PHYSICAL REVIEW D 100, 106010 (2019)

Towards expectations for couplings to axionlike particles

James Halverson, Cody Long, Brent Nelson, and Gustavo Salinas Department of , Northeastern University Boston, Massachusetts 02115-5000, USA

(Received 25 September 2019; published 21 November 2019)

Axionlike particle (ALP)–photon couplings are modeled in large ensembles of string vacua and random matrix theories. In all cases, the effective coupling increases polynomially in the number of ALPs, of which hundreds or thousands are expected in the string ensembles, many of which are ultralight. The expected −12 −1 −10 −1 value of the couplings gaγγ ≃ 10 GeV − 10 GeV provide viable targets for future x-ray telescopes and axion helioscopes, and in some cases are already in tension with existing data.

DOI: 10.1103/PhysRevD.100.106010

I. INTRODUCTION window, we will focus on analyses relevant for experiments that do not require ALP dark matter and only set limits If string theory is the correct theory of , below a fixed mass threshold. one of its vacua must realize the photon of classical For such experiments, the strongest bounds on the electromagnetism. Uncharged spin zero particles must cou- coupling arise at low mass, m ≤ 10−2 eV, where ple to the electromagnetic field strength, since all couplings a results [1] from the axion helioscope CAST require in string theory are determined by vacuum expectationvalues ≤ 7 10−11 −1 (VEVs) of scalar fields. This applies not only to the usual gaγγ × GeV . The CAST result already signifi- μν cantly outperforms projected collider bounds for m ≃ GeV parity even operator FμνF , but also the parity odd operator, a scale ALPs, leading us to focus on the low mass range. For requiring the existence of a coupling −12 even lower masses, ma ≤ 10 eV, observations from the 1 ≤ 8 10−13 −1 ˜ μν x-ray telescope Chandra require gaγγ × GeV L ⊃− g γγaFμνF ð1Þ 4 a [10]. The future helioscope IAXO and satellite STROBE-X −12 −1 are projected to probe g γγ ≃ 2 × 10 GeV for m ≤ ˜ μν μνρσ a a in the effective Lagrangian, where F ¼ ϵ Fρσ.The −2 −14 −1 10 eV [3] and gaγγ ≃ 8 × 10 GeV for ma ≤ pseudoscalar a is an axionlike particle (ALP), which is 10−12 eV [14], respectively. not necessarily the QCD axion, and it may be a nontrivial The primary focus of this work is to understand the linear combination of the hundreds or thousands of ALPs dependence of gaγγ on the number N of ALPs in both expected from studies of string vacua. models of string vacua and random matrix effective field Numerous ground-based experiments [1–4] and satellite ¼ 1 – theories. Though in the N case (using mild assump- observations [5 11] place constraints on ALP-photon tions described in the text) we have interactions, probing widely different regimes for the axion mass m and coupling strength g γγ. Existing limits are 1 a a ∶ ¼ pffiffiffi ð Þ already remarkable, within a few orders of magnitude of Single ALP gaγγ ; 2 3M grand unified theory (GUT) scale decay constants p ≡ −1 faγγ gaγγ. However, the exclusions depend critically on independent of the details of the compactification geometry whether the ALP is assumed to be a sizable fraction of the (including the string scale), it is reasonable to imagine that dark matter and also experimental limitations affecting the the introduction of additional ALPs will increase the accessible mass range. Since dark matter in string theory is effective coupling to . Concretely, we study often multicomponent (e.g., [12,13]) and many ALPs are whether the bulk of the gaγγ distribution at large N is in expected to be ultralight, but not yet in any fixed mass a range probed by current or future experiments. This is of interest because most vacua are expected to arise at the largest values of N afforded by the given ensemble. We will Published by the American Physical Society under the terms of refer to the mean of the distribution at this value of N as the the Creative Commons Attribution 4.0 International license. g γγ Further distribution of this work must maintain attribution to expected value of a . the author(s) and the published article’s title, journal citation, Our main result is that in all string ensembles and 3 and DOI. Funded by SCOAP . random matrix theories that we study, gaγγ increases

