#  Non-minimally assisted chaotic inflation

Sang Chul Hyun<EMAIL_ADDRESS>Department of Physics & IPAP & Lab for
Dark Universe, Yonsei University, Seoul 03722, Korea  Jinsu Kim
<EMAIL_ADDRESS>Theoretical Physics Department, CERN, 1211 Geneva 23,
Switzerland  Seong Chan Park<EMAIL_ADDRESS>Department of Physics &
IPAP & Lab for Dark Universe, Yonsei University, Seoul 03722, Korea  Korea
Institute for Advanced Study, Seoul 02455, Republic of Korea  Tomo Takahashi
<EMAIL_ADDRESS>Department of Physics, Saga University, Saga 840-8502,
Japan

###### Abstract

Conventional wisdom says that a chaotic inflation model with a power-law
potential is ruled out by the recent Planck-BICEP/Keck results. We find,
however, that the model can be assisted by a non-minimally coupled scalar
field and still provides a successful inflation. Considering a power-law
chaotic inflation model of the type $V\sim\varphi^{n}$ with
$n=\\{2,4/3,1,2/3,1/3\\}$, we show that $n=1/3$ ($n=\\{2/3,1/3\\}$) may be
revived with the help of the quadratic (quartic) non-minimal coupling of the
assistant field to gravity.

††preprint: YHEP-COS22-01 KIAS-P22015 CERN-TH-2022-040

##  I Introduction

Chaotic inflation with a power-law potential $V\sim\varphi^{n}$ has been
regarded as an attractive inflationary framework for a long time since 1983
[1]. The power-law chaotic inflation model predicts the scalar spectral index
$n_{s}$ and the tensor-to-scalar ratio $r$ as $n_{s}\approx 1-2(n+2)/(n+4N)$
and $r\approx 16n/(n+4N)$, respectively, where $N$ is the number of $e$-folds.
However, the recent Planck [2] and BICEP/Keck [3] results put a stringent
bound on the tensor-to-scalar ratio as $r_{0.05}<0.036$ (95% C.L.), while the
bound on the spectral index is given by $0.958\lesssim n_{s}\lesssim 0.975$
(95% C.L.). It indicates that the power-law chaotic inflation model is ruled
out for every $n$, which is not necessarily an integer, with $N=50$ or 60,
residing outside the $2\sigma$ acceptable range of $(n_{s},r)$. We are curious
if the power-law chaotic inflation can resurrect by extending the original
setup.

One known way is introducing a non-minimal coupling $\Omega^{2}(\varphi)$ with
the Ricci curvature $R$ in the Jordan frame [4, 5, 6, 7]. The potential in the
Einstein frame $V_{\rm E}$ becomes flat in the large-field limit as long as
the asymptotic ratio of the Jordan-frame potential $V(\varphi)$ and the
squared non-minimal coupling term becomes constant since $V_{\rm E}\sim
V(\varphi)/\Omega^{4}(\varphi)$ [7], thereby supporting successful slow-roll
inflation. A nice example is the Higgs inflation [8, 9] especially in the
vicinity of a critical point [10, 11].111 The addition of a $R^{2}$ term [12,
13, 14, 15, 16, 17, 18, 19] further improves its high energy behavior as the
scalaron emerges and unitarizes the theory [20, 21, 22, 23]; see Ref. [24] for
a recent review on various aspects of Higgs-$R^{2}$ inflation. For a
supersymmetric version of the Higgs inflation, see e.g., Refs. [25, 26, 27,
28, 29, 30, 31, 32, 33, 34].

Having this success in our minds, we would like to explore another
possibility. There may exist another scalar field $s$ (an assistant field)
which does not have a direct coupling to the inflaton field $\varphi$ but non-
minimally couples to the Ricci curvature $R$ as $s^{m}R$ with a power $m>0$.
On the other hand, we consider the case where the inflaton field $\varphi$ is
still minimally coupled to gravity. Although many studies are devoted to
investigate the case where the inflaton field is non-minimally coupled to
gravity 222 Cases of an arbitrary power for the chaotic inflation with the
inflaton non-minimal coupling are studied in e.g. Refs. [35, 36]. , given that
multiple scalar fields can naturally arise in high energy theories such as
superstring theories, scenarios with a non-minimal coupling between another
scalar field and $R$ should also be possible. Since the assistant field is
non-minimally coupled, the Einstein-frame potential and the inflationary
dynamics become non-trivial. We want to examine if chaotic inflation models
can move back to observationally acceptable ranges. For definiteness and also
for simplicity, in the current study, we assume that the energy density of the
assistant field is negligible compared to that of the inflaton field which
allows us to approximate the Jordan-frame potential as $V=V(\varphi)$. One
may, of course, easily extend our setup to a more general case, but we leave
the extensions for future studies; see, e.g., Ref. [37].

The rest of the paper is organized as follows: In Sec. II, we introduce in
detail chaotic inflation with a power-law potential and a non-minimally
coupled assistant field. We then analyze the two-field setup in the Einstein
frame and compute cosmological observables such as the spectral index, the
tensor-to-scalar ratio, and the local-type nonlinearity parameter, employing
the $\delta N$ formalism in Sec. III. In Sec. IV, we perform a numerical
analysis on the cosmological observables and check the compatibility with the
latest Planck-BICEP/Keck results for various powers of the inflaton potential
with the quadratic and quartic non-minimal couplings of the assistant field to
gravity. We show that a subclass of the power-law chaotic inflation models may
be revived with the help of the assistant field. We conclude in Sec. V.

##  II Model

The action for the inflaton field $\varphi$ and the assistant field $s$ is
introduced in the Jordan frame as

$\displaystyle S_{\rm J}=\int d^{4}x\,\sqrt{-g_{\rm J}}\,\left[\frac{M_{\rm
P}^{2}}{2}\Omega^{2}(s)R_{\rm J}-\frac{1}{2}g_{\rm
J}^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}g_{\rm
J}^{\mu\nu}\partial_{\mu}s\partial_{\nu}s-V_{\rm J}(\varphi)\right]\,,$ (1)

where $M_{\rm P}\equiv 1/\sqrt{8\pi G}=2.44\times 10^{18}$ GeV is the reduced
Planck mass, and the subscript J indicates that the action is written in the
Jordan frame. We note that only the assistant field $s$ couples to gravity
non-minimally, while the inflaton field $\varphi$ remains minimally coupled,
i.e., $\Omega^{2}=\Omega^{2}(s)$. We assume that the energy density of the
assistant field is negligible compared to that of the inflaton field.
Moreover, the additional scalar field $s$ is assumed to have no direct
coupling to the inflaton field $\varphi$. Thus, to a good approximation, the
scalar potential in the Jordan frame is given by $V_{\rm J}=V_{\rm
J}(\varphi)$. We consider the power-law potential,

$\displaystyle V_{\rm J}(\varphi)=\lambda_{\varphi}M_{\rm
P}^{4}\left(\frac{\varphi}{M_{\rm P}}\right)^{n}\,.$ (2)

The power $n$ does not necessarily take an integer value, and we consider
various cases with $n=\\{2,4/3,1,2/3,1/3\\}$. The cases with fractional power
are motivated by axion monodromy scenario [38, 39, 40, 41]; see also Ref.
[42]. In this parametrization, the self-coupling $\lambda_{\varphi}$ is a
dimensionless parameter regardless of the power $n$.

We expand the conformal factor $\Omega^{2}$ as

$\displaystyle\Omega^{2}=1+\xi_{2}\left(\frac{s}{M_{\rm
P}}\right)^{2}+\xi_{4}\left(\frac{s}{M_{\rm P}}\right)^{4}+\cdots\,,$ (3)

where $1$ corresponds to the Einstein-Hilbert action, and $\xi_{i}$
($i=2,4,6,\cdots$) are all dimensionless coefficients. We focus on the regime
$\xi_{i}(s/M_{\rm P})^{i}\ll 1$ so that the expansion (3) remains to be
valid.333 In general, a certain mass scale $\mu$ may exist, and for $s\ll\mu$,
we can Taylor-expand the conformal factor as
$\Omega^{2}=1+a_{m}(s/\mu)^{m}+\cdots$, where $a_{m}=\mathcal{O}(1)$ is the
leading-order term. Defining $\xi_{m}\equiv a_{m}(M_{\rm P}/\mu)^{m}$, we
recover Eq. (3).  We implicitly assume $\mathbb{Z}_{2}$ symmetry $s\to-s$ so
that $\xi_{i={\rm odd}}=0$ for all odd terms, and the leading-order term is
either $\xi_{2}$ or $\xi_{4}$ when $\xi_{2}$ is negligible. From now on, we
only keep the leading-order term in the $\xi_{m}$ expansion, and thus, we take
$\Omega^{2}\equiv 1+\xi_{m}(s/M_{P})^{m}$ with $m=2$ or $m=4$ (when
$\xi_{2}=0$). We will comment on the role of the higher-order terms later.

