††institutetext:  1Department of Physics, Kyushu University, 744 Motooka,
Nishi-ku, Fukuoka 819-0395, Japan††institutetext:  2Depertment of Physics &
Research and Education Center for Natural Sciences, Keio University, Hiyoshi
4-1-1, Yokohama, Kanagawa 223-8521, Japan

# Stringy constraints on primordial electromagnetic fields in axion inflation

Hajime Otsuka1 and  Ryo Yokokura2<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study primordial electromagnetic fields in effective actions of string
theory. In contrast to a conventional scenario of producing primordial
electromagnetic fields induced by the axion inflation, we deal with the Dirac-
Born-Infeld action as a non-linear generation of Maxwell theory. It turns out
that the intensity of generated electromagnetic fields is bounded from above
by the string scale which can also be rewritten in terms of supersymmetry
breaking scale in the context of type IIB Large Volume Scenario. The
instability parameter $\xi$ is constrained by the tadpole cancellation
condition of D3-branes and a realization of hierarchy between the string scale
and the Hubble scale of inflation.

††preprint:

##  1 Introduction

Inflationary models in the early Universe have been extensively studied in
past decades. Among others, axion inflation Freese:1990rb ; Adams:1992bn ;
McAllister:2008hb , i.e., inflation with an axion inflaton has several
features from the theoretical and observational viewpoints. One is that the
flatness of the inflaton potential can be originated from a classical shift
symmetry of the axion, because the axion can be understood as a Nambu-
Goldstone boson. The shift symmetry can be explicitly broken by some effects
such as non-perturbative effects. It is possible to generate a potential which
is consistent with observational data Planck:2018jri .

Another feature is that the axion inflation models can generate $U(1)$ gauge
fields tachyonically. There can be topological couplings between the axions
and $U(1)$ gauge fields. The couplings can be originated from a chiral anomaly
of the shift symmetry of the axions. In the presence of the non-zero axion
velocity, the topological couplings become first order spatial derivatives,
and give tachyonic modes in infrared region Carroll:1989vb ; Anber:2006xt .
The generated electromagnetic fields can be stable because it has a
topological charge called the magnetic helicity (see e.g., Ref. Davidson2001
). This mechanism is called the chiral instability, which has been discussed
and generalized in various fields in modern physics Akamatsu:2013pjd ;
Ohnishi:2014uea ; Yamamoto:2023uzq  (see also a recent review Kamada:2022nyt
). If we consider the chiral instability in the axion inflation models, the
maximum of the energy density for the electromagnetic fields can be greater
than the Hubble scale, and this mechanism has been applied to explain the
primordial magnetic fields Garretson:1992vt ; Barnaby:2011qe ; Fujita:2015iga
; Anber:2015yca ; Adshead:2016iae  or baryogenesis Giovannini:1997eg ;
Bamba:2006km ; Anber:2015yca ; Fujita:2016igl ; Kamada:2016eeb ;
Kamada:2016cnb ; Cado:2016kdp ; Jimenez:2017cdr  (see, e.g., Refs.
Durrer:2013pga ; Subramanian:2015lua  for reviews of primordial magnetic
fields).

Furthermore, the axion can be ubiquitous in effective theories derived from
string theory. String theory is one of the plausible candidates for quantum
gravity, and it may give us an ultra-violet completion of inflationary models.
In string theory, there are various anti-symmetric tensor fields, and these
zero-modes may be regarded as axions after the compactification of the extra
dimensions. The low-energy effective actions of the axions and $U(1)$ gauge
fields can be obtained in the weak coupling regime (see, e.g., Refs.
Blumenhagen:2006ci ; Ibanez:2012zz ).

If we construct an axion inflation model from string theory, we have several
constraints on the hierarchy of energy scales such as

$M_{\rm axion}<H<V_{\rm max}^{1/4}<M_{\rm comp.}<M_{s}<M_{\rm Pl}.$ (1.1)

Here, $M_{\rm axion}$, $H$, $V_{\rm max}^{1/4}$, $M_{\rm comp.}$, $M_{s}$, and
$M_{\rm Pl}$ denote a mass of the axion inflaton, a Hubble scale during the
inflation, a maximum energy scale of the inflaton potential, a scale of the
compactification for the extra dimensions, string scale, and the reduced
Planck mass, respectively. This hierarchy, in particular the compactification
scale, may constrain the energy scale of the produced electromagnetic fields
via the chiral instability so that the field theoretical description works.
Therefore, it will be a non-trivial question whether the stringy axion
inflation models can lead to the tachyonic production of the electromagnetic
fields whose energy scale is much greater than the Hubble scale.

In this paper, we discuss the constraints on the production of the
electromagnetic fields via the chiral instability in axion inflation models
from the viewpoint of string theory. We consider the inflationary models
obtained from type IIB string theory, where realistic cosmological models can
be successfully obtained. In Sec. 2, we first review the generation of
primordial electromagnetic fields induced by the non-zero axion velocity. In
the conventional scenario, we deal with a model with quadratic kinetic terms
of the axion and photon with the second order of the field strengths. We
discuss the constraints on the produced electromagnetic fields from string
theory in Sec. 3. From the viewpoint of string theory incorporating open
strings, such a model will be regarded as a low-energy description of Dirac-
Born-Infeld (DBI) action as reviewed in Sec. 3.1. In Sec. 3.2, we show that
the scale of intensity of electromagnetic fields is bounded from above by the
string scale; otherwise one has to deal with a non-linear generalization of
Maxwell theory which is a regime beyond the validity of the corresponding
effective field theory. The upper bound on the gauge field strength can be
rewritten in terms of the Hubble scale during the inflation and the
supersymmetry breaking scale in type IIB Large Volume Scenario
Balasubramanian:2005zx ; Conlon:2005ki  in Sec. 3.3. Since all the energy
scales in string theory depend on the string moduli, an amount of primordial
electromagnetic fields is related to the sparticle masses constrained by the
collider experiments. In Sec. 3.4, we analyze the backreaction of
electromagnetic fields induced by the axion inflation on the basis of the DBI
action. So far, we have not specified concrete axion inflationary models. In
Sec. 4, we consider models obtained from type IIB string theory, and examine
whether the above constraint can be satisfied or not. We put the constraints
on the intensity of electromagnetic fields in the stringy models. Since the
ratio the gauge coupling to the axion decay constant is uniquely determined in
weakly-coupled string theory Choi:1985je ; Banks:2003sx ; Svrcek:2006yi , the
intensity of electromagnetic fields can be fixed by a coefficient of axion-
photon coupling and the slow-roll parameter. Interestingly, the coefficient of
axion-photon coupling is restricted to satisfy the cancellation condition of
D-brane charges. Finally, Sec. 5 is devoted to the conclusions and
discussions.

##  2 Primordial electromagnetic fields in axion inflation

In this section, we review the generation of electromagnetic fields induced by
the axion inflation, paying attention to the dynamics of a $U(1)$ gauge field
coupled with the axion topologically. In this paper, we do not specify whether
the $U(1)$ gauge field is the photon of the Standard Model or a hidden $U(1)$
gauge field. Meanwhile, we consider both cases with and without massless
fermions charged under the $U(1)$ gauge field, which necessarily exist in the
Standard Model, but do not necessarily in the hidden sector.

First, we consider the generation of the electromagnetic fields in the absence
of $U(1)$ charged massless fermions. In the conventional scenario to produce
the gauge field on the de Sitter background, the relevant action is

$\displaystyle S$ $\displaystyle=S_{\rm axion}+S_{\rm em},$ (2.1)

with

$\displaystyle S_{\rm axion}$ $\displaystyle=\int
d^{4}x\left\\{\sqrt{-\det(g_{\mu\nu})}\biggl{[}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\biggl{]}\right\\},$
$\displaystyle S_{\rm em}$ $\displaystyle=\int
d^{4}x\left\\{\sqrt{-\det(g_{\mu\nu})}\left[-\frac{1}{4g^{2}}F_{\mu\nu}F^{\mu\nu}\right]+\frac{L\phi}{16\pi^{2}f}F_{\mu\nu}\tilde{F}^{\mu\nu}\right\\}.$
(2.2)

Here, $\phi$ is an axion field with the decay constant $f$,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength
of the $U(1)$ gauge field $A_{\mu}=(-A^{0},\mathbf{A})$ with the gauge
coupling $g$,
$\tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}$ with
$\epsilon^{0123}=+1$ is the Hodge dual of the field strength, $L$ is an
integer parameter and assumed to be $L>0$ without loss of generality, and
$V(\phi)$ is an axion inflaton potential. Note that
$F_{\mu\nu}\tilde{F}^{\mu\nu}=-4\mathbf{E}\cdot\mathbf{B}$.