2470-0010=2019=100(10)=106010(13) 106010-1 Published by the American Physical Society HALVERSON, LONG, NELSON, and SALINAS PHYS. REV. D 100, 106010 (2019) polynomially in N. We perform detailed studies of two are very light, as we will soon discuss. The Lagrangians of string geometry ensembles that we refer to as the tree interest, focusing on the ALP sector, take the form ensemble and the hypersurface ensemble, using the mass −12 1 1 threshold m ≤ 10 eV so that our results can be com- μν μ i j a L ¼ − F Fμν − δ ð∂ ϕ Þð∂μϕ Þ pared to numerous experiments. In the tree ensemble at the 4 2 ij ¼ 2483 1 expected value of N , we find that the mean of the − 2ðϕiÞ2 − ϕi ˜ μν ð Þ −12 −1 mi ci F Fμν: 3 projected gaγγ distribution is gaγγ ¼ 3.2 × 10 GeV .In 4 the hypersurface ensemble N ¼ 491 is expected, and we −10 −1 Here Fμν is the electromagnetic field strength, with find the expected value g γγ ¼ 2.0 × 10 GeV . We also a ˜ μν ¼ ϵμνγσ ϕi study the F-theory geometry with the most flux vacua [15], F Fγσ, and are the ALPs, with masses mi. which is very constrained due to the existence of only a From a low-energy perspective these parameters are gen- single divisor that can support the , given erally unconstrained; however, we will find that UV −12 −1 structure from string theory actually constrains the low- our assumptions; it yields gaγγ ¼ 3.47 × 10 GeV .All of these couplings are in range of future experiments, and in energy physics, and introduces correlations and patterns, some cases are already in tension with data; see the that will result in interesting structure in both the ALP discussion. Furthermore, by removing the mass threshold masses and the ALP-photon couplings. For concreteness, we will study the dependence of ALP- the expected value of g γγ does not change significantly, a photon couplings on the number of ALPs N in string implying that ALPs in string ensembles that we study will compactifications and random matrix models. In the latter not be seen in searches at LHC or future colliders. The context, we consider compactifications of type IIB string random matrix results suggest that g γγ should increase a theory/F-theory on a suitable Kähler manifold B, which N significantly with , even if our vacuum does not arise in yields a four-dimensional (4D) N ¼ 1 effective field one of the studied ensembles. theory (EFT). To date, such compactifications encompass We emphasize at the outset that we are modeling ALP- the largest known portion of the N ¼ 1 landscape [15,18– photon couplings using data from string theory. A complete 21]. Some of the most generic phenomena in this region, calculation, with engineered Standard Models and full such as large gauge sectors and numbers of ALPs, correlate moduli stabilization, is computationally intractable given strongly with moving away from weakly coupled limits current techniques, which can be formalized in the lan- [22]. The ability to make such statements away from weak guage of computational complexity [16]. At small N, string coupling relies critically on the holomorphy implicit however, more complete calculations can be performed, in algebraic geometry. including partial moduli stabilization; see, e.g., [17]. The data of such an F-theory compactification, in Instead, we model ALP-photon couplings by intelligently addition to the choice of B, are an elliptic fibration over sampling the Calabi-Yau and utilizing knowl- B that encodes the data of the gauge group; see [23] for edge of how realistic gauge sectors arise, without trying to further details. The ALPs θi that we study arise from the concretely engineer them or stabilize moduli. This allows dimensional reduction of the Ramond-Ramond four-form for the study of large ensembles at large N, but also C4 and become the imaginary parts of complexified Kähler motivates further studies once techniques for the complete moduli, written as calculations become available. The models that we use are Z   described thoroughly in the text and are accurately sum- 1 i ¼ ∧ þ ≡ τi þ θi ð Þ marized in the discussion. T 2 J J iC4 i : 4 This paper is organized as follows. In Sec. II we discuss Di ALP-photon couplings from the perspective of string Here the D are a basis of divisors (4-cycles) in B, numbering theory. In Sec. III we introduce the ensembles of string i 1;1ð Þ compactifications and random matrix effective theories. In h B , and J is the Kähler form on B, which can be written in terms of parameters t as J ¼ t ωi, with ωi ∈ H1;1ðBÞ. Sec. IV we present the g γγ distributions computed in the i i a τi ensembles. We review constraints and projections of The parametrize the volumes of the divisors in B. A typical 1;1 3 existing and proposed experiments in Sec. V and discuss B has h ðBÞ ∼ Oð10 Þ [19,24]; that is, in these ensembles, our results in light of them in Sec. VI. Interesting future thousands of ALPs are expected. directions are also discussed. In addition to the many ALPs, EFTs arising from string theory often contain many gauge sectors. Throughout we will take N to be the number of ALPs. In the context of II. ALP-PHOTON COUPLINGS F-theory, our focus will be on when this gauge sector, IN STRING THEORY indexed by α, is supported on a stack of seven- Low-energy effective field theories from string theory wrapping a cycle Qα, or from a nontrivial Mordell-Weil regularly have a large number of ALPs, and under certain group element of the elliptic fibration Qα. The ALP-gauge assumptions related to control of the theory many of them portion of the EFT takes the form

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2 μ i j L ¼ −MpKijð∂ θ Þð∂μθ Þ − VðθÞ future experimental bounds. We will investigate whether X large N effects lead to a significant enhancement of the − αðτi μν þ θi ˜ μν Þ ð Þ Qi Gα Gαμν Gα Gαμν : 5 ϕi α coupling. Note that if the masses of all the that appeared i nontrivially in ciϕ were the same, then jcj would simply be the coupling of the canonically normalized field Here Mp is the Planck mass, VðθÞ is the nonperturbative i φ ≡ c ϕ =jcj to Fμν. In general this will not be the case; ALP potential, and Kij is the metric on moduli space, i which at tree level is derived from the Kähler potential however, in string compactifications with N large we K ¼ −2 log V, where V is the volume of B, which is expect that some of the ALPs will be essentially massless, expressed as which we will now review. Z ALP masses depend critically on the fact that, in the 1 absence of sources, the θi enjoy a continuous shift V ¼ ∧ ∧ ¼ κijk ð Þ J J J titjtk; 6 symmetry to all orders in perturbation theory, broken to B 6 a discrete shift symmetry by nonperturbative effects. and κijk are the triple intersection numbers of B. In particular, the superpotential W is known to receive To justify our use of the tree-level Kähler potential, we corrections from stringy and/or strong gauge note that F-theory can be viewed as the dimensional dynamics that generate masses for the ALPs [27]. Due to reduction of type IIB SUGRA in the presence of a spatially the shift symmetry of the ALP and the holomorphy of the varying axio-. The type IIB SUGRA action is unique superpotential, any nonperturbative contribution to the up to field redefinitions and higher-derivative corrections, superpotential takes the schematic form and so any physical correction to the 4D EFT must come with additional powers of appropriate volumes of cycles. −2π ˜ ðτiþ θiÞ Δ ∼ Qi i ð Þ At large enough volume we therefore expect that using W e ; 9 K ¼ −2 log V should be a good approximation to the ˜ i Kähler potential. It is not known how large is large enough for rational Qi. Therefore, for a given θ , if the correspond- in this case, but since the volumes in compactifications with ing τi is very large, then any nonperturbative contribution to many cycles tend to be very large [25], we believe this W involving θi will be negligible, and θi is expected to be approximation should be valid for our study. essentially massless (the same is not true of the τi, as they The leading-order Kähler potential is independent of the do not enjoy a similar shift symmetry and can receive ALPs, so one can move to a canonically normalized frame masses via perturbative corrections to the Kähler potential ≡ in which the ALP mass matrix is diagonal. Let F1μν Fμν [28,29]). The central observation of [25] is that in com- be the electromagnetic field strength, corresponding to a pactifications with large N, the region of moduli space Q homology class (subtleties associated with electroweak where the EFT is expected to be valid (known as the symmetry breaking will be discussed momentarily). The stretched Kähler cone) is quite narrow. Restricting to the Lagrangian of interest then takes the form stretched Kähler cone for the sake of control forces some of 1 1 the τi to be very large, which in turn forces some of the θi to L ¼ − δ ð∂μϕiÞð∂ ϕjÞ − μν 2 ij μ 4 F Fμν be extremely light. This is the key result that (in this 1 context) puts numerous ALPs in the sub-eV mass range − ϕi ˜ μν − 2ðϕiÞ2 ð Þ 4 ci F Fμν mi : 7 relevant for the experiments we will discuss. In this work we make the technical assumption that the Standard Model In terms of geometric data, we can write [24] sector does not generate a large mass term for the ALP ≳10−12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( eV). This assumption is quite mild given the −1 scales of the potential generated by SUð2Þ instantons, j⃗j¼ Q · K · Q ð Þ L c ; 8 as well as current expectations for the QCD axion mass 2MpQ · τ [30–32]. where K−1 is the inverse Kähler metric on field space. If the ALP masses are much lower than the typical Equation (8) is independent of homogeneous scaling in the energies of an experiment, then they can be safely → λ neglected and the ALPs taken to be massless. We will Kähler cone J J, and therefore only depends on the −12 angle in the Kähler cone. assume that the typical experiment energies are ≳10 eV, ≪ 10−12 It is interesting top noteffiffiffi that if N ¼ 1, then this coupling is and so any ALP with a mass eV can be treated as massless. In this case we can define a single ALP fixed to be c ¼ 1=ð 3MpÞ independent of the details of the ≡ ϕi j j geometry [26]. This is important because in the case a ci = c that couples to Fμν, with strength gaγγ. Here of a single ALP it determines the photon coupling, the sum includes only the ϕi that have negligible mass. The gaγγ ≃ 1=Mp, which is well below current and (projected) relevant terms in Eq. (7) become

