# Status of deep subbarrier $\mathbf{{}^{12}{\rm C}+{}^{12}{\rm C}}$ fusion
and advancing the Trojan horse method

A. M. Mukhamedzanov  Cyclotron Institute, Texas A$\&$M University, College
Station, Texas, 77843, USA
E-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, I will update the current status of the carbon-carbon fusion
research taking into account that after the latest analysis [Beck et al. Eur.
Phys. J. A 56, 97 (2020), Letter to the Editor] new important experimental and
theoretical results had been published and will discuss how to advance new THM
measurements to extract the low-energy astrophysical $S$-factors.

##  I Introduction

Carbon plays a key role in astrophysics, and information about nuclear
reactions involving carbon is essential for understanding nucleosynthesis in
stars. Moreover, two nuclear astrophysical processes involving carbon,
${}^{12}{C}(\alpha,\gamma){}^{16}{\rm O}$ and ${}^{12}{\rm C}+{}^{12}{\rm C}$
fusion remain one of the main focus of scientists for many years. While our
knowledge about the first reaction has been significantly improved, the
implication of the carbon-carbon fusion reaction on stellar evolution is still
under debate despite extensive research, both theoretical and experimental,
see Mori ; Becker ; Kettner ; Aguilera ; Spillane ; Tumino ; Jiang ; Patterson
; Esbensen ; Wiescher ; Bonasera ; Godbey ; Wen ; SPP ; Chen ; Khoa ; Notani ;
Jiang13 ; Assuncao ; Zhang2019 ; Nagatani .

The ${}^{12}{\rm C}+{}^{12}{\rm C}$ fusion reaction has different exit
channels but five exothermic channels attract the attention of the scientists.
These are

$\displaystyle{}^{12}{\rm C}+{}^{12}{\rm C}\to{}^{23}{\rm
Na}(3/2^{+},\,E_{x}=0.0\,{\rm MeV})+p_{0},\quad Q=2.241\,{\rm MeV},$ (1)
$\displaystyle{}^{12}{\rm C}+{}^{12}{\rm C}\to{}^{23}{\rm
Na}(5/2^{+},\,E_{x}=0.44\,{\rm MeV})+p_{1},\quad Q=1.801\,{\rm MeV},$ (2)
$\displaystyle{}^{12}{\rm C}+{}^{12}{\rm C}\to{}^{20}{\rm
Ne}(0^{+},\,E_{x}=0.0\,{\rm MeV})+\alpha_{0},\quad Q=4.617\,{\rm MeV},$ (3)
$\displaystyle{}^{12}{\rm C}+{}^{12}{\rm C}\to{}^{20}{\rm
Ne}(3/2^{+},\,E_{x}=1.634\,{\rm MeV})+p_{0},\quad Q=2.983\,{\rm MeV},$ (4)
$\displaystyle{}^{12}{\rm C}+{}^{12}{\rm C}\to{}^{24}{\rm
Mg}(0^{+},\,E_{x}=0.0\,{\rm MeV})+\gamma,\quad Q=13.933\,{\rm MeV}.$ (5)

For stars with masses $(8-10)m_{\odot}$ the effective energy interval (Gamow
window) for the carbon-carbon fusion is $E\sim 1.2-1.8$ MeV, where $E\equiv
E_{{}^{12}{\rm C}\,{}^{12}{\rm C}}$ is the relative ${}^{12}{\rm
C}-{}^{12}{\rm C}$ kinetic energy. It corresponds to the temperature interval
$(0.35-0.64)T_{9}$ (temperature $T_{9}=10^{9}$ Kelvin). For heavier stars with
masses $\sim 25\,m_{\odot}$ the effective energy interval is $E\sim 1.8-2.6$
MeV corresponding to $(0.64-1.15)T_{9}$.

The Coulomb barrier for the carbon-carbon head-on collision (calculated for
the carbon radius $1.4\,\times 12^{1/3}$ fm neglecting the nuclear
interaction) is $\approx 6.6$ MeV making the astrophysically relevant energies
deep sub-Coulomb. The lowest measured resonance energy $E\approx 2.14$ MeV was
reported in Spillane . However, large uncertainties made it impossible to
determine the strength of this resonance. Although it is very probable that
the accuracy of direct measurements in the region $E\sim 2.0$ MeV soon will be
improved, it is not feasible, at least in the near future, that direct
measurements can cover the whole astrophysically relevant for masses
$(8-10)m_{\odot}$ energy interval $1.2-1.8$ MeV.

The first experimental breakthrough was presented in Tumino . In this work,
the indirect Trojan horse method (THM) utilizing the reaction generated by the
collision of ${}^{14}{\rm N}$ with ${}^{13}{\rm C}$ was used to obtain the
astrophysical $S$-factor for the carbon-carbon fusion down to $E=0.8$ MeV. The
measured astrophysical factor showed pronounced low-energy resonances at
$E=0.88,\,0.98$ and $1.5$ MeV accompanied by quite a few more minor resonances
demonstrating the power of the THM. The advantage of the THM is the absence of
the Coulomb penetrability factor in the $\,{}^{12}{\rm C}+{}^{12}{\rm C}\,$
channel because the nucleus $\,{}^{12}{\rm C}\,$ emerging from the projectile
$\,{}^{14}{\rm N}\,$ is the off-the-energy shell (off-shell). It allows one to
extract the low-energy $S$-factor avoiding the challenging problems appearing
in direct measurements. However, the analysis of the THM reactions is
complicated and is not as straightforward as for direct measurements. The
sharp rise of the astrophysical factor toward low energies obtained in Tumino
was the result of neglecting the Coulomb interaction in the initial and,
especially, the intermediate and final states of the THM reactions muk2019 ;
muk2020 ; Beck , see also discussion below. Taking into account the Coulomb-
nuclear distortions decreases the extracted in Tumino  astrophysical factors,
including the resonance peaks by up to three orders of magnitude.

From the theoretical point of view, existing of the resonances in the
astrophysical factors of the ${}^{12}{\rm C}+{}^{12}{\rm C}$ fusion is one of
the most exciting topics in nuclear theory related with nuclear astrophysics.
These resonances are believed to be molecular ${}^{12}{\rm C}+{}^{12}{\rm C}$
configurations in the compound ${}^{24}{\rm Mg}$. Given the importance of the
issue, it is not surprising that there have been so many theoretical papers
devoted to carbon-carbon fusion. Below I will briefly discuss some most
advanced publications related to carbon-carbon fusion.

In Jiang , different methods of extrapolation of astrophysical factors down to
astrophysically relevant energies for $\,{}^{12}{\rm C}\,$ and $\,{}^{16}{\rm
O}\,$ fusion reactions were exploited. The authors came to the conclusion that
the fusion hindrance (decrease of the $S$-factor toward low energies) should
be taken into account. However, the physical nature of the hindrance effect
was not clarified.

The paper Esbensen  presents one of the most advanced studies of carbon
isotopes fusion. The fusion cross-sections were calculated by solving the
coupled-channel equations using the Woods-Saxon or M3Y potentials augmented
with a repulsive core allowing one to take into account nuclear
incompressibility. The mutual excitations of both colliding nuclei were taken
into account. The nuclear excitations of both colliding carbon nuclei were
obtained from the surface excitations, which is described by the radial
deformation depending on deformation amplitudes. Two different results
corresponding to ten and twelve coupled channels were presented. The fusion
cross-sections demonstrated smooth behavior and exceeded experimental data,
especially for twelve channels. They can be considered as an upper limit of
the experimental cross-sections. These calculations were not intended to
reproduce resonance structure because the cluster effects were not included in
the formalism.

In Wiescher , the quantum dynamical model based on the solving of the time-
dependent Schrödinger equation with a collective Hamiltonian. The model used
the Time-Dependent Wave-Packet method. The calculated $S$-factor, in contrast
to the potential and coupled-channel methods, shows three resonances at $E>4$
MeV but smooth behavior at lower energies. The absence of resonance structures
at low energies was due to the lack of cluster effects.

In Bonasera  the carbon-carbon fusion cross-section was also calculated using
the TDHF method based on the Feynman Path Integral Method. The general trend
of the modified astrophysical factor showed a smooth decrease toward low
energies. No resonance structure was obtained because the model did not
include cluster effects. To reproduce the low-energy resonances, the authors
engaged the Bass ion-ion potential, which contains nuclear, Coulomb, and
centrifugal terms. Its strength was adjusted to produce the resonances found
in the THM experiment Tumino . These resonance can be considered as
phenomenological fit of the experimental data. It should be underscored that
the potential model was utterly independent of the basic TDHF approach, and
the obtained resonances were added to the TDHF calculations.

To analyze the ${}^{12}{\rm C}+{}^{12}{\rm C}$ fusion two microscopic methods
based on density-constrained static Hartree-Fock and density-constrained TDHF
methods were applied in Godbey . The bare ion-ion potentials were derived
using the density energy functional for each method. The calculated
$S$-factors using different microscopic approaches showed similar behavior
with an apparent rise toward low energies contradicting the hindrance model
Jiang . It was underscored that the $\alpha$-clustering effects in
${}^{12}{\rm C}$, which were not taken into account, can influence the fusion
process.

In Ref. Wen , published two years later, the hindrance effect was observed by
solving the coupled-channels approach with improved incoming boundary
conditions. The ion-ion potential included Coulomb and Woods-Saxon nuclear
potentials. The paper presented some insight into the hindrance phenomenon.
Again, a smooth energy behavior of the modified $S$-factor without any
resonance structure with a drop toward low energies was found.

The origin of the resonance structure of the ${}^{12}{\rm C}+{}^{12}{\rm C}$
fusion $S$-factor was analyzed in Jiang13 . The fusion of these two carbon
nuclei is compared with fusion of ${}^{12}{\rm C}+{}^{13}{\rm C}$ and
${}^{13}{\rm C}+{}^{13}{\rm C}$ systems. The conclusion of the paper was that
nonoverlapping compound states of ${}^{24}{\rm Mg}$ at energies $E<7$ MeV (the
scantiness of the resonances in the ${}^{12}{\rm C}+{}^{12}{\rm C}$ system and
the narrow resonance widths make them nonoverlapping) caused the resonance
structure and suppression of the deep sub-barrier fusion of ${}^{12}{\rm
C}+{}^{12}{\rm C}$. The calculated $S$-factors for the Woods-Saxon and the
coupled channels method using the M3Y+repulsion potential did not show any
resonance structure.

Another coupled-channel method was exploited in Assuncao . The ${}^{12}{\rm
C}+{}^{12}{\rm C}$ fusion modified $S$-factors were calculated using the
multichannel folding model. Up to four excited states of ${}^{12}{\rm C}$,
$(0_{1}^{+},\,2^{+},\,0_{2}^{+},\,3^{-})$, had been taken into account. The
folding potential was constructed using the $M3Y$ $\,NN\,$ potential and
${}^{12}{\rm C}$ densities using the resonating group method. For the number
of the excited states $\leq 2$ the calculated modified $S$-factors showed
monotonic increase as energy decreases. No resonance structure appeared
because the approach included no cluster effects.

