# High precision proton angular distribution measurements of 12C(p,p’) for
determination of the $E0$ decay branching ratio of the Hoyle state

K.J. Cook<EMAIL_ADDRESS>A. Chevis  Facility for Rare Isotope Beams,
Michigan State University, East Lansing, MI 48824, USA  Department of Physics
and Astronomy, Michigan State University, East Lansing, MI, 48824, USA  T.K.
Eriksen  Present address: Department of Physics, University of Oslo, N-0316
Oslo, Norway  E.C. Simpson  T. Kibédi  L.T. Bezzina  A.C. Berriman  J. Buete
I.P. Carter  M. Dasgupta  D.J. Hinde  D.Y. Jeung  P. McGlynn  Department of
Nuclear Physics, Research School of Physics, The Australian National
University, Canberra, ACT 2601, Australia  S. Parker-Steele  Present address:
King Solomon Academy, London, NW16RX, UK  Department of Nuclear Physics,
Research School of Physics, The Australian National University, Canberra, ACT
2601, Australia  Department of Physics, University of Surrey, Guildford GU2
7XH, United Kingdom  B.M.A. Swinton-Bland  Department of Nuclear Physics,
Research School of Physics, The Australian National University, Canberra, ACT
2601, Australia  T. Tanaka  Department of Nuclear Physics, Research School of
Physics, The Australian National University, Canberra, ACT 2601, Australia  W.
Wojtaczka  Present address: Institute for Nuclear and Radiation Physics, KU
Leuven, B-3001 Leuven, Belgium  Department of Nuclear Physics, Research School
of Physics, The Australian National University, Canberra, ACT 2601, Australia
Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom

###### Abstract

Background In stellar environments, carbon is produced exclusively via the
$3\alpha$ process, where three $\alpha$ particles fuse to form 12C in the
excited Hoyle state, which can then decay to the ground state. The rate of
carbon production in stars depends on the radiative width of the Hoyle state.
While not directly measurable, the radiative width can be deduced by combining
three separately measured quantities, one of which is the $E0$ decay branching
ratio. The $E0$ branching ratio can be measured by exciting the Hoyle state in
the 12C$(p,p^{\prime})$ reaction and measuring the pair decay of both the
Hoyle state and the first $2^{+}$ state of 12C.

Purpose To reduce the uncertainties in the carbon production rate in the
universe by measuring a set of proton angular distributions for the population
of the Hoyle state ($0^{+}_{2}$) and $2^{+}_{1}$ state in 12C in
12C$(p,p^{\prime})$ reactions between 10.20 and 10.70 MeV, used in the
determination of the $E0$ branching ratio of the Hoyle state.

Method Proton angular distributions populating the ground, first $2^{+}$, and
the Hoyle states in 12C were measured in 12C(p,p’) reactions with a silicon
detector array covering $22^{\circ}<\theta<158^{\circ}$ in 14 small energy
steps between 10.20 and 10.70 MeV with a thin ($60\ \mu$g/cm2) natC target.

Results Total cross-sections for each state were extracted and the population
ratio between the $2^{+}_{1}$ and Hoyle state determined at each energy step.
By appropriately averaging these cross-sections and taking their ratio, the
equivalent population ratio can be extracted applicable for any thick 12C
target that may be used in pair-conversion measurements. This equivalent ratio
agreed with a direct measurement performed with a thick target.

Conclusions We present a general data set of high-precision
12C$(p,p^{\prime})$ cross-sections that make uncertainties resulting from the
population of the $2^{+}_{1}$ and $0^{+}_{2}$ states by proton inelastic
scattering negligible for any future measurements of the $E0$ branching ratio
in 12C. Implications for future measurements are discussed, as well as
possible applications of this data set for investigating cluster structures in
13N.

##  I Introduction

In stars undergoing hydrogen burning, the synthesis of elements heavier than
hydrogen begins with $pp$-chain reactions and the CNO cycle, where four
protons are ultimately converted to one $\alpha$ particle. Formation of any
heavier elements during hydrogen burning is inhibited by the temperature of
the star being too low to induce $\alpha$ fusion as well as the lack of stable
$A=5$ and $A=8$ nuclei. As a result, no heavier elements are formed in stars
in this time. At the end of the hydrogen burning stage, the temperature of the
star increases allowing $\alpha$ fusion. At this point, the “$3\alpha$
process” becomes significant, which allows carbon to be produced in stars that
have an equilibrium concentration of 8Be ($t_{1/2}=10^{-16}$ s)Salpeter
(1952). The small probability of a third $\alpha$ particle to fuse with the
8Be before it decays enables the production of 12C,
$(\alpha+\alpha\rightarrow^{8}\\!\mathrm{Be})+\alpha\rightarrow^{12}\\!\mathrm{C}^{*}$.
The existence of the $0^{+}_{2}$ state at 7.65 MeV above the ground state of
12C is crucial for the $3\alpha$ process. Lying just above the $\alpha$
threshold, it acts as a resonance for $s$-wave $\alpha$ capture at stellar
temperatures. Without this resonance, the cross-section for the $3\alpha$
process would be far too small to produce the observed carbon abundance in the
universe. Hoyle predicted the existence of this state via the abundance ratios
of 16O:12C:4He in the universe, prior to its experimental observation Dunbar
_et al._ (1953); Hoyle (1954); Cook  _et al._ (1957), and it is thus commonly
known as the Hoyle state.

Figure 1: The decay of 12C from the Hoyle state primarily occurs via
sequential $\alpha$ decay, into 8Be$(\rightarrow\alpha+\alpha)+\alpha$. A very
small fraction of $\alpha$ decays proceed via direct $3-\alpha$ decay
($<0.019-0.043\%$)Smith  _et al._ (2017); Dell’Aquila  _et al._ (2017); Rana
_et al._ (2019); Bishop  _et al._ (2020). Stable carbon is produced when 12C
decays electromagnetically ($0.04-0.06\%$ of decays) via a cascade of two $E2$
transitions, or one $E0$ transition Freer and Fynbo (2014); Kibédi  _et al._
(2020). The branching ratios indicated in the figure arise from ref. Kibédi
_et al._ (2020).

The Hoyle state disintegrates back to 8Be$+\alpha$ or $3\alpha$ with
probability $>99.94\%$ Freer and Fynbo (2014); Kibédi  _et al._ (2020). Stable
carbon is only produced when the Hoyle state instead electromagnetically
decays directly to the ground state via an electric monopole ($E0$)
transition, or via the $2^{+}_{1}$ state by two electric quadrupole ($E2$)
transitions. The 12C production rate in the universe is therefore closely
related to the decay properties of the Hoyle state. The different pathways for
reaction outcomes following the $3\alpha$ process are shown in Fig. 1.

The carbon production rate in a star, $r_{3\alpha}$ can be described by the
resonance reaction equation Rolfs and Rodney (1988)

$r_{3\alpha}=4\sqrt{27}\frac{N_{\alpha}^{3}\pi^{3}\hbar^{5}}{M_{\alpha}^{3}k_{B}^{3}T^{3}}\frac{\Gamma_{\alpha}\Gamma_{rad}}{\Gamma}e^{-Q_{3}\alpha/k_{B}T}.$
(1)

The reaction rate depends on stellar properties: the number density of the
$\alpha$ particles $N_{\alpha}$, and the temperature $T$, as well as nuclear
properties: the mass of the $\alpha$ particle $M_{\alpha}$, the total
$\Gamma$, alpha $\Gamma_{\alpha}$, and electromagnetic (radiative)
$\Gamma_{rad}$ decay widths of the Hoyle state, and the Q-value for the three
$\alpha$ breakup of the Hoyle state, $Q_{3\alpha}$. Since the Hoyle state
dominantly decays via $\alpha$ emission, $\Gamma\approx\Gamma_{\alpha}$, and
$r_{3\alpha}$ can be simplified to:

$r_{3\alpha}\propto\frac{\Gamma_{rad}}{T^{3}}\times e^{-Q_{3\alpha}/k_{B}T}.$
(2)

We thus find that the carbon production rate depends linearly on the radiative
width of the Hoyle state, $\Gamma_{rad}$. $\Gamma_{rad}$ is made up of
contributions from the $3.21$ MeV $E2$ and $7.65$ MeV $E0$ transitions. Since
the contributions from electron conversion are negligible, we can write
$\Gamma_{rad}$ as the sum of contributions from photon ($\gamma$) and pair
conversion ($\pi$),
$\Gamma_{rad}=\Gamma^{E2}_{\gamma}+\Gamma^{E2}_{\pi}+\Gamma^{E0}_{\pi}$. Since
the $3\alpha$ process is sequential, and 8Be has a short half-life,
$\Gamma_{rad}$ cannot be directly measured. It is usually deduced from three
independently measured quantities (shown in brackets)

$\Gamma_{rad}=\left(\frac{\Gamma_{rad}}{\Gamma}\right)\times\left(\frac{\Gamma}{\Gamma^{E0}_{\pi}}\right)\times\left(\Gamma^{E0}_{\pi}\right).$
(3)

Precisely determining $\Gamma_{rad}$, and so the rate of carbon production in
the $3\alpha$ process, requires determination of all three quantities. To this
end, there have been years of continuous effort to reduce experimental
uncertainties. Two recent experiments have determined new values for
$\Gamma^{E0}_{\pi}/\Gamma$ ($14\%$ higher than the previous adopted value)
Eriksen  _et al._ (2020) and $\Gamma_{rad}/\Gamma$ ($50\%$ higher than the
previous adopted value) Kibédi  _et al._ (2020). When combined with the
currently adopted $\Gamma_{\pi}^{E0}$ value, these new experiments resulted in
$\Gamma_{rad}=5.1(6)\times 10^{-3}$ eV. This is approximately $34\%$ higher
than the previously adopted value, significantly impacting models of stellar
evolution Woosley and Heger (2021). In contrast, a very recent measurement
using coincident detection of 12C$+p$ nuclei implies a rate of 12C consistent
with the previous values Tsumura  _et al._ (2021), though it is sensitive to
the exact form of the background. Ref. Kibédi  _et al._ (2020) highlighted the
urgent need for new, independent, high resolution measurements to resolve the
discrepancies between different measurements.