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1 1 1 μ μν ˜ μν than causal. It would be interesting to explore this further in L ⊃− ð∂ aÞð∂μaÞ − F Fμν − g γγaF Fμν; ð10Þ 2 4 4 a future work. Our goal is to compute gaγγ in large ensembles of string where compactifications, as well as random matrix models, in rXffiffiffiffiffiffiffiffiffiffiffiffi order to model the distribution of such couplings that one ¼ 2 ð Þ expects from string theory and to understand the behavior gaγγ ci ; 11 i as the number of ALPs N grows large. in terms of the couplings ci in Eq. (7). Note that we can III. ENSEMBLES always redefine our ALP a to make g γγ non-negative, as a Having reviewed the EFTs expected from string theory, we do henceforth. the specific contexts in which we will study them, and the It is useful to express Eq. (11) in terms of the original presence of ultralight ALPs, we now introduce the ensem- geometric quantities that appear in Eq. (5). To do so, we bles in which we perform this study. determine the massless axions by finding the linear combinations of axions that receive a non-negligible mass term (see below for a discussion) in the canonically A. The tree ensemble normalized frame. Let the massive axions be specified F-theory is a nonperturbative generalization of type IIB a θa ≡ by a (generally nonfull rank) matrix Mi , such that string theory that allows for regions of strong string aθi Mi receives a non-negligible superpotential contribution coupling and has a more general gauge spectrum than in Eq. (9). We can move to a canonically normalized frame its IIB counterpart. In F-theory, the internal space B for by writing K ¼ ST · f · f · S, where S is a matrix of the compactification determines a minimal (geometric) gauge orthonormal eigenvectors of K, and f is a diagonal matrix structure from non-Higgsable seven-branes, whose pres- of the square roots of the eigenvalues. The massless axions ence requires no tuning in complex structure moduli space. (in the canonically normalized frame) are then specified by In [19] a lower bound of the number of bases B suitable the matrix for F-theory compactifications was determined to be 4=3 × 2.96 × 10755 via the discovery of a construction algorithm D ¼ KerðM · ST · f−1Þ; ð12Þ for an ensemble of B known as the tree ensemble. This ensemble represents a large graph, where nodes are i.e., the massless linear combination of canonically nor- geometries and edges are simple topological transitions malized axions is encoded in the rows or columns. (known as blowups) that may take place between the We can express these axions in the geometric θ basis by geometries. More specifically, each node contains a para- writing metric family (the complex structure moduli space) of geometries that have the same topological type, but for QM ¼ D · f · S; ð13Þ some representatives of the family the space becomes singular enough to allow for a topological transition. where the superscript refers to “massless,” and we may This means that at leading order in the physics, the space therefore rewrite gaγγ in Eq. (11) as can be explored by movement along flat directions of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi scalar potential to reach the singular point, from which a QM · K−1 · QM transition may be made to a different topological type g γγ ¼ ; ð14Þ (node) of geometry and accordingly a different EFT. a 2M Q τ p · Mathematically, this process may be done continuously in moduli space, as theorems relating canonical singular- for a fixed gauge group specified by the homology class Q. ities and the Weil-Petersson metric ensure that the paths Finally,p weffiffiffiffi comment briefly on the string scale through the graph are at finite distance in moduli space. See ≃ V Ms Mp= . For reasons that we will discuss, obtaining [19,33–35] for further discussion. control over the string effective theory at large N requires The tree ensemble, while enormous in size, has certain going to regions in Kähler moduli space where a nontrivial tractable aspects, due to detailed understanding of the number of four-cycle volumes is large. This correlates construction algorithm. In particular, the geometric gauge strongly with a large overall volume V, which generally group can be determined to high accuracy. Such a set gives rise to an intermediate string scale at large N in our provides a rich ensemble in which to address questions 12 15 ensembles, Ms ≃ 10 –10 GeV. However, since it is about distributions of effective field theories in string divisor volumes that more readily appear in our gaγγ theory. The tree ensemble is constructed by starting with calculations and the geometry does not necessarily have a weak Fano toric variety B and performing blowups of B the Swiss cheese property, we prefer to think of the that satisfy sufficient conditions to remain at finite distance relationship between Ms and gaγγ as correlative, rather in moduli space. Such blowups can be performed over toric

106010-4 TOWARDS STRING THEORY EXPECTATIONS FOR PHOTON … PHYS. REV. D 100, 106010 (2019) points or curves, and each sequence of blowups of a In case 1, where we compute the contribution to gaγγ ð2Þ particular toric point or curve in X is called a tree. A from SU L, which we denote gaWW, the coupling to the typical EFT from the tree ensemble has a minimal gauge physical photon is computed as group of the form 2 2 gaγγ ¼ gaWW sin θw þ gaYY cos θw; ð16Þ

