# On dual formulation of Axion solution to strong-CP problem

Otari Sakhelashvili  Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germany

###### Abstract

An exact duality between axion with arbitrary potential and antisymmetric
form-field has been derived some time ago. Using this duality, the axion
solution to strong-CP problem has been formulated as a gauge invariant theory
of forms. In this description, the QCD axion is represented by a Kalb-Ramond
field which is eaten-up by the Chern-Simons $3$-form of QCD, thereby making it
massive. This ensures the CP-invariance of the vacuum. Although viewed as an
effective low energy theory, this formulation accomplishes the same goal as
ordinary Peccei-Quinn mechanism, due to its gauge invariance, it is protected
against unwanted UV-corrections. In the previous work it has been shown that
dual formulation is insensitive to UV-physics in the sense that the
corrections to CP-conserving vacuum from arbitrary massive sources are
strictly zero. By going carefully through duality transformations and source-
resolution, we reproduce this curious result and give some further consistency
checks. We apply similar analysis to other approaches to naturalness problems
based on form fields and axions, such as cosmological relaxation of the
standard model Higgs boson mass via the attractor mechanism.

##  1 Introduction

The dualities involving form-fields play important role in descriptions of
physical systems. In [1] it was shown that there is an exact duality between a
massive pseudoscalar $a$ (axion) with arbitrary potential $V(a)$ and a gauge-
invariant theory of coupled antisymmetric $2$\- and $3$-forms. Using this
connection, in the above work, the strong-CP puzzle and its axion solution
were reformulated as the gauge theory of forms. The $3$-form gauge field $C$
is an organic part of QCD in form of the Chern-Simons three form. In the
absence of axions (or chiral massless quarks111In a chiral limit of a massless
quark, QCD contains a built-in axion in form of $\eta^{\prime}$-meson [1].),
the $3$-form is massless. Such a field has no propagating degrees of freedom.
However, its “electric” field-strength, $F\equiv dC$, can assume arbitrary
constant value in the vacuum. In the absence of axion, this electric field
${}^{*}E=F$ cannot be changed neither classically nor quantum-mechanically.
The different values of $E$ form different superselection sectors. Vacua with
$E\neq 0$ are CP non-conserving. Therefore, the vacua with different values of
$E$ represent alternative description of familiar $\theta$-vacua [2] of QCD.
In this description, $E$ plays the same role in parameterizing CP-breaking, as
the vacuum angle $\theta$ plays in the conventional picture.

In formulation of [1], the QCD axion is introduced in the theory in form of a
Kalb-Ramond antisymmetric $2$-form $B$. The ordinary Peccei-Quinn global
symmetry under the shift of axion, is replaced by gauge symmetry of $B$ and
$C$ forms. The $2$-form axion $B$ becomes a longitudinal (Stückelberg)
component of the $3$-form $C$, forming a massive $3$-form field. The massive
$3$-form gauge field can no longer sustain a non-zero constant electric field
$E$ in the vacuum, which becomes screened. Thus, in the presence of $B$-axion,
we end up with an unique CP-conserving vacuum $E=0$. In ordinary language,
this is equivalent to relaxing $\theta=0$.

At the level of low energy effective theory, the dual formulation of [1]
accomplishes exactly the same goal as the conventional axion [3] of Peccei-
Quinn scenario [4]. However, the sensitivities of the two formulations with
respect to UV-physics can be drastically different. In the language of
original Peccei-Quinn formulation [4] the UV-corrections are essentially
uncontrollable, due to the fact that an explicit breaking of global Peccei-
Quinn symmetry is not forbidden by any fundamental principle. For example, one
can argue that in gravity such a breaking is triggered by high dimensional
operators suppressed by Planck scale [5].

In contrast, the gauge symmetry of the dual formulation must be respected by
arbitrary UV-completion of the theory [1]. This puts the potentially dangerous
corrections under much stricter control as compared to ordinary formulation.

As shown in [1], due to its gauge invariance, the dual formulation is fully
insensitive to UV-physics. Namely, it was proven that, provided new physics
contains no long-range $3$-form correlators (i.e., no massless $3$-form
fields), the CP-invariance of the vacuum ($E=0$), is not affected at all. That
is, CP-violating corrections to the vacuum, coming from arbitrary massive
modes, are exactly zero. The proof is based on effective field theory methods
relying on gauge invariance and the analysis of the pole structure of the
propagators brought by new physics.

From the first look, the above sounds like a rather strong statement. However,
the formulation in terms of $3$-form gauge theory, makes the story
sufficiently transparent. In the present paper we shall reproduce this result
by explicitly resolving heavy sources and shall further elaborate on it.

As discussed in [1], the insensitivity of gauge formulation to massive physics
gives useful tool for parameterizing and avoiding unwanted corrections to
axion solution of strong-CP from the sources such as gravity. Namely, the UV-
insensitivity criterion tells us that, in dual formulation, the unwanted
corrections can only appear if the theory contains additional massless
$3$-forms that can also mix with the QCD axion. If such massless forms exist,
they can “disrupt” axion from fully screening the QCD Cern-Simons $3$-form
field. For example, in gravity such a potential danger to axion can come
uniquely from a gravitational Chern-Simons $3$-form, provided the latter can
give a long-range correlator.

As explained in [1], even if gravity contains such unwanted corrections, the
dual gauge-invariant formulation offers a way out. A disruption from
gravitational Chern-Simons can be easily avoided if the theory contains an
additional chiral symmetry that is anomalous with respect to gravity. In fact,
the role of a “protector” for axion can be played by a chiral symmetry of a
light fermion, such as neutrino [1, 6] 222Possible phenomenological
implications of this effect for neutrino masses were explored in [7]..

