KANAZAWA-21-05

January, 2022

Subcritical hybrid inflation
in a generalized superconformal model

Yoshihiro Gunji and Koji Ishiwata

Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192,
Japan

##  1 Introduction

Observations of the cosmic microwave background (CMB) radiation have provided
important clues to solving the mysteries of the Universe. One of the prominent
facts revealed by CMB observations is the indication of inflation at the early
stages of the Universe. Inflation is a paradigm that not only solves the
horizon and flatness problems but also gives the primordial curvature
perturbation leading to the large-scale structure of the present Universe (if
dark matter exists). Many inflation models have been proposed so far, and some
of them have already been excluded by the CMB observations. The latest results
by the Planck Collaboration on the scalar spectral index $n_{s}$, the tensor-
to-scalar ratio $r$, and the scalar amplitude $A_{s}$ are [1, 2]

$\displaystyle n_{s}$ $\displaystyle=0.9649\pm
0.0042\,(68\%\,\mathrm{C.L.})\,,$ (1) $\displaystyle r$
$\displaystyle<0.10\,(95\%\,\mathrm{C.L.})\,,$ (2) $\displaystyle A_{s}$
$\displaystyle=2.100\pm 0.030\times 10^{-9}\,(68\%\,\mathrm{C.L.})\,.$ (3)

Among the inflation models, the $R^{2}$ Starobinsky model [3, 4] is a
traditional and representative model that has good agreement with the
observations. It is also known that similar predictions for $n_{s}$ and $r$
are obtained in the $\alpha$ attractor model with small $\alpha$ [5].

Recently $D$-term hybrid inflation in a supersymmetric model has been
revisited, and new features of this model have been unveiled. It was found
that the $\alpha$ attractor appears in the superconformal version of the model
[6], while a chaotic regime was discovered in the subcritical regime, where
the inflaton field value gets smaller than the critical-point value of the
hybrid inflation [7, 8].#1#1#1References .[9, 10, 11] also point out that
inflation lasts below the critical point. In the model of the literature, the
inflation is induced by a slow-rolling waterfall field. On top of that, the
superconformal version of the model has turned out to have the subcritical
regime, and it has both the feature of the $\alpha$ attractor and natural
inflation [12]. Such multiple characteristics in the subcritical regime of
$D$-term hybrid inflation are controlled by (approximate) symmetries of the
Kähler potential and the superpotential. This may relate to the geometry of
the metric. The inflation model under conformal symmetry has recently been
studied in metric-affine geometry instead of Riemannian geometry. It was shown
that the $\alpha$ attractor and natural inflation emerge depending on the
global symmetry imposed on the model [13]. Furthermore, $k$-inflation is
studied in the conformal metric-affine geometry [14].

The subcritical hybrid inflation has other phenomenological features. It is
free from the cosmic string problem due to the fact that the inflation
continues long enough during the subcritical regime [7, 8]. Besides, it can be
embedded to the minimal supersymmetric standard model (and its extension) to
give rich phenomenology, such as producing baryon asymmetry and dark matter
and a characteristic pattern of neutrino masses [15]. Therefore, it is worth
investigating the subcritical regime in the broad class of the inflation
model.

In our paper, we study the subcritical regime of $D$-term hybrid inflation in
a generalized superconformal model. While the model is formulated in the
Einstein frame in the framework of the supergravity model, we show that it can
be formulated in the extended form of the canonical superconformal
supergravity model. Then we discuss the dynamics of the inflaton and waterfall
fields. It is shown that the subcritical regime is found in a wide range of
parameter space, and that various forms of the effective potential for
inflation are derived. Consequently, the tensor-to-scalar ratio turns out to
be larger than $\order{10^{-4}\,(10^{-3})}$ for 60 (50) $e$-folds before the
end of inflation. In addition, it is found that the symmetry-enhanced
points—namely, the Kähler potential without an explicit superconformal
breaking term or integer value for $3\alpha$ that relates to the dimension of
the compactified space in superstring theory—are consistent with the Planck
observation.

The results indicate further possibility to build a more phenomenologically
viable model. One example is the minimal supersymmetric standard model (MSSM)
augmented by the right-handed neutrinos, which was partly analyzed in Ref.
[15]. In this model, one of the right-handed sneutrinos plays the role of an
inflaton field, while the right-handed (s)neutrino generates the baryon number
of the Universe. In the analysis of the reference, however, the neutrino
oscillation data are not fully taken into account; meanwhile, the neutrino
sector has been intensively studied by neutrino oscillation experiments [16,
17, 18, 19, 20, 21] and cosmological observations [22, 1]. Those data not only
constrain such a model but also may give important hints for the new symmetry
of the flavor. Non-Abelian discrete symmetry, such as $S_{3}$, $A_{4}$,
$S_{4}$, and $A_{5}$, is one of the viable possibilities in that direction; it
has been rigorously studied [23, 24, 25, 26, 27, 28, 29] to explain the
mysterious pattern of the neutrino mixing and mass hierarchy. Thus, the
results obtained in the present work would give a hint towards unveiling the
underlying symmetry among inflation, baryogenesis, and the neutrino sector.

This paper is organized as follows: In Sec. 2, the $D$-term hybrid inflation
in the generalized superconformal model is formulated in a Jordan frame and in
the Einstein frame. Dynamics of the inflaton and waterfall fields are
discussed in Sec. 3, and consequently the effective inflaton potential in the
subcritical regime is derived. The cosmological consequences are discussed in
Sec. 4. Section 5 contains our conclusions and discussion for future work.
Throughout this paper, we use the Planck unit, $M_{\rm pl}=1$, unless
otherwise stated, and the metric tensor $g_{\mu\nu}$ that gives
$\eta_{\mu\nu}={\rm diag}(-1,1,1,1)$ in the flat limit.

##  2 The model

We consider a generalized version of the canonical superconformal supergravity
(CSS) model. The CSS model is proposed in Ref. [30]. The model is
characterized by two components, the superconformal Kähler potential ${\cal
N}$ and superconformal superpotential ${\cal W}$. In our paper, we introduce
an additional parameter $\alpha\,(>0)$ in the superconformal Kähler potential:

$\displaystyle{\cal
N}=-|X^{0}|^{2}\left[1-\frac{|S_{+}|^{2}+|S_{-}|^{2}+|N|^{2}}{|X^{0}|^{2}}-\frac{\chi}{2|X^{0}|^{2}}\Bigl{(}\frac{N^{2}\bar{X^{0}}}{X^{0}}+\frac{\bar{N}^{2}X^{0}}{\bar{X}^{0}}\Bigr{)}\right]^{\alpha}\,,$
(4)

where we have introduced a real constant $\chi$.#2#2#2In general, the term
proportional to $\chi$ can be more complicated form as shown in Ref. [30]. A
similar form of ${\cal N}$, but with $\chi=0$, is proposed in the context of
the superconformal $\alpha$ attractor [5]. Here $X^{0}$, $S_{\pm}$, and $N$
are chiral superfields that have the local U(1) charges 0, $\pm q$ $(q>0)$,
and $0$, respectively. In our paper, we use the same symbol for a chiral
superfield and its scalar field unless otherwise noticed. ($N$ and $S_{+}$
will be identified as the inflaton and waterfall fields, respectively.) A
nonzero $\chi$ explicitly breaks the superconformal symmetry. For the
superconformal superpotential, on the other hand, we consider a renormalizable
Yukawa interaction,

$\displaystyle{\cal W}=\lambda S_{+}S_{-}N\,,$ (5)

where $\lambda$ is a dimensionless constant. Here we ignore possible gauge-
invariant and renormalizable terms, such as $S_{+}S_{-}X^{0}$, $N^{3}$,
$X^{0}N^{2}$, etc., and stick to the simple model.#3#3#3Such extension would
be interesting in phenomenological point of view. For example, $N$ can be
identified as one of right-handed neutrino that has Majorana mass in Ref. [15]
when $\alpha=1$. After gauge fixing $X^{0}=\bar{X}^{0}=\sqrt{3}$ of the
superconformal symmetry, the Lagrangian (in a Jordan frame) is obtained as

$\displaystyle\frac{{\cal L}_{J}}{\sqrt{-g_{J}}}={\cal
N}\Bigl{(}-\frac{1}{6}R_{J}+{\cal A}^{2}_{\mu}\Bigr{)}-{\cal
N}_{\beta\bar{\beta}}g^{\mu\nu}_{J}{\cal D}_{\mu}z^{\beta}{\cal
D}_{\nu}\bar{z}^{\bar{\beta}}-V_{J}\,,$ (6)

where $R_{J}$ is the Ricci scalar, ${\cal
N}_{\beta\bar{\beta}}\equiv\partial^{2}{\cal N}/\partial
z^{\beta}\partial\bar{z}^{\bar{\beta}}$ ($z^{\beta}=S_{\pm},\,N$),
$g_{J\mu\nu}$ is the metric tensor, and ${\cal
D}_{\mu}\equiv\partial_{\mu}-igQA_{\mu}$ is the covariant derivative.
Meanwhile, the auxiliary gauge field ${\cal A}_{\mu}$ defined in Refs. [31,
30] has been introduced, and ${\cal A}_{\mu}$ can be taken to be zero when the
scalar part is discussed as described in the literature. $A_{\mu}$, $g$, and
$Q$ are the gauge field, the coupling, and the charge of the U(1) gauge.
$V_{J}=V_{J}^{F}+V_{J}^{D}$ is the scalar potential in the Jordan frame, where

$\displaystyle V_{J}^{F}$ $\displaystyle={\cal N}^{\beta\bar{\beta}}{\cal
W}_{\beta}\overline{{\cal W}}_{\bar{\beta}}\,,$ (7) $\displaystyle V_{J}^{D}$
$\displaystyle=\frac{1}{2}(\real f)^{ab}{\cal P}_{a}{\cal P}_{b}\,.$ (8)

Here, ${\cal N}^{\beta\bar{\beta}}$ is the $(\beta,\,\bar{\beta})$ component
of the inverse of ${\cal N}_{I\bar{J}}$ ($X^{I}=X^{0},\,S_{\pm},\,N$), which
is defined before gauge fixing; ${\cal W}_{\beta}\equiv\partial{\cal
W}/\partial z^{\beta}$; $f$ is the gauge kinetic function; and ${\cal
P}_{a}=-\eta^{\beta}_{a}{\cal N}_{\beta}-\tilde{\xi}$ (${\cal
N}_{\beta}\equiv\partial{\cal N}/\partial z^{\beta}$). $\eta^{\beta}_{a}$ is
the Killing vector, and in the present case $\eta^{\beta}_{a}=Qgz^{\beta}$ and
$f=1$. Since we consider U(1) theory, we omit the index $a$ hereafter. Note
that we have introduced the Fayet-Iliopoulos (FI) term $\tilde{\xi}$ by
adopting the procedure in Ref. [32]. According to the literature, an
additional term in the Lagrangian is considered:

$\displaystyle\frac{\Delta{\cal L}_{J}}{\sqrt{-g_{J}}}=g\frac{-{\cal
N}\xi}{3}{\cal P}\,,$ (9)

where $\xi$ is a constant. We take $\xi>0$ without the loss of generality.
This term gives $\tilde{\xi}=g{\cal N}\xi/3$ to get ${\cal
P}=-gQz^{\beta}{\cal N}_{\beta}-g{\cal N}\xi/3$.

The Lagrangian in the Einstein frame is obtained by the Weyl transformation,

$\displaystyle g_{J\mu\nu}=\Bigl{(}-\frac{{\cal
N}}{3}\Bigr{)}^{-1}g_{E\mu\nu}\,,$ (10)

where $g_{E\mu\nu}$ is the metric in the Einstein frame. Then we obtain

$\displaystyle\frac{\mathcal{L}_{E}}{\sqrt{-g_{E}}}=\frac{1}{2}R_{E}-K_{\beta\bar{\beta}}g^{\mu\nu}_{E}\mathcal{D}_{\mu}z^{\beta}\mathcal{D}_{\nu}\bar{z}^{\bar{\beta}}-V_{E}\,,$
(11)

where $R_{E}$ is the Ricci scalar in the Einstein frame, and

$\displaystyle K$
$\displaystyle=-3\alpha\ln\Bigl(-\frac{\Phi}{3}\Bigr{missing})\,,$ (12)
$\displaystyle\Phi$
$\displaystyle=-3+|S_{+}|^{2}+|S_{-}|^{2}+|N|^{2}+\frac{\chi}{2}(N^{2}+\bar{N}^{2})\,,$
(13) $\displaystyle V_{E}$ $\displaystyle=\Bigl{(}-\frac{{\cal
N}}{3}\Bigr{)}^{-2}V_{J}\,,$ (14)

and $K_{\beta\bar{\beta}}\equiv\partial^{2}K/\partial
z^{\beta}\partial\bar{z}^{\bar{\beta}}$.

Although we have derived it from a Jordan frame, one can derive the Lagrangian
in the Einstein frame starting from the Kähler potential [Eq. (12)] and
superpotential [Eq. (5)] in the supergravity model. Then the $F$ and $D$ terms
are derived from them as

$\displaystyle V_{E}^{F}$
$\displaystyle=e^{K}(K^{\beta\bar{\beta}}D_{\beta}{\cal
W}D_{\bar{\beta}}\overline{{\cal W}}-3|{\cal W}|^{2})$
$\displaystyle=\Bigl{(}-\frac{\Phi}{3}\Bigr{)}^{1-3\alpha}\frac{1}{\alpha}\Big{[}\delta^{\beta\bar{\beta}}{\cal
W}_{\beta}\overline{{\cal
W}}_{\bar{\beta}}+\frac{1}{\Delta}\big{|}\delta^{\beta\bar{\beta}}{\cal
W}_{\beta}\Phi_{\bar{\beta}}-3\alpha{\cal
W}\big{|}^{2}+\frac{9\alpha}{\Phi}(1-\alpha)|{\cal W}|^{2}\Big{]}\,,$ (15)
$\displaystyle V_{E}^{D}$
$\displaystyle=\frac{1}{2}D^{2}=\frac{g^{2}}{2}\big{(}K_{\beta}Qz^{\beta}-\xi\big{)}^{2}$
$\displaystyle=\frac{g^{2}}{2}\Big{[}\Bigl{(}-\frac{\Phi}{3}\Bigr{)}^{-1}\alpha
q(|S_{+}|^{2}-|S_{-}|^{2})-\xi\Big{]}^{2}\,.$ (16)

Here, $K^{\beta\bar{\beta}}$ is the inverse of $K_{\beta\bar{\beta}}$,
$D_{\beta}{\cal W}\equiv{\cal W}_{\beta}+K_{\beta}{\cal W}$, ${\cal
W}_{\beta}\equiv\partial{\cal W}/\partial z^{\beta}$,
$\Phi_{\beta}\equiv\partial\Phi/\partial z^{\beta}$, and
$\Delta\equiv\Phi-\delta^{\beta\bar{\beta}}\Phi_{\beta}\Phi_{\bar{\beta}}$. We
have explicitly checked that the sum of Eqs. (15) and (16) coincides with Eq.
(14). We note that when $\alpha=1$, this model reduces to one studied in Refs.
[32, 6, 33], as expected.