To summarise, the key point is that all the above-cited works did not include
the cluster degrees of freedom in ${}^{12}{\rm C}$. The resulting
astrophysical factors demonstrated a smooth energy behavior, which depends on
the adopted model. The trends toward low energies varied from increase to
hindrance. The dependence of the modified $S$-factors on the number of the
included excited states was demonstrated. I mean that neither of the referred
models was capable of reproducing the resonance structure of the ${}^{12}{\rm
C}+{}^{12}{\rm C}\to\alpha+{}^{20}{\rm Ne}$ and ${}^{12}{\rm C}+{}^{12}{\rm
C}\to p+{}^{23}{\rm Na}$ reactions, because these reactions are many-nucleon
rearrangement ones proceeding through the intermediate resonance ${}^{24}{\rm
Mg}^{*}$. The study of resonances in these reactions requires a microscopic
approach, which considers nucleon degrees of freedom and channel coupling.

In 2021, the first fully microscopic approach based on the antisymmetrized
molecular dynamic (AMD) had been published TaniguchiKimura . It was a real
breakthrough in the long-stalled theory of carbon-carbon fusion. The authors
presented the first fully microscopic treatment of the ${}^{12}{\rm
C}+{}^{12}{\rm C}$ fusion without any adjustable parameters. The AMD nuclear
wave function is given by the parity-projected Slater determinant, whose
elements are nucleon Gaussian wave packets. The superposition of the wave
functions of different rearrangement channels formed as the result of the
carbon-carbon fusion described the resonances of the compound nucleus
${}^{24}{\rm Mg}$. The resonances in the $S$-factor appear only due to the
channel coupling and disappear if only one channel ${}^{12}{\rm C}+{}^{12}{\rm
C}$ is taken into account.

In this paper, the current status of the carbon-carbon fusion is updated,
taking into account that after the latest analysis Beck  new important
experimental and theoretical results had been published. I will discuss how to
advance new THM measurements to extract the low-energy astrophysical
$S$-factors discussing the kinematics of two THM reactions: ${}^{12}{\rm
C}({}^{14}{\rm N},\,d){}^{24}{\rm Mg}^{*}$ and ${}^{12}{\rm C}({}^{13}{\rm
C},\,n){}^{24}{\rm Mg}^{*}$.

In what follows the system of units in which $\hbar=c=1$ is used.

##  II Latest experimental and theoretical results

Fig. 1 shows the modified astrophysical factors for the carbon-carbon fusion
published during the last three years. First of all, in 2020, two important
direct measurements of the carbon-carbon fusion were published by STELLA
collaboration STELLA  and Notre Dame-Mexico Tan . These experiments show some
disagreement between both measurements, see Fig. 1. The lowest measured
energies in both experiments are $E\sim 2.2$ MeV. Thus, since 2007, when the
energy of $E=2.14$ MeV was reached in Spillane , direct measurements were not
able to measure the $S$-factors at energies below 2.1 MeV. Hence it is
unlikely that soon direct measurements will reach energies down to $E\sim 1.5$
MeV.

Figure 1:  Modified astrophysical factors for the ${}^{12}{\rm C}+{}^{12}{\rm
C}$ fusion. The navy rectangles and the magenta diamonds are the astrophysical
factors from STELLA  (STELLA) and Tan  (Notre-Dame-Mexico), black dotted line
is the THM $S^{*}(E)$-factor Tumino  (THM (Nature)), the navy dash-dotted line
is the Coulomb-nuclear renormalized astrophysical factor muk2019 ; muk2020
normalized to the $S$-factor from STELLA  at $E=2.54$ MeV (THMCN(1)), the
magenta solid line is the Coulomb-nuclear renormalized astrophysical factor
normalized to the $S$-factor from Tan  at $E=2.64$ MeV (THMCN(2)), the cyan
dash line is the AMD microscopic calculations from TaniguchiKimura .

Also in Fig. 1 is presented the first modified astrophysical $S^{*}(E)$-factor
obtained using the indirect THM measurements Tumino , which reached the lowest
energy $\,E=0.8$ MeV. Since the width of the Gamow window for the carbon-
carbon is comparable or wider than the resonance widths, the averaged modified
astrophysical factor

$\displaystyle S^{*}(E)=\sigma(E)\,E\,e^{(\frac{87.21}{\sqrt{E}}+0.46\,E)},$
(6)

where $\sigma(E)$ is the fusion cross-section, is a relevant approach for the
astrophysical application. As one can see, the original THM data show three
significant resonance peaks with a sharp increase toward small energies.
However, as we have discussed, the analysis of the THM data was performed in
the plane-wave approximation (PWA), neglecting the Coulomb rescattering of the
nuclei in the initial and, especially, in the intermediate and final channels.
At the normalization energy of the THM data to the direct ones $E=2.67\,$ MeV
the relative $\,d-{}^{24}{\rm Mg}\,$ kinetic energy is $0.92$ MeV with the
Coulomb barrier about $3$ MeV. Moreover, although the relative ${}^{14}{\rm
N}-{}^{12}{\rm C}$ kinetic energy $13.84$ MeV in the entry channel of the THM
reaction is higher than the Coulomb barrier of $10$ MeV, the Coulomb parameter
in the entry channel is $4.4$. Such a strong Coulomb interaction in the
initial channel and the deep sub-Coulomb energy in the intermediate channel
explains the failure of the PWA used in Tumino . Including the Coulomb-nuclear
distorted waves causes a drop of the low-energy astrophysical factor by about
of factor $10^{3}$, see Fig. 1.

In Fig. 1 are also presented the two THM modified astrophysical factors
$S^{*}(E)$-factors obtained from data in Tumino  including the Coulomb-nuclear
distortions. These distortions are taken into account by replacing the PWA
transfer differential cross-sections used in Tumino  with the DWBA one. The
dash-dotted navy curve is normalized to the $S^{*}$-factor from STELLA  at
$E=2.54$ MeV (THMCN(1)), while the solid magenta line is the same Coulomb-
renormalized THM astrophysical factor but normalized to the astrophysical
factor from Tan  at $E=2.64$ MeV (THMCN(2)).

The calculated AMD $S^{*}$-factor TaniguchiKimura , cyan dash line, shows
strong resonances at $0.94$ MeV and $1.5$ MeV and a resonance at $2.2$ MeV,
see the cyan dash curve in Fig.1. Moreover, the AMD astrophysical factor
qualitatively agrees with the THMCN $S^{*}$-factors, especially with THMCN(1).
Further improvement of the indirect THM measurements and updated AMD approach
can converge.

##  III Advancing indirect THM

###  III.1 Energy-momentum equations for THM

The first indirect THM measurements Tumino  demonstrated that using this
technique scientists can measure the astrophysical factor for ${}^{12}{\rm
C}+{}^{12}{\rm C}$ down to $E=0.8$ MeV covering the whole energy interval
relevant for nuclear astrophysics. The THM experiment’s positive outcome is
discovering the resonances at energies $E<2$ MeV. However, the first indirect
experiment and its analysis encountered a few weighty drawbacks muk2019 ;
muk2020 ; Beck . It resulted in a wrong energy track of the THM astrophysical
factor and resonance spins assignment. In what follows, I will discuss how to
advance the indirect THM to finally solve the problem of the low-energy
astrophysical factor for the carbon-carbon fusion. To help the reader, I
present some essential equations needed for the THM application. Let us
consider the THM reaction

$\displaystyle a+A\to s+F^{*}\to s+b+B,$ (7)

where $a=(s\,x)$ is the Trojan horse (TH) particle, which brings the particle
$x$, hidden inside, behind the $a+A$ barrier allowing it to interact with the
target $A$ while particle $s$ leaves as a spectator, and $F^{*}$ is the
resonance in the subsystem $F=(xA)$. The idea of the THM is to extract the
information about the binary resonant subreaction

$\displaystyle x+A\to b+B.$ (8)

The THM reaction is a two-step reaction proceeding through the intermediate
resonance. Its mechanism is described by the diagram depicted in Fig. 2.

Figure 2: The mechanism of the THM in the PWA.

Note that when calculating the THM reaction amplitude, a priori, the distorted
waves in the initial, intermediate and final states should be taken into
account mukkad2019 .

The first step is the transfer reaction $a+A\to s+F^{*}$ populating the
resonance state $F^{*}$, which on the second stage decays into the two-body
channel $b+B$. In what follows, I first recall some basic equations and
definitions which are used in the THM analysis.

The energy conservation in the center-of-mass (c. m.) of the TH reaction reads

$\displaystyle E_{aA}-\varepsilon_{sx}=E_{sF}+E=E_{sF}+E_{bB}-Q_{if},$ (9)

where $E\equiv E_{xA},\,$ $\,E_{j\,t}=k_{j\,t}^{2}/(2\,\mu_{j\,t}),\,$
$\,Q_{if}=m_{x}+m_{A}-m_{b}-m_{B}\,$, $\,E_{j\,t}$, $\,{\rm{\bf k}}_{j\,t}\,$
and $\,\mu_{j\,t}\,$ are the relative kinetic energy, on-the-energy shell
relative momentum and reduced mass of the particles $j$ and $t$, $\,m_{j}$ is
the mass of the particle $j$, $\,\varepsilon_{sx}\,$ is the binding energy of
the particles $s$ and $x$ in the TH particle $\,a=(sx)$.
$\;E_{R(xA)}=E_{0(xA)}-i\,\Gamma/2,\,$ is a resonance energy in the subsystem
$x+A$, $\,E_{0(xA)}\,$ is the real part of the resonance energy in the channel
$x+A$, $\,\Gamma\,$ is the total resonance width of the resonance $F^{*}$
populated in the transfer reaction.

We consider a two-state coupled channel problem in which the resonance formed
in the channel $i=x+A$ decays into a different channel $f=b+B$. Therefore,
when in the channel $i$ $\;E\to E_{R(xA)}\,$ the relative energy $E_{bB}$
approaches the resonance energy $\,E_{R(bB)}\,$ in the channel $f$:
$\;E_{R(bB)}=E_{0(bB)}-i\,\Gamma/2$.

For $E\to E_{R(xA)}$, due to energy conservation, see (9), one gets that
$E_{sF}\to E_{R}$, where

$\displaystyle E_{R}=E_{0(sF)}-i\,\Gamma/2.$ (10)

Here

$\displaystyle
E_{0(sF)}=E_{aA}-\varepsilon_{sx}-E_{0(xA)}=E_{aA}-\varepsilon_{sx}+Q_{if}-E_{0(bB)}$
(11)

is the real part of the resonance energy in the system $s+F$.

From the energy-momentum conservation in the three-ray vertices $a\to s+x$ and
$x+A\to F^{*}$ in the diagram of Fig. 2 follows that $xA$ relative kinetic
energy $E$ is

$\displaystyle
E=\frac{p_{xA}^{2}}{2\,\mu_{xA}}-\frac{p_{sx}^{2}}{2\,\mu_{sx}}-\varepsilon_{sx}.$
(12)

In the system $k_{a}=0$ Eq. (12) reduces to

$\displaystyle
E=\frac{m_{x}}{m_{xA}}\,E_{A}-\frac{k_{s}^{2}}{2\,\mu_{sF}}+\frac{{\rm{\bf
k}}_{s}\cdot{\rm{\bf k}}_{A}}{m_{xA}}-\varepsilon_{sx}.$ (13)

${\rm{\bf p}}_{j\,t}$ is the $j-t$ the Galilean-invariant relative momentum
when one of the particles $j$ or $t$ is virtual (off-the-energy shell).

