APCTP Pre2021-012, CTPU-PTC-21-27
Aligned Natural Inflation in the Large Volume Scenario

An axion-like particle $a$ can undergo natural inflation [10, 11]. This scenario solves $\eta$-problem of inflation since the axion is a pseudo-Goldstone boson associated with a shift symmetry, and thus the potential is protected from radiative corrections. The axion potential is generated only by non-perturbative effects,

$$V=\Lambda^{4}\left(1-\cos\frac{a}{f}\right)\,,$$

(4)

with decay constant $f$ measured in units of the reduced Planck mass, $M_{\rm Pl}$. The axion may roll down the potential, with slow-roll dynamics characterized by the parameters [17]

	$\displaystyle\epsilon$	$\displaystyle\equiv\frac{M_{\rm Pl}^{2}}{2}\left(\frac{V^{\prime}(x)}{V(x)}\right)^{2}\,,$		(5)
	$\displaystyle\eta$	$\displaystyle\equiv M_{\rm Pl}^{2}\frac{V^{\prime\prime}(x)}{V(x)}\,.$		(6)

Observables relevant to inflation, of which we quote the Planck 2018 results in the $\Lambda$CDM model (Planck+lensing+BAO) [5], include the spectral index (68% confidence level),

$$n_{s}\simeq 1-6\epsilon+2\eta=1-f^{-2}-4\epsilon=0.9665\pm 0.0038\,,$$

(7)

and the tensor-to-scalar ratio,

$$r_{0.002}\simeq 16\epsilon=4\left(1-n_{s}-f^{-2}\right)<0.106\,.$$

(8)

The spectral index (7) is relatively robust, and obtaining sufficiently large $n_{s}$ in natural inflation requires a large enough decay constant,

$$f\gtrsim 4M_{\rm Pl}\,.$$

(9)

An important question is whether we can obtain such a large $f$ [12] in a controlled model. Since quantum corrections in string theory are suppressed only in a parameter regime below the Planck mass, this presents a challenge for string-theory model building. One candidate resolution is to generate effectively super-Planckian decay constants using only sub-Planckian parameters via an alignment mechanism, which we introduce in the following subsection.

Another useful fact is that the power spectrum of scalar perturbations ${\cal P}_{\zeta}$ determines the height of the inflationary potential [34],

$$\frac{\Lambda^{4}}{M_{\rm Pl}^{4}}\simeq\frac{12\pi^{2}}{(2q^{2}+1)e^{N_{e}/f^{2}}}{\cal P}_{\zeta}\,,$$

(10)

where $N_{e}$ is the $e$-folding number. The latest Planck measurement of the scalar power spectrum amplitude gives ${\cal P}_{\zeta}\simeq 2.1\times 10^{-10}$ [5]. Assuming $50<N_{e}<60$, for a wide range of decay constants, $3<f<10$, this translates into $\Lambda^{4}\simeq 10^{-9}M_{\rm Pl}^{4},$ corresponding to an inflation scale of $\Lambda\simeq 10^{16}\rm~{}GeV$.

We now review the axion alignment mechanism from a bottom-up field theoretical perspective. Later we will consider an explicit embedding into string theory, which will provide a more fundamental justification of this structure as well as an interpretation of the various parameters in terms of the underlying geometry. Consistency conditions such as supersymmetry will provide further constraints.

When multiple axions are present, a linear combination of them may obtain an effective decay constant which is much larger than those of the individual axions. Consider the case of an anomalous symmetry containing Abelian factors $U(1)_{j},j=1,\dots,N$, as well non-Abelian gauge groups $G_{I},I=1,\dots,M$, giving rise to a Lagrangian

$${\cal L}=k_{ij}\partial_{\mu}\vartheta_{i}\partial^{\mu}\vartheta_{j}-\sum_{I=1}^{M}\Lambda_{I}^{4}\left[1-\cos\left(Q_{Ij}\vartheta_{j}\right)\right]\,,$$

(11)

where $k_{ij}$ is a real, symmetric Kähler metric and $\Lambda_{I}$ are coefficients depending on the theory in question. Here $Q_{Ij}$ are the coefficients of the $G_{I}$-$G_{I}$-$U(1)_{j}$ anomaly,

$$\partial_{\mu}J_{j}^{5}=\frac{Q_{Ij}}{32\pi^{2}}\operatorname{tr}\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu,I}F_{\rho\sigma,I}\,,\qquad Q_{Ij}=\operatorname{tr}t_{I}^{2}q_{j}\,,$$

(12)

which are integrally quantized with normalization $\operatorname{tr}t^{a}t^{b}=\delta^{ab}$, in the case of the fundamental representation of $SU(N)$. It follows that the axion fields have well-defined periodicity, $\vartheta_{i}\to\vartheta_{i}+2\pi$.

For now and in the remainder of this discussion, we focus on the minimal configuration that captures the essential physics, which is the case of two instanton contributions. With a diagonal Kähler metric, $k_{ij}=\text{diag}(f_{1}^{2},f_{2}^{2})/2$, the canonically normalized fields are

$$\phi_{1}=f_{1}\vartheta_{1}^{\text{diag}}\,,\qquad{\phi_{2}}=f_{2}\vartheta_{2}^{\text{diag}}\,.$$

(13)

The Lagrangian (11) provides a lattice basis, which can be expanded as

$${\cal L}=\frac{1}{2}\partial_{\mu}\phi_{1}\partial^{\mu}\phi_{1}+\frac{1}{2}\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}-V\,,$$

(14)

where

$$V=\Lambda_{1}^{4}\left[1-\cos\left(\frac{Q_{11}}{f_{1}}\phi_{1}+\frac{Q_{12}}{f_{2}}\phi_{2}\right)\right]-\Lambda_{2}^{4}\left[1-\cos\left(\frac{Q_{21}}{f_{1}}\phi_{1}+\frac{Q_{22}}{f_{2}}\phi_{2}\right)\right]\,.$$

(15)

In the alignment limit [14],

$$Q_{11}:Q_{12}=Q_{21}:Q_{22}\,,$$

(16)

the two axions become coincident because the arguments of the two potentials in (15) become proportional. Since the total degrees of freedom should be preserved, there should be an orthogonal axion whose potential is flat. This is the basis of the Kim–Nilles–Peloso (KNP) mechanism, which makes use of the fact that the orthogonal axion direction obtains an effectively large decay constant [14].