From the scalar potential, the masses of scalar part of the canonically
normalized $S_{\pm}$ are given by

$\displaystyle
m_{\pm}^{2}=\Bigl{(}-\frac{\Phi}{3}\Bigr{)}^{2-3\alpha}\frac{\lambda^{2}}{\alpha^{2}}|N|^{2}\mp
qg^{2}\xi\,.$ (17)

As with canonical $D$-term hybrid inflation, $S_{+}$ acquires a tachyonic
instability depending on the value of $N$, while $S_{-}$ is stabilized at the
origin. Hereafter, we take $S_{-}=0$. The critical-point value of $N$ where
$S_{+}$ becomes tachyonic is determined by $m_{+}=0$. Since both the real and
imaginary parts of $N$ can play the role of inflaton in the model, we take
$\phi\equiv\sqrt{2}\mathrm{Re}\,N$ as the inflaton field without the loss of
generality. #4#4#4To be explicit, the results for the case where
$\imaginary\,N$ are the inflaton field is obtained by replacing $\chi$ with
$-\chi$. In contrast to the previous studies [8, 33], we do not restrict our
present study to the case of $\chi\simeq-1$ ($+1$) where $\real\,N$
($\imaginary\,N$) has an approximate shift symmetry. Here, “approximate” means
that the shift symmetry in the Kähler potential is broken by the
superpotential. Therefore, the mass of $N$ appears in general [see Eq. (30)].
In addition, since the scalar potential depends on $|S_{+}|$, we define a
field $s\equiv\sqrt{2}|S_{+}|$ that we refer to as the waterfall field. Then,
defining $\Phi(\phi,s)$ as

$\displaystyle\Phi(\phi,s)$
$\displaystyle\equiv\Phi|_{\sqrt{2}N=\sqrt{2}\bar{N}=\phi,\,\sqrt{2}|S_{+}|=s,\,S_{-}=0}$
$\displaystyle=-3+\frac{1}{2}\big{(}s^{2}+(1+\chi)\phi^{2}\big{)}\,,$ (18)

the critical-point value $\phi_{c}$ should satisfy

$\displaystyle\Bigl{(}-\frac{\Phi_{c}}{3}\Bigr{)}^{2-3\alpha}\phi_{c}^{2}$
$\displaystyle=\frac{2\alpha^{2}}{k}\,,$ (19)

where $\Phi_{c}\equiv\Phi(\phi_{c},0)$ and

$\displaystyle k\equiv\lambda^{2}/qg^{2}\xi\,.$ (20)

The number of solutions of Eq. (19) depends on the values of $\alpha$ and
$\chi$. If it has multiple solutions, the potential gets complicated, and it
becomes different from the potential in the canonical hybrid inflation. In our
study, we focus on the case where there is one critical point. In that case,
the valid parameter space is #5#5#5When $\chi=-1$,
$\phi^{2}_{c}=2\alpha^{2}/k$ for any value of $\alpha$. This case corresponds
to the subcritical hybrid inflation with shift symmetry, which is already
studied in Refs. [7, 8].

$\displaystyle\left\\{\begin{array}[]{ll}(\mathrm{i})&~{}\chi<-1~{}{\rm
and}~{}(0<)\,\alpha\leq 1\\\ (\mathrm{ii})&~{}\chi>-1~{}{\rm and}~{}\alpha\geq
2/3\end{array}\right.\,.$ (23)

In the parameter space, $\alpha=1$ and $2/3$ are special values, since the
critical-point values can be obtained analytically as

$\displaystyle\phi_{c}=\left\\{\begin{array}[]{ll}\sqrt{6/(3k+1+\chi)}&~{}~{}{\rm
for~{}}\alpha=1\\\\[5.69054pt] (2/3)\sqrt{2/k}&~{}~{}{\rm
for~{}}\alpha=2/3\end{array}\right.\,.$ (26)

It is obvious that $k$ has a bound:

$\displaystyle k>\left\\{\begin{array}[]{ll}-(1+\chi)/3&~{}~{}{\rm
for}~{}\alpha=1~{}{\rm and}~{}\chi<-1\\\\[5.69054pt] 4(1+\chi)/27&~{}~{}{\rm
for}~{}\alpha=2/3~{}{\rm and}~{}\chi>-1\end{array}\right.\,.$ (29)

The latter is given by $\phi_{c}<\phi_{\rm max}\equiv\sqrt{6/(1+\chi)}$, where
$\phi_{\rm max}$ is determined by $\Phi(\phi_{\rm max},0)=0$.

For successful inflation, the stabilization of the imaginary part of $N$,
which we define as $\tau\equiv\sqrt{2}\imaginary N$, should be guaranteed in
the subcritical region where $\phi<\phi_{c}$. The mass of $\tau$ in the region
is given by

$\displaystyle
m^{2}_{\tau}=\frac{g^{2}\xi^{2}k}{\alpha^{2}}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}^{1-3\alpha}\Bigl{(}1-\Psi(\phi)\Bigr{)}\Bigl{[}1-\frac{\phi^{2}}{6}\bigl{\\{}3-\chi+3\alpha(\chi-1)\bigr{\\}}\Bigr{]}\,,$
(30)

where $\Phi_{0}\equiv\Phi(\phi,0)$ and

$\displaystyle\Psi(\phi)\equiv\Bigl{(}\frac{\Phi_{0}}{\Phi_{c}}\Bigr{)}^{2-3\alpha}\frac{\phi^{2}}{\phi_{c}^{2}}=\frac{k}{2\alpha^{2}}\Bigl{(}-\frac{\Phi_{0}}{3}\Bigr{)}^{2-3\alpha}\phi^{2}\,,$
(31)

which satisfies $\Psi(\phi_{c})=1$. Here $s^{2}\ll\phi^{2}$ has been used,
which will be validated in the later discussion. To satisfy $m^{2}_{\tau}>0$
in the valid parameter space above, we find the following preferred regions:

  1. (i)

$\chi<-1$:

$\displaystyle\begin{cases}1/3+2/3(1-\chi)<\alpha\leq 1\,.\\\
\alpha<1/3+2/3(1-\chi),~{}{\rm
depending~{}on~{}other~{}parameters.}\end{cases}$ (32)

  2. (ii)

$\chi>-1$:

$\displaystyle\begin{cases}\chi<1\,.\\\ \chi\gg 1~{}{\rm
and~{}small}~{}\alpha~{}({\rm but}~{}\geq 2/3).\end{cases}$ (33)

In our numerical analysis, we will compute the cosmological consequences in
the parameter space given in Eq. (23) and see consistency with the above
regions. We will see that $m^{2}_{\tau}>0$ gives a constraint for the
$\chi\gtrsim 5$ case. When $\tau$ is stabilized to the origin, the scalar
potential after the critical point is given by

$\displaystyle V_{\mathrm{tot}}(\phi,s)$ $\displaystyle=V_{E}^{F}+V_{E}^{D}$
$\displaystyle=\Big{(}-\frac{\Phi(\phi,s)}{3}\Big{)}^{1-3\alpha}\frac{\lambda^{2}}{4\alpha}\phi^{2}s^{2}+\frac{g^{2}}{8}\bigg{[}\Big{(}-\frac{\Phi(\phi,s)}{3}\Big{)}^{-1}\alpha
qs^{2}-2\xi\bigg{]}^{2}\,.$ (34)

Even if $\tau$ is stabilized at the origin, it has a quantum fluctuation
during inflation, which might cause another instability. In the model, $-\Phi$
must be positive. Therefore, this requirement leads to a constraint on the
amplitude of $\tau$:

$\displaystyle\tau^{2}<\frac{6}{1-\chi}-\frac{1+\chi}{1-\chi}\phi^{2}\,,$ (35)

for $\chi<1$. (There is no constraint when $\chi\geq 1$.) The quantum
fluctuation during inflation is estimated as

$\displaystyle\tau^{2}\sim\frac{H^{4}}{m_{\tau}^{2}}\sim g^{2}\xi^{2}\,,$ (36)

where $H$ is the Hubble parameter. Here we have used Eq. (30) and $H\sim
g^{2}\xi^{2}$. Though $m_{\tau}=0$ at the critical point, there is a mass term
that arises at loop level (see the next section). Even in that case, the
estimate for the mass is roughly the same except for an extra loop factor. In
the allowed parameter space that we will show in the later analysis, this
quantity is extremely smaller than unity. Therefore, the condition for Eq.
(35) is always satisfied. One may worry about the isocurvature induced by
$\tau$. We note that since $\tau$ has the same order of decay width as $\phi$
has, $\tau$ decays at the time of reheating. Assuming that $\tau$ dominantly
decays to a radiation bath, the isocurvature that $\tau$ produces has no
effect on the later thermal history. In addition, the energy density of $\tau$
during inflation is estimated as $m_{\tau}^{2}\tau^{2}\sim H^{4}$, which is
much smaller than the inflaton energy density. Then, inflation driven by the
$\phi$-$s$ system is not affected by the quantum fluctuation of $\tau$. Thus,
there is no constraint due to the quantum fluctuation of $\tau$.#6#6#6We are
indebted to Tomo Takahashi for private communication on this issue.

##  3 Dynamics of inflaton and waterfall fields

We discuss the dynamics of inflaton and waterfall fields around and after the
critical point. In the typical hybrid inflation model, inflation ends at the
critical point where the waterfall field becomes tachyonic. In the subcritical
hybrid inflation, by contrast, inflation continues after the critical point.
The crucial point here is a suppressed $\lambda$, as pointed out in Ref. [7].
If $\lambda\ll 1$, therefore, the subcritical regime is expected to emerge in
the generalized framework of the superconformal hybrid inflation model. In
this section, we confirm this and give the effective potential for inflation
after the critical point. See Refs. [32, 8, 33] for details.

Before it reaches to the critical point, we assume that the inflaton field
slowly rolls down to the critical point while the other scalar fields are
initially stabilized at the origin. Since the tree-level potential is
constant, the motion of the inflaton field is driven by the Coleman-Weinberg
potential [34, 7], which is given by

$\displaystyle V_{1L}=\frac{g^{4}q^{2}\xi^{2}}{32\pi^{2}}L(\Psi)\,,$ (37)

where $L(x)\equiv(x-1)^{2}\ln(x-1)+(x+1)^{2}\ln(x+1)-2x^{2}\ln x-\ln 16$. The
velocity of the inflaton field at the critical point $\dot{\phi}_{c}$ is given
by

$\displaystyle\dot{\phi}_{c}=-\frac{1}{3H_{c}}\partialderivative{V_{1L}}{\phi}\evaluated{}_{\phi=\phi_{c}}\,,$
(38)

where $H_{c}$ is the Hubble parameter at the critical point and the dot
signifies a time derivative. After the critical point, the tachyonic growth of
the waterfall field begins. In order to describe the dynamics, we use the
canonically normalized waterfall field. Taking $\hat{s}_{k}$ as a Fourier mode
of the canonically normalized waterfall field, the equation of motion is given
by [32, 33, 35] #7#7#7Here, $k$ stands for momentum. Do not confuse this with
the dimensionless parameter $k$ given in Eq. (20).

$\displaystyle\ddot{\hat{s}}_{k}+\left(k^{2}e^{-2H_{c}t}-\frac{9}{4}H_{c}^{2}-\hat{d}^{3}t\right)\hat{s}_{k}=0\,,$
(39)

where $\hat{d}$ is obtained as

$\displaystyle\hat{d}^{3}=\frac{g^{2}q\xi(2-(1-\alpha)(1+\chi)\phi^{2}_{c})}{\phi_{c}(-\Phi_{c}/3)}|\dot{\phi}_{c}|\,.$
(40)

It is noted that $\hat{d}>0$ and $-\dot{\phi}_{c}>0$ are satisfied in the
parameter space of Eq. (23). By solving the equation of motion, we get the
variance $\expectationvalue{\hat{s}^{2}(t)}$. After the decoherence time
$t_{\rm dec}$ defined in Ref. [7], the variance is matched to the classical
field, and the time evolutions of the waterfall field and inflaton field are
determined by the classical equations of motion. Namely, we solve the
classical equations of motion of $\phi$ and $s$ with a boundary condition
$s(t_{\rm dec})=\sqrt{\expectationvalue{\hat{s}^{2}(t_{\rm
dec})}/K_{S_{+}\bar{S}_{+},\,c}}$:

$\displaystyle\dot{\phi}+\frac{1}{3HK_{N\bar{N}}}\partialderivative{V_{\rm
tot}}{\phi}=0\,,$ (41)
$\displaystyle\dot{s}+\frac{1}{3HK_{S_{+}\bar{S}_{+}}}\partialderivative{V_{\rm
tot}}{s}=0\,,$ (42)

where

$\displaystyle H=\sqrt{V_{\rm tot}(\phi,s)/3}\,,$ (43) $\displaystyle
K_{N\bar{N}}=\alpha\frac{1+\frac{1}{6}\chi(1+\chi)\phi^{2}-\frac{1}{6}s^{2}}{(-\Phi(\phi,s)/3)^{2}}\,,$
(44) $\displaystyle
K_{S_{+}\bar{S}_{+}}=\alpha\frac{1-\frac{1}{6}(1+\chi)\phi^{2}}{(-\Phi(\phi,s)/3)^{2}}\,.$
(45)

Here $K_{S_{+}\bar{S}_{+},\,c}$ is the value at the critical point.

