Couplings between QCD axion and photon from string compactification

The QCD axion couplings of various invisible axion models are presented. In particular, the exact global symmetry U(1)$_{\rm PQ}$ in the superpotential is possible for the anomalous U(1) from string compactification, broken only by the gauge anomalies at one loop level, and is shown to have the resultant invisible axion coupling to photon, $c_{a\gamma\gamma}\ge \frac83-c_{a\gamma\gamma}^{\rm ch\,br}$ where $c_{a\gamma\gamma}^{\rm ch\,br}\simeq 2$. However, this bound is not applicable in approximate U(1)$_{\rm PQ}$ models with sufficiently suppressed U(1)$_{\rm PQ}$-breaking superpotential terms. We also present a simple method to obtain $c_{a\gamma\gamma}^0$ which is the value obtained above the electroweak scale.


Introduction
The detection possibility of the invisible axions [1][2][3] chiefly relies on its appreciable couplings to photon c aγ γ which appears in the Lagrangian as L axion coupling = − a 32π 2 f a c 3 g 2 3 G aGa + c aγ γ e 2 F emF em , (1) where The axionic domain-wall number [4] is |c 3 + c 2 | where c 2 is the contribution from the standard model quarks [5]. With this normalization from the QCD sector, the coupling c aγ γ is defined and is composed of two parts, c aγ γ = c 0 aγ γ + c ch br aγ γ c 0 where c 0 aγ γ is the one obtained above the electroweak scale and c ch br aγ γ is the contribution obtained below the QCD chiral phase transition scale. Since the mass ratio of up and down quarks is very * Corresponding author.
E-mail address: jihnekim@gmail.com (J.E. Kim). close to 0.5 [6], we use the value m u /m d = 0.5 below. In this case, c ch br aγ γ is −2 (a bit smaller value −1.98, including the strange quark contribution) [7]. The early summary on the axion-photon-photon couplings was summarized in [5,8]. 1 An invisible axion is a pseudo-Goldstone boson whose mother symmetry is the Peccei-Quinn (PQ) symmetry [10]. The symmetry breaking scale relevant for the axion detection experiments [11,12] is the intermediate scale 10 10 GeV f a 10 12 GeV, which can be achieved by the vacuum expectation value (VEV) of an SU(2) × U(1) Y singlet σ [1]. But, the global symmetry which is broken at the intermediate scale has to be fine-tuned to avoid the gravity spoil of global symmetries. This difficulty has been appreciated [13] after realizing that even the classical gravity does not necessarily preserve global symmetries due to the topology change via wormholes [14,15]. The wormhole taking out gauge charges is depicted in Fig. 1. An observer in the almost flat space O notices that some gauge charges are flown to the shadow world S. To see the effect in his own space only, he cuts the wormhole, then notices that the escaped gauge charges are recovered to O. This conservation of gauge charges in the space O is due to the long range electric flux lines. For the global charges, there is no such flux lines and hence the escaped global charges are not considered to be recovered to O if he cuts the wormhole. Thus, global charges are broken if we consider the topology change. Related to this effect, at field theory level within supersymmetric (SUSY) framework, a host of discrete symmetries are considered [16][17][18][19][20]. Some discrete symmetries can lead to acceptable approximate global symmetries [21,22].
In this paper, we attempt to obtain a region of the parameter space from string compactified 4-dimensional (4D) effective field theory. The 4D models from string compactification do not allow global symmetries but allow some discrete symmetries [23]. The minimal supersymmetric standard model supplied with singlets σ (to house the invisible axion as pointed out in [1]) will be called σ MSSM. In the σ MSSM, we calculate the couplings between axion and photon.