The action (1) can be brought to the Einstein frame, where the gravity part
takes the standard Einstein-Hilbert term, via the Weyl rescaling $g_{{\rm
J}\mu\nu}\rightarrow g_{{\rm E}\mu\nu}=\Omega^{2}g_{{\rm J}\mu\nu}$. The
resultant Einstein-frame action is given by

$\displaystyle S_{\rm E}=\int d^{4}x\,\sqrt{-g_{\rm E}}\left[\frac{M_{\rm
P}^{2}}{2}R_{\rm E}-\frac{1}{2}\mathcal{K}_{1}g_{\rm
E}^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}\mathcal{K}_{2}g_{\rm
E}^{\mu\nu}\partial_{\mu}s\partial_{\nu}s-V_{\rm E}(\varphi,s)\right]\,,$ (4)

where

$\displaystyle\mathcal{K}_{1}$ $\displaystyle=\frac{1}{1+\xi_{m}s^{m}/M_{\rm
P}^{m}}\,,$ (5) $\displaystyle\mathcal{K}_{2}$
$\displaystyle=\frac{1+\xi_{m}s^{m}/M_{\rm
P}^{m}+(3/2)m^{2}\xi_{m}^{2}(s/M_{\rm P})^{2m-2}}{(1+\xi_{m}s^{m}/M_{\rm
P}^{m})^{2}}\,,$ (6)

and

$\displaystyle V_{\rm E}(\varphi,s)=\frac{V_{\rm
J}(\varphi)}{(1+\xi_{m}s^{m}/M_{\rm P}^{m})^{2}}\equiv F(\varphi)K(s)\,.$ (7)

Here, we have defined $F(\varphi)\equiv V_{\rm J}(\varphi)$ and $K(s)\equiv
1/(1+\xi_{m}s^{m}/M_{\rm P}^{m})^{2}$. Note that
$\mathcal{K}_{1,2}=\mathcal{K}_{1,2}(s)$ are functions of the $s$ field only.
Henceforth, we omit the subscript ${\rm E}$ for brevity.

Let us introduce a canonically normalized field $\sigma$, which is defined by

$\displaystyle\left(\frac{\partial\sigma}{\partial
s}\right)^{2}=\mathcal{K}_{2}\,.$ (8)

Then, we have

$\displaystyle S=\int d^{4}x\,\sqrt{-g}\left[\frac{M_{\rm
P}^{2}}{2}R-\frac{1}{2}(\partial\sigma)^{2}-\frac{1}{2}e^{2b}(\partial\varphi)^{2}-V(\varphi,s)\right]\,,$
(9)

where we have defined $b=b(\sigma(s))$ via

$\displaystyle e^{2b}\equiv\mathcal{K}_{1}=\frac{1}{1+\xi_{m}s^{m}/M_{\rm
P}^{m}}\,,$ (10)

or, equivalently, $b\equiv-(1/2)\ln(1+\xi_{m}s^{m}/M_{\rm P}^{m})$. We note
that the action (9) takes the same form as the one studied in Refs. [43, 44].

A few remarks are in order. As we are interested in the case where
$\xi_{m}s^{m}/M_{\rm P}^{m}\ll 1$, the scalar potential in the Einstein frame
may be expanded as

$\displaystyle V(\varphi,s)=\frac{\lambda_{\varphi}M_{\rm
P}^{4}(\varphi/M_{\rm P})^{n}}{(1+\xi_{m}s^{m}/M_{\rm
P}^{m})^{2}}\approx\lambda_{\varphi}M_{\rm P}^{4}\left(\frac{\varphi}{M_{\rm
P}}\right)^{n}\left(1-2\xi_{m}\frac{s^{m}}{M_{\rm P}^{m}}\right)\,.$ (11)

Thus, the $(n,m)=(2,2)$ case, for example, contains the Higgs-portal-type
interaction, $\varphi^{2}s^{2}$. Furthermore, the scalar potential for the
$m=2$ case, up to the leading order in $\xi_{2}s^{2}/M_{\rm P}^{2}$, can be
approximated as

$\displaystyle V(\varphi,s)\approx\frac{\lambda_{\varphi}M_{\rm
P}^{4}}{2}\left(\frac{\varphi}{M_{\rm
P}}\right)^{n}\left[1+\cos\left(\frac{s}{f}\right)\right]\,,$ (12)

with $f=(2\sqrt{2\xi_{2}})^{-1}M_{\rm P}$, provided $\xi_{2}>0$. In other
words, the Einstein-frame potential is a product of the chaotic inflation
model and the natural inflation model. We note that the natural inflation is
also disfavored by the recent BICEP/Keck observations. Similarly, for the
$m=4$ case, the potential in the Einstein frame is given by

$\displaystyle V(\varphi,s)\approx\frac{\lambda_{\varphi}M_{\rm
P}^{4}}{2}\left(\frac{\varphi}{M_{\rm
P}}\right)^{n}\left[1-2\xi_{4}\left(\frac{s}{M_{\rm P}}\right)^{4}\right]\,,$
(13)

which is a product of the chaotic inflation model and the hilltop quartic
inflation model. Our model can thus be viewed as a phenomenological model that
connects between the chaotic inflation-type models and natural inflation and
hilltop inflation. We stress at this point that we did not impose any direct
interaction between the fields $\varphi$ and $s$. Couplings between the two
fields are due to the presence of the non-minimal coupling of the assistant
field $s$ to gravity.

Finally, we briefly comment on the role of the higher-order terms in the
conformal factor. We first note that the potential of
$V=\lambda_{\varphi}M_{\rm P}^{4}(\varphi/M_{\rm
P})^{n}/(1+\xi_{m}s^{m}/M_{\rm P}^{m})^{2}$ along the $s$-field direction is
unstable, i.e., a runaway potential, for $\xi_{m}>0$, while the potential
develops a pole when $\xi_{m}<0$. However, this can be viewed as an artifact
of the fact that we truncated the non-minimal potential at the leading order.
In general, one has the higher-order terms in the conformal factor
$\Omega^{2}$, in which case, the potential may become stable without a pole.
During inflation, however, the higher-order terms have negligible effects.
Thus, we do not consider those higher-order terms in our analysis
below.444See, for example, Refs. [45, 46]. See also Ref. [47] for higher
curvature terms $R^{m>3}$.

##  III Cosmological Observables

For the Einstein-frame action (9), the background equations of motion are
given by

$\displaystyle H^{2}$ $\displaystyle=\frac{1}{3M_{\rm
P}^{2}}\left(\frac{1}{2}\dot{\sigma}^{2}+\frac{1}{2}e^{2b}\dot{\varphi}^{2}+V\right)\,,$
(14) $\displaystyle 0$
$\displaystyle=\ddot{\sigma}+3H\dot{\sigma}+V_{,\sigma}-b_{,\sigma}e^{2b}\dot{\varphi}^{2}\,,$
(15) $\displaystyle 0$
$\displaystyle=\ddot{\varphi}+(3H+2b_{,\sigma}\dot{\sigma})\dot{\varphi}+e^{-2b}V_{,\varphi}\,,$
(16)

where the dot represents the derivative with respect to the cosmic time and
${}_{,i}\equiv\partial/\partial\phi^{i}$ for $i=\\{\sigma,\varphi\\}$.

We define the following slow-roll parameters:

$\displaystyle\epsilon^{\sigma}\equiv\frac{M_{\rm
P}^{2}}{2}\left(\frac{V_{,\sigma}}{V}\right)^{2}=\frac{M_{\rm
P}^{2}}{2}\left(\frac{K_{,\sigma}}{K}\right)^{2}\,,\qquad\epsilon^{\varphi}\equiv\frac{M_{\rm
P}^{2}}{2}\left(\frac{V_{,\varphi}}{V}e^{-b}\right)^{2}=\frac{M_{\rm
P}^{2}}{2}\left(\frac{F_{,\varphi}}{F}e^{-b}\right)^{2}\,,$
$\displaystyle\eta^{\sigma\sigma}\equiv M_{\rm
P}^{2}\frac{V_{,\sigma\sigma}}{V}=M_{\rm
P}^{2}\frac{K_{,\sigma\sigma}}{K}\,,\qquad\eta^{\varphi\varphi}\equiv M_{\rm
P}^{2}\frac{V_{,\varphi\varphi}}{V}e^{-2b}=M_{\rm
P}^{2}\frac{F_{,\varphi\varphi}}{F}e^{-2b}\,,$
$\displaystyle\eta^{\varphi\sigma}\equiv M_{\rm
P}^{2}\frac{V_{,\varphi\sigma}}{V}e^{-b}\,,\qquad\epsilon^{b}\equiv 8M_{\rm
P}^{2}b_{,\sigma}^{2}\,.$ (17)

