An Ultralight Axion in Supersymmetry and Strings and Cosmology at Small Scales

TL;DR

This paper explores mechanisms within supergravity, string theory, and cosmology to generate ultralight axions around 10^{-21} to 10^{-22} eV, which could serve as dark matter affecting small-scale structures.

Contribution

It introduces a general effective operator framework for ultralight axions and analyzes their realization in supersymmetric and string models, maintaining hierarchy with the weak scale.

Findings

Ultralight axions can be generated via instanton effects in supersymmetry.

Higher dimensional operators naturally produce ultralight axions.

Large Volume Scenario in string theory achieves the desired hierarchy.

Abstract

Dynamical mechanisms to generate an ultralight axion of mass $\sim 10^{-21}-10^{-22}$ eV in supergravity and strings are discussed. An ultralight particle of this mass provides a candidate for dark matter that may play a role for cosmology at scales $10\, {\rm kpc}$ or less. An effective operator approach for the axion mass provides a general framework for models of ultralight axions, and in one case recovers the scale $10^{-21}-10^{-22}$ eV as the electroweak scale times the square of the hierarchy with an $O(1)$ Wilson coefficient. We discuss several classes of models realizing this framework where an ultralight axion of the necessary size can be generated. In one class of supersymmetric models an ultralight axion is generated by instanton like effects. In the second class higher dimensional operators involving couplings of Higgs, standard model singlets, and axion fields naturally lead to an ultralight axion. Further, for the class of models considered the hierarchy between the ultralight scale and the weak scale is maintained. We also discuss the generation of an ultralight scale within string based models. Here it is shown that in the single modulus KKLT moduli stabilization scheme an ultralight axion would require an ultra-low weak scale. However, within the Large Volume Scenario, the desired hierarchy between the axion scale and the weak scale is achieved. A general analysis of couplings of Higgs fields to instantons within the string framework is discussed and it is shown that the condition necessary for achieving such couplings is the existence of vector-like zero modes of the instanton. Some of the phenomenological aspects of these models are also discussed.

Figures

Equations

$$V_a=\tilde{A}{\mathcal{O}}_H{\mathcal{O}}_V\cos (a/F),$$

$$V_a=A\frac{s^{2m}(h^{†}h{)}^{2k}}{{\Lambda }^{4k+2m-4}}\cos (a/F).$$

$$m_a=A^{\frac{1}{2}}⟨h⟩{\left(\frac{⟨s⟩}{⟨h⟩}\right)}^m{\left(\frac{⟨h⟩}{\Lambda }\right)}^{n-1}\left(\frac{\Lambda }{F}\right),$$

$$m_a\simeq {\Lambda }_{EW}{\left(\frac{{\Lambda }_{EW}}{\Lambda }\right)}^{n-1}.$$

$$m_a$$

$$S_1\to e^{i\lambda }{S}_1,S_2\to e^{-i\lambda }{S}_2,$$

$$W_s={\mu }_0{S}_1{S}_2+\frac{{\lambda }_s}{2M}(S_1{S}_2{)}^2.$$

$$S_i$$

$${\varphi }_i=({\rho }_i^0+{\rho }_i)e^{i{a}_i/{\rho }_i^0},i=1,2,$$

$${\mu }_0+\left(\frac{{\lambda }_s}{M}\right)({\rho }_1^0{\rho }_2^0)=0.$$

$${\varphi }_i=(F+{\rho }_i)e^{i{a}_i/F},i=1,2.$$

$$a_{\pm }=\frac{1}{\sqrt{2}}(a_1\pm a_2).$$

$$V$$

$$W$$

$$W_n$$

$${\mu }_{0}F+(\frac{{\lambda }_s}{M})F^3-\alpha A{e}^{-\alpha F}=0.$$

$$V(a_{-})=2{\alpha }^2{A}^2{e}^{-2\alpha F}{e}^{-\alpha F\cos (a_{-}/\sqrt{2}F)}[1-\cos (\alpha F\sin (a_{-}/\sqrt{2}F))].$$

$$m_{a_{-}}\simeq {\alpha }^{2}A{e}^{-\alpha F}.$$

$$W=W_{vis}+W_{hid}+W_{vh},$$

$$W_{vis}$$

$$W_{hid}$$

$$W_{vh}$$

$${\mu }_0{\rho }_1^0+\frac{{\lambda }_s}{M}({\rho }_1^0{)}^2{\rho }_2^0+\frac{\lambda }{M}{\rho }_1^0{v}_1{v}_2+\frac{c}{M^{n-2}}{v}_0^n=0,$$

$${\mu }_0{\rho }_2^0+\frac{{\lambda }_s}{M}{\rho }_1^0({\rho }_2^0{)}^2+\frac{\lambda }{M}{\rho }_2^0{v}_1{v}_2+\frac{c}{M^{n-2}}{v}_0^n=0,$$

$${\mu }_0^{2}F+\frac{{\lambda }_s}{M}{F}^3+\frac{\lambda }{M}F{v}_1{v}_2+\frac{c}{M^{n-2}}{v}_0^n=0.$$

$$V(a_{-})=4c^2{\left(\frac{v_0^n}{M^{n-2}}\right)}^2\left(1-\cos (\frac{a_{-}}{\sqrt{2}F})\right).$$

$$m_{a_{-}}$$

$$={\Lambda }_{\mathrm{EW}}(\frac{{\Lambda }_{\mathrm{EW}}}{\Lambda }{)}^{n-1},$$

$$W(S)$$

$$2{\mu }_s{S}_0+{\lambda }_0{v}_1{v}_2+\frac{nc}{M^{n-2}}({\rho }_1^0+{\rho }_2^0)S_0^{n-1}=0.$$

$$V_S(a_1,a_2)=|2{\mu }_0{S}_0+{\lambda }_0{v}_1{v}_2+\frac{nc}{M^{n-2}}F(e^{i{a}_1/F}+e^{i{a}_2/F})S_0^{n-1}{|}^2.$$

Full text

**An Ultralight Axion in Supersymmetry and Strings

and Cosmology at Small Scales**

James Halverson111j.halverson@neu.edu  ,   Cody Long222co.long@neu.edu  ,   and Pran Nath333p.nath@neu.edu

 

Department of Physics, Northeastern University, Boston, MA 02115-5000, USA

Dynamical mechanisms to generate an ultralight axion of mass $\sim 10^{-21}-10^{-22}\,{\rm eV}$ in supergravity and strings are discussed. An ultralight particle of this mass provides a candidate for dark matter that may play a role for cosmology at scales $10{\rm kpc}$ or less. An effective operator approach for the axion mass provides a general framework for models of ultralight axions, and in one case recovers the scale $10^{-21}-10^{-22}\,{\rm eV}$ as the electroweak scale times the square of the hierarchy with an $O(1)$ Wilson coefficient. We discuss several classes of models realizing this framework where an ultralight axion of the necessary size can be generated. In one class of supersymmetric models an ultralight axion is generated by instanton like effects. In the second class higher dimensional operators involving couplings of Higgs, standard model singlets, and axion fields naturally lead to an ultralight axion. Further, for the class of models considered the hierarchy between the ultralight scale and the weak scale is maintained. We also discuss the generation of an ultralight scale within string based models. Here it is shown that in the single modulus KKLT moduli stabilization scheme an ultralight axion would require an ultra-low weak scale. However, within the Large Volume Scenario, the desired hierarchy between the axion scale and the weak scale is achieved. A general analysis of couplings of Higgs fields to instantons within the string framework is discussed and it is shown that the condition necessary for achieving such couplings is the existence of vector-like zero modes of the instanton. Some of the phenomenological aspects of these models are also discussed.