In Ref. Eriksen  _et al._ (2020), $\Gamma^{E0}_{\pi}/\Gamma$ was determined
using the 12C$(p,p^{\prime})$ reaction through:

$\frac{\Gamma_{\pi}^{E0}}{\Gamma}=\frac{N_{\pi}^{E0}}{N_{\pi}^{E2}}\times\frac{N_{p}(2_{1}^{+})}{N_{p}(0_{2}^{+})}\times\frac{\epsilon_{\pi}^{E2}}{\epsilon_{\pi}^{E0}}\times\frac{\alpha_{\pi}}{1+\alpha_{\pi}},$
(4)

where $N_{\pi}^{E0,E2}$ are the number of experimentally measured $E0$, $E2$
electron-positron pairs, $N_{p}(0_{2}^{+})$, $N_{p}(2_{1}^{+})$, are the
number of events populating the Hoyle and the $2^{+}_{1}$ states,
$\epsilon_{\pi}^{E0,E2}$ is the pair detection efficiency for each transition,
and $\alpha_{\pi}$ is the theoretical pair conversion coefficient,
$\alpha_{\pi}=\Gamma_{\pi}^{E2}/\Gamma_{\gamma}^{E2}$. As a result, the
determination of $\Gamma^{E0}_{\pi}/\Gamma$ requires the measurement of
inelastic proton scattering in the two excited states, as well as the angular
distribution of the $E2$ $\gamma$-decay to account for the alignment of the
$2_{1}^{+}$ state.

Here we present results of the ratio of the number of protons populating the
$2^{+}_{1}$ and $0^{+}_{2}$ states, $N_{p}(2_{1}^{+})/N_{p}(0_{2}^{+})$, that
were used in Ref. Eriksen  _et al._ (2020) to determine
$\Gamma^{E0}_{\pi}/\Gamma$. This paper is intended to describe fully the
$\Gamma^{E0}_{\pi}/\Gamma$ measurement of Ref. Eriksen  _et al._ (2020), as
well as to enable future experimental efforts at measuring $\Gamma_{rad}$ (as
urged by Ref. Kibédi  _et al._ (2020)) without having to do separate proton
angular distribution measurements. By measuring absolute cross-sections with a
thin target with comprehensive angular coverage, and covering the range of
energies relevant for these experiments, we aim to provide a high-precision
general data set for any future use. This data set extends beyond that in the
literature, which is either for a specific target thickness at a specific
energy Alburger (1977), or has limited angular coverage Swint  _et al._ (1966)
meaning that total cross-sections cannot be reliably extracted.

Figure 2: (a) Laboratory total energy vs $\theta_{\rm lab}$ spectrum for
particles detected at 10.30 MeV in the DSSD array. (b) Total energy spectrum
in a $76^{\circ}<\theta_{\rm lab}<85^{\circ}$ slice, corresponding to a single
detector arc in the BALiN array. The features of these spectra are discussed
in the text. Figure 3:  Differential cross-sections (blue squares) in the
laboratory frame for 12C$(p,p^{\prime})$ populating the $0^{+}_{2}$ state at
10.30 MeV (a) and 10.48 MeV (b), and the $2^{+}_{1}$ state at at the same
energies (c),(d). Other energies are shown in Appendix B. Error bars, smaller
than the points, are purely statistical. Magenta circles indicate cross-
sections from Ref. Swint  _et al._ (1966) which have been digitized and
converted to the laboratory frame. In panel (b), the cross-section from Ref.
Swint  _et al._ (1966) at $25.54^{\circ}$ (indicated by the magenta arrow)
lies off-scale at 113(4) mb/deg (discussed in the text). Light blue curves
represent $1\sigma$ confidence intervals of fifth order Legendre polynomial
fits to the differential cross-sections $d\sigma/d\Omega(\theta)$.

##  II Experimental Methods

The experiment was performed at the Australian National University Heavy Ion
Accelerator Facility (HIAF). Proton beams were delivered by the 14UD
electrostatic accelerator at 14 energies between $E_{\rm beam}=10.20$ and
$10.70$ MeV impinging on thin natC ($98.94\%$ 12C, $1.06\%$ 13C) targets of
thicknesses 50 $\mu$g/cm2 and 60 $\mu$g/cm2. The targets were located on the
same ladder, and oriented at 45 degrees to the beam axis. Here, we report the
results from the 60 $\mu$g/cm2 target, which had a thin Au “flash” (1.3
$\mu$g/cm2) evaporated on the side oriented upstream of the target material,
thus enabling absolute cross-sections to be obtained via normalization to
$p+$Au elastic scattering at forward angles. Partial results from measurements
with the 50 $\mu$g/cm2 target were presented in Ref. Eriksen  _et al._ (2020),
which used relative yields to extract $N_{p}(2_{1}^{+})/N_{p}(0_{2}^{+})$
appropriate for that pair conversion experiment.

Thin targets were chosen to minimize energy loss, with the total energy loss
through the target being between 3.45 keV at $E_{\rm beam}=10.20$ MeV and 3.33
keV at $E_{\rm beam}=10.70$ MeV. Due to this small energy loss, in this paper
we will quote incident proton energies ($E_{\rm beam}$) rather than mid-target
energies. The energy analysing magnet after the 14UD was recycled to reduce
differential hysteresis effects, providing precise energy definition,
estimated to be within $0.01\%$ (1 keV) of the required beam energy Spear  _et
al._ (1977). Measurements were performed in 20 keV steps between 10.40 and
10.50 MeV (inclusive) to span the energy loss of $E_{beam}=10.50$ MeV protons
incident on the $1$ mg/cm2 target used in Ref. Eriksen  _et al._ (2020) with
high granularity. For 10.20 – 10.40 and 10.50 – 10.70 MeV, 50 keV steps in
beam energy were taken to increase the range in beam energy that could be
covered.

Reaction products were measured using the Breakup Array for Light Nuclei
(BALiN), an array comprised of $60^{\circ}$ wedge-shaped Double Sided Silicon
Detectors (DSSDs) segmented into 16 arcs and 8 sectors Rafiei  _et al._
(2010); Cook  _et al._ (2018). In this experiment, the array was configured as
two $\Delta E-E$ telescopes consisting of 400 $\mu$m ($\Delta E$) and 500
$\mu$m ($E_{\rm res}$) stages placed on either side of the target. The DSSD
thicknesses ensured that the elastically scattered protons stopped in the
active volume of the silicon detectors, while the inelastic scattering
populating the $2^{+}_{1}$ and $0^{+}_{2}$ states stopped in the $\Delta E$
stage. The array provided continuous coverage of scattering angles
$22^{\circ}<\theta_{\rm lab}<158^{\circ}$ (with a small overlap around
$90^{\circ}$), with azimuthal acceptance $99^{\circ}<\phi<166^{\circ}$ and
$270^{\circ}<\phi<336^{\circ}$. The $\Delta\phi$ coverage was almost $\theta$
independent. A $5^{\circ}$ segment of detector centered at $33.9^{\circ}$ was
used for beam normalization.

The extended angular coverage and granularity of the detectors enabled “single
shot” measurements (taking about 50 minutes) of proton scattering angular
distributions from the $2_{1}^{+}$, $0_{2}^{+}$ (Hoyle) states, as well as
from the ground-state of 12C and 197Au (the latter being used for
normalization).

The resulting total energy ($\Delta E+E_{\rm res}$) vs $\theta_{\rm lab}$
distribution at $E_{\rm beam}=10.30$ MeV is shown in Fig. 2(a), with a
projection of the total energy between $76^{\circ}<\theta_{\rm
lab}<85^{\circ}$ shown in Fig. 2(b). Important features include elastic
scattering from the 197Au flash and the 12C target material, as well as much
less intense peaks from 1H,16O,28Si target impurities. The energy calibration
was optimised for the $2^{+}_{1}$ and $0^{+}_{2}$ curves. The small increase
in the energy of the elastic lines around $90^{\circ}$ and $160^{\circ}$ is
due to imperfect energy matching between the $\Delta E$ and $E_{\rm res}$
stages. Since this does not influence the yield determination, this does not
impact the results of this study in any way. Elastic scattering from the small
contribution of 13C cannot be distinguished from the 12C elastic scattering
events, and contributes a $1.06\%$ systematic error on the elastic scattering
cross-sections. This systematic error does not apply to the $2^{+}_{1}$ and
$0^{+}_{2}$ Hoyle state cross-sections.