Note that $\eta^{\varphi\sigma}\sim\sqrt{\epsilon^{\sigma}\epsilon^{\varphi}}$
in our case as the Einstein-frame potential is product-separable, i.e.,
$V(\varphi,s)=F(\varphi)K(s)$. Requiring the slow-roll conditions,
$\\{\epsilon^{i},|\eta^{ij}|,\epsilon^{b}\\}\ll 1$
($i,j=\\{\sigma,\varphi\\}$), the equations of motion (14)–(16) become

$\displaystyle H^{2}\approx\frac{V}{3M_{\rm P}^{2}}\,,\quad
3H\dot{\sigma}\approx-V_{,\sigma}\,,\quad
3H\dot{\varphi}\approx-e^{-2b}V_{,\varphi}\,.$ (18)

We note that, under the slow-roll approximation,
$\epsilon^{\sigma}\approx\epsilon\cos^{2}\theta$ and
$\epsilon^{\varphi}\approx\epsilon\sin^{2}\theta$, and thus
$\epsilon\approx\epsilon^{\sigma}+\epsilon^{\varphi}$, where
$\epsilon\equiv-\dot{H}/H^{2}$ and $\theta$ is defined through

$\displaystyle\cos\theta=\frac{\dot{\sigma}}{\sqrt{\dot{\sigma}^{2}+e^{2b}\dot{\varphi}^{2}}}\,,\quad\sin\theta=\frac{\dot{\varphi}e^{b}}{\sqrt{\dot{\sigma}^{2}+e^{2b}\dot{\varphi}^{2}}}\,.$
(19)

For later convenience, we also define

$\displaystyle\eta^{b}\equiv 16M_{\rm P}^{2}b_{,\sigma\sigma}\,.$ (20)

To compute cosmological observables such as the curvature power spectrum
$\mathcal{P}_{\zeta}$, scalar spectral index $n_{s}$, tensor-to-scalar ratio
$r$, and the local-type nonlinearity parameter $f_{\rm NL}^{\rm(local)}$, we
adopt the $\delta N$ formalism [48, 49, 50, 51, 52], where the curvature
perturbation is given by the difference of the number of $e$-folds $N$ between
the initial flat hypersurface and final uniform-density hypersurface, i.e.,
$\zeta=\delta N$. For small enough perturbations $\delta\phi^{i}$
($\phi^{i}=\\{\sigma,\varphi\\}$), one may Taylor-expand $\delta N$ to obtain

$\displaystyle\zeta=\delta N=\frac{\partial
N}{\partial\phi^{i}}\delta\phi^{i}+\frac{1}{2}\frac{\partial^{2}N}{\partial\phi^{i}\partial\phi^{j}}\delta\phi^{i}\delta\phi^{j}+\cdots$
(21)

Here, we summarize the expressions for the cosmological observables in the
$\delta N$ formalism (see Refs. [50, 51, 53, 54, 55, 43, 44] for details).
First, the curvature power spectrum is given by

$\displaystyle\mathcal{P}_{\zeta}=\left(\frac{H}{2\pi}\right)^{2}G^{ij}N_{,i}N_{,j}\,,$
(22)

where $G^{ij}$ is the inverse metric of the field space and
$N_{,i}\equiv\partial N/\partial\varphi^{i}$. The spectral index is

$\displaystyle n_{s}-1=-2\epsilon-2\frac{1+N_{,k}(\frac{M_{\rm
P}^{6}}{3}R^{kmnl}V_{,m}V_{,n}/V^{2}-M_{\rm
P}^{4}V^{;kl}/V)N_{,l}}{G^{ij}N_{,i}N_{,j}M_{\rm P}^{2}}\,,$ (23)

where the semicolon denotes the covariant derivative in the field space, and
$R^{kmnl}$ is the Riemann tensor in the field space whose non-zero components,
in our case, are given by

$\displaystyle
R^{\sigma\varphi\sigma\varphi}=R^{\varphi\sigma\varphi\sigma}=-R^{\sigma\varphi\varphi\sigma}=-R^{\varphi\sigma\sigma\varphi}=-e^{-2b}\left(b_{,\sigma\sigma}+b_{,\sigma}^{2}\right)\,.$
(24)

The tensor-to-scalar ratio is given by

$\displaystyle r=\frac{8/M_{\rm P}^{2}}{G^{ij}N_{,i}N_{,j}}\,.$ (25)

Finally, the local-type (shape-independent) nonlinearity parameter is obtained
as

$\displaystyle-\frac{6}{5}f_{\rm
NL}^{\rm(local)}=\frac{G^{ij}G^{mn}N_{,i}N_{,m}N_{,jn}}{(G^{kl}N_{,k}N_{,l})^{2}}\,.$
(26)

The quantities are to be evaluated at the horizon crossing, i.e., when a mode
exits the Hubble radius, $k=aH$. We denote the horizon-crossing point by
super- or sub-script $\ast$ below. Similarly, the sub- or super-script $e$
denotes the end of inflation.

The number of $e$-folds is given by

$\displaystyle N=-\int_{t_{e}}^{t_{*}}H\,dt\approx\frac{1}{M_{\rm
P}^{2}}\int_{\sigma_{e}}^{\sigma_{*}}\frac{K}{K_{,\sigma}}d\sigma\,,$ (27)

where the slow roll is assumed. The first and second derivatives of the number
of $e$-folds, $N_{,i}$ and $N_{,ij}$, have been worked out in, e.g., Ref. [44]
for the action (9). The resultant expressions for $N_{,i}$ are given as
follows:

$\displaystyle M_{\rm P}\frac{\partial N}{\partial\sigma_{*}}$
$\displaystyle=\frac{1}{\sqrt{2}}{\rm
sgn}\left(\frac{K^{*}}{K_{,\sigma}^{*}}\right)\frac{1}{\sqrt{\epsilon_{*}^{\sigma}}}\left(1-\frac{\epsilon_{e}^{\varphi}}{\epsilon_{e}}e^{2b^{e}-2b^{*}}\right)\,,$
(28) $\displaystyle M_{\rm P}\frac{\partial N}{\partial\varphi_{*}}$
$\displaystyle=\frac{1}{\sqrt{2}}{\rm
sgn}\left(\frac{F^{*}}{F_{,\varphi}^{*}}\right)\frac{1}{\sqrt{\epsilon_{*}^{\varphi}}}\left(\frac{\epsilon_{e}^{\varphi}}{\epsilon_{e}}\right)e^{2b^{e}-b^{*}}\,.$
(29)

Here, we have used $\epsilon\approx\epsilon^{\sigma}+\epsilon^{\varphi}$.
Positivity of the scalar potential for each field allows us to write ${\rm
sgn}(K/K_{,\sigma})={\rm sgn}(V_{,\sigma})$ and ${\rm
sgn}(F/F_{,\varphi})={\rm sgn}(V_{,\varphi})$. We shall thus use
$s^{\sigma}\equiv{\rm sgn}(V_{,\sigma})$ and $s^{\varphi}\equiv{\rm
sgn}(V_{,\varphi})$ in the following. Similarly, we define $s^{b}\equiv{\rm
sgn}(b_{,\sigma})$. The expressions for $N_{,ij}$ are