In order to track the inflation dynamics in the $\phi$-$s$ system, we define
the slow-roll parameters by the Hubble parameter,

$\displaystyle\epsilon_{H}\equiv-\frac{\dot{H}}{H^{2}}\,,~{}~{}~{}\eta_{H}\equiv\epsilon_{H}-\frac{\ddot{H}}{2H\dot{H}}\,.$
(46)

We have confirmed that inflation continues after crossing the critical point
in the parameter space we are interested in. In addition, it is found that the
dynamics after the critical point is effectively described by the single
field. To be explicit, the waterfall field is relaxed to the local minimum
$s_{\rm min}$ after up to a few Hubble times. During this period, the inflaton
field merely moves and stays at the critical point. The local minimum value is
approximately given by #8#8#8 To be precise, the local minimum should be
determined numerically by solving $\partial V_{\rm tot}(\phi,s)/\partial s=0$.
However, it turns out that Eq. (47) agrees well with the exact solution when
$\xi\ll 1$, which is the parameter region on which we focus.

$\displaystyle s_{\mathrm{min}}^{2}(\phi)$
$\displaystyle=-\frac{\Phi_{0}}{3}\frac{2\xi}{q\alpha}\Big{(}1-\Psi(\phi)\Big{)}\,.$
(47)

We have numerically checked that mass of $s$ is larger than the Hubble in the
subcritical region. Namely, the same situation in Refs. [7, 8, 33] is
realized. Putting $s_{\rm min}$ into $V_{\rm tot}(\phi,s)$ and ignoring
parametrically unimportant terms suppressed by $\xi$, the effective potential
in the subcritical regime is obtained as

$\displaystyle V(\phi)$
$\displaystyle=g^{2}\xi^{2}\Psi(\phi)\Big{(}1-\frac{1}{2}\Psi(\phi)\Big{)}\,.$
(48)

We have confirmed that after the waterfall field relaxes to the local minimum,
$\epsilon_{H}$ and $\eta_{H}$ coincide with $\epsilon(\phi)$ and $\eta(\phi)$
defined in Eq. (50), respectively, derived from the single-field effective
potential $V(\phi)$. In later analysis, we use the effective potential to
discuss the cosmological consequences. We will see in the next section that
the typical inflaton field value that is canonically normalized is super-
Planckian. However, the predicted tensor-to-scalar ratio can be much smaller
than unity.

Careful readers may worry about the effect of the waterfall field on the
adiabatic curvature perturbation. It is expected to be negligible since, as we
will see in the next section, the trajectory of the subcritical inflation is
almost straight along the inflaton field. Such trajectory was already studied
and analyzed in Ref. [7]. #9#9#9The inflation trajectory below the critical
point is also analyzed in Refs .[9, 10, 11]. In that case, the trajectory
during inflation is almost the waterfall field direction.  The effect can be
evaluated by calculating a quantity $e^{\beta}=1+4\eta_{\perp}^{2}H^{2}/M^{2}$
given in Ref. [36] (see also Ref. [37].) Here we follow the notation of Ref.
[36]. (Do not confuse their $\beta$ with the one defined in this paper.) A
variable $\eta_{\perp}$ describes the curvature of the trajectory. Namely,
$\eta_{\perp}=0$ means that the trajectory is straight. $M$ is the mass of the
field perpendicular to the inflaton direction. The value $e^{\beta}$ has an
impact on the scalar amplitude as

$\displaystyle A_{s}\to A_{s}^{\prime}=e^{\beta}A_{s}\,.$ (49)

Using the formula given in Ref. [36], we have found that
$\eta_{\perp}^{2}\sim\order{10^{-6}}$ around 60 $e$-folds in the valid
parameter regions that will be shown in Fig. 1. In addition,
$H^{2}/M^{2}\sim\xi$, which is suppressed as $\order{10^{-4}}$ (see Figs. 2,
5, and 8). Consequently, $e^{\beta}-1\sim\order{10^{-10}}$, which is
sufficiently small. Therefore, the effective description in the subcritical
regime is valid.

##  4 Cosmological consequences

Now we are ready to discuss cosmological consequences of this model. We
compute the scalar spectral index, the tensor-to-scalar ratio, and the scalar
amplitude and compare them with the latest observational results [Eqs.
(1)–(3)].

###  4.1 Cosmological parameters and overview of the results

The slow-roll parameters defined by the effective potential are

$\displaystyle\epsilon(\phi)\equiv\frac{1}{2}\Big{(}\frac{V^{\prime}}{V}\Big{)}^{2}\,,\quad\eta(\phi)\equiv\frac{V^{\prime\prime}}{V}\,.$
(50)

Here, $V^{\prime}=dV/d\hat{\phi}$ and
$V^{\prime\prime}=d^{2}V/d\hat{\phi}^{2}$, and $\hat{\phi}$ is the canonically
normalized inflaton field, which is defined by

$\displaystyle\frac{d\phi}{d\hat{\phi}}=K_{N\bar{N}}^{-1/2}\evaluated{}_{s=s_{\rm
min}}\,.$ (51)

Since the field value of the waterfall field is found to be parametrically
much smaller than the inflaton field value, it can be approximated as
$s_{\mathrm{min}}\simeq 0$ during inflation. Therefore,

$\displaystyle
K_{N\bar{N}}\simeq\frac{3\alpha}{-\Phi_{0}}\Bigl{[}1+\frac{(1+\chi)^{2}\phi^{2}}{-2\Phi_{0}}\Bigr{]}\,.$
(52)

Inflation ends at

$\displaystyle\phi=\phi_{\rm end}\equiv{\rm
Max}\\{\phi_{\epsilon},\phi_{\eta}\\}\,.$ (53)

$\phi_{\epsilon}$ and $\phi_{\eta}$ are determined by
$\epsilon(\phi_{\epsilon})=1$ and $|\eta(\phi_{\eta})|=1$, respectively.
$n_{s}$, $r$, and $A_{s}$ are determined by the slow-roll parameters as

$\displaystyle n_{s}$ $\displaystyle=1+2\eta(\phi_{*})-6\epsilon(\phi_{*})\,,$
(54) $\displaystyle r$ $\displaystyle=16\epsilon(\phi_{*})\,,$ (55)
$\displaystyle A_{s}$
$\displaystyle=\frac{V(\phi_{*})}{24\pi^{2}\epsilon(\phi_{*})}\,.$ (56)

Here, $\phi_{*}$ is determined by the number of $e$-folds before the end of
inflation:

$\displaystyle N_{*}=\int_{\hat{\phi}_{\rm
end}}^{\hat{\phi}_{*}}d\hat{\phi}\frac{V}{V^{\prime}}=\int_{\phi_{\rm
end}}^{\phi_{*}}d\phi\sqrt{\frac{K_{N\bar{N}}}{2\epsilon(\phi)}}\,,$ (57)

where $\hat{\phi}_{*}$ and $\hat{\phi}_{\rm end}$ are canonically normalized
field values corresponding to $\phi_{*}$ and $\phi_{\rm end}$, respectively.
In our numerical analysis, we take $q=g=1$ without the loss of generality.
This is due to the fact that when $\xi\ll 1$, which is the case we are
interested in, $q$ and $g$ can be absorbed into $\lambda$ and $\xi$ by
redefining these as $\bar{\lambda}\equiv\lambda/\sqrt{qg}$ and
$\bar{\xi}\equiv g\xi$, respectively. This is easily seen in Eq. (48). Since
$\xi$ is cancelled in the slow-roll parameters, $\phi_{\rm end}$ and
$\phi_{*}$ are determined for $\alpha,\,\chi,\,k$, and $N_{*}$, and they give
$n_{s}$ and $r$. The median value of the observed scalar amplitude determines
the value of $\lambda$ and $\xi$ (also using the value of $k$).

Figure 1: Allowed region for $\alpha$ and $\chi$ for $N_{*}=60$ (top) and $50$
(bottom) by imposing $m^{2}_{\tau}>0$ and Eqs. (1)–(3). The right panels are
the same as the left ones but the range of $-1.5\leq\chi\leq-0.5$ is
magnified. The color map indicates the minimum value of the predicted tensor-
to-scalar ratio, $r_{\mathrm{min}}$.

First of all, we give the allowed region on the $(\alpha,\,\chi)$ plane in
Fig. 1. We impose $m^{2}_{\tau}>0$ and the observed results given in Eqs.
(1)–(3). In the plot $N_{*}=60$ and $50$ are taken, and each dot corresponds
to an allowed point. For a given set of $\alpha$, $\chi$, and $N_{*}$, the
values of $n_{s}$ and $r$ are given as a function of $k$. When $k$ has
solutions such that the predicted $n_{s}$ and $r$ are within the range of Eqs.
(1) and (2), a dot is plotted on the $\alpha$-$\chi$ plane. The color map
shows the minimum value of $r$ in the range of Eqs. (1) and (2). It is found
that selective regions are allowed, and the behavior changes around $\chi=-1$.
In addition, the allowed value of $\alpha$ saturates for $|\chi|\gtrsim 5$.
When $N_{*}=60$, for instance, the allowed regions are $\alpha\simeq 1$ for
$\chi\lesssim-5$, $2/3\leq\alpha\leq 1$ for $\chi\simeq-1$, and $\alpha\simeq
2/3$ for $\chi\gtrsim 5$, and the predicted $r$ changes by orders of
magnitude. It is found that Eq. (32) does not give constraints on the
parameters for $\chi<-1$. On the contrary, the obtained allowed region roughly
tracks the region indicated in Eq. (33) for $\chi>-1$. To investigate further,
we categorize the parameter space into two regions:

$\displaystyle\begin{cases}|\chi|\gtrsim 5\\\ \chi\simeq-1\end{cases}.$

We examine the dynamics of the inflaton and waterfall fields and their
consequences in detail in Secs. 4.2 and 4.3. In addition, we further
investigate the specific cases $\alpha=1$, $2/3$, and $\chi=0$ in Sec. 4.4.
These cases are motivated by theoretical models beyond supergravity. In
superstring theory, $3\alpha$ corresponds to the dimension of the compactified
space. Therefore, it is supposed to be an integer. When $\chi=0$, on the other
hand, the superconformal invariance is exact in the Kähler potential, and it
corresponds to a symmetry enhanced point.

###  4.2 $|\chi|\gtrsim 5$

Figure 2: Left: Predicted scalar spectral index and tensor-to-scalar ratiofor
fixed $e$-folds, $N_{*}=60$ (solid curves) and $50$ (dotted curves), and
various values of $\alpha$. In the top and bottom panels, $\chi$ is taken to
$-5$ and $5$, respectively. We impose $m^{2}_{\tau}>0$, and 1$\sigma$ (dark
shaded) and 2$\sigma$ (light shaded) regions from the Planck Collaboration [2]
are also shown. Right: Allowed regions for $\lambda$ and $\sqrt{\xi}$ for
$\chi=-5$ (top) and $5$ (bottom), by imposing Eqs. (1)–(3).

Summary of the predictions. — The predictions for $n_{s}$ and $r$ are shown in
the left panels of Fig. 2 for $\chi=-5$ (top) and $5$ (bottom) with various
values of $\alpha$. Here the stability condition for $\tau$ is imposed. In the
right panels, the parameters ($\sqrt{\xi}$, $\lambda$) that are consistent
with the Planck observations are plotted for $N_{*}=60$ and $50$. For a given
$\alpha$, it is found that $n_{s}$ and $r$ get larger for smaller $k$. For
$\chi=-5$, $r$ tends to be smaller when $\alpha\to 1$; meanwhile, it can be as
large as $\order{0.1}$. This is the opposite behavior compared to the $\alpha$
attractor model [5]. As a consequence, a lower bound is obtained as $r\gtrsim
10^{-3}$. For $\chi=5$, on the other hand, $r$ is found to be suppressed as
$10^{-4}\lesssim r\lesssim 10^{-3}$, and it gets smaller as $\alpha$
decreases. The parameter space that is consistent with the Planck data turns
out to be $10^{-3}\lesssim\lambda\lesssim 10^{-2}$ and
$\sqrt{\xi}\sim\order{10^{16}\,{\rm GeV}}$. The results for $\chi<-5$
($\chi>5$) behave almost the same as those for $\chi=-5$ ($\chi=5$).
#10#10#10We find that the preferred value of $\lambda$ becomes larger, but it
is less than $10^{-2}$.