The background metric $g_{\mu\nu}$ ($\mu,\nu=0,1,2,3$) is chosen as the
Friedmann-Lemaître-Robertson-Walker background with vanishing curvature:

$\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)d\mathbf{x}^{2}=-a^{2}(\eta)(d\eta^{2}-d{\bf x}^{2}),$
(2.3)

where $a$ is the scale factor and $\eta$ is the conformal time. Since the
inflaton develops a time-dependent vacuum expectation value, the conformal
invariance of the gauge field is lost; thereby the gauge field quanta will be
produced during the inflation. It will be convenient to rewrite the Maxwell
term with the canonically normalized kinetic term,

$\displaystyle S=\int
d^{4}x\left\\{\sqrt{-\det(g_{\mu\nu})}\biggl{[}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)-\frac{1}{4}g^{\mu\rho}g^{\nu\sigma}\hat{F}_{\mu\nu}\hat{F}_{\rho\sigma}\biggl{]}+\frac{\alpha
L}{4\pi f}\phi\hat{F}_{\mu\nu}\hat{\tilde{F}}^{\mu\nu}\right\\}.$ (2.4)

Here, $\hat{A}_{\mu}$ is a $U(1)$ gauge field with the canonically normalized
kinetic term, whose index is raised and/or lowered by the metric $g^{\mu\nu}$.
The parameter $\alpha=\frac{g^{2}}{4\pi}$ is the fine structure constant. When
we redefine the gauge field in the comoving coordinate:

$\displaystyle\hat{A}_{\mu}=(-A_{0},{\bf
A})=:A_{\mu},\qquad\hat{A}^{\mu}=\left(\frac{A^{0}}{a^{2}},\frac{{\bf
A}}{a^{2}}\right)=:\frac{A^{\mu}}{a^{2}},$ (2.5)

the action is simplified as

$\displaystyle S=\int
d^{4}x\left\\{\sqrt{-\det(g_{\mu\nu})}\biggl{[}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\biggl{]}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\alpha
L}{4\pi f}\phi F_{\mu\nu}\tilde{F}^{\mu\nu}\right\\},$ (2.6)

where it is notable that the index of comoving (physical) field $A$
($\hat{A}$) is raised and/or lowered by the metric $\eta^{\mu\nu}$
($g^{\mu\nu}$).

The equation of motion for the $U(1)$ gauge field is

$\displaystyle\partial_{\mu}\left(F^{\mu\nu}+\partial^{\nu}A^{\mu}\right)=\partial_{\mu}\left(\frac{\alpha
L}{\pi f}\phi\tilde{F}^{\mu\nu}\right),$ (2.7)

where we adopt the Feynman gauge. When the axion motion is negligible, the
gauge field is quantized in the usual way:

$\displaystyle A_{\mu}(x)=\int\frac{d^{3}k}{(2\pi)^{3/2}}\sum_{\sigma=\pm,{\rm
L},{\rm S}}\biggl{[}\epsilon_{\mu}^{(\sigma)}a_{\bf
k}^{(\sigma)}A_{\sigma}(\eta,{\bf k})e^{i{\bf k}\cdot{\bf x}}+{\rm
h.c.}\biggl{]},$ (2.8)

where $\epsilon_{\mu}^{(\sigma)}$ is the polarization vector, and the operator
$\\{a_{\bf k}^{(\pm)},a_{\bf k}^{(\pm)\dagger}\\}$ satisfies the commutation
relation $[a_{\bf k}^{(\pm)},a_{\bf q}^{(\pm)\dagger}]=\delta^{3}({\bf k}-{\bf
q})$. Then, the equation of motion for $A_{\pm}$ reduces to

$\displaystyle\partial_{\eta}^{2}A_{\pm}(\eta,{\bf k})+k(k\pm 2\lambda\xi
aH)A_{\pm}(\eta,{\bf k})=0$ (2.9)

with $\lambda={\rm sign}(\dot{\phi})$ and

$\displaystyle\xi=\frac{L\alpha\lambda\dot{\phi}}{2\pi fH},$ (2.10)

where $\dot{\phi}=\frac{d\phi}{dt}$ and $H=a^{-1}\frac{da}{dt}$. Thus, one of
the helicities of the gauge field will be a tachyonic mode for $k<2aH\xi$. If
we take $\lambda>0$, the tachyonic mode is $A_{-}$. As we will see below, the
parameter $\xi$ characterizes the intensity of the electromagnetic fields.
This parameter is called the instability parameter (see e.g., Ref.
Jimenez:2017cdr ), which plays an important role in discussing the generation
of the electromagnetic fields by the instability as we will see below. Note
that we assume that the instability parameter can be approximately a constant,
which will be justified during and at the end of the inflation.

By imposing the plane wave for $A_{\pm}\propto e^{-i\omega_{k}\eta}$ with
$\omega_{k}^{2}=k(k\pm 2\lambda aH)$ in an enough inside the horizon
($-k\eta\gg 1$), the gauge field is obtained as

$\displaystyle A_{-\lambda}(\eta,{\bf
k})=\frac{e^{\pi\xi/2}}{\sqrt{2k}}W_{-i\lambda\xi,1/2}(2ik\eta).$ (2.11)

As a result, the physical electric and magnetic fields are estimated as
Jimenez:2017cdr

$\displaystyle\langle\hat{E}^{2}\rangle$
$\displaystyle=\frac{e^{2\pi\xi}}{|\xi|^{3}}H^{4}\left(\frac{|\xi|^{3}}{4\pi^{2}}e^{-\pi\xi}\int_{0}^{\kappa_{\rm
UV}}d\kappa\,\kappa^{3}\frac{\partial}{\partial\kappa}|W_{-i\lambda\xi,1/2}(-2i\kappa)|^{2}\right)$
$\displaystyle\simeq 2.6\times 10^{-4}\frac{e^{2\pi\xi}}{|\xi|^{3}}H^{4},$
$\displaystyle\langle\hat{B}^{2}\rangle$
$\displaystyle=\frac{e^{2\pi\xi}}{|\xi|^{5}}H^{4}\left(\frac{|\xi|^{5}}{4\pi^{2}}e^{-\pi\xi}\int_{0}^{\kappa_{\rm
UV}}d\kappa\,\kappa^{3}\frac{\partial}{\partial\kappa}|W_{-i\lambda\xi,1/2}(-2i\kappa)|^{2}\right)$
$\displaystyle\simeq 3.0\times 10^{-4}\frac{e^{2\pi\xi}}{|\xi|^{5}}H^{4},$
$\displaystyle\langle\mathbf{E}\cdot\mathbf{B}\rangle$
$\displaystyle=\lambda\frac{e^{2\pi\xi}}{|\xi|^{4}}H^{4}\left(-\frac{|\xi|^{4}}{8\pi^{2}}e^{-\pi\xi}\int_{0}^{\kappa_{\rm
UV}}d\kappa\,\kappa^{3}\frac{\partial}{\partial\kappa}|W_{-i\lambda\xi,1/2}(-2i\kappa)|^{2}\right)$
$\displaystyle\simeq 2.6\times
10^{-4}\lambda\frac{e^{2\pi\xi}}{|\xi|^{4}}H^{4},$ (2.12)

where we take $\kappa=-k\eta=k/(aH)$ and $\kappa_{\rm UV}\geq 6$. The
intensity of electromagnetic fields is

$\displaystyle\langle\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}\rangle=2(\langle\hat{B}^{2}\rangle-\langle\hat{E}^{2}\rangle)\simeq
6\times
10^{-4}e^{2\pi\xi}H^{4}\left(\frac{1}{|\xi|^{5}}-\frac{1}{|\xi|^{3}}\right),$
(2.13)

which is highly dependent on $\xi$. When $|\xi|\gtrsim{\cal O}(1)$, one of the
gauge field is amplified due to the tachyonic instability, and the large
production of electromagnetic fields leaves a measurable imprint on
cosmological observables Anber:2006xt ; Jimenez:2017cdr , while we should be
careful about the screening by charged particles Domcke:2018eki ;
Sobol:2019xls ; Fujita:2022fwc  or backreactions Sobol:2019xls ;
vonEckardstein:2023gwk .

We should remark that the scale of the intensity or the energy scale of the
electromagnetic fields can exceed the Hubble scale. However, the energy of the
electromagnetic fields cannot exceed the maximum value of the inflaton
potential $V_{\rm max}^{1/4}$. This constraint further implies that it cannot
be greater than $M_{\rm comp.}$ and $M_{s}$ to preserve the hierarchy (1.1) if
we consider four-dimensional (4D) models using string theory.

Next, we include massless fermions charged under the $U(1)$ gauge field
Domcke:2018eki . Since the electromagnetic fields produce the massless
fermions by the Schwinger mechanism, and the generation of the electromagnetic
fields is non-trivially changed. For simplicity, we consider a Dirac fermion
$\psi$ described by the Lagrangian,

${\cal L}_{\rm
f}=i\bar{\psi}\gamma^{a}e_{a}{}^{\mu}\left(\partial_{\mu}+\frac{1}{4}\omega_{\mu}{}^{ab}\Sigma_{ab}+iA_{\mu}\right)\psi,$
(2.14)

where $\gamma_{a}$, ($a=0,1,2,3$) is the Gamma matrix, with
$\\{\gamma_{a},\gamma_{b}\\}=-2\eta_{ab}$,
$\Sigma_{ab}=\frac{1}{2}[\gamma_{a},\gamma_{b}]$ is the generator of the
Lorentz group acting on the Dirac fermion, $e_{a}{}^{\mu}$ is the vielbein
satisfying $\eta_{ab}=e_{a}{}^{\mu}e_{b}{}^{\nu}g_{\mu\nu}$, and
$\omega_{\mu}{}^{ab}$ is the spin connection made of the vielbein. Since the
Dirac fermion is massless, there is classically the axial symmetry denoted as
$U(1)_{A}$ described by the conserved current
$j_{A}^{\mu}=\bar{\psi}\gamma^{\mu}\gamma_{5}\psi$, but it is violated by the
axial anomaly, $\partial_{\mu}\langle
j_{A}^{\mu}\rangle=\frac{1}{2\pi^{2}}\mathbf{E}\cdot\mathbf{B}$. This shows
that the parallel electric and magnetic fields can be a source of the fermion.