≥ 10 18 9 H2 H3 H4 ð Þ 2 G × × U × F4 × × A1 ; 15 where θw is the Weinberg angle (sin θw ≃ 0.23), and gaYY ð1Þ is the coupling to U Y. In only this case it is possible that g ≠ g where U is a B-dependent gauge group and H are aYY aWW. However, as modeling the hypercharge in i our ensembles is nontrivial, we simply assume that the computable B-dependent integers that are almost always hypercharge contribution to g γγ does not lead to significant nonzero. Other gauge groups can be tuned, but the group a given in Eq. (15) is required by the geometry (though some cancellation. This would require gaWW and gaYY to be of of the gauge factors could be broken by the introduction of similar order of magnitude and opposite sign. As long as this is not the case, which we find plausible, jgaWWj gives G4 flux). j j In addition, an overwhelming fraction of EFTs from the an approximate lower bound to gaγγ . tree ensemble has a large number of ALPs. The expected number of ALPs is N ¼ 2483. This value is determined by B. Calabi-Yau hypersurfaces in toric varieties a bubble cosmology model on the tree ensemble, where the In addition to the tree ensemble, we consider Calabi-Yau vacuum transitions are modeled by the topological tran- hypersurfaces in toric varieties [36,37]. Such a hyper- sitions between the geometries [35] (drawing from a flat surface combinatorially corresponds to a triangulated distribution gives similar results, with the preferred value reflexive 4D polytope. Systematically orientifolding a large of N ¼ 2015). number of Calabi-Yau threefolds is beyond the scope of this From the expected gauge structure of the tree ensemble work, and so we use the geometric data obtained from the discussed in the previous section it is clear that there are in Calabi-Yau itself as a model for the appropriate N ¼ 1 principle many ways to realize the Standard Model in the data, namely in constructing K−1. For the hypersurfaces tree ensemble. In addition to the minimal gauge structure case, we assume that the gauge group (see the listed options given in Eq. (15) one can tune additional gauge groups, and in Sec. III A) is supported on the restriction of a divisor Q additional Uð1Þ factors could be realized by sections in the that is a linear combination of toric divisors to the hyper- elliptic fibration. surfaces. There is, a priori, an infinite number of choices For understanding the couplings of the ultralight ALPs to for such a linear combination, but physical considerations ð2Þ ð1Þ the photon, the relevant gauge sector is SU L × U Y. will render this set finite and computable, as we will discuss Since pure Abelian factors have not been studied in the tree in Sec. IV. ensemble, we will consider two cases of gauge sectors. ð2Þ ð1Þ First, one could try to realize SU L × U Y by embed- C. Random matrix EFTs ding it in a larger non-Abelian gauge group, for instance a For the sake of comparison to our string results, we will GUT, in which case we will model the coupling g γγ a also compute ALP-photon couplings in certain random directly as the coupling of the ALP to the larger group, matrix (RM) effective field theories. We emphasize, assuming it breaks to the Standard Model in one of the though, that we do not currently have a reason to believe canonical ways. The second case is realizing SUð2Þ L that the random matrix ensembles that we study accurately directly from the geometrically determined SUð2Þ gauge represent actual string data [38]. Instead, we simply wish to symmetry on the seven-, in which case we will compare and also to demonstrate that they can give rise to compute the contribution to g γγ from SUð2Þ . The cases a L growing ALP-photon couplings as a function of N. This are summarized as follows: lends some further credence to the idea that g γγ should (1) G ¼ SUð2Þ arises directly on a seven-brane. a L increase with N. (2) G is a geometric gauge group on a seven-brane such For our random (RMT) analysis we will that G → SUð2Þ × Uð1Þ × G0,butG0 ⊅ SUð3Þ; L Y consider the simplified Lagrangian of massless ALPs and i.e., QCD comes from a different seven-brane. gauge fields, (3) G is a gauge group on seven-brane such that G → SUð2Þ × Uð1Þ × SUð3Þ; this includes some M2 1 1 L Y L ¼ − p ð∂μθiÞð∂ θjÞ − μν − θi ˜ μν common GUT scenarios. 2 Kij μ 4 F Fμν 4 Qi F Fμν; In the last option the ALP linear combination that couples ð Þ ð2Þ 17 to SU L is the QCD axion, while in the first two options this is not necessarily the case. Since all three possibilities where Fμν is the electromagnetic field strength. In a lead to similar results, the forthcoming plots of ALP- canonically normalized frame the ALP-photon coupling photon couplings take into account all three. is simply