$\displaystyle{\rm{\bf p}}_{xA}=-{\rm{\bf
k}}_{A}+\frac{m_{A}}{m_{xA}}\,{\rm{\bf k}}_{F},$ $\displaystyle{\rm{\bf
p}}_{sx}={\rm{\bf k}}_{s}-\frac{m_{s}}{m_{sx}}\,{\rm{\bf k}}_{a}.$ (14)

$\,m_{j\,t}=m_{j}+m_{t}$, $\,\mu_{j\,t}=m_{j}\,m_{t}/m_{j\,t},\,$ ${\rm{\bf
k}}_{j}$ is on-the-energy shell momentum of particle $j$. In the c. m. of the
THM reaction ${\rm{\bf k}}_{a}=-{\rm{\bf k}}_{A}$ and ${\rm{\bf
k}}_{s}=-{\rm{\bf k}}_{F}$ and ${\rm{\bf p}}_{xA}={\rm{\bf
k}}_{a}-\frac{m_{A}}{m_{xA}}\,{\rm{\bf k}}_{s}$.

For on-the-energy shell particles

$\displaystyle{\rm{\bf k}}_{j\,t}=\frac{m_{t}\,{\rm{\bf
k}}_{j}-m_{j}\,{\rm{\bf k}}_{t}}{m_{j\,t}}$ (15)

and ${\rm{\bf k}}_{aA}={\rm{\bf k}}_{a},\;\;{\rm{\bf k}}_{sF}={\rm{\bf
k}}_{s}$.

We can express $p_{sx}$ and $p_{xA}$ in terms of $E$. In the c. m. of the THM
reaction we get from Eq. (14)

$\displaystyle
p_{sx}=\sqrt{2\,\mu_{sF}\,(E_{aA}-E-\varepsilon_{sx})-2\,\frac{m_{s}}{m_{a}}\,\cos(\theta)\,\sqrt{2\,\mu_{sF}\,(E_{aA}-E-\varepsilon_{sx})}\,\sqrt{2\,\mu_{aA}\,E_{aA}}+\big{(}\frac{m_{s}}{m_{a}}\big{)}^{2}\,2\,\mu_{aA}\,E_{aA}},$
$\displaystyle
p_{xA}=\sqrt{2\,\mu_{aA}\,E_{aA}-2\,\frac{m_{A}}{m_{xA}}\,\cos(\theta)\,\sqrt{2\,\mu_{sF}\,(E_{aA}-E-\varepsilon_{sx})}\,\sqrt{2\,\mu_{aA}\,E_{aA}}+\big{(}\frac{m_{A}}{m_{xA}}\big{)}^{2}\,2\,\mu_{sF}\,(E_{aA}-E-\varepsilon_{sx})}.$
(16)

where $\cos(\theta)={\rm{\bf{\hat{k}}}}_{s}\cdot{\rm{\bf{\hat{k}}}}_{a}$,
$\,{\rm{\bf{\hat{k}}}}={\rm{\bf k}}/k$.

Equation (13) shows that at fixed $\,{\rm{\bf k}}_{A}\,$ to vary $\,E\,$ one
needs to vary ${\rm{\bf k}}_{s}$.

###  III.2  THM triple differential cross section

Let me consider now another important equations, such as differential cross
sections (DCSs), triple, double and single, and THM astrophysical factor. I
start from the general equation for the triple DCS for the reaction $2\,\,{\rm
particles}\to 3\,\,{\rm particles}$. The kinematics of these reactions is
determined by the four momentum-energy conservation equations:

$\displaystyle{\rm{\bf k}}_{a}+{\rm{\bf k}}_{A}={\rm{\bf k}}_{s}+{\rm{\bf
k}}_{b}+{\rm{\bf k}}_{B},$ $\displaystyle
E_{a}+E_{A}+m_{a}+m_{A}=E_{s}+E_{b}+E_{B}+m_{s}+m_{b}+m_{B}.$ (17)

The total number of the variables describing three particles in the final
state is nine. Subtracting four variables from Eqs. (17) we get five
independent variables in the THM reactions. The incident beam selects a
direction (axis) in the space. The angle describing the rotation around this
axis can be dropped. Then there are only four significant independent
Galilean-invariant variables, which are needed to describe the THM triple
differential cross-section. These four variables, for example, can be the
energies and scattering angles of two particles. They provide the full 3-body
kinematics.

Here we use another set of independent variables using the fact that the THM
reaction is the two-step process. As independent variables we select
$E_{sF},\,$ solid angles $\,\Omega_{{\rm{\bf k}}_{sF}},\,$ and
$\,\Omega_{{\rm{\bf k}}_{bB}}.\,$ Then the triple DCS is given by mukkad2019 ;
Dolinski

$\displaystyle\frac{{{{\rm d}^{3}}\sigma}}{{{\rm d}{\Omega_{{k_{bB}}}}{\rm
d}{\Omega_{{k_{sF}}}}{\rm
d}{E_{sF}}}}=|N_{C}|^{2}\,\frac{\mu_{aA}\,\mu_{sF}}{{{{(2\pi)}^{3}}}}\,\frac{k_{sF}}{k_{aA}}\,\frac{k_{bB}}{\mu_{bB}}\,{\overline{\big{|}{\cal
M}_{R}\big{|}^{2}}},$ (18)

where $k_{sF}=\sqrt{2\,\mu_{sF}\,E_{sF}}$.

####  III.2.1 Multi-level, two-channel case

To write done explicitly ${\overline{\big{|}{\cal M}_{R}\big{|}^{2}}}$, first,
I consider a multilevel, two-channel case. This case should be considered when
there are overlapping resonances with the same quantum numbers. Such
resonances interfere and the best way to analyze such cases is to use the
$R$-matrix method:

$\displaystyle\overline{\big{|}{\cal
M}_{R}{\big{|}^{2}}}=\,{{\widehat{J}}_{a}{\widehat{J}}_{A}}\sum\limits_{{M_{B}}{M_{b}}{M_{s}}{M_{a}}{M_{A}}}\,\Big{|}\sum\limits_{M_{F_{\nu}}\,M_{F_{\tau}}}\sum\limits_{\nu\,\tau=1}^{N}\,$
$\displaystyle\times W_{\nu\,M_{B}M_{b}}^{M_{F_{\nu}}}({\rm{\bf
k}}_{bB})\,[{\bf
A}]_{\nu\,\tau}^{-1}\,M_{\tau\,M_{F_{\tau}}M_{s};M_{A}M_{a}}({\rm{\bf{k}}}_{sF},\,{\rm{\bf
k}}_{aA})\Big{|}^{2}.$ (19)

$\,J_{i}(M_{i})$ is the spin (its projection) of nucleus $i$,
$\,{\hat{J}}_{i}=2\,J_{i}+1.\,$ $N$ is the number of $R$-matrix levels and
${\bf A}$ is the $R$-matrix level matrix, which provides the coupling of the
different levels and channels:

$\displaystyle{\bf
A}_{\nu\,\tau}(E)=\big{(}E_{\nu}-E\big{)}\,\delta_{\nu\,\tau}-\sum\limits_{c}\,\gamma_{\nu\,c}\,\gamma_{\tau\,c}\Big{[}{\hat{S}}_{c}(E)-B_{c}$
$\displaystyle+i\,P_{c}(E,\,R_{ch})\Big{]}.$ (20)

Here $\sum\limits_{c}$ is the sum over the included open channels.
$\,\gamma_{\tau\,c}\,$ is the formal $R$-matrix reduced width amplitude of the
level $\tau$ in the channel $c$, ${\hat{S}}_{c}(E),\,$ $B_{c}$ and
$P_{c}(E,\,R_{ch})$ are the $R$-matrix level shift, boundary condition and
penetrability factor in the channel $c,\,$ respectively. $E_{\tau}$ is the
$\tau$-th level energy.

The formal $R$-matrix reduced width amplitude $\,\gamma_{\tau\,c}$, which is a
fitting parameter, is related to the formal $R$-matrix resonance width
$\Gamma_{\tau\,c}(E)$:

$\displaystyle\Gamma_{\tau\,c}(E)=2\,P_{c}(E,R_{ch})\,\gamma_{\tau\,c}^{2},$
(21) $\displaystyle
P_{l_{c}}(E,\,R_{ch})=k_{c}\,R_{ch}\,|O_{l_{c}}(k_{c}\,R_{ch})|^{-2},$ (22)

where $l_{c}$ and $\,k_{c}$ are the relative orbital angular momentum and the
relative momentum in the channel $c$. I need to add a few words about the
$R$-matrix energy levels $E_{\tau}$. One can adopt one of the energy levels
equal to known resonance energy, and then all other energy levels will fitting
parameters

Note that the observable ${\tilde{\Gamma}}_{\tau\,c}(E_{0(c)})$ and the formal
$\Gamma_{\tau\,c}(E_{0(c)})$ partial resonance widths at the real part of the
resonance energy $E_{0(c)}$ in the channel $c$ are related by

$\displaystyle{\tilde{\Gamma}}_{\tau\,c}(E_{0(c)})=\frac{\Gamma_{\tau\,c}(E_{0(c)})}{1+\sum\limits_{c}\,\gamma_{\tau\,c}^{2}\,\frac{{\rm
d}{\hat{S}}_{c}}{{\rm d}E}|_{E=E_{0(c)}}}.$ (23)

$M_{\tau\,M_{F_{\tau}}M_{s};M_{A}M_{a}}({\rm{\bf k}}_{sF},{\rm{\bf k}}_{aA})$
is the amplitude of the transfer reaction $a+A\to s+F_{\tau}^{*}$ populating
the resonance state $F_{\tau}^{*}$:

$\displaystyle M_{\tau\,M_{F_{\tau}}\,{M_{s}};M_{A}\,M_{a}}({\rm{\bf
k}}_{sF},\,{\rm{\bf
k}}_{aA})={i^{-{l_{xA}}}}{e^{-i\,{\delta_{l_{xA}}^{p}}(k_{xA})}}\,\frac{1}{2\,\mu_{xA}}\,\sqrt{\frac{\Gamma_{\tau(xA)}\,\mu_{xA}}{\,k_{xA}}}\,O_{{l_{xA}}}({k_{xA}}{R_{ch}})\,j_{{l_{xA}}}({k_{xA}}{R_{ch}})$
$\displaystyle\times{\cal
W}_{l_{xA}}\,{Y_{{l_{xA}},{m_{{l_{xA}}}}}}({\widehat{\bf{k}}_{xA}})\,M_{{\tau\,M_{F_{\tau}}}{M_{s}};{M_{A}}{M_{a}}}^{DWZR(prior)}({\rm{\bf
k}}_{sF},\,{\rm{\bf k}}_{aA}),$ (24)

where the zero-range prior DWBA amplitude is

$\displaystyle
M_{\tau\,M_{F_{\tau}}{M_{s}};{M_{A}}{M_{a}}}^{DWZR(prior)}({{\bf{k}}_{sF}},{{\bf{k}}_{aA}})=\sum\limits_{m_{s_{xA}}\,m_{l_{xA}}M_{x}}\,C_{s_{xA}m_{s_{xA}}\,\,l_{xA}m_{l_{xA}}}^{J_{F}M_{F_{\tau}}}$
$\displaystyle\times
C_{J_{x}M_{x}\,\,J_{A}M_{A}}^{s_{xA}m_{s_{xA}}}C_{s_{sx}m_{s_{sx}}\,l_{sx}m_{l_{sx}}}^{J_{a}M_{a}}\,C_{J_{s}M_{s}\,\,J_{x}M_{x}}^{s_{sx}m_{s_{sx}}}\,{{\cal
M}^{DWZR(prior)}},$ (25)

$l_{i\,j}$ is the orbital angular momentum of the bound state $(i\,j)$,
$\,\Gamma_{\tau(xA)}\,$ is the partial resonance width for the level $\tau$ in
the channel $x+A$.