While it is in general difficult to discuss the fermion production exactly, it
has been shown in Ref. Domcke:2018eki  that we can approximately compute the
production under the assumption that the electromagnetic fields are constant.
This can be justified because the fermion production is much faster than the
tachyonic production of the electromagnetic fields. Under the assumption, we
can show that the production of the electric current
$j^{\mu}=\bar{\psi}\gamma^{\mu}\psi$ is

$\frac{g}{a^{3}}\langle
j^{3}\rangle\simeq\frac{g^{3}}{6\pi^{2}}\frac{\hat{E}\hat{B}}{H}\coth\left(\frac{\pi\hat{B}}{\hat{E}}\right),$
(2.15)

where we have chosen the direction of the electromagnetic fields as the
$x^{3}$-direction, $\hat{\mathbf{E}}=(0,0,\hat{E})$ and
$\hat{\mathbf{B}}=(0,0,\lambda\hat{B})$. The conservation law of the
electromagnetic fields
$\rho_{A}=\frac{1}{2}(\hat{\mathbf{E}}{}^{2}+\hat{\mathbf{B}}{}^{2})$ from the
Maxwell-Ampère law, $\dot{\rho}_{A}=-4H\rho_{A}+2\xi
H\hat{\mathbf{E}}\cdot\hat{\mathbf{B}}-g\hat{\mathbf{E}}_{i}\cdot\langle\mathbf{j}_{i}\rangle$,
shows that the instability parameter $\xi$ is effectively modified as

$\xi_{\rm
eff}:=\xi-\frac{g^{3}}{12\pi^{2}}\frac{\hat{E}}{H^{2}}\coth\left(\frac{\pi\hat{B}}{\hat{E}}\right).$
(2.16)

The modification of the instability parameter gives us the upper bound on the
electromagnetic fields. The upper bound can be estimated by imposing the
assumption that the time evolution of the electromagnetic energy is
approximately zero, $\dot{\rho}_{A}\simeq 0$ Domcke:2018eki . If $\xi_{\rm
eff}$ is dominated by $\xi\gg 1$, the condition
$\dot{\rho}_{A}=-2H(\hat{\mathbf{E}}{}^{2}+\hat{\mathbf{B}}{}^{2})+2\xi_{\rm
eff}H\hat{\mathbf{E}}\cdot\hat{\mathbf{B}}\simeq 0$ leads to the linear
relation between $\hat{\mathbf{E}}$ and $\hat{\mathbf{B}}$,
$\hat{\mathbf{E}}\simeq\xi\hat{\mathbf{B}}$ with
$|\hat{\mathbf{E}}|>|\hat{\mathbf{B}}|$. On the other hand, this linear
relation is violated when the correction by the Schwinger mechanism cannot be
neglected. The upper bound of the electric field can be roughly estimated
around this region,
$\xi\sim\frac{g^{3}}{12\pi^{2}}\frac{\hat{E}}{H^{2}}\coth\left(\frac{\pi\hat{B}}{\hat{E}}\right)$,
which implies that

$\hat{E}_{\rm max}\sim\frac{12\pi^{2}}{g^{3}}\xi H^{2},$ (2.17)

with the assumption $\coth\left(\frac{\pi\hat{B}}{\hat{E}}\right)\sim 1$. The
upper bound on the intensity of the electromagnetic fields is

$|\langle\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}\rangle|=|2(\langle\hat{B}^{2}\rangle-\langle\hat{E}^{2}\rangle)|<2H^{4}\left(\frac{12\pi^{2}\xi}{g^{3}}\right)^{2}.$
(2.18)

Thus, the exponential growth of the electromagnetic fields by the instability
parameter can be suppressed in the presence of the massless fermion.

In the following sections, we will discuss the possible constraints of the
energy scale of the electromagnetic fields characterized by the instability
parameter $\xi$ from the viewpoint of string theory. One is the constraint
from the DBI action, and another is from the tadpole constraints Bena:2020xrh
; Ishiguro:2023jjc .

##  3 DBI constraints on electromagnetic fields

In this section, we will show that the DBI action gives us some constraints
from string theory. One is the possible value of the coupling constant $g$,
which is determined by the string coupling constant and the string tension.
Another is the upper bound on the generated electromagnetic fields so that the
expansion of the DBI action is under control. We show that the upper bound
will be related to the Planck mass and the volume of the extra dimensions.

###  3.1 Review of DBI action

First, we briefly review the DBI action as known as a non-linear
generalization of Maxwell theory *1*1$*$1Note that other higher-order
corrections for the electromagnetic fields are possible. For example, we can
take the Euler-Heisenberg action, which is a 1-loop effective action after
integrating out a massive field coupled with the gauge field Domcke:2019qmm .
In this case, the constraint on the magnetic field means that it should be
sufficiently small so that the dynamics of the massive mode can be neglected..
(See, Refs. Ibanez:2012zz ; Blumenhagen:2013fgp , in more details.) In the
context of string theory, the DBI action describes the dynamics of the
massless open string modes involving the electromagnetic fields which couple
to the bulk fields in the Neveu-Schwarz sector, i.e., the dilaton $\Phi$, the
metric, and the Kalb-Ramond $B$-field. In particular, for a single D$p$-brane
with $p<9$, the bosonic part in the string frame is described by

$\displaystyle S_{\rm DBI}=-T_{p}\int d^{p+1}\xi
e^{-\Phi(X)}\sqrt{-\det(g_{\alpha\beta}(X)+2\pi\alpha^{\prime}{\cal
F}_{\alpha\beta}(X))},$ (3.1)

where $T_{p}=2\pi l_{s}^{-(p+1)}$ is the tension of D$p$-brane*2*2$*$2Here, we
consider type II string theory, but we need the extra factor $1/\sqrt{2}$ in
the case of type I string theory., $l_{s}=2\pi\sqrt{\alpha^{\prime}}$ is the
string length, and $2\pi\alpha^{\prime}{\cal F}=B+2\pi\alpha^{\prime}F$ is the
gauge invariant field strength including the $U(1)$ gauge field strength $F$.
Here and in what follows, we denote $A,B=0,1,...,9$ for 10-dimensional (10D)
spacetime indices, $\alpha,\beta=0,1,...,p$ for the brane world-volume
coordinates, and $i,j=p+1,...,9$ for the intrinsic world volume coordinates,
respectively. Since the embedding of the world volume ${\cal W}$ into 10D
spacetime is specified by the function $X^{M}(\xi)$, the pull-back metric
$g_{\alpha\beta}$ and $B_{\alpha\beta}$ are respectively given by

$\displaystyle
g_{\alpha\beta}=\partial_{\alpha}X^{A}\partial_{\beta}X^{B}G_{AB},\qquad
B_{\alpha\beta}=\partial_{\alpha}X^{A}\partial_{\beta}X^{B}B_{AB}.$ (3.2)

For simplicity, let us assume the $B$-field and fluctuations of D$p$-brane to
zero and adopt the static gauge*3*3$*$3The fluctuations of D$p$-brane, i.e.,
open string moduli, will be stabilized by four-form fluxes in the context of
M-theory and F-theory compactifications Becker:1996gj ; Sethi:1996es ;
Gukov:1999ya ; Denef:2005mm ; Honma:2017uzn .:

$\displaystyle X^{\alpha}=\xi^{\alpha},\qquad X^{i}=x^{i}.$ (3.3)

In this gauge, the DBI action is expanded as follows:

$\displaystyle S_{\rm DBI}$ $\displaystyle=-T_{p}\int d^{p+1}\xi
e^{-\Phi(X)}\sqrt{-\det(g_{\alpha\beta})}\sqrt{\det(1+2\pi\alpha^{\prime}g^{\alpha\beta}{\cal
F}_{\alpha\beta})}$ $\displaystyle=-T_{p}\int d^{p+1}\xi
e^{-\Phi(X)}\sqrt{-\det(g_{\alpha\beta})}\biggl{[}1+\frac{(2\pi\alpha^{\prime})^{2}}{4}F^{a}_{\alpha\beta}F^{a\alpha\beta}-\frac{(2\pi\alpha^{\prime})^{4}}{12}{\cal
O}(F^{4})\biggl{]},$ (3.4)

with ${\rm tr}(T^{a}T^{b})=\delta^{ab}$.