To get a better understanding of the results, it is legitimate to describe the
effective potential in terms of the canonically normalized field $\hat{\phi}$.
In addition, we consider a large field value limit for $\phi$ to give
analytical expressions for the effective potential. Even though the analytical
expressions are not always valid, they are used to help understand the
numerical results for $n_{s}$ and $r$ qualitatively. When $|\chi|\gg 1$ and in
a large field limit, the second term in the parenthesis in Eq. (52) can be
neglected. In that case, the Kähler metric is approximately given by

$\displaystyle
K_{N\bar{N}}\simeq\frac{3\alpha(1+\chi)^{2}\phi^{2}}{2\Phi^{2}_{0}}\,,$ (58)

and consequently Eq. (51) can be solved analytically as to give $\phi$. Using
the result, $\Phi_{0}$ and $\Psi$ are determined. The results are

$\displaystyle\phi^{2}$
$\displaystyle\simeq\begin{cases}\displaystyle\frac{1}{\beta}(Ce^{\sqrt{\frac{2}{3\alpha}}\hat{\phi}}-1)&~{}~{}~{}(\chi<-1)\\\\[8.53581pt]
\displaystyle\frac{1}{-\beta}(1-Ce^{-\sqrt{\frac{2}{3\alpha}}\hat{\phi}})&~{}~{}~{}(\chi>-1)\end{cases}\,,$
(59) $\displaystyle-\frac{\Phi_{0}}{3}$
$\displaystyle\simeq\begin{cases}\displaystyle
Ce^{\sqrt{\frac{2}{3\alpha}}\hat{\phi}}\displaystyle&~{}~{}~{}(\chi<-1)\\\\[5.69054pt]
Ce^{-\sqrt{\frac{2}{3\alpha}}\hat{\phi}}&~{}~{}~{}(\chi>-1)\end{cases}\,,$
(60) $\displaystyle\Psi$
$\displaystyle\simeq\begin{cases}\displaystyle\frac{k}{2\alpha^{2}\beta}C^{2-3\alpha}e^{\sqrt{\frac{2}{3\alpha}}(2-3\alpha)\hat{\phi}}(Ce^{\sqrt{\frac{2}{3\alpha}}\hat{\phi}}-1)&~{}~{}~{}(\chi<-1)\\\\[8.53581pt]
\displaystyle\frac{k}{-2\alpha^{2}\beta}C^{2-3\alpha}e^{-\sqrt{\frac{2}{3\alpha}}(2-3\alpha)\hat{\phi}}(1-Ce^{-\sqrt{\frac{2}{3\alpha}}\hat{\phi}})&~{}~{}~{}(\chi>-1)\end{cases}\,,$
(61)

where $\beta\equiv-(1+\chi)/6$ and $C$ is a positive constant. $C$ can be
determined by, for example, $\Psi(\hat{\phi}_{c})=1$. The results are
summarized in Table 1. For the cases of $\chi<-1$ & $\alpha\neq 1$ and
$\chi>-1$ & $\alpha\neq 2/3$, $2\alpha^{2}|\beta|/k\gg 1$ is further assumed,
which will be discussed soon. The approximated expressions for $\Psi$ have
similarity even for $\chi<-1$ and $\chi>-1$. Based on the expressions, we
categorize the parameter space into two cases:

$\displaystyle\left\\{\begin{array}[]{l}\chi<-1\,\&\,\alpha\neq
1\,\text{or}\,\chi>-1\,\&\,\alpha\neq 2/3\\\
\chi<-1\,\&\,\alpha=1\,\text{or}\,\chi>-1\,\&\,\alpha=2/3\end{array}\right.\,,$

and we discuss the numerical results in Fig. 2.

Table 1: Approximated expression of $\Psi$ in the limit $|(1+\chi)^{2}\phi^{2}/(-2\Phi_{0})|\gg 1$. For cases of $\chi<-1$ & $\alpha\neq 1$ and $\chi>-1$ & $\alpha\neq 2/3$, $2\alpha^{2}|\beta|/k\gg 1$ is further assumed. The parameter space where the approximated expression is valid is given in the right column. (See also Fig. 1 for the allowed region for $\alpha$ and $\chi$.) The effective potential can be obtained from $\Psi$ by using Eq. (48). Cases | Approximated expression for $\Psi$ | Valid parameters
---|---|---
$\chi<-1\,$& $\alpha\neq 1$ | $e^{\sqrt{\frac{6}{\alpha}}(1-\alpha)(\hat{\phi}-\hat{\phi}_{c})}$ |  $\chi\lesssim-20$ & $\alpha\lesssim 2.8/3$
$\chi>-1$ & $\alpha\neq 2/3$ | $e^{\sqrt{\frac{2}{3\alpha}}(3\alpha-2)(\hat{\phi}-\hat{\phi}_{c})}$ |  $\chi\gtrsim-0.5$ & $\alpha\gtrsim 2.2/3$
|  $\chi<-1$ & $\alpha=1$
---
$\chi>-1$ & $\alpha=2/3$
$\frac{k}{2\alpha^{2}|\beta|}\left[1-\Bigl{(}1-\frac{2\alpha^{2}|\beta|}{k}\Bigr{)}e^{-\sqrt{\frac{2}{3\alpha}}(\hat{\phi}-\hat{\phi}_{c})}\right]$ |  | $\chi\lesssim-10$
---
$\chi\gtrsim 2$

####  4.2.1 $\chi<-1$ & $\alpha\neq 1$ or $\chi>-1$ & $\alpha\neq 2/3$

As mentioned above, we further assume

$\displaystyle 2\alpha^{2}|\beta|/k\gg 1$ (62)

to derive the expressions, which are shown in the first row of Table 1. If the
above condition is satisfied, then the second term in the parenthesis of Eq.
(59) can be ignored. (The condition can be rewritten as
$|\beta|\phi_{c}^{2}\gg 1$.) We note here that this approximation is only
valid for an $\alpha$ which is not much closer to 1 or 2/3. Ignoring this term
under the limit of $\alpha\to 1$ or 2/3, one cannot determine a consistent $C$
from the analytic expressions.

To get a very rough picture, let us make more simplifications. If we further
assume that the effective potential is determined by $V\sim g^{2}\xi^{2}\Psi$,
then $n_{s}$ and $r$ are given by $(n_{s}\,,r)\sim(1-p^{2},\,8p^{2})$, where
$p\equiv\sqrt{6/\alpha}(1-\alpha)$ for $\chi<-1$ and
$p\equiv\sqrt{2/3\alpha}(3\alpha-2)$ for $\chi>-1$. The expressions indicate
that for $\chi<-1$ ($\chi>-1$), $n_{s}$ and $r$ get larger and smaller,
respectively, when $\alpha$ approaches unity (2/3). This estimation is
consistent with the results shown in Fig. 2. Although the qualitative behavior
can be understood from this rough estimation, quantitative discussion is found
to be more complicated.

For the $\chi<-1$ case, the approximation is found to be good for
$\chi\lesssim-20$ and $\alpha\lesssim 2.8/3$. If $\alpha$ becomes larger, we
find that the approximated expression of $\Psi$ does not give the correct
values of $n_{s}$ and $r$ even though Eq. (62) is satisfied. The maximum
values of $n_{s}$ and $r$ come from saturated values of $\phi_{*}$ in the
limit $k\to 0$.

For the $\chi>-1$ case, on the other hand, the expression is found to be valid
for $\chi\gtrsim-0.5$ and $\alpha\gtrsim 2.2/3$. However, we have found that
the upper bound for $n_{s}$ (and $r$) is given by the stability of $\tau$.
This is expected from Eq. (33). Therefore, it is difficult to understand the
predicted value of $n_{s}$ and $r$ based on the simple approximation.

Therefore, to see the $\alpha$ dependence on $n_{s}$ and $r$ for cases of
$\chi<-1$ and $\chi>-1$, it is more intuitive to plot the effective potential
numerically. Figure 3 shows the effective potential $\hat{V}$ (normalized by
the value at the critical point) plotted for various values of $\alpha$. Here
we have derived $\hat{\phi}$ by numerically solving Eq. (51). One can see that
$\hat{\phi}_{c}$ (and $\hat{\phi}_{*}$) becomes smaller as $\alpha$ approaches
$1$. Consequently smaller $r$ is obtained, which is consistent with the
results shown in Fig. 2.

####  4.2.2 $\chi<-1$ & $\alpha=1$ or $\chi>-1$ & $\alpha=2/3$

As seen in Table 1, $\Psi$ is given in more complicated expressions compared
to the previous case; meanwhile, both the $\alpha=1$ and $2/3$ cases give a
similar $\Psi$ expression. This expression is valid when
$|(1+\chi)^{2}\phi^{2}/(-2\Phi_{0})|\gg 1$ is satisfied. We have found that
the approximated $\Psi$ is valid for $\chi\lesssim-10$ and $\chi\gtrsim 2$
when $\alpha=1$ and $2/3$, respectively. Besides this, recall that $k$ has a
lower bound, given in Eq. (29). Aside from the exponential factor, $\Psi$ is
given as a function of $2\alpha^{2}|\beta|/k$, and it has the same upper
bound:

$\displaystyle 2\alpha^{2}|\beta|/k<1\,.$ (63)

Therefore, the predicted values of $n_{s}$ and $r$ for $\alpha=1$ and $2/3$
are expected to be similar to each other. #11#11#11The model with $\alpha=1$
is the same model studied in Ref. [33]. The predictions for $(n_{s},\,r)$ are
different from those in the literature, because now we consider $\chi$ away
from $-1$.  In fact, Fig. 2 shows the expected results. We have found that the
lowest value of $k$ determines the maximum values for $n_{s}$ and $r$, and
that the results resemble each other. One can see this fact from the plot of
the effective potential shown in Fig. 3. We will come back to $\alpha=1$ and
$2/3$ cases with various values of $\chi$ in Sec. 4.4.

In a nutshell, for $|\chi|\gg 1$, the potential becomes flatter as $\alpha$
approaches unity or 2/3. This allows $\hat{\phi}_{*}$ to have smaller values,
and consequently, we obtain smaller $r$ values.

Figure 3: Normalized potential $\hat{V}$ as a function of the canonically
normalized inflaton field $\hat{\phi}$. Only the subcritical regime is shown,
and the field values of $\hat{\phi}$ at the end of inflation and at 60
$e$-folds are indicated as $\hat{\phi}_{\rm end}$ and $\hat{\phi}_{*}$,
respectively. Color codes are the same as in Fig. 2, and $\lambda$ and
$\sqrt{\xi}$ are chosen to give the best-fit value of the observed $n_{s}$.

Figure 4: Trajectory of the waterfall field (red dotted curve) as a function
of the canonically normalized inflaton field $\hat{\phi}$. As a reference, the
normalized potential $\hat{V}$ (blue solid curve) is also plotted. The field
values of $\hat{\phi}$ at the end of inflation and at 60 $e$-folds are
indicated as $\hat{\phi}_{\rm end}$ and $\hat{\phi}_{*}$, respectively. In the
top (bottom) panels, $\chi$ is taken to $-5$ ($5$). The other parameters are
$\alpha=1$, $\lambda=3.1\times 10^{-3}$, $\sqrt{\xi}=6.5\times 10^{15}$ GeV
(top-left); $\alpha=2.8/3$, $\lambda=3.3\times 10^{-3}$, $\sqrt{\xi}=2.0\times
10^{16}$ GeV (top-right); $\alpha=2/3$, $\lambda=2.2\times 10^{-3}$,
$\sqrt{\xi}=5.5\times 10^{15}$ GeV (bottom-left); and $\alpha=2.04/3$,
$\lambda=2.4\times 10^{-3}$, $\sqrt{\xi}=6.6\times 10^{15}$ GeV (bottom-
right). With these parameters, the median value of the observed $n_{s}$ is
obtained, and the tensor-to-scalar ratio is predicted to be $7.9\times
10^{-4}$ (top-left), $6.4\times 10^{-2}$ (top-right), $4.2\times 10^{-4}$
(bottom-left), and $8.5\times 10^{-4}$ (bottom-right).

Finally, we plot the trajectory of the waterfall field as a function of
$\hat{\phi}$ in Fig. 4. The potential is found to be almost flat at $60$
$e$-folds from the end of inflation except for the $\alpha=2.8/3$ and
$\chi=-5$ case. In this case, the waterfall field grows much larger than the
global minimum value due to a factor

$\displaystyle-\frac{\Phi_{0}}{3}(1-\Psi)\sim-\Bigl{(}\frac{2\alpha^{2}\beta}{k}\Bigr{)}^{\frac{1}{3(1-\alpha)}}\sqrt{\frac{6}{\alpha}}(1-\alpha)(\hat{\phi}-\hat{\phi}_{c})$
(64)

where $(2\alpha^{2}\beta/k)^{1/3(1-\alpha)}\sim\order{10^{5}}$ after the
critical point. Due to this enhancement, the effective potential is distorted
to have convection points. It is worth noticing that the subcritical regime of
the superconformal hybrid inflation reduces to such a potential and that
predicted $n_{s}$ and $r$ values can be consistent with the observations. In
the other cases, on the other hand, there is no such enhancement after the
critical point, and the waterfall field grows slowly as shown in the figure.

###  4.3 $\chi\simeq-1$

Figure 5: Same as Fig. 2, but taking $\chi=-1.03$ (top) and $\chi=-0.97$
(bottom) and different values of $\alpha$ accordingly. In the right panels, we
take $N_{*}=60$ (solid curves) and $N_{*}=55$ (dot-dashed curves).

Figure 6: Same as Fig. 3 but with $\chi=-1.03$ and $-0.97$. Color codes are
the same as in Fig. 5, and $\lambda$ and $\sqrt{\xi}$ are chosen to give the
best-fit value for $n_{s}$.

Summary of the predictions. — Figure 5 is the same as Fig. 2, but taking
$\chi=-1.03$ (top) and $\chi=-0.97$ (bottom) and various values of $\alpha$.
In the right panels, the results are given for $N_{*}=60$ and $55$ (since the
allowed region for $N_{*}=50$ is very limited). As in the case of
$|\chi|\gtrsim 5$, $n_{s}$ and $r$ become larger for smaller $k$. It is found
that $10^{-2}\lesssim r\lesssim 0.1$ for both cases in the region consistent
with the Planck data. While the result with $\chi=-1.03$ is similar to one
given in Ref. [33], $r$ becomes larger for smaller $\alpha$, and it is
eventually out of the preferred region by the Planck observations. The results
of $\chi=-0.97$ look similar. However, $r$ becomes smaller for smaller
$\alpha$. The allowed region is found to be $10^{-4}\lesssim\lambda\lesssim
10^{-3}$ and $\sqrt{\xi}\sim\order{10^{16}\,{\rm GeV}}$ for both cases.