In this paper we take some care to separate axion charges and decay constants, such that we can define alignment purely in terms of charges. To see how the alignment manifests, redefine the basis by an $SO(2)$ rotation

$$\left(\begin{array}[]{c}\phi_{\xi}\\ \psi_{\psi}\end{array}\right)=\frac{1}{\sqrt{f_{1}^{2}Q_{12}^{2}+f_{2}^{2}Q_{11}^{2}}}\left(\begin{array}[]{cc}f_{1}Q_{12}&-f_{2}Q_{11}\\ f_{2}Q_{11}&f_{1}Q_{12}\end{array}\right)\left(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\end{array}\right)\,.$$

(17)

The first term of the potential (15) then becomes a function of $\phi_{\xi}$ only. For each of $\phi_{\xi}$ and $\phi_{\psi}$, we can define the respective decay constants as

$$f_{\psi}=\frac{f_{1}f_{2}\sqrt{Q_{11}^{2}+Q_{12}^{2}}}{\sqrt{f_{1}^{2}Q_{12}^{2}+f_{2}^{2}Q_{11}^{2}}}\,,\qquad f_{\xi}=\frac{\sqrt{f_{1}^{2}Q_{12}^{2}+f_{2}^{2}Q_{11}^{2}}}{\sqrt{Q_{11}^{2}+Q_{12}^{2}}}\,.$$

(18)

Note that there is no special hierarchy between $\{f_{\psi},f_{\xi}\}$ and $\{f_{1},f_{2}\}$. In this basis, the potential can be expressed as

$$\begin{split}V=\Lambda_{1}^{4}\left[1-\cos\left(\frac{Q_{1\psi}}{f_{\psi}}\phi_{\psi}\right)\right]+\Lambda_{2}^{4}\left[1-\cos\left(\frac{Q_{\xi}}{f_{\xi}}\phi_{\xi}+\frac{Q_{2\psi}}{f_{\psi}}\phi_{\psi}\right)\right]\end{split}\,,$$

(19)

with the charges

$$\begin{split}Q_{1\psi}&=\sqrt{Q_{11}^{2}+Q_{12}^{2}}\,,\qquad Q_{\xi}=\displaystyle\frac{Q_{21}Q_{12}-Q_{11}Q_{22}}{\sqrt{Q_{11}^{2}+Q_{12}^{2}}}\,,\\ Q_{2\psi}&=\left(\frac{f_{1}^{2}Q_{12}Q_{22}+f_{2}^{2}Q_{11}Q_{21}}{f_{1}^{2}Q_{12}^{2}+f_{2}^{2}Q_{11}^{2}}\right)Q_{1\psi}\,.\end{split}$$

(20)

Thus, the alignment limit (16) can be achieved by choosing $Q_{Ii}$ to satisfy

$$|Q_{\xi}|\ll 1\,.$$

(21)

As a result, the effective decay constant of $\phi_{\xi}$, $f_{\rm eff}$, from the second term of the potential, becomes hierarchically larger than $f_{\xi}$ as

$$f_{\rm eff}=\frac{f_{\xi}}{|Q_{\xi}|}=\frac{\sqrt{f_{1}^{2}Q_{12}^{2}+f_{2}^{2}Q_{11}^{2}}}{|Q_{21}Q_{12}-Q_{11}Q_{22}|}\,.$$

(22)

One can simplify the dynamics of the light axion by integrating out the heavy degree of freedom. Depending on the origin of the terms in the potential, there can be a natural hierarchy between $\Lambda_{1}^{4}$ and $\Lambda_{2}^{4}$. Without loss of generality, by assuming $\Lambda_{1}^{4}\gg\Lambda_{2}^{4}$ we can easily integrate out the heavy axion ($a_{H}\simeq\phi_{\psi}$) from the first term of the potential,

$$\left\langle\frac{Q_{1\psi}\phi_{\psi}}{f_{\psi}}\right\rangle=2\pi k+{\cal O}\left(\frac{\Lambda_{2}^{4}Q_{2\psi}^{2}}{\Lambda_{1}^{4}Q_{1\psi}^{2}}\right)\quad{\rm for}\ k\in\mathbb{Z}\,.$$

The resulting effective Lagrangian for the light axion ($a\simeq\phi_{\xi}$) becomes

$$\begin{split}{\cal L}_{\rm eff}&=\frac{1}{2}(\partial_{\mu}a)^{2}-\Lambda_{2}^{4}\left[1-\cos\left(\frac{a}{f_{\rm eff}}+\frac{Q_{2\psi}}{Q_{1\psi}}2\pi k\right)\right]\\ &\quad+\frac{\Lambda_{2}^{8}Q_{2\psi}^{2}}{2\Lambda_{1}^{4}Q_{1\psi}^{2}}\sin^{2}\left(\frac{a}{f_{\rm eff}}+\frac{Q_{2\psi}}{Q_{1\psi}}2\pi k\right)+{\cal O}\left(\frac{\Lambda_{2}^{12}Q_{2\psi}^{4}}{\Lambda_{1}^{8}Q_{1\psi}^{4}}\right)\,.\end{split}$$

(23)

The leading term of the potential is simplified to the form of (4), with decay constant $f_{\rm eff}\gg f_{1,2}$. In the practical analysis to obtain cosmological parameters, we take into account the dynamics of both axions without approximations. Note that the existence of such a light direction does not rely on a hierarchy between $\Lambda_{1}^{4}$ and $\Lambda_{2}^{4}$.