In what follows, we assume that $\,{\widehat{\bf{k}}_{xA}}\,$ is directed
along the axis $\,z$, that is,
$\,{Y_{{l_{xA}},{m_{{l_{xA}}}}}}\big{(}{\widehat{\bf{k}}_{xA}}\big{)}=\sqrt{\frac{{\hat{l}}_{xA}}{4\,\pi}}\,\delta_{m_{l_{xA}}\,0}$.

$\displaystyle{\cal M}^{DWZR(prior)}=\int{d{{\rm{\bf
r}}_{sx}}}\,\Psi_{-{{\rm{\bf k}}_{sF}}}^{(+)}({{\rm{\bf
r}}_{sx}})\,{\phi_{sx}}({{\rm{\bf r}}_{sx}})$
$\displaystyle\times\Psi_{{\rm{\bf
k_{aA}}}}^{(+)}(\frac{{{m_{s}}}}{{{m_{a}}}}{\rm{\bf r}}_{sx}).$ (26)

is the DWBA amplitude, which does not depend on the resonant wave function of
the resonance state $F_{\tau}^{*}$ and on the $V_{xA}$ potential.
$\,\Psi_{{\rm{\bf k_{aA}}}}^{(+)}(\frac{{{m_{s}}}}{{{m_{a}}}}{\rm{\bf
r}}_{sx})\,$ and $\,\Psi_{-{{\rm{\bf k}}_{sF}}}^{(+)}({{\rm{\bf r}}_{sx}})\,$
are the distorted waves in the channels $a+A$ and $s+F$, respectively. The
final-state distorted wave calculated at
$\,k_{sF}=\sqrt{2\,\mu_{sF}\,E_{sF}}$. For simplicity, we assume that
$l_{sx}=0,$ which usually is the case for the THM. Equation (26) looks like
the zero-range DWBA (ZRDWBA). The presence of the distorted waves in the
initial and final states in the DWBA amplitude ${\cal M}^{DWZR(prior)}$ has a
substantial impact on the angular distribution of $s$ and the absolute value
of the transfer reaction amplitude. We get the PWA amplitude if we replace the
distorted waves with the plane waves.

$\displaystyle{\cal
W}_{l_{xA}}=\Big{[}R_{ch}\,\frac{{\partial{\ln[O_{{l_{xA}}}}({k_{xA}}{r_{xA}})]}}{{\partial{r_{xA}}}}$
$\displaystyle-1-R_{ch}\,\frac{\partial{{\ln\,j_{{l_{xA}}}}({k_{xA}}{r_{xA}})}}{{\partial{r_{xA}}}}\Big{]}{\Big{|}_{{r_{xA}}={R_{ch}}}}$
(27)

is the off-shell factor, which appears because the transferred particle $x$,
see Fig. 2, is off-shell. $O_{{l_{xA}}}({k_{xA}}{R_{ch}})$ is outgoing
scattered wave and $\,j_{{l_{xA}}}({k_{xA}}{R_{ch}})\,$ is the spherical
Bessel function.

$W_{M_{B}M_{b}}^{\nu\,M_{F_{v}}}({\rm{\bf k}}_{bB})$ is the vertex form factor
for decay $F_{\nu}\to b+B$:

$\displaystyle W_{M_{B}M_{b}}^{\nu\,M_{F_{\nu}}}({\rm{\bf
k}}_{bB})=\sum\limits_{s_{bB}\,l_{bB}\,m_{s_{bB}}\,m_{l_{bB}}}\,C_{s_{bB}\,m_{s_{bB}}\,l_{bB}\,m_{l_{bB}}}^{J_{F}\,M_{F_{\nu}}}\,$
$\displaystyle\times
C_{J_{b}\,M_{b}\,\,J_{B}\,M_{B}}^{s_{bB}m_{s_{bB}}}\,Y_{l_{bB}\,m_{l_{bB}}}({\rm{\bf
k}}_{bB})\,e^{i\,\delta_{l_{bB}}^{p}(k_{bB})}\,\sqrt{\frac{\mu_{bB}\,\Gamma_{\nu(bB)}}{k_{bB}}}.$
(28)

Assembling Eqs. (24) - (28) we get

$\displaystyle\overline{\big{|}{\cal
M}_{R}{\big{|}^{2}}}=\,\frac{{\hat{J}}_{F}}{{\widehat{J}}_{A}\,{\hat{J}}_{x}}\,\frac{{\hat{l}}_{xA}}{16\,\pi}\,\,\frac{1}{\mu_{xA}}\,\frac{1}{k_{xA}}\,\sum\limits_{M}(C_{s\,M\,\,l_{xA}\,0}^{J_{F}\,M_{F}})^{2}$
$\displaystyle\times j_{l_{xA}}^{2}(k_{xA}\,R_{ch})\,{\cal
W}_{l_{xA}}^{2}\,\big{|}{\cal M}^{DWZR(prior)}\big{|}^{2}$
$\displaystyle\times\sum\limits_{M_{b}\,M_{B}}\,\Big{|}\,\sum\limits_{M_{F_{\nu}}}\,\sum\limits_{\nu\,\tau=1}^{N}\,W_{M_{B}M_{b}}^{\nu\,M_{F_{\nu}}}({{\rm{\bf
k}}_{bB}})\,[{\bf A}]_{\nu\,\tau}^{-1}\Gamma_{\tau\,(xA)}^{1/2}\Big{|}^{2}.$
(29)

####  III.2.2 Single-level, two-channel case

For the single-level case and $k_{sF}\to k_{0(sF)}$

$\displaystyle\overline{\big{|}{\cal M}_{R}{\big{|}^{2}}}=$
$\displaystyle\frac{1}{{\hat{J}}_{a}{\hat{J}}_{A}}\sum\limits_{M_{F}M_{F}^{\prime}M_{A}M_{a}M_{s}}\,M_{M_{F}M_{s};M_{A}M_{a}}({\rm{\bf{k}}}_{sF},{\rm{\bf
k}}_{aA})\,[M_{M_{F}^{\prime}M_{s};M_{A}M_{a}}({\rm{\bf{k}}}_{sF},{\rm{\bf
k}}_{aA})]^{*}$
$\displaystyle\times\frac{1}{(E_{0(sF)}-E_{sF})^{2}+{\Gamma^{2}}/{4}}\sum\limits_{M_{B}M_{b}}W_{M_{B}M_{b}}^{M_{F}}({\rm{\bf
k}}_{bB})\,\left[W_{M_{B}M_{b}}^{M_{F}^{\prime}}({\rm{\bf
k}}_{bB})\right]^{*}.$ (30)

Here $\,l_{bB}$ $(m_{l_{bB}})$ is the $b-B$ relative orbital angular momentum
(its projection) in the resonance $F^{*}$, $s_{bB}$ $(m_{bB})$ is the $b+B$
channel spin (its projection) in the resonance state and
$\delta_{l_{bB}}^{p}(k_{0(bB)})$ is the potential scattering phase shift in
the $b+B$ channel, $k_{0(bB)}=\sqrt{2\,\mu_{bB}\,E_{0(bB)}}$.

####  III.2.3 Coulomb factor

Taking into account that

$\displaystyle\big{|}\Gamma[1+i\,\eta]\big{|}^{2}=\frac{\pi\,\eta}{\sinh(\pi\,\eta)}$
(31)

we get the Coulomb factor for narrow resonances mukkad2019 :

$\displaystyle|N_{C}|^{2}=$
$\displaystyle\frac{{\sinh[\pi({\eta_{sb}}+{\eta_{sB}})]}}{{\sinh(\pi{\eta_{sb}})\sinh(\pi{\eta_{sB}})}}\,\frac{{\pi{\eta_{sb}}{\eta_{sB}}}}{{({\eta_{sb}}+{\eta_{sB}})}}\frac{{\pi{\eta_{\zeta}}}}{{\sinh(\pi{\eta_{\zeta}})}}$
$\displaystyle\times|F(-i{\eta_{sB}},-i{\eta_{sb}},1;-1){|^{2}}$
$\displaystyle\times{\exp{\left[2\zeta\arctan\frac{{2({E_{0(bB)}}-{E_{bB}})}}{\Gamma}\right]}},$
(32)

where

$\displaystyle\zeta=\eta_{sb}+\eta_{sB}-\eta_{{0}},$ (33)

$\eta_{j\,t}=(Z_{j}\,Z_{t}/137)\,\mu_{j\,t}/k_{j\,t}$,
$\eta_{0}=Z_{s}\,Z_{F}\,\mu_{sF}/k_{0},$ $\,Z_{j}\,e$ is the charge of
particle $j$. The presence of the factor $|N_{C}|^{2}$ shifts the resonance
peak and the resonance shape line in the system $b+B$ due to the intermediate
$s+F^{*}$ and the three-body ($s+b+B$) final-state Coulomb interactions. The
analytical expression for $N_{C}$ can be derived only for narrow resonances.
The complications are caused by the final-state three-body Coulomb
interactions mukkad2019 .

It is convenient to integrate the triple DCS over $\,\Omega_{{\rm{\bf
k}}_{bB}}\,$ to get the double DCS Dolinski , which is expressed in terms of
the DCS of the reaction $\,a+A\to s+F^{*}\,$ corresponding to the first step
of the TH reaction. However, in the case under consideration, due to the
presence of the Coulomb factor $|N_{C}|^{2}$, the DCS obtained from
integrating the triple DCS over $\,\Omega_{{\rm{\bf k}}_{bB}}\,$ cannot be
expressed in terms of the DCS of the first step. The reason is that $N_{C}$
depends on the integration variable $\,\Omega_{{\rm{\bf k}}_{bB}}$. However,
in the following cases one can neglect this dependence:

  1. 1.

When $\,|\eta_{sb}|\ll 1\,$ and $\,\eta_{sB}\approx\eta_{{0}}.\,$ In this
case, $\,|N_{C}|\approx 1\,$ and the integration over $\,\Omega_{{\rm{\bf
k}}_{sF}}\,$ can be performed without any complications.

  2. 2.