$\displaystyle M_{\rm P}^{2}\frac{\partial^{2}N}{\partial\sigma_{*}^{2}}$
$\displaystyle=\left(1-\frac{\eta_{*}^{\sigma\sigma}}{2\epsilon_{*}^{\sigma}}\right)\left(1-\frac{\epsilon_{e}^{\varphi}}{\epsilon_{e}}e^{2b^{e}-2b^{*}}\right)+\frac{1}{2}s_{*}^{b}s_{*}^{\sigma}\sqrt{\frac{\epsilon_{*}^{b}}{\epsilon_{*}^{\sigma}}}\frac{\epsilon_{e}^{\varphi}}{\epsilon_{e}}e^{2b^{e}-2b^{*}}$
$\displaystyle\qquad+e^{4b^{e}-4b^{*}}\frac{\epsilon_{e}^{\varphi}\epsilon_{e}^{\sigma}}{\epsilon_{*}^{\sigma}\epsilon_{e}^{2}}\left[\frac{\epsilon_{e}^{\sigma}\eta_{e}^{\varphi\varphi}+\epsilon_{e}^{\varphi}\eta_{e}^{\sigma\sigma}}{\epsilon_{e}}-4\frac{\epsilon_{e}^{\varphi}\epsilon_{e}^{\sigma}}{\epsilon_{e}}-\frac{1}{2}s_{e}^{b}s_{e}^{\sigma}\sqrt{\frac{\epsilon_{e}^{b}}{\epsilon_{e}^{\sigma}}}\frac{(\epsilon_{e}^{\varphi})^{2}}{\epsilon_{e}}\right]\,,$
(30) $\displaystyle M_{\rm
P}^{2}\frac{\partial^{2}N}{\partial\varphi_{*}^{2}}$
$\displaystyle=\left(1-\frac{\eta_{*}^{\varphi\varphi}}{2\epsilon_{*}^{\varphi}}\right)\frac{\epsilon_{e}^{\varphi}}{\epsilon_{e}}e^{2b^{e}}$
$\displaystyle\qquad+e^{4b^{e}-2b^{*}}\frac{\epsilon_{e}^{\varphi}\epsilon_{e}^{\sigma}}{\epsilon_{*}^{\varphi}\epsilon_{e}^{2}}\left[\frac{\epsilon_{e}^{\sigma}\eta_{e}^{\varphi\varphi}+\epsilon_{e}^{\varphi}\eta_{e}^{\sigma\sigma}}{\epsilon_{e}}-4\frac{\epsilon_{e}^{\varphi}\epsilon_{e}^{\sigma}}{\epsilon_{e}}-\frac{1}{2}s_{e}^{b}s_{e}^{\sigma}\sqrt{\frac{\epsilon_{e}^{b}}{\epsilon_{e}^{\sigma}}}\frac{(\epsilon_{e}^{\varphi})^{2}}{\epsilon_{e}}\right]\,,$
(31) $\displaystyle M_{\rm
P}^{2}\frac{\partial^{2}N}{\partial\varphi_{*}\partial\sigma}$
$\displaystyle=-s_{*}^{\varphi}s_{*}^{\sigma}e^{4b^{e}-3b^{*}}\frac{\epsilon_{e}^{\varphi}\epsilon_{e}^{\sigma}}{\epsilon_{e}^{2}\sqrt{\epsilon_{*}^{\sigma}\epsilon_{*}^{\varphi}}}\left[\frac{\epsilon_{e}^{\sigma}\eta_{e}^{\varphi\varphi}+\epsilon_{e}^{\varphi}\eta_{e}^{\sigma\sigma}}{\epsilon_{e}}-4\frac{\epsilon_{e}^{\varphi}\epsilon_{e}^{\sigma}}{\epsilon_{e}}-\frac{1}{2}s_{e}^{b}s_{e}^{\sigma}\sqrt{\frac{\epsilon_{e}^{b}}{\epsilon_{e}^{\sigma}}}\frac{(\epsilon_{e}^{\varphi})^{2}}{\epsilon_{e}}\right]\,.$
(32)

Putting the expressions for the first and second derivatives of $N$ into Eqs.
(33)–(26), we obtain

$\displaystyle\mathcal{P}_{\zeta}$
$\displaystyle=\frac{H_{*}^{2}}{8\pi^{2}M_{\rm
P}^{2}}e^{2X}\left(\frac{u^{2}\alpha^{2}}{\epsilon_{*}^{\sigma}}+\frac{v^{2}}{\epsilon_{*}^{\varphi}}\right)\,,$
(33) $\displaystyle n_{s}$
$\displaystyle=1-2\epsilon_{*}-\frac{4e^{-2X}}{u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\varphi}}-\frac{1}{12}\frac{\eta_{*}^{b}+2\epsilon_{*}^{b}}{u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\varphi}}\left(u\alpha\sqrt{\frac{\epsilon_{*}^{\varphi}}{\epsilon_{*}^{\sigma}}}-v\sqrt{\frac{\epsilon_{*}^{\sigma}}{\epsilon_{*}^{\varphi}}}\right)^{2}$
$\displaystyle\quad+\frac{2}{u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\varphi}}\left[u^{2}\alpha^{2}\frac{\eta_{*}^{\sigma\sigma}}{\epsilon_{*}^{\sigma}}+v^{2}\frac{\eta_{*}^{\varphi\varphi}}{\epsilon_{*}^{\varphi}}+4uv\alpha+\frac{1}{2}s_{*}^{b}s_{*}^{\sigma}\sqrt{\epsilon_{*}^{b}\epsilon_{*}^{\sigma}}v\left(\frac{v}{\epsilon_{*}^{\varphi}}-\frac{2u\alpha}{\epsilon_{*}^{\sigma}}\right)\right]\,,$
(34) $\displaystyle r$
$\displaystyle=\frac{16e^{-2X}}{u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\varphi}}\,,$
(35) $\displaystyle-\frac{6}{5}f_{\rm NL}^{\rm(local)}$
$\displaystyle=\frac{2e^{-X}}{(u^{2}\alpha^{2}/\epsilon_{*}^{\sigma}+v^{2}/\epsilon_{*}^{\varphi})^{2}}\Bigg{[}\left(1-\frac{\eta_{*}^{\sigma\sigma}}{2\epsilon_{*}^{\sigma}}\right)\frac{u^{3}\alpha^{3}}{\epsilon_{*}^{\sigma}}+\left(1-\frac{\eta_{*}^{\varphi\varphi}}{2\epsilon_{*}^{\varphi}}\right)\frac{v^{3}}{\epsilon_{*}^{\varphi}}$
$\displaystyle\qquad+\frac{1}{2}s_{*}^{b}s_{*}^{\sigma}\frac{u^{2}v\alpha^{2}}{\epsilon_{*}^{\sigma}}\sqrt{\frac{\epsilon_{*}^{b}}{\epsilon_{*}^{\sigma}}}+\left(\frac{u\alpha}{\epsilon_{*}^{\sigma}}-\frac{v}{\epsilon_{*}^{\varphi}}\right)^{2}e^{X}\mathcal{C}\Bigg{]}\,,$
(36)

where we have defined

$\displaystyle u$
$\displaystyle\equiv\frac{\epsilon_{e}^{\sigma}}{\epsilon_{e}}\,,\quad
v\equiv\frac{\epsilon_{e}^{\varphi}}{\epsilon_{e}}\,,\quad X\equiv
2b^{e}-2b^{*}\,,$ $\displaystyle\mathcal{C}$
$\displaystyle\equiv\frac{\epsilon_{e}^{\sigma}\epsilon_{e}^{\varphi}}{\epsilon_{e}^{2}}\left(\frac{\epsilon_{e}^{\sigma}\eta_{e}^{\varphi\varphi}+\epsilon_{e}^{\varphi}\eta_{e}^{\sigma\sigma}}{\epsilon_{e}}-4\frac{\epsilon_{e}^{\varphi}\epsilon_{e}^{\sigma}}{\epsilon_{e}}-\frac{1}{2}s_{e}^{\sigma}s_{e}^{b}\sqrt{\frac{\epsilon_{e}^{b}}{\epsilon_{e}^{\sigma}}}\frac{(\epsilon_{e}^{\varphi})^{2}}{\epsilon_{e}}\right)\,,$
$\displaystyle\alpha$ $\displaystyle\equiv
e^{2b^{*}-2b^{e}}\left[1+\frac{\epsilon_{e}^{\varphi}}{\epsilon_{e}^{\sigma}}\left(1-e^{2b^{e}-2b^{*}}\right)\right]\,.$
(37)

We perform a numerical analysis to obtain the cosmological observables for our
model, exploiting Eqs. (33)–(37).

##  IV Results

The number of $e$-folds (27) for the system (9) is given by

$N=\left\\{\begin{array}[]{lcc}\dfrac{3}{4}\ln\left(\dfrac{M_{\rm
P}^{2}+\xi_{2}s_{e}^{2}}{M_{\rm
P}^{2}+\xi_{2}s_{\ast}^{2}}\right)+\dfrac{1}{4\xi_{2}}\ln\left(\dfrac{s_{e}}{s_{\ast}}\right)&\text{for
}m=2\,,\\\ \\\ \dfrac{3}{4}\ln\left(\dfrac{M_{\rm
P}^{4}+\xi_{4}s_{e}^{4}}{M_{\rm
P}^{4}+\xi_{4}s_{\ast}^{4}}\right)+\dfrac{M_{\rm
P}^{2}}{16\xi_{4}}\left(\dfrac{1}{s_{\ast}^{2}}-\dfrac{1}{s_{e}^{2}}\right)&\text{for
}m=4\,.\\\ \end{array}\right.$

For a given set of values of $\\{m,\xi_{m}\\}$, the number of $e$-folds
becomes a function of $s_{*}$ and $s_{e}$. We treat the value of the $s$ field
at the CMB pivot scale, $s_{*}$, as a parameter. Then, once $s_{\ast}$ is
specified, $s_{e}$ can be given in terms of $s_{*}$ and $N$.

The evolution of the $\varphi$ field is governed by

$\displaystyle\frac{d\varphi}{dN}\approx-e^{-2b}\frac{V_{,\varphi}}{3H^{2}}=-\frac{nM_{\rm
P}^{2}}{\varphi}\left(1+\xi_{m}\frac{s^{m}}{M_{\rm P}^{m}}\right)\,,$ (38)

where the slow-roll approximation is used; see Eq. (18). Inserting the
evolution of the $s$ field obtained from the number of $e$-folds above and
integrating the $\varphi$ evolution equation, we obtain an expression of the
$\varphi$-field value at the end of inflation, $\varphi_{e}$, as a function of
$s_{*}$, $\varphi_{*}$, and $N$, for a given value of $n$.