For a non-diagonal Kähler metric, we can follow essentially the same procedure. Since $k_{ij}$ is a real, symmetric matrix, we may diagonalize it via an orthogonal matrix $P^{-1}=P^{\top}$ such that

$$P_{ik}^{\top}(k_{kl})P_{lj}=\frac{1}{2}\begin{pmatrix}f_{1}^{2}&0\\ 0&f_{2}^{2}\end{pmatrix}\,,\qquad\vartheta^{\text{diag}}_{i}=P^{\top}_{ij}\vartheta_{j}\,,$$

where the eigenvalues are positive semi-definite. After canonically normalizing as in (13), we obtain

$$\begin{split}\ {\cal L}&=\frac{1}{2}\partial_{\mu}\phi_{1}\partial^{\mu}\phi_{1}+\frac{1}{2}\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}\\ &\quad-\sum_{I=1,2}\Lambda_{I}^{4}\left[1-\cos\left(\frac{Q_{Ik}P_{k1}}{f_{1}}\phi_{1}+\frac{Q_{Ik}P_{k2}}{f_{2}}\phi_{2}\right)\right]\,.\end{split}$$

(24)

Redefining $Q_{Ik}P_{kj}\equiv Q^{\prime}_{Ij}$, the Lagrangian formally reduces to (15). Obviously, for a general Kähler metric the potential is not necessarily periodic with $\phi_{i}\to\phi_{i}+2\pi f_{i}$ because the elements $P_{kj}$ may not be integers. Rather, in this expression the periodicity of $\vartheta_{i}$ is hidden. Nevertheless, each quantity $f_{i}/Q_{Ii}^{\prime}$ still plays the same role as $f_{i}/Q_{Ii}$ in (15) when we consider the dynamics of the axions. Therefore, the alignment condition for the primed charges can be discussed in the same way. In fact, for any $P_{kj}$, the alignment condition reduces to that of the unprimed charges $Q_{Ik}$. More explicitly, from (20) we find that the effective charge $Q_{\xi}^{\prime}$ is

$$Q_{\xi}^{\prime}=\frac{Q_{21}^{\prime}Q_{12}^{\prime}-Q_{11}^{\prime}Q_{22}^{\prime}}{\sqrt{Q_{11}^{\prime 2}+Q_{12}^{\prime 2}}}=Q_{\xi}\,.$$

The alignment limit $|Q_{\xi}|\ll 1$ is thus equivalent to $|Q_{\xi}^{\prime}|\ll 1$.

Alternatively, we may define the matrix-valued decay constant ${\bf f}$ as one satisfying

$$2k_{ij}=({\bf f}^{\top}{\bf f})_{ij}={\bf f}_{ki}{\bf f}_{kj}\,.$$

(25)

It is known that we may always find a symmetric matrix ${\bf f}$ with this property [18]. Then the canonically normalized axion is defined as

$$\phi_{i}\equiv{\bf f}_{ij}\vartheta_{j}\,.$$

(26)

The matrix-valued decay constant ${\bf f}$ is related to $P$ as ${\bf f}_{ij}=f_{i}P^{\top}_{ij}$ (no sum over $i$). The potential can be written as

$$\begin{split}V&=\sum_{I=1,2}\Lambda_{I}^{4}\Big{[}1-\cos\left(Q_{Ik}{\bf f}^{-1}_{k1}\phi_{1}+Q_{Ik}{\bf f}^{-1}_{k2}\phi_{2}\right)\Big{]}\,.\end{split}$$

(27)

Note that we have again separated the charges and the decay constants. Still in this case, the alignment limit can be expressed purely in terms of the charges as in (16). This is essentially because the decay constant is well-defined as dependent on the axion flavor $i$ as in (26), not the instanton flavor $I$. We stress that each of them rotates covariantly under a basis transformation and the size of the decay constants remain of the same order, verifying that the alignment arises from the special combination of charges. Following the same logic as used to obtain (23), the general expression for the effective decay rate of the light axion $\phi_{\xi}$ becomes

$$f_{\xi}=\frac{\sqrt{({\bf f}_{ij}\epsilon_{jk}Q_{1k})({\bf f}_{ij^{\prime}}\epsilon_{j^{\prime}k^{\prime}}Q_{1k^{\prime}})}}{\sqrt{Q_{11}^{2}+Q_{12}^{2}}}\,,\qquad f_{\rm eff}=\frac{f_{\xi}}{Q_{\xi}}\,,$$

(28)

where $\epsilon_{12}=-\epsilon_{21}=1$.

Finally, there is a room for further improvement by utilizing off-diagonal terms of the Kähler metric. In the two-axion case, if the rotation matrix becomes maximal,

$$P=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\ -1&1\end{pmatrix}\,.$$

(29)

then (28) becomes

$$f_{\rm eff}=\frac{\sqrt{f_{1}^{2}\left(\frac{Q_{11}+Q_{12}}{\sqrt{2}}\right)^{2}+f_{2}^{2}\left(\frac{Q_{11}-Q_{12}}{\sqrt{2}}\right)^{2}}}{|Q_{21}Q_{12}-Q_{11}Q_{22}|}\,.$$

(30)

Comparing (30) with the simple diagonal example (22), we see that for $f_{1}=f_{2}$, the two expressions are the same regardless of the detailed charge assignment. However, if there is some hierarchy between the eigenvalues of ${\bf f}$, such as $f_{1}\gg f_{2}$, a clever charge assignment can provide an additional enhancement factor. For instance, if $Q_{11}=Q_{12}$, (22) becomes $f_{\rm eff}\simeq f_{1}/|Q_{21}-Q_{22}|$, while (30) becomes $f_{\rm eff}\simeq\sqrt{2}f_{1}/|Q_{21}-Q_{22}|$. This is an example of kinetic alignment [18, 26], in which alignment between the kinetic metric and the charges is imposed.