As in the previous subsection, we derive the effective potential in terms of
the canonically normalized field. In the present case, the second term in the
parenthesis on the rhs in Eq. (52) can be ignored due to $\chi+1\simeq 0$ to
get

$\displaystyle K_{N\bar{N}}\simeq-\frac{3\alpha}{\Phi_{0}}\,.$ (65)

Consequently, $\phi$ and $\Phi_{0}$ are given by

$\displaystyle\phi^{2}$
$\displaystyle\simeq\begin{cases}\displaystyle\frac{1}{\beta}\sinh^{2}\sqrt{\frac{\beta}{\alpha}}\hat{\phi}&~{}~{}~{}(\chi<-1)\\\\[8.53581pt]
\displaystyle\frac{1}{-\beta}\sin^{2}\sqrt{\frac{-\beta}{\alpha}}\hat{\phi}&~{}~{}~{}(\chi>-1)\end{cases}\,,$
(66) $\displaystyle-\frac{\Phi_{0}}{3}$
$\displaystyle\simeq\begin{cases}\displaystyle\cosh^{2}\sqrt{\frac{\beta}{\alpha}}\hat{\phi}&~{}~{}~{}(\chi<-1)\\\\[8.53581pt]
\displaystyle\cos^{2}\sqrt{\frac{-\beta}{\alpha}}\hat{\phi}&~{}~{}~{}(\chi>-1)\end{cases}\,,$
(67)

and $\Psi$ is given in Table 2. Here we have taken the boundary condition
$\hat{\phi}=0$ at $\phi=0$. One can see that both cases give similar
expressions. In fact, both are exactly the same around $\phi=0$, and the
qualitative difference appears at large field values. #12#12#12As mentioned
earlier, the $\chi=-1$ case corresponds to the model studied in Refs. [7, 8]
when potential is written in terms of $\hat{\phi}$. We find that the
approximated expression is valid in $|1+\chi|\lesssim 0.05$ and
$|1+\chi|\lesssim 0.01$ for $\chi<-1$ and $\chi>-1$, respectively. Since the
valid parameter region is limited, it is better to see $\alpha$ dependence
numerically by computing the potential, which is shown in Fig. 6. From the
figure, one can see that the potential becomes flatter as $\alpha$ approaches
$1$ ($2/3$) and gives smaller values of $r$ for $\chi=-1.03$ ($-0.97$). This
behavior is similar to that seen in Sec. 4.2, and it can be understood
qualitatively as follows. Expanding the effective potential in terms of
$\hat{\phi}$, the tensor-to-scalar ratio is approximately given by

$\displaystyle
r\sim\frac{32}{\hat{\phi}^{2}_{*}}+\frac{32(9\alpha-7)(1+\chi)}{9\alpha}\,.$
(68)

Due to the second term, $r$ becomes smaller (larger) as $\alpha$ gets larger
when $\chi<-1$ ($\chi>-1$). We note that the above rough estimate cannot give
the precise value for $r$; however, it is enough to understand the response to
the value of $\alpha$. In addition, we find that $\alpha=2/3$ is not allowed
for $\chi=-0.97$. This comes from the lower bound for $k$ given in Eq. (29).
We will discuss this in the next subsection in detail by comparing the results
with $\alpha=1$. Except for $\alpha=2/3$ or $1$, we find that the endpoint of
$n_{s}$ and $r$ comes from saturated value of $\phi_{*}$ under $k\to 0$.

Table 2: Approximated expression of $\Psi$ in the limit $|(1+\chi)^{2}\phi^{2}/(-2\Phi_{0})|\ll 1$. The parameter space where the approximated expression is valid is given in the right column. The effective potential can be obtained from $\Psi$ by using Eq. (48). Cases | Approximated expression for $\Psi$ | Valid parameters
---|---|---
$\chi<-1$ | $\displaystyle\frac{k}{2\alpha^{2}\beta}\cosh^{2(2-3\alpha)}\sqrt{\frac{\beta}{\alpha}}\hat{\phi}\times\sinh^{2}\sqrt{\frac{\beta}{\alpha}}\hat{\phi}$ | $|1+\chi|\lesssim 0.05$
$\chi>-1$ | $\displaystyle\frac{k}{2\alpha^{2}|\beta|}\cos^{2(2-3\alpha)}\sqrt{\frac{|\beta|}{\alpha}}\hat{\phi}\times\sin^{2}\sqrt{\frac{|\beta|}{\alpha}}\hat{\phi}$ | $|1+\chi|\lesssim 0.01$

Figure 7: Same as Fig. 4, but taking $\chi=-1.03$ (top) and $\chi=-0.97$
(bottom). The other parameters in each panel are $\alpha=1$,
$\lambda=7.6\times 10^{-4}$, $\sqrt{\xi}=1.8\times 10^{16}$ GeV (top-left);
$\alpha=2.6/3$, $\lambda=5.5\times 10^{-4}$, $\sqrt{\xi}=2.1\times 10^{16}$
GeV (top-right); $\alpha=2.1/3$, $\lambda=4.1\times 10^{-4}$,
$\sqrt{\xi}=1.9\times 10^{16}$ GeV (bottom-left); $\alpha=2.3/3$,
$\lambda=4.3\times 10^{-4}$, $\sqrt{\xi}=2.3\times 10^{16}$ GeV (bottom-
right). The tensor-to-scalar ratio is found to be $3.5\times 10^{-2}$ (top-
left), $5.9\times 10^{-2}$ (top-right), $4.5\times 10^{-2}$ (bottom-left), and
$r=9.5\times 10^{-2}$ (bottom-right).

For comparison with the $|\chi|\gtrsim 5$ case, Fig. 7 shows the trajectory of
the waterfall field. Although the potential is not so flat, a large field
value keeps the inflaton slow-roll. For $\chi<-1$, the waterfall field grows
to its global minimum value just after entering the subcritical regime, but it
does not overshoot much greater than the global minimum.

###  4.4 Specific cases

Figure 8: Left: Predicted values of $n_{s}$ and $r$ for $N_{*}=60$ (solid
curves) and $50$ (dotted curves) and various values of $\chi$. $\alpha$ is
taken to be $1$ (top) and $2/3$ (bottom). Right: Allowed parameters for
$N_{*}=60$. Here, $\alpha=1$ (top) and $2/3$ (bottom), and various values of
$\chi$ are taken.

Figure 9: Normalized potential as a function of the canonically normalized
inflaton field. Color codes are the same as in Fig. 8, and $\lambda$ and
$\sqrt{\xi}$ are chosen to give the best-fit value for $n_{s}$.

Finally, we focus on the specific cases where $\alpha=1$, $\alpha=2/3$, and
$\chi=0$. Such values of the parameters are motivated by superstring theory or
superconformal symmetry. Therefore, it is worth analyzing these cases in
detail, although part of the results shown in this subsection are already
presented in the previous subsections. As is found in the previous
subsections, the results for $\alpha=1$ and $2/3$ share some behaviors. Thus
we categorize the contents as $\alpha=1$ or $2/3$, and $\chi=0$.

####  4.4.1 $\alpha=1$ or $2/3$

Summary of the predictions. — Figure 8 shows the predicted $n_{s}$ and $r$
(left panels) and the allowed region (right panels) for $\alpha=1$ (top) and
$\alpha=2/3$ (bottom) and various values of $\chi$. The allowed parameters are
roughly $\lambda\sim\order{10^{-3}}$ and $\sqrt{\xi}\sim\order{10^{16}\,{\rm
GeV}}$ and mildly depend on $\chi$. For $\alpha=1$, $\chi<-1$ gives consistent
results with the Planck observations for $N_{*}=60$, while $N_{*}=50$ is found
to be disfavored. For $\alpha=2/3$, $n_{s}$ and $r$ are consistent with the
observed data when $\chi\gtrsim 0$ or $\chi\simeq-1$. In the latter case,
$r\sim 0.1$, and $N_{*}=60$ is preferred. As in $\alpha=1$ case, $N_{*}=50$
has a tension with the observed data.

It can be seen the resultant $n_{s}$ and $r$ for $\alpha=1$ and $2/3$ are
similar when $\chi\simeq-1$ and $|\chi|\gg 1$. The similarity at
$\chi\simeq-1$ is due to the fact that the effective potential reduces to the
same potential as shown in Table 2. In the $|\chi|\gg 1$ case, $\Psi$ has the
same form of function but with a different exponent of the exponential factor
(see Table 1) and the valid domain of the parameter. That is why the behavior
is quite similar. However, they give quantitatively different predictions for
$n_{s}$ and $r$, which comes from the different exponent in $\Psi$. The
quantitative difference is prominent in the other value of $\chi$: i.e.,
$\chi\lesssim-1$ and $\chi\gtrsim-1$.

Fig. 9 shows $\chi$ dependence on the potential. It is clearly seen that the
potential gets flatter for larger value of $|\chi|$ for both $\alpha=1$ and
$2/3$, and consequently $r$ is suppressed.

Figure 10: Same as Fig. 2, but taking $\chi=0$ and different values of
$\alpha$ accordingly. In the right panels, we take $N_{*}$=60 (solid curves)
and $N_{*}=55$ (dot-dashed curves).  Figure 11: Same as Fig. 3 but with
$\chi=0$. Color codes are the same as in Fig. 10, and $\lambda$ and
$\sqrt{\xi}$ are chosen to give the best-fit value for $n_{s}$.

Figure 12: Same as Fig. 4, but taking $\chi=0$ and $\alpha=2/3$ (left) and
$2.16/3$ (right). In the left panel, $\lambda=9.6\times 10^{-4}$ and
$\sqrt{\xi}=6.0\times 10^{15}$ GeV. In the right panel, $\lambda=1.3\times
10^{-3}$ and $\sqrt{\xi}=2.1\times 10^{16}$ GeV. The tensor-to-scalar ratio is
$6.1\times 10^{-4}$ (left) and $7.0\times 10^{-2}$ (right).

####  4.4.2 $\chi=0$

Summary of the predictions. — Fig. 10 shows the $n_{s}$, $r$, and the allowed
parameter space for $\chi=0$. Roughly speaking, the results are found to be
similar to those in the $\chi\gtrsim 5$ case (see Fig. 2). It is found that
$\alpha\lesssim 2.16/3$ gives a consistent result with the observations for
both $N_{*}=60$ and $50$. A notable difference is that $r$ can be as large as
$\order{10^{-2}}$.

In the $\chi=0$ case, it is easy to derive the potential analytically, since
the Kähler metric [Eq. (52)] takes a simple form without any approximation:

$\displaystyle K_{N\bar{N}}=\frac{36\alpha}{(6-\phi^{2})^{2}}\,.$ (69)

As a result, Eq. (51) can be solved analytically to give

$\displaystyle\phi^{2}$
$\displaystyle=6\tanh^{2}\frac{\hat{\phi}}{\sqrt{6\alpha}}\,,$ (70)
$\displaystyle-\frac{\Phi_{0}}{3}$
$\displaystyle=\cosh^{-2}\frac{\hat{\phi}}{\sqrt{6\alpha}}\,,$ (71)
$\displaystyle\Psi$
$\displaystyle=\frac{3k}{\alpha^{2}}\tanh^{2}\frac{\hat{\phi}}{\sqrt{6\alpha}}\times\cosh^{2(3\alpha-2)}\frac{\hat{\phi}}{\sqrt{6\alpha}}\,.$
(72)

Here, we have taken the boundary condition $\hat{\phi}=0$ at $\phi=0$.
Although the effective potential is given exactly, it behaves nontrivially as
a function of $\alpha$. When $\alpha$ is away from $2/3$, the discussion for
$\chi\gg-1$ case in Sec. 4.2 can be applied. Namely, when $\alpha$ gets
smaller, $r$ becomes smaller. If $\alpha$ goes much closer to $2/3$, on the
other hand, Eq. (62) is no longer satisfied. Instead, we can derive the
effective potential for $\alpha=2/3$ as

$\displaystyle
V=\frac{27g^{2}\xi^{2}k}{4}\tanh^{2}\frac{\hat{\phi}}{2}\times\Bigl{(}1-\frac{27k}{8}\tanh^{2}\frac{\hat{\phi}}{2}\Bigr{)}\,.$
(73)

The largest values of $n_{s}$ and $r$ are obtained from the lower bound for
$k$. The resultant $n_{s}$ and $r$ that are consistent with the observed data
are found for $N_{*}=60$ and $50$. In order to see $\alpha$ dependence on
$n_{s}$ and $r$, it is more intuitive to plot the effective potential, which
is shown in Fig. 11. As $\alpha$ approaches 2/3, it can be seen that the
potential becomes flatter, and smaller values of $r$ are obtained.

For completeness, Fig. 12 shows the same plot as Fig. 4 but takes $\chi=0$ and
$\alpha=1$, $2.16/3$. It can be seen that the shape of the potential changes
nontrivially depending on $\alpha$. That is why the prediction for $r$ changes
by orders of magnitude for an $\alpha$ that even slightly deviates from $2/3$.

##  5 Conclusions and discussion

We have studied the subcritical regime of $D$-term hybrid inflation in the
generalized framework of a superconformal model. The model is characterized by
the superconformal Kähler potential and superconformal superpotential. The
former contains a parameter $\alpha$ and an explicit superconformal breaking
term that is turned on by nonzero $\chi$. The latter is given by the Yukawa
interaction of the inflaton and waterfall fields with a coupling $\lambda$. In
addition, we introduce the Fayet-Iliopoulos term $\xi$ that appears after
gauge fixing of the superconformal symmetry. In this framework, we focus on
the parameter space $\lambda\ll 1$, which is supported by an approximate shift
symmetry of the inflaton field, and $\alpha\leq 1$ ($\alpha\geq 2/3$) for
$\chi<-1$ ($\chi>-1$) to give a single critical point.

In the parameter space, it has been found that inflation continues in the
subcritical regime of the inflaton field, and that the inflaton potential in
the subcritical regime changes drastically depending on $\alpha$ and $\chi$.
The latest Planck data prefer $2/3\leq\alpha\leq 1$ for $\chi\simeq-1$ and
$\alpha\simeq 1$ ($2/3$) for $\chi\lesssim-1$ ($\chi\gtrsim-1$) for 60
$e$-folds, while the preferred parameter space is limited for $50$ $e$-folds
as $2.5/3\lesssim\alpha\leq 1$ ($2/3\leq\alpha\lesssim 2.2/3$) for $\chi<-1$
($\chi>-1$). The tensor-to-scalar ratio $r$ turns out to be
$r>\order{10^{-2}}$ for $\chi\simeq-1$, $r>\order{10^{-3}}$ for
$\chi\lesssim-1$, and $r\sim\order{10^{-4}}$ for $\chi\gtrsim-1$ for $60$
$e$-folds. Roughly speaking, $r$ tends to be suppressed when $\alpha$
approaches $1$ for $\chi<-1$, or $2/3$ for $\chi>-1$ or $|\chi|\gg 1$.

The other parameters, $\lambda$ and $\xi$, on the other hand, are
$\order{10^{-4}}<\lambda<\order{10^{-2}}$ and
$\sqrt{\xi}\sim\order{10^{16}\,{\rm GeV}}$. This result indicates that the FI
term is determined to be around the GUT scale, which might be a clue for
further phenomenological study of the GUT, neutrino sector, and inflation [38,
39, 40, 15].