Gauge transformation of model-independent axion
Four dimensional pseudoscalars in the σ MSSM from string compactification appear from B MN (= 10D antisymmetric tensor field with M, N ∈ {1, 2, · · · , 10}) and from the matter supermultiplets in the σ MSSM. In 10D, B MN is a gauge field satisfying the gauge transformation, where M are gauge functions. If both M and N take the internal space coordinates i, j = {5, 6, · · · , 10}, B ij is a pseudoscalar in 4D. From the 4D point of view, the original gauge transformation is not local, not carrying the 4D index μ = {1, 2, 3, 4}. These pseudoscalars are the model-dependent (MD) axion [24], which is known to generate superpotential terms [25]. So, the MD axions are not the useful candidates for solutions of the strong CP problem. On the other hand the model-independent (MI) axion [26,27], where both M and N of B MN take 4D indices μ, ν, is a good candidate of 4D gauge transformation. Namely, the gauge transformation (4) is still a gauge transformation in 4D. So, B μν is not spoiled by gravity after the compactification.

Anomaly-free U(1) gauge symmetry
If the MI axion is not behaving as a longitudinal degree of a gauge boson, then the axion potential is generated and the bosonic collective motion behaves as cold dark matter (CDM) [12].
But, then the axion decay constant is near the string scale, f a > 10 16 GeV [28], and a fine-tuning is needed, or the anthropic scenario must be invoked [29]. The coupling c aγ γ is the same as the one considered in the following subsection with anomalous U(1) without extra charged singlets.

Anomalous U(1) gauge symmetry
If the compactification produces an anomalous U(1) anom gauge symmetry, the corresponding U(1) anom gauge boson obtains a large mass, at a slightly lower scale than the string scale. The presence of a Fayet-Illiopoulos D-term connects the MI axion with the anomalous U(1) anom gauge boson [30], which is a kind of the Higgs mechanism providing the longitudinal degree of the gauge boson. The generator of the anomalous U(1) anom belongs to the E 8 × E 8 algebra, and matter fields have the U(1) anom charges. The field B μν or the MI axion does not couple to matter fields. So below the U(1) anom gauge boson mass scale, the U(1) anom charge of matter fields becomes a global charge which can be called a U(1) PQ charge. In this way, a global symmetry free of the gravity obstruction is created below the U(1) anom gauge boson mass scale. Since the mother U(1) anom is a gauge symmetry, there is no gravity obstruction of this U(1) PQ global symmetry. In string compactification, it has been explicitly shown that the anomalies are the same for all gauge groups both for non-Abelian and (properly normalized) Abelian gauge fields [31,32]. However, this statement holds only for orbifold compactifications, where there is a single axion, whose shift must cancel all anomalies. In smooth compactifications one has many axion fields and therefore universality of anomalies does not hold anymore. There are such examples in the compactification of Type-I and Type-IIB string, where three anomalous U(1)'s were constructed [38]. Therefore, our present study has a limitation arising from the orbifold compactification of the heterotic string. Nevertheless, since the full phenomenologically acceptable σ MSSM spectra of matter fields have been so far presented in the heterotic compactification, our study of anomalous U(1) gauge symmetries can be gates to low energy gravity-safe global symmetries. Now, we restrict to the case of B μν from the heterotic string.
Let the anomalous gauge symmetry be U(1) anom . Its charge operator and the coupling constant be and e , respectively. The potential which is invariant under U(1) anom is also invariant under the global symmetry U(1) whose charge generator is also . To see the effective global symmetry below the anomalous scale, therefore, it is sufficient to see how the local transformation is described. Since the longitudinal degree of the U(1) anom gauge boson is solely provided by B μν , matter scalars having the nonvanishing charge do not develop VEVs. To see the U(1) anom gauge transformation of a complex scalar , consider the kinetic energy term If we consider the global transformation U(1) below the anomalous scale, only the first term survives in the above equation, where we expressed the U(1) anom gauge boson as Z . Below the anomalous scale, it describes a global symmetry U(1) coupling with the heavy anomalous gauge boson with the same charge . In the potential V , this gauge boson coupling respects the global symmetry also. Thus, we obtain an exact global symmetry U(1) below the anomalous scale.
Thus, an intermediate scale global symmetry is from the anomalous U(1) anom in the compactification process of the heterotic string. Since the anomalous U(1) anom has the same coupling to gauge fields, we have the following MI axion coupling, Table 1 The axion-photon-photon coupling for several invisible axion models. The third row in superstring corresponds to the exact global symmetry U(1) PQ = U(1) anom in the superpotential W . The MI axion with f a > 10 16 GeV [28] has the same value as that of the U(1) anom string. The non-SUSY DFSZ models have a fine-tuning problem. One related cosmological problem even within SUSY framework was pointed out in [33].
where Y 1 is the properly normalized U(1) gauge field. Note that . Note that C 2 = 5 3 in the SU(5) model. Here, we used Therefore, we obtain the following coupling for the anomalous case, where we used m u