For $N$ axions, except for charge alignment scenarios like the clockwork mechanism [15, 35, 36], we may also obtain an enhancement factor $\sqrt{N}$ simply from the kinetic metric, due to the number of summed terms appearing inside the square root in (28), as in the N-flation scenario. Strategic alignment between the kinetic metric and the charges can also provide additional enhancement if the eigenvalues of the decay constant have a hierarchical structure. The distribution of enhancement factors for randomly generated Kähler metrics, as well as the possibility of both kinds of enhancement, was studied in Ref. [25].

There is further room for enhancement of the axion decay constant. If the axion is obtained from gaugino condensation in a supersymmetric gauge theory, as will be discussed in detail in Section 3.1, the decay constant may increase by a factor of $N$, the rank of the $U(N)$ gauge group.

In gaugino condensation of a strongly coupled $U(N)$ gauge theory, the axion arises as the imaginary part of a complex field $T$, $\vartheta={\rm Im}T$, appearing in the gaugino condensate superpotential and its resulting scalar potential,

$$W\sim e^{-T/N}\to V\sim\cos\frac{\vartheta}{N}=\cos\frac{a}{Nf}\,.$$

It appears that this would lead to enhancement of the decay constant $f$ by a factor of $N$. However, it is not immediately obvious that this would happen in practice, since there may be degenerate vacua which may lead to tunneling between the different branches of the axion potential. Similarly, in the presence of multiple axions, any tunneling between different minima would ruin the separation between the axions, in turn ruining the alignment. This presents a general problem for the alignment scenario. Here we analyze this possibility by computing the tunneling rate between adjacent branches of the axion potential.

As a warmup, we first analyze the enhancement using a simplified aligned two-axion model, which captures the essential physics of tunneling between different branches. We take a Lagrangian of the form

$${\cal L}=\frac{1}{2}f_{1}^{2}(\partial_{\mu}\vartheta_{1})^{2}+\frac{1}{2}f_{2}^{2}(\partial_{\mu}\vartheta_{2})^{2}+\Lambda_{H}^{4}\cos(N\vartheta_{1}-\vartheta_{2})+\Lambda_{L}^{4}\cos(\vartheta_{1})\,,$$

(31)

where the index “$H$” implies that the corresponding potential is the dominant source making one of the axions heavy, while “$L$” corresponds to the potential that is relevant for the dynamics of the light axion. In the alignment limit, integrating out the heavy axion ($a_{H}\simeq f_{1}\vartheta_{1}$) from the first potential term induces the following effective Lagrangian of the light axion:

$$\begin{split}{\cal L}_{\rm eff}&=\frac{1}{2}\left(f_{2}^{2}+\frac{f_{1}^{2}}{N^{2}}\right)(\partial_{\mu}\vartheta_{2})^{2}+\Lambda_{L}^{4}\cos\left(\frac{\vartheta_{2}}{N}+\frac{2\pi k}{N}\right)\\ &=\frac{1}{2}(\partial_{\mu}a)^{2}+\Lambda_{L}^{4}\cos\left(\frac{a}{Nf_{a}}+\frac{2\pi k}{N}\right)\quad{\rm for}\ k\in\mathbb{Z}\,.\end{split}$$

(32)

Canonical normalization determines the light axion, $a\equiv f_{a}\vartheta_{2}$, with

$$f_{\rm eff}=Nf_{a}=\sqrt{f_{1}^{2}+N^{2}f_{2}^{2}}\,.$$

(33)

Here $f_{\rm eff}$ is enhanced by a factor $N$ compared to $f_{a}$, while there are $N$ branches labeled by $k=0,\cdots,N-1$. For a given branch (e.g. $k=0$), during an excursion of the axion in the range $0\leq a\leq\pi f_{\rm eff}$, quantum tunneling may allow a rapid transition to other branches with lower values of the potential. In such a case, the axion, as the inflaton, can only travel slowly for a distance of ${\cal O}(f_{a})$ due to the deformed scalar potential, cancelling the enhancement factor $N$.

To obtain the tunneling rate between different branches (for instance, from $k=0$ to $-1$), we consider the nucleation rate for the critical bubble which connects two local minima along the direction of the heavy axion with the field displacement $\Delta a_{H}\simeq f_{1}\Delta\vartheta_{1}=2\pi f_{1}/N$. Tunneling can take place whenever the potential at $k=-1$ is lower than the potential at $k=0$. The potential difference between the two positions is given as

$$\Delta V_{L}\equiv V_{L}(k=0)-V_{L}(k=-1)>0\,,$$

(34)

where $V_{L}$ is the light axion potential from (32). The height of the potential barrier between the two local minima along the heavy axion direction is of ${\cal O}(\Lambda_{H}^{4})$, and it is hierarchically larger than $\Delta V_{L}$, i.e. $\Lambda_{H}^{4}\gg\Lambda_{L}^{4}\gtrsim\Delta V_{L}$. Therefore, one can use the thin wall approximation to evaluate the bounce solution following [37]. Ignoring gravitational effects, the transition rate is given by

$$\Gamma_{\rm tunnel}\propto e^{-S_{E}}\,,$$

(35)

where $S_{E}$ is the bounce action

$$S_{E}=\frac{27\pi^{2}}{2}\frac{\sigma_{B}^{4}}{(\Delta V_{L})^{3}}\,.$$

(36)

Here $\sigma_{B}$ is the tension of the bubble wall, to be calculated shortly, and $\Delta V_{L}$ is the potential difference between the branches (34).