Besides this, the cases of integer $3\alpha$ (namely, $2$ or $3$ in the
present model) and $\chi=0$ are motivated by the compactification of the extra
dimensions in superstring theory and the superconformal symmetry,
respectively. We have found the allowed parameter spaces for such cases. This
may bring another clue to investigating the relation between the symmetry of
the compactified space and extension of the minimal supersymmetric standard
model that accommodates the inflaton sector.

As mentioned in the Introduction, non-Abelian discrete symmetry can be one of
such symmetries. Recently, the modular symmetry has drawn a lot of attention
[41] giving a nice fit with the experimental results of neutrino
oscillations—for example, under the modular $S_{3}$ [42], $A_{4}$ [41, 42, 43,
44, 45, 46, 47, 48, 49, 50], $S_{4}$ [51, 52, 53], and $A_{5}$ [54, 55].
Furthermore, the study of the modular symmetry has been applied to solve the
cosmological issues. Reference [56] has studied leptogenesis with the modular
$A_{4}$ and showed that right-handed neutrinos with a mass scale of $10^{13}$
GeV can account for the observed baryon asymmetry of the Universe. This
coincides with the mass scale of right-handed sneutrinos that plays the role
of inflaton and can be a source of baryon asymmetry in the superconformal
framework embedded into the MSSM [33, 15]. Since the model proposed in Ref.
[15] predicts that one of the light neutrinos is massless, the model may give
completely different consequences on inflation and the leptogenesis if the
modular symmetry rules the lepton sector. Additionally, it was pointed out
that higher-dimension operators that are allowed by the modular symmetry in
the Kähler potential have the possibility to spoil the success of fitting with
the neutrino oscillation data [57]. To avoid such a danger, a large volume
limit in superstring theory [58, 59] is considered. For instance, the model
with an additional gauge singlet Higgs in the large volume limit can give a
consistent result with the neutrino oscillation data [60]. However, whether it
is compatible with the cosmological issues needs further investigation. Our
results, especially on the symmetry-enhanced points, would be another
possibility to provide a phenomenologically and cosmologically acceptable
scenario [61].

## Acknowledgments

We are grateful to Tatsuo Kobayashi and Tomo Takahashi for valuable
discussions. We also thank Takashi Shimomura for hosting the “Miyazaki
Workshop on Particle Physics and Cosmology 2020”, where this work was
initiated. This work was supported by JSPS KAKENHI Grant No. JP17K14278, No.
JP17H02875, No. JP18H05542, and No. JP20H01894 (K.I.).

## References

  * [1] Planck, N. Aghanim et al.,  Astron. Astrophys. 641, A6 (2020), arXiv:1807.06209.
  * [2] Planck, Y. Akrami et al.,  Astron. Astrophys. 641, A10 (2020), arXiv:1807.06211.
  * [3] A. A. Starobinsky,  Phys. Lett. B 91, 99 (1980).
  * [4] V. F. Mukhanov and G. V. Chibisov,  JETP Lett. 33, 532 (1981).
  * [5] R. Kallosh, A. Linde, and D. Roest,  JHEP 11, 198 (2013), arXiv:1311.0472.
  * [6] W. Buchmuller, V. Domcke, and K. Kamada,  Phys. Lett. B 726, 467 (2013), arXiv:1306.3471.
  * [7] W. Buchmuller, V. Domcke, and K. Schmitz,  JCAP 11, 006 (2014), arXiv:1406.6300.
  * [8] W. Buchmuller and K. Ishiwata,  Phys. Rev. D 91, 081302 (2015), arXiv:1412.3764.
  * [9] S. Clesse,  Phys. Rev. D 83, 063518 (2011), arXiv:1006.4522.
  * [10] H. Kodama, K. Kohri, and K. Nakayama,  Prog. Theor. Phys. 126, 331 (2011), arXiv:1102.5612.
  * [11] S. Clesse and B. Garbrecht,  Phys. Rev. D 86, 023525 (2012), arXiv:1204.3540.
  * [12] K. Freese, J. A. Frieman, and A. V. Olinto,  Phys. Rev. Lett. 65, 3233 (1990).
  * [13] Y. Mikura, Y. Tada, and S. Yokoyama,  EPL 132, 3 (2020), arXiv:2008.00628.
  * [14] Y. Mikura, Y. Tada, and S. Yokoyama,  (2021), arXiv:2103.13045.
  * [15] Y. Gunji and K. Ishiwata,  JHEP 09, 065 (2019), arXiv:1906.04530.
  * [16] MINOS, P. Adamson et al.,  Phys. Rev. Lett. 110, 251801 (2013), arXiv:1304.6335.
  * [17] MINOS, P. Adamson et al.,  Phys. Rev. Lett. 110, 171801 (2013), arXiv:1301.4581.
  * [18] T2K, K. Abe et al.,  Phys. Rev. D 96, 092006 (2017), arXiv:1707.01048,  [Erratum: Phys.Rev.D 98, 019902 (2018)].
  * [19] T2K, K. Abe et al.,  Phys. Rev. Lett. 121, 171802 (2018), arXiv:1807.07891.
  * [20] NOvA, P. Adamson et al.,  Phys. Rev. Lett. 118, 231801 (2017), arXiv:1703.03328.
  * [21] NOvA, M. A. Acero et al.,  Phys. Rev. D 98, 032012 (2018), arXiv:1806.00096.
  * [22] S. Vagnozzi et al.,  Phys. Rev. D 96, 123503 (2017), arXiv:1701.08172.
  * [23] E. Ma and G. Rajasekaran,  Phys. Rev. D 64, 113012 (2001), arXiv:hep-ph/0106291.
  * [24] K. S. Babu, E. Ma, and J. W. F. Valle,  Phys. Lett. B 552, 207 (2003), arXiv:hep-ph/0206292.
  * [25] G. Altarelli and F. Feruglio,  Nucl. Phys. B 720, 64 (2005), arXiv:hep-ph/0504165.
  * [26] G. Altarelli and F. Feruglio,  Rev. Mod. Phys. 82, 2701 (2010), arXiv:1002.0211.
  * [27] H. Ishimori et al.,  Prog. Theor. Phys. Suppl. 183, 1 (2010), arXiv:1003.3552.
  * [28] S. F. King and C. Luhn,  Rept. Prog. Phys. 76, 056201 (2013), arXiv:1301.1340.
  * [29] S. F. King, A. Merle, S. Morisi, Y. Shimizu, and M. Tanimoto,  New J. Phys. 16, 045018 (2014), arXiv:1402.4271.
  * [30] S. Ferrara, R. Kallosh, A. Linde, A. Marrani, and A. Van Proeyen,  Phys. Rev. D 83, 025008 (2011), arXiv:1008.2942.
  * [31] S. Ferrara, R. Kallosh, A. Linde, A. Marrani, and A. Van Proeyen,  Phys. Rev. D 82, 045003 (2010), arXiv:1004.0712.
  * [32] W. Buchmüller, V. Domcke, and K. Schmitz,  JCAP 04, 019 (2013), arXiv:1210.4105.
  * [33] K. Ishiwata,  Phys. Lett. B 782, 367 (2018), arXiv:1803.08274.
  * [34] S. R. Coleman and E. J. Weinberg,  Phys. Rev. D 7, 1888 (1973).
  * [35] T. Asaka, W. Buchmuller, and L. Covi,  Phys. Lett. B 510, 271 (2001), arXiv:hep-ph/0104037.
  * [36] A. Achucarro, J.-O. Gong, S. Hardeman, G. A. Palma, and S. P. Patil,  JCAP 01, 030 (2011), arXiv:1010.3693.
  * [37] X. Chen and Y. Wang,  JCAP 04, 027 (2010), arXiv:0911.3380.
  * [38] V. Domcke, K. Schmitz, and T. T. Yanagida,  Nucl. Phys. B 891, 230 (2015), arXiv:1410.4641.
  * [39] V. Domcke and K. Schmitz,  Phys. Rev. D 95, 075020 (2017), arXiv:1702.02173.
  * [40] V. Domcke and K. Schmitz,  Phys. Rev. D 97, 115025 (2018), arXiv:1712.08121.
  * [41] F. Feruglio,  (2019), arXiv:1706.08749.
  * [42] T. Kobayashi, K. Tanaka, and T. H. Tatsuishi,  Phys. Rev. D 98, 016004 (2018), arXiv:1803.10391.
  * [43] J. C. Criado and F. Feruglio,  SciPost Phys. 5, 042 (2018), arXiv:1807.01125.
  * [44] T. Kobayashi et al.,  JHEP 11, 196 (2018), arXiv:1808.03012.
  * [45] P. P. Novichkov, S. T. Petcov, and M. Tanimoto,  Phys. Lett. B 793, 247 (2019), arXiv:1812.11289.
  * [46] T. Nomura and H. Okada,  Phys. Lett. B 797, 134799 (2019), arXiv:1904.03937.
  * [47] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi,  JHEP 02, 097 (2020), arXiv:1907.09141.
  * [48] G.-J. Ding, S. F. King, and X.-G. Liu,  JHEP 09, 074 (2019), arXiv:1907.11714.
  * [49] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, and T. H. Tatsuishi,  Phys. Rev. D 100, 115045 (2019), arXiv:1909.05139,  [Erratum: Phys.Rev.D 101, 039904 (2020)].
  * [50] T. Kobayashi, T. Nomura, and T. Shimomura,  Phys. Rev. D 102, 035019 (2020), arXiv:1912.00637.
  * [51] J. T. Penedo and S. T. Petcov,  Nucl. Phys. B 939, 292 (2019), arXiv:1806.11040.
  * [52] P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov,  JHEP 04, 005 (2019), arXiv:1811.04933.
  * [53] S. F. King and Y.-L. Zhou,  Phys. Rev. D 101, 015001 (2020), arXiv:1908.02770.
  * [54] P. P. Novichkov, J. T. Penedo, S. T. Petcov, and A. V. Titov,  JHEP 04, 174 (2019), arXiv:1812.02158.
  * [55] G.-J. Ding, S. F. King, and X.-G. Liu,  Phys. Rev. D 100, 115005 (2019), arXiv:1903.12588.
  * [56] T. Asaka, Y. Heo, T. H. Tatsuishi, and T. Yoshida,  JHEP 01, 144 (2020), arXiv:1909.06520.
  * [57] M.-C. Chen, S. Ramos-Sánchez, and M. Ratz,  Phys. Lett. B 801, 135153 (2020), arXiv:1909.06910.
  * [58] V. Kaplunovsky and J. Louis,  Nucl. Phys. B 444, 191 (1995), arXiv:hep-th/9502077.
  * [59] I. Antoniadis, E. Gava, K. S. Narain, and T. R. Taylor,  Nucl. Phys. B 432, 187 (1994), arXiv:hep-th/9405024.
  * [60] T. Asaka, Y. Heo, and T. Yoshida,  Phys. Lett. B 811, 135956 (2020), arXiv:2009.12120.
  * [61] Work in progress by Y. Gunji, K. Ishiwata, and T. Yoshida.


# PiCoGen: Generate Piano Covers with a Two-stage Approach

Chih-Pin Tan  National Taiwan University, KKCompany TechnologiesNo. 1, Sec. 4,
Roosevelt Rd.TaipeiTaiwan<EMAIL_ADDRESS>,  Shuen-Huei Guan
KKCompany TechnologiesTaipeiTaiwan<EMAIL_ADDRESS>and  Yi-Hsuan Yang
National Taiwan UniversityNo. 1, Sec. 4, Roosevelt Rd.TaipeiTaiwan
<EMAIL_ADDRESS>
(2024)

###### Abstract.

Cover song generation stands out as a popular way of music making in the
music-creative community. In this study, we introduce Piano Cover Generation
(PiCoGen), a two-stage approach for automatic cover song generation that
transcribes the melody line and chord progression of a song given its audio
recording, and then uses the resulting lead sheet as the condition to generate
a piano cover in the symbolic domain. This approach is advantageous in that it
does not required paired data of covers and their original songs for training.
Compared to an existing approach that demands such paired data, our evaluation
shows that PiCoGen demonstrates competitive or even superior performance
across songs of different musical genres.

Cover song generation, audio-to-symbolic music generation, controllable
generation, style transfer, transcription, Transformer

††journalyear: 2024††copyright: acmlicensed††conference: Proceedings of the
2024 International Conference on Multimedia Retrieval; June 10–14, 2024;
Phuket, Thailand††booktitle: Proceedings of the 2024 International Conference
on Multimedia Retrieval (ICMR ’24), June 10–14, 2024, Phuket, Thailand††doi:
10.1145/3652583.3657626††isbn: 979-8-4007-0619-6/24/06††ccs: Applied computing
Sound and music computing††ccs: Information systems Multimedia content
creation

##  1\. Introduction

Cover song generation, recreating or rearranging the musical elements from an
existing piece, is popular within the music-creative community. Musicians
craft a cover song with careful consideration of musical components such as
melody, chords, rhythm, and performance techniques, in a style or musical
genre that may be different from the original piece (e.g., Pop to Jazz), and
with different sets of instruments (e.g., using only the piano or the guitar).

Attempts have also been made to create cover songs automatically, relying on a
supervised approach trained on paired data of songs and their human-made cover
versions. Takamori _et al._ (Takamori et al., 2019) employ a regression model
with acoustic features and structural analysis to generate piano covers.
Song2Guitar (Ariga et al., 2017) uses a hidden Markov model to generate guitar
covers with fundamental frequency (f0), beat, and chord information. More
recently, Choi _et al._ present the Pop2Piano model (Choi and Lee, 2023),
which represents the first attempt that uses deep learning for cover song
generation. Specifically, it leverages the MT3 architecture (Gardner et al.,
2022), an encoder-decoder Transformer designed to convert an audio signal into
a sequence of discrete tokens, to create piano covers of music. In collecting
paired data needed for model training, Choi _et al._ find it important to
employ heuristics to improve the data quality, involving a matching routine to
identify covers of songs from the Internet, and an alignment routine to
synchronize the songs and their covers.

Figure 1. An overview of PiCoGen. The model generates a piano cover in two
stages: extracts firstly a lead sheet (i.e., melody line and chord
progression) from an audio recording of the original song via audio analysis
(i.e., transcription), and then turns the extracted lead sheet into a piano
performance via conditional symbolic-domain music generation.