Another application of $3$-form/axion system is Dvali-Vilenkin “attractor”
mechanism [8, 9]. Originally this mechanism was used for the cosmological
relaxation of the mass of the standard model Higgs boson. This scenario is
motivated by the hierarchy problem, an inexplicable smallness of the Higgs
mass relative to Planck scale, an ultimate cutoff of the theory. The
conventional approaches, such as low energy supersymmetry, or low scale
quantum gravity [10, 11], predict the existence of new physics not far from
the weak scale. The mechanism of [8, 9] intended to offer a potential way to
relax the Higgs mass (and the VEV) to acceptably low values, without the need
of low energy stabilizing physics. The idea is to use a $3$-form field as a
control parameter for the Higgs mass. The vacua with different values of the
Higgs mass are then actualized due to the change of the electric field $E$ by
its sources. Their role can be played by branes or by solitons (domain walls)
of heavy axion-like fields. The key ingredient of the mechanism is that the
step of the variation of $E$ diminishes towards a certain critical value
$E_{*}$. This results into a divergent density of vacuum states with values of
$E$ arbitrarily close to $E_{*}$. Due to this, the vacuum $E=E_{*}$ acts as an
attractor point of the cosmological evolution.

The $3$-form attractor mechanism has also been applied to the strong-CP
problem [12]. In this setup, the role of $C$ is played by the QCD Chern-Simons
and the attractor point is at the vanishing electric field $E_{*}=0$,
corresponding to CP-conserving vacuum. The idea is that the universe can be
driven to it by cosmic evolution without the need of the axion field.

In order to represent a legitimate solution of the naturalness problems, it is
important for the attractor mechanism not to be UV-sensitive. In particular,
the attractor point should not be sensitive to the details of short distance
physics such as the inner structure of the brane. For this, it is important to
understand under what circumstances the brane structure becomes important. In
the present paper we shall also address such issues.

The paper is orginized as follows. After carefully reviewing the duality
between $3$-forms and axion and a dual formulation of axion given in [1], we
study resolutions of the heavy sources. We observe that screening of $E$
(equivalently $\theta$) by axion in QCD is not sensitive to the presence of
the heavy sources and/or massive states. We thus confirm the findings of [1].

We then turn to the situations when no light axion is present in low energy
theory and $C$ is sourced exclusively by massive branes. In such a case $C$
remains massless and can produce a long-range electric field $E$. This
situation does not correspond to the case of QCD axion, but is relevant for
the attractor setups [8, 9, 12]. We compute the back reaction from the
electric field $E$ to the brane in the leading order and establish the time-
scale after which the inner structure of the brane is affected. Applying this
to the attractor mechanism, we shall find out that the physics near the
attractor point is insensitive to this resolution. As another sort of a back
reaction, the motion of branes leads to the particle-creation. In case of [8,
9] the radiated quanta are Higgs bosons. This is due to the change of the
Higgs boson mass triggered by the electric field $E$. Again, the effect
vanishes near the attractor point but can be significant when the system is
far from it.

##  2 Duality

The dualities between the form-fields have long history. In particular, it has
been known for a long time that a theory of a free massless $2$-form Kalb-
Ramond field $B_{\mu\nu}$, described by the Lagrangian,

$\mathcal{L}=\frac{1}{12}(dB)^{2},$ (1)

is dual to a free massless pseudoscalar $a$,

$\mathcal{L}=\frac{1}{2}(\partial a)^{2}\,,$ (2)

where $d$ denotes an exterior derivative. The proof of duality at the level of
free theory is straightforward. However, for some time, the dualities at the
level of massive interacting theories were a source of controversy. It has
even been suggested [13] that duality ceases to exist at the level of an
interacting theory and that interaction and mass terms break duality.

This issue was clarified in [1] where it was proven that duality continues to
hold for massive and interacting axion field with arbitrary scalar potential
$V(a)$. The caveat is that the non-derivative interaction terms of the axion
dualize to a non-trivial kinetic function of a massive $3$-form field
$C_{\mu\nu\alpha}$ in which the Kalb-Ramond $B_{\mu\nu}$ enters as the
longitudinal (Stückelberg) degree of freedom. This duality has smooth limits
both for the zero coupling and the zero mass. The number of degrees of freedom
remains in tact in both limits. We shall start by reviewing this duality
closely following [1].

Let us consider a theory of a massive pseudo-scalar axion with arbitrary
potential

$\mathcal{L}=\frac{1}{2}(\partial a)^{2}-V(a)\,.$ (3)

We wish to show that this theory is dual to the following theory of an
interacting massive $3$-form $C$ with the field strength
$F_{\mu\alpha\beta\gamma}=\partial_{[\mu}C_{\alpha\beta\gamma]}=\epsilon_{\mu\alpha\beta\gamma}E$,

$\mathcal{L}=\Lambda^{4}\mathcal{K}\left(\frac{E}{\Lambda^{2}}\right)+\frac{1}{2}m^{2}C^{2}\,,$
(4)

where $\Lambda$ and $m$ are parameters of mass dimensionality. The quantity
$E$ shall be referred to as the “electric” field. The
$\mathcal{K}\left(\frac{E}{\Lambda^{2}}\right)$ is a non-derivative function
of its argument $E$, which satisfies,

$V(a)=\frac{1}{\sqrt{6}}m\Lambda^{2}\int da\,{\rm\bf
inv}\,\mathcal{K}^{\prime}\left(\frac{ma}{\sqrt{6}\Lambda^{2}}\right)\,$ (5)

where, prime denotes a derivative with respect to the argument and inv stands
for an inverse function. For a simplest choice
$\mathcal{K}(x)=\frac{1}{2}x^{2}$, we get quadratic and canonically normalized
Lagrangian.