When $|\eta_{sb}|\ll 1$ and $m_{B}\gg m_{s},\,m_{b}$. Let us choose as
independent variables the Galilean momenta $\,{\rm{\bf k}}_{sF}\,$ and
$\,{\rm{\bf k}}_{bB}$. Then one can write

$\displaystyle{\rm{\bf k}}_{sB}=\frac{m_{B}\,M}{m_{sB}\,m_{bB}}\,{\rm{\bf
k}}_{sF}+\frac{m_{s}}{m_{sB}}\,{\rm{\bf k}}_{bB}\approx{\rm{\bf k}}_{sF}.$
(34)

Then $\eta_{sB}=(Z_{s}\,Z_{B}/137)\,\mu_{sB}/k_{sF}\,$ and $\,N_{C}\,$ does
not depend on $\,{\rm{\bf k}}_{bB}\,$ and integration over $\,\Omega_{{\rm{\bf
k}}_{bB}}\,$ can be performed in a straightforward way.

We assume that $|N_{C}|^{2}=1$.

###  III.3 Double differential cross section

Taking into account Eq. (29) and integrating the triple DCS (18) over
$\,\Omega_{{\rm{\bf k}}_{bB}}\,$ using the orthogonality of the Clebsch-Gordan
coefficients and the spherical harmonics we get the double THM DCS

$\displaystyle\frac{{{\rm d}^{2}\sigma^{THM}}}{{{\rm d}{\Omega_{{{\rm{\bf
k}}_{sF}}}}{\rm
d}{E_{sF}}}}=\frac{{\hat{J}}_{F}}{{\widehat{J}}_{A}\,{\hat{J}}_{x}}\,\frac{{\hat{l}}_{xA}}{16\,\pi}\,\frac{1}{(2\,\pi)^{3}}\,\frac{\mu_{aA}\,\mu_{sF}}{\mu_{xA}}\,\frac{k_{sF}}{k_{aA}\,k_{xA}}\,$
$\displaystyle\times\sum\limits_{M_{F}}\,(C_{s_{xA}\,M_{F}\,\,l_{xA}\,0}^{J_{F}\,M_{F}})^{2}\,j_{l_{xA}}^{2}(k_{xA}\,R_{ch})\,{\cal
W}_{l_{xA}}^{2}\,$ $\displaystyle\times\big{|}{\cal
M}^{DWZR(prior)}\big{|}^{2}\,\Big{|}\sum\limits_{\nu\,\tau=1}^{N}\,\Gamma_{\nu(bB)}^{1/2}\,[{\bf
A}]_{\nu\,\tau}^{-1}(E)\,\Gamma_{\tau\,(xA)}^{1/2}\Big{|}^{2}.$ (35)

Note that Eq. (35) is the THM double DCS proceeding through the intermediate
resonance states.

For the single-level case the double THM DCS takes the form

$\displaystyle\frac{{{\rm d}^{2}\sigma^{THM}}}{{{\rm d}{\Omega_{{{\rm{\bf
k}}_{sF}}}}{\rm
d}{E_{sF}}}}=\frac{1}{{2\,\pi}}\frac{{{\Gamma_{bB}}}}{{{{({E_{0(bB)}}-{E_{bB}})}^{2}}+{{{\Gamma^{2}}}}/{4}}}\frac{{d\sigma}}{{d{\Omega_{{{\rm{\bf
k}}_{sF}}}}}},$ (36)

where

$\displaystyle\frac{{{\rm d}\sigma}}{{{\rm d}{\Omega_{{{\rm{\bf
k}}_{sF}}}}}}=\frac{{{\mu_{aA}}{\mu_{sF}}}}{{4{\pi^{2}}}}\frac{{{k_{sF}}}}{{{k_{aA}}}}\sum\limits_{M_{F}M_{s}M_{A}M_{a}}$
$\displaystyle\times|{M_{{M_{F}}{M_{s}};{M_{A}}{M_{a}}}}({\rm{\bf
k}}_{sF},{\rm{\bf k}}_{aA})|^{2}$
$\displaystyle=\frac{\mu_{aA}\,\mu_{sF}}{(4\,\pi)^{3}}\,\frac{{\hat{J}}_{F}}{{\hat{J}}_{x}\,{\hat{J}}_{A}}\,\frac{{{k_{sF}}}}{{{k_{aA}}}}\,\frac{\Gamma_{xA}\,R_{ch}}{\mu_{xA}\,P_{l_{xA}}}\,j_{{l_{xA}}}^{2}({k_{xA}}{R_{ch}})$
$\displaystyle\times{\cal W}_{l_{xA}}^{2}\,|{\cal M}^{DWZR(prior)}({\rm{\bf
k}}_{sF},\,{\rm{\bf k}}_{aA})|^{2}$ (37)

is the DCS of the reaction $a+A\to s+F^{*}$.

In view of Eq. (21) it is clear that the penetrability factor $P_{l_{xA}}$ is
absent in the THM double DCS. It explains why the THM can be used to determine
the astrophysical factor down to astrophysically relevant energies.

Integrating the double DCS over $E_{sF}$ gives

$\displaystyle\int\limits_{0}^{\infty}\,{\rm d}{E_{sF}}\frac{{{\rm
d}\sigma}}{{{\rm
d}{\Omega_{{{\bf{k}}_{sF}}}}d{E_{sF}}}}=\frac{{{\Gamma_{bB}}}}{\Gamma}\frac{{d\sigma}}{{d{\Omega_{{{\bf{k}}_{sF}}}}}},$
(38)

where $\Gamma_{bB}$ is the partial resonance width for the decay of the
resonance to the channel $b+B$.

###  III.4 THM astrophysical factor

####  III.4.1 $S$-factor for multi-level, two-channel case

The astrophysical factor for the multilevel, two-channel case is (I remind
that $E=E_{xA}$)

$\displaystyle
S(E)=\frac{{\hat{J}}_{F}}{{\hat{J}}_{x}\,{\hat{J}}_{A}}\,\frac{\pi}{2\,\mu_{xA}}\,e^{2\,\pi\,\eta_{xA}}$
$\displaystyle\times\lambda_{N}^{2}\,m_{u}^{2}\,\Big{|}\sum\limits_{\nu,\,\tau=1}^{N}\,\sqrt{\Gamma_{\nu(bB)}}\,[{\bf
A}(E)]_{\nu\,\tau}^{-1}\,\sqrt{\Gamma_{\tau\,(xA)}}\Big{|}^{2}.$ (39)

Here $\lambda_{N}$ is the nucleon Compton wave length, the reduced mass
$\mu_{xA}$ is expressed in MeV, $\,m_{u}=931.5\,$ MeV is the atomic mass.

Singling out this $S$-factor from THM double DCS we get

$\displaystyle\frac{{{\rm d}^{2}\sigma}^{THM}}{{{\rm d}{\Omega_{{{\rm{\bf
k}}_{sF}}}}{\rm
d}{E_{sF}}}}=e^{-2\,\pi\,\eta_{xA}}\,{\hat{l}}_{xA}\,\frac{\mu_{aA}\,\mu_{sF}}{2\,(2\,\pi)^{5}}\,\frac{k_{sF}}{k_{aA}}\,R_{ch}\,$
$\displaystyle\times
P_{l_{xA}}^{-1}(E,R_{ch})\,\sum\limits_{M_{F}}\,(C_{s_{xA}\,M_{F}\,\,l_{xA}\,0}^{J_{F}\,M_{F}})^{2}\,j_{l_{xA}}^{2}(k_{xA}\,R_{ch})\,{\cal
W}_{l_{xA}}^{2}\,$
$\displaystyle\times\lambda_{N}^{-2}\,m_{u}^{-2}\,\big{|}{\cal
M}^{DWZR(prior)}\big{|}^{2}\,S(E).$ (40)

####  III.4.2 $S$-factor for the single-level, two channel case

For the single-level, two channel case

$\displaystyle
S(E)=\frac{{\hat{J}}_{F}}{{\hat{J}}_{x}\,{\hat{J}}_{A}}\,\frac{\pi}{2\,\mu_{xA}}e^{2\,\pi\,\eta_{xA}}\lambda_{N}^{2}\,m_{u}^{2}\,\frac{\Gamma_{bB}\,\Gamma_{xA}}{\big{(}E_{0(xA)}-E\big{)}^{2}+{\Gamma^{2}}/{4}}.$
(41)

This $S$-factor can be singled out from the THM double DCS.

$\displaystyle\frac{{{\rm d}^{2}\sigma}^{THM}}{{{\rm d}{\Omega_{{{\rm{\bf
k}}_{sF}}}}d{E_{sF}}}}=S(E)\,e^{-2\,\pi\,\eta_{xA}}\,P_{l_{xA}}^{-1}(E,R_{ch})\,\frac{{\hat{l}}_{xA}R_{ch}}{64\,\pi^{5}}\,$
$\displaystyle\times\lambda_{N}^{-2}\,m_{u}^{-2}\,\Big{|}{\cal
W}_{l_{xA}}(E,R_{ch})\Big{|}^{2}\;\frac{{{\rm d}\sigma}^{DWZR(prior)}}{{{\rm
d}{\Omega_{{{\rm{\bf k}}_{sF}}}}}}$ (42)

and

$\displaystyle\frac{{d\sigma}^{DWZR(prior)}}{{d{\Omega_{{k_{sF}}}}}}=\frac{{{\mu_{aA}}{\mu_{sF}}}}{{4{\pi^{2}}}}\frac{{{k_{sF}}}}{{{k_{aA}}}}$
$\displaystyle\times\sum\limits_{M_{F}M_{s}M_{A}M_{a}}\Big{|}M_{{M_{F}}{M_{s}};{M_{A}}{M_{a}}}^{DWZR(prior)}({k_{0}}{{\bf{\hat{k}}}_{sF}},{{\bf{k}}_{aA}}){\Big{|}^{2}}$
(43)

is the DCS of the reaction $\,a+A\to s+F^{*}\,$ populating the resonance state
$\,F^{*}$.

The renormalization factor presented in Fig. 5 is the ratio of the DWBA
differential cross section given by Eq. (43) to the PWA differential cross
section.

I presented a set of equations that can be used to analyze the THM reactions.
Now we are in a position to discuss two THM reactions, which provide an
indirect method to determine the astrophysical factor for the ${}^{12}{\rm
C}+{}^{12}{\rm C}$ fusion at the astrophysically relevant energies.

##  IV  THM reactions induced by collision $\boldsymbol{{}^{14}{\rm
N}+{}^{12}{\rm C}}$

The reactions

$\displaystyle{}^{14}{\rm N}+{}^{12}{\rm C}\to d+{}^{24}{\rm
Mg}^{*}\to\alpha_{0}(\alpha_{1})+{}^{20}{\rm Ne}({}^{20}{\rm Ne}^{*},$ (44)
$\displaystyle{}^{14}{\rm N}+{}^{12}{\rm C}\to d+{}^{24}{\rm Mg}^{*}\to
p_{0}(p_{1})+{}^{23}{\rm Na}({}^{23}{\rm Na}^{*})$ (45)

hade been used, see Tumino , to determine the astrophysical factor for the
carbon-carbon fusion at energies $E\geq 0.8$ MeV. For this reaction
$a={}^{14}{\rm N}\,\,s=d,\,\,x=A={}^{12}{\rm C},\,\,F={}^{24}{\rm
Mg},\,b=p,\,\alpha,\,\,B={}^{23}{\rm Na},\,{}^{20}{\rm Ne}.\,$ Also the
orbital angular momentum of the bound state ${}^{14}{\rm N}=(d\,{}^{12}{\rm
C})$ is $l_{d\,{}^{12}{\rm C}}=0$.