To get rid of the reliance on paired data and to supply deep learning models
with more training data, we explore instead in this paper a novel approach for
piano cover generation that does not need paired data at all. Our approach
draws inspiration from musical domain knowledge that a cover and its original
song tend to share a common melody line and chord progression, and therefore
uses the symbolic-domain _lead sheet_ (Liu and Yang, 2018; Wu and Yang, 2023)
(i.e., a MIDI-like representation of melody$+$chord)111The lead sheet is
strongly linked to the f0 contours and chroma features of music, both of which
play significant roles in cover song identification (Serra et al., 2010;
Yesiler et al., 2019; Silva et al., 2018). as the intermediate gateway
transferring a musical audio involving arbitrary instrumentation into a piano-
only rendition. Specifically, as depicted in Figure 1, we propose a two-stage
model that firstly extracts the lead sheet directly from the song audio, and
then generates a piano cover given the extracted lead sheet. We refer to the
resulting model as Piano Cover Generation (PiCoGen).

The previous Pop2Piano model (Choi and Lee, 2023) uses a _piano transcription_
-like approach and creates piano covers with a _single_ stage of audio-to-
symbolic conversion with no explainable intermediate representation, trained
using paired data of original music and piano covers. In contrast, PiCoGen
bypasses the need of such a paired data by decomposing the conversion task
into _two_ stages. PiCoGen needs other two types of paired data, one for each
stage, yet both of which have been available in the research community through
efforts on audio-to-symbolic lead sheet transcription (Ryynänen and Klapuri,
2008; Weil et al., 2009; Donahue et al., 2022) and symbolic-to-symbolic
conditional piano generation (Hsiao et al., 2021; Wu and Yang, 2023).

As its name implies, Pop2Piano is evaluated on Pop music only (Choi and Lee,
2023). However, we set forth to investigate how well PiCoGen compares to
Pop2Piano on generating the piano covers for up to ten different musical
genres, including cases where lead sheet transcription can be difficult. This
is to study to which extent the domain knowledge (or assumptions) incorporated
by PiCoGen is adequate, and to evaluate the generalizability of both Pop2Piano
and PiCoGen.

We invite readers to our project
website222https://tanchihpin0517.github.io/PiCoGen for audio examples of the
generated piano covers and the source code of this project.

##  2\. Background

Piano cover generation is a special class of cover song generation where the
target output is a piano-only rearrangement of a given music piece. It is
expected that the generated cover and the original piece can be identified as
different versions of the same music composition, and that the cover itself is
pleasant to listen to.

Due to the availability of high-quality piano synthesizers, piano cover
generation can be approached via generating piano music in a symbolic
representation such as Musical Instrument Digital Interface (MIDI), instead of
generating audio signals directly. This is the approach taken by Pop2Piano and
similarly this work. As such, piano cover generation is related to the
following three tasks.

_Automatic Music Transcription_ (AMT) involves converting music signals into
symbolic representations such as piano rolls. _Automatic piano transcription_
, for example, aims to generate a musical score-like transcription of all the
notes involved in an audio recording of piano performance (Benetos et al.,
2019; Toyama et al., 2023; Hawthorne et al., 2017, 2021; Gardner et al., 2022;
Kong et al., 2021). _Lead sheet transcription_ , as another instance of AMT,
transcribes the notes corresponding to the melody line and recognizes the
chord names involved in the harmonic progression (Ryynänen and Klapuri, 2008;
Weil et al., 2009; Donahue et al., 2022). Pop2Piano adopts MT3 (Gardner et
al., 2022), the state-of-the-art of piano transcription, as their model
backbone as both piano transcription and piano cover generation convert audio
into piano scores. A main difference here is that MT3 assumes its input to be
audio recordings of piano-only music, so Pop2Piano needs paired data of
{original songs, piano covers} for supervised training to cope with input with
arbitrary instrumentation.

_Symbolic-domain music generation_ aims to generate novel pieces of music in a
symbolic format, usually by representing a music piece as a sequence of
discrete tokens (Huang et al., 2019; Huang and Yang, 2020; Hsiao et al., 2021;
Wu and Yang, 2023; von Rütte et al., 2023; Dong et al., 2023; Lu et al.,
2023). The generation process can be either _unconditional_ (i.e., from-
scratch generation) or _conditional_. A key observation of our work is that
piano cover generation can be viewed as a conditional generation
task—generating token sequences of piano performances conditioned on pre-given
lead sheets. Such a symbolic-to-symbolic conditional piano generation problem
has been tackled before (Hsiao et al., 2021; Wu and Yang, 2023).

_Music style transfer_ involves the process of altering the style of a music
piece to match the style of a provided example, influenced by various
attributes such as orchestration, chord progression, and tonality (Roberts et
al., 2018; Cífka et al., 2020; Wu and Yang, 2021). Cover song generation can
be considered as a specific type of music style transfer in general.

##  3\. PiCoGen

Given an audio segment $A$ of a song, cover song generation establishes a
model $f:A\rightarrow S$ that creates an alternative version of the input. For
piano cover generation, the output $S$ is pure piano music, which can be
represented in the so-called symbolic domain with discrete tokens bearing
explicit musical meaning (Oore et al., 2018). While many token representations
for symbolic music have been proposed in the literature (Fradet et al., 2021),
Pop2Piano adopts the _MIDI-like_ representation (Huang et al., 2019), which
uses tokens that indicate the pitch, onset time, offset time, and velocity
(which is related to perceptual loudness) of each note involved in a piano
playing. Pop2Piano can therefore be viewed as a cross-domain sequence-to-
sequence model converting a segment of continuous audio waveform to a sequence
of discrete tokens.

PiCoGen is different from Pop2Piano mainly in two aspects: the model
architecture (two-stage vs. single-stage), and the adopted token
representation for symbolic music. We provide details below.

###  3.1. Proposed Two-stage Model

PiCoGen decomposes $f$ into two steps, $f=e\circ g$. The first step
$e:A\rightarrow L$ converts the audio input into an intermediate
representation $L$, while the second step $g:L\rightarrow S$ generates a cover
based only on $L$. In other words, this two-stage approach assumes $A$ and $S$
are conditional independent given $L$. As shown in Figure 1, a fundamental
assumption of PiCoGen is that the lead sheet can serve as such an intermediate
representation. When $A$ and $S$ share the same underling lead sheet, we
assume that they would sound like the same song. From a style transfer
viewpoint, lead sheet is regarded as the “content” to be reserved. Unlike
style transfer in general, the content here is a human-readable lead sheet,
facilitating interpretable and controllable music generation.

As shown in Figure 1, PiCoGen accordingly contains two sub-models, the
_Extractor_ that performs $e$ and outputs the corresponding lead sheet
sequence $L$ of the input audio $A$, and the _Performer_ that performs $g$ and
generates piano token sequence $S$ based on the transcribed lead sheet. Here
$e$ and $g$ can be viewed as two sequence-to-sequence problems. In both cases,
the objectives are to maximize the probability of the target sequence given
the source sequence.

To better inform the Performer of the temporal correspondence between the lead
sheet $L$ and the piano $S$, we follow the approach of Wu & Yang (Wu and Yang,
2023) and implement the Performer as a decoder-only Transformer that deals
with an interleaved sequence composed of a “bar-wise mix” of $L$ and $S$, as
shown in Figure 2. Specifically, we segment $L$ and $S$ respectively into
multiple sub-sequences $L=[L^{1},\dots,L^{B}]$ and $S=[S^{1},\dots,S^{B}]$,
where $L^{k}$ and $S^{k}$ are the lead sheet sub-sequence and piano sub-
sequence for the same, $k$-th bar (i.e., musical measure) of the input music,
and $B$ denotes the number of bars. We use the bar-wise mix
$[L^{1},S^{1},\dots,L^{B},S^{B}]$ to train the Performer, such that the
generation for the $k$-th bar of piano $S^{k}$ would depend on (i.e., can
attend to) the current and preceding sub-sequences of lead sheet
$[L^{1},L^{2},\dots L^{k}]$ and the preceding piano sub-sequences
$[S^{1},S^{2},\dots S^{k-1}]$.

Figure 2. For each bar (musical measure) $k$ of the input, the Extractor
transcribes from the input its lead sheet $L^{k}$ (a token sequence), and the
Performer generates autoregressively the piano performance $S^{k}$ (also a
token sequence) for the same bar given the current and preceding sequences of
lead sheet $[L^{1},L^{2},\dots L^{k}]$ and the preceding piano performances
$[S^{1},S^{2},\dots S^{k-1}]$ organized in an interleaving fashion.

###  3.2. Proposed Token Representation

A known weakness of the _MIDI-like_ tokens (Huang et al., 2019) used by
Pop2Piano to represent piano music $S$ is the overly large number of tokens
needed to represent a musical bar, making it hard to learn long-term
dependency (Hsiao et al., 2021). We instead adopt a modified version of
compound-word (CP) token representation (Hsiao et al., 2021) to make the token
sequence compact. The main idea of CP is to group related tokens into a “super
token.” As shown in Figure 2, tokens belonging to the same super token are
combined into a single embedding vector before feeding to the Transformer,
which in turn predicts the tokens belonging to the next super token
collectively at the same time using different heads. We refer to (Hsiao et
al., 2021) for details.

Specifically, CP uses the idea of “token classes” to organize the tokens
(Hsiao et al., 2021). While the original CP considers only two token classes,
metric and non-metric, we consider four token classes so as to have a unified
CP-like representation for not only the piano $S$ but also the lead sheet $L$.
They are Spec, Bar, Metric and Note. Spec class includes special tokens, e.g.,
[BOS] and [EOS] (i.e., beginning or ending of a sequence). Bar class contains
[bar_src] and [bar_tgt], which are used to distinguish tokens belongs to lead
sheet or piano cover. Metric class is composed with three sub-classes:
Position, Tempo, and Chord. Note class is composed with three sub-classes
describing a musical note: Pitch, Duration and Velocity.333Position indicates
the timing shift from the beginning of each bar, in the resolution of 16th
note. Tempo indicate the beat-per-minute (BPM), ranging from 32 to 244. Chord
is used to represent the chord condition, composed by {chord root, chord
quality} pair. Pitch ranges from A0 to C8. Duration indicates the length of
each notes, with the same resolution of Position. We limit the maximum
duration to the length of 8 quarter notes in this work. Velocity takes MIDI
value from 0 to 127.

###  3.3. Implementation Details

To implement the Extractor of PiCoGen, we employ SheetSage (Donahue et al.,
2022), the state-of-the-art for lead sheet transcription. Authors of SheetSage
have kindly released a model
checkpoint444https://github.com/chrisdonahue/sheetsage well-trained on the
Hook Theory dataset (Donahue et al., 2022), which consists of approximately
40,000 paired data of audio clips and lead sheets. We simply leverage this
checkpoint as our Extractor without fine-tuning.

As for the Performer, we adopt the decoder-only architecture of the CP
Transformer (Hsiao et al., 2021) and train it from scratch. The model has 8
self-attention layers, 8 heads for multi-head attention, 512 hidden
dimensions, a sequence length of 1,024 super tokens, and GeLU as the
activation function. At inference time, the lead sheet $L$ obtained from the
Extractor is converted to the proposed CP-like tokens and then fed to the
Performer to generate the piano cover $S$.

Figure 3. Mean opinion score in the metric OVL in the subjective evaluation,
for each of the ten considered genres.

##  4\. Experimental Setup

###  4.1. Datasets

Training the Performer requires paired data of lead sheets $\mathcal{D}_{l}$
and piano performances $\mathcal{D}_{p}$. For $\mathcal{D}_{p}$, we adopt the
Pop1k7 dataset (Hsiao et al., 2021), which encompasses around 1,700 piano
covers of various Japanese anime, Korean popular and Western popular songs. To
construct $\mathcal{D}_{l}$, we follow the approach of Wu & Yang (Wu and Yang,
2023) and employ symbolic-domain music analysis techniques (Uitdenbogerd and
Zobel, 1998) to recognize the lead sheet from each piano performance.

For model evaluation, we use the GTZAN dataset (Tzanetakis and Cook,
2002),555http://marsyas.info/download/data_sets a well-known public-domain
dataset containing 100 30-second song clips for each of 10 selected genres,
including Pop, Jazz, Classical, Rock, etc (see Figure 3 for the full list),
enabling evaluation beyond Pop music. Note that SheetSage assumes constant
tempo and would report errors for songs that violate this assumption. We
therefore discard the 117 songs of GTZAN that has this problem, using the
remaining 883 songs in our evaluation.

###  4.2. Baseline and Ablations

We consider Pop2Piano (Choi and Lee, 2023) as the baseline, as it represents
the state-of-the-art for piano cover generation. Specifically, we use the
model checkpoint released by the original authors for
testing.666https://github.com/sweetcocoa/pop2piano Moreover, we train and
evaluate two ablated versions of PiCoGen. The first version, denoted as “w/o
chord”, uses only the melody as the intermediate representation $L$, namely
dropping chord information entirely. This is to study the effect of chords.
The second version, denoted as “midi-like,” uses the MIDI-like tokens (Huang
et al., 2019) to represent the piano music, instead of the more advanced CP-
like representation. This is to use the same token representation as Pop2Piano
(Choi and Lee, 2023) so as to examine whether any possible performance
difference between Pop2Piano and PiCoGen is due to the two-stage architecture
or due to token representations. We note that these ablations involve changes
only to the intermediate data, and that both ablations adhere to the proposed
two-stage strategy.

###  4.3. Evaluation Metrics

For objective evaluation, we follow Pop2Piano and compute melody chroma
accuracy (MCA) (Choi and Lee, 2023). The idea is to compare the top line of
the piano MIDI generated by a model with the melody contour of the input
audio. According to Pop2Piano (Choi and Lee, 2023), the melody contour is
computed by separating the vocal of the input via a source separation model
(Hennequin et al., 2020), and then using pYIN (Mauch and Dixon, 2014) for f0
estimation from the separated vocal. MCA is $\in[0,1]$ and the higher the
better.