In order to prove the above, let us first decompose the massive field $C$ in
its transverse and longitudinal modes,

$C=C^{T}-dB\,$ (6)

The longitudinal mode, which is the only propagating mode of the massive
$3$-form field, is a Kalb-Ramond field. It also serves as the Stückelberg
field for explicitly maintaining the following gauge redundancy,

$C^{T}\rightarrow C^{T}+d\Omega\,,~{}~{}B\rightarrow B+\Omega\,.$ (7)

Here $\Omega$ is a gauge shift parameter, which represents an arbitrary two-
form function of spacetime coordinates.

Notice, (6) is just a decomposition, and does not amount to any modification
of the theory. The original theory (4), as well as its decomposed version,

$\mathcal{L}=\Lambda^{4}\mathcal{K}\left(\frac{E}{\Lambda^{2}}\right)+\frac{1}{2}m^{2}(C^{T}-dB)^{2}\,,$
(8)

are physically equivalent for arbitrary values of the mass, including the
limit $m\rightarrow 0$. This is obvious from the fact that they give identical
physical observables, such as the interactions mediated between arbitrary
external sources.

Following [1], we now perform dualization. As the first step, we treat
$dB\equiv X$ as a fundamental $3$-form, simultaneously imposing the Bianchi
identity ($dX=0$) as a constraint through a Lagrange multiplier $a$,

$\mathcal{L}=\Lambda^{4}\mathcal{K}\left(\frac{E}{\Lambda^{2}}\right)+\frac{1}{2}m^{2}(C^{T}-X)^{2}+\frac{1}{\sqrt{6}}m\,a\,\epsilon_{\mu\alpha\beta\gamma}\partial_{\mu}X_{\alpha\beta\gamma}\,.$
(9)

Integrating out $X$ through the equations of motion, we get the following
effective theory,

$\mathcal{L}=\Lambda^{4}\mathcal{K}\left(\frac{E}{\Lambda^{2}}\right)+\frac{1}{2}(\partial
a)^{2}-\frac{1}{\sqrt{6}}m\,a\,E.$ (10)

Notice that the gauge symmetry (7) of Kalb-Ramond field is replaced by the
global shift symmetry of the axion field by an arbitrary constant,

$a\rightarrow a+{\rm const}\,.$ (11)

The equation of motion for $C$ gives,

$\partial_{\mu}\left(\mathcal{K}^{\prime}\left(\frac{E}{\Lambda^{2}}\right)-\frac{m}{\sqrt{6}\Lambda^{2}}a\right)=0,$
(12)

whereas the one of axion is,

$\Box a=\frac{1}{\sqrt{6}}mE\,.$ (13)

Solving for $E$ as function of $a$ from(12) and taking into account (13) makes
it obvious that

$E(a)=\frac{\sqrt{6}}{m}\frac{dV(a)}{da}\,.$ (14)

This is the source of the relation (5). Thus, integrating out $E$-field, we
finally arrive to the theory (3) with the potential $V(a)$ determined by the
function $\mathcal{K}^{\prime}$ through the relation (5). Thus we reproduce
the result of [1] that theory of an axion field $a$ (3) with an arbitrary
potential, has an exact dual in form of a gauge theory (8) of $C^{T}$ and $B$,
where the axion potential is translated into the kinetic function
$\mathcal{K}$ of the $3$-form through the relation (5).

Next, using the above duality, in [1] the solution of the strong-CP problem
was formulated as the Higgs-like phase of the gauge theory of forms. This
formulation contains no reference to global chiral symmetry and only relies on
gauge redundancy. This dual formulation is described by the Lagrangian (8),
where $C^{T}$ represents a Chern-Simons $3$-form of QCD and $B$ is a Kalb-
Ramond dual of the axion $a$. The theory (8) eliminates strong-CP violation in
a gauge redundant formulation. In this formulation, the global Peccei-Quinn
symmetry, which acts on axion (11) is replaced by the gauge symmetry (7).
Viewed as low energy theories, the two formulations accomplish exact same
goal, but exhibit different sensitivities towards UV-physics. In particular,
in formulation (8) the vacuum of the theory exhibits zero UV-sensitivity.

An example of $\mathcal{K}(x)$ function,
$\mathcal{K}(x)\propto(x\arcsin{x}+\sqrt{1-x^{2}})$, provided in [1], gives a
simplest prototype potential $V(a)\propto\rm\cos(\frac{a}{f_{a}})$ (here
$f_{a}$ is an axion decay constant) commonly used in axion literature.
Conventionally, this potential is obtained by instanton calculation in dilute
gas approximation [14]. The $3$-form language shows that the vanishing of CP-
violation is insensitive to a precise form of $V(a)$.

In QCD, the function $\mathcal{K}$ is not known exactly, but important thing
is that none of the key aspects of the axion physics, such as the generation
of the axion mass gap and elimination of $\theta$, are sensitive to its exact
form. In particular, from (14) it is clear that $E=0$ at every extremum of the
axion potential, regardless the form of the function $\mathcal{K}$. This
constitutes an alternative support of Vafa-Witten theorem [15] and its
generalization to all extrema of $V(a)$.