106010-5 HALVERSON, LONG, NELSON, and SALINAS PHYS. REV. D 100, 106010 (2019) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −1 2. The Kähler cone gaγγ ¼ Q · K · Q: ð18Þ In a complete calculation, the VEVs of the τi are A natural model for K (K−1) is to draw it from an (inverse)- determined by moduli stabilization. We make the Wishart distribution, as we generally observe that the assumption that the τi are stabilized in a regime of non- entries of K (K−1) shrink (grow) as a function of N. perturbative control. In particular, we require the volumes Taking K to be a Wishart matrix, we have K ¼ A†A, with of all curves to be greater than or equal to unity. This region the entries of A drawn from a normal distribution Ωð0; σÞ is known as the stretched Kähler cone, and as observed in centered around zero with standard deviation (SD) σ. This [25], is very narrow at large h1;1ðBÞ.Asin[24], where the in turn makes K−1 an inverse-Wishart matrix, which in tree ensemble was explored in the context of axion practice is easier to generate directly than to invert K.We reheating, we evaluate the τi at the apex of the stretched take Q to be a unit vector and study two cases: the first Kähler cone, defined by minimizing the sum of the toric being the (unnormalized) entries of Q drawn randomly curves. Since the cone is narrow, and the couplings in from the distribution above, and the second taking Q to be a Eq. (8) are invariant under scaling out in the cone via unit vector pointing in a basis direction Q ¼ eˆi. J → λJ, we expect this point to be a good representative of the physics. In addition, as one scales out J → λJ the cycle IV. DISTRIBUTIONS OF ALP-PHOTON volumes increase, and so more ALPs will become light, COUPLINGS IN THE ENSEMBLES which would enhance the couplings of the gauge groups to the ultralight ALPs, and so performing this analysis should We wish to understand the distribution of g γγ in the a provide a lower bound on the size of the couplings in the generic case, i.e., when N is large. Computing g γγ for the a stretched Kähler cone. largest N regime of our ensembles in a large number of examples is computationally prohibitive, and so in order to explore the large N regime we will study gaγγ for moder- 3. ALP masses and gauge couplings ately large N, and then extrapolate to the largest N, where With the EFT data in hand, we proceed to compute gaγγ. the bulk of the geometries are believed to occur (this has in Since we are interested in the effects of the ultralight ALPs fact been demonstrated explicitly in a cosmological model we only need to determine which ALPs will have masses on the tree ensemble, and there is strong evidence for this in much less than any relevant experimental energy, which we the case of hypersurfaces [39]). We will find that in fact the take to be 10−12 eV. Recall that for any ALP θi, any term in relevant statistical quantities in the distributions of gaγγ the potential is accompanied by an exponentially sup- obey nice scaling properties. pressed prefactor expð−2πτiÞ, and therefore the θi who have τi ≫ 1 will have very small masses. In the full N ¼ 1 A. The EFT computation SUGRA potential such terms are accompanied by prefac- tors involving inverse powers of V, the constant term W0 1. Constructing the ensembles arising from the Gukov-Vafa-Witten (GVW) flux super- To construct the ensemble of geometries to study we will potential [40], and the various two- and four-cycle volumes, randomly draw geometries from the various ensembles. For and such prefactors are generally ≪ Oð1Þ in the stretched the tree ensemble, we construct 1000 geometries for one Kähler cone (see [25]). In general, linear combinations of through ten trees each, corresponding to h1;1ðBÞ ranging divisors can contribute to the superpotential, so to give an from 55 to 235, differing by jumps of Δh1;1ðBÞ¼20. For a upper bound for the ALP mass one should consider a fixed number of trees we randomly draw a tree configu- generating set of divisors such that all effective divisors can ration from the ensemble and blow up a randomly drawn be expressed as non-negative integer linear combinations of toric point with that sequence of blowups. The configura- that set. For a toric variety the toric divisors generate the tion of blowups fixes the topological properties of B. cone of effective divisors, known as the effective cone.1 We For the case of hypersurfaces, we randomly draw can therefore give an upper bound on the mass scale reflexive 4D polytopes from the Kreuzer-Skarke ensemble generated for an ALP θi of the form [37] and compute the pushing triangulation of each poly- 1 tope to calculate the relevant topological data for the 2 −2πτi ≲ =Ci ð Þ mi 2 e ; 19 corresponding Calabi-Yau hypersurface. We select 1000 fmin geometries for h1;1ðBÞ¼10, 20, 30, 40, 50, 80, 120, 160. 1;1 For h ¼ 200 there are only 706 polytopes; we utilized all where fmin is the smallest ALP decay constant (square root of them. of the smallest eigenvalue of K), the τi are the volumes of Having calculated the relevant topological data, in order to write down the effective field theory for the ALPs and 1In the hypersurface case we work under the assumption that i gauge sectors, we need to specify the VEVs of the τ . This the effective cone of the ambient space provides a good is done by choosing a point in the Kähler cone. approximation to the effective cone of the hypersurface.

106010-6 TOWARDS STRING THEORY EXPECTATIONS FOR PHOTON … PHYS. REV. D 100, 106010 (2019) the generators of the effective cone, and C are geometry c2 m3 i Γ ¼ ð Þ i;α i ð Þ dependent constants that range from 1 to 30 and are often ϕi→gg dim G 64π : 21 2 dual Coxeter numbers. The inverse factor of fmin provides the weakest upper bound of the canonical normalization The coupling may be written effects. In practice, many θi have their corresponding τi ≫ 1, and so we can treat such ALPs as massless.     1 3=2 8 1=2 −1 −1=2 eV min Concretely, this means that we compute the upper bounds c α ≈ð16.6 GeV Þ ×dimðGÞ : τ i; m Γ−1 on the masses using (19) and the values of fmin and i in a i ϕi→gg geometry, and call the ALP θ massless if the associated i ð22Þ bound is below a fixed mass threshold. For instance, in the tree ensemble the fraction of axions with masses −1 ≤10−12 eV grows from 0.37 at N ¼ 55 to 0.46 at N ¼ 195, Therefore, the coupling has to be Oð10Þ GeV for an at which it becomes approximately constant. eV ALP produced in the Sun to decay before it reaches the Keeping only these nearly massless ALPs, i.e., those Earth. In all of our ensembles, the couplings of ALPs to any with a mass upper bound below 10−12 eV, we canonically of the gauge sectors are orders of magnitude less than −1 −1 normalize the ALPs and gauge fields. Not every gauge 10 GeV , and therefore premature decays of ALPs will group is a viable candidate for the SM as mentioned in not be a concern. The ALPs will not decay before reaching Sec. III A. In particular, the gauge coupling in the UV must the Earth. be large enough to produce the correct low-energy gauge couplings. For a gauge group supported on a cycle given by B. Results Qi, we have the relation 1. Results for the tree ensemble 1 In Fig. 1 we show the normalized distributions of i ð Þ ¼ 55 ≃ Qiτ ; ð20Þ log10 gaγγ ×GeV for our smallest N and largest g 2 UV N ¼ 235. Clearly the distribution shifts to the right as N grows. In order to extrapolate to even larger values of N, which and so any cycle with Q τi very large gives rise to very i is often argued to be the location of the most vacua in the weak gauge couplings that are not consistent with well- string landscape, we will determine quantitative properties studied models and the observed gauge couplings. For α ≃ 0 03 of the N dependence of the distribution. In particular, we instance, the SUSY GUT value is UV . , which find that both the mean and the SD of log10ðg γγ × GeVÞ corresponds to Q τi ¼ 2.7. Leaving some room for model a i have well-behaved N scaling, as shown in Fig. 2, and are building, we demand that α is not more than 1 order of UV therefore suitable to use for such an extrapolation. magnitude smaller than the SUSY GUT value, and there- ð Þ i We fit a power law to the mean of log10 gaγγ ×GeV as a fore we impose the cutoff Qiτ ≤ 25. In particular, in the function of N. Expressed in terms of g γγ, we find hypersurface ensemble we study SM candidates on linear a combinations Qi of toric divisors whose volume satisfies i Qiτ ≤ 25, while in the tree ensemble we study SM candidates on toric divisors with non-Higgsable clusters with volume ≤ 25. For each gauge group satisfying this condition, we then compute the coupling of the corresponding ultralight ALP. We present the results in the next section.