1 Introduction

Recently it has been proposed [1, 2] that an ultralight boson candidate for dark matter (sometimes referred to as fuzzy dark matter (FDM)), with mass of $\mathcal{O}(10^{-22})\,{\rm eV}$, can properly explain cosmology at scales of 10 kpc or less444Alternative possibilities for cosmology at small scales include complex dynamics or baryonic physics. However, in this work we focus on the approach involving an ultralight boson.. Such an ultralight particle was identified with an axion555 For early work on axions see [3, 6, 4, 5, 8, 7, 9]. with a decay constant in the range $10^{16}\leq F\leq 10^{18}\,{\rm GeV}$. It was shown that an axion of the size needed could be generated via instanton effects. See [33]-[71] for recent works related to ultralight axions.

We emphasize, as did [1], that this ultralight axion is not the QCD axion. In the latter case the axion mass $m_{a}\simeq\Lambda_{QCD}^{2}/F$ depends on one parameter, since $\Lambda_{QCD}\simeq 200$ MeV is known. For relic QCD axions produced by misalignment, this sets an upper bound $F\lesssim\,10^{12}\,{\rm GeV}$. The axion considered here is another axion, perhaps a string axion (see e.g. [13]), that is not necessarily related to gauge dynamics in any way. Instead, its effective $\Lambda$ is set by non-perturbative effects, such as string instantons, and therefore $m_{a}\simeq\Lambda^{2}/F$ depends on two parameters. This allows for greater freedom in the axion mass and relic abundance, and such axions are ubiquitous in string theory [73].

In this work we discuss explicit models where an ultralight axion can arise. We will study the axion mass scale using effective operators and will account for the scale $O(10^{-22})\,{\rm eV}$ in terms of the electroweak scale and the hierarchy. We will also exhibit the emergence of such a light particle both in supergravity effective field theory and then in the framework of a specific class of string-motivated models. The outline of the paper is as follows: In Section 2 we discuss the general issue of the mass scale of the axion using an effective operator approach. In Section 3 we discuss field theoretic models based on supersymmetry and supergravity that lead to an ultralight axion. In Section 4 we discuss the possibility of realizing the axion within a more general string frameworks. Specifically we consider the KKLT and Large Volume Scenario (LVS) moduli stabilization schemes, and show that the desired hierarchy between the axion scale and the weak scale can be achieved in the latter but not the former. In Section 5 we discuss conditions within string theory that allow the possibility of coupling axions with higher dimensional Higgs operators via D-brane ≠instantons. Phenomenology of these models is discussed in Section 6 and we conclude in Section 7.

2 The mass scale of the ultralight axion

An apparent conspiracy of scales exists [1] between the observed dark matter relic abundance, astrophysical observations, and common properties of axions in string theory. Specifically, if one considers an axion in string theory with string scale decay constant $O(10^{16})$ GeV and demands that misalignment produces an axion relic abundance matching the observed dark matter relic abundance $\Omega h^{2}=.12$, then the axion must be ultralight with mass $m_{a}$ of $O(10^{-22})\,{\rm eV}$. This is the relevant mass scale for accounting for a variety of astrophysical observations, as discussed in [1].

From an ultraviolet perspective, however, it is preferable to turn this logic around: if the mass scale $m_{a}\simeq 10^{-22}{\rm eV}$ could be motivated by theoretical considerations, then a misalignment-produced axion, with string scale decay constant, would give a derivation of the observed relic abundance. In [1] this was achieved by tuning an instanton action to obtain the mass, which is possible in string theory but depends critically on moduli stabilization. In this section we will instead study axion masses utilizing symmetry arguments and effective field theory, motivating $m_{a}\simeq 10^{-22}\,{\rm eV}$.

The effective operator $V_{a}$ in the scalar potential that gives the axion its mass must respect all of the symmetries of the theory. In particular, the axion itself has a perturbative continuous shift symmetry that is expected to be (and typically is in concrete constructions) broken to a discrete shift symmetry by instantons. This consideration leads to $V_{a}\sim\cos(a/F)$. The coefficient of this periodic term must also respect all symmetries of the theory. Since any theory consistent with observations respects at least standard model gauge invariance, it is natural to decompose $V_{a}$ as

[TABLE]

where $\mathcal{O}_{H}$ is a hidden sector operator, and $\mathcal{O}_{V}$ is a visible sector operator that contains only standard model (or MSSM) fields or a standard model singlet $s$ that couple to the Higgs. For this term to give the axion a mass, both coefficient operators must receive vacuum expectation values (VEVs), where one or both could be the identity operator. Defining $A:=\tilde{A}\langle\mathcal{O}_{H}\rangle$ and recognizing that if $\mathcal{O}_{V}$ obtains a VEV it can666In the MSSM we could use $(h_{u,d}^{\dagger}h_{u,d})^{k}$ and similar conclusions would hold. only involve powers of $s$ and $(h^{\dagger}h)$, we write777In supergravity and strings with strong dynamics a fermion condensate of appropriate power could replace the $s^{2m}(h^{\dagger}h)^{2k}$ factor in Equation 2.

[TABLE]

We note that in supersymmetric formulations higher dimensional operators with integer powers in the superpotential will naturally lead to even integer powered higher dimensional operators. For this reason we take the powers of $s$ and of ($h^{\dagger}h)$ to be $(2m,2k)$ where $(m,k)$ are integer or half-integer.

Equation 2 gives rise to an axion mass

[TABLE]

where $n=2k+m$ and where $\Lambda$ is some ultraviolet cutoff. The precise axion mass depends on model dependent details that determine the precise values of $A$, $F$, and $\Lambda$, but if $A$ is not too small and $F$ is near the high scale cutoff, as motivated by string theory, and also $\langle s\rangle\sim\langle h\rangle\sim\Lambda_{\rm EW}$, we have the approximate mass equation

[TABLE]

For high scale cutoff $\Lambda\simeq 10^{18}\,{\rm GeV}$ and $\Lambda_{EW}\simeq 10^{2}\,{\rm GeV}$, this gives

[TABLE]

and we therefore have four different regimes for axion masses: high scale, electroweak scale, neutrino scale, and ultralight scale. Note that the mass scale relevant for ultralight axion dark matter has arisen out of known mass scales in nature. In Section 3 we will show that a potential of the form Equation 2 arises naturally from a superpotential, in which case the appearance of the singlet is related to having integral powers of superfields.

Concrete analyses of some the possibilities discussed above for various values of $A$, $F$, and $\Lambda$ will be presented in Section 3, but we would like to make some brief comments here. One critical aspect of the Section 6 analysis will address the fact that $A$ in string theory is typically exponentially suppressed by the volume of an internal cycle in a Calabi-Yau manifold. From this perspective, [1] used the $n=0$ case and fine-tuned this exponential to obtain the axion mass $O(10^{-22})\,{\rm eV}$. This requires a large internal cycle and depends on moduli stabilization. We are simply proposing that the same small scale can be obtained by trading instanton suppression for the electroweak hierarchy. In particular, we will see that reasonable values of $A$ in string theory can be accommodated in this framework. In Section 5 we will discuss how operators of the schematic form Equation 1 may arise from D-brane instanton corrections to the superpotential in which vector-like instanton zero-modes play a crucial role.