In principle, since the potential is not a local minimum for the light axion either before or after tunneling, we should also consider the kinetic energy of $a$ and the field displacement $\Delta a$ during tunneling. However, owing to the fact that the light axion is slowly rolling during inflation, we can safely ignore the effect of the axion kinetic energy. Furthermore, the possibility of nonzero displacement $\Delta a$ of the light axion leads to a larger total field distance $\sqrt{\Delta a_{H}^{2}+\Delta a^{2}}$ traversed during the tunneling process. As $\Delta a$ increases, the bubble wall tension increases more significantly compared to $\Delta V_{L}$, so that the corresponding bounce action also increases. Therefore, in order to obtain a conservative bound on the tunneling rate and observe its parametric dependence, it is sufficient to consider the heavy-axion field direction only. For more concrete discussion about the possible effect of the light field dynamics, see [38]. We then estimate the tension as $\sigma_{\rm B}\simeq\Lambda_{H}^{2}\Delta a_{H}$, giving

$$\Delta V_{L}\simeq\frac{2\pi}{N}\Lambda_{L}^{4}\sin(a_{\rm inf}/f_{\rm eff})\,,\qquad\sigma_{\rm B}=\frac{c_{B}\pi}{N}\Lambda_{H}^{2}f_{1}\,,$$

(37)

where $a_{\rm inf}$ is the axion value during inflation, and $c_{B}$ is an ${\cal O}(1)$ number. Therefore, the bounce action is

$$S_{E}\simeq\frac{27c_{B}^{4}\pi^{3}}{16\sin(a_{\rm inf}/f_{\rm eff})^{3}N}\left(\frac{f_{1}}{\Lambda_{L}}\right)^{4}\left(\frac{\Lambda_{H}}{\Lambda_{L}}\right)^{8}\,.$$

(38)

It is not difficult to obtain $S_{E}\geq{\cal O}(100)$ either from the hierarchy between $f_{1}$ and $\Lambda_{L}$ or between $\Lambda_{H}$ and $\Lambda_{L}$, thus tunneling is suppressed and the decay constant may be enhanced, $f_{\rm eff}=Nf_{a}$. Since $N$ appears in the denominator, in effect this restricts $N$ to be small. To estimate the allowed value, we consider conservative values: $\sin(a_{\rm inf}/f_{\rm eff})\simeq 1$, $c_{B}^{4}\simeq 0.01$, $\Lambda_{H}/\Lambda_{L}\simeq 10$, $f_{1}/\Lambda\simeq 0.1$, giving $S_{E}\simeq 5232/N$. This shows that typical values, $N\lesssim 50$, are robust.

We can use the same analysis on the multi-axion model discussed in the previous section, where we allowed general instanton charges $Q_{Ij}$ as the coefficients of the respective axions $\vartheta_{j}$. Again we focus on the two-axion case. In the alignment limit, the heavy axion $\phi_{\psi}$ may be integrated out, leading to the possibility of degenerate vacua depending on the ratio of charges. Then although we have a light axion with a large effective decay constant, again there can potentially be tunneling between different minima, ruining the alignment.

Before integrating out heavy axion $\phi_{\psi}$, the Lagrangian (19) shows that its excursion distance is $\Delta\phi_{\psi}\simeq 2\pi f_{\psi}/Q_{1\psi},$ giving the bubble tension

$$\sigma_{B}=\frac{c_{B}\pi}{\sqrt{Q_{11}^{2}+Q_{12}^{2}}}\Lambda_{1}^{2}f_{\psi}.$$

Note that in the alignment limit, $Q_{2\psi}/Q_{1\psi}\to Q_{21}/Q_{11}$. The local minima of the first potential gives $\langle Q_{1\psi}\phi_{\psi}/f_{\psi}\rangle=2\pi k$, which leads to $Q_{11}/{\rm gcd}(Q_{11},Q_{21})\equiv N^{\prime}$ number of branches for the light axion. In most cases, $Q_{11},Q_{21}$ are unrelated and they can be easily relatively prime so that $N^{\prime}=Q_{11}.$ In the worst case, we have $N^{\prime}=2$. If $1/N^{\prime}$ is an integer, we do not have degenerate vacua. For the effective Lagrangian (23), the transition between the branches $k=0$ and $k=-1$ gives

$$\Delta V_{L}\simeq\frac{2\pi}{N^{\prime}}\Lambda_{2}^{4}\sin\frac{a_{\rm inf}}{f_{\rm eff}},\quad$$

(39)

Then, the bounce action is

$$S_{E}\simeq\frac{27c_{B}^{4}\pi^{3}N^{\prime 3}}{16\sin(a_{\rm inf}/f_{\rm eff})^{3}(Q_{11}^{2}+Q_{12}^{2})^{2}}\left(\frac{f_{\psi}}{\Lambda_{2}}\right)^{4}\left(\frac{\Lambda_{1}}{\Lambda_{2}}\right)^{8}.$$

(40)

This time, the tunneling may prohibit large charges $Q_{11},Q_{12}$. Assuming $Q_{11}\simeq Q_{12}\equiv Q$, we estimate similarly as above $S_{E}\simeq 1308N^{\prime 3}/Q^{4}$, requiring $Q_{Ij}\lesssim 3N^{\prime 3/4}$.