Thus, the above duality allows to understand the generation of axion mass in
QCD as the $3$-form Higgs effect. Notice, this description is exact [1], as
the entire information about the axion mass and its potential is contained in
non-derivative function $\mathcal{K}$. None of the high derivative
contributions affect either the mass or the potential, since such
contributions vanish in zero momentum limit. Thus, as long as no additional
massless $3$-form fields are added to QCD, the generation of the axion mass
with simultaneous elimination of the $\theta$-vacua, can be understood in the
language of the Lagrangian (8) (equivalently (4)) or its dual (3).

This language also makes it transparent why the existence of an additional
potential $\tilde{V}(a)$ (not generated via a QCD $3$-form) un-Higgses the
$3$-form and re-generates $\theta$-vacua. Indeed, in a theory with an
additional potential

$\mathcal{L}=\Lambda^{4}\mathcal{K}\left(\frac{E}{\Lambda^{2}}\right)+\frac{1}{2}(\partial
a)^{2}-\frac{1}{\sqrt{6}}m\,a\,E-\tilde{V}(a)\,.$ (15)

the $3$-form remains massless. The way to understand this in the $3$-form
Higgs language is to notice that the additional potential can be traded for an
additional $3$-form [1]. It then becomes clear that only one superposition of
the two $3$-forms can become massive by eating up a single scalar field. This
is the reason why any external explicit breaking of the axion shift symmetry
(PQ symmetry) in QCD brings back the observable CP-violation.

##  3 UV-in-sensitivity of Dual Axion Mechanism

We now wish to discuss the point of [1] that the axion solution, when
formulated as $3$-form gauge theory (8), is insensitive towards UV physics.
That is, the generation of the mass gap cannot be affected by the presence of
the heavy fields at some mass scale $M_{f}$. In particular, coupling the QCD
$3$-form to arbitrary massive branes, cannot regenerate a non-zero CP
violation. A curious thing about this statement is that it is meant to be
exact.

Naively, one would think that the introduction of massive states at some high
scale $M_{f}$, can affect the low energy physical observables by small but
non-zero amount. The corrections, while possibly exponentially small or
suppressed by powers of the scale $M_{f}$, are usually non-zero. In case of
the gauge formulation (4) of the QCD axion, this would imply that coupling the
QCD Cern-Simons term to heavy banes would perturb the observable
$\theta$-parameter by small amount and trigger the CP-violation.

However, this is not what is happening. The CP-violating order parameter $E$
remains exactly zero, as long as new physics does not come with new massless
poles in $3$-form correlators. The proof, which can be found in [1], is based
on integration out of the heavy physics and writing down the effective
propagator for the $3$-form $C$. We shall not repeat it here. However, we
shall do a cross check in the example below.

We can test the above claim by using the explicit resolution of the brane,
analogous to the one in [9]. For this, let us assume the $C$ form in the
Lagrangian (10) to stand for a QCD Chern-Simons field. Let us now add its
couplings to a heavy brane coming from some UV-physics. This heavy brane can
be resolved in form of a soliton of a heavy axion $b$ with the effective
potential $V(b)$. The theory now becomes,

$\mathcal{L}=-\frac{1}{48}F^{2}-\frac{1}{2\pi}\partial_{\mu}(qa+q_{b}b)\epsilon_{\mu\alpha\beta\gamma}C_{\alpha\beta\gamma}+\frac{1}{2}f_{a}^{2}(\partial
a)^{2}\,+\frac{1}{2}f_{b}^{2}(\partial b)^{2}-V_{b}(b)\,.$ (16)

Here $f_{a}$ and $f_{b}$ are decay constants of $a$ and $b$ axions which are
not canonically normalized. $\mathcal{K}(x)=\frac{1}{2}x^{2}$ and we also
parameterize the mixing between $3$-form and axions with constants $q$ and
$q_{b}$ with mass dimension-2. The connection between the parameters of (16)
and (10) is straightforward.

Solving the equations of motion shows that, despite the presence of the heavy
field $b$, in the vacuum state the electric field $E$ of $C$ remains exactly
zero. This is immediately apparent from the equation,

$f_{a}^{2}\Box a=\frac{q}{2\pi}E\,,$ (17)

which shows that no vacuum can be reached for any non-zero value of $E$. This
zero sensitivity of $E=0$ vacuum towards heavy physics (in this case heavy
axion $b$) is again very transparent in the Language of $3$-form Higgs effect.
Using the exact duality, we can rewrite the potential for axion $b$ as a Higgs
effect with respect to a second $3$-form $C^{b}$. The Lagrangian then becomes,

$\begin{split}\mathcal{L}=-\frac{1}{48}F^{2}-\frac{1}{2\pi}\partial_{\mu}(qa+q_{b}b)\epsilon_{\mu\alpha\beta\gamma}C_{\alpha\beta\gamma}+\frac{1}{2}f_{a}^{2}(\partial
a)^{2}\,+\\\ +\frac{1}{2}f_{b}^{2}(\partial
b)^{2}-\frac{q_{bb}}{2\pi}\partial_{\mu}b\epsilon_{\mu\alpha\beta\gamma}C^{b}_{\alpha\beta\gamma}+\Lambda^{4}\mathcal{K}_{b}\left(\frac{E^{b}}{\Lambda^{2}}\right)\,\end{split}$
(18)

where $\mathcal{K}_{b}$ satisfies the same condition (5) with respect to
$V_{b}(b)$. This theory describes a Higgs effect with two gauge three-forms
($C$ and $C_{b}$) and two axions ($a$ and $b$). As a result, no massless
degree of freedom is remaining in the theory and there exist no electric field
in the vacuum with respect to any $3$-form. Thus, all CP-violating order
parameters are exactly zero.

