# Deeply-virtual and photoproduction of mesons at higher-order and higher-
twist ††thanks: Presented at XXIX Cracow Epiphany Conference on Physics at the
Electron-Ion Collider and Future Facilities, Cracow, Poland, 16-19 January
2023

K. Passek-K.
Division of Theoretical Physics, Rudjer Bošković Institute, Zagreb, Croatia

###### Abstract

Both deeply-virtual and photoproduction of mesons offer promising access to
generalized parton distributions and complementary description of different
kinematical regions. The higher-order contributions offer stabilizing effect
with respect to the dependence on renormalization scales, while higher-twist
effects have been identified as especially important in the case of the
production of pseudo-scalar mesons. This was confirmed by recent evaluation of
the complete twist-3 contribution to $\pi$ and $\eta$/$\eta^{\prime}$
photoproduction and its confrontation with experimental data.

##  1 Introduction

Historically, most of the information about the high-energy nucleon structure
came from the deeply inelastic scattering (DIS). From DIS data one extracts
the parton distribution functions (PDFs) being the probabilities that a
certain parton is found in a nucleon with certain longitudinal momentum
fraction of the nucleon momentum. Through PDFs the one dimensional structure
of the nucleon is thus revealed. The hard-exclusive processes offer insight
into the transverse distribution of partons and corresponding generalized
parton distributions (GPDs) give access to nucleon 3D structure. GPDs are
functions of three variables: $x$, the parton’s ”average” longitudinal
momentum fraction, $\xi$, the longitudinal momentum transfer (skewness
parameter), and $t$, momentum transfer squared, while their evolution with
energy is encapsulated in the dependence on the factorization scale. At
leading twist-2 there are eight quark GPDs and eight gluon GPDs classified
according to different quantum numbers (parity, chirality), as well as
different GPDs for different quark flavours. To reveal their form is thus not
an easy task and information from several processes have to be combined.

For the description of the hard exclusive processes one employs the handbag
mechanism in which only one quark from the incoming nucleon and one from the
outgoing nucleon participate in the hard subprocess while all other partons
are spectators. The simplest and well-investigated process to which this
approach has been applied is Compton scattering
$\gamma^{(*)}N\rightarrow\gamma N$, while meson electroproduction
$\gamma^{(*)}N\rightarrow MN^{\prime}$ represents the natural extension and
offers access to quark flavours. A prerequisite for the handbag mechanism is
the presence of at least one large scale, which allows for the use of
perturbative expansion in the strong coupling constant and the power, i.e.,
twist, expansion. Two kinematic regions have been extensively studied: the
deeply virtual (DV) region, where the virtuality $Q^{2}$ of the incoming
photon is large, and the momentum transfer $(-t)$ from the incoming to the
outgoing nucleon is small; and the wide-angle (WA) region, where $(-t)$,
$(-u)$, and $s$ are all large, while $Q^{2}$ is smaller than $(-t)$ ($Q^{2}=0$
in the case of photoproduction). Factorization proofs exist for all orders for
DV Compton scattering (DVCS) [1] and DV meson production (DVMP) [2], with the
process amplitudes factorizing into hard perturbatively calculable subprocess
amplitudes and GPDs that encapsulate the soft hadron-parton transitions and
the hadron structure. However, general factorization proofs are still lacking
for WA processes, although it has been shown that factorization holds to next-
to-leading order in the strong coupling for WA Compton scattering (WACS) [3,
4] and to leading order for WA meson production (WAMP) [5]. It is argued that
in the symmetric frame where skewness is zero, the amplitudes can be
represented as a product of subprocess amplitudes and form factors that
represent $1/x$ moments of GPDs at zero-skewness.

Both DVCS and WACS were widely investigated in the last decades and the
handbag factorization achieved a good description of the experimental data.
The leading twist-2 description of DV vector meson production only considers
the contributions of longitudinally polarized photons, specifically
$\gamma^{*}_{L}N\rightarrow V_{L}N^{\prime}$. This description has been
observed to be in relatively good agreement with the current experimental data
(see [6, 7] and the references therein). However, there is still a lack of
systematic separation between longitudinal and transverse experimental data.
The contributions of transversely polarized photons
$\gamma^{(*)}_{T}N\rightarrow V_{L,T}N^{\prime}$ have also been investigated
by including the twist-3 corrections to the meson state [8, 9]. On the other
hand, the experimental data for DV pion production [10, 11, 12, 13] suggest
the high significance of transversely polarized photons, which are not
accounted for by the leading twist-2 $\gamma^{*}_{L}N\rightarrow\pi
N^{\prime}$ contributions. As in the vector meson case, a twist-3 calculation
has been proposed, which incorporates twist-2 chiral-odd, i.e., transversity
(parton helicity flip), GPDs and twist-3 pion corrections. The calculation
including only the twist-3 2-body pion Fock component (Wandzura-Wilczek
approximation) has already achieved a successful agreement with the data [14].
Experimental data for WA pion production [15, 16, 17] also indicate that the
twist-2 contributions [5] are not sufficient. But unlike DVMP, the twist-3
contribution to pion photoproduction was found to vanish in the commonly used
Wandzura-Wilczek approximation. In [19], both 2- and 3-body twist-3 Fock
components of $\pi_{0}$ were considered and successfully fitted to CLAS data
[17]. This work was extended to photoproduction of $\eta$ and $\eta^{\prime}$
mesons [20] and WA electroproduction of $\pi^{\pm},\pi^{0}$ [21]. The
application of the latter analytical results for the subprocess amplitudes to
the DVMP subprocess amplitudes is straightforward, and the phenomenological
analysis is underway.