3 The axion in supersymmetry and supergravity models

In this section we construct explicit supersymmetric models that generate an ultralight axion. The ultralight nature of the axion is due to a perturbatively-exact shift symmetry which is broken by a small amount (relative to other scales in the model) by non-perturbative effects such as instantons. Construction of a superpotential at the perturbative level that respects invariance under a $U(1)$ shift symmetry $S\to e^{i\lambda}S$ for a field $S$ can be achieved with extra matter charged under the standard model and $U(1)$, and in this case terms in the superpotential involving $S$ and the extra matter can be written such that the superpotential is neutral under the shift symmetry [7, 8]. Alternately one may make the MSSM fields charged under $U(1)$ and introduce terms in the superpotential involving $S$ and the MSSM fields [6].

Here we take an alternative approach where we introduce two fields $S_{1}$ and $S_{2}$ which are $SU(3)\times SU(2)_{L}\times U(1)_{Y}$ singlets but are oppositely charged under the global $U(1)$ symmetry, i.e., under a global $U(1)$ transformation one has

[TABLE]

so that $S_{1}S_{2}$ is neutral under the $U(1)$. We consider a superpotential of the form

[TABLE]

The superfields $S_{i}$ (i=1,2) have the expansion

[TABLE]

where $\phi_{i}$ is a complex scalar containing the axion and the saxion, $\xi_{i}$ is the axino and $F_{i}$ the auxiliary field. Here we write

[TABLE]

where the $\rho_{i}$ are expansions about the VEVs $\rho_{i}^{0}$. The higher dimensional operator in Equation 7 is needed to give a VEV to the scalar component of $\phi_{i}$. The F-term equations of motion give the constraint888We will assume throughout this section that the axions are stabilized at zero, which we will find to be a consistent assumption.

[TABLE]

Further one finds $F\equiv\rho_{1}^{0}=\rho_{2}^{0}$. Thus we may write $\phi_{i}$ in the form

[TABLE]

It is useful to define the combination of axion fields $a_{1}$ and $a_{2}$ so that

[TABLE]

Here one finds that Equation 7 leads to the following potential for $a_{+}$

[TABLE]

Equation 13 gives $a_{+}$ a mass $m_{a_{+}}=2\sqrt{2}\mu_{0}$. One may also check that the saxion field $\rho_{+}$ defined so that $\rho_{\pm}=(\rho_{1}\pm\rho_{2})/\sqrt{2}$ and the axino fields $\xi_{{}_{+}}$ where $\xi_{{}_{+}}=(\xi_{1}\pm\xi_{2})/\sqrt{2}$ also have exactly the same mass. Thus the superpotential in Equation 7 gives rise to an entire massive chiral multiplet $\rho_{+},a_{+},\xi_{+}$, as required by supersymmetry. We also note that the axion $a_{-}$ still possesses a continuous shift symmetry, and thus no potential is generated for $a_{-}$ and so it remain massless. The same applies to $\rho_{-}$ and $\xi_{-}$. Thus one combination of the original chiral fields become massive while the orthogonal combination remains massless. We now turn to generation of a mass for $a_{-}$. To give $a_{-}$ mass we need to include contributions in the superpotential which break the continuous shift symmetry. We will discuss two classes of models. For one class we will use an instanton type contribution and for the other class we will use higher dimensional operators, which couple the Higgs fields and standard model singlets to the axion fields, which breaks the continuous shift symmetry.

We begin by considering models of the first type. Here we take a superpotential of the form

[TABLE]

where $W_{s}$ is as defined by Equation 7 and $W_{n}$ violates the shift symmetry. In this case the equations of motion give

[TABLE]

Retaining only the dependence on $a_{-}$ the axion potential takes the form

[TABLE]

We note that the form of the axion potential is not of the standard $\cos(ca)$. However, it reduces to it when we expand $\sin(a_{-}/\sqrt{2}F)$ about $a_{-}=0$ and retain the first term in the expansion. Thus an expansion of the potential, and using the condition $\alpha F\gg 1$, is needed to simulate an instanton-like effect and leads to a mass term for $a_{-}$ of the form

[TABLE]

Using numbers consistent with [1], i.e., $F=10^{17}\,{\rm GeV},~{}\alpha^{2}A=10^{12}\,{\rm GeV}$, $\alpha F=99$, one finds $m_{a_{-}}=10^{-21}\,{\rm eV}$. A similar analysis holds for the saxion $\rho_{-}$ and the axino $\xi_{-}$ which develop a mass of similar size. We assume that $\mu_{0}$ is electroweak scale. Since $F=10^{17}$ GeV, this requires $\lambda_{s}$ to be $O(10^{-12})$.999This choice of $\lambda_{s}$, though small, is protected from renormalization by supersymmetry. We note also that this size of $\lambda_{s}$ can be generated in string perturbation theory; e.g. in type IIA disc instantons can generate suppressions of the form $e^{-A}$, where $A$ is the disc area. This effect is distinct from the Euclidean D-brane instantons that we consider elsewhere.

Next we discuss the case when the shift symmetry is broken by a higher dimensional operator involving couplings to the Higgs, standard model singlets, and the axion fields. As an organizing principle we consider supersymmetric models with three sectors: visible, hidden and an overlap sector between the hidden and the visible sectors with interactions suppressed by Planck mass101010Supersymmetric models of this sort with three sectors have been considered in previous works, see, e.g, [10]. so that

[TABLE]

where $W_{vis}$ contains fields in the visible sector, $W_{hid}$ contains fields in the hidden sector and $W_{vih}$ contains the overlap. In this analysis we assume that $W_{vis}$ contains the fields $H_{1},H_{2}$, and $S$, where $S$ is a standard model singlet like the one used in the nMSSM and does not possess any shift symmetry and $W_{hid}$ contains the axion fields $S_{1},S_{2}$ discussed above. Here we take

[TABLE]

We assume that the Higgs fields develop VEVs due to sources in the visible sector not considered here. The effects of $W_{vh}$ on the VEVs of $S,S_{1},S_{2}$ are small because of Planck mass suppression. Thus, to the lowest order, one can see that the minimization condition in the S sector gives $\langle S\rangle\sim\lambda_{0}\langle v_{1}v_{2}\rangle/\mu_{s}$. We assume $\mu_{s}$ to be electroweak size which implies $v_{0}\equiv\langle S\rangle$ is electroweak size.

Next we focus on the F-term equations in the $S_{1}$ and $S_{2}$ sectors. Here we find

[TABLE]

From Equation 20 we deduce $F=\rho_{1}^{0}=\rho_{2}^{0}$, which results in the constraint

[TABLE]

The axion potential results from the term $\sum_{i=1,2}|\partial W/\partial{S_{i}}|^{2}$. Retaining only the dependence on $a_{-}$ we find

[TABLE]

Equation 22 leads to mass for $a_{-}$ of the form

[TABLE]

where $v_{0}\sim\Lambda_{\rm EW}$, $\Lambda=(FM^{n-2})^{1/n-1}$.

We now show that the term $|\partial W/\partial{S}|^{2}$ does not contribute to the $a_{-}$ mass. The $S$ dependent terms in the superpotential are given by

[TABLE]

The F-term equation in this sector reads

[TABLE]

Using the result deduced above that $\rho_{a}^{0}=F=\rho_{2}^{0}$, the axion potential from this this sector is given by

[TABLE]

Applying Equation 25 in Equation 26 we have

[TABLE]

From the above we deduce that $a_{-}$-dependent part of the potential is

[TABLE]

which gives a vanishing mass for $a_{-}$. Therefore $|F_{S}|^{2}$ does not contribute to the mass of $a_{-}$. Finally we consider the potential for $a_{-}$ generated by the terms $\sum_{i=1,2}|\frac{\partial W}{\partial H_{i}}|^{2}$. Here we find

[TABLE]

which gives a vanishing contribution to $V(a_{-})$. Superpotentials of the type considered in $W_{vh}$ in Equation 19 can be generated in string models as discussed in section 5.