Since we are interested in the electromagnetic fields, we assume that the
gauge group has a $U(1)$ subgroup. The $U(1)$ gauge coupling on the D$p$-brane
is then extracted as

$\displaystyle g_{{\rm
D}p}^{2}=\frac{e^{\langle\Phi\rangle}}{T_{p}(2\pi\alpha^{\prime})^{2}}=g_{s}(2\pi)^{p-2}(\alpha^{\prime})^{\frac{p-3}{2}},$
(3.5)

where $g_{s}=e^{\langle\Phi\rangle}$ represents the string coupling. Thus, the
Maxwell term as well as the higher-order corrections for the gauge fields are
given by

$\displaystyle S_{\rm DBI}\supset-\int
d^{p+1}\xi\sqrt{-\det(g_{\alpha\beta})}\frac{1}{4g_{{\rm
D}p}^{2}}\biggl{[}F_{\alpha\beta}F^{\alpha\beta}-\frac{(2\pi\alpha^{\prime})^{2}}{3}{\cal
O}(F^{4})\biggl{]}.$ (3.6)

###  3.2 Upper bound on electromagnetic fields from DBI action

Now, we are ready to discuss a constraint on the electromagnetic fields
originating from the DBI action. In the conventional scenario to generate the
4D electromagnetic fields in the early Universe, we often deal with the
quadratic (Maxwell) term of the $U(1)$ gauge field. However, the non-linear
generalization of the Maxwell theory is motivated in string theory. For
concreteness, we consider the DBI action for a D3-brane system in type IIB
string theory. *4*4$*$4When we start from the higher-dimensional DBI action,
one can introduce the electric or magnetic field in the extra-dimensional
space. They are typically of order unity in the unit of the string length,
otherwise the tadpole cancellation conditions will be violated. Thus, the
critical value of the 4D electromagnetic field strength (3.7) is still
controlled by the volume of extra-dimensional spaces, and the following
discussion is applicable to such a system. We will come back to this point in
Sec. 4. The 4D electromagnetic field strength is bounded from above as:

$\displaystyle F^{2}_{\rm
max}\simeq\frac{1}{(2\pi\alpha^{\prime})^{2}}\equiv(2\pi)^{2}M_{s}^{4},\quad\text{or
equivalently,}\quad\hat{F}^{2}_{\rm
max}\simeq\frac{1}{(2\pi\alpha^{\prime})^{2}g_{{\rm
D}3}^{2}}=2\pi\frac{M_{s}^{4}}{g_{s}}\simeq\frac{M_{\rm Pl}^{4}}{8\pi{\cal
V}^{2}};$ (3.7)

otherwise, the DBI action in (3.1) is not justified because the action becomes
imaginary. Note that the string scale
$M_{s}=(l_{s})^{-1}=(2\pi\sqrt{\alpha^{\prime}})^{-1}$ is written in terms of
the 4D reduced Planck mass $M_{\rm Pl}$ and the Einstein-frame volume of extra
six-dimensional space ${\cal V}$ in units of the string length:

$\displaystyle M_{s}=\frac{1}{l_{s}}\simeq\frac{g_{s}^{1/4}}{\sqrt{4\pi{\cal
V}}}M_{\rm Pl}.$ (3.8)

Note that this relation can be obtained from the reduction of the the 10D
Einstein-Hilbert action in the string frame to the 4D one, $S_{\rm
EH,10D}=\frac{2\pi}{g_{s}^{2}l_{s}^{8}}\int d^{10}x\sqrt{-g^{\rm 10D}}R^{\rm
10D}\to S_{\rm EH,4D}=\frac{2\pi{\cal V}_{s}}{g_{s}^{2}l_{s}^{2}}\int
d^{4}x\sqrt{-g}R^{\rm 4D}=\frac{M_{\rm Pl}^{2}}{2}\int d^{4}x\sqrt{-g}R^{\rm
4D}$. Here, $g^{\rm 10D}_{AB}$ and $g^{\rm 10D}$ are the metric in 10D and its
determinant, respectively. The symbols $R^{\rm 10D}$ and $R^{\rm 4D}$ denote
the Ricci scalar in 10D and 4D, respectively. ${\cal
V}_{s}=\frac{1}{l_{s}^{6}}\int d^{6}x\sqrt{g^{\rm 6D}}$ is the volume of the
extra dimensions in the string frame, which is related to ${\cal V}$ in the
Einstein frame as ${\cal V}_{s}=g_{s}^{3/2}{\cal V}$.

From the upper bound, we can find two constraints. One is the upper bound of
the instability parameter $\xi$. The other is the lower bound on the breaking
scale of the supersymmetry because the bound is related to the size of the
extra dimensions. In the following, we estimate the upper bound on the
parameter $\xi$, which characterizes the intensity of the electromagnetic
fields. When the dynamics of electromagnetic fields is governed by the
D3-brane action with the Chern-Simons term, the relevant action is

$\displaystyle S$ $\displaystyle=S_{\rm axion}+S_{\rm em},$ (3.9)

with

$\displaystyle S_{\rm axion}$ $\displaystyle=\int
d^{4}x\left\\{\sqrt{-\det(g_{\mu\nu})}\biggl{[}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\biggl{]}\right\\},$
$\displaystyle S_{\rm em}$ $\displaystyle=\int
d^{4}x\left\\{-T_{3}e^{-\Phi(x)}\sqrt{-\det(g_{\mu\nu})}\Biggl{[}\sqrt{1+\frac{(2\pi\alpha^{\prime})^{2}}{2}F_{\mu\nu}F^{\mu\nu}-\frac{(2\pi\alpha^{\prime})^{4}}{16}(F_{\mu\nu}\tilde{F}^{\mu\nu})^{2}}\,\Biggl{]}\right.$
$\displaystyle\hskip
56.9055pt+\left.\frac{\phi}{16\pi^{2}f}F_{\mu\nu}\tilde{F}^{\mu\nu}\right\\},$
(3.10)

where the axion field $\phi$, its the decay constant $f$, and the axion-photon
coupling can be derived from the kinetic terms of Ramond-Ramond (RR) fields,
and the Chern-Simons coupling on the D3-brane, which we will discuss in Sec.
4.1. $V(\phi)$ is an axion inflaton potential. Note that we have $L=1$ for the
D3-brane effective theories, because D3-branes are localized in the extra
dimensions. After canonically normalizing the gauge fields, the relevant
action of the gauge fields is

$\displaystyle S_{\rm DBI}$ $\displaystyle=-T_{3}\int
d^{4}\xi\sqrt{-\det(g_{\alpha\beta})}\sqrt{1+\frac{(2\pi\alpha^{\prime})^{2}}{2}F_{\alpha\beta}F^{\alpha\beta}-\frac{(2\pi\alpha^{\prime})^{4}}{16}(F_{\alpha\beta}\tilde{F}^{\alpha\beta})^{2}}$
$\displaystyle=-\int d^{4}\xi
e^{-\Phi(X)}\sqrt{-\det(g_{\alpha\beta})}\frac{1}{4}\biggl{[}\hat{F}_{\alpha\beta}\hat{F}^{\alpha\beta}-\frac{(2\pi\alpha^{\prime})^{2}g_{{\rm
D}3}^{2}}{3}{\cal O}(\hat{F}^{4})\biggl{]},$ (3.11)

with $\alpha,\beta=0,1,2,3$. We should remark that the expansion of the DBI
action up to the quadratic order of the field strength is meaningful as long
as $\hat{F}^{2}\ll M_{\rm Pl}^{4}/(8\pi{\cal V}^{2})$, otherwise the higher-
order term cannot be neglected. In the following subsections, we first discuss
the generation of $U(1)$ gauge field incorporating the DBI constraint (3.7) in
Sec. 3.3. In Sec. 3.4, we next examine the backreaction from the DBI action.

###  3.3 Analysis up to quadratic order of electromagnetic fields

After expanding the DBI action (3.10) up to a quadratic order of $F$, we can
read off the axion employed in the conventional scenario to produce the gauge
field on the de Sitter background:

$\displaystyle S_{\rm quad}=\int
d^{4}x\left\\{\sqrt{-\det(g_{\mu\nu})}\biggl{[}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)-\frac{1}{4}g^{\mu\rho}g^{\nu\sigma}\hat{F}_{\mu\nu}\hat{F}_{\rho\sigma}\biggl{]}+\frac{\alpha_{{\rm
D3}}}{4\pi f}\phi\hat{F}_{\mu\nu}\hat{\tilde{F}}^{\mu\nu}\right\\},$ (3.12)

where $\hat{A}_{\mu}$ is a $U(1)$ gauge field with the canonically normalized
kinetic term, and the index is raised and/or lowered by the metric
$g^{\mu\nu}$. The parameter $\alpha_{\rm D3}$ is the fine structure constant
given by $\alpha_{{\rm D3}}=\frac{g^{2}_{\rm D3}}{4\pi}$.

The effective action of string theory will be justified in the small string
coupling and the large volume expansions, as utilized in the Large Volume
Scenario Balasubramanian:2005zx ; Conlon:2005ki . Since the large volume of
the extra-dimensional space significantly reduces the critical value of the 4D
electromagnetic fields, it will be quite important to incorporate this DBI
bound in the context of the tachyonic production of the electromagnetic fields
in the early Universe. In particular, we focus on the generation of 4D
electromagnetic fields in generic axion inflation models.