###  IV.1 $\boldsymbol{E_{{}^{14}{\rm N}}=30}$ MeV

In Tumino  the energy of the ${}^{14}{\rm N}$ beam was $E_{{}^{14}{\rm N}}=30$
MeV corresponding to $E_{{}^{14}{\rm 14}\,{}^{12}{\rm C}}\approx 14$ MeV. Let
us analyze the kinematic conditions of the THM reaction at this incident beam
energy.
1\. $E_{{}^{14}{\rm N}\,{}^{12}{\rm C}}\approx 14$ MeV is higher than the
Coulomb barrier of $\approx 10$ MeV. This condition is necessary to avoid
suppression of the THM DCS due to the Coulomb barrier in the entry channel
${}^{14}{\rm N}+{}^{12}{\rm C}$. However, the relative energy
$E_{d\,{}^{24}{\rm Mg}}$ remains well below the Coulomb barrier, especially at
higher $E\equiv E_{{}^{12}{\rm C}\,{}^{12}{\rm C}}$. Large Coulomb parameter
in the initial state of the THM reaction, $4.4$, and sub-Coulomb energies in
the final state makes the PWA used in Tumino  invalid.
2\. The second necessary condition of the THM is to utilize kinematics the
most appropriate to measure the THM double DCS in the energy interval $0.8\leq
E\leq 2.66$ MeV (at energy $2.66$ MeV the THM $S$-factor is normalized to the
directly measured astrophysical fsctor) remaining close to the forward peak
over the scattering angle $\theta_{sF}$ of the DCS $\frac{{{\rm
d}\sigma}^{DWZR(prior)}}{{{\rm d}{\Omega_{{{\rm{\bf k}}_{sF}}}}}}$. This peak
corresponds to $p_{sx}<\kappa_{sx}$. To elaborate on the relationship between
$p_{sx}$ and the THM double DCS, let us consider a simple PWA, which can be
obtained by replacing in Eq. (26) the distorted waves by the plane waves. Then
it is straightforward to see that the amplitude ${\cal M}^{DWZR(prior)}$ is
proportional to $\,\phi_{sx}(p_{sx})$. Hence the THM double DCS in the PWA is
proportional to $\phi_{sx}^{2}(p_{sx})$. Moreover, if the THM reaction is
above the Coulomb barrier in the initial and final states, the $p_{sx}$
momentum distribution extracted from the DWBA DCS should be close to the
momentum distribution given by $\phi_{sx}^{2}(p_{sx})$ for
$p_{sx}<\kappa_{sx}.$

Fig. 3 shows the momentum distribution of $\phi_{d\,{}^{12}{\rm
C}}^{2}(p_{d\,{}^{12}{\rm C}})$, which, as expected, has a peak at
$p_{d\,{}^{12}{\rm C}}=0$. This peak is called a quasi-free (QF) one because
$p_{d\,{}^{12}{\rm C}}=0$ means that particles $s$ and $x$, being in the bound
state ${}^{14}{\rm N}=(d\,{}^{12}{\rm C})$, are moving like quasi-free (non-
interacting) particles with the relative zero velocity. The shape of the
$s$-wave peak depends on
$\,\kappa_{sx}=\sqrt{2\,\mu_{sx}\,\varepsilon_{sx}},\,$
$\,\varepsilon_{sx}=m_{s}+m_{x}-m_{a}$ is the binding energy of the bound
state $a=(sx)$. For the case under consideration the binding energy
$\,\varepsilon_{d\,{}^{12}{\rm C}}=10.7\,$ MeV is, hence,
$\kappa_{d\,{}^{12}{\rm C}}=1.7$ fm-1 are large and the maximum of the
momentum distribution of $\phi_{d\,{}^{12}{\rm C}}^{2}(p_{d\,{}^{12}{\rm C}})$
occurs at $p_{d\,{}^{12}{\rm C}}<<\kappa_{d\,{}^{12}{\rm C}}$, that is, in the
vicinity of $p_{d\,{}^{12}{\rm C}}=0$. For further analysis I assume that the
QF kinematics is also constrained by the condition $p_{d\,{}^{12}{\rm
C}}<<\kappa_{d\,{}^{12}{\rm C}}$. From the uncertainty principle follows that
for $p_{sx}<<\kappa_{sx}$ $\;r_{sx}>>\kappa_{sx}^{-1}$, that is, a vicinity of
the QF peak corresponds to larger distances between $d$ and ${}^{12}{\rm C}$,
and we can treat $d$ as a spectator with minimized impact on the ${}^{12}{\rm
C}-{}^{12}{\rm C}$ interaction.

Figure 3: Momentum distribution for the $(d\,{}^{12}{\rm C})$ bound state.

In Table 1 are presented $E_{d\,{}^{24}{\rm Mg}}$ and $p_{d\,{}^{12}{\rm C}}$
as functions of the $E$ at $\theta=0$. We conclude that to cover the energy
interval $0.8\leq E\leq 2.7\,{\rm MeV}$ at fixed $E_{{}^{14}{\rm
N}\,{}^{12}{\rm C}}$ one needs to vary $p_{d\,{}^{12}{\rm C}}$ in the interval
$0\leq p_{d\,{}^{12}{\rm C}}\leq 0.21\,$ fm-1. This interval is within the
vicinity of the QF peak and at $p_{d\,{}^{12}{\rm C}}=0.21\,$ the Fourier
tansform $\,\phi_{d\,{}^{12}{\rm C}}^{2}(p_{d\,{}^{12}{\rm C}})$ drops only by
a factor of $1.5$ from its peak value at $p_{d\,{}^{12}{\rm C}}=0$, see Fig.
3. Another essential observation follows from comparing the columns $4$ and
$5$. A substantial difference off-shell momentum $p_{{}^{12}{\rm
C}\,{}^{12}{\rm C}}\,$ and on-the-energy shell momentum $k_{{}^{12}{\rm
C}\,{}^{12}{\rm C}}=\sqrt{2\,\mu_{{}^{12}{\rm C}\,{}^{12}{\rm C}}\,E}$
underscores that off-the-energy shell effects in the THM reaction (factor
${\cal W}_{l_{xA}}(E,R_{ch})$, see Eq. (27) ) are very significant and on-the-
energy shell approximation for the THM double DCS is invalid.

Table 1: Kinematic energy-momentum calculations for $E_{{}^{14}{\rm N}}=30\,$ MeV. $E_{d\,{}^{24}{\rm Mg}}$ and $p_{d\,{}^{12}{\rm C}}$ as functions of the ${}^{12}{\rm C}-{}^{12}{\rm C}$ relative energy $E$. Equation (9) is used to calculate $E_{d\,{}^{24}{\rm Mg}}$. $p_{d\,{}^{12}{\rm C}}$ and $p_{{}^{12}{\rm C}\,{}^{12}{\rm C}}$ are calculated using Eq. (16) assuming that $\theta=\arccos({\rm{\bf{\hat{k}}}}_{d}\cdot{\rm{\bf{\hat{k}}}}_{{}^{14}{\rm N}})=0$. $k_{{}^{12}{\rm C}\,{}^{12}{\rm C}}$ is on-the-energy shell ${}^{12}{\rm C}-{}^{12}{\rm C}$ momentum. $E$ | $E_{d\,{}^{24}{\rm Mg}}\,$ | $p_{d\,{}^{12}{\rm C}}\,$ | $p_{{}^{12}{\rm C}\,{}^{12}{\rm C}}\,$ | $k_{{}^{12}{\rm C}\,{}^{12}{\rm C}}\,$
---|---|---|---|---
(MeV) | (MeV) | (fm-1) | (fm-1) | (fm-1)
0.8 | 2.93 | 0.211537 | 1.82631 | 0.479292
0.9 | 2.83 | 0.202781 | 1.83069 | 0.508366
1. | 2.73 | 0.193868 | 1.83515 | 0.535864
1.1 | 2.63 | 0.18479 | 1.83969 | 0.562019
1.2 | 2.53 | 0.175539 | 1.84431 | 0.58701
1.3 | 2.43 | 0.166102 | 1.84903 | 0.61098
1.4 | 2.33 | 0.156469 | 1.85385 | 0.634043
1.5 | 2.23 | 0.146627 | 1.85877 | 0.656297
1.6 | 2.13 | 0.136562 | 1.86381 | 0.677821
1.7 | 2.03 | 0.126258 | 1.86896 | 0.698682
1.8 | 1.93 | 0.115696 | 1.87424 | 0.718938
1.9 | 1.83 | 0.104858 | 1.87966 | 0.738638
2. | 1.73 | 0.0937186 | 1.88523 | 0.757827
2.1 | 1.63 | 0.0822526 | 1.89096 | 0.776541
2.2 | 1.53 | 0.0704291 | 1.89688 | 0.794815
2.3 | 1.43 | 0.0582125 | 1.90298 | 0.812679
2.4 | 1.33 | 0.0455608 | 1.90931 | 0.830158
2.5 | 1.23 | 0.0324238 | 1.91588 | 0.847276
2.6 | 1.13 | 0.0187409 | 1.92272 | 0.864056
2.7 | 1.03 | 0.00443808 | 1.92987 | 0.880515

$3$D plot in Fig. 4 shows the momentum $p_{p_{d\,{}^{12}{\rm C}}}$ as a
function of $E$ and $\theta$. It allows one to select optimal kinematic
conditions at each energy $E$: smaller angles $\theta$ (the angle between
${}{\rm{\bf k}}_{{}^{14}{\rm N}}$ and ${\rm{\bf k}}_{d}$) provide smaller
$p_{d\,{}^{12}{\rm C}}$, that is, higher THM double DCS. At $p_{d\,{}^{12}{\rm
C}}=0.3\,$ fm-1 $\,\phi_{d\,{}^{12}{\rm C}}^{2}(p_{d\,{}^{12}{\rm C}})$ drops
by about a factor of two compared to its peak value.

Figure 4: $p_{p_{d\,{}^{12}{\rm C}}}$ as a function of $E$ and $\theta$ for
$E_{{}^{14}{\rm N}}=30\,$ MeV.

Thus a comprehensive analysis show that the beam energy $\,E_{{}^{14}{\rm
N}}=30\,$ MeV could provide an optimal kinematics to cover the astrophysically
relevant energies $0.8\leq E\leq 2$ MeV for low $\,p_{d\,{}^{12}{\rm C}}$.
However, a very strong effect of the Coulomb-nuclear distortions neglected in
Tumino  completely changes both the absolute value of DCS, see Fig. 5, and the
deuteron angular distributions muk2019 ; muk2020 . These distortions is the
main obstacle to use the THM reaction at $\,E_{{}^{14}{\rm N}}=30\,$ MeV. Fig.
5 shows the renormalization factors of the modified THM astrophysical factors
at different beam energies. We see that at $E_{{}^{14}{\rm N}}=30$ MeV at low
$E$ the renormalization factor is about $0.001$. However, at the beam energy
of $35$ MeV at low $E$, the renormalization factor is about two orders of
magnitude larger than at $30$ MeV.