4. ALP decays to other sectors A generic ALP in our setup couples to a large number of gauge sectors (other than the visible sector) via terms of the i ˜ μν form ci;αϕ Gα Gαμν for fixed α (note that this coupling is written for canonically normalized fields). An experimental FIG. 1. The normalized distributions of log10ðgaγγ × GeVÞ for concern arises if the ALP decays too quickly to another our smallest N ¼ 55 and largest N ¼ 235, as well as the sector; for instance, in helioscope experiments the ALPs extrapolated distribution for the preferred value of N ¼ 2483 may in principle decay into a dark sector before it travels in the tree ensemble. There is a clear shift of the distribution from the sun to Earth. The decay rate of an ALP of mass mi towards larger values as N grows. Current (solid) and projected to a pair of dark in one of these sectors is given by (dashed) exclusion lines are presented for various experiments.

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(a)

FIG. 3. A plot of meanðlog10ðgaγγ ×GeVÞÞ versus meanðlog10 × ðλ ð −1ÞÞÞ ∼0 38 max K . The slope of the line is . .

from Eq. (8) that gaγγ is proportional to the norm of a vector Qi, computed with K−1. Since we are imposing Q · τ ≲ 25 so that the cycle in question can support realistic SM gauge couplings, the denominator of Eq. (14) is essentially a constant, ranging from 1 to 25, compared to the large hierarchies that may arise in the numerator. In (b) Fig. 3, we show a plot of meanðlog10ðgaγγ ×GeVÞÞ versus ð ðλ ð −1ÞÞÞ λ ð −1Þ FIG. 2. Top: The mean of log10ðgaγγ × GeVÞ in the tree mean log10 max K , where max K is the largest ensemble as a function of N, with a best-fit curve. Bottom: eigenvalue of K−1. Figure 3 shows a clear correlation, ð Þ The SD of log10 gaγγ × GeV in the tree ensemble as a function of which can be explained as follows: if the largest eigenvalue N, with a best-fit curve. of K−1 grows with N, and if Qi has nontrivial overlap with the corresponding eigenvector that does not shrink too quickly with N, then we will find N-dependent growth of ð Þ¼2 73 10−18 1.77 −1 ð Þ mean gaγγ . × × N GeV ; 23 gaγγ with N. This is indeed the case. In Fig. 4 we show the ð ðλ ð −1ÞÞÞ N dependence of mean log10 max K with N. Clearly which gives the dependence of the coupling on the number λ ð −1Þ max K grows rapidly with N. In addition, we find that of ALPs. We note here that “mean” indicates that we have i ð Þ the alignment of the Q with the corresponding eigenvector fit the mean of log10 gaγγ ×GeV , as opposed to gaγγ itself. ˆ ¼ 55 vmax shrinks slowly as a function of N:atN we find Given the excellent fit, it is reasonable to extrapolate the ð ðˆ ˆ ÞÞ ¼ −1 72 ¼ 235 ð Þ mean log10 vmax · Q . , while at N we find mean of the distribution of log10 gaγγ ×GeV to the ˆ meanðlog10ðvˆ · QÞÞ ¼ −2.59. Therefore, the growth of preferred value of N in the tree ensemble, at N ¼ 2483. max λ ðK−1Þ dominates over the decreasing vˆ · Qˆ , explain- At this value the predicted mean of log10ðg γγ ×GeVÞ is max max a ing the observed correlation. −11.50, which is much larger than the mean at small-to- moderate N. In order to estimate the distribution itself at N ¼ 2483 we assume that the distribution is modeled by a Gaussian, with the mean and standard deviation obtained by extrapolating the curves in Fig. 2 to N ¼ 2483. The expected distribution is shown in Fig. 1, along with its smaller-N counterparts, for the sake of comparison. For the tree ensemble at large N, the distribution sits right on the edge of experimental sensitivity, depending slightly on the assumptions and experiment. We will present a thorough analysis of our results relative to experimental results and prospects in the discussion, since it will be useful to also compare results across ensembles. We would like to understand the origin of this result. It is ð ðλ ð −1ÞÞÞ clear from our analysis so far, and from Fig. 1, that as N FIG. 4. A plot of mean log10 max K versus N, with a increases so does the mean of log10ðgaγγ ×GeVÞ. Recall best-fit line to demonstrate the correlation.

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From a random matrix perspective, constructing a RMT For the hypersurface at large N ¼ 491, the mean of the ensemble whose largest eigenvalue reproduces the scaling projected distribution is at log10ðgaγγ ×GeVÞ¼−9.71, and in Fig. 4 seems reasonable, as one could choose a Wishart so a significant portion of the distribution is already in ensemble whose distribution of entries reproduced such tension with data; see the discussion below. a scaling. However, given that Qˆ tend to point along the standard basis eˆ directions in the natural geometric basis, it ˆ is clear that the corresponding eigenvector vmax does not 3. Results for RMT obey standard eigenvector delocalization,p asffiffiffiffi we would As described in Sec. III C we model K−1 as an inverse ˆ 1 expect the entries of vmax to be roughly = N. Wishart matrix (or K as a Wishart matrix) drawn from Ωð0; σÞ, for some σ, and Q a unit vector either pointing in a 2. Results for hypersurfaces basis direction or with entries drawn from Ωð0; σÞ. We find We perform the same analysis for hypersurfaces and find that the different models for Q produce essentially the quite similar results. In particular, we find that the mean of same behavior, so we will focus on the case that Q ¼ð1; 0; …; 0Þ. In the simplified model the coupling log10ðg γγ ×GeVÞ increases as a function of N, even more a of ALP to the photon is given by Eq. (18). rapidly than in the tree ensemble, and such an increase can −1 In Fig. 7 we show log10ðgaγγ ×GeVÞ computed in the be correlated with the maximal eigenvalue of K . −1 RMT models versus meanðlog10ðλ ðK ÞÞÞ, to demon- In Fig. 5 we show the distributions for our smallest and max largest N that we analyze in the hypersurface case, which strate the similar behavior to the string ensembles studied ð ðλ ð −1ÞÞÞ are N ¼ 10 and N ¼ 200, respectively. In the same figure above. Implicitly in the plot mean log10 max K is we also show the projected distribution for the largest N in increasing with N, and each data point is at a different N, the hypersurface ensemble, which is N ¼ 491. There is a ranging from 0 to 1500. Comparing the growth of ð ð ÞÞ striking feature in N dependence of the distribution in the mean log10 gaγγ ×GeV with respect to both N and the hypersurface case as compared to the tree ensemble: while maximum eigenvalue of K−1, we note that the inverse 2 the standard deviation of the distribution of log10ðgaγγÞ Wishart ensembles with σ ¼ 1=N and σ ¼ 1=N are the decreased as a function of N for the tree ensemble, in the closest fits to the actual string data, though both differ from case of hypersurfaces it actually increases, leading to the it nontrivially. largest spread for large N. This could be due to the This lends further credence to the central idea of this rich intersection structure of Calabi-Yau threefolds as work: gaγγ should grow significantly with N. In particular, compared to toric threefolds, whose intersection numbers if some other string ensemble has a different ALP-photon are relatively tame. We fit a power law to the mean of coupling scaling compared to the ones that we have log10ðgaγγ ×GeVÞ as a function of N. Expressed in terms of studied, these EFT considerations suggest that the coupling gaγγ, we find should nevertheless increase polynomially in N, as shown in Fig. 6. We therefore believe that the central result likely −22 4.22 −1 meanðgaγγÞ¼8.52 × 10 × N GeV : ð24Þ extends to other string ensembles.

