðNÞ, I∈axions hence Uð1Þ and SUðNÞ are considered simultaneously. The nonanomalous gauge bosons can become massive as the anomalous ones. Then, the repulsive forces mediated by such massive gauge bosons will not contribute to the WGC. Hereafter, we suppose that this mechanism works in the quiver gauge theories studied in this paper, and these terms are ignored otherwise stated.

7 anomalous ¼ P Once anomalous gauge fields are written as Aμ ðiÞ ’ ’ i biAμ , bi s would be related to ci s in Eq. (3) through the orthogonality among Uð1Þ’s. If there exists a large hierarchy μ ðiÞ among bi’sinbið∂ θÞAμ , some ci’s would become very large. FIG. 1. A quiver diagram with k nodes.

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We suppose that the anomalies are canceled as in Sec. II B and then (non)anomalous gauge fields get massive in a gauge invariant form. Quiver gauge theories associated with deconstructed extra dimension [41,42] could relate our approach to the weak coupling conjecture [29]. In quiver gauge theories with Uð1Þk of our interest, the action is written by

Z  Xk 4 1 ðjÞ ð Þμν S ¼ d x − Fμν F j 4 2 FIG. 2. Three nodes quiver diagram. ¼1 gj j  þ ψ¯ γμð∂ þ ðjÞ − ðjþ1ÞÞψ þ ð Þ j;jþ1i μ iAμ iAμ j;jþ1 ; 20 A. Uð1Þ2k − 1 We consider quiver gauge theories with Uð1Þ2k−1 groups where ellipsis shows gravity and interaction terms among as shown in Fig. 1. These types of (supersymmetric) fermions which we have neglected. We assume that kinetic models have often been studied in D-brane models on mixings among gauge fields are absent at the tree level for or intersecting/magnetized D-brane models. simplicity, and will ignore them in this paper. The gauge They are used also to realize realistic Yukawa couplings ðjÞ or higher order couplings. We hereafter focus just on field of Uð1Þ is denoted by Aμ and ψ þ1 is a left-handed j j;j fermions producing anomalies. As seen below, these ðþ1 −1Þ ð ð1Þ spinor with a charge of ; against the U j; theories can have an unique anomaly-free Uð1Þ. ð1Þ Þ U jþ1 gauge group as noted above. The index runs as j ¼ 1; 2; …;kand satisfies k þ 1 ≡ 1. There will exist a 1. Uð1Þ3 8 → þ 1 symmetry that shifts labels simultaneously as j j : One of the simplest case is the quiver gauge theory with 3 9 Uð1Þ ¼ Uð1Þ1 × Uð1Þ2 × Uð1Þ3 groups in Fig. 2.Asin ðjÞ ðjþ1Þ ψ gj → gjþ1;Aμ → Aμ ; ψ j;jþ1 → ψ jþ1;jþ2; Eq. (20), there exist three left-handed chiral fermions Li (i ¼ 1, 2, 3), which have charges of ð1; −1; 0Þ; ð0; 1; −1Þ ð21Þ and ð−1; 0; 1Þ against ðUð1Þ1;Uð1Þ2;Uð1Þ3Þ respectively. This model will have a Z3 symmetry as noted above, and when we treat the gauge couplings as spurion fields, which there is no other choices to connect each node. The are expected to be moduli fields in the string theory. This divergences of Uð1Þ3 chiral currents jiμ (i ¼ 1, 2, 3) are Z can be regarded as a k symmetry acting on k nodes with a given by element of 8 ∂ 1 ¼ − 0 1 <> · j Q2 Q3 0100 0 2 ∂ · j ¼ Q3 − Q1 ð23Þ B C :> B 0010 0 C 3 B C ∂ · j ¼ Q1 − Q2; B 0001 0 C B C: ð22Þ B . . C ∂ i ¼ ∂ iμ @ . A where · j μj and Qi is the topological charge ¼ 1 ϵμνρσ ðiÞ ðiÞ 1000 0 density, Qi 32π2 Fμν Fρσ . Thus we define the anomaly-free Uð1Þ by We can study anomalies and identify anomaly-free Uð1Þ’s ð1Þ ≔ ð1Þ þ ð1Þ þ ð1Þ ð Þ systematically owing to this symmetry as seen below. This U X c1U 1 c2U 2 c3U 3; 24 symmetry will be broken in the low energies when an anomaly-free gauge group is given by a linear combination and impose the divergence of its current to vanish of UV Uð1Þ’s. So, interactions of axions to gauge fields are X expected to violate this discrete symmetry. X i ∂ · j ¼ ci∂ · j ¼ð−c2 þ c3ÞQ1 þðc1 − c3ÞQ2 In terms of particle phenomenology, this theory may be i¼1;2;3 the hidden sector for dark matter apart from the visible þð−c1 þ c2ÞQ3 ≡ 0: ð25Þ sector [43]. In Appendix B, we discussed also several quiver models not shown in this section.

8See also Refs. [39,50,51]. 9In a supersymmetric case, we have a Yukawa coupling.

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FIG. 3. Quiver diagrams with five nodes. Both diagrams have a Z5 cyclic symmetry among each node. In the right diagram, all nodes are connected with arrows.

Then the solution is be hard to survive in the low energy limit while discrete gauge symmetries originating from the anomalous Uð1Þ’s ð1Þ ¼ ð1Þ þ ð1Þ þ ð1Þ ð Þ U X U 1 U 2 U 3: 26 can survive if any. If global symmetries survive, this model is in the swampland. It will be necessary to embed this In this model, the anomaly-free gauge group is determined model into string theory in order to know what kind of uniquely (up to overall normalization of the charges), and symmetries survives. This is beyond the scope of the paper its gauge coupling is given by and left for future work. 1 1 1 1 ¼ þ þ ð Þ 2k − 1 2 2 2 2 : 27 2. Uð1Þ eX g1 g2 g3 We consider quiver gauge theories with more general 2k−1 Here, the anomaly-free gauge coupling eX is written so that Uð1Þ groups. Figure 3 shows quiver diagrams with the the gauge kinetic term becomes the canonical form: five nodes, and it is noted that the number of incoming arrows is equal to that of outgoing ones at each node and X 1 1 ðjÞ ð Þμν ðXÞ ð Þμν both diagrams have a Z5 cyclic symmetry among each − Fμν F j ¼ − Fμν F X 4 2 4 2 j gj eX node. As in Eq. (20) and in the left diagram of Fig. 3, we have five left-handed fermions charged against þðanomalous gauge fieldsÞ: ð28Þ ðUð1Þ1;Uð1Þ2;Uð1Þ3;Uð1Þ4;Uð1Þ5Þ. The divergences of Uð1Þ5 chiral currents are given by Thus, the anomaly-free gauge coupling eX can be smaller 0 1 0 10 1 than the original Uð1Þ gauge couplings gi’s. 1 j 0100−1 Q1 It is noted that all the matters are then neutral under B 2 C B CB C ð1Þ ∀ ¼ 0 B j C B −10 1 0 0CB Q2 C this anomaly-free U X, i.e., qX . It seems that this B C B CB C B 3 C B CB C model may not be naively applied to the WGC, but the ∂ · B j C ¼ B 0 −10 1 0CB Q3 C: presence of global symmetries is important. The low energy B 4 C B CB C @ j A @ 00−10 1A@ Q4 A Lagrangian will be given by 5 j 100−10 Q5 X 1 L ¼ ψ ∂ψ − ð ðXÞÞ2 þ ð Þ ð30Þ i Li= Li 4 2 Fμν ; 29 i∈matter eX The number of anomaly-free Uð1Þ’s is given by that of zero ðXÞ if anomaly-free gauge boson Aμ survives in low energy eigenvalues of this coefficient matrix, and we find only one limit. Ellipsis includes interactions among fermions and zero eigenvalue in this model. The anomaly-free Uð1Þ is anomaly-free gauge boson and there will additionally exist given by the corresponding eigenvector kinetic mixings such as Kijψ iL=∂ψ jL and Majorana mass − ψ C ψ Uð1Þ ¼ Uð1Þ1 þ Uð1Þ2 þ Uð1Þ3 terms of Mij iL jL in low energy limit after anomalous anomaly-free massive bosons are integrated out. These terms will violate þ Uð1Þ4 þ Uð1Þ5: ð31Þ invariance under phase rotations of fermions. Now the ðXÞ ðXÞ original Z3 symmetry acts as eX → eX, Aμ → Aμ and Thus all matters are again neutral under this anomaly-free ð1Þ ψ Li → ψ Liþ1, but whether this low energy theory has the U and this system will not simply be applied to the Z3 symmetry depends on parameters for fermions. Since all WGC. The situation is similar to the three quivers model in ð1Þ fermions are neutral under U X, global symmetries will Sec. III A 1.