When supersymmetry is promoted to supergravity [11, 12] and supersymmetry breaking is taken into account, one will generate soft terms and the potential will have the form

[TABLE]

where $\phi_{i}$ are all the fields that enter in the superpotential and $V_{\rm soft}$ are terms such as $m_{0}^{2}\sum_{i}\phi_{i}\phi^{\dagger}$ and trilinear terms. In this case one finds that the dominant term that contributes to the axion $a_{-}$ mass is

[TABLE]

where $q$ is an $\mathcal{O}(1)$ number, and we assume $\mu_{0}\sim s\sim h$. Taking $h\sim\Lambda_{EW}$, we then have

[TABLE]

where $m=(n+1)/2$. Here $m=3$ requires $n=5$.

3.1 Models with higher dimensional Higgs-axion couplings

Next we discuss the case when the shift symmetry is broken by a higher dimensional operator involving couplings of the Higgs and $S_{i}$. Here we assume a superpotential of the form

[TABLE]

Next using the superpotential of Equation 33 and after spontaneous breaking which gives VEVs to $S_{i}$ and also assuming that $H_{i}$ develop VEVS, axion $a_{-}$ potential can be obtained as discussed in the previous analysis and one gets

[TABLE]

For the case $k=2$ the first term in the brace on the right hand side of Equation 34 is small relative to the second which gives an axion mass

[TABLE]

This is of the form Equation 4 with $n=2k-1$ and for $k=2$ one has $n=3$ which gives the ultralight axion. We note that after soft terms are taken into account we will have a result similar to Eq (32).

As a final example we consider a model where the axion couples directly to the Higgs fields, via a non-perturbative term in the superpotential. We present this model because it is a very simple realization of the organizing principle of Section 2 involving higher dimensional Higgs-axion couplings. In this example the axion $a$ is the imaginary part of a complex modulus $T=\tau+i\,a$, whose potential is generated non-perturbatively. This class of models is ubiquitous in string theory, and we will explore the details of string embeddings in Sections 4 and 5. We consider a superpotential of the form

[TABLE]

where $W_{0}$ is a constant obtained from integrating out heavy fields. The axion appears in the potential only via the $H_{1}$ and $H_{2}$ F-terms, and a quick calculation shows the mass of $a$ takes the form

[TABLE]

Taking $F\sim\Lambda$ to be a high scale and $h\sim\mu\sim\Lambda_{EW}$, we have

[TABLE]

Furthermore, if we take $\Lambda e^{-\tau}\sim\Lambda_{EW}$, we find

[TABLE]

Here taking $n=2$ provides the desired ultralight mass for the axion. In many string models [21, 22, 23] the $\mu$-term in the superpotential is generated non-perturbatively, so we find it plausible that additional non-perturbative effects could generate this coupling at the same scale. Alternatively, it may be possible for the instanton that generates the higher order Higgs coupling to be in the same homology class as the instanton that generates the $\mu$-term; in this case the relationship $\Lambda e^{-\tau}\sim\Lambda_{EW}$ is automatic. We leave the study of these important global issues to future work.

4 Axions in simplified string models

The authors of [1] suggest that the FDM model of dark matter could be embedded in a string compactification, and the necessary mass and axion decay constant are natural from a stringy point of view. To make a precise statement one should scan over an ensemble of vacua and use the distribution of axion masses and decay constants to estimate the frequency in which parameters consistent with FDM occur. Unfortunately, while it is well-known how to calculate axion decay constants even when the number of moduli is large (c.f. [14]), calculating the masses requires intimate knowledge of non-perturbative effects, which are currently only partially calculable. In addition, moduli stabilization with a large number of moduli is notoriously difficult.

It is therefore our goal to find a realistic simplified model to demonstrate that embedding FDM in string theory is consistent with moduli stabilization, and does not remove us from the regime of validity of the effective theory.

A typical 4d effective SUGRA theory constructed from a string compactification has scalar fields known as moduli. These fields arise from reducing the metric and various $p$-form gauge fields along appropriate $p$-cycles in the internal space $X$. A virtually universal class of moduli are the Kähler moduli, whose vacuum expectation values parameterize complexified volumes of holomorphic cycles in $X$. We consider a compactification of IIB string theory on a Calabi-Yau orientifold $X$, which yields an effective $\mathcal{N}=1$ SUGRA theory in 4d. Type IIB string theory has a four-form gauge field $C_{4}$ in 10d, and dimensionally reducing $C_{4}$ along a holomorphic four-cycle (divisor) in $X$ yields an axion in the 4d theory. This axion pairs with the volume modulus of the four-cycle in a complex scalar field, which is the lowest component of a chiral superfield. The Kähler moduli $T^{i}$ are written as

[TABLE]

where $D^{i}$ is the corresponding divisor with volume modulus $\tau^{i}$ and axion $\theta^{i}$, and $J$ is the Kähler form on $X$. The theory typically has other moduli besides Kähler moduli, including the complex structure moduli $U$ and the holomorphic axio-dilation $S=e^{-\phi}+i\,C_{0}\equiv S_{1}+i\,S_{2}$.111111Here the variables $S$, $S_{1}$, and $S_{2}$ are not to be confused with the ones from Section 3. The tree-level Kähler potential takes the form

[TABLE]

The complex structure moduli and holomorphic axio-dilaton acquire masses via the tree level flux superpotential [15]

[TABLE]

where $G_{3}$ is a particular flux on $X$, and $\Omega$ is the holomorphic $(3,0)$-form. We will assume that $S$ and $U$ are stabilized at a high scale by $W_{\text{Tree}}$. The Kähler moduli, on the other hand, only appear in the superpotential non-perturbatively [20]. Including these non-perturbative effects, the superpotential then takes the form

[TABLE]

where $W_{0}=\langle W_{\text{Tree}}\rangle$, and the matrix $q^{a}_{\,i}$ is a matrix of rational numbers.

4.1 KKLT moduli stabilization

In this section we discuss the KKLT moduli stabilization scheme [16], in which the classical superpotential is balanced against an exponentially small non-perturbative effect in order to stabilize the Kähler moduli. We wish to see if an ultralight axion can be generated within the KKLT scheme. We consider the $D=4,\,\mathcal{N}=1$ supergravity (SUGRA) potential [11, 12]:

[TABLE]

In the analysis below we set $\kappa=1$. In the case of a single Kähler modulus the Kähler potential can be written as121212In this section we suppress the dependence on $S$ and $U$ as they will only contribute an overall scale.

[TABLE]

and the superpotential takes the form

[TABLE]

where $A$ and $W_{0}$ is independent of $T$. Without loss of generality we assume $A$ and $W_{0}$ are real. Let us now expand $V$ in the following form

[TABLE]

Using the decomposition

[TABLE]

$V$ takes the form

[TABLE]

Solving the F-term equations $DW=0$ one finds131313We note that $DW=0$ gives exactly the same condition for the critical point as the minimization of the potential in this case.

[TABLE]

where $\tau_{0}=\langle\tau\rangle$. We expand around the critical point so that $\tau=\tau_{0}+\tau^{\prime},\langle\theta\rangle=0$. The kinetic energy then takes the form

[TABLE]

We define the canonically-normalized fields

[TABLE]

for which the kinetic energy takes the canonical form. We have

[TABLE]

At the AdS minimum, the mass of $a$ can be written as

[TABLE]

where we have used Equation 50 to evaluate the mass at the minimum of the potential. In gravity mediated breaking of supersymmetry (see [12] and the references therein) the weak scale $m_{s}$ is related to the hidden sector $W_{0}$ so that $m_{s}=e^{K/2}|W_{0}|$ . Setting $q=2\pi,A=1$ and stabilizing the modulus $T$ one finds that an axion mass of $10^{-22}\,{\rm eV}$ requires the string scale to be far below the electroweak scale.