Situation would change drastically, if the new physics, coupled to ordinary
axion, would come in form of a $3$-form without its own axion partner,

$\mathcal{L}=-\frac{1}{48}F^{2}-\frac{1}{2\pi}\partial_{\mu}a\epsilon_{\mu\alpha\beta\gamma}(qC_{\alpha\beta\gamma}+q_{b}C^{b}_{\alpha\beta\gamma})-\frac{1}{48}F_{b}^{2}+\frac{1}{2}f_{a}^{2}(\partial
a)^{2}\,$ (19)

This is an example considered in [1] as violating the criterion of absence of
massless $3$-forms. In such a case, the new physics will be represented solely
by $C_{b}$ which has no axion partner. Correspondingly, there is a massless
pole that new physics brings in. Integrating out $C_{b}$ and $a$, we recover
(up to notations) the equation (35) of [1],

$\mathcal{L}=-\frac{1}{48}F^{2}-\frac{1}{48}\frac{q^{2}}{q_{b}^{2}}F_{\mu\alpha\beta\gamma}\frac{m^{2}}{\Box+m^{2}}F_{\mu\alpha\beta\gamma}\,.$
(20)

Where $m^{2}=\frac{q_{b}^{2}}{4\pi^{2}f_{a}^{2}}$. Since in zero momentum
limit ($\Box\rightarrow 0$) the only effect of the second term is that it
corrects kinetic term normalization, the theory contains a long-range
correletor in form of a massless $3$-form field. Correspondingly, there exists
a vacuum solution with a constant $E$, which violates CP-symmetry.

At the same time, if we perform analogous integration in (18), assuming
$\mathcal{K}_{b}(x)=\frac{1}{2}x^{2}$, the effective theory for $C$ will be,

$\mathcal{L}=\frac{1}{2}C_{\alpha\beta\gamma}\left(\Box+m^{2}_{a}+\left(\frac{q_{b}}{q_{bb}}\right)^{2}\frac{m^{2}_{b}\Box}{\Box+m^{2}_{b}}\right)\Pi_{\alpha\mu}C_{\mu\beta\gamma}\,,$
(21)

Where
$\Pi_{\alpha\mu}=\eta_{\alpha\mu}-\frac{\partial_{\alpha}\partial_{\mu}}{\Box}$
is a transverse projector, $m_{a}^{2}=\frac{q^{2}}{4\pi^{2}f_{a}^{2}}$ and
$m_{b}^{2}=\frac{q_{bb}^{2}}{4\pi^{2}f_{b}^{2}}$. This theory describes a
theory with mass gap in which the longitudinal mode has been integrated out.
There is no constant electric field in the vacuum and correspondingly no CP-
violation. This confirms the statement of [1] about the insensitivity of the
dual formulation of axion with respect to heavy physics.

##  4 More on Resolving Sources

Our next steps are the following. First we consider a three form sourced by a
brane in the zero width approximation. We then resolve the brane in form of a
soliton as this was done in [9]. As shown there, we obtain that the
topological charge of the soliton in low energy theory acts as an electric
(Noether) charge for the $3$-form field. We then take into account the back
reaction that branes experience from the $3$-form dynamics and we ask how
sensitive the resolution is to this back reaction.

For resolving the brane as a soliton of heavy axion like field $a$ we choose
the Lagrangian in the following form [9],

$\mathcal{L}=-\frac{1}{48}F^{2}-\frac{q}{2\pi}\partial_{\mu}a\epsilon_{\mu\alpha\beta\gamma}C_{\alpha\beta\gamma}+\frac{1}{2}f_{a}^{2}(\partial
a)^{2}-{V}(a),$ (22)

the field $a$ is a non-canonically normalized axion, with decay constant
$f_{a}$ and a periodic potential ${V}(a)$, which we choose in the form,

$V(a)=V_{0}[1-cos(a)]\,,$ (23)

where $V_{0}$ is constant with the dimension of the energy density. The
parameter $q$ is a coupling between the $3$-form and the axion. The connection
between the parameters of the above and (15) is straightforward. This is a
typical sine-Gordon potential, which we choose as a simple prototype, but our
conclusion hold for arbitrary periodic potential.

Equation of motions for $C$ field will give

$-\partial_{\mu}F_{\mu\alpha\beta\gamma}=J_{\alpha\beta\gamma},$ (24)

Where $J$ is axionic current

$J_{\alpha\beta\gamma}=-\frac{q}{2\pi}\epsilon_{\mu\alpha\beta\gamma}\partial_{\mu}a\,.$
(25)

This current is trivially conserved due to its topological nature.

For the full treatment we must solve the coupled system of equations of
motion. We shall however proceed iteratively, by first ignoring the back-
reaction from $F$ to $a$. The field $a$ will be replaced by the sine-Gordon
soliton,

$a(z)=4\tan^{-1}e^{\frac{\sqrt{V_{0}}(z-z_{0})}{f_{a}}}+2\pi N,$ (26)

which depends on a single coordinate $z$. This soliton describes a
topologically non-trivial configuration in which the axion field $a$
interpolates between the two nearest neighbouring minima of the periodic
potential,

$\begin{array}[]{ll}a=2\pi N,&z=-\infty\\\ a=2\pi(N+1),&z=\infty,\end{array}$
(27)

where $N$ is an integer. The parameter $z_{0}$ is arbitrary and represents a
collective coordinate of the soliton. The configuration (26) solves the
equation motion for $a$ in the limit $q=0$, when the back reaction from the
$3$-form can be ignored.