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The result is not changed by adding five chiral fermions to this model as in the right diagram of Fig. 3. Then their action is additionally given by

X5 Z 4 μ ðjÞ ðjþ2Þ S ¼ d x½ψ¯ j;jþ2iγ ð∂μ þ iAμ − iAμ Þψ j;jþ2 þ: j¼1 ð32Þ

The anomaly coefficient matrix reads 0 1 011−1 −1 FIG. 4. A quiver diagram of three nodes connected with five B C nodes by a pair of two arrows of vectorlike matters. B −10 1 1−1 C B C B C B −1 −1011C: ð33Þ B C where we assume that the bifundamental vectorlike matters @ 1 −1 −10 1A ð1Þ ð1Þ are charged under the gauge groups of U 1 × U 2k and 11−1 −10 that the mass m remains nonzero in the weak gauge coupling limit. In this case, the discrete symmetry is Thus, the anomaly-free Uð1Þ is similarly given by Eq. (31). explicitly broken since ψ 1;2k is transformed to ψ 2;2kþ1 that So far we have discussed specific quiver models with is originally absent. For instance, we focus on Uð1Þ3 × three and five nodes, but the result can be simply extended Uð1Þ5 theory with vectorlike matter, which is the case of to general models with odd number nodes as shown in k ¼ 2 and l ¼ 3. Since vectorlike matter does not contrib- Fig. 1. Since the entries of the anomaly coefficient matrix ute to the chiral anomalies, we have two anomaly-free are composed of the same number of 1 and −1 as above, the ð1Þ’ ð1Þ ð1Þ3 U s as mentioned above: one denotes U X from U ð1Þ 5 anomaly-free U is uniquely determined as ð1Þ 0 ð1Þ and another denotes U X from U . Here, 2Xk−1 X3 X8 Uð1Þ ¼ Uð1Þ ; ð34Þ anomaly-free i Uð1Þ ¼ Uð1Þ ;Uð1Þ 0 ¼ Uð1Þ ; ð37Þ i X i X i i¼1 i¼4 and the gauge coupling is given by hence charges of vectorlike matters are ðþ1; −1Þ and ð−1 þ1Þ ð ð1Þ ð1Þ Þ 1 2Xk−1 1 ; for U X;U X0 and other chiral matters ¼ ð Þ 2 2 : 35 are neutral for them. The respective gauge couplings are e i¼1 gi given by

In concrete models, these are easily verified and it is 1 X3 1 1 X8 1 ¼ ¼ ð Þ checked also that the result does not change for models 2 2 ; 2 2 : 38 with odd nodes and full diagonal lines that are similar to the eX i¼1 gi eX0 i¼4 gi right diagram of Fig. 3. As noted previously, however, there exist no charged chiral matters for this anomaly-free Uð1Þ. These can be weaker than the original gauge couplings of gi’s. Then the WGC for the vectorlike matter reads 3. Uð1Þ2k − 1 ×Uð1Þ2l − 1 with vectorlike matters m2 ð1Þ2k−1 2 þ 2 ≥ ð Þ We shall consider quiver gauge theories with U × eX eX0 2 : 39 2 −1 2M Uð1Þ l in the presence of vectorlike matters. As in Pl Fig. 4, the correspoinding diagram is composed of two In the large limit of k and l with a given m and g ’s, we find diagrams with odd nodes which are connected by a pair of i two arrows of vectorlike matters. Action is given by two 2 2 g g ð1Þ2k−1 ð1Þ2l−1 2 þ 2 ∼ i þ i → 0 ð Þ kinds of Eq. (20) showing U × U symmetry eX eX0 ; 40 and vectorlike part of k l Z and then the WGC can be violated since the couplings 4 μ ð1Þ ð2kÞ S ¼ d x½ψ¯ 1;2kiγ ð∂μ þ iAμ − iAμ Þψ 1;2k becomes very weak as long as the mass m remains nonzero in the limit of eX → 0 and eX0 → 0. Note that we now fix μ ð1Þ ð2kÞ g ’s but change only k and l. This indicates that there exists þ ψ¯ 2k;1iγ ð∂μ − iAμ þ iAμ Þψ 2k;1 i an upper bounds on the numbers of Uð1Þ gauge groups, k − ψ C ψ þ ð Þ M 2 m 2k;1 1;2k H:c: ; 36 þ ≲ ð PlÞ and l as k l gi m if the WGC is correct.

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ð1Þ TABLE I. The charges of fields for the anomaly-free U X × ð1Þ 0 U X group.

Fields qX qX0

ψ ab −1 þ c −1 − 1=c ψ da 1 − c 1 þ 1=c ψ bc 1 − c 1 þ 1=c ψ cd −1 þ c −1 − 1=c φ 00

Z FIG. 5. The 4 symmetric quiver diagram with four nodes. 1 2 1 2 1 ¼ c þ þ c þ ð Þ 2 2 2 2 2 ; 44 e g1 g2 g3 g4 B. Uð1Þ2k X We consider quiver gauge theories with Uð1Þ2k sym- 1 1 2 1 1 2 1 metry as shown in Fig. 1. These types of models are ¼ =c þ þ =c þ ð Þ 2 2 2 2 2 : 45 2k−1 also studied in D-brane models similarly to Uð1Þ cases. eX0 g1 g2 g3 g4 As the simplest model with chiral anomalies, we focus on Uð1Þ4 symmetry and this model has four left-handed In this model, the chiral fermions have nontrivial charges fermions as in Fig. 5. Divergences of each chiral current under these anomaly-free Uð1Þ’s as shown in Table I. In the are given by next section, we will numerically study the SWGC in this model by adding a complex scalar. 0 1 1 0 10 1 2k j 010−1 Q1 Extending this model to general theories with Uð1Þ is B 2 C B CB C simple, and we can verify that there exists at least two B j C B −1010CB Q2 C ∂ B C ¼ B CB C ð Þ ð1Þ’ · 3 : 41 anomaly-free U s in a concrete model. So it is expected @ j A @ 0 −10 1A@ Q3 A that in the large k limit with a given c and a fixed gi, 4 j 10−10 Q4 anomaly-free gauge couplings become very small as in cases of Uð1Þ2k−1 × Uð1Þ2l−1. Then there exists an upper Since the anomaly coefficient matrix have two zero eigen- bound on the number of Abelian gauge groups if the WGC state, this model has two independent anomaly-free Uð1Þ’s, is correct and a fermion mass remains nonzero in the large which are represented by the eigenvectors ð1; 0; 1; 0ÞT and k limit. ð0; 1; 0; 1ÞT. The former relates first node to third one, whereas the latter does second node to fourth one. The IV. A Uð1Þ4 MODEL AND THE SWGC independent anomaly-free Uð1Þ’s are generally given by In this section, we discuss the detail of Uð1Þ4 quiver ð1Þ ¼ ð1Þ þ ð1Þ þ ð1Þ þ ð1Þ ð Þ U X cU 1 U 2 cU 3 U 4; 42 gauge theory shown in the previous section and its application to the SWGC at the tree level in the presence 1 1 of a complex scalar field. The motivation for this is to study Uð1Þ 0 ¼ − Uð1Þ þ Uð1Þ − Uð1Þ þ Uð1Þ ; ð43Þ X c 1 2 c 3 4 SWGC in a more realistic (or string-inspired) model with a scalar field. The SWGC shows numerically constraints of a where c is a free parameter that depends on the D-brane smallness of gauge couplings against Yukawa couplings. configuration in concrete UV string models [19–21],10 and We also study a UV completion of 5D orbifold model for it. will be a rational number. Otherwise, there exists a global – symmetry [52 54]. The result does not change even if A. Constraints of the SWGC we add bifundamental vectorlike matters that are charged Figure 6 shows a quiver diagram of Uð1Þ4 model in the under only Uð1Þ1 × Uð1Þ3 or only Uð1Þ2 × Uð1Þ4.For ¼ 0 ð1Þ ¼ ð1Þ þ ð1Þ ð1Þ ¼ presence of a complex scalar φ, whose charge is ðþ1; −1Þ c , we find U X U 2 U 4 and U X0 ð ð1Þ ð1Þ Þ 11 for U b;U d . Due to this scalar field, we have Uð1Þ1 þ Uð1Þ3. It is noted that for a general c a linear 2 Yukawa couplings of combination can violate the Z4 to Z2 exchanging 1 ↔ 3 and 2 ↔ 4. The gauge couplings relevant to the anomaly-free 11 Uð1Þ’s read The direction of the dashed arrow shows the scalar charge same as chiral fermions. We changed the names of gauge groups ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ from U 1 × U 2 × U 3 × U 4 to U a × U b × 10 ð1Þ ð1Þ A gauge boson in one of the two Uð1Þ’s could be massive in U c × U d and accordingly those of left-handed fermions UV models. But, the behavior of gauge coupling will not change from ðψ 1;2; ψ 2;3; ψ 3;4; ψ 4;1Þ to ðψ ab; ψ bc; ψ cd; ψ daÞ for the latter in the large k limit. convenience.