Thus, we see that single modulus KKLT is incompatible with an ultralight axion.

4.2 The Large Volume Scenario

In Section 4.1 we found that the single modulus KKLT realization of FDM had a separation of scales issue. In order to get around this we must modify the theory, by introducing more fields and/or by considering further corrections to the potential. A particularly simple way to introduce an additional scale is to consider the first non-vanishing $\alpha^{\prime}$-correction to the Kähler potential. This correction was computed in [17], and the corrected Kähler potential takes the form141414In this note we work exclusively in the Einstein frame.

[TABLE]

where $\alpha=\frac{1}{2}\xi S_{1}^{3/2}$, $\xi=\zeta(3)\chi/2(2\pi)^{3}$, and $\chi$ is the topological Euler characteristic of $X$. The Large Volume Scenario (LVS) [19] is a multi-modulus ($\geq 2$) stabilization scheme that uses the $\alpha^{\prime}$-correction, along with a non-perturbative effect, to realize a hierarchy of scales.

Here we will consider the simplest case, where the number of Kähler moduli, which is counted by the Hodge number $h^{1,1}(X)$, equals two. It was shown in [18] that the volume all $h^{1,1}=2$ Calabi-Yau manifolds can be written in the Strong Cheese form, such that

[TABLE]

Here $\tau_{b}$ is the big (or large) cycle, which controls the overall volume (size of the cheese), and $\tau_{s}$ is a small cycle (a hole in the cheese). The constant $\eta$ is typically an $\mathcal{O}(1)$ number, which depends on the intersection numbers of $X$. We will take $\eta=1/9\sqrt{2}$ for concreteness, as in the $\mathbb{P}^{4}_{1,1,1,6,9}$ Calabi-Yau hypersurface. Each of these volume moduli pairs with an axion, so we have two complex scalers $T_{s}=\tau_{s}+i\,\theta_{s}$ and $T_{b}=\tau_{b}+i\,\theta_{b}$.

In LVS the overall volume is taken the be large, with $\tau_{s}$ left small, so that $V\sim\tau_{b}^{3/2}$, and

[TABLE]

In this regime that Kähler potential can be expanded as

[TABLE]

In standard LVS the cycle $\tau_{b}$ is taken to be large enough to effectively ignore any non-perturbative effects that depend on $\tau_{b}$. The superpotential then takes the form

[TABLE]

The axion $\theta_{b}$ is massless in this approximation, as it does not appear in the potential. Of course, it is expected that a non-perturbative correction to the potential will generate a mass of $\theta_{b}$. In an $\mathcal{N}=1$ SUGRA model the mass for $\theta_{b}$ can be generated by a correction to either the superpotential or the Kähler potential (or both). Let us first consider a correction to the superpotential, of the form

[TABLE]

At large volume (large $\tau_{s}$) this correction is negligible compared to the terms in Equation 59, and will therefore not affect the stabilization of $\tau_{b}$, $\tau_{s}$, or $\theta_{s}$. However, Equation 60 provides the only term in $W$ that explicitly depends on $\theta_{b}$, and will therefore be the leading-order operator that generates a mass for $\theta_{b}$, in the absence of additional corrections. However, this term will be quite suppressed, and so one must consider whether this correction truly is leading order. Holomorphy, along with the shift-symmetry of the axion, constrain $\Delta W$ to take the form derived in [20]:

[TABLE]

where the $q^{i}_{\,j}$ are rational numbers, and $-q^{i}_{\,j}\tau^{j}$ is a positive rational multiple of the volume of a divisor.

However, holomorphy does not constrain the Kähler potential, and the corrections can take a more general form. It is beyond the scope of this work to explicitly calculate any such corrections; instead, we believe the following assumptions are well-motivated:

$\Delta K$ is periodic in $\theta_{b}$. 2. 2.

$\Delta K$ is generated by instantons that are charged under $C_{4}$; namely, Euclidean D3 and anti-D3 branes. 3. 3.

The nonperturbative correction preserves the logarithmic form of the Kähler potential.

If one assumes that the correction is generated by Euclidean D3 or anti-D3 brane, wrapping a cycle $\gamma$, then we expect the correction to the Kähler potential to take the form

[TABLE]

where $f$ is a periodic function of $\theta_{b}$. Here $S$ is the instanton action, which we expect to go roughly as the volume of the brane. In order to to solve the equations of motion $\gamma$ should be a locally volume minimizing representative of its class $[\gamma]$, with volume vol$(\gamma)$, and so $S\simeq\text{vol}(\gamma)$. However, since this instanton is correcting the Kähler potential, and not the superpotential, $\gamma$ does not need to have minimal volume in the class $[\gamma]$, as it is not necessarily a holomorphic representative. Therefore, $\text{vol}(\gamma)\geq\tau_{\gamma}$, where $\tau_{\gamma}$ is the minimal volume of $[\gamma]$. Without an explicit calculation we see no reason to assume that the inequality $\text{vol}(\gamma)\geq\tau_{\gamma}$ cannot be saturated by at least some corrections to the Kähler potential. If this is the case the correction in Equation 62 could provide corrections to $V$ of the same order as those in Equation 60. We will assume this is not the case, but it is important to understand these corrections further in the future.

Under the assumption that the correction to $W$ given in Equation 60 provides the leading order term for $\theta_{b}$, the scalar potential151515We set $K_{\text{cs}}=0$ for simplicity, and absorb any phase of $W_{0}$ into the axions. takes the form

[TABLE]

This form is derived in the appendix. The axions are stabilized at $\theta_{b}=\pi/a_{b},\theta_{s}=\pi/a_{s}$. A non-supersymmetric AdS minimum of the potential is found approximately at $a\tau_{s}\sim\text{ln}\mathcal{V}$. For a concrete example we consider the following parameters:

[TABLE]

These numbers are well-motivated in weakly-coupled IIB string theory. Calabi-Yau manifolds with a hierarchy in $h^{1,1}$ and $h^{2,1}$ are quite common, and the dual Coxeter number $6$ appearing in $a_{s}$ and $a_{b}$ corresponds to an SO(8) gauge group, which is consistent with our weak coupling assumption. Here we also have $S_{1}=1/g_{s}$, so in this example $g_{s}\approx 0.1$ is small. Inserting these parameters into Equation 4.2 and minimizing the potential, we find the volume161616In this section we express all of our volumes in the appropriate units of $\alpha^{{}^{\prime}}$. is stabilized at $\mathcal{V}=187$. The small cycle is stabilized at $\tau=32.5$. One might be concerned that a volume of $\mathcal{O}(100)$ is too small for the $1/\mathcal{V}$ expansion of the Kähler potential to be valid, but in this example the correction is at the percent level, so we expect the approximation to be good171717While the relative smallness of the perturbation to the Kähler potential is a necessary condition for the LVS approximation to be valid, it is not sufficient, due to the non-trivial Kähler geometry. We have checked that the higher order terms are subleading.. Using the parameters in Equation 64 we find a light axion mass of $3.9\times 10^{-22}$ eV. The mass of the other axion is approximately $26$ TeV, and the masses of the saxions are 590 GeV and 280 TeV. The fermions masses are 13 TeV and 26 TeV. Both axion decay constants are $\mathcal{O}(10^{16})$ GeV. Importantly, the gravitino mass, which is the order parameter for SUSY breaking, is not too large, at approximately 13 TeV. It would be difficult to argue for SUSY as a solution to the hierarchy problem if the gravitino mass was near the Planck scale.