####  3.3.1 Without massless fermion

Let us examine whether the produced electromagnetic fields in the early
Universe satisfy the DBI bound (3.7). By utilizing Eq. (2.13), we arrive at
the condition such that the expansion of the DBI action can be controlled
against the higher-order corrections,

$\displaystyle\frac{e^{\pi\xi}}{|\xi|^{3/2}}$ $\displaystyle\ll
324\left(\frac{0.1}{g_{s}}\right)^{1/2}\left(\frac{M_{s}}{H}\right)^{2}\simeq
8.1\times 10^{5}\left(\frac{10^{7}}{{\cal
V}}\right)\left(\frac{10^{-6}\,M_{\rm Pl}}{H}\right)^{2},$ (3.13)

indicating that the large volume of the extra-dimensional space strongly
constrains the amount of gauge fields. Here, the string scale is chosen to be
close to the grand unification scale while respecting the hierarchy in Eq.
(1.1), $10^{14}\leq M_{s}\leq 10^{16}$ GeV, taking into account the
uncertainty of high-scale threshold corrections.

The volume range is now within the range $10^{3}\leq{\cal V}\leq 10^{7}$ for
$g_{s}\simeq 0.1$ due to the relation in (3.8). Such a large volume is
preferred in the string compactification due to the fact that the perturbative
string effective field theory is controlled by the volume expansion. As a non-
trivial consequence, it is possible to constrain the lower bound on the
supersymmetry breaking scale. Here, we consider the Large Volume Scenario,
where all moduli can be stabilized and de Sitter uplifting can be realized.
Indeed, in the Large Volume Scenario, the volume modulus is stabilized at the
supersymmetry breaking minimum with exponentially large volume. In particular,
the gravitino mass is given by Conlon:2005ki *5*5$*$5Here, the vacuum
expectation value of Kähler potential for the complex structure moduli is
chosen as $e^{\frac{K_{\rm cs}}{2M_{\rm Pl}^{2}}}=1$.

$\displaystyle
m_{3/2}=\left(\frac{g_{s}^{2}}{2\sqrt{2\pi}}\right)\frac{|W_{0}|}{\mathcal{V}}M_{\rm
Pl},$ (3.14)

where $W_{0}$ is the (dimensionless) vacuum expectation value of the flux-
induced superpotential $W$. Thus, the constraint (3.14) for electromagnetic
fields in the early Universe is also rewritten in terms of the supersymmetry
breaking scale rather than the volume:

$\displaystyle\frac{e^{\pi\xi}}{|\xi|^{3/2}}$ $\displaystyle\ll 8.1\times
10^{5}\left(\frac{50}{|W_{0}|}\right)\left(\frac{0.1}{g_{s}}\right)^{2}\left(\frac{m_{3/2}}{2.4\times
10^{10}\,{\rm GeV}}\right)\left(\frac{10^{-6}\,M_{\rm Pl}}{H}\right)^{2}.$
(3.15)

Note that the magnitude of the flux superpotential is statistically favored at
$|W_{0}|={\cal O}(10)$ Cicoli:2013swa . Of course, it has an option to
consider small $|W_{0}|$ required in the KKLT construction, but it will be a
limited corner of the string landscape. *6*6$*$6See for a realization of the
small flux superpotential by instanton effects, e.g., Refs. Demirtas:2019sip ;
Demirtas:2020ffz ; Alvarez-Garcia:2020pxd ; Honma:2021klo .

Let us apply the condition (3.15) to the production of the gauge field via the
chiral instability. To produce a large amount of gauge field quanta during the
axion inflation, for instance, $\frac{e^{\pi\xi}}{|\xi|^{3/2}}\simeq 6\times
10^{5}$ for $\xi\sim 5$ used in a recent analysis of Ref.
vonEckardstein:2023gwk , the above inequality indicates that the typical
supersymmetry breaking scale (i.e., the gravitino mass) is at least larger
than ${\cal O}(10^{10})$ GeV scale; otherwise one has to incorporate non-
linear effects.

Since the lower bound on the supersymmetry breaking scale can be estimated,
one can further discuss the typical mass scale of the sparticles. In string
models, the scale of soft supersymmetry breaking terms depends on the location
and type of Standard Model D-brane configuration. For illustrative purposes,
let us focus on the sequestered model Higaki:2012ar ; Aparicio:2014wxa  where
the Standard Model sector will be realized at a certain singularity of the
underlying Calabi-Yau orientifold. The explicit model building on concrete
Calabi-Yau orientifold with Minimal Supersymmetric Standard Model (MSSM)-like
D3-branes at singularities was performed in type IIB flux compactifications
with O3/O7-planes Cicoli:2012vw ; Cicoli:2013mpa ; Cicoli:2013cha ;
Cicoli:2017shd ; Cicoli:2021dhg . The sparticle spectrum depends on the
mechanism of uplifting the anti de-Sitter minimum in the Large Volume Scenario
to a de Sitter vacuum as analyzed in Ref. Aparicio:2014wxa . When we focus on
a specific uplifting scenario supported by E$(-1)$-instantons Cicoli:2012fh ,
one can realize both a typical MSSM-like and split supersymmetry spectra:

  1. 1.

Typical MSSM scenario

In the so-called ultra-local scenario, the gaugino mass $M_{1/2}$, soft scalar
masses $m_{0}$ are the same order:

$\displaystyle M_{1/2}\sim m_{0}\sim m_{3/2}\varepsilon\ll m_{3/2},$ (3.16)

with $\varepsilon\sim{\cal V}^{-1}$. Thus, even if the gravitino mass is of
${\cal O}(10^{10})$ GeV, one can realize ${\cal O}(1)$ TeV soft masses due to
the large volume suppression ${\cal V}\simeq 10^{7}$. However, this scenario
will be in tension with bounds on charge and color breaking vacua in the MSSM
Cicoli:2023gbd .

  2. 2.

Split supersymmetry scenario

In the so-called local model, the gaugino and soft scalar masses are also
suppressed by the Calabi-Yau volume:

$\displaystyle M_{1/2}$ $\displaystyle\sim m_{3/2}\varepsilon,$ $\displaystyle
m_{0}$ $\displaystyle\sim m_{3/2}\sqrt{\varepsilon}.$ (3.17)

The large volume ${\cal V}\simeq 10^{7}$ and ${\cal O}(10^{10})$ GeV gravitino
mass lead to the spectra of split supersymmetry: $M_{1/2}\sim{\cal
O}(1)\,\text{TeV},m_{0}\sim{\cal O}(100)\,\text{TeV}$.

In both scenarios, Eq. (3.15) indicates that the supersymmetry breaking scale,
more specifically the sparticle masses constrained by the collider experiments
put a bound on the amount of electric and magnetic fields.

####  3.3.2 With massless fermion

Next, we consider the upper bound by the DBI action in the presence of the
massless fermion. The upper bound in (3.7) with the intensity of the
electromagnetic fields in (2.18) shows that the intensity should be
sufficiently small,

$2(12\pi^{2}\xi)^{2}\ll(2\pi)^{4}g_{s}^{2}\frac{M_{s}^{4}}{H^{4}}\simeq
15.6\left(\frac{g_{s}}{0.1}\right)^{2}\left(\frac{M_{s}}{H}\right)^{4},$
(3.18)

where we have used $g^{2}=g_{\rm D3}^{2}=2\pi g_{s}$. The constraints on $\xi$
in this case is not so strict if we take the mass hierarchy into account,
e.g., $\xi\ll 2.4\times 10^{4}$ for $M_{s}\simeq 10^{3}H$.

###  3.4 Analysis beyond quadratic order of electromagnetic fields

So far, we have analyzed the $U(1)$ gauge field governed by the DBI action up
to a quadratic order. It is however non-trivial how higher order terms affect
our discussion. To see this, we deal with the DBI action by fluctuating the
gauge fields at a given time:

$\displaystyle F_{\mu\nu}\rightarrow\langle F_{\mu\nu}\rangle+\delta
F_{\mu\nu},$ (3.19)

where $\langle F_{\mu\nu}\rangle$ is the background gauge field strength
obtained in the previous section. For simplicity, in the following analysis,
we set the background gauge field to time-independent; the gauge field quanta
is evaluated at a given time.

To analytically evaluate the backreaction, we start from the DBI action in the
limit of $\alpha^{\prime}\ll 1$:

$\displaystyle S_{\rm DBI}$ $\displaystyle=-T_{3}e^{-\langle\Phi\rangle}\int
d^{4}x\sqrt{-\det(g_{\mu\nu})}\sqrt{1+\frac{(2\pi\alpha^{\prime})^{2}}{2}F_{\mu\nu}F^{\mu\nu}}.$
(3.20)

By fluctuating the gauge field (3.19), one can derive the action of $\delta
F_{\mu\nu}$ up to a quadratic order:

$\displaystyle S_{\rm DBI}$ $\displaystyle=-T_{3}e^{-\langle\Phi\rangle}\int
d^{4}x\sqrt{-\det(g_{\mu\nu})}\sqrt{1+\frac{(2\pi\alpha^{\prime})^{2}}{2}F_{\mu\nu}F^{\mu\nu}}$
$\displaystyle=-T_{3}e^{-\langle\Phi\rangle}\int
d^{4}x\sqrt{-\det(g_{\mu\nu})}\sqrt{1+\frac{(2\pi\alpha^{\prime})^{2}}{2}\left(\langle
F_{\mu\nu}\rangle+\delta F_{\mu\nu}\right)\left(\langle
F^{\mu\nu}\rangle+\delta F^{\mu\nu}\right)}$ $\displaystyle=-\int
d^{4}x\sqrt{-\det(g_{\mu\nu})}\frac{1}{4g_{{\rm
D}3}^{2}\sqrt{1+\frac{(2\pi\alpha^{\prime})^{2}}{2}\langle
F_{\mu\nu}F^{\mu\nu}\rangle}}\delta F_{\mu\nu}\delta F^{\mu\nu}+{\cal
O}(\delta F^{4}).$ (3.21)