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scalar φ is neutral under anomaly-free Uð1Þ’s and will not be considered for the SWGC. The scalar potential will be neglected hereafter with an assumption that φ is sufficiently light at energy scales of our interest since the scalar potential will be model-dependent. To check strong SWGC [11] for φ is an interesting issue, but this is left for future work and we focus on the SWGC for fermions with nontrivial anomaly-free gauge charges. Here the anomaly-free gauge couplings are given by

2 2 FIG. 6. A quiver diagram of Uð1Þ4 model including a complex 1 c 1 c 1 ¼ þ þ þ ; ð Þ scalar. 2 2 2 2 2 50 eX ga gb gc gd 1 1 2 1 1 2 1 L ¼ − φψ C ψ − 0φ†ψ C ψ þ ð Þ ¼ =c þ þ =c þ ð Þ Yukawa y ab da y bc cd H:c: 46 2 2 2 2 2 : 51 eX0 ga gb gc gd This model is inspired by intersecting brane models φ [30,32].NoZ4 symmetry exists. This is because φ can In the presence of a very light , the SWGC can be expressed as be written as φbd in the view point of the Uð1Þ charges and hence φ is transformed to φ that is originally absent.   bd ca 1 2 2 2 There could exist Z2 that simultaneously exchanges the 2 2 2 M Y ð−1 þ cÞ e þ 1 þ e 0 ≥ þ ; ð52Þ 0 X X 2 2 2 labels as a ↔ c and b ↔ d for y ¼ y , if we can identify c MPl φ ¼ φ† bd db. As seen in the previous section, two anomaly- ð1Þ’ for a test fermion. Here, Y ¼ y and M ¼ yReðφÞ for a Dirac free U s are given by 0 0 fermion of ψ ab þ ψ da, whereas Y ¼ y and M ¼ y ReðφÞ ð1Þ ¼ ð1Þ þ ð1Þ þ ð1Þ þ ð1Þ ð Þ for ψ þ ψ . A factor Y2=2 is obtained because of the U X cU a U b cU c U d; 47 bc cd canonical normalization of ReðφÞ, and ImðφÞ contributes to 1 1 the spin-dependent interaction that is not 1=r2-force. If the Uð1Þ 0 ¼ − Uð1Þ þ Uð1Þ − Uð1Þ þ Uð1Þ ; ð48Þ X c a b c c d scalar is sufficiently heavy, Y does not contribute to the SWGC condition owing to exponentially damping force where a free parameter c is a rational number and can be and hence the WGC can be easily satisfied. To reduce the fixed in concrete models by the brane configuration in the number of parameters, we will set gb ¼ gd ≕ g for sim- string theory. The charges of the fields are summarized in plicity. In the next subsection, we will study this situation Table I. realized in the 5D orbifold model. Thus this equation can be The effective Lagrangian showing two anomaly-free rewritten as Uð1Þ’s may read ð1 − Þ2 ð1 þ 1 Þ2 X c =c 1 ð Þ 1 ð 0Þ þ L¼ ψ = ψ −j∂ φj2 − ð X Þ2 − ð X Þ2 2ð 2 2 þ 2 2Þþ2 ð1 2Þð 2 2 þ 2 2Þþ2 i ID I μ 2 Fμν 2 Fμν c g =ga g =gc =c g =ga g =gc 4e 4e 0   I¼ab;bc;cd;da X X 1 Y 2 ≳ : ð53Þ −½ φψ C ψ þ 0φ†ψ C ψ þ þ ð Þ 2 y ab da y bc cd H:c: ; 49 g

ð Þ ð 0Þ ≪ 1 X X Here, the masses are neglected because M=MPl is where Dμ ¼ ∂μ þ iqXAμ þ iqX0 Aμ , gauge bosons rel- evant to two anomalous Uð1Þ’s are neglected since they numerically expected in the effective field theory. Indeed, there is almost no change in appearance of the plots for become massive if the Green-Schwarz mechanism works. ≲ 0 1 Yukawa couplings between the complex scalar and chiral M=gMPl . , where gMPl is expected as a cutoff scale 0 [3], when the scalar is massless. It is noted also that a gauge fermions are denoted by y and y , which are not the same in ð1Þ ð1Þ boson in either U X or U X0 may be massive owing to general. It is noted that vectorlike pairs of ψ ab þ ψ da and ψ þ ψ will constitute the Dirac spinors.12 Now the Stückelberg coupling and then either eX or eX0 vanishes bc cd in Eq. (53). In the top panels of Fig. 7, we show the plots of the 12 ð1Þ’ From the view point of the anomaly-free U s, there may SWGC (53) in the ðX; Y=gÞ-, ðc; Y=gÞ- and ðc; XÞ-planes, ψ C ψ ψ C ψ 2 2 2 2 exist Yukawa couplings including ab bc and cd da, but they where X ≔ g =g þ g =g . The each line saturates Eq. (53), are supposed to be much smaller than y and y0 here. Similarly, a c there may exist Dirac masses to these fermions because φ is hence the allowed region exists below them. In the presence singlet for the anomaly-free Uð1Þ’s in the low energy, but the of the mass, the SWGC is violated on each line. Note that masses would be negligibly small against the Planck scale. these plots are symmetric under c → −1=c owing to the

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FIG. 7. Plots of the SWGC constraints in ðX; Y=gÞ-, ðc; Y=gÞ- and ðc; XÞ-planes. The top panels plot the constraints of Eq. (53). The 0 ¼ 0 ¼ 0 ð1Þ ð1Þ middle (bottom) figures show the similar plots with eX (eX ), when U X (U X0 ) gauge group survives in low energy limit. The condition is saturated on each lines, below which there exist an allowed region. In the presence of mass, the SWGC is violated on each line.