While the potential in Equation 4.2 is a toy-model for a real string compactification, with all relevant corrections computed, our analysis demonstrates the a mass scale for the lightest axion of $\mathcal{O}(10^{-22})$ eV is arguably consistent with moduli stabilization and a realistic electroweak scale. Of course, further study of both non-perturbative and perturbative corrections to the Kähler potential, such as those in [72], and superpotential is important in understanding how FDM could be embedded in string theory.

5 Ultralight Axion Couplings to the Higgs in String Theory

In this section we discuss how operators of the form $(h^{\dagger}h)^{n}cos(a/F)$ may arise in string theory, focusing on non-perturbative corrections to the superpotential [20]. Some of the concepts implicit in previous sections will be repeated here in order to present a more complete picture of instanton corrections to the superpotential in string theory.

Non-perturbative corrections to the superpotential may arise from gauge dynamics, Euclidean D-brane instantons, M2-brane instantons, or worldsheet instantons, depending on the situation. For example, in type IIB compactifications, in particular in KKLT and LVS, Euclidean D3 (ED3) instantons may generate such corrections, and Euclidean D2 (ED2) instantons and M2-brane instantons provide similar corrections in type IIA and M-theory compactifications. The non-perturbative contribution to the superpotential from a single instanton is typically written in the schematic form

[TABLE]

where $T$ is a modulus appropriate to the compactification, e.g. a Kähler modulus in type IIb compactifications, where $\langle\text{Re}(T)\rangle=vol(D)$, with $D$ the internal cycle wrapped by the instanton, and the axion $a$ is $\text{Im}(T)$. $A(\phi)$ is an instanton prefactor that depends on other moduli. These couplings do not couple $a$ to the Higgs, and therefore are not of the desired type.

More general classes of brane instantons exist [21, 22, 23] in which the instanton prefactor may also contain gauge invariant combinations of chiral supermultiplets charged under gauge groups. Such corrections arise due to the presence of additional instanton zero modes when $D$ intersects some other cycle $D^{\prime}$ wrapped by spacetime-filling branes that carry non-trivial gauge sectors. We write the general form of these corrections as

[TABLE]

where the visible sector operator $\mathcal{O}_{V}$ contains only MSSM superfields, whereas $\mathcal{O}_{H}$ may have charged fields beyond the MSSM, which could live in a hidden sector separated from the visible sector in the extra dimensions. One important aspect of these instantons is that they may generate the leading coupling in $\mathcal{O}_{H}\mathcal{O}_{V}$, if $\mathcal{O}_{H}\mathcal{O}_{V}$ on its own is forbidden by an anomalous $U(1)$ symmetry. For example, in weakly-coupled type II compactifications the top-quark Yukawa Coupling $10\,10\,5$ of a Georgi-Glashow $SU(5)$ GUT is always forbidden in perturbation theory, as are the flavor-diagonal Majorana mass terms for right-handed neutrinos. Obtaining these superpotential couplings therefore requires non-perturbative effects, such as the ones described.

For concreteness, we will restrict our attention to ED3 instantons in type IIb compactifications, though similar statements regarding vector-like zero modes and higher dimensional operators should hold in other contexts as well.

We would like to study situations under which an ultralight axion mass can arise from an effective operator of the schematic form (1), which itself arises from an instanton contribution to the superpotential. For this to happen, holomorphy and gauge invariance dictate that the non-perturbative superpotential contains a term181818One could easily incorporate the field $S$, considered in Section 2, in this effect, but we omit it here for simplicity of discussion.

[TABLE]

Whether or not such a term exists depends on the detailed structure of the instanton zero modes. These include ED3-ED3 zero modes, as well as ED3-D7 zero modes that arise from ED3 intersections with spacetime filling D7-branes that give rise to the Higgs fields $H_{1}$ and $H_{2}$. Of particular important are the fermionic zero modes, so-called $\lambda$-modes, in the ED3-D7 sector.

For example, if the $\mu$-term $H_{1}H_{2}$ is forbidden by an anomalous $U(1)$ symmetry, a non-perturbative effective of the form

[TABLE]

may generate it non-perturbatively [21, 22, 23], where the effective $\mu$ parameter $\mu_{eff}=AM_{s}e^{-\langle Re(T)\rangle}$ may be at the electroweak scale depending on the expectation value of the stabilized field $T$. In this way, ED3-instantons give a solution to the $\mu$-problem. Generating such an operator that is forbidden in perturbation theory by an anomalous $U(1)$ symmetry requires a chiral excess of $\lambda$-modes and an associated shift of $T$ under the anomalous $U(1)$, so that the entire operator is gauge invariant. In such a case the axion in $T$ becomes the longitudinal component of the massive $Z^{\prime}$ boson associated to the anomalous $U(1)$, which has a string scale mass via the Stückelberg mechanism. See [27, 28] for systematic phenomenological studies in this context.

For an ultralight axion to appear in $\eqref{Wax}$, it is necessary for it to not be eaten via the Stückelberg mechanism. Therefore the operators $(H_{1}H_{2})^{n}$ must not be forbidden by an anomalous $U(1)$, and correspondingly the instanton must have at most vector-like $\lambda$-modes, i.e. the modes have index zero. Using the instanton calculus of [21], an instanton on a divisor $D$ with Kähler modulus $T$ and a single vector-like pair $\lambda\overline{\lambda}$ with an appropriate structure of ED3-ED3 zero modes generates an effective operator of the form

[TABLE]

which is precisely $W_{ax}$ in the $n=1$ case. More generally, there may be $n$ pairs of vector-like zero-modes $\lambda_{i}\overline{\lambda}_{i}$, in which case there are more Grassmann integrals, and we have

[TABLE]

which is precisely $W_{ax}$. Thus, we see that a superpotential operator $W_{ax}$ of the desired form may be generated if there is an instanton with $n$ pairs of vector-like zero modes $\lambda\overline{\lambda}$. The $n=2$ case is quite similar to the non-perturbative Weinberg operator $LH_{2}LH_{2}$ studied in [29], since $L$ and $H_{1}$ have the same quantum numbers under the MSSM gauge group.

The appearance $W_{ax}$, then, depends crucially on the structure of vector-like instanton zero modes, and we would like to consider when such zero modes exist.

Suppose that an ED3 and a D7-brane (or a stack of D7-branes) wrap divisors $D$ and $D^{\prime}$ in a smooth Calabi-Yau threefold $X$ that intersect along a curve $C:=D\cdot D^{\prime}$. Both the instanton and the D7-brane may carry $(1,1)$-form worldvolume fluxes (or more generally holomorphic vector bundles), which may be written in terms of line bundles $\mathcal{L}_{D}$ and $\mathcal{L}_{D^{\prime}}$ on $D$ and $D^{\prime}$, respectively. Then the ED3-D7 instanton zero-modes at the intersection are counted by the cohomology $h^{i}(C,K_{C}^{1/2}\otimes\mathcal{L})$, where $\mathcal{L}:=\mathcal{L}_{D}|_{C}\otimes\mathcal{L}^{-1}_{D^{\prime}}|_{C}$.

As discussed, a necessary condition for obtaining couplings of the desired type is that there is no chiral excess of ED3-D7 zero modes on $C$, i.e.