The DV and WA regions enable complementary access to GPDs at small and large
$(-t)$, respectively. A vast amount of experimental data needs to be
confronted with reliable theoretical predictions, which should include higher-
order perturbative predictions as well as higher-twist contributions. Here, we
provide a brief overview of some recent developments.

##  2 Deeply-virtual meson production at twist-2 and NLO

The DVMP amplitude $\gamma^{*}N\to MN^{\prime}$ can be expressed through, so-
called, transition form factors

${}^{a}\mathcal{T}(\xi,t,Q^{2})=\int_{-1}^{1}\frac{{\rm
d}x}{2\xi}\;\int_{0}^{1}{\rm
d}u\;T^{a}(x,\xi,u,\mu)\;F^{a}(x,\xi,t,\mu)\;\phi(u,\mu)\,$ (1)

with $a$ denoting quark and gluon contributions, and $u$ the longitudinal
momentum fraction of the meson’s parton. The factorization scale $\mu$
separates the short-distance dynamics, represented by the subprocess
amplitudes $T^{a}$, from the long-distance dynamics represented by the hadron
wave functions: the GPD $F^{a}$ and the meson distribution amplitude (DA)
$\phi$.

Transition form factors ${}^{a}\mathcal{T}$ have the similar role in DVMP as
Compton form factors in DVCS, but they additionally depend on meson DA, i.e.,
meson structure, making the analysis of the process both more challenging, as
well as, potentially more rewarding. In contrast to DVCS, DVMP enables easy
access to GPDs of different quark flavours, and offers the natural distinction
of GPDs of different parity: at twist-2 chiral-even GPDs $H^{q}$, $E^{q}$
contribute to the production of longitudinally polarized vector mesons
($V_{L}$) and scalar ($S$) mesons, while $\widetilde{H}^{q}$ and
$\widetilde{E}^{q}$ appear in the production of pseudoscalar ($P$) and axial-
vector ($A_{L}$) mesons. Moreover, the contribution of gluon GPDs $H^{g}$,
$E^{g}$ ($\widetilde{H}^{g}$, $\widetilde{E}^{g}$) to the production of
neutral $V_{L}$ ($A_{L}$) mesons is more significant since, unlike in DVCS,
they contribute already at the leading order. Therefore, their form is
phenomenologically more accessible.

The twist-2 DVMP subprocess amplitudes $\gamma^{*}_{L}q\to(q\bar{q})q$ and
$\gamma^{*}_{L}g\to(q\bar{q})g$ are calculated perturbatively order by order
in the strong coupling constant

$T^{a}(x,\xi,u,\mu)=\frac{\alpha_{s}(\mu_{R})}{4\pi}T^{a(1)}(x,\xi,u)+\frac{\alpha_{s}^{2}(\mu_{R})}{(4\pi)^{2}}T^{a(2)}(x,\xi,u,\mu_{R},\mu)+\cdots$
(2)

and they have been determined to next-to-leading order (NLO) for flavour non-
singlet and singlet $P$ and $V_{L}$ mesons [22, 23, 24], as well as, for the
(crossed) production of $S$ and $A_{L}$ mesons [25, 24]. Predictions at finite
order are inherently dependent on the renormalization scale $\mu_{R}$ and
scheme, introducing additional theoretical uncertainty. Therefore, the
inclusion of higher-order corrections is crucial to reduce this dependence and
stabilize predictions. Although meson DAs ($\phi$) and GPDs ($F^{a}$) are
intrinsically nonperturbative quantities, their evolution can be calculated
perturbatively. The complete closed form is known to NLO order [26], and more
recently, NNLO contributions to the evolution kernels have been obtained [27].

Figure 1: Relative NLO corrections to the imaginary part of the flavor singlet
TFF (solid) broken down to the gluon (dashed), pure singlet quark (dash-
dotted) and ‘non-singlet’ quark (dotted) contributions (Ref. [25]).  Figure 2:
Skewness ratio for GPD $H$ (preliminary K. Kumerički, Transversity 2022).