As well as inelastic scattering populating the $2^{+}_{1}$ and $0^{+}_{2}$
states in 12C, inelastic scattering populating various states in 13C,16O,28Si
is seen in Fig. 2. The intensities of these peaks are very low, and they are
easily distinguished from the the $2^{+}_{1}$ and $0^{+}_{2}$ states in 12C
via their different energy-angle relationships. Thus, the presence of these
inelastic scattering peaks does not impact our analysis. At $\sim 40^{\circ}$
and $\sim 60^{\circ}$ the 1H$(p,p)$ kinematic curve intersects with that of
the 12C $2^{+}_{1}$ and $0^{+}_{2}$ states respectively; the subtraction of
these events is discussed in Appendix A. Also observed are events with low
energy and angle, corresponding to recoiling 12C particles from elastic
scattering. Finally, there is a diffuse band (marked $E_{\rm res}=0$) of
points corresponding to events where an event was recorded in the $\Delta E$
stage of the detector but not the $E_{\rm res}$ stage, due to the imperfect
detector overlap and the energy threshold in the $E_{\rm res}$ telescope. The
‘tail’ present below the 12C elastic scattering peak (most intense at $\sim
90^{\circ}$ and $\sim 150^{\circ}$) arises similarly from the energy threshold
in the $E_{\rm res}$ telescope. The correction for these events (only
impacting the elastic scattering yields) will be discussed in Appendix A.

Differential cross-sections for 12C($p,p^{\prime}$)${}^{12}C(x)$ populating
state $x$ at a laboratory angle of $\theta$ were determined via the following
relation:

$\frac{d\sigma_{p+^{12}C(x)}}{d\Omega}(\theta)=\frac{Y_{p+^{12}C(x)}(\theta)}{Y_{p+Au}(\theta_{M})}\frac{N^{Au}}{N^{C}}\frac{d\sigma_{p+Au}}{d\Omega}(\theta_{M})\frac{d\Omega(\theta_{M})}{d\Omega(\theta)}.$
(5)

Here, $Y_{p+^{12}C(x)}(\theta)$ is the yield of protons scattered from carbon
in state $x$ in angle bin $\theta$, $Y_{p+Au}(\theta_{M})$ is the yield of
protons elastically scattered from Au in the monitor angular region
$\theta_{M}=31.4^{\circ}\leq\theta\leq 36.4^{\circ}$, where
$\frac{d\sigma_{p+Au}}{d\Omega}(\theta_{M})$ is the associated average elastic
cross-section, $\frac{N^{Au}}{N^{C}}$ is the relative number density between
the 12C target material and Au flash, and
$\frac{d\Omega(\theta_{M})}{d\Omega(\theta)}$ is the ratio of the solid angles
between the monitor region and the $\theta$ bin. The determination of each
quantity is described in Appendix A.

The main purpose of these measurements is to provide a set of cross-sections
that can be energy-averaged to find the equivalent
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ ratio for any thick target used for pair
conversion measurements. Such targets are required to obtain sufficient
statistics. Therefore, to assess the fidelity of energy integrated thin-target
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ ratios when applied to thick targets, a
measurement was also performed with the same $1$ mg/cm2 target used in Ref.
Eriksen  _et al._ (2020) for pair conversion measurements. The energy loss of
the 10.5 MeV beam through this target, placed at $45^{\circ}$ to the beam
axis, totals 57 keV. Lacking a Au flash for beam normalization, we do not
present absolute cross-sections for this target, instead we present the
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ ratio. The ratio for the thick target was
extracted including the same detector geometry term $d\Omega(\theta)$ as in
Eq. 5.

##  III Results and Discussion

Differential cross-sections ($d\sigma/d\theta_{\rm lab}$) are shown by the
blue points in Fig. 3 for 12C$(p,p^{\prime})$ reactions populating the
$2^{+}_{1}$ and $0^{+}_{2}$ states at two representative energies $E_{\rm
beam}=10.30$ MeV and $10.48$ MeV. Differential cross-sections for other
energies, are shown in Appendix B for the $0^{+}_{2}$ state in Fig. 5, and the
$2^{+}_{1}$ state in Fig. 6. The elastic scattering data are shown in Fig. 7
in Appendix B.

Digitized cross-sections from plots in Ref. Swint  _et al._ (1966),
transformed to the laboratory frame, are shown by the magenta circles in Fig.
3. The error bars include only the statistical errors given by Ref. Swint  _et
al._ (1966) and do not include a systematic error induced by the digitization
process, which may add another few $\%$ uncertainty. For the most part, the
correspondence between the data from Ref. Swint  _et al._ (1966) and the
present work is excellent, the exception being the $0^{+}_{2}$ state cross-
sections measured at $25.54^{\circ}$ (panels (a) and (b)), which far exceed
those of the present work. In panel (b) it is off-scale, at 113(4) mb/deg,
indicated by the arrow. This anomaly is very likely due to the fact that the
kinematic curve for protons produced from the $0^{+}_{2}$ state coincides with
that of recoiling elastically scattered 12C at this angle, leading to a
spuriously large yield that was incorrectly assigned to protons from the
$0^{+}_{2}$ state at this angle in Ref. Swint  _et al._ (1966). In the present
experiment, the much larger energy loss of the 12C nuclei compared to protons
in the detector dead-layers separated the peaks from the $0^{+}_{2}$ state and
the recoiling 12C, allowing clean separation of the $0^{+}_{2}$ state, as seen
in Fig. 2(a).

Figure 4: Total cross-sections for (a) the $0^{+}_{2}$ and (b) $2^{+}_{1}$
states between $E_{lab}=10.20$ and $10.70$ MeV (blue points). The error bars
arise from the $1\sigma$ confidence limits of the fits shown in Figs. 3, 5 and
6. Cross-sections for the $2^{+}_{1}$ state from Ref. Kiener  _et al._ (1998)
are shown in panel (b) by the orange squares.
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ ratios are shown in (c), zoomed around
10.38-10.50 MeV in panel (d). The value of Ref. Alburger (1977) is shown by
the magenta square. The ratio from the 1 mg/cm2 target measurement (oriented
at $45^{\circ}$) is shown by the purple diamond. Taking the weighted average
of the thin target (blue) points over the energy loss expected in a 1 mg/cm2
target oriented at $45^{\circ}$ yields the green triangle. For the ratios
extracted for thick targets, the energy range covered is indicated by the
horizontal bar. The agreement of the ‘equivalent target’ with the actual thick
target measurement is excellent.

To obtain total cross-sections, we extrapolated our $d\sigma/d\Omega$
distributions beyond the angular range of the detectors via fifth order
Legendre polynomial fits, prior to conversion to $d\sigma/d\theta$. Fifth
order fits were the lowest order required to adequately fit the experimental
data at all beam energies. The resulting fits are shown by the light blue
curves in Fig. 3 for two representative energies and in Figs. 5 and 6 in
Appendix B for all other energies. One-$\sigma$ confidence intervals are
indicated by the shaded regions, which are small due to both high statistics
and the extensive angular coverage and granularity of the array, thus strongly
constraining the fit parameters. After integrating the fitted functions, the
resulting total cross-sections are shown in Fig. 4, for the $0^{+}_{2}$ (a),
and $2^{+}_{1}$ states (b). The data are also tabulated in Table 1 (Appendix
C). Error bars were extracted from the $1\sigma$ confidence intervals of the
fits. The extrapolated regions ($\theta<22^{\circ}$, $\theta>158^{\circ}$)
contribute an average of ($6.5\pm 1)\%$ of the total cross-sections. Cross-
sections for $2^{+}_{1}$ population measured in 12C(p,p$\gamma_{4.44}$)
reactions in Ref. Kiener  _et al._ (1998) are shown by the orange squares in
Fig. 3(b). While the cross-sections from Ref. Kiener  _et al._ (1998) follow a
very similar trend in energy, they lie $\sim 7\%$ above those in the present
work. Those cross-sections were normalized to a previous measurement Dyer  _et
al._ (1981) which were themselves additionally normalized to a separate
12C$(p,p^{\prime})$ measurement at a single energy, due to target non-
uniformity in their $\gamma$ measurement. We consider it likely that this
multi-step normalization process could lead to this discrepancy. On the other
hand, the normalization of the present data was achieved via simultaneous
elastic scattering measurements at every energy. Additionally, our data agree
very well with those of Ref. Swint  _et al._ (1966), as seen in Fig. 3,
lending additional confidence to our analysis. We wish to emphasize that for
the purposes of extraction of the ratio $N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$,
any overall normalization errors in the cross-sections would cancel out.

The population ratio $N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ at each energy,
required for measurement of the $E0$ branching ratio using Eq. 4 are shown in
Fig. 4(c). The error bars in the population ratios presented here are lower
than that quoted in Ref. Eriksen  _et al._ (2020) from the same experiment. A
more complete statistical analysis of the fits yielded a much smaller error
bar, though the central values are the same within error. The uncertainty in
the $E0$ branching ratio extracted in Ref. Eriksen  _et al._ (2020) was
dominated by the pair conversion measurement, and does not change with these
new smaller uncertainties. Also shown is the reported
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ ratio of $3.74\pm 0.18$ from Ref. Alburger
(1977), (magenta square) which was extracted from averaging across five
measurements using a 120 $\mu$g/cm2 12C target in energy increments of 25 keV
from 10.4 to 10.5 MeV. The energy range over which the ratio was averaged is
indicated by the horizontal bar. The agreement with the present data is
excellent. However, the measurement of Alburger (1977) cannot be applied to a
target of a different thickness or a different bombarding energy. An
appropriate weighted average of the present thin-target data can provide
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ for various target thicknesses and
bombarding energies between 10.20 and 10.70 MeV.