[TABLE]

Computing this index by applying the Hirzebruch-Riemann-Roch theorem, we have

[TABLE]

and we see the index is zero when $c_{1}(\mathcal{L})=0.$ By this we see that if $c_{1}(\mathcal{L}_{D}|_{C})=c_{1}(\mathcal{L}_{D^{\prime}}|_{C})$ then $\chi(C,K_{C}^{1/2}\otimes\mathcal{L})=0$, i.e. we have at most vector-like instanton zero modes on $C$.

In such a case, determining whether there actually are vector-like instanton zero modes requires computing the cohomology, not just the index. This computation can be done by a variety of means, but as an existence proof we would like to present a simple example.

Consider the case where a divisor $D=\mathbb{P}^{1}\times\mathbb{P}^{1}$ is wrapped by an ED3 instanton that intersects a space-time filling D7-brane on another divisor $D^{\prime}$ at a degree $(m,n)$ curve $C\subset D$, and there are no worldvolume fluxes, i.e. $\mathcal{L}_{D}=\mathcal{O}_{D}$ and $\mathcal{L}_{D^{\prime}}=\mathcal{O}_{D^{\prime}}$. The zero modes are counted by $h^{i}(C,K_{C}^{1/2})$, which has index zero, where $K_{C}=(K_{D}+\mathcal{O}(C))|_{C}=\mathcal{O}(m-2,n-2)|_{C}$. Taking the square root, a Koszul sequence for $K_{C}^{1/2}$ is given by

[TABLE]

By Serre duality, $h^{i}(D,\mathcal{O}_{D}(\frac{m}{2}-1,\frac{n}{2}-1))=h^{2-i}(D,\mathcal{O}_{D}(-\frac{m}{2}-1,-\frac{n}{2}-1))$. Since a degree $l$ line bundle on $\mathbb{P}^{1}$ has $l+1$ global sections, and therefore a degree $(k-1,l-1)$ line bundle on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ has $kl$ global sections, $h^{i}(D,\mathcal{O}_{D}(\frac{m}{2}-1,\frac{n}{2}-1))=(mn/4,0,0)$. Using the long exact sequence in cohomology associated to the Koszul sequence (73), we obtain

[TABLE]

which shows that there are vector-like instanton zero modes for general even $m$ and $n$. For a more in depth introduction to this type of computation, see e.g. [25, 26].

6 Phenomenology

As discussed in [1], the relic density of the ultralight axion arises from misalignment, where after inflation the axion begins to oscillate around its minimum. Initially the axion field is assumed to have a value close to the decay constant, which leads to a relic density

[TABLE]

consistent with WMAP [30] and Planck [31] if $m_{a}\simeq 10^{-22}{\rm eV}$ and $F\simeq 10^{17}{\rm GeV}$. If $\langle s\rangle\simeq\langle h\rangle$ or $m=0$ (see Equation 3), the effective operator Equation 2 of section 2 accounts for this mass scale in the $n=3$ case with Wilson coefficient $A=1$ and ultraviolet cutoff and axion decay constant of size $\Lambda=F=M_{pl}$, in which case

[TABLE]

From (75), we see that with this axion decay constant the relic abundance is oversaturated.

A sub-Planckian axion decay constant and suppressed coefficient $A$ may give rise to the correct relic abundance and relevant axion mass, though, and this is well-motivated by ultraviolet considerations. The analysis is simplified by the assumption that $\langle s\rangle\simeq\langle h\rangle$ or $m=0$, in which case the axion mass in Equation 3 becomes

[TABLE]

Given this simplifying assumption, in Figure 1 we plot the axion mass and relic abundance as a function of $F$ and $A$ in the cases $n=3$ and $n=0$. In the $n=0$ ($n=3$) case the relic abundance $\Omega_{a}h^{2}=\Omega_{obs}h^{2}=.12$ and axion mass $m_{a}=10^{-22}\,{\rm eV}$ arise for $F=2\times 10^{17}\,{\rm GeV}$ and $A\simeq 10^{-100}$ ($A\simeq 5\times 10^{-4}$). From the perspective of this effective operator, [1] studied the $n=0$ case and used a large instanton suppression to account for the relic abundance and ultralight axion. We see that the $n=3$ case may also do so by utilizing the electroweak hierarchy to account for the small mass scale, rather than a very large instanton suppression. From the figure we also see that smaller values of $A$ and $F$ are also permitted in the case that the ultralight axion is a subcomponent of the dark matter.

As discussed in section (3), Equation 23 gives an axion mass of the desired size for the case $n=3$ and from Equation 75 we find that the same mass then gives the desired relic density. Thus as mentioned in section (3) one may call this the $n=3$ miracle. We discuss now the remaining fields arising from $S,S_{1},S_{2}$ that appear in section 3. The field $S$ has cubic interactions with the Higgs fields and assuming its mass to be larger than the Higgs it decays into MSSM fields and does not contribute to the relic density. We are then left left with the fields $a_{+},\rho_{+},\xi_{+}$ and $\rho_{-},\xi_{-}$. To discuss their disposition we need to look at their couplings to the Higgs given in Equation 19191919Another term bilinear in the MSSM fields which can be added to the superpotential is $\frac{\lambda^{\prime}}{M}S_{1}S_{2}LH_{2}$. However, in the analysis here we focus on the term exhibited in Equation 19.. After $S_{1}$ and $S_{2}$ develop VEVs, the interaction $S_{1}S_{2}H_{1}H_{2}$ in Equation 19 will generate an effective $\mu H_{1}H_{2}$ term where $\mu=(\lambda_{0}S_{0}+\frac{\lambda F^{2}}{M})$. For any reasonable phenomenology $\mu$ must be electroweak size. Noting the size of $F$ as given in section 3 we infer $\lambda\sim 10^{-12}$. While we have no fundamental explanation for the smallness of $\lambda$, we note that the desired size is technically natural202020A $\lambda$ this size may be generated by the same mechanism as discussed in footnote 5..

Every supersymmetric model of an ultralight axion will be accompanied by a scalar saxion $\rho$ and fermion partner axino. In general, the saxion may give rise to cosmological problems if it dominates the energy density of the universe through the time of BBN. However, many UV completions give rise to Planck suppressed operators that lead to a saxion decay rate

[TABLE]

As is well known, if $m_{\phi}\gtrsim 50\,{\rm TeV}$ then the saxion decays prior to BBN. Throughout, we assume that UV completions of our models give rise to such operators and scalars of this mass, in order to avoid spoiling BBN.

Let us discuss these ideas in the specific case of three of the models of Section 3, which have heavy fields $\rho_{+},a_{+},\xi_{+}$. Here we assume that they, as well as $\rho_{-}$ (which acquires a mass through soft breaking), have masses of size $10^{5}$ GeV. Such a mass assures their decay before the BBN time. For specificity let us discuss the $\rho_{+}$ decay. Here the relevant term arises from the couplings in Equation 19 and is

[TABLE]

The interaction above allow for the decay $\rho_{+}\to\tilde{H}_{1}\tilde{H}_{2}$ with a lifetime consistent with the BBN constraints. The lifetime for $a_{+}$ and for the axino $\xi_{+}$ are of similar size. Thus the fields $\rho_{+},a_{+},\xi_{+}$ all decay consistent with the BBN constrains and do not play a role in any further discussion. To decay $\rho_{-}$ we consider the coupling

[TABLE]

where $f_{r}(S_{-})$ is the real part of the kinetic energy function in supergravity [11, 12, 32]. Using the interaction of Equation 80 the decay width of $\rho_{-}$ to gauge bosons is given by [32]

[TABLE]

where $d_{f}\sim 1$; note that this effective operator has realized a decay rate of the form in Equation 78. There is an identical contribution arising from the decay into gauginos. For $n_{g}=4$ for the electroweak gauge bosons and for a $\rho_{-}$ mass of $10^{5}\,{\rm GeV}$ one gets a decay lifetime consistent with BBN.