Substituting the solution (26) into the topological current we get an external
source for $C$. This approximation amounts to a limit in which we take
$\frac{\sqrt{2V_{0}}}{f_{a}}\rightarrow\infty$ and keeping $q$ small. In this
approximation, the soliton field acts as an external source for the $3$-form,
while experiencing no back reaction from it. In fact in this limit, we have

$a^{\prime}(z)=2\pi\delta(z-z_{0})\,,$ (28)

where ′ denotes a derivative with respect to $z$ and the sine-Gordon soliton
becomes effectively a delta-function source.

The equation of motion (24) and are gauge redundant and require gauge fixing.
We fix the Coulomb gauge. Then using the substitution of
$C_{\alpha\beta\gamma}=\epsilon_{\mu\alpha\beta\gamma}C_{\mu}$ and taking into
account (28), the above equation reduces to

$\epsilon_{z\alpha\beta\gamma}C_{z}^{\prime\prime}=J_{\alpha\beta\gamma}=-q\delta(z-z_{0})\epsilon_{z\alpha\beta\gamma}.$
(29)

This equation is identical to an equation of an electrostatic field produced
by a static point charge in one space dimension with coordinate $z$.

The solution for $C_{z}$ is,

$C_{z}=-\frac{1}{2}q|z-z_{0}|-4E_{0}(z-z_{0})+c_{2}\,,$ (30)

where $E_{0}$ and $c_{2}$ are integration constants. The electric field
corresponding to this solution has the form,

$E=\frac{1}{8}q\,sign(z-z_{0})+E_{0}.$ (31)

Thus, the topological charge of the soliton effectively acts as a electric
Noether charge for the $3$-form gauge field.

###  4.1 Back Reaction

We now wish to take into account the leading back reaction in $q$ on the
soliton from the electric field. In particular, the back-reaction is expected
to take place because of the difference between the electric fields on the two
sides of the kink. This creates an energy difference which shall accelerate
the kink. In other words, the kink carries a charge under the $3$-form
electric field and is pushed towards infinity.

This effect can be taken into account by considering an effective theory for a
time-dependent fluctuation of the collective coordinate,

$z_{0}\rightarrow z_{0}+\delta z_{0}(t)\,.$ (32)

Inserting this anzats back into the Lagrangian together with three-form
solution, we get an effective Lagrangian for the fluctuation

$\mathcal{L}=-\frac{1}{48}F^{2}-\frac{q}{2\pi}aE+\frac{1}{2}f_{a}^{2}a^{\prime
2}\dot{\delta z_{0}}^{2}-\frac{1}{2}f_{a}^{2}a^{\prime 2}-V(a).$ (33)

In the above Lagrangian, first term is constant and the last two terms give a
total derivative, since the fluctuation of Kink does not change its
topological properties. We are therefore left with a simpler Lagrangian
density,

$L=\int dz\mathcal{L}=\int dz\left(\frac{1}{2}f_{a}^{2}a^{\prime 2}\dot{\delta
z_{0}}^{2}-\frac{q}{8\pi}a^{\prime}C_{z}\right),$ (34)

Next we integrate over $z$ coordinate. In order to resolve the square of the
delta-function in the first term, we explicitly take into account the profile
(26) of the sine-Gordon soliton and integrate over it. In the second term, we
can perform the integration within delta function approximation (28). Choosing
proper normalization we get following Lagrangian,

$L=\frac{1}{2}M\dot{\delta
z_{0}}^{2}+\frac{1}{f_{a}^{2}}\left(\frac{1}{8}q^{2}|\delta
z_{0}|+qE_{0}\delta z_{0}\right),$ (35)

this system can be understood as a point charge with mass
$M=8\frac{\sqrt{V_{0}}}{f_{a}}$ and coordinate $\delta z_{0}$ in one
dimensional external electric field [17], For $\delta z_{0}>0$ the above
system has solutions in which the soliton (domain wall) is moving accelerated

$\delta
z_{0}(t)=\frac{1}{2Mf_{a}^{2}}\left(\frac{1}{8}q^{2}+qE_{0}\right)t^{2}+vt+\delta
z_{0}(0)\,.$ (36)

where $v$ is the initial velocity. Without loss of generality, let us assume
that the initial coordinate and initial velocity are zero.

Let us try to identify the validity time-scale of the above approximation.
This is the time-scale after which the effective theory in which we treat
soliton without changing internal structure breaks down.

Most conservatively, this can be estimated by demanding that the variation of
the collective coordinate should not exceed the width of the soliton. This
gives a condition,

$\frac{f_{a}}{\sqrt{V_{0}}}\gtrsim|\delta z_{0}|,$ (37)

which applied to the solution (36) implies,

$\frac{f_{a}}{\sqrt{V_{0}}}\gtrsim\frac{1}{16\sqrt{V_{0}}f_{a}}\left|\frac{1}{8}q^{2}+qE_{0}\right|t^{2}\,.$
(38)

This leads us to the following upper bound on the validity time-scale,

$t\lesssim\frac{4f_{a}}{|q|\sqrt{\left|\frac{1}{8}+\frac{E_{0}}{q}\right|}}\equiv
t_{*}\,.$ (39)

After this time, the internal structure of the soliton is affected by the
action of the electric field. Of course, for a long distance observer the
description of the soliton as of a point particle accelerated by a constant
electric field, is still valid. The full relativistic solution for such a
particle is well-known [17].