From the end-of-inflation condition, which we choose to be $\epsilon=1$, one
may relate $\varphi_{e}$ and $s_{e}$. Since $\varphi_{e}$ and $s_{e}$ are
given in terms of $s_{*}$, $\varphi_{*}$, and $N$, we obtain a relation
between $s_{*}$ and $\varphi_{*}$. Since we are treating $s_{*}$ as a
parameter, all the other quantities, such as $s_{e}$, $\varphi_{e}$, and
$\varphi_{*}$, are functions of $s_{*}$ together with $N$. In our analysis, we
take $N=60$.

We then use Eqs. (34)–(37) to compute the spectral index $n_{s}$, the tensor-
to-scalar ratio $r$, and the local-type nonlinearity parameter $f_{\rm
NL}^{\rm(local)}$. We examine the power-law potential with
$n=\\{2,4/3,1,2/3,1/3\\}$ for $m=2$ and $m=4$ cases. One may notice that the
model parameter $\lambda_{\varphi}$ does not enter in the expressions of
$n_{s}$, $r$, and $f_{\rm NL}^{\rm(local)}$ and that only the curvature power
spectrum (33) depends on $\lambda_{\varphi}$. We use this degree of freedom to
match the Planck normalization, namely $\mathcal{P}_{\zeta}\simeq 2\times
10^{-9}$ at the CMB scale. Therefore, there remain only two free parameters,
$\xi_{m}$ and $s_{*}$. We explore the behavior of $n_{s}$, $r$, and $f_{\rm
NL}^{\rm(local)}$ by varying $\xi_{m}$ and $s_{*}$.

Figure 1:  Effects of the quadratic non-minimal coupling $\xi_{2}$ of the
assistant field on the cosmological observables in the $n_{s}$ – $r$ plane
(left) and in the $n_{s}$ – $f_{\rm NL}^{\rm(local)}$ plane (right). The
power-law potential is considered with $n=2$ (red), $n=4/3$ (blue), $n=1$
(magenta), $n=2/3$ (cyan), and $n=1/3$ (green). The points represent the
predictions of the standard power-law chaotic inflation models which is
recovered when $\xi_{2}=0$, while the $n_{s}\simeq 0.95$ points correspond to
$\xi_{2}\simeq 0.01$ ($0.02$) for $s_{*}=10^{-1}M_{\rm P}$ ($10^{-3}\,M_{\rm
P}$). The dashed (solid) lines correspond to the $s_{*}=10^{-3}\,M_{\rm P}$
($10^{-1}\,M_{\rm P}$) case. As $\xi_{2}$ increases, the spectral index
$n_{s}$ and the tensor-to-scalar ratio $r$ decrease. On the other hand, the
nonlinearity parameter $f_{\rm NL}^{\rm(local)}$ increases as $\xi_{2}$ grows,
while remaining compatible with the Planck 2$\sigma$ bound [56]. The Planck
[2] (Planck-BICEP/Keck [3]) 1$\sigma$ and 2$\sigma$ bounds on the $n_{s}$–$r$
plane are depicted by the gray (black) solid and gray (black) dashed lines,
respectively. The $n=1/3$ may be revived with the help of the assistant field.
The $n=2/3$ is marginally ruled out and the other higher powers remain to be
ruled out by the Planck-BICEP/Keck results.

We present our numerical analysis in Fig. 1 for the quadratic ($m=2$) non-
minimal coupling and in Fig. 2 for the quartic ($m=4$) non-minimal coupling.
In both Figs. 1 and 2, we present by varying $\xi_{m}$ the behavior of the
cosmological observables in the $n_{s}$ – $r$ plane (left panels) and in the
$n_{s}$ – $f_{\rm NL}^{\rm(local)}$ plane (right panels), for $n=2$ (red),
$n=4/3$ (blue), $n=1$ (magenta), $n=2/3$ (cyan), and $n=1/3$ (green). For the
$m=2$ case, we consider $s_{*}=10^{-3}\,M_{\rm P}$ (dashed) and
$s_{*}=10^{-1}\,M_{\rm P}$ (solid). For the $m=4$ case, we take
$s_{*}=10^{-1}\,M_{\rm P}$ (solid) and $s_{*}=M_{\rm P}$ (dashed). In the
$n_{s}$ – $r$ plane, we overlay the Planck 1$\sigma$ (solid gray) and
2$\sigma$ (dashed gray) bounds as well as the Planck-BICEP/Keck 1$\sigma$
(solid black) and 2$\sigma$ (dashed black) bounds. The dots correspond to the
standard power-law chaotic inflation predictions, namely the $\xi_{m}=0$ case.
We clearly see that they sit outside the Planck-BICEP/Keck bounds.

In the left panel of Fig. 1, one may see the effect of the assistant field $s$
on $n_{s}$ and $r$ for the $m=2$ case. While recovering the standard
predictions of the power-law chaotic inflation models when $\xi_{2}=0$, the
presence of the assistant field that couples only to gravity decreases both
the spectral index $n_{s}$ and the tensor-to-scalar ratio $r$. As a result,
the $n=1/3$ case becomes compatible with the latest Planck-BICEP/Keck results.
The $n=2/3$ case is marginally ruled out, and the higher powers,
$n=\\{2,4/3,1\\}$, remain to be ruled out. The tendency of the local-type
nonlinearity parameter $f_{\rm NL}^{\rm(local)}$ is shown in the right panel
of Fig. 1 for the $m=2$ case. We observe that the nonlinearity parameters are
small for the $s_{*}=10^{-1}\,M_{\rm P}$. The nonlinearity parameters may
become sizable for the $s_{*}=10^{-3}\,M_{\rm P}$, while residing inside
Planck 2$\sigma$ bound, $-11.1<f_{\rm NL}^{\rm(local)}<9.3$.555 The Planck
1$\sigma$ bound for the local-type nonlinearity parameter corresponds to
$f_{\rm NL}^{\rm(local)}=-0.9\pm 5.1$ [56].

Figure 2:  Effects of the quartic non-minimal coupling $\xi_{4}$ of the
assistant field on the cosmological observables in the $n_{s}$ – $r$ plane
(left) and in the $n_{s}$ – $f_{\rm NL}^{\rm(local)}$ plane (right). The
power-law potential is considered with $n=2$ (red), $n=4/3$ (blue), $n=1$
(magenta), $n=2/3$ (cyan), and $n=1/3$ (green). The points represent the
predictions of the standard power-law chaotic inflation models which is
recovered when $\xi_{4}=0$, while the $n_{s}\simeq 0.95$ points correspond to
$\xi_{4}\simeq 0.1$ ($0.001$) for $s_{*}=10^{-1}\,M_{\rm P}$ ($M_{\rm P}$).
The dashed (solid) lines correspond to the $s_{*}=M_{\rm P}$ ($10^{-1}\,M_{\rm
P}$) case. As $\xi_{4}$ increases, the spectral index $n_{s}$ and the tensor-
to-scalar ratio $r$ decrease. On the other hand, the nonlinearity parameter
$f_{\rm NL}^{\rm(local)}$ tends to increase as $\xi_{4}$ increases. For the
$s=10^{-1}\,M_{\rm P}$ case, the nonlinearity parameter goes outside the
Planck 2$\sigma$ bound [56], $-11.1<f_{\rm NL}^{\rm(local)}<9.3$. The region
that is incompatible with this bound is grayed out in the $n_{s}$ – $r$ plot.
The $s_{*}=M_{\rm P}$ case is, however, compatible with the Planck 2$\sigma$
bound on the local-type nonlinearity parameter. The Planck [2] (Planck-
BICEP/Keck [3]) 1$\sigma$ and 2$\sigma$ bounds on the $n_{s}$–$r$ plane are
depicted by the gray (black) solid and gray (black) dashed lines,
respectively. In the case of the quartic non-minimal coupling, both the
$n=1/3$ and $n=2/3$ powers may be revived with the help of the assistant
field. The other higher powers remain to be ruled out by the Planck-BICEP/Keck
results.

Similarly, the left panel of Fig. 2 shows how the presence of the assistant
field $s$ affects the $n_{s}$ and $r$ for the $m=4$ case. Again, as $\xi_{4}$
increases, both $n_{s}$ and $r$ decrease from the standard predictions marked
by points which correspond to $\xi_{4}=0$. Consequently, both the powers of
$n=1/3$ and $n=2/3$ may become compatible with the latest Planck-BICEP/Keck
results. The higher powers, $n=\\{2,4/3,1\\}$, remain to be ruled out. We
observe from the right panel of Fig. 2 that the local-type nonlinearity
parameter $f_{\rm NL}^{\rm(local)}$ tends to increase as $\xi_{4}$ increases.
While the values of $f_{\rm NL}^{\rm(local)}$ are within the Planck 2$\sigma$
bound, $-11.1<f_{\rm NL}^{\rm(local)}<9.3$, for the $s_{*}=M_{\rm P}$ case,
they may become too large for the $s_{*}=10^{-1}\,M_{\rm P}$ case. The region
that is incompatible with the Planck 2$\sigma$ bound on the local-type
nonlinearity parameter is grayed out in the $n_{s}$ – $r$ plot in the left
panel of Fig. 2.