For subjective evaluation, we conduct an online user study involving 30
volunteers consisting of 10 amateurs, 17 pro-ams(Sai Vanka et al., 2023), and
3 pros. We generate 31 distinct testing sets, distributing them to the human
subjects according to their birth dates. In each testing set, we randomly
select 5 genres from the GTZAN dataset and designate one song per genre as the
testing target. Consequently, each testing set comprises 5 songs, and we
generate 4 results for each song, using Pop2Piano, PiCoGen and the two
ablation models, presented to the subject in random order. Subjects are
instructed to score the results with a Likert scale from 1 to 5 (the higher
the better) in the following five aspects:

  * •

Similarity (SI): How much the piano cover sounds similar to its original song
in general?

  * •

Melody similarity (SI${}_{\text{m}}$): The perceived similarity between the
melody lines of the piano cover and the original song.

  * •

Chord similarity (SI${}_{\text{c}}$): The similarity in chord progression.

  * •

Music fluency (FL): Does the piano cover sound fluent?

  * •

Overall (OVL): How much do you like the piano cover?

Model | _objective_ | _subjective evaluation_ $\uparrow$
---|---|---
MCA $\uparrow$ | SI | SI${}_{\text{m}}$ | SI${}_{\text{c}}$ | FL | OVL
Pop2Piano(Choi and Lee, 2023) | 0.25 | 2.65 | 2.85 | 2.65 | 2.60 | 2.55
PiCoGen | 0.17 | 3.20 | 3.35 | 3.05 | 3.20 | 3.35
–w/o chord | 0.14 | 3.10 | 3.25 | 2.95 | 3.10 | 3.05
–midi-like | 0.12 | 2.95 | 3.10 | 2.95 | 3.10 | 3.05
Table 1. Evaluation result on piano cover generation for only Pop music. Best
result per metric highlighted in bold.

##  5\. Results and Discussion

Table 1 displays the average MCA and mean opinion scores (MOS) from the user
study for Pop only, the genre targeted by Pop2Piano. We see that the proposed
PiCoGen and its variants show a lower MCA than Pop2Piano, but outperform it
greatly in every subjective metric. Although statistical test does not reveal
significant performance difference (due to large stanadard deviation), we view
this as an encouraging result since PiCoGen does not use paired data of music
and their piano covers at all. Moreover, although the two ablated variants
receive moderately lower MOS than PiCoGen, the result is in general
comparable, providing further evidences the effectiveness of the two-stage
approach. We also note that the MOS in OVL reduces from 3.35 to 3.05 when
dropping chords from the intermediate representation, suggesting the
superiority of using lead sheet instead of melody only in PiCoGen.

Table 1 also suggests a conflicting result between MCA and its subjective
counterpart SI${}_{\text{m}}$. PiCoGen underperforms Pop2Piano in MCA, but its
MOS in SI${}_{\text{m}}$ is higher. We conjecture that this might due to the
fact that MCA only considers vocal melody, neglecting the melody of the
leading instrument, as discussed in (Wang et al., 2020). We consider the
development of better objective metrics as a future work and focus on the
result of the user study here.

Figure 3 shows the MOS in OVL for all the ten genres in GTZAN, extending our
evaluation beyond Pop music. Figure 3 shows that PiCoGen outperforms Pop2Piano
not only for Pop but also for genres such as Blues, Disco, and Metal,
demonstrating the generalizability of PiCoGen to some extent. However, there
are also many genres for which Pop2Piano and PiCoGen have similar MOS. In
particular, both Pop2Piano and PiCoGen perform poorly for Hip-hop music. Upon
scrutinizing the attributes of the testing songs, we identify that the quality
of the lead sheets extracted by SheetSage is not satisfactory for some of the
genres, especially for Hip-hop. Moreover, the MOS of PiCoGen in Table 1 and
Figure 3 ranges from 2.00 to 3.50 in general, leaving considerable room for
future improvement given that the scores are from 1 to 5.

There are many interesting avenues to be investigated in future work. First,
both Pop2Piano and PiCoGen can be potentially improved by collecting a larger
training dataset encompassing more genres. However, we note that Pop2Piano
demands both the piano covers and their original songs in diverse genres as
the paired data for training. In contrast, it is easier to expand the training
data of PiCoGen as we only need the piano covers (not the audio recordings of
the original songs). Second, besides data, better result for PiCoGen might be
obtained along with advances in lead sheet transcription, and by incorporating
intermediate representations of music other than the lead sheet (e.g.,
rhythmic elements) for musical genres with no dominant melody line. Finally,
it might also be interesting to develop a model that combines the strengths of
the music-informed design of PiCoGen and the data-driven design of Pop2Piano,
leveraging both unpaired and paired data for training.

##  6\. Conclusion

In this paper, we have presented PiCoGen, a generative model capable of
creating a piano cover of an input audio with two distinct steps: extracting a
lead sheet from the audio and generating a piano performance based on this
lead sheet. Treating the lead sheet as the common ground between the input
music and the target piano MIDI, PiCoGen bypasses the need of curating paired
data of covers and their original songs for training. Compared to the existing
single-stage model Pop2Piano, PiCoGen exhibits comparable or superior
performance for Pop music and some other genres. Directions for future
improvement have also been discussed.

## References

  * (1)
  * Ariga et al. (2017) Shunya Ariga, Satoru Fukayama, and Masataka Goto. 2017.  Song2Guitar: A difficulty-aware arrangement system for generating guitar solo covers from polyphonic audio of popular music.  _In Proc. International Society for Music Information Retrieval (ISMIR)_.
  * Benetos et al. (2019) Emmanouil Benetos, Simon Dixon, Zhiyao Duan, and Sebastian Ewert. 2019.  Automatic music transcription: An overview.  _In Proc. IEEE Signal Processing Magazine_ (2019).
  * Choi and Lee (2023) Jongho Choi and Kyogu Lee. 2023.  Pop2Piano: Pop audio-based piano cover generation.  _In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_.
  * Cífka et al. (2020) Ondřej Cífka, Umut Şimşekli, and Gaël Richard. 2020.  Groove2groove: One-shot music style transfer with supervision from synthetic data.  _In Proc. IEEE/ACM Transactions on Audio, Speech, and Language Processing (TASLP)_ (2020).
  * Donahue et al. (2022) Chris Donahue, John Thickstun, and Percy Liang. 2022.  Melody transcription via generative pre-training.  _In Proc. International Society for Music Information Retrieval (ISMIR)_.
  * Dong et al. (2023) Hao-Wen Dong, Ke Chen, Shlomo Dubnov, Julian McAuley, and Taylor Berg-Kirkpatrick. 2023.  Multitrack music Transformer.  _In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_.
  * Fradet et al. (2021) Nathan Fradet, Jean-Pierre Briot, Fabien Chhel, Amal El Fallah Seghrouchni, and Nicolas Gutowski. 2021.  MidiTok: A Python package for MIDI file tokenization.  _In Proc. International Society for Music Information Retrieval (ISMIR)_.
  * Gardner et al. (2022) Josh Gardner, Ian Simon, Ethan Manilow, Curtis Hawthorne, and Jesse Engel. 2022.  MT3: Multi-task multitrack music transcription.  _In Proc. International Conference on Learning Representations (ICLR)_.
  * Hawthorne et al. (2017) Curtis Hawthorne, Erich Elsen, Jialin Song, Adam Roberts, Ian Simon, Colin Raffel, Jesse Engel, Sageev Oore, and Douglas Eck. 2017.  Onsets and Frames: Dual-objective piano transcription.  arXiv:1710.11153
  * Hawthorne et al. (2021) Curtis Hawthorne, Ian Simon, Rigel Swavely, Ethan Manilow, and Jesse Engel. 2021.  Sequence-to-sequence piano transcription with Transformers.  _In Proc. International Society for Music Information Retrieval (ISMIR)_.
  * Hennequin et al. (2020) Romain Hennequin, Anis Khlif, Felix Voituret, and Manuel Moussallam. 2020.  Spleeter: A fast and efficient music source separation tool with pre-trained models.  _In Proc. Journal of Open Source Software_ (2020).
  * Hsiao et al. (2021) Wen-Yi Hsiao, Jen-Yu Liu, Yin-Cheng Yeh, and Yi-Hsuan Yang. 2021.  Compound word Transformer: Learning to compose full-song music over dynamic directed hypergraphs.  _In Proc. AAAI Conference on Artificial Intelligence (AAAI)_.
  * Huang et al. (2019) Cheng-Zhi Anna Huang, Ashish Vaswani, Jakob Uszkoreit, Ian Simon, Curtis Hawthorne, Noam Shazeer, Andrew M. Dai, Matthew D. Hoffman, Monica Dinculescu, and Douglas Eck. 2019.  Music Transformer.  _In Proc. International Conference on Learning Representations (ICLR)_.
  * Huang and Yang (2020) Yu-Siang Huang and Yi-Hsuan Yang. 2020.  Pop music Transformer: Beat-based modeling and generation of expressive pop piano compositions.  _In Proc. ACM Multimedia (ACM MM)_.
  * Kong et al. (2021) Qiuqiang Kong, Bochen Li, Xuchen Song, Yuan Wan, and Yuxuan Wang. 2021.  High-resolution piano transcription with pedals by regressing onset and offset times.  _In Proc. IEEE/ACM Transactions on Audio, Speech, and Language Processing (TASLP)_ (2021).
  * Liu and Yang (2018) Hao-Min Liu and Yi-Hsuan Yang. 2018.  Lead sheet generation and arrangement by conditional generative adversarial network. In _Proc. IEEE International Conference on Machine Learning and Applications (ICMLA)_. 722–727.
  * Lu et al. (2023) Peiling Lu, Xin Xu, Chenfei Kang, Botao Yu, Chengyi Xing, Xu Tan, and Jiang Bian. 2023.  MuseCoco: Generating symbolic music from text.  arXiv:2306.00110
  * Mauch and Dixon (2014) Matthias Mauch and Simon Dixon. 2014.  pYIN: A fundamental frequency estimator using probabilistic threshold distributions.  _In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_.
  * Oore et al. (2018) Sageev Oore, Ian Simon, Sander Dieleman, Douglas Eck, and Karen Simonyan. 2018.  This time with feeling: Learning expressive musical performance.  _In Proc. Neural Computing and Applications_ (2018).
  * Roberts et al. (2018) Adam Roberts, Jesse Engel, Colin Raffel, Curtis Hawthorne, and Douglas Eck. 2018.  A hierarchical latent vector model for learning long-term structure in music.  _In Proc. International Conference on Machine Learning (ICML)_.
  * Ryynänen and Klapuri (2008) Matti P. Ryynänen and Anssi P. Klapuri. 2008.  Automatic transcription of melody, bass line, and chords in polyphonic music.  _Computer Music Journal_ (2008).
  * Sai Vanka et al. (2023) Soumya Sai Vanka, Maryam Safi, Jean-Baptiste Rolland, and György Fazekas. 2023.  Adoption of AI technology in music mixing workflow: an investigation.  _Audio Engineering Society (AES)_.
  * Serra et al. (2010) Joan Serra, Emilia Gómez, and Perfecto Herrera. 2010.  Audio cover song identification and similarity: background, approaches, evaluation, and beyond.  _In Proc. Advances in music information retrieval_ (2010).
  * Silva et al. (2018) Diego Furtado Silva, Felipe Falcao, and Nazareno Andrade. 2018.  Summarizing and comparing music data and its application on cover song identification.  _In Proc. International Society for Music Information Retrieval (ISMIR)_.
  * Takamori et al. (2019) Hirofumi Takamori, Takayuki Nakatsuka, Satoru Fukayama, Masataka Goto, and Shigeo Morishima. 2019.  Audio-based automatic generation of a piano reduction score by considering the musical structure.  _In Proc. MultiMedia Modeling: 25th International Conference (MMM)_.
  * Toyama et al. (2023) Keisuke Toyama, Taketo Akama, Yukara Ikemiya, Yuhta Takida, Wei-Hsiang Liao, and Yuki Mitsufuji. 2023.  Automatic piano transcription with hierarchical frequency-time Transformer.  _In Proc. International Society for Music Information Retrieval (ISMIR)_.
  * Tzanetakis and Cook (2002) George Tzanetakis and Perry Cook. 2002.  Musical genre classification of audio signals.  _In Proc. IEEE Transactions on Speech and Audio Processing_ (2002).
  * Uitdenbogerd and Zobel (1998) Alexandra L Uitdenbogerd and Justin Zobel. 1998.  Manipulation of music for melody matching.  _In Proc. ACM Multimedia (ACM MM)_.
  * von Rütte et al. (2023) Dimitri von Rütte, Luca Biggio, Yannic Kilcher, and Thomas Hofmann. 2023.  FIGARO: Generating symbolic music with fine-grained artistic control.  _In Proc. International Conference on Learning Representations (ICLR)_.
  * Wang et al. (2020) Ziyu Wang, Ke Chen, Junyan Jiang, Yiyi Zhang, Maoran Xu, Shuqi Dai, Guxian Bin, and Gus Xia. 2020.  POP909: A pop-song dataset for music arrangement generation. In _In Proc. International Society for Music Information Retrieval (ISMIR)_.
  * Weil et al. (2009) Jan Weil, Thomas Sikora, Jean-Louis Durrieu, and Gaël Richard. 2009.  Automatic generation of lead Sheets from polyphonic music signals.  _In Proc. International Society for Music Information Retrieval (ISMIR)_.
  * Wu and Yang (2021) Shih-Lun Wu and Yi-Hsuan Yang. 2021.  MuseMorphose: Full-song and fine-grained piano music style transfer with one Transformer VAE.  _In Proc. IEEE/ACM Transactions on Audio, Speech, and Language Processing (TASLP)_ (2021).
  * Wu and Yang (2023) Shih-Lun Wu and Yi-Hsuan Yang. 2023.  Compose & Embellish: Well-structured piano performance generation via a two-Stage Approach.  _In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_.
  * Yesiler et al. (2019) Furkan Yesiler, Chris Tralie, Albin Andrew Correya, Diego F Silva, Philip Tovstogan, Emilia Gómez Gutiérrez, and Xavier Serra. 2019.  Da-TACOS: A dataset for cover song identification and understanding.  _In Proc. International Society for Music Information Retrieval (ISMIR)_.