4 1 definition of the anomaly-free Uð1Þ’s. A region for a B. A Uð1Þ model from S =Z2 orbifold and the SWGC large Yukawa coupling is excluded by the SWGC. For a ð2Þ ð1Þ We consider a 5D gauge theory with U × U a × large X, the constraint becomes tighter. In other words, 1 Uð1Þ on the S =Z2 orbifold for realizing chiral fermions. a big discrepancy between gauge couplings is disfavored. It c ¼1 The purpose of this subsection is to give a concrete Yukawa turns out that the constraint becomes stronger near c coupling associated with the gauge coupling and a relation e e 0 because either X or X vanishes then. We find also that the between gauge couplings as in the previous subsection via constraint is independent of c for special values of X ¼ 2 pffiffiffi the symmetry breaking of Uð2Þ → Uð1Þ × Uð1Þ by an ¼ 2 ¼ 2 b d and Y=g . This is because for X the left-handpffiffiffi side orbifold projection. The fields contents and their represen- of Eq. (53) becomes unity and hence for Y=g ≥ 2 the tations are exhibited in Table II. The 5D action is given by SWGC is then violated in the presence of the mass term. Z  We can find also that the constraint becomes weaker as the pffiffiffiffiffiffiffiffiffi 1 1 ¼ 4 − R − ð Þ2 c>0 increases in the top-left panel for X>2, because S5D d xdy G5 2 5 2 tr FMN 1 2κ 2ˆ M4×S =Z2 5 g2 either eX or eX0 gets stronger then whereas the constraint ≪ 1 1 ðaÞ 1 ðcÞ does not depend on c for X . It is noted that in the string − ðF Þ2 − ðF Þ2 4gˆ2 MN 4gˆ2 MN theory c depends on the D-brane configuration and the a c  couplings depend on moduli fields with the fixed configu- þ Ψ ð = − ÞΨ þ Ψ ð = − ÞΨ ð Þ ration. The middle (bottom) figures show the similar plots a iD Ma a c iD Mc c ; 54 0 ¼ 0 ¼ 0 ð1Þ with eX (eX ), when only a gauge boson of U X (Uð1Þ 0 ) remains massless and mediates the long-range X ¼ 0 1 2 3 ¼ ∇ þ þ ˆ ðaÞ þ repulsive force. In these cases, the condition of the SWGC where M ; ; ; ;y, DM M iAM iqaAM ˆ ðcÞ ˆ ˆ ð1Þ tends to give tighter constraints. iqcAM and qa and qc are the charges of U a and

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TABLE II. Table of the field contents and their charges in 5D ðbÞ 1 ð0Þ ð3Þ ðdÞ 1 ð0Þ ð3Þ where Aμ ≔ 2 ðAμ þ Aμ Þ, Aμ ≔ 2 ðAμ − Aμ Þ, φ ¼ model for realizing Uð1Þ4 gauge theory in 4D. Subscripts of Uð2Þ pffiffiffi −ð ð1Þ þ ð2ÞÞ 2 representation for fermions are Uð1Þ charges against the overall iAy Ay = is the complex scalar originating from 13 Uð1Þ ∈ Uð2Þ. the W-boson of y-direction, and the ψ L (ψ R) is the left- handed (right-handed) chiral fermion in 4D. It turns out that Fields Uð2Þ Uð1Þ Uð1Þ ð1Þ ð1Þ a c there exists the gauge symmetry of U a × U b × ð1Þ ð1Þ AM adj 0 0 U c × U d. As seen from the zero mode basis in the Ψa 21=2 −1 0 Uð2Þ gauge bosons, we find matter charges for the gauge Ψ 2¯ 0 þ1 ð ð1Þ ð1Þ ð1Þ ð1Þ Þ ψ C ≡ c −1=2 symmetry: As for U a;U b;U c;U d , a1R ðaÞ 0 adj 0 ψ ∶ðþ1 −1 0 0Þ ψ C ≡ψ ∶ð0 þ1 −1 0Þ ψ ≡ψ ∶ AM ab ; ; ; , c1R bc ; ; ; , c2L cd ð Þ c 0 0 adj ð0;0;1;−1Þ, ψ 2 ≡ ψ ∶ð−1; 0; 0; 1Þ and φ∶ð0; 1; 0; −1Þ. AM a L da This is the same field content as in the previous subsection, hence φ is a neutral scalar under anomaly-free Uð1Þ’s and ð1Þ U c respectively. 5D Chern Simons terms associated will not be considered for the SWGC. The scalar potential with 4D Green-Schwarz mechanism is neglected as already will be neglected as previously noted since the scalar noted. The field strengths are given by FMN ¼ ∂MAN − potential including radion will depend on the model and the ∂ þ ½ ða;cÞ ¼ ∂ ða;cÞ − ∂ ða;cÞ radion stabilization. Deriving the scalar potential and NAM i AM;AN and FMN MAN NAM for the non-Abelian gauge field and Abelian gauge fields checking the strong SWGC for this is left for future work. respectively. The normalization of generator of Uð2Þ is The 4D parameters are given by the 5D parameters with an assumption of hσi¼0. For the details, see Appendix C. chosen as trðTaTbÞ¼δab=2, hence the Uð2Þ gauge field 12 ð0Þ σa ðaÞ The 4D Planck mass is associated with the 5D gravitational is expanded as AM ¼ 2 A þ 2 A , where 12 is 2 × 2 M M coupling κ5 as identity matrix and σa’s(a ¼ 1, 2, 3) are the Pauli matrices. Ψ Ψ ∋ Since the covariant derivative is acting on a as DM a 2 πL ð0Þ ¼ ð Þ Ψ ¼ 12 Ψ þ Ψ 1 2 MPl 2 ; 59 iAM a i 2 AM a , a has = charge against the κ5 overall Uð1Þ. This is similar to Ψc, which has the opposite T Uð1Þ charge. Here, Ψa;c ¼ðψ a;c1; ψ a;c2Þ are doublets for and the gauge couplings in 4D are expressed by the SUð2Þ and ψ a;c are the 4D Dirac spinors. The metric of 1 2 −σ μ ν 2σ 2 2 1 1 2πL 1 πL 1 πL M4 × S =Z2 is written by ds ¼ e gμνdx dx þ e dy , ≔ ¼ ¼ ≔ ≔ ð Þ 5 2 2 2 ˆ2 ; 2 ˆ2 ; 2 ˆ2 : 60 where σ is the radion field, and gives 4D Einstein frame. g2 gb gd g2 ga ga gc gc 1 The size of S =Z2 is assumed to be πL and we take hσi¼0 L ¼ without loss of generality. The graviphoton gμy is dropped This is because we have the gauge kinetic term 2 ðbÞ 2 2 ðdÞ 2 since it is parity odd while the 4D gμν remains −2=4g2ðFμν Þ − 2=4g2ðFμν Þ via the symmetry breaking ð2Þ → ð1Þ ð1Þ massless. Then, the massive graviphoton mediates the of U U b × U d. With these, the anomaly-free short-range force among particles which have the Kaluza- couplings are defined as previously: Klein (KK) charges, and does not contribute to the SWGC 1 2 2 2 2 condition. On top of the usual periodic boundary condition ¼ c þ þ c þ ð Þ 1 2 2 2 2 2 ; 61 of S , the orbifold boundary conditon is given by eX ga g2 gc g2 ð − Þ −1 ¼ η ð Þ 1 1 2 2 1 2 2 PAM x; y P AAM x; y ; ¼ =c þ þ =c þ ð Þ 2 2 2 2 2 ; 62 ða;cÞ ða;cÞ 0 ð − Þ¼η ð Þ ð Þ eX ga g2 gc g2 AM x; y AAM x; y ; 55 where a free parameter c is a rational number. PΨ ðx; −yÞ¼γ5Ψ ðx; yÞ; ð56Þ a;d a;d The Yukawa couplings between φ and ψ’s are given by where P ¼ diagðþ1; −1Þ ∈ Uð2Þ, η ¼ 1 for M ¼ μ ¼ 0, A gˆ2 g2 1, 2, 3 and η ¼ −1 for M ¼ y. Thus 4D massless modes y ¼ y0 ¼ pffiffiffiffiffiffiffiffiffi ¼ pffiffiffi : ð63Þ A 2π 2 read L pffiffiffi 0  ðbÞ    Here, y and y are the same definition as in the previous Aμ iφ= 2 subsection. This equation relates the Yukawa coupling Aμ ¼ ;A¼ pffiffiffi ; ðdÞ y † ðφÞ Aμ −iφ = 2 to the gauge coupling. In the presence of a light Re ðaÞ ðcÞ Aμ ;Aμ ; ð57Þ 13Here we define the complex scalar by multiplying the þ − ψ ψ ψ ψ ð Þ ordinary W -boson by i so that the effective Lagrangian of a1R; a2L; c1R; c2L: 58 the zero modes reproduces Eq. (49) after this orbifold projection.