##  V Conclusion

A single-field chaotic inflation with a power-law potential $V\sim\varphi^{n}$
is known to reside outside of observationally acceptable range of $(n_{s},r)$
space regardless of the value of the power $n$. To remedy this problem, we
have considered an additional scalar field (an assistant field) $s$ which non-
minimally couples to the curvature $R$ in the form of $s^{m}R$ with some power
$m$.

As explicit examples, we have performed a numerical analysis of the two-field
setup with $m=2$ and $m=4$ for various powers of $n$, employing the $\delta N$
formalism. We have found that the model with $n=1/3$ for $m=2,4$ and $n=2/3$
for $m=4$ moves into the acceptable ranges and becomes compatible with the
latest Planck-BICEP/Keck results, even though the assistant field $s$ is
assumed to have no sizable potential in the Jordan frame and no direct
coupling between the inflaton field $\varphi$ and the assistant field $s$ is
introduced. In a multi-field setup, non-Gaussianities may become large. We
have computed the local-type nonlinearity parameter $f_{\rm NL}^{(\rm local)}$
and checked the agreement with the Planck data.

The resurrection of the potential with a higher power $n>2/3$ is found to be
difficult with the assistance of a non-minimally coupled field within the
simple setup we considered in this paper. Of course, one may easily extend our
setup e.g. by allowing a non-trivial potential for the assistant field in the
Jordan frame. We leave the extensions for future studies.

## Acknowledgments

The work is partly supported by the National Research Foundation of Korea
NRF-2021R1A4A2001897, NRF-2019R1A2C1089334, JSPS KAKENHI Grant Number
17H01131, 19K03874 and MEXT KAKENHI Grant Number 19H05110.

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#  Sub-barrier fusion hindrance and absence of neutron transfer channels

Vinay Singh1, Joydev Lahiri1, Partha Roy Chowdhury2 and D. N. Basu1 1Variable
Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700064, INDIA  2
Chandrakona Vidyasagar Mahavidyalaya, Chandrakona Town, Paschim Medinipur,
West Bengal 721201, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The sub-barrier fusion hindrance has been observed in the domain of very low
energies of astrophysical relevance. This phenomenon can be analyzed
effectively using an uncomplicated straightforward elegant mathematical
formula gleaned presuming diffused barrier with a Gaussian distribution. The
mathematical formula for cross section of nuclear fusion reaction has been
obtained by folding together a Gaussian function representing the fusion
barrier height distribution and the expression for classical cross section of
fusion assuming a fixed barrier. The variation of fusion cross section as a
function of energy, thus obtained, describes well the existing data on sub-
barrier heavy-ion fusion for lighter systems of astrophysical interest.
Employing this elegant formula, cross sections of interacting nuclei from 16O
+ 18O to 12C + 198Pt, all of which were measured down to $<$ 10 $\mu$b have
been analyzed. The agreement of the present analysis with the measured values
is comparable, if not better, than those calculated from more sophisticated
calculations. The three parameters of this formula varies rather smoothly
implying its usage in estimating the excitation function or extrapolating
cross sections for pairs of interacting nuclei which are yet to be measured.
Possible effects of neutron transfers on the hindrance in heavy-ion fusion
have been explored.

Keywords: Fusion reactions; Excitation function; Barrier distribution;
Nucleosynthesis.

###### pacs:

25.70.-z; 26.20.Np; 25.60.Pj; 97.10.Cv

##  I Introduction

The phenomenon of fusion hindrance in heavy-ion fusion reactions for lighter
systems may have important consequences on the nuclear processes taking place
in astrophysical scenarios. The hindrance can seriously affect the energy
generation by heavy-ion fusion reactions occurring in the region of extreme
sub-barrier energies, encompassing reactions involving lighter systems, such
as the reactions occurring in the stage of carbon and oxygen burning in heavy
stars Ro91 ; Da98 ; Wa97  and their evolution and elemental abundances. It is
well recognized that excitation functions for fusion of two colliding nuclei
can not be explained satisfactorily within the framework of a single barrier
penetration that is well-defined in its total potential energy. While
replicating the shapes of excitation functions for fusion reactions,
particularly in the domain of low energies near-threshold, coexistence of
different barriers must be assumed. Describing the fusion cross section
calculations within the framework of any theory of nuclear reactions which
involves coupling to several collective states We91 ; Ste95 ; Ti98 ; Tr01 ;
St06 ; St07 ; Den14 ; Den19 , this condition is naturally accounted.

The intent of the current exercise is to estimate the cross sections for
nuclear reactions in heavy-ion fusion of lighter systems of astrophysical
interest. The phenomenological description of the dependence of nuclear fusion
reactions on the collisional kinetic energy is accomplished by assuming the
Gaussian form of fusion barrier distribution and considering the mean height
of barriers and the variance as freely varying parameters. The cross sections
for fusion reactions have been obtained by folding a Gaussian distribution for
fusion barrier heights together with the classical expression of fixed single
barrier cross section for fusion. The effective radius, which is the distance
of the position of the potential barrier of interacting nuclei, is treated as
an additional free parameter. These three parameters can be extracted
distinctively for each reaction by fitting the theoretical predictions to the
experimental data. The excitation functions for fusion reactions, thus
obtained, describes well existing data of sub-barrier fusion and energy
dependence of capture reactions for lighter heavy-ion systems of astrophysical
interest.

##  II Distribution of fusion barriers

It has been observed that the excitation functions for fusion of two colliding
nuclei can not be well explained by invoking a well-defined single one-
dimensional potential barrier penetration model. The explanation of fusion
cross sections, particularly in case of heavy-ions, requires invoking of a
potential barrier distribution Ro91 . The quantum mechanical barrier
penetration smoothens out a set of distinct potential energy barriers to an
effective barrier distribution which is continuous. In order to replicate the
dependence of nuclear cross sections of fusion reaction on the collisional
kinetic energy, specifically measured at low energies near fusion threshold,
the assumption of a fusion barrier height distribution becomes necessary to
simulate the effects resulting from the coupling to other channels. In the
coupled-channel calculations it is naturally achieved which involve, in both
colliding nuclei, the coupling to collective states down to the lowest level.
The nuclear structure effects influencing the distribution of potential energy
barriers have been considered negligible and hence ignored in this work. A
Gaussian form $D(h)$ simulating the shape of the diffused barrier has been
conceptualized Wi04  for the fusion barrier height distribution. Therefore,
the distribution of barriers is provided by

$D(h)=\frac{1}{\sqrt{2\pi}\sigma_{h}}\exp\Big{[}-\frac{(h-h_{0})^{2}}{2\sigma_{h}^{2}}\Big{]}$
(1)

where for each individual reaction, the parameters $h_{0}$ (mean barrier
height) and $\sigma_{h}$ (width of barrier distribution) are to be determined
exclusively.

Figure 1: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 12C+24Mg.

Figure 2: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 12C+30Si.

##  III Fusion cross section calculation

In furtherance of providing a systematic analysis to the excitation function
measurements of nuclear fusion reactions, a mathematical formula for the cross
section can be derived Wi04  for surmounting the barrier arising due to
interacting nuclei. The energy dependency of the cross sections for nuclear
fusion reactions is accomplished by folding the Gaussian distribution for
fusion barriers Wi04 ; Ca11  given by Eq.(1) together with the classical
nuclear fusion reaction cross section expression which is provided by

$\displaystyle\sigma_{f}(h)=$ $\displaystyle\pi
R_{h}^{2}\Big{[}1-\frac{h}{E}\Big{]}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}h\leq E$ $\displaystyle=$ $\displaystyle
0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}h\geq E$ (2)

where $R_{h}$ marks, approximately, the position of the barrier (effective
radius corresponding to the relative distance), which results in the following
expression

$\displaystyle\sigma_{c}(E)=\int_{E_{0}}^{\infty}\sigma_{f}(h)D(h)dh$
$\displaystyle=\int_{E_{0}}^{h_{0}}\sigma_{f}(h)D(h)dh+\int_{h_{0}}^{E}\sigma_{f}(h)D(h)dh$
(3) $\displaystyle=\pi
R_{h}^{2}\frac{\sigma_{h}}{E\sqrt{2\pi}}\Big{[}\xi\sqrt{\pi}\Big{\\{}{\rm
erf}(\xi)+{\rm erf}(\xi_{0})\Big{\\}}+e^{-\xi^{2}}-e^{-\xi_{0}^{2}}\Big{]}$

where $E_{0}=0$ for positive $Q$ value reactions and $E_{0}=Q$ for negative
$Q$ value reactions, $Q$ value being the sum of the rest masses of fusing
nuclei minus rest mass of the resultant fused nucleus,

$\displaystyle\xi$ $\displaystyle=\frac{E-h_{0}}{\sigma_{h}\sqrt{2}}$
$\displaystyle\xi_{0}$ $\displaystyle=\frac{h_{0}-E_{0}}{\sigma_{h}\sqrt{2}}$
(4)

and erf($\xi$) is the Gaussian error integral for argument $\xi$. The three
parameters $R_{h}$, $h_{0}$ and $\sigma_{h}$ have to be determined by a least
square fitting of Eq.(3), while making use of Eq.(4), to the measured
excitation functions for fusion reactions. While deriving the formula of
Eq.(3), the quantal effects of barrier penetration have not been taken
explicitly into account. The structure of a given excitation function for
fusion reaction is, however, influenced by the sub-barrier tunneling which has
been included effectively through the parameter $\sigma_{h}$ which describes
the width of barrier distribution.