##  IV Integration for thick target measurements

In a pair conversion measurement for measuring the $E0$ decay branching ratio,
targets of thickness of order $1$ mg/cm2 must be used to obtain sufficient
statistics. The energy loss in the target is sufficiently high, and
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ varies sufficiently quickly with energy,
that the energy loss cannot be neglected. In Ref. Eriksen  _et al._ (2020), a
$1$ mg/cm2 target oriented at $45^{\circ}$ to the beam axis (equivalent
thickness 1.41 mg/cm2) was used for the pair conversion measurement. The
energy loss in this target is calculated to be 57 keV. Using this same target,
12C$(p,p^{\prime})$ measurements were performed under the same conditions as
for the $60~{}\mu$g/cm2 target. The resulting
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ value for this target is 3.94(3), and is
shown by the purple diamond in Fig. 4(c), and in more detail in panel (d),
where the beam energy range in the target is indicated by the width of the
horizontal bar. The equivalent ratio was constructed from the 60 $\mu$g/cm2
target measurements by linearly interpolating between each energy, averaging
the $0^{+}_{2}$ and $2^{+}_{1}$ cross-sections between 10.44 and 10.50 MeV,
and taking the ratio of the average cross-sections:

$\displaystyle
N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})=\frac{\int_{E_{1}}^{E_{2}}\sigma_{2_{1}^{+}}(E)dE}{\int_{E_{1}}^{E_{2}}\sigma_{0_{2}^{+}}(E)dE}.$
(6)

Using this method, the ratio found for the $1$ mg/cm2 target oriented at
$45^{\circ}$ to the beam axis is 3.94(2), precisely agreeing with the
experimental value from the thick target measurement. This point is shown in
green in Fig. 4(c,d). We are clearly able to extract the same
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ either with a thick target measurement
(requiring a separate measurement on each target used), or by averaging over
thin target measurements. This approach requires that absolute cross-sections
are extracted, since we take their weighted average before taking the ratio,
and is more reliable than taking the average of the ratios (which gives
3.96(2) in this instance) when the cross-sections vary significantly over the
energy range through the target.

In extracting this value, we assumed linear energy loss in the target, a
uniform target thickness, and no energy width in the beam. These assumptions
were justified for this case. The $1$ mg/cm2 target used in this experiment of
Ref. Eriksen  _et al._ (2020) was uniform, and the tandem beams used in this
experiment are extremely narrow in energy. If the target is non-uniform, very
thick, or the beam has substantial energy width, we recommend careful
consideration of the distribution of energies within the target when averaging
across the target thickness. For extreme cases, the method of Ref. Fisichella
_et al._ (2015) could be applied.

##  V Conclusions

In this work, we have performed measurements of inelastic proton scattering
from 12C at energies between 10.20 and 10.70 MeV, for the purpose of use in
experiments determining the $E0$ branching ratio of the Hoyle state in 12C.
This quantity is important for extracting the production rate of carbon (and
thus all heavier elements) in the universe, which occurs via the $3\alpha$
process. By measuring cross-sections in small energy steps with a thin target
and comprehensive angular coverage, we have, for the first time, extracted a
generally applicable data set for future measurements of $E0$ branching ratios
via pair measurements. The equivalent $N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$
ratio for a thick target can be found by taking the weighted average of
$N_{p}(2_{1}^{+})$ and $N_{p}(0^{+}_{2})$ across the energy range of the beam
in the target. Using this method, we find precise agreement between the ratio
extracted from a $1$ mg/cm2 target previously used in pair conversion
measurements Eriksen  _et al._ (2020).

It has been proposed that the radiative width of the Hoyle state may be
extracted from a direct measurement of the ratio of the pair transitions de-
exciting the Hoyle state, $\Gamma_{\pi}^{E2}/\Gamma_{\pi}^{E0}$ Kibédi  _et
al._ (2012). This method would allow the radiative width to be determined in
an independent way. In the recent experiments of Ref. Eriksen  _et al._ (2020)
undertaken at a beam energy of 10.5 MeV, there is significant random
background in the region around the 3.22 MeV $E2$ transition from the Hoyle
state that prevents this method from being applied. This background likely
arises from the 4.44 MeV $E2$ transition from the $2^{+}_{1}$ state Eriksen
_et al._ (2020). It was estimated that a successful measurement using this
method requires a factor of twenty reduction in the background in the vicinity
of the 3.22 MeV pair peak Eriksen  _et al._ (2020). These data show that we
cannot achieve such a reduction in the background by changing the beam energy
alone to reduce $N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$, but instead re-design of
the Super-e pair spectrometer at ANU would be required. However, running the
experiment at a beam energy of 10.60 MeV will reduce the background by
$~{}25\%$ for essentially the same $0^{+}_{2}$ cross-section and should be
strongly considered.

The cross-sections for populating the ground, $2_{1}^{+}$, and $0_{2}^{+}$
states may be used for applications beyond measuring the $E0$ branching ratio
of the Hoyle state. 13N, as well as the mirror nucleus 13C, have been
discussed as a candidate for showing cluster states of the form $3\alpha+p$
($3\alpha+n$) near their $\alpha$ thresholds von Oertzen  _et al._ (2006);
Rudchik  _et al._ (2019); Fujimura  _et al._ (2004); Milin and von Oertzen
(2002). The present experiments were conducted between 1.86-2.13 MeV above the
9B$+\alpha$ threshold (9.495 MeV) in 13N. Studies of other nuclei around 12C
have shown that R-matrix fits of cross-sections around these energies can
provide useful information on cluster configurations, e.g. Dell’Aquila (2020);
Freer  _et al._ (2012). Due to the low-lying proton removal threshold in 13N,
R-matrix fitting of this excitation energy region is a significant task,
beyond the scope of this paper. However we encourage the future application of
these cross-sections to this interesting question.

###### Acknowledgements.

This work was supported by Australian Research Council Grants DP140102896,
DP170101673, DP170102423, DP190100256, DP200100601. Operation of the ANU Heavy
Ion Accelerator Facility is supported by the NCRIS HIA capability. The support
from the technical staff of the ANU HIAF is greatly appreciated. Useful
discussions with Phil Adsley are gratefully acknowledged. KJC thanks the
Department of Nuclear Physics at the Australian National University for their
hospitality during her visit.

##  Appendix A Cross-section determination

Yields $Y_{p+^{12}C(0_{1}^{+})}(\theta)$, $Y_{p+^{12}C(2_{1}^{+})}(\theta)$,
$Y_{p+^{12}C(0_{2}^{+})}(\theta)$, and $Y_{p+Au}(\theta_{M})$ were background
subtracted using third order polynomial fitting of the backgrounds. The
backgrounds were typically $10^{3}$ below the peak values for the $2^{+}_{1}$
state, and $2\times 10^{2}$ below the peak values for the $0^{+}_{2}$ state.
The exception to this was for p+p scattering from hydrogen impurities in the
target material, which produces a larger background where the kinematic curve
intersects that of the $2^{+}_{1}$ state at $40^{\circ}$ and the $0^{+}_{2}$
state at $60^{\circ}$.

Determination of the relative number densities of the 12C target material and
Au flash, $\frac{N^{Au}}{N^{C}}$, as well as verification of the the solid
angle $\frac{d\Omega(\theta_{M})}{d\Omega(\theta)}$ of the detector array was
achieved using a 16O beam at a below-barrier energy of 14.63 MeV, where
scattering is expected to be purely Rutherford.

For natC(p,p) and 197Au(p,p) elastic scattering, the proton energies were
above the punch-through energies in the $\Delta E$ stage, and a small (about
$5\%$) portion of events have a $\Delta E$ but not $E_{\rm res}$ (residual
energy) signal due to (a) the imperfect geometric overlap of the $\Delta E$
and $E_{residual}$ stages and (b) events with total energy slightly above the
punch-through thresholds not giving a signal in the $E_{residual}$ stage.
These events are marked $E_{\rm res}=0$ in Fig. 2. Corrections were made on
the basis of the number of $\Delta E$ signals with the correct energy loss
compared to that in total energy. At $90^{\circ}$, where the correction was
$>15\%$, the elastic scattering data is not presented. These corrections were
necessary only for the elastic scattering events. No such corrections were
necessary for the $2_{1}^{+}$ and $0_{2}^{+}$ states, since the protons all
stop in the $\Delta E$ stage.

To normalize the 12C$(p,p^{\prime})$ cross-sections to $p+Au$ elastic
scattering reactions via Eq. 5, elastic scattering cross-sections
$\frac{d\sigma_{p+Au}}{d\Omega}(\theta_{M})$, were calculated using FRESCO
Thompson (1988) with optical potentials from refs. Becchetti and Greenlees
(1969); Rathmell and Haeberli (1972). For both potentials, the proton elastic
scattering cross-section at $33.9^{\circ}$ (the angle selected for
normalization of the 12C$(p,p^{\prime})$ data) deviated by $<1\%$ from the
Rutherford scattering formula at these energies, making the choice of
potential insignificant to the overall normalization of the p+12C cross-
sections.

To ensure accuracy of cross-sections at very forward angles where the cross-
sections change rapidly across a single pixel of the detector array, a set of
events were simulated from the elastic cross-sections using Monte-Carlo
techniques. The detector response was simulated by randomizing the position of
the elastic scattering events in each pixel of the detector array, and the
simulated events were passed through the same analysis pipeline as the
experimental data. This assured correct cross-section normalization using the
197Au$(p,p^{\prime})$ data.