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FIG. 8. Left panel: plots of the SWGC conditons (65) in the ðc; XÞ-planes. The condition is saturated on each lines, below which there ð1Þ exist an allowed region. In the presence of mass, the SWGC is violated on each line. Right panel: a similar plot with only U X (red) ð1Þ and one with only U X0 (blue). and the radion, the SWGC inequality for zero mode forces and all bosons including scalar zero mode are neutral fermions reads under anomaly-free Uð1Þ’s and fermions with nontrivial     gauge charges are considered for the SWGC. Parity odd 2 2 2 ψ ψ ψ ψ 2 2 1 2 y 1 1 M fermions of a1L; a2R; c1L and a2R have opposite ð1 − cÞ e þ 1 þ e 0 ≥ þ þ ð64Þ X c X 2 2 6 M2 charges to zero mode fermions. KK modes of a field have Pl the same charge as that of the lightest mode. Yukawa couplings that are invariant under Z2 projection are given where M has the same definition as in the previous ϕ ψ ψ ϕ ψ ψ ϕ ψ ψ 14 1 6 by even even even, even odd odd, odd odd even, where subsectionpffiffi and = comes from the radion exchange ϕ ϕ ψ ψ −σ 6 even ( odd) is an even (odd) parity scalar and even ( odd) c= MPl ðφÞψψ¯ via ye pReffiffiffiffiffiffiffiffi with the canonically normalized is an even (odd) parity fermion. As massive scalars σ ¼ 3 2 σ radion c = MPl . We have neglected momentum- do not contribute to a long-range force, Yukawa dependent terms induced by the radion exchange with couplings relevant to the SWGC are associated with φ: ψγ¯ μψ∂ σ ðφÞ φψ ψ φψ ψ terms of μ .IfRe is sufficiently heavy, the even even; odd odd. After integration over the extra 0 Yukawa interaction in this equation can be neglected dimension, we will find Yukawa couplings of yφψnψ n in 0 0 and the WGC can be then easily satisfied. Gravitational addition to KK mass terms ðn=LÞψ¯ nψ n þðn=LÞψ nψ n for interactions including radion exchange will be numerically n-th KK modes with the canonically normalized kinetic neglected below as in the previous subsection owing to the terms (up to the radion dependence). Then nth KK mass Planck-suppressed interaction within the effective field eigenstates will have mass M2 ¼ðn=L yReðφÞÞ2. As the ≲ 0 1 ≲ 0 1 theory. For M=g2MPl . and p=g2MPl . , where p SWGC could be violated by heavy KK modes, it is is the momentum of a test fermion, there are not signi- necessary to check whether lighter modes including the ficant differences compared to the plots shown below. zero modes satisfy the SWGC. Substituting the above couplings given by Eqs. (61)–(63) to Figure 8 shows the plots of the SWGC (65) in the ðc; XÞ Eq. (64), we then find the SWGC condition plane. The each line saturates Eq. (65), hence the allowed region exists below them. In the presence of mass, the ð1 − Þ2 ð1 þ 1 Þ2 c þ =c SWGC is violated on each line. The these plots are ðc2=2Þðg2=g2 þ g2=g2Þþ2 ð1=2c2Þðg2=g2 þ g2=g2Þþ2 symmetric under c → −1=c. The left panel shows the 2 a 2 c 2 a 2 c ð1Þ’ 1 constraint when there exists two anomaly-free U s. ≳ : ð65Þ This is very similar to the top-right one of Fig. 7 for a 4 ð1Þ small value of Yukawa. For a large X, not only U b × Uð1Þ gauge coupling but also Yukawa coupling ∝ g2 This is also obtained when the parameters in Eq. (53) are d become much stronger than g or g , and hence there exist replaced as g2 → g2=2 and ðY=gÞ2 → 1=2. This gives a a c 2 an upper bound on X. It is noted that in the context of the ≔ 2 2 þ 2 2 constraint between c and X g2=ga g2=gc. string theory a large X may imply a big discrepancy among As for the KK modes or the massive parity odd ones, a moduli vacuum expectation values. In the vincity of similar equation to Eq. (64) will be hold. It is noted that c ¼1, matter becomes neutral against either one of the massive gauge bosons do not contribute to long-range anomaly-free Uð1Þ’s, then the constraint becomes tighter. In the right panel, plots show the SWGC constraint with 14 −σ We have factored out the common radion dependence e . eX ¼ 0 or eX0 ¼ 0, when only the gauge boson of either

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ð1Þ ð1Þ U X or U X0 remains massless owing to a Stückelberg scalar potential in addition to the radion. Hence it may be coupling as often seen in concrete string models. an interesting challenge to check the strong SWGC within a fixed model. This is left for future work. V. CONCLUSION It will be worth to investigate the (S)WGC in theories with more general gauge groups. In actual string compac- We have studied the (S)WGC in several types of quiver tifications, the number of closed string axions is known to ð1Þk gauge theories with U gauge symmetry in the presence be finite and depends on the Hodge number of compacti- of bi-fundamental chiral fermions leading to the chiral fication manifold. Some of the axions play an important anomalies, which is supposed to be canceled by the Green- role to cancel anomalies through the Green-Schwarz Schwarz mechanism. The theories which we consider mechanism. Hence, the number of anomalous Uð1Þ gauge Z possesses a cyclic k symmetry associated with a shift theories, which is k − 1 or k − 2 in our cases, should be of the label of the gauge groups. As a consequence of this, constrained by the number of such axions. If the anomalies we can study anomalies in the models systematically and are independent among the anomalous theories, the number the (S)WGC constraints on the gauge couplings. We of anomalous Uð1Þ’s can be less than that of axions for ð1Þ’ identified concretely the anomaly-free U s and their anomaly cancellation. Also in the string theory, the con- gauge couplings obtained via linear combinations of the jecture would constrain brane configuration and moduli ð1Þ’ Z original U s. Then, k symmetry can be broken in values. If one starts with 10D super Yang-Mills theory, 4D general. In the large k limit, an anomaly-free gauge effective action including an anomaly-free Uð1Þ may be −1=2 coupling becomes very weak as e ∼ k , and there exists given by [55,56] an upper bound on k if the WGC is correct and the mass for a test particle remains in the large k limit. This may be ϑðτÞ L ¼ − S ð Þ2 − pffiffiffi ϕψψ¯ þ ð Þ regarded as an example of the weak coupling conjecture. c 4 Fμν ; 66 2k−1 S For quiver theories with UPð1Þ , an unique anomaly-free ð1Þ 2k−1 ð1Þ U is proportional to i¼1 U i and all matters are where S is the 4D , τ is a complex structure modulus, neutral under the anomaly-free Uð1Þ. There exist charged and a rational number c originates from a linear combi- matters in the presence of vectorlike pairs, and Z2k−1 nation of Uð1Þ’s depends on brane configuration of the symmetry is broken then. For quiver theories with Uð1Þ2k number of branes and magnetic fluxes. The SWGC of e2 ≳ gauge symmetry, there exist two anomaly-free Uð1Þ’s and y2 (up to mass term) for matter fermion may read charged matters under these gauge groups, and Z2k symmetry is broken in general. Even if the gauge couplings 1 c ≲ : ð67Þ of the anomaly-free Uð1Þ’s receive quantum corrections, jϑðτÞj2 the IR couplings will remain very weak in the large k limit since Uð1Þ couplings are generally asymptotic nonfree as However, it will be required a deep understanding of far as the perturbation theory is valid. the string theory or concrete effective field theories includ- We have numerically studied also the SWGC in Uð1Þ4 ing (non-Abelian) Dirac-Born-Infeld action to study the theory in the presence of a complex scalar field, and SWGC constraints on for consistent gauge construct a similar model based on a 5D orbifold. It turns theories in the presence of the Green-Schwarz mechanism. out that a much larger Yukawa coupling than gauge This is also left for future works. couplings is forbidden and also that a big discrepancy among gauge couplings is disfavored. A special linear ACKNOWLEDGMENTS combination for realizing the anomaly-free Uð1Þ’s can be also be disfavored, since matter charge becomes small then. The authors thank Toshifumi Noumi and Tatsuo i jμν Kobayashi for useful discussions and comments. The So far, we neglected kinetic mixings χijFμνF among authors thank also Yoshiyuki Tatsuta at the early stage gauge fields. If we have such terms,P we may have a kinetic χ X X0μν χ ∼ k χ 0 of collaboration. The work of Y. A. is supported by JSPS mixing of FμνPF , where i;j ijciPcj, for anomaly- ð1Þ ¼ ð1Þ ð1Þ ¼ 0 ð1Þ Grant-in-Aid for Scientific Research, No. JP20J11901. free U X i ciU i and U X0 i ciU i with 3=2−α ci’s ¼ Oð1Þ. If the mixing χ is at most of Oðk Þ (α > 0) in the large k limit, the WGC can still be violated as in APPENDIX A: ANOMALIES IN STRING- Sec. III. This is because the canonically normalized mixing INSPIRED (SUSY) GAUGE THEORIES 0 χ is given by eXeX and hence an induced coupling of a We discuss anomaly-free Uð1Þ’sinUðNÞk quiver gauge fermion to an anomaly-free gauge field is proportional to theories inspired by the string theory. We focus only on 2 2 −α 0 χ χ eXeX or eXeX0 that are scaling as k then. However, if certain types of quiver theories considered in Sec. III in this 2 χ ¼ Oðk Þ in the large k limit, the WGC can be satisfied. section. Hereafter, Na denotes the rank of the gauge group In the Sec. IV, the scalar φ is a singlet under the anomaly- of UðNaÞ at the a-node, and nab shows the number of free gauge groups, and we did not discuss the detail of the bifundamental matter fields which correspond to that of