FIG. 5. The normalized distributions of log10ðgaγγ ×GeVÞ for the hypersurfaces. Shown are the calculated distributions for N ¼ 10 and N ¼ 200, as well as the extrapolated distribution ð Þ for the largest hypersurfaces at N ¼ 491. There is a clear shift of FIG. 6. The mean ofpffiffiffiffi log10 gaγγ × GeV in the inverse Wishart 2 3 the distribution towards larger values as N grows. Current (solid) model for σ ¼ 1; 1= N; 1=N; 1=N ; 1=N as a function of N, ∝ x and projected (dashed) exclusion lines are presented for various with a best-fit line of the form mean(gaγγ) N . The models with experiments. σ ¼ 1=N2 and σ ¼ 1=N are the best for the string data.

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computation for reasons of complexity, it is interesting to check what the distribution of gaγγ would be if we were able to extrapolate outside of the tree ensemble to such an boundary point of the skeleton ensemble. We extrapolate using the fit of the related tree ensemble in Fig. 8. Another important example is the geometry Bmax, which is the F-theory base thought to house the largest number of flux vacua [15]. In this geometry the number of ALPs 9 8 is 98, and the non-Higgsable gauge group is E8 × F4 × 16 ðG2 × SUð2ÞÞ . At the apex of the stretched Kähler cone there is only a single cycle with volume τ that supports a gauge group and satisfies τ ≤ 25. This cycle supports an E8 group that could provide a GUT after flux breaking. For −12 −1 this gauge group we find gaγγ ¼ 3.47 × 10 GeV .

FIG. 7. log10ðgaγγ × GeVÞ, computed in the RMT models, This coupling is also projected to be probed at future −1 versus meanðlog10ðλmaxðK ÞÞÞ, with varying σ. Note that for experiments. −1 all models both log10ðgaγγ × GeVÞ and meanðlog10ðλmaxðK ÞÞÞ increase as a function of N. V. CURRENT AND FUTURE EXPERIMENTS Even though they are not relevant for our analysis, for the 4. Remarks on other ensembles sake of thoroughness we will begin by discussing experi- While the tree ensemble of F-theory bases and the ments that assume that ALPs contribute nontrivially to the Kreuzer-Skarke (KS) ensemble of Calabi-Yau (CY) hyper- dark matter and/or are sensitive only to ALPs in some finite surfaces are the largest-to-date explicit ensembles of string mass region. Currently, the best limits on the existence of ≲ 1 geometries, there are other important examples that we now light (ma eV) axionlike particles comes from resonant- discuss. cavity haloscope experiments. The bounds are restricted to ≃ 2 5 The first is that of the skeleton ensemble [21], which narrow mass windows centered around ma . × 10−6 ≃ 2 5 10−5 generalizes the tree ensemble beyond the simple sufficient eV for ADMX [41] and ma . × eV for criteria to remain at finite distance in moduli space. HAYSTAC [42]. Both experiments place an impressive ≲ 10−15 −1 Actually, in this case the construction of the geometries limit on the coupling of gaγγ GeV . However, the B is a less-restrictive version of the tree ensemble, and so effectiveness of these searches is highly contingent on the we expect the results to exhibit similar scaling. However, assumption that the ALP in question comprises the totality the largest number N of ALPs found in this ensemble is of the local dark matter halo, as the signal scales linearly 16103, which is a great deal larger than the maximal N with the local halo density of ALPs. String theory generally found in the tree ensemble. While this result should be provides a wealth of potential dark matter candidates, and taken with a grain of salt, since we do not perform the even if an ALP were to represent the bulk of the local dark matter density, it would be a remarkable coincidence if this particle was to also couple to photons in an appreciable manner. Considering the above shortcoming, we are thus led to consider experiments which are both broadband in sensi- tivity and independent of assumptions about the ALP contribution to local dark matter. The so-called “light- shining-through-walls” (LSW) experiments achieve both, with the DESY-based ALPS [43] and the CERN-based −8 −1 OSQAR [44] achieving a limit of gaγγ ≲ 6 × 10 GeV for ALPs with vanishing masses up to a threshold of −4 ma ≃ 10 eV. The proposed ALPS-II experiment [45] hopes to achieve a sensitivity to couplings of order gaγγ ≲ 2 × 10−11 GeV−1 in the very near future. Such a limit will be competitive with the current bound set by the CERN FIG. 8. The normalized predicted distribution of log10ðgaγγ × helioscope CAST, which constrains the coupling to g γγ ≲ GeVÞ for the largest known value of N ¼ 16103 in the skeleton a 7 10−11 −1 ensemble, contrasted with that predicted for the largest N ¼ 2483 × GeV for ALPs with vanishing masses up to a −2 in the tree ensemble. Current (solid) and projected (dashed) threshold of ma ≃ 10 eV. This is the current best limit for exclusion lines are presented for various experiments. ALPs in the broad mass range of interest to this paper.