The mathematical expression of Eq.(3), for the interaction cross section of
overcoming the barrier arising due to potential-energy, is achieved by the use
of a diffused-barrier. This formula provides a very elegant parametrization
for such cross sections. Hence, for the predictions and analysis of excitation
functions for fusion reactions, particularly in the span of energies below
barrier, it may be used effectively for systems involving light, medium or
moderately heavy ions.

In case of systems involving light or medium heavy ions, fusion is
automatically guaranteed that leads to compound nucleus formation, once it
surmounts the barrier of the colliding nuclei. The word ‘capture’ refers to
the action of surmounting the potential barrier of colliding ions that follows
a composite system formation. In general, there is a probability $f$ that
composite nucleus experiences fusion in an event of target nucleus capturing
the projectile. This probability approaches unity for light and medium
systems. Under this condition ($f\sim 1$) fusion ensues for most of the
capture events leading to capture cross sections being practically identical
to fusion cross sections. On the contrary, there is only a meager probability
($f<1$) that events leading to capture would eventually proceed to fusion in
case of very heavy systems. In these cases most of the events remaining re-
separate before equilibration. For such cases distinguishing fusion from
capture becomes necessary. Therefore, in the event of quite heavy systems,
estimations obtained using Eq.(3) will produce capture cross sections where
fusion is not automatically guaranteed once the barrier penetration is
complete.

Figure 3: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 12C+198Pt.

Figure 4: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 16O+18O.

##  IV  Results and discussion

###  IV.1  The fusion excitation functions

The excitation functions for nuclear reactions of heavy-ion fusion for lighter
systems of astrophysical interest at sub-barrier energies have been analyzed
theoretically. This has been facilitated through Eq.(3) which is obtained by
folding an elegant diffused barrier distribution [provided in Eq.(1)] of
Gaussian shape together with the classical mathematical fusion cross section
expression for single fixed barrier. Identical combinations of projectile-
target system involved in heavy ion sub-barrier fusion reactions have been
chosen which have been recently Mo20 ; Mo97 ; Sh16 ; We21 ; Ra20  studied. By
using the method of least-square fitting, the values of three parameters
$h_{0}$ (mean barrier height), $\sigma_{h}$ (width) and $R_{h}$ (effective
radius) have been extracted. The values of these are tabulated in Table-I
arranging it in the ascending order of projectile masses.

Figure 5: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 28Si+64Ni.

Figure 6: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 58Ni+58Ni.

The task of estimating cross section $\sigma_{c}(E)$ for a particular reaction
rests upon predicting the parameter values for $h_{0}$ (mean barrier height),
$\sigma_{h}$ (width) and $R_{h}$ (effective radius) reliably. Since $h_{0}$ is
necessarily the mean barrier height, it should be a function of
$z=Z_{1}Z_{2}/(A_{1}^{1/3}+A_{2}^{1/3})$ (Coulomb parameter) in neighborhood
of the fusion barrier. The third entity, the effective barrier radius $R_{h}$,
unquestionably should depend upon $r_{0}(A_{1}^{1/3}+A_{2}^{1/3})=r_{0}A_{12}$
where $r_{0}$ is the nuclear radius parameter. The extrapolation of the
tendencies of $\sigma_{h}$ is extra difficult, which basically arise due to
nuclear deformation, vibrations and quantum mechanical barrier tunneling
probability. It may be observed from Table-I that the mean barrier height
$h_{0}$ increases with the Coulomb parameter $z$ for all cases while the
effective radius $R_{h}$ also increases with $A_{12}$ except for 16O+18O and
28Si+64Ni and interestingly, as may be seen from Figs.-1-7, for 16O+18O case,
the theoretical calculations are slightly off at lower energies as well.

Table 1:  The extracted values of $h_{0}$ (mean barrier height), $\sigma_{h}$ (width) and $R_{h}$ (effective radius), obtained from the analyses of the measured fusion excitation functions. The table is arranged in order of increasing projectile mass. Reaction | Refs. | $z$ | $A_{12}$ | $\sigma_{h}$ | $h_{0}$ | $R_{h}$
---|---|---|---|---|---|---
|  |  |  | [MeV] | [MeV] | [fm]
12C+24Mg | Mo20  | 13.916 | 5.174 | 0.815 | 11.483 | 6.646
12C+30Si | Mo97  | 15.565 | 5.397 | 1.090 | 13.540 | 8.300
12C+198Pt | Sh16  | 57.650 | 8.118 | 1.749 | 55.140 | 11.179
16O+18O | Mo20  | 12.450 | 5.141 | 0.859 | 9.797 | 7.743
28Si+64Ni | We21  | 55.709 | 7.037 | 1.402 | 50.403 | 7.182
58Ni+58Ni | Ra20  | 101.269 | 7.742 | 2.275 | 98.278 | 8.550
64Ni+64Ni | Ra20  | 98.000 | 8.000 | 1.466 | 92.646 | 8.862

In Figs.-1-7, the fusion excitation functions which are measured
experimentally have been depicted by full circles and compared with the
predicted estimates obtained using the diffused fusion barrier formula
represented by the continuous lines. Results for the colliding 12C+24Mg,
12C+30Si, 12C+198Pt, 16O+18O, 28Si+64Ni, 58Ni+58Ni and 64Ni+64Ni systems are
illustrated in Figs.-1-7. It may be easily identified from the plots that
accurately measured excitation functions for fusion reactions yield a
systematized information on the cardinal attributes of the nucleus-nucleus
interaction potential, viz. $h_{0}$ (mean barrier height) and $\sigma_{h}$
(width) of its distribution for collisions between two nuclei. The capture or
fusion cross sections for planning experiments can be also guessed using
Eq.(3) along with the theoretically extracted values of $h_{0}$ and
$\sigma_{h}$ parameters.

As can be visualized from the Figs.-1-7, the theoretical estimates facilitated
by the diffused barrier formula described in this work resulted in good fits
to the measured experimental data. This observation obviously infers that
almost all the events leading to capture proceed to fusion for the chosen set
of nuclei resulting in capture cross sections being essentially identical to
the fusion cross sections. It may be further imply that the for the barrier
distribution, the choice of Gaussian form describes quite well the nuclear
cross sections for fusion reactions at energies below the barrier. This fact
justifies the model ‘beyond single barrier’ which arises out of vibration and
deformation of nuclei and more importantly tunneling. Whereas theoretically
‘barrier distribution’ is a valid concept under a few approximations, the fact
that fits to the experimentally measured data are good implies certainly that
in fusion reactions involving heavy-ions, it remains a meaningful concept at
least for lighter systems of astrophysical interest.

Figure 7: Plots of the analytical estimates (solid line) and the measured
values (full circles) of the capture excitation functions for 64Ni+64Ni.

It is pertinent to mention here that although the single-Gaussian
parameterization [Eq.(1)] for barrier distribution is reasonably successful in
providing a good description of fusion process in general, neither the formula
derived for fusion cross-section nor the method using the barrier distribution
can be put to use for all fusing systems. One can visualize from Eq.(3) that
the excitation function is all the time a monotonically rising function of
energy. This puts a limitation on Eqs.(1,2) which can not be used for
describing fusion reactions at higher energies when incomplete fusion as well
as deep-inelastic scattering can cause a lowering of the fusion cross section.
Similar limitation arises for lighter systems (e.g. 12C+12C, 12C+16O, 16O+16O
etc.) as well when excitation functions possess oscillations and resonance
structures. The possibility of better agreement to data may further be
explored by opting a more intricate formula for the barrier distribution,
which, nonetheless, will bring in more additional adjustable parameters than
just three used in the present work. Such refinements for the barrier
distribution of Eq.(1) may include distributions having different widths on
lower or higher energy sides, certain moderation of the exponent in Gaussian
distribution form or multi-component distributions.