The weak mixing angle in GUTs with extra U(1)'s
If the electromagnetic charge operator is embedded in a simple GUT group SU(N), the charge operator on the fundamental representation is a traceless matrix, Q em (N) = diag.(a, a, a, 0, −1, b 6 , · · · ), In the Georgi-Glashow (GG) SU(5) model [36], a = 1 3 and b i = 0.
If Q em is completed by the simple SU(N) generators, the information on the fundamental representation is enough since the other higher dimensional representations can be constructed in terms of direct products of fundamentals. At the GUT scale three gauge couplings are the same, and the mixing angle is defined as sin θ W = e/g 2 . For properly normalized generators Q 1 and T 3 in the fundamental representation, the trace is 1 2 , and we have where g 1 = g 2 at the GUT scale. Thus, we obtain For the SU (5) The SU(7) model of Ref. [37] gives sin 2 θ W = 3/20.
. (15) If a vectorlike representation of the form 5 −a + 5 a is present in the model, then Eq. (12) is still applicable [32]. But, if 5 or 5 does not appear in the anomaly-free combination as that from 16, that fundamental representation cannot be used in Eq. (12). The Higgs if one fundamental representation is enough to calculate c 0 aγ γ . Because the SM, represented in Eq. (14), has the contribution from three families the extra charged singlets will make this contribution larger. Therefore, GUTs predict c 0 Thus, we have the excluded region for the case of anomalous U(1) being U(1) PQ in Fig. 2. However, if U(1) PQ is approximate as calculated in a string compactification [34], this bound does not apply.

A simple calculation of c 0 aγ γ from quantum numbers
In this section, we show a simple method for calculating the entries in the DFSZ models in Table 1. Let the invisible axion is housed in the complex singlet σ [1]. The DFSZ model connects the PQ charges of H u and H d to that of σ . One possible connection is H u H d σ 2 , (19) and the PQ charge of σ is assigned to be +1. The mass terms of the up-and down-type quarks are (20) where q L and L are SU (2)

Conclusion
For the exact global symmetry U(1) PQ from string compactification, we obtained the lower bound, 8 3 − c ch br aγ γ , for the axionphoton-photon coupling c aγ γ , where c ch br aγ γ 2, and presented it in Table 1 and Fig. 2 together with other models. This bound is free from the gravity obstruction of global symmetries. However, if U(1) PQ is approximate, this bound does not apply. We based our argument from the orbifold compactification, which has some limitation as stated in Sec. 2. Finally, let us present a caveat. In the effective field theory of orbifold models, due to the Fayet-Iliopoulos term associated to the anomalous U(1) symmetry, one has to give VEVs to some chiral multiplets in order to cancel the Fayet-Iliopoulos piece. On the geometrical side, this can be understood as an instability of the orbifold point and the fields having VEVs as resolution modes smoothening the orbifold singularities. Whether the U(1) PQ is broken or not depends on the number of required VEVed fields to smooth out the orbifold singularity. If there were one gauged U(1) and one global U(1), and only one VEVed field is needed, then a global symmetry remains unbroken by the so-called 't Hooft mechanism [40]. Thus, if the needed number of independent VEVed fields, to smooth out the orbifold singularities, is the same or less than the number of unbroken U(1) gauge groups, then the U(1) PQ survives down to the low energy scale.
Center for Theoretical Physics, for the invitation to the Bethe Forum on "Axions and the Low Energy Frontier" (7-18 March 2016), where this work has been finished. J.E.K. is supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MEST) (NRF-2015R1D1A1A01058449) and the IBS (IBS-R017-D1-2016-a00), and S.N is supported in part by NRF-2013R1A1A2004538.