Figure 5: Renormalization factor of the generalized THM astrophysical factor
caused by Coulomb-nuclear distortions for three different $E_{{}^{14}{\rm N}}$
energies: $30,\,33\,$ and $35\,$ MeV. There are two lines for each energy: the
low line is the renormalization factor caused by the pure Coulomb distortions
and the upper line is caused by the Coulomb-nuclear distortions.

###  IV.2 $\boldsymbol{E_{{}^{14}{\rm N}}=35}$ MeV

Since the $30\,$ MeV beam is too low, I will repeat all the previous
calculations at $E_{{}^{14}{\rm N}}=35\,$ MeV.

Table 2: Kinematic energy-momentum calculations for $E_{{}^{14}{\rm N}}=35\,$ MeV. Notations are the same as in Table 1.  $E$ | $E_{d\,{}^{24}{\rm Mg}}\,$ | $p_{d\,{}^{12}{\rm C}}\,$ | $p_{{}^{12}{\rm C}\,{}^{12}{\rm C}}\,$ | $k_{{}^{12}{\rm C}\,{}^{12}{\rm C}}\,$
---|---|---|---|---
(MeV) | (MeV) | (fm-1) | (fm-1) | (fm-1)
0.8 | 5.08 | 0.352861 | 1.90051 | 0.479292
0.9 | 4.98 | 0.346213 | 1.90381 | 0.508366
1. | 4.88 | 0.339497 | 1.90714 | 0.535864
1.1 | 4.78 | 0.332712 | 1.9105 | 0.562019
1.2 | 4.68 | 0.325857 | 1.9139 | 0.58701
1.3 | 4.58 | 0.318927 | 1.91734 | 0.61098
1.4 | 4.48 | 0.311921 | 1.92082 | 0.634043
1.5 | 4.38 | 0.304837 | 1.92433 | 0.656297
1.6 | 4.28 | 0.297671 | 1.92789 | 0.677821
1.7 | 4.18 | 0.290422 | 1.93148 | 0.698682
1.8 | 4.08 | 0.283084 | 1.93512 | 0.718938
1.9 | 3.98 | 0.275657 | 1.93881 | 0.738638
2. | 3.88 | 0.268135 | 1.94254 | 0.757827
2.1 | 3.78 | 0.260516 | 1.94632 | 0.776541
2.2 | 3.68 | 0.252796 | 1.95015 | 0.794815
2.3 | 3.58 | 0.24497 | 1.95404 | 0.812679
2.4 | 3.48 | 0.237033 | 1.95798 | 0.830158
2.5 | 3.38 | 0.228982 | 1.96197 | 0.847276
2.6 | 3.28 | 0.220811 | 1.96603 | 0.864056
2.7 | 3.18 | 0.212514 | 1.97014 | 0.880515

The second column of Table 2 shows that at $\,E_{{}^{14}{\rm N}}=35\,$ MeV all
$E_{d\,{}^{24}{\rm 24}}$ are higher than the Coulomb barrier (only on the
higher end of $E\;$ energy $E_{d\,{}^{24}{\rm 24}}$ approaches the Coulomb
barrier of $3\,$ MeV). Taking into account that the initial energy
$E_{{}^{14}{\rm N}\,{}^{12}{\rm C}}=16.15\,$ MeV is higher than the initial
Coulomb barrier $10\,$ MeV we conclude that the angular distribution of the
deuterons is forward peaked. Hence the experimental momentum distribution of
the deuterons extracted from the THM double DCS for all $E$ and
$p_{d\,{}^{12}{\rm C}}$ from Table 2 should be similar to the momentum
distribution in Fig. 3.

$3$D plot in Fig. 6 shows the momentum $p_{d\,{}^{12}{\rm C}}$ as a function
of $E$ and $\theta$ for $E_{{}^{14}{\rm N}}=35\,$ MeV.

Let us select two important resonance energies: $E=0.9\,$ and $1.5\,$ MeV.
From Tables 1 and 2 follows that for these resonances: for $\,E_{{}^{14}{\rm
N}}=30\,$ MeV $p_{d\,{}^{12}{\rm C}}=0.20$ and $0.15$ fm-1, respectively; for
$\,E_{{}^{14}{\rm N}}=35\,$ MeV $\,p_{d\,{}^{12}{\rm C}}=0.35\,$ and $0.30$
fm-1, respectively. From Fig. 3 we find that $\phi_{d\,{}^{12}{\rm
C}}^{2}(0.35\,{\rm fm}^{-1})/\phi_{d\,{}^{12}{\rm C}}^{2}(0.2\,{\rm
fm}^{-1})=0.43$ and $\phi_{d\,{}^{12}{\rm C}}^{2}(0.30\,{\rm
fm}^{-1})/\phi_{d\,{}^{12}{\rm C}}^{2}(0.15\,{\rm fm}^{-1})=0.53$,
respectively. These ratios show the decrease of the THM double DCS at two
critical resonance energies when the energy of the incident beam of
${}^{14}{\rm N}$ increases from $30$ MeV to $35$ MeV. Meantime, from Fig. 5
follows that the ratio of the renormalization factors $R_{35\,{\rm
MeV}}(0.9\,{\rm MeV})R_{30\,{\rm MeV}}(0.9\,{\rm MeB})=333.3$ and $R_{35\,{\rm
MeV}}(1.5\,{\rm MeV})R_{30\,{\rm MeV}}(1.5\,{\rm MeB})=115.6$. Thus the drop
of $\phi_{d\,{}^{12}{\rm C}}^{2}(p_{d\,{}^{12}{\rm C}})$ with increase of the
energy $\,E_{{}^{14}{\rm N}}\,$ is well compensated by the increase of the
renormalization factor: the total gain for the energy $\,E_{{}^{14}{\rm N}}\,$
increase from $30\,$ to $35\,$ MeV is $\,110\,$ for $\,E=0.9\,$ MeV and
$\,61\,$ for $\,E=1.5\,$ MeV. Thus an increase of the beam energy from
$\,30\,$ MeV Tumino  to $\,35\,$ MeV will increase the THM double DCS because
the outgoing deuterons become above the Coulomb barrier. Besides, the angular
distribution of the deuterons will be forward peaked and the THM experiment
can be repeated at $\,E_{{}^{14}{\rm N}}=35\,$ MeV avoiding the problems
appeared for $E_{{}^{14}{\rm N}}=30\,$ MeV.

Figure 6: $p_{p_{d\,{}^{12}{\rm C}}}$ as a function of $E$ and $\theta$ for
$E_{{}^{14}{\rm N}}=35\,$ MeV.

##  V  THM reaction induced by collision $\boldsymbol{{}^{13}{\rm
C}+{}^{12}{\rm C}}$

Now I consider another THM reactions suitable for determination of the
$S$-factors for the carbon-carbon fusion:

$\displaystyle{}^{13}{\rm C}+{}^{12}{\rm C}\to n+{}^{24}{\rm
Mg}^{*}\to\alpha_{0}(\alpha_{1})+{}^{20}{\rm Ne}({}^{20}{\rm Ne}^{*}),$ (46)
$\displaystyle{}^{13}{\rm C}+{}^{12}{\rm C}\to n+{}^{24}{\rm Mg}^{*}\to
p_{0}(p_{1})+{}^{23}{\rm Na}({}^{23}{\rm Na}^{*}).$ (47)

In the previous section notations we need to replace $a$ and $s$ with
$a={}^{13}{\rm C}$ and $s=n$; also now the orbital angular momentum of the
bound state ${}^{13}{\rm C}=(n\,{}^{12}{\rm C})\,$ $\,l_{n\,{}^{12}{\rm C}}=1$
and the bound-state wave number $\kappa_{n\,{}^{12}{\rm C}}=0.47$. Thus these
will be the first THM reactions in which the Trojan horse particle is the
$\,p\,$-wave bound state. Since the spectator in reactions (46) and (47) is
neutron, there are no Coulomb interactions in the intermediate and final state
of the THM reactions, which suppressed the THM double DCS in the previous
section with the deuteron as a spectator.

The momentum distribution of $\,\phi_{n\,{}^{12}{\rm C}}^{2}(p_{n\,{}^{12}{\rm
C}})\,$ shown in Fig. 7, due to $\,l_{n\,{}^{12}{\rm C}}=1,\,$ is peaked at
$\,p_{n\,{}^{12}{\rm C}}=0.4\,$ fm${}^{-1}\,$ rather then at
$\,p_{d\,{}^{12}{\rm C}}=0\,$ for the $s$-wave bound state.

Figure 7: Momentum distribution for the $(n\,{}^{12}{\rm C})$ bound state.

In Table 3 are presented $E_{n\,{}^{24}{\rm Mg}}$ and $p_{n\,{}^{12}{\rm C}}$
as functions of $E$ at $\theta=0$ at $E_{{}^{13}{\rm C}}=29\,$ MeV. We
conclude that to cover the energy interval $0.8\leq E\leq 2.7\,{\rm MeV}$ at
fixed $E_{{}^{13}{\rm C}}=29\,$ MeV one needs to vary $p_{n\,{}^{12}{\rm C}}$
in the interval $0.38\leq p_{n\,{}^{12}{\rm C}}\leq 0.46\,$ fm-1. This
interval is lower than $\kappa_{n\,{}^{12}{\rm C}}=0.47$ fm${}^{-1}.\,$ Within
this interval the Fourier transform $\,\phi_{d\,{}^{12}{\rm
C}}^{2}(p_{n\,{}^{12}{\rm C}})$ changes very little, see Fig. 7.

Also, as it was the case for ${}^{14}{\rm N}+{}^{12}{\rm C}$, from comparing
the columns $4$ and $5$ we see that the off-shell effects are very
significant.

Table 3: Kinematic energy-momentum calculations for $E_{{}^{13}{\rm C}}=29\,$ MeV. $E_{n\,{}^{24}{\rm Mg}}$ and $p_{n\,{}^{12}{\rm C}}$ as functions of the ${}^{12}{\rm C}-{}^{12}{\rm C}$ relative energy $E$. Equation (9) is used to calculate $E_{n\,{}^{24}{\rm Mg}}$. $p_{n\,{}^{12}{\rm C}}$ and $p_{{}^{12}{\rm C}\,{}^{12}{\rm C}}$ are calculated using Eq. (16) assuming that $\theta=\arccos({\rm{\bf{\hat{k}}}}_{n}\cdot{\rm{\bf{\hat{k}}}}_{{}^{13}{\rm C}})=0$. $k_{{}^{12}{\rm C}\,{}^{12}{\rm C}}$ is on-the-energy shell ${}^{12}{\rm C}-{}^{12}{\rm C}$ momentum. $E$ | $E_{n\,{}^{24}{\rm Mg}}\,$ | $p_{n\,{}^{12}{\rm C}}\,$ | $p_{{}^{12}{\rm C}\,{}^{12}{\rm C}}\,$ | $k_{{}^{12}{\rm C}\,{}^{12}{\rm C}}\,$
---|---|---|---|---
(MeV) | (MeV) | (fm-1) | (fm-1) | (fm-1)
0.8 | 8.15 | 0.457724 | 1.73267 | 0.479292
0.9 | 8.05 | 0.453942 | 1.73454 | 0.508366
1. | 7.95 | 0.450137 | 1.73642 | 0.535864
1.1 | 7.85 | 0.446308 | 1.73832 | 0.562019
1.2 | 7.75 | 0.442455 | 1.74022 | 0.58701
1.3 | 7.65 | 0.438577 | 1.74214 | 0.61098
1.4 | 7.55 | 0.434673 | 1.74407 | 0.634043
1.5 | 7.45 | 0.430743 | 1.74602 | 0.656297
1.6 | 7.35 | 0.426787 | 1.74797 | 0.677821
1.7 | 7.25 | 0.422804 | 1.74994 | 0.698682
1.8 | 7.15 | 0.418794 | 1.75193 | 0.718938
1.9 | 7.05 | 0.414755 | 1.75393 | 0.738638
2. | 6.95 | 0.410687 | 1.75594 | 0.757827
2.1 | 6.85 | 0.40659 | 1.75797 | 0.776541
2.2 | 6.75 | 0.402463 | 1.76001 | 0.794815
2.3 | 6.65 | 0.398305 | 1.76207 | 0.812679
2.4 | 6.55 | 0.394116 | 1.76414 | 0.830158
2.5 | 6.45 | 0.389895 | 1.76623 | 0.847276
2.6 | 6.35 | 0.385641 | 1.76833 | 0.864056
2.7 | 6.25 | 0.381354 | 1.77045 | 0.880515

$3$D plot in Fig. 8 shows that for small neutron scattering angles
$\,p_{n\,{}^{12}{\rm C}}<\kappa_{n\,{}^{12}{\rm C}}.\,$

Figure 8: $p_{n\,{}^{12}{\rm C}}$ as a function of $E$ and $\theta$ for
$E_{{}^{13}{\rm C}}=29\,$ MeV.