##  Appendix B Angular distributions for all energies

Fig. 5 shows angular distributions and Legendre polynomial fits for
12C$(p,p^{\prime})$ populating the $0^{+}_{2}$ state at all other energies
(besides 10.30 and 10.48 MeV, presented in Fig. 3). Similarly, Fig. 6 shows
all other angular distributions for 12C$(p,p^{\prime})$ populating the
$2^{+}_{1}$ state. Fig. 7 shows the natC$(p,p)$ elastic scattering cross-
sections at each energy. Contributions from 13C$(p,p)$ ($1.06\%$ abundance)
could not be separated from 12C$(p,p)$ ($98.94\%$ abundance). Therefore, these
cross-sections can be regarded as 12C$(p,p)$ cross-sections with a $1.06\%$
systematic error. If the 13C$(p,p)$ cross-sections were identical to
12C$(p,p)$, the systematic error would reduce to zero.

Figure 5:  Angular distributions for 12C$(p,p^{\prime})$ populating the
$0^{+}_{2}$ state. Error bars (smaller than the points) are purely
statistical. Light blue curves represent fifth order Legendre polynomial fits
to $d\sigma/d\Omega(\theta)$ to enable extrapolation beyond the detection
region. The width of the light blue curves show the $1\sigma$ confidence
interval of the fit. The results for $E_{\rm{Beam}}=10.30$ and 10.48 MeV are
shown in Fig. 3(a) and (b). Figure 6: Angular distributions for
12C$(p,p^{\prime})$ populating the $2^{+}_{1}$ state. Error bars (smaller than
the points) are purely statistical. Light blue curves represent fifth order
Legendre polynomial fits to $d\sigma/d\Omega(\theta)$ to enable extrapolation
beyond the detection region. The width of the light blue curves show the
$1\sigma$ confidence interval of the fit. The results for
$E_{\rm{Beam}}=10.30$ and 10.48 MeV are shown in Fig. 3(c) and (d). Figure 7:
Elastic scattering angular distributions for natC$(p,p)$ reactions. Error bars
(smaller than the points) are purely statistical. These may also be regarded
as 12C$(p,p)$ cross-sections for which there is an additional systematic error
of up to $1.06\%$ at each angle due to the contribution of 13C$(p,p)$ that
could not be separated.

##  Appendix C Tabulated data

Tabulated cross-sections for 12C$(p,p^{\prime})$ reactions populating the
$0^{+}_{2}$ and $2^{+}_{1}$ states, and their ratio at each energy
$N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ are shown in Table 1.

Table 1: Total cross-sections for 12C$(p,p^{\prime})$ reactions populating the $0^{+}_{2}$ and $2^{+}_{1}$ states, and their ratio at each energy $N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$ Energy |  $\sigma_{0_{2}^{+}}$ |  $\sigma_{2_{1}^{+}}$ |  $N_{p}(2_{1}^{+})/N_{p}(0^{+}_{2})$
---|---|---|---
(MeV) |  (mb) |  (mb) |
10.20 |  $52.7\pm 0.3$ |  $288.9\pm 0.9$ |  $5.48\pm 0.04$
10.25 |  $54.4\pm 0.4$ |  $289.1\pm 0.9$ |  $5.32\pm 0.04$
10.30 |  $61.9\pm 0.3$ |  $303.7\pm 1.0$ |  $4.91\pm 0.03$
10.35 |  $71.4\pm 0.4$ |  $305.7\pm 0.9$ |  $4.28\pm 0.03$
10.40 |  $79.0\pm 0.4$ |  $302.5\pm 1.1$ |  $3.83\pm 0.03$
10.42 |  $82.7\pm 0.4$ |  $307.1\pm 1.1$ |  $3.72\pm 0.02$
10.44 |  $84.7\pm 0.5$ |  $312.5\pm 1.4$ |  $3.69\pm 0.03$
10.46 |  $89.6\pm 0.5$ |  $342.1\pm 1.7$ |  $3.82\pm 0.03$
10.48 |  $88.9\pm 0.5$ |  $359.1\pm 1.7$ |  $4.04\pm 0.03$
10.50 |  $89.4\pm 0.5$ |  $377.4\pm 1.8$ |  $4.22\pm 0.03$
10.55 |  $87.0\pm 0.4$ |  $314.8\pm 1.0$ |  $3.62\pm 0.02$
10.60 |  $86.3\pm 0.4$ |  $271.4\pm 0.9$ |  $3.15\pm 0.02$
10.65 |  $81.0\pm 0.4$ |  $283.6\pm 1.0$ |  $3.50\pm 0.02$
10.70 |  $69.4\pm 0.3$ |  $298.1\pm 1.1$ |  $4.30\pm 0.03$

## References

  * Salpeter (1952) E. E. Salpeter, Astrophys. J. 115, 326 (1952).
  * Dunbar  _et al._ (1953) D. N. F. Dunbar, R. E. Pixley, W. A. Wenzel, and W. Whaling, Phys. Rev. 92, 649 (1953).
  * Hoyle (1954) F. Hoyle, APJ S. 1, 121 (1954).
  * Cook  _et al._ (1957) C. W. Cook, W. A. Fowler, C. C. Lauritsen, and T. Lauritsen, Phys. Rev. 107, 508 (1957).
  * Smith  _et al._ (2017) R. Smith, T. Kokalova, C. Wheldon, J. E. Bishop, M. Freer, N. Curtis, and D. J. Parker, Phys. Rev. Lett. 119, 132502 (2017).
  * Dell’Aquila  _et al._ (2017) D. Dell’Aquila, I. Lombardo, G. Verde, M. Vigilante, L. Acosta, C. Agodi, F. Cappuzzello, D. Carbone, M. Cavallaro, S. Cherubini, A. Cvetinovic, G. D’Agata, L. Francalanza, G. L. Guardo, M. Gulino, I. Indelicato, M. La Cognata, L. Lamia, A. Ordine, R. G. Pizzone, S. M. R. Puglia, G. G. Rapisarda, S. Romano, G. Santagati, R. Spartà, G. Spadaccini, C. Spitaleri, and A. Tumino, Phys. Rev. Lett. 119, 132501 (2017).
  * Rana  _et al._ (2019) T. Rana, S. Bhattacharya, C. Bhattacharya, S. Manna, S. Kundu, K. Banerjee, R. Pandey, P. Roy, A. Dhal, G. Mukherjee, V. Srivastava, A. Dey, A. Chaudhuri, T. Ghosh, A. Sen, M. Asgar, T. Roy, J. Sahoo, J. Meena, A. Saha, R. Saha, M. Sinha, and A. Roy, Physics Letters B 793, 130 (2019).
  * Bishop  _et al._ (2020) J. Bishop, G. V. Rogachev, S. Ahn, E. Aboud, M. Barbui, A. Bosh, C. Hunt, H. Jayatissa, E. Koshchiy, R. Malecek, S. T. Marley, E. C. Pollacco, C. D. Pruitt, B. T. Roeder, A. Saastamoinen, L. G. Sobotka, and S. Upadhyayula, Phys. Rev. C 102, 041303(R) (2020).
  * Freer and Fynbo (2014) M. Freer and H. Fynbo, Progress in Particle and Nuclear Physics 78, 1 (2014).
  * Kibédi  _et al._ (2020) T. Kibédi, B. Alshahrani, A. E. Stuchbery, A. C. Larsen, A. Görgen, S. Siem, M. Guttormsen, F. Giacoppo, A. I. Morales, E. Sahin, G. M. Tveten, F. L. B. Garrote, L. C. Campo, T. K. Eriksen, M. Klintefjord, S. Maharramova, H.-T. Nyhus, T. G. Tornyi, T. Renstrøm, and W. Paulsen, Phys. Rev. Lett. 125, 182701 (2020).
  * Rolfs and Rodney (1988) C. E. Rolfs and W. S. Rodney,  _Cauldrons in the cosmos: nuclear astrophysics_ (University of Chicago Press, 1988).
  * Eriksen  _et al._ (2020) T. K. Eriksen, T. Kibédi, M. W. Reed, A. E. Stuchbery, K. J. Cook, A. Akber, B. Alshahrani, A. A. Avaa, K. Banerjee, A. C. Berriman, L. T. Bezzina, L. Bignell, J. Buete, I. P. Carter, B. J. Coombes, J. T. H. Dowie, M. Dasgupta, L. J. Evitts, A. B. Garnsworthy, M. S. M. Gerathy, T. J. Gray, D. J. Hinde, T. H. Hoang, S. S. Hota, E. Ideguchi, P. Jones, G. J. Lane, B. P. McCormick, A. J. Mitchell, N. Palalani, T. Palazzo, M. Ripper, E. C. Simpson, J. Smallcombe, B. M. A. Swinton-Bland, T. Tanaka, T. G. Tornyi, and M. O. de Vries, Phys. Rev. C 102, 024320 (2020).
  * Woosley and Heger (2021) S. E. Woosley and A. Heger, The Astrophysical Journal Letters 912, L31 (2021).
  * Tsumura  _et al._ (2021) M. Tsumura, T. Kawabata, Y. Takahashi, S. Adachi, H. Akimune, S. Ashikaga, T. Baba, Y. Fujikawa, H. Fujimura, H. Fujioka, T. Furuno, T. Hashimoto, T. Harada, M. Ichikawa, K. Inaba, Y. Ishii, N. Itagaki, M. Itoh, C. Iwamoto, N. Kobayashi, A. Koshikawa, S. Kubono, Y. Maeda, Y. Matsuda, S. Matsumoto, K. Miki, T. Morimoto, M. Murata, T. Nanamura, I. Ou, S. Sakaguchi, A. Sakaue, M. Sferrazza, K. Suzuki, T. Takeda, A. Tamii, K. Watanabe, Y. Watanabe, H. Yoshida, and J. Zenihiro, Physics Letters B 817, 136283 (2021).
  * Alburger (1977) D. E. Alburger, Phys. Rev. C 16, 2394 (1977).
  * Swint  _et al._ (1966) J. Swint, A. Barnard, T. Clegg, and J. Weil, Nuclear Physics 86, 119 (1966).
  * Spear  _et al._ (1977) R. Spear, D. Kean, M. Esat, A. Joye, and M. Fewell, Nuclear Instruments and Methods 147, 455 (1977).
  * Rafiei  _et al._ (2010) R. Rafiei, R. du Rietz, D. H. Luong, D. J. Hinde, M. Dasgupta, M. Evers, and A. Diaz-Torres, Phys. Rev. C 81, 024601 (2010).
  * Cook  _et al._ (2018) K. J. Cook, I. P. Carter, E. C. Simpson, M. Dasgupta, D. J. Hinde, L. T. Bezzina, S. Kalkal, C. Sengupta, C. Simenel, B. M. A. Swinton-Bland, K. Vo-Phuoc, and E. Williams, Phys. Rev. C 97, 021601(R) (2018).
  * Kiener  _et al._ (1998) J. Kiener, M. Berheide, N. L. Achouri, A. Boughrara, A. Coc, A. Lefebvre, F. de Oliveira Santos, and C. Vieu, Phys. Rev. C 58, 2174 (1998).
  * Dyer  _et al._ (1981) P. Dyer, D. Bodansky, A. G. Seamster, E. B. Norman, and D. R. Maxson, Phys. Rev. C 23, 1865 (1981).
  * Fisichella  _et al._ (2015) M. Fisichella, A. C. Shotter, A. Di Pietro, P. Figuera, M. Lattuada, C. Marchetta, V. Privitera, L. Romano, C. Ruiz, and M. Zadro, Phys. Rev. C 92, 064611 (2015).
  * Kibédi  _et al._ (2012) T. Kibédi, A. E. Stuchbery, G. D. Dracoulis, and K. A. Robertson, EPJ Web Conf. 65, 06001 (2012).
  * von Oertzen  _et al._ (2006) W. von Oertzen, M. Freer, and Y. Kanada-En’yo, Physics Reports 432, 43 (2006).
  * Rudchik  _et al._ (2019) A. Rudchik, A. Rudchik, O. Kutsyk, K. Kemper, K. Rusek, E. Piasecki, A. Trzcińska, S. Kliczewski, E. Koshchy, V. Pirnak, O. Ponkratenko, I. Strojek, V. Plujko, S. Sakuta, R. Siudak, A. Ilyin, Y. Stepanenko, Y. Shyrma, and V. Uleshchenko, Nuclear Physics A 992, 121638 (2019).
  * Fujimura  _et al._ (2004) H. Fujimura, H. Akimune, I. Daito, M. Fujiwara, K. Hara, K. Y. Hara, M. N. Harakeh, F. Ihara, T. Inomata, K. Ishibashi, T. Ishikawa, T. Kawabata, A. Tamii, M. Tanaka, H. Toyokawa, T. Yamanaka, and M. Yosoi, Phys. Rev. C 69, 064327 (2004).
  * Milin and von Oertzen (2002) M. Milin and W. von Oertzen, Eur. Phys. J. A 14, 295 (2002).
  * Dell’Aquila (2020) D. Dell’Aquila, Eur. Phys. J. Plus 135, 165 (2020).
  * Freer  _et al._ (2012) M. Freer, N. L. Achouri, C. Angulo, N. I. Ashwood, D. W. Bardayan, S. Brown, W. N. Catford, K. A. Chipps, N. Curtis, P. Demaret, C. Harlin, B. Laurent, J. D. Malcolm, M. Milin, T. Munoz-Britton, N. A. Orr, S. D. Pain, D. Price, R. Raabe, N. Soić, J. S. Thomas, C. Wheldon, G. Wilson, and V. A. Ziman, Phys. Rev. C 85, 014304 (2012).
  * Thompson (1988) I. J. Thompson, Computer Physics Reports 7, 167 (1988).
  * Becchetti and Greenlees (1969) F. D. Becchetti and G. W. Greenlees, Phys. Rev. 182, 1190 (1969).
  * Rathmell and Haeberli (1972) R. Rathmell and W. Haeberli, Nuclear Physics A 178, 458 (1972).