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Ti’s. These include anomalies of Uð1Þ3, Uð1ÞSUðNÞ2 and the mixed anomalies between the gravity and Uð1Þ’s, which are denoted by A ð1Þ 2. We impose that they are U a;b;cG vanishing:

A ð1Þ 2 ∝ N ðn N − n N Þ ≡ 0; ðA8Þ U aG a ab b ca c

A ð1Þ 2 ∝ N ðn N − n N Þ ≡ 0; ðA9Þ U bG b bc c ab a

3 ð Þ A ð1Þ 2 ∝ N ðn N − n N Þ ≡ 0: ðA10Þ FIG. 9. A quiver diagram of U N gauge theory. U cG c ca a bc b

This is the same condition as in the non-Abelian arrows connecting between a-node and b-node in the anomalies. There are not existing charged fields under ð ð1Þ ð1Þ ð1Þ Þ quiver diagram. all U a;U b;U c , then the anomaly between ð1Þ ð1Þ ð1Þ U aU bU c vanishes automatically. Then we find UðNÞ3 1. a ∂ · j ≡ NaðNbnabQb − NcnacQcÞ; ðA11Þ We consider a UðNÞ3 quiver gauge theory as shown in ð1Þ Fig. 9, and identify an anomaly-free U . To this end, we ∂ · jb ≡ N ðN n Q − N n Q Þ; ðA12Þ calculate chiral anomalies and mixed anomalies. Then, we b c bc c a ab a find the constraints on the ranks of gauge groups and the ∂ jc ≡ N ðN n Q − N n Q Þ: ð Þ numbers of generations. For a consistent theory, the non- · c a ca a b bc b A13 Abelian cubic anomalies A ð Þ3 give the following SU Na;b;c To identify the anomaly-free Uð1Þ we define it as constraints, c c c Uð1Þ ≔ a Uð1Þ þ b Uð1Þ þ c Uð1Þ ; ðA14Þ A ð Þ3 ∝ ðnabNb − ncaNcÞ ≡ 0; ðA1Þ X a b c SU Na Na Nb Nc

A 3 ∝ðn N − n N Þ ≡ 0; ðA2Þ SUðNbÞ bc c ab a and we impose that the divergence of the current associated ð1Þ with U X vanishes A 3 ∝ð − Þ ≡ 0 ð Þ SUðN Þ ncaNa nbcNb : A3 X c c ∂ · jX ¼ x ∂ · jx With these equations, the ranks of the gauge groups are x¼a;b;c Nx related as ¼ Naðccnca − cbnabÞQa þ Nbðcanab − ccnbcÞQb n n bc ca þ Ncðcbnbc − cancaÞQc ≡ 0: ðA15Þ Na ¼ Nc ∈ N;Nb ¼ Nc ∈ N: ðA4Þ nab nab From this equation, the coefficients satisfy the following ¼ ¼ ¼ ¼ Thus, we find Na Nb Nc for nab nbc nca. The conditions a;b;c ð1Þ divergences of the chiral currents j for U a;b;c are given by nbc nca ca ¼ cc;cb ¼ cc: ðA16Þ nab nab a ∂ · j ¼ NaðNbnabQb − NcnacQcÞ We take cc ¼ 1 and use Eqs. (A4) and (A16), then the þ NaðnabNb − ncaNcÞQa þ A ð1Þ 2 ; ðA5Þ U aG anomaly-free Uð1Þ is given by ∂ · jb ¼ N ðN n Q − N n Q Þ b c bc c a ab a ð1Þ ¼ cc ð ð1Þ þ ð1Þ þ ð1Þ Þ ð Þ U X U a U b U c : A17 þ N ðn N − n N ÞQ þ A ð1Þ 2 ; ðA6Þ Nc b bc c ab a b U bG

∂ · jc ¼ N ðN n Q − N n Q Þ It is noted that all fields are neutral matter under this c a ca a b bc b anomaly-free Uð1Þ. The anomaly-free gauge coupling is þ N ðn N − n N ÞQ þ A ð1Þ 2 ; ðA7Þ c ca a bc b c U cG given by

ð1Þ 1 ð Þ 1 μνρσ ðxÞ ðxÞ 1 1 1 1 ¼ U x þ SU Nx Gx ¼ ϵ ð μν ρσ Þ where Qx Q N Q , Q 16π2 tr F F ¼ þ þ : ðA18Þ x e2 g2 g2 g2 for x ¼ a, b, c, and trðTiTjÞ¼δij=2 for UðNÞ generators X a b c

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2. UðNÞ4 In this case, we impose that the anomaly coefficients of non-Abelian cubic anomaly are vanishing:

A 3 ∝ ðn N − n N Þ ≡ 0; ðA19Þ SUðNaÞ ab b da d

A 3 ∝ð−n N þ n N þ n N Þ ≡ 0; ðA20Þ SUðNbÞ ab a bc c bd d

A 3 ∝ð−n N þ n N Þ ≡ 0; ðA21Þ SUðNcÞ bc b cd d FIG. 10. UðNÞ4 quiver diagram. A 3 ∝ðn N − n N − n N Þ ≡ 0: ðA22Þ SUðNdÞ da a bd b cd c X c x ∂ x ¼ ð− þ Þ Solving these equations, we find that the ranks of gauge · j Na cbnab ndnda Qa Nx groups and the numbers of generations have the following x¼a;b;c;d relations, þ Nbðcanab − ccnbc − cdnbdÞQb     þ Ncðcbnbc − cdncdÞQc ¼ nda ∈ N ¼ nda − nbd ∈ N Nb Nd ;Nc Na Nd ; þ N ð−c n þ c n þ c n ÞQ nab ncd nab d a da b bd c cd d nda ncd ≡ 0: ð Þ ¼ : ðA23Þ A33 nab nbc Solving these equations for the coefficients cx, we get the The cancellation of the mixed anomaly between the gravity following relations and Uð1Þ’s imposes the same constraints as above:     nda nda nbd nda ncd A 2 ∝ ð − Þ ≡ 0 ð Þ ¼ ¼ − ¼ Uð1Þ G Na nabNb ndaNd ; A24 cb cd;cc ca cd ; : a nab ncd nab nab nbc ðA34Þ A ð1Þ 2 ∝ N ð−n N þ n N þ n N Þ ≡ 0; ðA25Þ U bG b ab a bc c bd d