Thus, we arrive at the action of the fluctuation $\delta F$:

$\displaystyle S_{\rm em}^{(\delta F)}=-\int
d^{4}x\sqrt{-\det(g_{\mu\nu})}\frac{1}{4g_{{\rm
D}3}^{2}\sqrt{1+\frac{(2\pi\alpha^{\prime})^{2}}{2}\langle
F_{\mu\nu}F^{\mu\nu}\rangle}}\delta F_{\mu\nu}\delta
F^{\mu\nu}+\frac{\langle\phi\rangle}{4f}\delta
F_{\mu\nu}\delta\tilde{F}^{\mu\nu},$ (3.22)

where the axion $\phi$ is assumed to be the background value. (See for the
backreaction of the axion field, e.g., Ref. Domcke:2020zez .) The equation of
motion for the fluctuation $\delta F$ is the same as its background, except
for the gauge coupling. The corrected gauge coupling is

$\displaystyle G_{\rm D3}^{2}$ $\displaystyle:=g_{\rm
D3}^{2}\sqrt{1+\frac{(2\pi\alpha^{\prime})^{2}}{2}\langle
F_{\mu\nu}F^{\mu\nu}\rangle}=g_{\rm
D3}^{2}\sqrt{1+(2\pi\alpha^{\prime})^{2}(\langle\hat{B}^{2}\rangle-\langle\hat{E}^{2}\rangle)}$
$\displaystyle\simeq g_{\rm D3}^{2}\sqrt{1+\frac{3\times
10^{-4}}{(2\pi)^{2}}e^{2\pi\xi}\left(\frac{H}{M_{s}}\right)^{4}\left(\frac{1}{|\xi|^{5}}-\frac{1}{|\xi|^{3}}\right)}$
$\displaystyle\simeq g_{\rm D3}^{2}\sqrt{1-7.6\times
10^{-6}\frac{e^{2\pi\xi}}{|\xi|^{3}}\left(\frac{H}{M_{s}}\right)^{4}},$ (3.23)

where we use Eq. (2.13) with $\xi\geq{\cal O}(1)$ in the absence of the
charged fermion. Since the effective coupling constant $G_{{\rm D}3}$ is not
so different from the original one $g_{{\rm D}3}$ for the parameter region
$\xi\lesssim 5$ if the mass hierarchy $H<M_{s}$ is maintained, our discussion
can be valid if the higher-order corrections are taken into account.

##  4 Constraints on electromagnetic fields from D-brane models with axion
inflation

In this section, we show that inflationary models obtained from string theory
satisfy the constraints on the electromagnetic fields. If we consider stringy
inflationary models, several parameters can be related to the quantities in
string theory or constrained by the mass hierarchy. In particular, the
coupling constant $g$ of the electromagnetic fields, the axion decay constant
$f$, the coefficient of the axion-photon coupling $L$, and the cut-off scale
of effective field theories, will be determined or constrained by string
theory. In the following discussion, we show that there are model-independent
constraints on the produced magnetic fields, and confirm the constraints using
some models.

###  4.1 Universal expression of the instability parameter

First, we start with a model-independent analysis of estimating a typical
value of the instability parameter $\xi$ in string axion inflation. By
introducing the so-called slow-roll parameter:

$\displaystyle\epsilon$
$\displaystyle=-\frac{\dot{H}}{H^{2}}=\frac{\dot{\phi}^{2}}{2H^{2}M_{\text{Pl}}^{2}},$
(4.1)

the axion velocity is found to be

$\displaystyle\dot{\phi}^{2}=2\epsilon H^{2}M_{\rm Pl}^{2}.$ (4.2)

Therefore, the instability parameter $\xi$ can be expressed as

$|\xi|=\frac{L\alpha|\dot{\phi}|}{2\pi fH}=L\cdot\frac{M_{\rm
Pl}}{\sqrt{2}\pi}\cdot\frac{\alpha}{f}\cdot\sqrt{\epsilon}.$ (4.3)

Since the electromagnetic fields are amplified toward the end of the
inflation, i.e., $\epsilon\sim{\cal O}(1)$, the instability parameter $\xi$
will be determined by the integer $L$ and the ratio $\frac{\alpha}{f}$. We
will show that the ratio $\frac{\alpha}{f}$ will be universally determined up
to an ${\cal O}(1)$ factor, and $L$ is bounded from above by ${\cal O}(1)$ in
typical toroidal orientifolds.

First, we find that the ratio $\frac{\alpha}{f}$ at the tree-level is
universally determined if the axion-photon couplings are obtained from the
Chern-Simons terms of the D-brane effective theories as previously shown in
Ref. Svrcek:2006yi . Here, the origin of the photon and the axion can be the
gauge fields on the D-branes and the RR $q$-from gauge fields $C_{q}$ in the
10D bulk as

$\tilde{\phi}=\frac{2\pi}{l_{s}^{q}}\int_{\Sigma_{q}}C_{q},$ (4.4)

with a $q$-cycle $\Sigma_{q}$ in the extra dimensions wrapped by the D-branes,
respectively. Here, the axion field $\tilde{\phi}$ is dimensionless. Note that
the rank $q$ cannot be greater than the dimension of D$p$-branes along the
extra dimensions $q\leq p-3$ so that the D-branes can wrap $\Sigma_{q}$. We
will use the inequality later.

When we construct inflationary models using D-branes, $\alpha$ and $f$ depend
on the volume moduli of extra dimensions on which the D-branes wrap. The
volume dependence can be found by the kinetic terms of the axion and photon in
4D effective theories,

$\int d^{4}x\left\\{\sqrt{-\det(g_{\mu\nu})}\left[-\frac{f^{2}_{4{\rm
D}}}{2}\partial_{\mu}\tilde{\phi}\partial^{\mu}\tilde{\phi}-\frac{V_{{\rm
D}p}}{4g^{2}_{{\rm D}p}}F_{\mu\nu}F^{\mu\nu}\right]+\frac{L_{{\rm
D}p}}{16\pi^{2}}\tilde{\phi}F_{\mu\nu}\tilde{F}^{\mu\nu}+\cdots\right\\}.$
(4.5)

Here, $f_{4{\rm D}}$ denotes the axion decay constant derived from the RR
gauge fields. $V_{{\rm D}p}$ is the volume of the extra dimensions on which
the D$p$-branes wrap, and the integer $L_{{\rm D}p}$ is obtained by the
magnetic fluxes in the extra dimensions. The dimensionless axion field
$\tilde{\phi}$ and the axion field $\phi$ in the previous discussions are
related as $f_{\rm 4D}\tilde{\phi}=\phi$. We have omitted other irrelevant
terms. In the following, we will specify these parameters.

The kinetic term of the axion with the decay constant $f_{4{\rm D}}$ can be
obtained from the kinetic term of RR $q$-form gauge field in 10D bulk (see
e.g., Ref. Blumenhagen:2013fgp ),

$S_{{\rm RR},q}=-\frac{2\pi}{l_{s}^{8}g_{s}^{(4-q)/2}}\int
d^{10}x\sqrt{-g^{\rm
10D}}\frac{1}{2(q+1)!}F_{A_{1}...A_{q+1}}F^{A_{1}...A_{q+1}},$ (4.6)

with the field strength
$\frac{1}{(q+1)!}F_{A_{1}...A_{q+1}}dx^{A_{1}}\wedge\cdots\wedge
dx^{A_{q+1}}=dC_{q}$, while the kinetic term of the photon can be originated
from the DBI action on ${\rm D}p$-branes in (3.1). The 4D axion-photon
coupling can be derived from topological couplings to the RR gauge fields on
the D$p$-branes:

$\displaystyle S_{\rm CS}=T_{p}\int\sum_{q}C_{q}\wedge
e^{2\pi\alpha^{\prime}{\cal F}}\wedge\hat{A}(R),$ (4.7)

where $C_{q}$ is pulled back onto the brane world-volume, and $\hat{A}(R)$ is
the so-called A-roof polynomial depending on the curvature 2-form. Through
Kaluza-Klein reduction, the 4D Chern-Simons coupling depends on the D$p$-brane
worldvolume fluxes in some cases whose typical values are of ${\cal O}(1)$
denoted by $L_{{\rm D}p}$, as will be discussed below.