##  VI Summary

The update of the current status of the modified astrophysical
$S^{*}(E)$-factors for the carbon-carbon fusion is given. The latest two
direct measurements in STELLA  and Tan  show disagreement at energies $E\geq
2.1\,$ MeV. It is not feasible in the near future that direct measurements can
reach the Gamow window $1.2\leq E\leq 1.8$ MeV. Nowadays, the only way to
reach these energies is to use the indirect THM. In the THM only the energy
dependence of the astrophysical factor is extracted. To get their absolute
value one needs to normalize the THM data to direct ones at higher energies
where reliable direct measurements are available. For the carbon-carbon fusion
the normalization energy was $E\approx 2.6\,$ MeV. Thus the THM measurements
should cover quite a broad energy interval, $0.8\leq E\leq 2.7\,$ MeV.
Reconciliation of direct measurements at $E=2.1\,$ MeV would help THM
experiments: the normalization to direct measurements can be performed at
$2.1\,$ MeV.

Some important THM equations are presented, which are needed to apply the THM.
To determine the $S^{*}$-factors for the carbon-carbon fusion for the
astrophysically relevant energy interval $0.8\leq E\leq 2.0\,$ MeV different
THM reactions have been analyzed. Among them are two THM reactions induced by
the collision $\,{}^{14}{\rm N}+{}^{12}{\rm C}\,$ at $E_{{}^{14}{\rm N}}=30\,$
MeV Tumino . and $E_{{}^{14}{\rm N}}=35\,$ MeV. It is shown that the higher
energy beam allows one to avoid difficulties in the THM experiment Tumino  at
$E_{{}^{14}{\rm N}}=30\,$ MeV and can be used to extract the $S^{*}$-factors
for the carbon-carbon fusion.

Also, the kinematics of the THM reaction induced by the ${}^{13}{\rm
C}+{}^{12}{\rm C}$ collision is analyzed and is shown that this reaction can
be used to determine the $S^{*}$-factors for the carbon-carbon fusion.

###### Acknowledgements.

A. M. M. acknowledges a support from the U.S. DOE Grant No. DE-FG02-93ER40773,
and the NNSA Grant No. DENA000384.

## References

  * (1) K. Mori et al., MNRAS: Letters 482, L70 (2019); arXiv: 1810.01025 (2018).
  * (2) H. W. Becker et al., Z. Phys. A 303, 305 (1981).
  * (3) K. U. Kettner, H. Lorenz-Wirzba, C. Rolfs, Z.Phys. A, 298, 65 (1980).
  * (4) E. F. Aguilera et al., Phys.Rev. C 73, 064601 (2006).
  * (5) T. Spillane et al., Phys.Rev.Lett. 98, 122501 (2007).
  * (6) J. R. Patterson, H. Winkler, and C. S. Zaidins, Astrophys. J. 157, 367 (1969).
  * (7) A. Tumino et al., Nature, 557, 687 (2018).
  * (8) C. L. Jiang et al., Phys. Rev. C 75, 015803 (2007).
  * (9) H. Esbensen, X. Tang, and C. L. Jiang, Phys. Rev. C 84, 064613 (2011).
  * (10) A. Diaz-Torres and M. Wiescher, Phys.Rev. C 97, 055802 (2018).
  * (11) A. Bonasera, J. Natowitz, Phys. Rv. C 102, 061602 (2020).
  * (12) K. Godbey, C. Simenel, and A. S. Umar, Phys. Rev. C 100, 024619 (2019).
  * (13) P. W. Wen et al., Phys. Rev. C 103, 054601 (2021).
  * (14) L. R. Gasques et al., Phys. Rev. C 76, 035802 (2007).
  * (15) Le Hoang Chien et al., Phys. Rev. C, 98, 064604 (2018).
  * (16) Dao T. Khoa et al., Nucl. Sci. Tech., 29, 182 (2018).
  * (17) M. Notani et al., Phys. Rev. C 85, 014607 (2012).
  * (18) C. L. Jiang et al., Phys. Rev. Lett. 110, 072701 (2013).
  * (19) M. Assuncao, P. Descouvemont, Phys. lett. B 723, 355 (2013).
  * (20) N. T. Zhang et al., Phys. Lett. B 801, 135170 (2020).
  * (21) K. Nagatani, Phys. Lett. B 35, 286 (1971).
  * (22) A. M. Mukhamedzhanov, D. Y. Pang and A. S. Kadyrov, Phys. Rev C 99, 064618 (2019).
  * (23) A. M. Mukhamedzhanov, A. S. Kadyrov and D. Y. Pang, Eur. Phys. J. A 56, 233 (2020).
  * (24) C. Beck, A. M. Mukhamedzhanov and X. Tang, Eur. Phys. J. A 56, 97 (2020), Letter to the Editor.
  * (25) G. Fruet et al., Phys. Rev. Lett. 124, 192701 (2020).
  * (26) W. P. Tan et al., Phys. Rev. Lett. 124, 192702 (2020).
  * (27) Y. Taniguchi, M. Kimura, Phys. Lett. B 823, 136790 (2021).
  * (28) A. M. Mukhmedzhanov and A. S. Kadyrov, Few-Body Syst. 60, 27 (2019).
  * (29) E. I. Dolinski, P. O. Dzhamalov and A. M. Mukhamedzhanov, Nucl. Phys. A 202, 97 (1973).


* Spooren et al. (2017) Jan Spooren, Davy Preuveneers, and Wouter Joosen. 2017\.  Leveraging Battery Usage from Mobile Devices for Active Authentication.  _Mobile Information Systems_ 2017 (2017), 1367064:1–1367064:14.  https://doi.org/10.1155/2017/1367064
  * Starov et al. (2019) Oleksii Starov, Pierre Laperdrix, Alexandros Kapravelos, and Nick Nikiforakis. 2019.  Unnecessarily Identifiable: Quantifying the Fingerprintability of Browser Extensions Due to Bloat. In _The World Wide Web Conference_ _(WWW ’19)_. ACM, New York, NY, USA, 3244–3250.  https://doi.org/10.1145/3308558.3313458
  * Starov and Nikiforakis (2017) Oleksii Starov and Nick Nikiforakis. 2017.  XHOUND: Quantifying the Fingerprintability of Browser Extensions. In _38th IEEE Symposium on Security and Privacy (S &P 2017)_. San Jose, United States.
  * Takei et al. (2015) N. Takei, T. Saito, K. Takasu, and T. Yamada. 2015\.  Web Browser Fingerprinting Using Only Cascading Style Sheets. In _2015 10th International Conference on Broadband and Wireless Computing, Communication and Applications (BWCCA)_. 57–63.  https://doi.org/10.1109/BWCCA.2015.105
  * Torres et al. (2015) Christof Torres, Hugo Jonker, and Sjouke Mauw. 2015\.  FP-Block: usable web privacy by controlling browser fingerprinting. In _Proceedings of the 20th European Symposium on Research in Computer Security (ESORICS 2015)_.
  * Trickel et al. (2019) Erik Trickel, Oleksii Starov, Alexandros Kapravelos, Nick Nikiforakis, and Adam Doupé. 2019\.  Everyone is Different: Client-side Diversification for Defending Against Extension Fingerprinting. In _28th USENIX Security Symposium (USENIX Security 19)_. USENIX Association, Santa Clara, CA, 1679–1696.  https://www.usenix.org/conference/usenixsecurity19/presentation/trickel
  * Unger et al. (2013) T. Unger, M. Mulazzani, D. Frühwirt, M. Huber, S. Schrittwieser, and E. Weippl. 2013\.  SHPF: Enhancing HTTP(S) Session Security with Browser Fingerprinting. In _2013 International Conference on Availability, Reliability and Security_. 255–261.  https://doi.org/10.1109/ARES.2013.33
  * Vastel et al. (2018a) Antoine Vastel, Pierre Laperdrix, Walter Rudametkin, and Romain Rouvoy. 2018a.  FP-Scanner: The Privacy Implications of Browser Fingerprint Inconsistencies. In _27th USENIX Security Symposium (USENIX Security 18)_. USENIX Association, Baltimore, MD, 135–150.  https://www.usenix.org/conference/usenixsecurity18/presentation/vastel
  * Vastel et al. (2018b) Antoine Vastel, Pierre Laperdrix, Walter Rudametkin, and Romain Rouvoy. 2018b.  FP-STALKER: Tracking Browser Fingerprint Evolutions. In _39th IEEE Symposium on Security and Privacy (S &P 2018)_. San Fransisco, United States.
  * Wu et al. (2019) Shujiang Wu, Song Li, Yinzhi Cao, and Ningfei Wang. 2019\.  Rendered Private: Making GLSL Execution Uniform to Prevent WebGL-based Browser Fingerprinting. In _28th USENIX Security Symposium (USENIX Security 19)_. USENIX Association, Santa Clara, CA, 1645–1660.  https://www.usenix.org/conference/usenixsecurity19/presentation/wu
  * Wu et al. (2016) W. Wu, J. Wu, Y. Wang, Z. Ling, and M. Yang. 2016.  Efficient Fingerprinting-Based Android Device Identification With Zero-Permission Identifiers.  _IEEE Access_ 4 (2016), 8073–8083.  https://doi.org/10.1109/ACCESS.2016.2626395
  * Yu et al. (2016) Zhonghao Yu, Sam Macbeth, Konark Modi, and Josep M. Pujol. 2016\.  Tracking the Trackers. In _Proceedings of the 25th International Conference on World Wide Web_ _(WWW ’16)_. International World Wide Web Conferences Steering Committee, Republic and Canton of Geneva, Switzerland, 121–132.  https://doi.org/10.1145/2872427.2883028