From Eqs. (A23) and (A34), the coefficient ca (or cb)isa A ð1Þ 2 ∝ Ncð−nbcNb þ ncdNdÞ ≡ 0; ðA26Þ U cG free parameter. In order to solve these equations, we shall impose some assumptions. Here we will list some examples A ð1Þ 2 ∝ N ðn N − n N − n N Þ ≡ 0: ðA27Þ U dG d da a bd b cd c satisfying these equations. (i) ∀ N ¼ 1 The divergences of the Uð1Þ currents are expressed as – nbd ¼ 0 A solution is a ∂ · j ≡ NaðNbnabQb − NdndaQdÞ; ðA28Þ N ¼ N ¼ N ¼ N ¼ 1; b a b c d ∂ · j ≡ Nbð−NanabQa þ NbnbcQc þ NdnbdQdÞ; ðA29Þ nab ¼ nbc ¼ ncd ¼ nda; c ¼ 0 ð Þ ∂ · j ≡ Ncð−NbnbcQb þ NdncdQdÞ; ðA30Þ nbd : A35

d ∂ · j ≡ NdðNandaQa − NbnbdQb − NcncdQcÞ; ðA31Þ This is similar to the quiver gauge theory shown in Fig. 5. The two independent anomaly-free where we used vanishing conditions of non-Abelian Uð1Þ’s are generally given by Eqs. (42) and anomalies. We define the anomaly-free Uð1Þ by the (43), and the corresponding gauge couplings following equation as in the previous subsection are expressed as Eq. (44) and (45). – nbd ¼ 2 ð1Þ ≔ ca ð1Þ þ cb ð1Þ þ cc ð1Þ þ cd ð1Þ A solution is given by U X U a U b U c U d; Na Nb Nc Nd ¼ ¼ ¼ ¼ 1 ðA32Þ Na Nb Nc Nd ;

nab ¼ nda ¼ −nbc ¼ −ncd ¼ 1;nbd ¼ 2: and impose the current divergence associated with this ð1Þ ðA36Þ U X is vanishing

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The minus sign represents the opposite arrow of bd matter is neutral for these anomaly-free gauge Fig. 10. The independent anomaly-free Uð1Þ’s groups. The gauge couplings are given by are defined by 1 c2 1=2 ð1 − cÞ2 1=2 ¼ þ þ ; ðA44Þ Uð1Þ ¼ cUð1Þ þ Uð1Þ þð2 − cÞUð1Þ 2 2 2 2 2 X a b c eX ga gb gc gd þ Uð1Þ ; ðA37Þ     d 1 2 − 3 2 1 1 2 2 þ 1 2 1 1 2 ¼ c þ = þ c þ = 2 2 2 2 2 : c − 3 c þ 1 0 4c − 2 4c − 2 ð1Þ ¼ ð1Þ þ ð1Þ þ ð1Þ eX ga gb gc gd U X0 − 1 U a U b − 1 U c c c ðA45Þ þ ð1Þ ð Þ U d: A38 It is noted that a coefficient of 1=g2 is given by For these anomaly-free Uð1Þ’s, bd matters are b;d 1=2 ¼ N ð1=2Þ2 N ¼ 2 neutral. The anomaly-free gauge couplings are b;d · for b;d . given by

1 2 1 ð2 − Þ2 1 APPENDIX B: MODELS INSPIRED BY THE SM ¼ c þ þ c þ ð Þ 2 2 2 2 2 ; A39 ð1Þ4 ð1Þ5 eX ga gb gc gd We consider two quiver models with U and U     symmetries inspired by the SM. These are different from 1 − 3 2 1 1 þ 1 2 1 1 ¼ c þ þ c þ the models exhibited in the Sec. III in terms of chiral 2 − 1 2 2 − 1 2 2 : fermions. We show just that the anomaly-free gauge eX0 c ga gb c gc gd couplings are still given by a linear combination of the ð Þ A40 original couplings. The SM might not originate from a gauge symmetry that has too many Uð1Þ’s. (ii) ∀ jnj¼1 A solution is given by 1. A model inspired by Pati-Salam ¼ ¼ 2 ¼ ¼ 1 4 Nb Nd ;Na Nc ; We shall consider the Uð1Þ gauge theory shown in the right panel of Fig. 11. It is noted that we have two left- nab ¼ nda ¼ nbd ¼ −nbc ¼ −ncd ¼ 1: ðA41Þ handed fermions charged only under Uð1Þ3 × Uð1Þ4, and The independent anomaly-free Uð1Þ’s for this sol- there exist six chiral fermions and two complex scalars. ution is defined as This model is obtained from three nodes model of Uð2Þ × Uð1Þ3 × Uð1Þ4 in the left panel of Fig. 11 by the Higgs 1 ð2Þ Uð1Þ ¼ cUð1Þ þ Uð1Þ þð1 − cÞUð1Þ mechanism of U complex adiont scalar whose vacuum X a 2 b c expectation value is given by hΦi¼diagðv; −vÞ. This can 1 þ ð1Þ ð Þ be regard as a toy model of left-right symmetric theory 2 U d; A42 obtained from the Pati-Salam model [57–59] as in Fig. 12. The Uð1Þ4 model has two anomaly-free Uð1Þ’s and 2 − 3 1 2 þ 1 ð1Þ ¼ c ð1Þ þ ð1Þ þ c ð1Þ nontrivial charged matter fields, but we focus only on the U X0 U a U b U c 4c − 2 2 4c − 2 relevant gauge couplings. The detail of the anomaly 1 þ Uð1Þ : ðA43Þ cancellation is discussed in Appendix A. The divergences 2 d of Uð1Þ currents are given by

FIG. 11. The left panel: Uð2Þ × Uð1Þ1 × Uð1Þ2 quiver diagram. The right panel: Uð1Þ1 × Uð1Þ2 × Uð1Þ3 × Uð1Þ4 quiver diagram obtained from Uð2Þ → Uð1Þ1 × Uð1Þ2 by the Higgs mechanism. The dashed quiver shows bifundamental scalars arising from this symmetry breaking.

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FIG. 13. A diagram of quiver gauge theory inspired by the SM. FIG. 12. A quiver diagram of left-right symmetric Pati-Salam model.

0 1 1 0 10 1 j 001−1 Q1 B 2 C B CB C B j C B 001−1 CB Q2 C ∂ B C ¼ B CB C ð Þ · 3 ; B1 @ j A @ −1 −10 2A@ Q3 A 4 j 11−20 Q4 and we define two independent anomaly-free Uð1Þ’s with a free parameter c as

ð1Þ ¼ ð1Þ þð2 − Þ ð1Þ þ ð1Þ þ ð1Þ U X cU 1 c U 2 U 3 U 4; ðB2Þ FIG. 14. A quiver diagram of the SM-like model [33]. 0 1 0 10 1 3 − c 1 þ c 1 ð1Þ ¼ ð1Þ − ð1Þ þ ð1Þ þ ð1Þ j 0 −21 0 1 Q1 U X0 U 1 U 2 U 3 U 4: 1 − c 1 − c B 2 C B CB C B j C B 20−22−2 CB Q2 C ð Þ B C B CB C B3 B 3 C B CB C ∂ · B j C ¼ B −12 0−10CB Q3 C: B 4 C B CB C Two chiral fermions charged only under Uð1Þ3 × Uð1Þ4 is @ j A @ 0 −21 0 1A@ Q4 A still neutral but other fermions have nontrivial charges 5 j −12 0−10 Q5 under these anomaly-free Uð1Þ gauge groups. The corre- sponding anomaly-free gauge couplings read ðB6Þ