Finally, we may combine both effects. The restriction on the rank of the gauge group from the initial toy model can be directly applied to the aligned axion potential. As a result, each instanton charge $Q_{Ij}$ is effectively rescaled by the rank $N_{I}$ of the corresponding gauge group,

$$Q_{Ij}\to Q_{Ij}/N_{I}\,.$$

(41)

Then the bounce action is rescaled as $S\to SN_{1}^{4}$, which allows larger charges $Q_{Ii}$ due to the effective suppression by $N_{I}$ in (41), giving the less stringent bound

$$Q_{Ij}\lesssim 3N_{I}N^{\prime 3/4}\,.$$

(42)

Consider type IIB string theory compactified on a Calabi–Yau orientifold $X$. At energies below the compactification scale, the physics should be described by an effective four-dimensional $\mathcal{N}=1$ supergravity theory with various massless scalar fields: the axio-dilaton $S=e^{-\phi}+iC_{0}$, where $g_{s}=e^{\phi}$ gives the string coupling, $C_{0}$ is the RR scalar, Kähler moduli $T_{i}=\tau_{i}+ib_{i}$ that we define below, and complex structure moduli $U_{a}=u_{a}+ic_{a}$ associated with the shape of the internal manifold. Here and in what follows, dimensionless lengths are measured in the string scale $\ell_{s}=2\pi\sqrt{\alpha^{\prime}}$ with Regge slope $\alpha^{\prime}$.

The Kähler form of $X$ may be expanded in the generators $D_{i}$ of $H^{1,1}(X,{\mathbb{Z}})$,

$$J=t^{i}D_{i}\,.$$

It follows that the volume ${\cal V}$ of $X$ may be expressed as [41, 42]

$${\cal V}=\frac{1}{6}\int_{X}J\wedge J\wedge J=\frac{1}{6}\kappa_{ijk}t^{i}t^{j}t^{k}\,.$$

(43)

Using the Poincaré dual four cycles $D_{i}$ in $H_{4}(X,{\mathbb{Z}})$,⁶⁶6We denote the Poincaré dual pair of forms and cycles using the same notation. which are divisors of $X$, we define the triple intersection numbers $\kappa_{ijk}\equiv D_{i}\cdot D_{j}\cdot D_{k}$. To leading order, the Kähler potential $K$ is given by

$$K=K_{0}-2\ln{\cal V}\equiv\frac{\hat{K}}{M_{\rm Pl}^{2}}\,,$$

(44)

where $K_{0}$ encapsulates the $S$ and $U_{a}$ dependence and we have restored the mass dimension in $\hat{K}$.

Further natural quantities are the volumes of the four-cycles $D_{i}$,

$$\tau_{i}=\frac{1}{2}\int_{D_{i}}J\wedge J=\frac{1}{2}\kappa_{ijk}t^{j}t^{k}\,.$$

(45)

These form the complexified Kähler moduli, $T_{i}=\tau_{i}+ib_{i},i=1,\dots,h^{1,1}(X)$, along with the shift-symmetric axions

$$b_{i}=\int_{D_{i}}C_{(4)}\,,$$

(46)

where $C_{(4)}$ is the RR four-form.

The continuous shift symmetry is perturbatively exact and is only broken by non-perturbative effects. Here it is broken to a discrete symmetry by branes wrapped on the corresponding four-cycles $D_{i}$. The normalization in (46) gives the periodicity $b_{i}\to b_{i}+1$, such that the generators $D_{i}$ form a basis of the integral homology.

We consider a number $M$ of stacks of D7- or Euclidian D3-branes. Each stack wraps cycles $D_{j}$ an integer $Q_{Ij}$ number of times,

$$D_{I}=Q_{Ij}D_{j}\,,\quad Q_{Ij}\in{\mathbb{Z}}\,,\quad I=1,\dots,M\,.$$

(47)

Along these branes we can define new Kähler moduli generalizing (45) and (46),

$$\int_{D_{I}}\left(\frac{1}{2}J\wedge J+iC_{(4)}\right)=Q_{Ij}T_{j}\,.$$

(48)

The non-perturbative superpotential that obeys the above periodicity is

$$W=W_{0}+\sum_{I=1}^{M}A_{I}e^{-c_{I}Q_{Ij}T_{j}}\equiv\frac{\sqrt{4\pi}}{g_{s}^{3/2}M_{\rm Pl}^{3}}\hat{W}\,,$$

(49)

where $W_{0}$ is the order-one tree-level superpotential, to be discussed in (99), $A_{I}$ are order-one coefficients and

$$c_{I}=\begin{cases}2\pi\,,\qquad\;\text{ED3}\,,\\ 2\pi/N_{I}\,,\quad\text{D7}\,.\\ \end{cases}$$

(50)

In the former case, a D3-brane wrapped on a four-cycle becomes an instanton and the superpotential describes the instanton effect. In the latter, a stack of $N_{I}$ D7-branes gives rise to a $U(N_{I})$ gauge theory, and the corresponding superpotential is the well-known one from gaugino condensation, with gauge coupling $\tau_{i}$.

Consideration of the D7-gauge anomaly shows that indeed this winding number $Q_{Ij}$ is nothing but the anomaly coefficient (12). For the D3-worldvolume gauge theory the same interpretation follows, which can be extrapolated to the case of a Euclidian D3-brane. In this brane setup, negative $Q_{ij}$ is also allowed and corresponds to D-branes winding in the opposite direction. To ensure stability of the D-brane setup we need supersymmetry, which in turn requires a special relation between the charges. This relation is also subject to global consistency conditions. In this work, we focus on the dynamics of the axions and leave the complete construction including the Standard Model to future work.

The range of the moduli fields $\tau_{i},t^{i}$ are further restricted because volumes should be positive-definite and the instanton action should be positive. For all holomorphic curves, the the Käher form should be positive definite,

$$\int_{C}J>0\,,$$

(51)

such that the resulting volumes are non-negative: that is, $J$ should lie in the Kähler cone. We may express this condition in terms of the set of effective curves $C$, by demanding that $C\cdot D\geq 0$ for any divisors $D$ of $X$. This set of curves forms the Mori cone. Writing the generators of the Mori cone as $C_{i}$, the Kähler cone condition becomes

$$\int_{C_{i}}J=t^{j}\int_{C_{i}}D_{j}\geq 0\,.$$

(52)

The conditions on $t^{j}$ can in turn be translated into conditions on $\tau_{i}$ through the relation (45). This not only guarantees the positivity of the volumes $\tau_{i}$, but restricts their allowed ranges.