We assume that in the MSSM sector there exists a term which is R-parity violating which makes the neutralino unstable. Thus the only remaining dark matter particles are the axion $a_{-}$ and the axino $\xi_{-}$. There is no efficient production mechanism to generate the relic density for $\xi_{-}$ comparable to the $a_{-}$ and thus dark matter is dominated by the ultralight axion whose relic density is given by Equation 75.

7 Conclusion

Recently it has been proposed that a boson of deBroglie wavelength $1{\rm kpc}$ may help resolve problems in cosmology at scales order $10{\rm kpc}$. A possible candidate is an ultralight axion of mass in the range $10^{-21}-10^{-22}\,{\rm eV}$. In this work we discussed models within the framework of supersymmetry, supergravity and strings where an ultralight axion of the desired mass may arise. In Section 2 we presented an effective operator analysis of the axion mass, noting its possible dependence upon the expectation values of the Higgs field and singlets $s$ that couple to the Higgs. We saw that for one effective operator the relevant axion mass arises as the electroweak scale times the square of the electroweak hierarchy; this arose in the case $n=3$, where $n$ is an integer parameter in the effective operator. In Section 3 we discussed two classes of supersymmetric models. In one class the shift symmetry is broken by instanton type effects and the instanton action can be fine tuned to generate the desired axion mass. In the second class of models it is shown that higher dimensional operators constructed out of Higgs fields, standard model singlet fields and the axion fields which violate the shift symmetry naturally lead to an ultralight axion of size $10^{-21}-10^{-22}\,{\rm eV}$. Quite remarkably it is shown in [1] that such an ultralight axion leads to the relic density consistent with WMAP [30]. In the analysis given in section (3) it is shown that for the case when the shift symmetry is broken by higher dimensional operators involving Higgs fields, standard model singlet fields and the axion fields both the mass of the axion and correspondingly the relic density consistent with WMAP arise naturally for the case $n=3$. The possibility of generating an ultralight axion within string based models was also discussed. It is shown that within the KKLT moduli stabilization the axion scale and the weak SUSY scale are related. However, it is shown that within the Large Volume Scenario a hierarchy between the axion scale and the weak SUSY scale can be achieved. To ensure that higher dimensional Higgs-instanton operators which violate shift symmetry can be generated in string theory, conditions necessary for the coupling of instanton to Higgs fields were discussed. It was shown that the conditions require the existence of vector like zero modes of the instanton. An illustrative example was given where such vector like zero modes can arise. Some phenomenological aspects of the models analyzed were discussed, including the dependence of the axion mass and relic abundance on the Wilson coefficient $A$ and axion decay constant $F$, as well as cosmologically relevant decay channels.

The concrete models discussed here for the realization of ultra light dark matter may help in further investigations.

**Acknowledgments: ** We thank Brent Nelson and Jonathan Carifio for discussions. JH and CL thank the Center for Theoretical Physics at MIT for its hospitality. This research was supported in part by NSF Grants PHY-1620526 and PHY-1314774.

Appendix A Derivation of the LVS Potential

In this section we derive the form of the LVS potential, using the $\alpha^{\prime}$ corrected Kähler potential $K=-2\,\text{log}(\mathcal{V}+c)$, where $c$ is independent of the Kähler moduli. We first work with the classical Kähler potential $\tilde{K}$ given by $c=0$, and then treat $c$ as a perturbation, where $c/\mathcal{V}\ll 1$. The good Kähler coordinates on moduli space of $X$ are the complexified divisor volumes $T^{i}=\tau^{i}+i\,\theta^{i}$. However, the volume is most naturally expressed in terms of the dual coordinates $t_{i}$:

[TABLE]

Here the $\kappa^{ijk}$ are the divisor triple intersection numbers of $X$. The relationship between $\tau^{i}$ and the $t_{j}$ is given by

[TABLE]

It is also useful to define the following symmetric matrix:

[TABLE]

We will denote the inverse of $A^{ij}$ by $A_{ij}$, such that $A_{ik}A^{kj}=\delta^{i}_{\,j}$. We also note the useful identities:

[TABLE]

The metric on Kähler moduli space is given by

[TABLE]

However, since $\mathcal{V}$ only depends on the real parts of the $T^{i}$ we can replace the holomorphic and anti-holomorphic derivatives with real derivatives via

[TABLE]

and similarly for the anti-holomorphic derivatives. The Kähler connection is given by

[TABLE]

The metric takes the form

[TABLE]

and the inverse metric is then

[TABLE]

In the $\mathcal{N}=1$ SUGRA potential contains the contractions $\tilde{K}^{i\bar{j}}\tilde{K}_{\bar{j}}$. We have

[TABLE]

Therefore

[TABLE]

In the large volume limit $\mathcal{V}\gg c$, we can write

[TABLE]

Here $\Delta K$ can be treated as a perturbation to the classical Kähler potential $\tilde{K}$. The correction to the Kähler connection is then

[TABLE]

The correction to the Kähler metric is then

[TABLE]

From this we can infer the correction to the inverse Kähler metric, via

[TABLE]

where in the last equality we have dropped terms of $\mathcal{O}(\Delta K_{i\bar{j}}^{2})$. We then have

[TABLE]

To evaluate this, we first calculate

[TABLE]

We then have

[TABLE]

We now calculate the $\mathcal{N}=1$ SUGRA potential for our specific example. We will first use the tree-level Kähler potential, and then add in the $\alpha^{\prime}$-correction after. We consider a superpotential of the form

[TABLE]

We then have $\partial_{i}W=-a_{i}A_{i}e^{a_{i}T_{i}}=-a_{i}W_{i}$, where there is no sum on $i\in\{T_{b},T_{s}\}$. In addition, we will take $W_{0}$ to be much larger than the non-perturbative contributions to the superpotential, so that $K_{i}W\approx K_{i}W_{0}$. Using the no-scale structure $K^{i\bar{j}}K_{i}K_{\bar{j}}=3$, we then have

[TABLE]

where $K_{\text{total}}$ is the Kähler potential for all the moduli. In the large volume limit the relevant terms are

[TABLE]

In LVS we have $K^{s\bar{s}}\approx-4\mathcal{V}A^{s\bar{s}}$, and for our particular example $K^{s\bar{b}}=4\tau^{s}\tau^{\bar{b}}$. Taking a volume of the form

[TABLE]

we have

[TABLE]

.

Taking $K_{\text{cs}}=0$, we have

[TABLE]

and so we can write the potential as

[TABLE]

where we have absorbed any phase of $W_{0}$ into the axions. We now calculate the $\alpha^{\prime}$-correction to the SUGRA potential, whose presence is crucial for the existence of a large volume minimum. The term that is important in LVS is given by the leading-order breaking of the no-scale structure, given schematically by $\Delta(K^{i\bar{j}}K_{i}K_{\bar{j}})|W_{0}|^{2}$. We have

[TABLE]

We will calculate this term-by-term. First, we have:

[TABLE]

We also have

[TABLE]

Putting it all together, and including the non-trivial factor of $e^{K_{\text{total}}}$, we have

[TABLE]

Plugging in $c=\frac{1}{2}\xi S_{1}^{3/2}$, we find

[TABLE]

The full potential then takes the form Equation 4.2.