1 2 ð2 − Þ2 1 1 We find three anomaly-free Uð1Þ’s and they can generally ¼ c þ c þ þ ð Þ 2 2 2 2 2 ; B4 be written as eX g1 g2 g3 g4     2 2 1 3 − 1 1 þ 1 1 1 Uð1Þ ¼ c1Uð1Þ þ Uð1Þ þ c2Uð1Þ þð2 − c1ÞUð1Þ ¼ c þ c þ þ ð Þ X 1 2 3 4 2 1 − 2 1 − 2 2 2 : B5 eX0 c g1 c g2 g3 g4 þð2 − c2ÞUð1Þ5; ðB7Þ

These gauge couplings are given by linear combinations of with two free parameters of c1 and c2 which will be a ð1Þ ð1Þ the original ones, and when U 1 × U 2 is unified to rational numbers. The parameters of c ’s are taken as a ð2Þ ¼ i U we find g1 g2. gauge group is orthogonal to each other. Then the fermions have nontrivial charge in this anomaly-free Uð1Þ’s, but we 2. A model inspired by the SM focus only on the gauge couplings. The relevant gauge Another example is a model in Fig. 13 that is inspired by coupling is given by 15 the SM-like model in Fig. 14. The divergences of chiral 1 2 1 2 ð2 − Þ2 ð2 − Þ2 ¼ c1 þ þ c2 þ c1 þ c2 ð Þ currents are given by 2 2 2 2 2 2 : B8 eX g1 g2 g3 g4 g5 15The authors of Ref. [33] discussed the world-volume theory on a stack of D3-branes reproducing the field content of the As mentioned earlier, an anomaly-free coupling can contain minimal supersymmetric standard model with extended Higgs more of the original couplings as the number of Uð1Þ’sina sector in a quiver extension. theory increases.

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APPENDIX C: ORBIFOLD COMPACTIFICATION where A and a represent 5D and 4D local Lorentz indices a respectively, and e μ is the 4D vielbein. The off-diagonal Let us dimensionally reduce the 5D action in Eq. (54) and show the gauge couplings and Yukawa couplings element of the vielbein is absent because the orbifold 1 Z 2 ¼ projection prohibits the graviphoton. It is noted that 5D in 4D. The metric of M4 × S = 2 is written by ds5 ¯ −σ μ ν 2σ 2 fermion kinetic term is given by ΨΓAE MD Ψ,where e gμνdx dx þ e dy . Thus, the vielbein is given as A M E M is the inverse matrix of EA . Using these equations,   A M −σ=2 a we obtain 4D action for massless modes in Eqs. (57) A e e μ E M ¼ ; ðC1Þ and (58): eσ

Z  pffiffiffiffiffiffiffiffi π 3π ¼ 4 − L R − L ð∂ σÞ2 S4D d x g4 2 4 2 μ 2κ5 4κ5 1 π σ 1 π σ 1 π σ 1 π σ π −2σ − Le ð ðaÞÞ2 − 2 Le ð ðbÞÞ2 − Le ð ðcÞÞ2 − 2 Le ð ðdÞÞ2 − Le j φj2 4 2 Fμν 4 2 Fμν 4 2 Fμν 4 2 Fμν 2 Dμ gˆa gˆ2 gˆc gˆ2 gˆ2 −σ=2 −σ=2 −σ=2 −σ=2 þ πLe iψ abD= ψ ab þ πLe iψ daD= ψ da þ πLe iψ cdD= ψ cd þ πLe iψ bcD= ψ bc πLe−2σ πLie−2σ − pffiffiffi φ ðψ C ψ þ ψ ψ C Þ − pffiffiffi φ ðψ C ψ − ψ ψ C Þ 2 R ab da da ab 2 I ab da da ab  πLe−2σ πLie−2σ − pffiffiffi φ ðψ C ψ þ ψ ψ C Þþ pffiffiffi φ ðψ C ψ − ψ ψ C Þ − Vðσ; φÞ ; ðC2Þ 2 R bc cd cd bc 2 I bc cd cd bc

P ð Þ j −σ=2 where Dμ ¼ ∂μ þ i ¼ qψ Aμ is the covariant deri- gˆ2e j a;b;c;d L ¼ − pffiffiffiffiffiffiffiffiffi φ ðψ C ψ þ ψ ψ C Þ ð1Þ ð1Þ 4D;Yukawa R da da vative associated with the gauge group U a × U b × 2πL ab ab ð1Þ ð1Þ ψ U c × U d, and the chiral fermions ij are defined in −σ=2 gˆ2ie Sec. IV B. The four dimensional Ricci scalar is denoted by − pffiffiffiffiffiffiffiffiffi φ ðψ C ψ − ψ ψ C Þ 2π I ab da da ab R4, and φ ¼ φR þ iφI is the complex scalar. We introduce L ðσ φÞ 16 −σ=2 the scalar potential V ; formally. Thus, the gauge gˆ2e pffiffiffiffiffiffiffiffiffi C C − φRðψ ψ cd þ ψ cdψ Þ couplings are given as in Eq. (60). 2πL bc bc In order to find the relation between the Yukawa −σ=2 gˆ2ie coupling and the gauge coupling, we canonically normalize pffiffiffiffiffiffiffiffiffi C C þ φIðψ ψ cd − ψ cdψ Þ: ðC5Þ the fermion and the complex scalar as 2πL bc bc

σ=4 ˆ σ We introduce the Dirac fermions as pe ffiffiffiffiffiffi pg2ffiffiffiffiffiffie ψ ij → ψ ij; φ → φ: ðC3Þ πL πL     ψ da ψ cd ψ ¼ :ψ ¼ : ðC6Þ a ψ C c ψ C Then, the kinetic term is rewritten as ab bc

With these Dirac fermions, the kinetic terms of the i π −σ=2 ψ = ψ → ψ = ψ − ψ γμψ ∂ σ ð Þ anomaly-free sector read Le i ijD ij i ijD ij 2 ij ij μ : C4

1 ðXÞ 2 1 ðX0Þ 2 2 L4 ¼ − ðFμν Þ − ðFμν Þ − ð∂μφ Þ D;KT 4 2 4 2 R Hereafter, we will ignore the derivative coupling of the eX eX0 2 radion to the fermions. The Yukawa interactions are − ð∂μφIÞ þ iψ¯aD= ψ a þ iψ¯cD= ψ c; ðC7Þ expressed as where we neglected gauge bosons in anomalous Uð1Þ’s, the 16 σ φ At the classical level, the scalars and do not have anomaly-free gauge couplings are defined by Eqs. (61) and potential due to the gauge symmetries, but the potential can be generated by the radiative corrections. In addition, we (62). As shown in Table I, the covariant derivatives of the assume the radion field develops the VEV of hσi¼0 around Dirac fermions associated with anomaly-free Uð1Þ’s are the radius L. expressed as

065004-18 IMPLICATIONS OF THE WEAK GRAVITY CONJECTURE IN … PHYS. REV. D 102, 065004 (2020)     −σ 2 −σ 2 1 0 L ¼ − = φ ψ ψ − = φ ψ γ ψ ðXÞ ðX Þ 4D;Yukawa ye R a a iye I a 5 a Dμψ a ¼ ∂μ þ ið−1 þ cÞAμ − i 1 þ Aμ ψ a; ðC8Þ c −σ=2 −σ=2 − ye φRψ cψ c þ iye φIψ cγ5ψ c;     0 ð Þ ðXÞ 1 ðX Þ C10 Dμψ ¼ ∂μ − ið−1 þ cÞAμ þ i 1 þ Aμ ψ : ðC9Þ c c c where the 4D Yukawa coupling is defined as ψ ψ ð1Þ ð1Þ The charges of a and c under U X and U X0 are opposite to each other due to the 4D anomaly-free con- gˆ2 y ¼ pffiffiffiffiffiffiffiffiffi : ðC11Þ ditions. With the Dirac spinor, the Yukawa terms in this 2πL Lagrangian are rewritten as below:

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