###  IV.2  The fusion barrier distribution parameters

The anticipation of the swing of $\sigma_{h}$ is easier said than done. The
reason lies in the fact that $\sigma_{h}$ crops up mainly due to nuclear
vibrations, deformation and quantum mechanical barrier tunneling probability.
A nucleus (tagged $i$) with a static deformation of magnitude $\beta_{2}(i)$
can have all possible orientations which lead to a standard deviation (SD) of
$\Delta R_{i}$ Es81  in the effective radius $R_{i}$ given by

$\Delta R_{i}=\frac{\beta_{2}(i)R_{i}}{\sqrt{4\pi}}$ (5)

where except for the quadrupole, all other higher multipoles have been
neglected. Thus, for a fixed distance between the centers of mass of two
nuclei, the distribution of the sub-sequential surface-surface distance
results in the SD $\sigma_{i}$ for the barrier height distribution given by

$\displaystyle\sigma_{i}=$ $\displaystyle\bigg{|}\frac{\partial V}{\partial
r}\bigg{|}_{r=R_{h}}\Delta R_{i}$ $\displaystyle=$
$\displaystyle\frac{Z_{1}Z_{2}e^{2}}{R_{h}}\frac{\beta_{2}(i)}{\sqrt{4\pi}}\frac{R_{i}}{R_{h}}\Big{[}1-\frac{3R_{i}}{5R_{h}}\Big{]}$
(6)

Therefore, $\sigma_{h}$ can be given by

$\sigma_{h}=\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{0}^{2}}$ (7)

where $R_{h}=R_{1}+R_{2}=r_{0}[A_{1}^{1/3}+A_{2}^{1/3}]$, $\Delta R_{1}$ and
$\Delta R_{2}$ are the SDs of the radius vectors specifying the surfaces of
the target as well as the projectile nuclei having mean radii $R_{1}$ and
$R_{2}$ and quadrupole deformation parameters $\beta_{2}(1)$ and
$\beta_{2}(2)$, respectively. The quantity $\sigma_{0}$ in the above Eq.(7) is
an adjustable parameter which, at least approximately, takes into account the
nuclear vibrations and quantum mechanical barrier tunneling probability.
Manifestly, for semi-magic as well as magic nuclei, $\sigma_{1}=\sigma_{2}=0$
and then $\sigma_{h}=\sigma_{0}$.

The fusion $Q$ values, being the sum of the rest masses of fusing nuclei minus
rest mass of the resultant fused nucleus, have been calculated using atomic
mass excesses Au03 . In Table-II, the fusion $Q$ values, the effective radius
parameter $r_{0}$ and the quantities $\sigma_{1}$, $\sigma_{2}$ and
$\sigma_{0}$, obtained from the analyses of the measured fusion excitation
functions, have been listed. For obtaining the values of $\sigma_{1}$ and
$\sigma_{2}$, the theoretical values of static deformation $\beta_{2}(i)$ Mo95
from recent tabulation Mo16  have been used while the measured nuclear
deformations are available in Ref.Ra01 .

Table 2:  The list of extracted values of effective radius parameter $r_{0}$ and the quantities $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{0}$, obtained from the analyses of the measured fusion excitation functions is presented. The table is arranged in order of increasing projectile mass along with the fusion $Q$ values. The one neutron and two neutron transfer $Q$-values from projectile to target nuclei $Q_{1n}$ and $Q_{2n}$, respectively, are also listed while neutron transfers from target to projectile nuclei are provided within parentheses. Reaction |  $Q$ (fusion) | $Q_{1n}$ | $Q_{2n}$ | $r_{0}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{0}$
---|---|---|---|---|---|---|---
| [MeV] | [MeV] | [MeV] | [fm] | [MeV] | [MeV] | [MeV]
12C+24Mg | 16.298 | -11.391 (-3.866) | -13.418 (-0.739) | 1.285 | 0.0000 | 0.6417 | 0.5025
12C+30Si | 14.114 | -12.134 (-5.663) | -16.051 (-5.960) | 1.538 | 0.0000 | 0.3656 | 1.0268
12C+198Pt | -13.955 | -13.166 (-2.611) | -19.004 (-0.281) | 1.377 | 0.0000 | 0.7992 | 1.5557
16O+18O | 24.413 | -11.709 (-3.910) | -17.323 (0.0) | 1.506 | 0.0116 | 0.0119 | 0.8588
28Si+64Ni | -1.787 | -11.082 (-1.185) | -15.441 (2.587) | 1.021 | 2.5738 | 0.7806 | 0.0000
58Ni+58Ni | -65.855 | -3.218 (-3.218) | -2.079 (-2.079) | 1.104 | 0.0000 | 0.0000 | 2.2750
64Ni+64Ni | -48.797 | -3.560 (-3.560) | -1.446 (-1.446) | 1.108 | 1.1823 | 1.1823 | 0.0000

###  IV.3  Fusion hindrance and neutron transfer $Q$-values

The one neutron and two neutron transfer $Q$-values from projectile to target
nuclei $Q_{1n}$ and $Q_{2n}$, respectively, and those from target to
projectile nuclei (except for 28Si+64Ni) are all negative. The reactions
studied in the present work, the transfer effects are, therefore, absent
leading to fusion hindrance. Thence the width $\sigma_{h}$ of the distribution
function is not enhanced due to any mixture of transfer and deformation
effects. It is worthwhile to mention here that for reaction 32S + 110Pd for
which neutron transfer $Q$-values are positive (and hence a mixture of
transfer and deformation effects) leading to large $\sigma_{h}$ = 3.10 MeV
which is significantly greater than $\sigma_{h}$ = 1.92 MeV for the reaction
36S + 110Pd for which neutron transfer $Q$-values are negative (and only pure
deformation effect due to the deformed Pd) At14 . In general, for positive
fusion $Q$-values, the fusion cross section should be more. But for the fusion
reactions 12C+24Mg, 12C+30Si and 16O+18O with positive $Q$-values, fusion
hindrance remains owing to particularly small $\sigma_{h}$ values as is
evident from Eqs.(3,4). Such an effect is due to the saturation properties of
nuclear matter, which hinders density build up and prevents substantial
overlap of light nuclei participating in reactions causing hindrance in
quantum tunneling. This leads to rapid decrease in fusion cross section
characterizing a major impact on the estimations of thermonuclear reaction
rates which play a very significant role in stellar evolution studies.

##  V  Summary and conclusion

In the region of sub-barrier energies, the fusion reaction cross sections have
been estimated spanning a broad energy range. In order to envision the
conditions of overcoming the potential barrier in nuclear collisions and to
have a systematic knowledge on the essential characteristics, viz. $h_{0}$
(mean barrier height), $\sigma_{h}$ (width) and $R_{h}$ (effective radius), of
the interacting potential, a set of accurately measured excitation functions
of fusion reactions has been studied for two colliding nuclei. A Gaussian
distribution function for the barrier heights is assumed to derive a simple
diffused-barrier formula. The values of the essential parameters $h_{0}$ (mean
barrier height), $\sigma_{h}$ (width) and $R_{h}$ (effective radius) are
determined using the method of least-square fit. In the fusion reactions
studied here, the transfer effects are absent leading to fusion hindrance. The
widths of the barrier distribution are not enhanced due to any mixture of
transfer and deformation effects. Even for positive fusion $Q$-values, the
fusion hindrance remains because of small $\sigma_{h}$. This effect may be
attributed to the saturation properties of nuclear matter, which prevents
substantial overlap of light nuclei participating in the fusion reactions
causing hindrance in quantum tunneling.

The present formula of cross section for fusion reactions can be used to
calculate the cross sections for surmounting the barrier in collisions of
moderately heavy systems for a given projectile-target combination. For
calculating the production cross sections of superheavy nuclei, the prediction
of the capture excitation functions or sticking can be used in the sticking-
diffusion-survival model An93  as one of three basic ingredients. The
reasonably good fit to the experimental data provided by the theory described
above implies two principal facts that for the investigated set of nucleus-
nucleus systems almost all the events leading to capture ultimately proceeds
to fusion and the idea of the Gaussian distribution of barrier provides
excellent description of cross sections for fusion reactions in the domain of
the sub-barrier energies. Although the single-Gaussian parametrization for
barrier distribution is reasonably successful in providing a good description
of fusion process in general, neither the formula derived for fusion cross-
section nor the method using the barrier distribution can be put to use for
all fusing systems. One can visualize from Eq.(3) that the excitation function
is all the time a monotonically rising function of energy. This puts a
limitation on Eqs.(1,2) which can not be used for describing fusion reactions
at higher energies when incomplete fusion as well as deep-inelastic scattering
can cause a lowering of the fusion cross section. Similar limitation can arise
for lighter systems also when excitation functions possess oscillations and
resonance structures. Possibility of better compliance to measured data may be
explored by opting a more intricate formula for barrier distribution, which,
however, will bring in more additional adjustable parameters than just three
used in the present work. Such improvements for the barrier distribution can
be realized through distributions having distinctive widths on the lower or
higher energy sides, multi-component distributions or a modification of the
exponent appearing in distribution represented by a Gaussian form.

###### Acknowledgements.

One of the authors (DNB) acknowledges support from Science and Engineering
Research Board, Department of Science and Technology, Government of India,
through Grant No. CRG/2021/007333.

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