When the volume of $(p-3)$-cycle $V_{{\rm D}p}$ on which ${\rm D}p$-branes
wrap is isotropic, i.e., $V_{{\rm D}p}=R^{p-3}$, the 4D gauge coupling
constant is described by

$\displaystyle g_{4{\rm D}}^{2}=\frac{g_{{\rm D}p}^{2}}{V_{{\rm D}p}}=2\pi
g_{s}\left(\frac{l_{s}}{R}\right)^{p-3}.$ (4.8)

The decay constant of the axion, in particular zero-modes of the $q$-form RR
field $C_{q}$, can be read off from its kinetic term Svrcek:2006yi

$\displaystyle\frac{f_{4{\rm D}}}{M_{\rm
Pl}}=\frac{g_{s}}{\sqrt{8\pi^{2}}}\left(\frac{l_{s}}{R}\right)^{q},$ (4.9)

Thus, the ratio $\alpha_{4{\rm D}}=g_{4{\rm D}}^{2}/(4\pi)$ to $f_{4{\rm D}}$
can be determined as

$\displaystyle\frac{\alpha_{4{\rm D}}}{f_{4{\rm D}}}=\frac{\sqrt{2}\pi}{M_{\rm
Pl}}\left(\frac{l_{s}}{R}\right)^{p-3-q}.$ (4.10)

Since $q\leq p-3$, the ratio must be $\frac{\alpha_{4{\rm D}}}{f_{4{\rm
D}}}\leq\frac{\sqrt{2}\pi}{M_{\rm Pl}}$ as long as the hierarchy $l_{s}<R$,
i.e., $M_{\rm comp.}<M_{s}$ in (1.1) holds. Since we are considering the upper
bound on the instability parameter, we focus on the case $p-3=q$, where the
ratio is universally determined as

$\displaystyle\frac{\alpha_{4{\rm D}}}{f_{4{\rm D}}}=\frac{\sqrt{2}\pi}{M_{\rm
Pl}}.$ (4.11)

By substituting the ratio to Eq. (4.3), we obtain

$|\xi|=L_{{\rm D}p}\sqrt{\epsilon}.$ (4.12)

Note that even in an anisotropic case, one can obtain the similar bound, as
discussed in Ref. Svrcek:2006yi . Here, we have focused on type II string
theory, but the ratio $\alpha$ to $f$ is the same order in heterotic string
theory. Thus, if we realize the axion inflation in heterotic string theory, as
discussed in e.g., Refs. Abe:2014pwa ; Ali:2014mra , one can derive the same
expression for $\xi$ in the same manner as in type II string theory.

Before going into the detail about the constraint on $L_{{\rm D}p}$, let us
comment on the ratio $\frac{\alpha}{f}$. We have focused on the ratio
$\frac{\alpha}{f}$ for the axion which appears in the gauge kinetic function
at the tree-level. However, gauge kinetic functions receive moduli-dependent
threshold corrections Dixon:1990pc . For instance, it was known that in type
IIB string on toroidal orientifolds, axions associated with the complex
structure moduli couple to the photon at the one-loop level Blumenhagen:2006ci
.*7*7$*$7In the case of type IIA string, see, Ref. Lust:2003ky . In that case,
the value of gauge coupling will be determined by the vacuum expectation value
of Kähler moduli, and the volume dependence of $\alpha$ and $f$ for the axion
of the complex structure moduli is different from each other. It results in
the non-universal ratio $\frac{\alpha}{f}$. For the axion decay constant for
the complex structure moduli, it was studied in a broad class of complex
structure moduli space e.g., Refs. Conlon:2016aea ; Honda:2016jnd , which
admit the small value of axion decay constant in a certain moduli space. Thus,
the instability parameter $\xi$ will be enhanced by the small axion decay
constant for the axion associated with the complex structure moduli.

###  4.2 Bound on the instability parameter in the D-brane system without
charged fermions

In this subsection, we discuss a maximum value of $L$ when the charged
fermions under $U(1)$ are absent, as in the system of a single D-brane. For a
quantitative description of the DBI bound, we define a critical value of the
electromagnetic field strength:

$\displaystyle\langle\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}\rangle\bigl{|}_{\rm
cr}=\zeta\langle\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}\rangle\bigl{|}_{\rm max},$
(4.13)

where $\langle\hat{F}_{\mu\nu}\hat{F}^{\mu\nu}\rangle\bigl{|}_{\rm max}$ is
given in (3.7). Here, we introduce a dimensionless parameter $\zeta$, which
varies from $10^{-2}$ to $10^{-4}$ in the following numerical analysis. Note
that since the slow-roll parameter $\epsilon$ approaches to ${\cal O}(1)$
value, the instability parameter is expressed by $L_{{\rm D}p}$, (4.13) is
rewritten in terms of the critical value of $L_{{\rm D}p}$, $L_{{\rm
D}p}^{({\rm cr})}$,

$\displaystyle\frac{e^{\pi L_{{\rm D}p}^{({\rm cr})}}}{|L_{{\rm D}p}^{({\rm
cr})}|^{3/2}}=324\zeta\left(\frac{0.1}{g_{s}}\right)^{1/2}\left(\frac{M_{s}}{H}\right)^{2}.$
(4.14)

From Table 1, $L_{{\rm D}p}^{({\rm cr})}$ is bounded from above by the DBI
constraint, and the maximum value is of ${\cal O}(5)$. Since the string scale
is smaller than the Planck scale, one cannot take a large hierarchy between
$M_{s}$ and $H$. Such an ${\cal O}(1)$ value of $L_{{\rm D}p}$ is consistent
with the string setup, as discussed below.

$\zeta$ $\setminus$ $\frac{M_{s}}{H}$ | $10^{2}$ | $10^{3}$ | $10^{4}$
---|---|---|---
$10^{-2}$ | 3 | 5 | 7
$10^{-3}$ | 3 | 4 | 8
$10^{-4}$ | 2 | 3 | 5
Table 1: Critical value of $L_{{\rm D}p}$ for $g_{s}=0.1$. Here, we show a
maximum integer for $L_{{\rm D}p}$.

###  4.3 Bound on the instability parameter in the D-brane system with charged
fermions

In this subsection, we move to the system with charged massless fermion under
$U(1)$, which can be realized in magnetized D-branes. By using the critical
value of the electromagnetic fields (4.13), (3.18) is rewritten in terms of
$L_{{\rm D}p}$ for the slow-roll parameter $\epsilon={\cal O}(1)$,

$2(12\pi^{2}L_{{\rm D}p}^{({\rm
cr})})^{2}=15.6\zeta\left(\frac{g_{s}}{0.1}\right)^{2}\left(\frac{M_{s}}{H}\right)^{4}.$
(4.15)

In contrast to the system without charged fermions, one can consider a large
value of $L_{{\rm D}p}^{({\rm cr})}$ for $M_{s}\gg H$. However, a large value
of $L_{{\rm D}p}^{({\rm cr})}$ induces a large amount of D-brane charges,
which will be in tension with the tadpole cancellation condition of
D$p$-branes. To clarify this statement, we discuss the constraint on $L_{{\rm
D}p}$ from a different point of view.

##### Constraints on axion-photon coupling $L$

Let us consider the $T^{6}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{\prime})$
orientifold with O3/O7-planes. (For a detailed discussion, see, Refs.
Blumenhagen:2006ci ; Ibanez:2012zz .) For $N_{a}$ stacks of magnetized
D$7_{a}$-branes, wrapping numbers $n_{a}^{i}$ and magnetic fluxes $m_{a}^{j}$
on $(T^{2})_{i}$, with $i=1,2,3$, are defined as*8*8$*$8Here, $n_{a}^{i}$ and
$m_{a}^{i}$ for each $a,i$ are chosen as coprime numbers. This field strength
should be described as  $F_{a}^{i}=\left(\begin{matrix}\frac{2\pi
m_{a}^{i}}{n_{a}^{i}}\boldsymbol{1}_{n_{a}^{i}}&\\\
&\boldsymbol{0}_{N_{a}-n_{a}^{i}}\end{matrix}\right)$ The Dirac quantization
condition for the $U(1)$ factor is $\int_{(T^{2})_{i}}{\rm tr}F_{a}^{i}=2\pi
m_{a}^{i}$.

$\displaystyle\frac{n_{a}^{i}}{2\pi}\int_{(T^{2})_{i}}F_{a}^{i}=m_{a}^{i},$
(4.16)

where $F_{a}$ corresponds to $U(1)_{a}$ gauge field strength on
D$7_{a}$-branes. Such topological numbers appear in a gauge kinetic function
on D$7_{a}$-branes wrapping $(T^{2})_{j}\times(T^{2})_{k}$:

$\displaystyle 2\pi f^{{\rm
D}7_{a}}=m_{a}^{j}m_{a}^{k}S-n_{a}^{j}n_{a}^{k}T_{i},\qquad i\neq j\neq k,$
(4.17)

where the axionic component of the axio-dilaton $S$ and the Kähler moduli
$T_{i}$ (determining the volume of $(T^{2})_{j}\times(T^{2})_{k}$) originate
from the Kaluza-Klein reduction of RR fields $C_{0}$ and $C_{4}$,
respectively. Thus, a product of wrapping numbers $n^{j}$ and magnetic fluxes
$m^{j}$ controls the magnitude of topological coupling $L_{{\rm D}p}$.
However, these topological numbers are severely restricted to satisfy the
cancellation of D-brane charge:

$\displaystyle\sum_{a}N_{a}m_{a}^{1}m_{a}^{2}m_{a}^{3}+\frac{N_{\rm
flux}}{2}=16,$ (4.18)

for D3-brane and

$\displaystyle\sum_{a}N_{a}m_{a}^{i}n_{a}^{j}n_{a}^{k}=-16,$ (4.19)

for D$7_{a}$-branes wrapping $(T^{2})_{j}\times(T^{2})_{k}$. Here, $N_{\rm
flux}$ denotes the tadpole charge determined by imaginary self-dual three-form
fluxes. Note that the 64 O3-plane and 4 O7-planes contribute to RR charges
$-32$ in corresponding homology classes. Since the generation number of quarks
and leptons is determined by the combination of wrapping numbers and magnetic
fluxes, we have to be careful about the choice of these topological numbers.