11institutetext: 11email<EMAIL_ADDRESS>
# Harnessing Deep Q-Learning for Enhanced Statistical Arbitrage in High-
Frequency Trading: A Comprehensive Exploration

Soumyadip Sarkar

###### Abstract

The realm of High-Frequency Trading (HFT) is characterized by rapid decision-
making processes that capitalize on fleeting market inefficiencies. As the
financial markets become increasingly competitive, there is a pressing need
for innovative strategies that can adapt and evolve with changing market
dynamics. Enter Reinforcement Learning (RL), a branch of machine learning
where agents learn by interacting with their environment, making it an
intriguing candidate for HFT applications. This paper dives deep into the
integration of RL in statistical arbitrage strategies tailored for HFT
scenarios. By leveraging the adaptive learning capabilities of RL, we explore
its potential to unearth patterns and devise trading strategies that
traditional methods might overlook. We delve into the intricate exploration-
exploitation trade-offs inherent in RL and how they manifest in the volatile
world of HFT. Furthermore, we confront the challenges of applying RL in non-
stationary environments, typical of financial markets, and investigate
methodologies to mitigate associated risks. Through extensive simulations and
backtests, our research reveals that RL not only enhances the adaptability of
trading strategies but also shows promise in improving profitability metrics
and risk-adjusted returns. This paper, therefore, positions RL as a pivotal
tool for the next generation of HFT-based statistical arbitrage, offering
insights for both researchers and practitioners in the field.

###### Keywords:

Reinforcement Learning Deep Q Learning Statistical Arbitrage High Frequency
Trading

##  1 Introduction

The landscape of financial markets has witnessed revolutionary changes over
the past few decades, driven by technological advancements, increased
computational power, and the proliferation of electronic trading platforms.
One such transformative force that emerged from this confluence is High-
Frequency Trading (HFT), a form of algorithmic trading characterized by high
speeds, high turnover rates, and high order-to-trade ratios [1]. HFT
strategies often revolve around executing large numbers of orders at extremely
fast speeds, capitalizing on minuscule price inefficiencies that may exist
only for milliseconds [3].

Statistical arbitrage, a quantitatively driven approach to trading, seeks to
exploit relative price deviations among related financial instruments and has
been a staple in the toolkit of many quantitative traders and hedge funds [2].
In the HFT environment, the traditional paradigms of statistical arbitrage are
constantly being challenged due to the rapidity of trades and the need for
real-time adaptability. Given this backdrop, there is a growing interest in
the application of machine learning techniques to enhance, optimize, and
innovate trading strategies tailored for the HFT milieu.

Enter Reinforcement Learning (RL), a subfield of machine learning wherein
agents learn to make decisions by taking actions in an environment to maximize
a notion of cumulative reward [4]. Unlike supervised learning, where learning
is guided by a labeled dataset, RL operates in a feedback loop where the agent
learns from the consequences of its actions, making it well-suited for dynamic
environments where the optimal decision or strategy is not known a priori.
Given the inherent uncertainties and dynamism of financial markets, RL’s
approach to learning and decision-making offers an intriguing proposition for
statistical arbitrage in HFT [5].

Historically, the application of machine learning in finance has revolved
around supervised techniques, primarily for forecasting prices or returns [6].
However, the financial decision-making process, especially in trading, is
inherently sequential and involves a series of actions over time, with rewards
(or penalties) that may be delayed. This sequential decision-making nature
aligns closely with the principles of RL, which focuses on learning optimal
sequences of actions in interactive environments. Moreover, as markets evolve,
the adaptability of RL algorithms, which continuously update and refine their
strategies based on new data and feedback, can be a significant asset [7].

The potential of RL in finance is not just theoretical. Some pioneering works
have demonstrated its applicability in various domains of finance. For
instance, Deng et al. [8] showcased how deep reinforcement learning, an
amalgamation of RL with deep neural networks, could be employed for portfolio
optimization. Similarly, RL’s principles have been applied to areas like
option pricing and risk management [9].

However, while the promise of RL in finance is evident, its application in the
specific context of statistical arbitrage for HFT remains an area ripe for
exploration. The challenges are multifaceted. Financial markets are
notoriously noisy, making the reward signals in RL ambiguous. Additionally,
the non-stationarity of financial data, where statistical properties change
over time, can pose significant challenges for RL models that inherently
assume stationary environments [10].

This paper seeks to bridge this gap, providing a comprehensive exploration
into the integration of RL in statistical arbitrage strategies tailored for
HFT scenarios. Through systematic methodologies and empirical analyses, we aim
to offer insights that can guide both researchers and practitioners in
harnessing the potential of RL in the fast-paced world of HFT.

##  2 Model

###  2.1 Model Description

Deep Q-Learning, an innovative fusion of traditional Q-Learning and deep
neural networks, has recently emerged as a powerful tool in the domain of
reinforcement learning. At its core, Q-Learning is a model-free reinforcement
learning algorithm that seeks to learn the optimal action-selection policy for
a given finite Markov decision process [11]. This is achieved by iteratively
updating the Q-values, representations of the expected cumulative rewards for
taking specific actions in particular states, using the Bellman equation.