##  5 Implications for Cosmic Attractor

We shall now apply the above results to the attractor mechanism [8, 9]. The
idea of the attractor is that the charge of the brane $q$ with respect to a
massless $3$-form $C$, through a certain chain of influences, effectively
depends on the electric field $q(E)$. An important point is that for certain
critical value $E_{*}$, it vanishes. That is $q(E_{*})=0$. Since the change of
the field $E$ among the neighbouring vacua is given by the charge,

$\Delta E\propto q(E)\,,$ (40)

the step vanishes when we approach $E_{*}$. That is, the number of distinct
vacua required to traverse through for reaching $E_{*}$, is infinite. In other
words, there exist infinitely many vacua with values of the electric field
that are arbitrarily close to the attractor value $E_{*}$.

One of the requirements to the attractor mechanism is that it should not be
sensitive to UV physics. That is, the singularity in number of vacua at the
attractor point $E_{*}$, should be trusted without knowing the internal
structure of the brane. We can now explicitly verify this by evaluating the
scaling of the critical time $t_{*}$ near the attractor point. Since at the
attractor point the electric field is finite $E_{*}$, whereas the brane change
$q(E_{*})$ vanishes, it is clear from (39) that $t_{*}\rightarrow\infty$. That
is, closer we are to the attractor vacuum, longer is takes for the thin-brain
approximation to break down.

###  5.1 Effect of Particle Creation

An important ingredient of the attractor solution to the hierarchy problem [8,
9], is that the value of the Higgs mass and the VEV is determined by the
$3$-form electric field. In this way, the Higgs mass changes from vacuum to
vacuum.

Thus, in the brane background the Higgs mass is effectively dependent on the
space-time coordinates. The part of the Lagrangian in which this information
is encoded has the following form

$\mathcal{L}=\frac{1}{2}(\partial
h)^{2}-\frac{1}{2}h^{2}\left(m_{h}^{2}+\frac{F^{2}}{48M_{f}^{2}}\right),$ (41)

where $h$ is a Higgs field, $m_{h}$ is its “bare” mass and $M_{f}$ is some
fundamental scale, which will be assumed to be large.

Taking in the account solution for electric field (31), we get effectively a
mass term for Higgs which depends on a position on $z$-axis. Basically, since
the electric field changes across the brane, so does the Higgs mass.
Correspondingly, we can define two different masses $M_{-}$ and $M_{+}$,

$M_{\pm}^{2}=m_{h}^{2}-\frac{1}{128}\frac{q^{2}}{M_{f}^{2}}-\frac{E_{0}^{2}}{2M_{f}^{2}}\pm\frac{1}{8}\frac{E_{0}q}{M_{f}^{2}}\,,$
(42)

on the left and right sides of the brane respectively.

Without loss of generality, we assume that the constant part of the electric
field is positive. For the opposite case we could just change the labeling.
Let us assume that for some initial time $t=0$, on both sides of the wall the
Higgs field is in corresponding vacuum states. That is, no Higgs particles are
excited.

Now, since the wall is moving accelerated (36), the electric field changes in
time, and correspondingly changes the vacuum of the Higgs field. This leads to
a particle creation.

In order to compute the rate of particle-creation, let us start with a region
in the vacuum which corresponds to $M_{+}$. After the brane passes-by this
region, the mass decreases to $M_{-}$.

In this case the number operator of $M_{-}$-mass particles is defined in the
following way

$N=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2\omega_{-}}a^{+}_{-}(\vec{p})a_{-}(\vec{p})\,,$
(43)

where $\omega_{\pm}^{2}=p^{2}+M_{\pm}^{2}$, and the subscript on the ladder
operators has the same meaning. Performing a Bogoliubov transformation (56)
and averaging in the vacuum $\ket{0_{+}}$ (which is the vacuum in this case),
we get following expression,

$N=SL\frac{1}{4}\int
d^{3}p\frac{\omega_{+}}{\omega_{-}}\left(\frac{\omega_{-}}{\omega_{+}}-1\right)^{2}\,,$
(44)

where $S$ is the surface area of the brane and $L$ is the length traveled by
it in $z$ direction. Taking into account that

$\omega_{+}^{2}-\omega_{-}^{2}=\frac{1}{16}\frac{E_{0}q}{M_{f}^{2}}\,,$ (45)

for $M_{f}^{2}\gg\frac{E_{0}q}{M_{-}^{2}}$ we get

$\omega_{+}\approx\omega_{-}+\frac{1}{32}\frac{E_{0}q}{M_{f}^{2}\omega_{-}}\,.$
(46)

At the lowest order, the produced particle number per unit surface is given by

$\frac{N}{S}=L\frac{1}{4096}\frac{E_{0}^{2}q^{2}}{M_{f}^{4}}\int
d^{3}p\frac{1}{\omega_{-}^{4}}\,.$ (47)

After integration we obtain,

$\frac{N}{S}=L\frac{\pi^{2}}{4096}\frac{E_{0}^{2}q^{2}}{M_{f}^{4}M_{-}}\,.$
(48)

Substituting the distance traveled by the domain wall in time $t$, we get the
final result

$\frac{N}{S}=\left(\frac{1}{16\sqrt{V_{0}}f_{a}}\left(\frac{1}{8}q^{2}+qE_{0}\right)t^{2}+vt\right)\frac{\pi^{2}}{4096}\frac{E_{0}^{2}q^{2}}{M_{f}^{4}M_{-}}.$
(49)

The above expression gives us the amount of Higgs particles created by a
passing-by domain wall.