For instance, for the local supersymmetric brane configurations incorporating
the Standard-Model spectra Marchesano:2004xz *9*9$*$9Note that the
$U(1)_{B-L}$ is identified as a linear combination of $U(1)_{a}$ and
$U(1)_{d}$, and the other $U(1)$ becomes massive through the Green-Schwarz
mechanism. The hypercharge $U(1)_{Y}$ can be realized as a spontaneous
symmetry breaking of $SU(2)_{R}\times U(1)_{B-L}$., the topological numbers
are specified as

$N_{\alpha}$ | Gauge group | ($m_{\alpha}^{1},n_{\alpha}^{1}$) | ($m_{\alpha}^{2},n_{\alpha}^{2}$) | ($m_{\alpha}^{3},n_{\alpha}^{3}$)
---|---|---|---|---
$N_{a}=6$ | $U(3)_{C}$ | (1,0) | ($m_{\rm flux},1$) | ($m_{\rm flux},-1$)
$N_{b}=2$ | $SU(2)_{L}$ | (0,1) | ($1,0$) | ($0,-1$)
$N_{c}=2$ | $SU(2)_{R}$ | (0,1) | ($0,-1$) | ($1,0$)
$N_{d}=2$ | $U(1)_{d}$ | (1,0) | ($m_{\rm flux},1$) | ($m_{\rm flux},-1$)

Then, the gauge kinetic functions of D7-branes are given by

$\displaystyle{\rm D}7_{a}$ $\displaystyle:2\pi f_{a}=m_{\rm
flux}^{2}S+T_{1},$ $\displaystyle{\rm D}7_{b}$ $\displaystyle:2\pi
f_{b}=T_{2},$ $\displaystyle{\rm D}7_{c}$ $\displaystyle:2\pi f_{c}=T_{3},$
$\displaystyle{\rm D}7_{d}$ $\displaystyle:2\pi f_{d}=m_{\rm
flux}^{2}S+T_{1}.$ (4.20)

It indicates that topological couplings to RR fields $L_{{\rm D}p}$ can be
enhanced by the magnetic flux $m_{\rm flux}$. However, the possible choices of
magnetic flux $m_{\rm flux}$ is of ${\cal O}(1)$ through the tadpole
cancellation condition of D$3$-branes:

$\displaystyle m_{\rm flux}^{2}=2-\frac{N_{\rm flux}}{16},$ (4.21)

where it is notable that $N_{\rm flux}$ is positive due to the imaginary self-
dual condition. Thus, the magnitude of topological couplings to RR coupling
$L_{{\rm D}p}$ is of ${\cal O}(1)$.

So far, we have focused on a specific toroidal orientifold compactification
with Standard-Model spectra, but it can be generalized to D$p$-branes wrapping
holomorphic divisors in Calabi-Yau manifolds. As an extension of toroidal
orientifolds, let us consider generic type IIB orientifolds with O3/O7-planes,
which can be uplifted to F-theory on Calabi-Yau fourfolds. It was known that
the D3-brane charges should be canceled to satisfy the so-called tadpole
cancellation condition Becker:1996gj ; Sethi:1996es ; Haack:2001jz :

$\displaystyle N_{\rm D3}+\frac{N_{\rm flux}}{2}+N_{{\rm D}7}^{\rm
flux}=\frac{\chi(Y_{4})}{24},$ (4.22)

where $N_{\rm D3}$, $N_{\rm flux}$, $N_{{\rm D}7}^{\rm flux}$ and
$\chi(Y_{4})$ denote the number of D3-branes, D3-brane charge induced by
three-form fluxes and magnetic fluxes on D7-branes, and Euler characteristics
of Calabi-Yau fourfold, respectively.

When we consider $U(1)$ magnetic fluxes on $N$ stacks of D7-branes, the gauge
flux can be expanded in the basis of D7-brane divisors $D_{a}$ with
$a=1,2,...,h^{1,1}(D_{a})$. Let us denote $c_{1}(l_{a})$ as the first Chern
class of line bundle $l_{a}$ with structure group $U(1)$, which is quantized
as $c_{1}(l_{a})=m_{a}[D_{a}]$ on each divisor. Note that $m_{a}$ determines
the magnitude of the axion-photon coupling, which appears in the gauge kinetic
function of D7-branes, as in (4.20). Then, the contribution of gauge flux
$N_{{\rm D}7}^{\rm flux}$ is rewritten as Blumenhagen:2008zz

$\displaystyle N_{{\rm D}7}^{\rm flux}$
$\displaystyle=\frac{1}{2}\sum_{a}N_{a}m_{a}^{2}\kappa_{aaa},$ (4.23)

where $N_{a}$ is the number of D7-branes wrapping holomorphic divisor $D_{a}$,
and $\kappa_{aaa}$ is defined as
$\kappa_{aaa}=-\int[D_{a}]\wedge[D_{a}]\wedge[D_{a}]$. As discussed in Ref.
Ishiguro:2023jjc , the $N_{{\rm D}7}^{\rm flux}$ is bounded from above by
using the Cauchy-Schwarz inequality:

$\displaystyle|m_{\rm max}|^{2}\leq\frac{\chi(Y_{4})}{12\,{\rm
min}_{a}(N_{a}\kappa_{aaa})}-(h^{1,1}(D_{a})-1),$ (4.24)

where we define $m_{\rm max}$ such that $|m_{a}|\leq{\rm
max}_{a}(m_{a}):=|m_{\rm max}|$ for all $a$, and we assume $m_{a}^{i}\neq 0$
for all $a$.

In a similar to the toroidal case, the magnitude of topological couplings to
RR fields is mutually related with the tadpole cancellation condition
Ishiguro:2023jjc . In the context of IIB/F-theory, the possible value of flux
quanta depends on the Euler characteristics of Calabi-Yau fourfolds. Since the
largest value of $\chi(Y_{4})$ is known as 1,820,448 Candelas:1997eh ;
Taylor:2015xtz , it seems to consider the large value of flux quanta, in other
words the large value of axion-photon coupling $L_{{\rm D}p}$, in such a
Calabi-Yau compactification. However, there exists the large number of moduli
fields on Calabi-Yau fourfolds with large value of $\chi(Y_{4})$. Of course,
one can consider the large value of background fluxes in Calabi-Yau manifolds
with large number of moduli, but it would be difficult to stabilize all the
moduli fields at the stable vacua, as in the tadpole conjecture Bena:2020xrh .
Furthermore, from the phenomenological viewpoint, the value of magnetic fluxes
is related to the number of chiral zero-modes on D7-branes through the index
theorem. It indicates that the small value of magnetic fluxes is also favored
to be consistent with the low-energy effective theory including the Standard
Model. We conclude that it is in general difficult to realize the
electromagnetic fields that are much greater than the Hubble scale, which is
consistent with the mass hierarchy involving the volume of the extra
dimensions and the string scale.

##  5 Conclusions

A tachyonic production of gauge field helicities during the axion inflation
has many attractive features such as the production of density perturbations
and gravitational waves through the topological coupling between the
electromagnetic fields and the axion. The magnitude of one gauge field
helicity during the axion inflation can be controlled by the parameter $\xi$,
which can be expected to be ${\cal O}(1)$ to probe the axion dynamics by the
cosmological observations.

In this paper, we examined stringy constraints for produced electromagnetic
fields induced by the non-zero axion velocity during the inflation. First, we
discuss the validity of the model with conventional quadratic kinetic terms of
axion and photon. Since the action of axion electrodynamics will be regarded
as the low-energy effective description of DBI action in string theory, the
intensity of electromagnetic fields is bounded from above by the string scale
to have control over the effective field theory. It gives the constraint on
$\xi$ as shown in Eq. (3.13), whose bound can be rewritten in terms of
supersymmetry breaking scale in the context of type IIB Large Volume Scenario.
Second, we take into account the tadpole constraints. Given that the ratio the
gauge coupling to the axion decay constant is constant in weakly-coupled
string theories, we found that the parameter $\xi$ was given by the slow-roll
parameter $\epsilon$: $|\xi|\simeq L_{{\rm D}p}\sqrt{\epsilon}$ where we use
the slow-roll approximation in the axion inflation. In the analysis of Sec.
4.2 where the charged fermions under $U(1)$ absent in the system, the
coefficient of axion-photon coupling $L_{{\rm D}p}$ is restricted to ${\cal
O}(1)$ due to the hierarchy between the string scale and the inflation scale.
They indicate that it will be in general difficult to realize ${\cal O}(1)$
value of $\xi$ in the string effective theory. On the other hand, the
existence of charged fermions allows us a large value of the instability
parameter, but it is constrained to satisfy the tadpole cancellation condition
as well as the tadpole conjecture Bena:2020xrh , as discussed in Sec. 4.3.

###### Acknowledgements.

We thank K. Kamada and K. Mukaida for helpful discussions and comments. This
work was supported by JSPS KAKENHI Grant Numbers JP20K14477 (H. O.),
JP21J00480 (R. Y.), JP21K13928 (R. Y.), JP23H04512 (H. O).

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