PHYSICAL REVIEW D 100, 106010 (2019) Towards string theory expectations for photon couplings to axionlike particles James Halverson, Cody Long, Brent Nelson, and Gustavo Salinas Department of Physics, 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 quantum gravity, 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 photons. 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 Standard Model, 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 moduli space 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.