The leading axion-dependent terms in the scalar potential are

$$V\supset-e^{K}K^{j\bar{\imath}}\sum_{I=1}^{M}\left[A_{I}c_{I}Q_{Ij}e^{-c_{I}Q_{Ik}T_{k}}\overline{W}_{0}\partial_{\bar{\imath}}K+\overline{A}_{I}c_{I}Q_{I{\bar{\imath}}}e^{-c_{I}Q_{Ik}\overline{T}_{k}}W_{0}\partial_{j}K\right]\,.$$

(53)

Assuming $W_{0}$ to be $\mathcal{O}(1)$, further terms such as $e^{-c_{I}Q_{Ij}T_{j}-c_{I}Q_{Ij}T^{*}_{j}}$ are suppressed, since we are considering the region ${\rm Re}T_{i}>1$ and $c_{I}Q_{Ij}=2\pi Q_{Ij}/N_{I}\simeq 2\pi\gg 0$, in which the exponential suppression is sufficiently strong. Using the relation $K^{j\bar{\imath}}\partial_{\bar{\imath}}K=-2\tau_{j},K^{j\bar{\imath}}\partial_{j}K=-2\tau_{i}$, the leading order scalar potential becomes

$$V=4e^{K}W_{0}\sum_{I=1}^{M}A_{I}S_{I}e^{-S_{I}}\cos(c_{I}Q_{Ij}b_{j})\,,$$

(54)

with the instanton actions

$$S_{I}=c_{I}Q_{Ij}\tau_{j}\,.$$

(55)

Note that we may complexify the axion and obtain the above superpotential purely from supersymmetry. Thus, it is supersymmetry which is responsible for fixing the instanton action in terms of the Kähler moduli. Naturally, the superpotential makes sense if the instanton action is positive-definite. Note also that is the leading order expression. Furthermore, to complete the scenario we need moduli stabilization and correlation functions between the various fields, which should be related to supersymmetry breaking and uplifting.

The physics of axions is highly dependent on the internal space. Axions are paired with the volume moduli of four-cycles, and thus the geometrical structure of the four-cycles determines the axion potential and decay constants through the superpotential and Kähler metric, respectively. It then stands to reason that an appropriate choice of Calabi–Yau orientifold is necessary for realizing axion alignment.

A general Calabi–Yau orientifold gives rise to multiple $h^{1,1}>1$ axions. In principle, alignment is a restriction on charges as in (16), and regardless of the sizes of the decay constants, all of these axions can participate in the alignment. Here we will argue that

The axions taking part in the alignment should be those associated with anisotropic bulk four-cycles.

By bulk cycles we mean those with hierarchically large volumes in the sense of moduli stabilization by means of the Large Volume Scenario (LVS). By anisotropic, we mean that the bulk volume is locally factorizable as a product of those bulk cycles, such that multiple light axions remain after moduli stabilization. This nontrivial requirement arises because the axions associated with small cycles are decoupled at high energy scales by moduli stabilization. Furthermore, as we will see in section 3.5, the weak gravity conjecture favors large cycles with similar sizes. In other words, we consider anisotropic bulk geometry in the near-isotropic limit. The importance of anisotropic geometry has been stressed in many works [46, 47, 48, 55].

A minimal choice on which we will focus our analysis in this paper is the well-known K3 fibration over $\mathbb{CP}^{1}$. In this case, the bulk volume is determined by two moduli $\tau_{1}$ and $\tau_{2}$,

$${\cal V}=\alpha\sqrt{\tau_{1}}\tau_{2}\,.$$

(56)

Here the overall normalization $\alpha$ is determined by a concrete realization of the geometry; however, shortly we see that the key physics is independent of $\alpha$. One example of this type of geometry can be realized as a degree-12 hypersurface in ${\mathbb{C}}{\mathbb{P}}^{4}_{[1,1,2,2,6]}$ [41] under a suitable redefinition. In our explicit calculations we will mostly refer to this example — the geometric construction is summarized in the Appendix.

We will stabilize the moduli using Large Volume Scenario. This mechanism relies on contributions from additional small blow-up cycles. To this end, we also supplement our construction with “swiss cheese” geometry by including small internal cycles. For the case of ${\mathbb{C}}{\mathbb{P}}^{4}_{[1,1,2,2,6]}(12)$, we introduce a further small cycle denoted $\tau_{s}$. The overall volume is now of the form

$${\cal V}=\alpha(\tau_{1}^{1/2}\tau_{2}-\gamma\tau_{s}^{3/2})\,,$$

(57)

where $\gamma$ depends on the details of the blow-up.

Other examples can yield similar geometries. The manifold $\mathbb{CP}^{4}_{[1,1,2,2,2]}(8)$ obtains a similar form,

$${\cal V}=\alpha(\tau_{1}^{1/2}\tau_{2}-\gamma_{3}\tau_{3}^{3/2}-\gamma_{4}\tau_{4}^{3/2})\,,$$

(58)

with $\alpha=\sqrt{2}/12$ and $\gamma_{3}=\gamma_{4}=2\sqrt{2}/3$. Also the geometry ${\mathbb{C}}{\mathbb{P}}^{4}_{[1,1,1,1,4]}(8)$ with three blow-ups yields a volume [48]

$${\cal V}=\alpha(\sqrt{\tau_{1}\tau_{2}\tau_{3}}-\gamma\tau_{s}^{3/2})\,,$$

(59)

with $\alpha=\sqrt{2}/4$ and $\gamma=2\sqrt{2}/3$, leading to three large cycles and one small cycle. Although the three manifolds introduced here appear structurally different, all three are K3 fibrations over a two-base and lead to the same general structure,

$${\cal V}=nt^{a}\tau_{a}-\alpha\sum_{\tau_{i}\text{ small}}\gamma_{i}\tau_{i}^{3/2}\quad\text{($a$ not summed)},$$

(60)

where $\tau_{a}$ is the volume of the K3 fiber, $t^{a}$ is that of the two-base and $n$ is the integer power to which $t^{a}$ appears in the bulk volume, ${\cal V}\propto(t^{a})^{n}$. There are also requirements on the small cycles. In particular, in order to have a stable minimum, at least one of them should be a blown-up cycle resolving a singularity [41].