However, in environments with vast and continuous state or action spaces, such
as those encountered in High-Frequency Trading (HFT), representing Q-values in
a tabular format becomes infeasible. This challenge is addressed by Deep
Q-Learning. Instead of a table, Deep Q-Learning employs deep neural networks
to approximate these Q-values [12]. The neural network, parameterized by
weights, takes the environment’s state as input and outputs the estimated
Q-values for all possible actions. This approach not only allows for
generalization across states but also enables the handling of high-dimensional
input spaces, a characteristic feature of financial datasets in HFT.

The amalgamation of the Q-Learning algorithm’s strengths with the function
approximation capabilities of deep neural networks makes Deep Q-Learning
particularly promising for applications in the dynamic and complex world of
statistical arbitrage in HFT.

###  2.2 Model Fundamentals

Deep Q Learning has garnered considerable attention in recent years due to its
capacity to handle complex, high-dimensional state and action spaces. As a
sophisticated extension of Q-Learning, this method elegantly fuses the
principles of traditional reinforcement learning with the prowess of deep
neural networks.

At its essence, Q-Learning is a renowned model-free, off-policy reinforcement
learning algorithm. Its primary objective is to determine an optimal action-
selection policy, which guides the agent in choosing actions that maximize
cumulative rewards over time. This is realized within the context of a finite
Markov decision process, a mathematical approach for modeling decision-making
in situations where outcomes are partly uncertain, but possess a degree of
probabilistic determination [13]. The Q in Q-Learning represents the ’quality’
of a particular action in a given state, capturing the expected future rewards
for that action-state pairing. This is updated iteratively using the Bellman
equation, ensuring that the agent progressively refines its understanding of
the environment and improves its decision-making capabilities [11].

However, when applied to environments like those in High-Frequency Trading,
which are marked by vast and continuous state spaces, tabular Q-Learning
encounters scalability issues. This is where Deep Q-Learning comes into play.
By introducing deep neural networks to approximate the Q-value function, the
model can generalize across a multitude of states, making it apt for handling
the intricate and high-dimensional datasets frequently encountered in
financial markets [12]. The neural network’s architecture, often consisting of
multiple layers, is trained to predict Q-values based on input states, thereby
bypassing the need for a tabular representation.

In the context of High-Frequency Trading, the adaptive learning capabilities
of Deep Q-Learning can be particularly advantageous. The financial markets,
characterized by their dynamic nature and non-stationarities, require models
that can evolve in real-time, responding adeptly to the rapid fluctuations and
nuanced patterns inherent in trading data.

###  2.3 Q Learning

Q-Learning stands as one of the cornerstones of reinforcement learning, laying
foundational principles that have been pivotal in advancing the field. Deep
Q-Learning, with its integration of deep neural networks, builds upon the
traditional Q-Learning algorithm, enhancing its ability to operate in high-
dimensional state spaces like those encountered in High-Frequency Trading
(HFT).

At the heart of Q-Learning lies the objective of determining an optimal
action-selection policy. This policy guides an agent’s decisions in an
environment, aiming to maximize long-term rewards. The mechanism to achieve
this is through the learning of the action-value function, denoted as
$Q(s,a)$. This function provides an estimate of the expected cumulative reward
for taking action $a$ in state $s$. The "Q" in Q-Learning is often interpreted
as the ’quality’ of a certain action in a particular state [11].

The iterative nature of Q-Learning ensures that the agent refines its
estimates of the Q-values over time. The updating process relies on the
Bellman equation, which is expressed as:

$Q(s,a)\leftarrow
Q(s,a)+\alpha\left(r+\gamma\max_{a^{\prime}}Q(s^{\prime},a^{\prime})-Q(s,a)\right)$

In this equation:

  * •

$s$ represents the agent’s current state.

  * •

$a$ corresponds to the action undertaken by the agent in state $s$.

  * •

$r$ denotes the immediate reward reaped after executing action $a$ in state
$s$.

  * •

$s^{\prime}$ stands for the subsequent state the agent transitions into post-
action.

  * •

$\alpha$, the learning rate, dictates the extent to which the new Q-value
estimate will overwrite the previous estimate. It essentially balances the
trade-off between exploration (trying out new actions) and exploitation
(relying on known information).

  * •

$\gamma$ is the discount factor, capturing the agent’s degree of consideration
for future rewards. A value close to 1 makes the agent prioritize long-term
reward, whereas a value closer to 0 makes it focus on immediate rewards [4].

For HFT scenarios, the rapidity and dynamism of market movements accentuate
the importance of an agent’s ability to quickly and accurately estimate
Q-values. The temporal dependencies and non-stationarities inherent in
financial data make the iterative learning process of Q-Learning especially
relevant. By constantly updating its Q-value estimates based on the most
recent market feedback, the agent can potentially adapt its trading strategy
in real-time, capturing statistical arbitrage opportunities as they emerge.

Furthermore, while Q-Learning provides a robust framework, its tabular
representation becomes impractical in high-dimensional spaces. This limitation
catalyzed the evolution to Deep Q-Learning, where deep neural networks
approximate the Q-value function, allowing the algorithm to generalize across
states and handle the intricate data structures prevalent in HFT.

##  3 Neural Network Approximation

In the neural network approximation method approach, rather than using a table
to store Q-values, a neural network is employed to approximate the Q-value
function. The network, parameterized by weights $\theta$, offers a function
approximation mechanism, allowing the model to generalize across a myriad of
states. The strength of neural networks in capturing complex, non-linear
relationships makes them particularly suited for this task [15].

Mathematically, the neural network can be represented as:

$Q(s,a;\theta)=f(s;\theta)$

Here:

  * •

$Q(s,a;\theta)$ denotes the approximated Q-value for action $a$ in state $s$,
given the current parameters $\theta$.

  * •

$f$ is the function represented by the neural network.

  * •

The input $s$ is the state, which in the context of financial markets could be
a vector containing various market indicators, historical price data, trading
volumes, and more.

  * •

The output is a vector representing the Q-values for each possible action,
given the current state.

The training of this network involves adjusting the weights $\theta$ to
minimize the difference between the predicted Q-values and the target
Q-values. This is typically done using gradient descent optimization
techniques [16].

By employing neural networks as function approximators, Deep Q-Learning
leverages their capability to handle vast input spaces and model complex
relationships, making it particularly apt for financial markets. In the realm
of High-Frequency Trading (HFT), where decisions need to be made rapidly based
on vast amounts of data, the ability of the neural network to quickly provide
Q-value estimates for given market states becomes invaluable.

Moreover, the adaptive nature of neural networks, where they can continuously
learn and adjust to new data, aligns well with the dynamic and ever-evolving
landscape of financial markets. This ensures that the trading strategy remains
responsive to market shifts and can adapt in real-time, potentially maximizing
profits and minimizing losses [12].

##  4 Loss function

The primary objective during the training of the neural network in Deep
Q-Learning is to minimize the discrepancy between the predicted Q-values
(obtained from the current neural network) and the target Q-values (often
derived from a separate target network). The rationale behind this separation
is to provide a stable learning process, mitigating the risk of oscillations
or divergence in the learning updates [12].

The most commonly employed loss function in this context is the Mean Squared
Error (MSE). Mathematically, the MSE for Q-value approximation is represented
as:

$L(\theta)=\mathbb{E}\left[\left(r+\gamma\max_{a^{\prime}}Q(s^{\prime},a^{\prime};\theta^{-})-Q(s,a;\theta)\right)^{2}\right]$

In this equation:

  * •

$L(\theta)$ is the loss function given the current network parameters
$\theta$.

  * •

$r$ represents the immediate reward received after executing a certain action
in a particular state.

  * •

$\gamma$ is the discount factor, emphasizing the agent’s consideration for
future rewards.

  * •

$Q(s^{\prime},a^{\prime};\theta^{-})$ is the Q-value derived from the target
network with parameters $\theta^{-}$ for the next state $s^{\prime}$ and
action $a^{\prime}$.

  * •

$Q(s,a;\theta)$ is the Q-value predicted by the current network for the
present state $s$ and action $a$.

The term $\theta^{-}$ denotes the parameters of the target network, which is a
periodically updated version of the main Q-network. This periodic update,
rather than continuous synchronization with the main network, provides the
necessary stability to the learning process, ensuring that the updates do not
chase a constantly moving target, a phenomenon that could lead to unstable
learning dynamics.

Given the rapid pace of market movements and the potential for significant
financial implications based on trading decisions, ensuring that the neural
network accurately predicts Q-values is crucial. By minimizing the proposed
loss function, the Deep Q-Learning algorithm ensures that the trading strategy
it develops is based on accurate estimations of future rewards, thus enhancing
the potential for profitable trades.

##  5 Experience Replay

Experience Replay, a pivotal component in the domain of Deep Q-Learning,
serves as a mechanism to improve the stability and efficiency of the learning
process. Originating from the domain of control theory, its integration into
reinforcement learning has significantly bolstered the effectiveness of
algorithms like Deep Q-Learning, especially in complex environments such as
High-Frequency Trading (HFT).

In reinforcement learning, agents learn from interactions with the
environment. Traditionally, agents process these interactions sequentially,
learning from each experience as it occurs. However, this approach has
inherent limitations. Sequential processing can lead to strong correlations
between consecutive experiences, which, when used to train neural networks,
can result in oscillations or divergence in the learning process.