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Both of the techniques mentioned above should see Our main results in each ensemble are that the mean improvements in future years. The International Axion value of the gaγγ distribution (gaγγ) scales polynomially in Observatory (IAXO), a next-generation helioscope, should N, and its extrapolation to large N, where most vacua are −12 −1 reach gaγγ ¼ 2 × 10 GeV [3], with a prototype taking argued to occur, is near the sensitivity of current or data within the next few years, while next-generation proposed experiments. We take each point in turn. superconducting radiofrequency cavities hope to reach a In both string ensembles we studied,pffiffiffiffi the scaling gaγγ was similar limit on the same timescale [4]. significantly stronger than the N scaling that one might Finally, we remark on the ability of astrophysical naively expect from Eq. (11). This is due to at least two observations to constrain ALP couplings to photons. effects. First, the number of ALPs contributing to the Oscillations in the spectrum of observed x-ray and effective gaγγ depends not only on N, but also the fraction of gamma-ray photons from extragalactic sources may be ALPs with masses below the chosen threshold; the latter induced by photon-ALP conversion in the presence of large itself increases with N [25]. Second, due to canonical galactic magnetic fields. This technique can be applied for normalization, g γγ depends critically on the largest eigen- ≲ 10−9 a very low ALP masses (ma eV), with the best limits value of K−1, which grows significantly with N; see Figs. 3 ≲ 10−12 coming from x-ray astronomy for ma eV. The and 4. Specifically, the two string ensembles that we ≲ 8 10−13 −1 most stringent current limits of gaγγ × GeV studied were called the tree ensemble and the hypersurface come from observations of a number of point sources by 1.77 4.22 ensemble, and they exhibited gaγγ ∝ N and gaγγ ∝ N , the Chandra satellite [10]. In the next decade, both the respectively. We note that g γγ can also be fit accurately to a European Space Agency (ATHENA) [11] and NASA a weak exponential, but in the absence of any reason to expect (STROBE-X) [14] have proposed missions to improve exponential growth, we opt for the more conservative choice. these limits by an additional order of magnitude. In both string ensembles that we studied we found that gaγγ, extrapolated to the expected large value of N in each VI. DISCUSSION ensemble, is within range of current and/or proposed experiments. Specifically, with contributing ALPs having In this paper we studied distributions of ALP-photon −12 m ≤ 10 eV, couplings in ensembles of string compactifications and a random matrix theories. Since hundreds or thousands −12 −1 tree∶ g γγ ¼ 3.2 × 10 GeV at N ¼ 2483; of axionlike particles typically arise in string compactifi- a ∶ ¼ 2 0 10−10 −1 ¼ 491 ð Þ cations, we studied the scaling of the effective gaγγ hyper: gaγγ . × GeV at N : 25 distribution with N and its extrapolation to the expected value of N in a given ensemble. We also computed gaγγ in the F-theory geometry with the Before stating the main results, we review the setup. most flux vacua, where the requirement τ ≤ 25 within the The details of string geometry, which in our cases are stretched Kähler cone allows the SM to exist only on a Calabi-Yau manifolds in IIB compactifications or Kähler single divisor, which carries geometric gauge group E8.In −12 −1 threefold bases of F-theory compactifications, determine that example, gaγγ ¼ 3.47 × 10 GeV . the coupling of a linear combination of ALPs to gauge These results should be compared to the strongest sectors arising on a four-manifold parametrized by an current and projected constraints from experiments con- integral vector Q. Specifically, the string data that enter sistent with the assumptions of our analysis: that the ALPs the calculation are the choice of manifold from the do not have to (but could) comprise the dark matter, and ensemble, taken to be at the apex of the stretched that the masses are below a threshold rather than in a fixed Kähler cone, the computation of the Kähler metric Kij window. For experiments satisfying these assumptions, the on ALP kinetic terms that plays a crucial physical role in strongest current constraints are canonically normalized couplings, and choices of Q that in −11 −1 −2 principle allow for the Standard Model gauge couplings. CAST∶ gaγγ ≤ 7.0 × 10 GeV for ma ≤ 10 eV; ALP masses are generated by corrections to the −13 −1 −12 Chandra∶ g γγ ≤ 8.0 × 10 GeV for m ≤ 10 eV; superpotential, and we utilize a natural geometric pro- a a cedure that gives an upper bound on the ALP mass. The ð26Þ effective ALP-photon couplings that we study include only the contributions from individual ALPs whose mass upper and the strongest projected sensitivities in proposed experi- bound is below an experimentally relevant threshold ments are 10−12 eV. This conservative choice allows our calculations ∶ ≤ 2 10−12 −1 ≤ 10−2 to also apply to experiments that are sensitive to ALPs IAXO gaγγ × GeV for ma eV; below masses of 10−6 eV (e.g., [4]) and 10−2 eV (e.g., −14 −1 −12 STROBE-X∶ gaγγ ≤ 8 × 10 GeV for ma ≤ 10 eV: helioscopes). Further details are, of course, provided in the ð Þ main text. 27

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We note the proximity of the computed gaγγ in models of Nevertheless, this result, taken together with the ubiquity ALP-photon couplings in string compactifications to the of ALPs in string compactifications and the necessity of the experimental sensitivities. photon in the correct theory of quantum gravity, provides Our study provides the first concrete evidence that excellent motivation for future work. ultralight ALPs in string compactifications could very well have detectable interactions with photons at large N. Given this strong claim, we would like to clearly ACKNOWLEDGMENTS emphasize three potential caveats. First, this is a large N We thank Joe Conlon, Caterina Doglioni, Yoni Kahn, effect, and though there are compelling arguments that most vacua exist at large N, it could be that our vacuum is Sven Krippendorf, Liam McAllister, Kerstin Perez, and Ben Safdi for valuable discussions. Portions of this work realized in the small N regime. Second, the quoted gaγγ values are the mean of a projected distribution, and the were completed at the Aspen Center for Physics, which is photon associated with our vacuum could be realized in the supported by National Science Foundation Grant No. PHY- tail. Third, the precise details of the distribution could be 1607611. J. H. is supported by NSF Grant No. PHY- modified in more precise studies that attempt to explicitly 1620526 and NSF CAREER Grant No. PHY-1848089. realize Standard Model sectors and stabilize moduli, rather B. N. and G. S. are supported by National Science than modeling them, as we have. Each of these caveats is Foundation Grant No. PHY-1913328. deserving of careful study.

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