However, we should remember that we can trust the above analysis until the
time (39). During this time, we get following amount of particles created per
unit surface

$\frac{N}{S}=\frac{f_{a}}{\sqrt{V_{0}}}\frac{\pi^{2}}{4096}\frac{E_{0}^{2}q^{2}}{M_{f}^{4}M_{-}}\,.$
(50)

Far from the attractor, where $q$ is large, the effect of particle creation
can be significant and affect the dynamics of the system. However, not
surprisingly, since the effect is proportional to $q^{2}$, it is vanishingly
small near the attractor point. Again, the attractor behave is largely
unaffected by the high energy effects.

##  6 Conclusions

The dual formulation of the axion solution of the strong-CP problem in form of
a gauge theory of a $3$-form [1], by the power of gauge invariance, gives a
possibility of controlling the unwanted UV-corrections. In particular, it has
been shown that corrections from arbitrary massive physics to CP-invariance of
the vacuum are exactly zero. In the present paper we reproduced this result
and performed consistency checks on some explicit examples of heavy physics.
In accordance with the proof given in [1], we find that the only possibility
of destabilizing the CP-invariant vacuum is through the appearance of
additional massless $3$-form fields coupled to axion. In their absence, the
QCD vacuum remains exactly CP-conserving. 333We note that, perhaps, an
alternative physical way of understanding this UV-insensitivity is that the
generation of axion mass from $3$-form Higgs effect can be understood in
purely topological terms, as discussed in [16].

We also gave a resolution of brane sources coupled to massless $3$-form fields
and estimated a back reaction from the $3$ electric field to leading order.
Such setup has been applied to the solutions of the hierarchy [8, 9] and
strong-CP [12] problems via the attractor mechanism. We observe that the
existence of the attractor point is not sensitive to the brane resolution or
to back reaction. We also estimated particle-creation in the background of
time dependent $3$-form electric field.

## Acknowledgements

We would like to thank Gia Dvali for many illuminating discussions. We also
thank Lasha Berezhiani and Georgios Karananas for valuable discussions. We
would like to mention that general discussions about the strong CP problem
with Cesar Gomez and Goran Senjanovic were important.

##  Appendix A Bogolyubov Transformations

In the above section, we have a scalar field the mass of which effectively
depends on time. This means that this field does not have a fixed vacuum.
Using Bogolyubov transformations, we can properly account for the change of
the vacuum. Let us consider a free scalar field $\phi$, in Schrödinger picture

$\phi(\vec{x})=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2\omega(p,t)}(a_{t}(\vec{p})e^{i\vec{p}\vec{x}}+a^{+}_{t}(-\vec{p})e^{i\vec{p}\vec{x}})\,,$
(51)

where $\omega(\vec{p},t)$ is the time depended frequency. Therefore, ladder
operators depend on time as well (despite being described in Schrödinger
picture). The corresponding canonical momenta have the following form

$\pi(\vec{x})=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{-i}{2}(a_{t}(\vec{p})e^{i\vec{p}\vec{x}}-a^{+}_{t}(-\vec{p})e^{i\vec{p}\vec{x}})\,.$
(52)

The important point here is that the field and the canonical momenta do not
depend on time explicitly. Thus, for any given time we can write

$\displaystyle\frac{1}{2\omega(p,t)}\left(a(\vec{p})_{t}+a^{+}(-\vec{p})_{t}\right)=\int
d^{3}xe^{-i\vec{p}\vec{x}}\phi(\vec{x})$
$\displaystyle\frac{-i}{2}\left(a(\vec{p})_{t}-a^{+}(-\vec{p})_{t}\right)=\int
d^{3}xe^{-i\vec{p}\vec{x}}\pi(\vec{x})\,.$ (53)

From this equation, immediately follows the following transformation

$\displaystyle
a(\vec{p})=\frac{1}{2}\left(\frac{\omega}{\omega_{0}}+1\right)a_{0}(\vec{p})+\frac{1}{2}\left(\frac{\omega}{\omega_{0}}-1\right)a_{0}^{+}(-\vec{p})$
$\displaystyle
a(-\vec{p})^{+}=\frac{1}{2}\left(\frac{\omega}{\omega_{0}}+1\right)a_{0}(-\vec{p})^{+}+\frac{1}{2}\left(\frac{\omega}{\omega_{0}}-1\right)a_{0}(\vec{p}),$
(54)

where subscript $0$ means that the quantities are evaluated at $t_{0}=0$. The
quantities without subscript are evaluated at time $t$ and all frequencies are
evaluated for momentum $p$. We can compute different quantities after we get
connection between these operators. One example is the number operator, which
at the time $t$ has following form

$N=\int\frac{d^{3}p}{(2\pi)^{2}}\frac{1}{2\omega}a^{+}(\vec{p})a(\vec{p}).$
(55)

Rewriting it in the terms of operators in time $t_{0}=0$, we get

$\begin{split}N=\frac{1}{4}\int\frac{d^{3}p}{(2\pi)^{2}}\frac{1}{2\omega}\left[\left(\frac{\omega}{\omega_{0}}+1\right)a_{0}(\vec{p})^{+}+\left(\frac{\omega}{\omega_{0}}-1\right)a_{0}(-\vec{p})\right]\\\
\left[\left(\frac{\omega}{\omega_{0}}-1\right)a_{0}(-\vec{p})^{+}+\left(\frac{\omega}{\omega_{0}}+1\right)a_{0}(\vec{p})\right]\,,\end{split}$
(56)

which shows that in vacuum, corresponding $t_{0}=0$ after certain time
particles will be created.

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