Experience Replay addresses this limitation. The core idea is to store each
experience, represented as a tuple $(s,a,r,s^{\prime})$, where $s$ is the
current state, $a$ is the action taken, $r$ is the received reward, and
$s^{\prime}$ is the subsequent state, in a data structure known as the replay
buffer. Rather than learning from experiences in a sequential manner, the
algorithm periodically samples a mini-batch of experiences from this buffer
for training. This random sampling ensures that the experiences used for
learning are decorrelated, providing two primary benefits:

  1. 1.

Diverse Training Samples: By sampling randomly from the buffer, the neural
network is exposed to a broader range of experiences during training,
enhancing its ability to generalize across different states and actions.

  2. 2.

Efficient Use of Past Experiences: Storing experiences in a replay buffer
allows the algorithm to revisit and learn from them multiple times. This not
only ensures that no experience is wasted but also aids in the stabilization
of the learning process by providing a more uniform distribution of
experiences [17].

Mathematically, the update using a mini-batch from the replay buffer can be
represented as:

$\Delta\theta=\mathbb{E}_{(s,a,r,s^{\prime})\sim
U(D)}\left[\nabla_{\theta}\left(r+\gamma\max_{a^{\prime}}Q(s^{\prime},a^{\prime};\theta^{-})-Q(s,a;\theta)\right)^{2}\right]$

Where $U(D)$ denotes uniform sampling from the replay buffer $D$, and
$\Delta\theta$ represents the change in network parameters during training.

Experience Replay ensures that the Deep Q-Learning agent remains robust. By
drawing from a diverse set of past market scenarios, the agent becomes better
equipped to handle the complexities and intricacies of financial markets,
making more informed and profitable trading decisions.

##  6 Application to HFT Statistical Arbitrage

###  6.1 State $s$:

In reinforcement learning, the concept of state is pivotal. It encapsulates
the current situation or environment in which the agent operates. In HFT, the
state is a rich tapestry of financial data and indicators:

  1. 1.

Recent Price Movements and Returns of Assets: These reflect the short-term
trends and momentum in the market. Capturing this data can provide insights
into the immediate direction of the market or a particular asset [18].

  2. 2.

Relevant Technical Indicators: These are mathematical calculations based on
historical price, volume, or open interest information that aims to forecast
financial market direction. Common indicators include Moving Averages,
Bollinger Bands, and the Relative Strength Index (RSI) [19].

  3. 3.

Market Microstructure Variables: These encompass details like trading volume,
which can indicate market sentiment, and bid-ask spread, which can reflect the
liquidity and potential transaction costs. Understanding market microstructure
can provide a deeper insight into market dynamics and potential price
movements [20].

Mathematically, the state can be represented as a vector:

$s=[p_{1},p_{2},\dots,p_{n},i_{1},i_{2},\dots,i_{m},v,b]$

Where $p$ represents price movements, $i$ denotes technical indicators, $v$ is
the trading volume, and $b$ is the bid-ask spread.

###  6.2 Action $a$:

In the context of trading, the action determines the trading decision. The
action space, though seemingly simple, carries profound implications:

  1. 1.

Buy: Acquiring assets, anticipating a price increase.

  2. 2.

Hold: Retaining the current portfolio without making new trades, often taken
when the market direction is uncertain.

  3. 3.

Sell: Offloading assets, either to realize profits or to mitigate potential
losses.

###  6.3 Reward $r$:

The reward structure is what drives the learning in reinforcement algorithms.
In trading:

  1. 1.

Profit or Loss from Trading: The immediate feedback from a trade, whether it
results in profit or loss, forms the primary component of the reward.

  2. 2.

Risk-Adjusted Reward: Simply focusing on profit can lead to high-risk
strategies. Therefore, rewards are often adjusted for risk, ensuring that the
strategy doesn’t adopt an overly aggressive stance for minimal gains [21].

Mathematically, the risk-adjusted reward can be defined as:

$r=\frac{\text{Expected Return}}{\text{Standard Deviation of Returns}}$

Where the numerator captures the profit or loss, while the denominator
represents the risk.

By continuously updating Q-values based on market feedback and utilizing the
deep neural network’s ability to generalize across various market states, the
Deep Q-Learning algorithm can dynamically adapt its trading strategy. This
adaptability is especially crucial in HFT, where market conditions can change
rapidly, and the algorithm needs to respond in real-time to capture fleeting
statistical arbitrage opportunities.

##  7 Conclusion

The world of High-Frequency Trading (HFT) represents one of the most dynamic
and complex arenas in the financial domain. Within this fast-paced
environment, Statistical Arbitrage stands out as a sophisticated strategy,
aiming to exploit temporary market inefficiencies through advanced
computational and statistical techniques. The incorporation of Deep Q-Learning
into this space underscores a promising evolution, marrying the strengths of
reinforcement learning with the demands of HFT.

Throughout this exploration, we delved deep into the mechanics of Q-Learning
and its neural network-based extension, Deep Q-Learning. The integration of
neural network approximators, experience replay, and a meticulous reward
structure has showcased the potential for superior adaptability and precision
in trading decisions. By leveraging a neural network’s prowess in capturing
intricate patterns and relationships, Deep Q-Learning offers the ability to
navigate the vast and high-dimensional state spaces typical of financial
markets.

Experience replay, with its emphasis on random sampling and decorrelation of
experiences, enhances the stability of the learning process. Such stability
becomes paramount in HFT, where market dynamics shift rapidly, and the
algorithm needs to ensure consistent and reliable performance.

Furthermore, the meticulous design of states and actions within the HFT
context ensures that the agent possesses a comprehensive understanding of the
market environment. By continuously updating based on market feedback and
employing a neural network to generalize across states, Deep Q-Learning
exhibits the potential to adapt trading strategies in real-time, capturing
fleeting statistical arbitrage opportunities.

In wrapping up, it’s evident that the fusion of Deep Q-Learning with HFT
Statistical Arbitrage heralds a new era in algorithmic trading. As financial
markets continue to evolve and become increasingly complex, the need for
adaptive, data-driven strategies becomes paramount. Deep Q-Learning, with its
blend of deep learning and reinforcement learning principles, offers a beacon
of promise in this pursuit. As we move forward, it will be intriguing to
witness the real-world applications and refinements of this approach,
potentially setting new benchmarks in the domain of algorithmic trading.

## References

  * [1] Aldridge, I. (2013). High-frequency trading: a practical guide to algorithmic strategies and trading systems. John Wiley & Sons.
  * [2] Avellaneda, M., & Lee, J. H. (2010). Statistical arbitrage in the US equities market. Quantitative Finance, 10(7), 761-782.
  * [3] Chaboud, A. P., Chiquoine, B., Hjalmarsson, E., & Vega, C. (2014). Rise of the machines: Algorithmic trading in the foreign exchange market. The Journal of Finance, 69(5), 2045-2084.
  * [4] Sutton, R. S., & Barto, A. G. (2018). Reinforcement learning: An introduction. MIT press.
  * [5] Nevmyvaka, Y., Feng, Y., & Kearns, M. (2006). Reinforcement learning for optimized trade execution. In Proceedings of the 23rd international conference on Machine learning (pp. 673-680).
  * [6] Atsalakis, G. S., & Valavanis, K. P. (2009). Surveying stock market forecasting techniques–Part II: Soft computing methods. Expert Systems with Applications, 36(3), 5932-5941.
  * [7] Moody, J., & Saffell, M. (2001). Learning to trade via direct reinforcement. IEEE transactions on neural networks, 12(4), 875-889.
  * [8] Deng, Y., Bao, F., Kong, Y., Ren, Z., & Dai, Q. (2017). Deep direct reinforcement learning for financial signal representation and trading. IEEE transactions on neural networks and learning systems, 28(3), 653-664.
  * [9] García, J. E., & Fernández, F. (2015). A comprehensive survey on safe reinforcement learning. Journal of Machine Learning Research, 16(1), 1437-1480.
  * [10] Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2), 223-236.
  * [11] Watkins, C. J., & Dayan, P. (1992). Q-learning. Machine Learning, 8(3-4), 279-292.
  * [12] Mnih, V., Kavukcuoglu, K., Silver, D., Graves, A., Antonoglou, I., Wierstra, D., & Riedmiller, M. (2015). Human-level control through deep reinforcement learning. Nature, 518(7540), 529-533.
  * [13] Puterman, M. L. (1994). Markov decision processes: Discrete stochastic dynamic programming. John Wiley & Sons.
  * [14] Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep learning. MIT press.
  * [15] Schmidhuber, J. (2015). Deep learning in neural networks: An overview. Neural Networks, 61, 85-117.
  * [16] Ruder, S. (2016). An overview of gradient descent optimization algorithms. arXiv preprint arXiv:1609.04747.
  * [17] Lin, L. J. (1992). Self-improving reactive agents based on reinforcement learning, planning and teaching. Machine Learning, 8(3-4), 293-321.
  * [18] Bouchaud, J. P., Farmer, J. D., & Lillo, F. (2018). How markets slowly digest changes in supply and demand. Handbook of Financial Markets: Dynamics and Evolution, 57-156.
  * [19] Murphy, J. J. (1999). Technical analysis of the financial markets: A comprehensive guide to trading methods and applications. Penguin.
  * [20] O’Hara, M. (1995). Market microstructure theory. Blackwell.
  * [21] Sharpe, W. F. (1994). The Sharpe Ratio. The Journal of Portfolio Management, 21(1), 49-58.