One could also consider manifolds with more cycles, but in the present work we limit our discussion to small $h^{1,1}\lesssim 4$. This is because the Kähler cone condition restricts the range of moduli space, prohibiting large decay constants [43]; moreover, with many cycles the cone becomes very narrow, yielding large cycle volumes $\tau_{i}\sim(h^{1,1})^{p}$ [44]. Although the KNP mechanism works well regardless of the axion decay constants, for consistent moduli stabilization other cycles should be sufficiently small, such that we may expect their relatively heavy axions to decouple.

Note that in all of the above geometries, the volumes $\tau_{i}$ are not those of the cycles of integral cohomology $H^{1,1}(X,{\mathbb{Z}})$. We need a suitable redefinition of the diagonal form of the volume, which depends on the details of the construction. This imposes an important constraint on the range of the Kähler parameter $t^{i}$ to satisfy the Kähler cone condition (52). For example,

	$\displaystyle{\mathbb{C}}{\mathbb{P}}^{4}_{[1,1,2,2,6]}(12)$	$\displaystyle:\tau_{2}\geq\frac{4}{3}\tau_{1}>0,$		(61)
	$\displaystyle{\mathbb{C}}{\mathbb{P}}^{4}_{[1,1,2,2,2]}(8)$	$\displaystyle:\tau_{2}\geq 4\tau_{1}>0,\ \tau_{3}\geq\tau_{4}\geq\frac{1}{2}\tau_{1}>0,$		(62)
	$\displaystyle{\mathbb{C}}{\mathbb{P}}^{4}_{[1,1,1,2,4]}(8)$	$\displaystyle:\frac{2\tau_{s}}{\tau_{i}}<\frac{\tau_{j}}{\tau_{k}}<\frac{\tau_{i}}{2\tau_{s}},\quad i\neq j\neq k.$		(63)

Finally, the Large Volume Scenario makes use of an ${\cal O}(\alpha^{\prime 3})$ correction in order to realize the large-volume minimum, which should induce a positive contribution to the potential. We will see that requiring this term to be positive implies that the Euler number should be negative, $\chi(X)\equiv 2h^{1,1}-2h^{2,1}<0.$

The axion decay constants are determined by the geometry and encoded in the Käher metric. The Kähler metric $K_{i\bar{\jmath}}\equiv\partial^{2}K/(\partial T_{i}\partial T_{j}^{*})=4\partial^{2}K/(\partial\tau_{i}\partial\tau_{j})$ is a real, symmetric matrix, giving the kinetic terms

$${\cal L}_{\rm kin}=\sum_{i,j}K_{i\bar{\jmath}}\partial_{\mu}T_{i}\partial^{\mu}T_{j}^{*}\,.$$

(64)

If the Kähler metric is diagonal, each entry is the square of the axion decay constant, in the normalization that the scalar fields have periodicities $2\pi$,

$${\cal L}_{\rm kin}=\sum_{i}\left[\frac{f_{i}^{2}}{2}\partial_{\mu}(2\pi b_{i})\partial^{\mu}(2\pi b_{i})+\frac{f_{i}^{2}}{2}\partial_{\mu}(2\pi\tau_{i})\partial^{\mu}(2\pi\tau_{i})\right]\,.$$

(Note the different normalization from $\vartheta$ in Section 2). Comparing these two expressions, we may read off the decay constants,

$$f_{i}^{2}=\frac{1}{2\pi^{2}}\langle K^{\rm diag}_{i\bar{\imath}}\rangle\quad(\text{no summation)}\,.$$

For a non-diagonal Kähler metric, we may define the matrix-valued decay constant ${\bf f}$ as in (25),

$${\bf f}_{ki}{\bf f}_{kj}=\frac{1}{2\pi^{2}}\langle K_{i\bar{\jmath}}\rangle\,.$$

(65)

The moduli dependence of decay constants can be understood by considering a generic volume of the form⁷⁷7We assume three or fewer bulk four-cycles. Calabi–Yau manifiolds in general are non-simplicial and we cannot express the volume simply in terms of $\tau_{i}$s.

$${\cal V}=\alpha(\tau_{1}^{p_{1}}\tau_{2}^{p_{2}}\tau_{3}^{p_{3}}+\beta\tau_{1}^{q_{1}}\tau_{2}^{q_{2}}\tau_{3}^{q_{3}})\equiv\alpha({\cal V}_{f}+\beta{\cal V}_{s})\,.$$

(66)

Here due to the intersection structure (43), the powers $p_{i},q_{i}$ can take only non-negative half-integer values. Noting that $\tau_{i}$ are the volumes of four-cycles, the powers must additionally satisfy $p_{1}+p_{2}+p_{3}=q_{1}+q_{2}+q_{3}=3/2=\dim_{\mathbb{C}}X/2$, with $dim_{\mathbb{C}}X=3$ the complex dimension of $X$. In the simple case $p_{1}=p_{2}=p_{3}=1/2,\beta=0$, (66) describes the volume of a complex threefold as a product of three Riemann surfaces.