The evolution is simpler to implement in the conformal momentum
representation. Conformal moments are analogous to Mellin moments in DIS and
represent the moments with respect to the eigenfunctions of the leading order
evolution kernels, i.e., with respect to Gegenbauer polynomials $C_{n}^{3/2}$
and $C_{n}^{5/2}$ for quarks and gluon, respectively. The convolution over $x$
and $u$ in transition form factors (1) is thus replaced by the summation over
conformal moments, and consequently the series is summed using the Mellin-
Barnes integral over complex conformal moment $j$ [25]

$\displaystyle{}^{a}\mathcal{T}(\xi,t,Q^{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2i}\int_{c-i\infty}^{c+i\infty}\\!dj\left[i\pm\left\\{{\tan\atop\cot}\right\\}\left(\frac{\pi
j}{2}\right)\right]\,\xi^{-j-1}$ (3)
$\displaystyle\times\left[T^{a}_{jk}(\mu)\stackrel{{\scriptstyle
k}}{{\otimes}}\phi_{k}(\mu)\right]F^{a}_{j}(\xi,t,\mu)\,.$

This approach has been developed and extensively applied to DVCS [28], and
then extended to DVMP. Regardless of whether one considers Compton or
transition form factors in momentum fraction (1) or conformal momentum space
(3), complete deconvolution is impossible, and GPD access is only possible
through different modelling approaches. Dedicated software is now available:
PARTONS [29] and Gepard [30] for analysis in momentum fraction and conformal
momentum space, respectively. The DV$V_{L}$P is included in the latter.

While there has been a lot of interest in the DVCS process, there are
relatively few NLO phenomenological analyses of the DVMP process [22, 31, 25],
despite the availability of experimental data. The complete set of $x$ and $j$
space analytical results for all meson channels can be found in [25, 24]. The
numerical analysis performed there shows that NLO corrections are important
and model-dependent (Fig.1). The effects of LO GPD and DA evolution are
significant and for NLO calculations one should include also NLO evolution.
Gluon corrections play a significant role in small $\xi$ production of vector
mesons, and there may be a need for resummation of the large logarithmic
$\ln(1/\xi)$ terms observed in both gluon evolution and gluon coefficient
function. Finally, the choice of meson distribution amplitude is found to have
a significant impact on the results.

Since GPDs are process-independent quantities, the simultaneous description
and global fits of GPDs to DIS, DVCS and DVMP data represent the next
necessary step. Through these one hopes to gain additional information on the
importance and stability of NLO predictions and validity of different models.
Using the conformal momentum representation [25] the first global fits on DIS,
DVCS and DV$V_{L}$P small-x HERA collider data have been performed at LO [32]
($\chi^{2}/n_{\mathrm{d.o.f}}\approx 2$), and at NLO [33] (Bayesian analysis).
The recent NLO analysis using corrected NLO analytical results [24] and Gepard
software shows promising agreement of theory and experiment
($\chi^{2}/n_{\mathrm{d.o.f}}$ = 254.3/231) and indicates that a global
description of DVCS and DV$V_{L}$P is reachable at NLO (Fig. 2).

##  3 Pseudoscalar meson production at higher-twist

Figure 3: Generic diagrams for 2- and 3-body subprocess amplitudes.

The twist-3 LO prediction for the electroproduction of the pseudoscalar meson
$P$, which includes 2- and 3-body meson Fock states, was first calculated for
the WA region [21]. Here, we review the findings and their confrontation with
experimental data for photoproduction ($Q^{2}=0$). The analytical expressions
obtained for the subprocess amplitudes can also be applied for the DV$P$P
analysis ($t\to 0$).

The helicity amplitudes for $\gamma^{(*)}N\rightarrow PN^{\prime}$ process in
the WA angle region can be expressed in terms of the subprocess amplitudes
${\cal H}$ multiplied by the soft form factors, $R_{i}^{P}$ and $S_{i}^{P}$,
which represent of $1/x$-moments of zero-skewness GPDs
$\int_{0}^{1}\frac{dx}{x}\,F_{i}^{a}(x,t)$. The $R$-type form factors are
related to the helicity non-flip GPDs $H$, $\widetilde{H}$ and $E$. The
$S$-type form factors are related to the helicity-flip or transversity GPDs
$H_{T}$, $\bar{E}_{T}$ and $\widetilde{H}_{T}$111The GPDs $\tilde{E}$ and
$\tilde{E}_{T}$ and their associated form factors decouple at zero skewness..

The amplitudes ${\cal H}$ correspond to the subprocesses
$\gamma^{(*)}q\rightarrow Pq^{\prime}$ and they are calculated using handbag
diagrams as the ones depicted on Fig. 3. The meson $P$ is replaced by an
appropriate 2- or 3-body Fock state. The projector $\pi\rightarrow q\bar{q}$
contributes to the subprocess amplitudes corresponding to the diagrams
depicted on Fig. 3a and its structure is given by

${\cal P}_{2}^{P}\sim
f_{\pi}\,\Big{\\{}\gamma_{5}\,p\\!/\phi_{\pi}(u,\mu)+\mu_{\pi}(\mu)\gamma_{5}\Big{[}\,\phi_{\pi
p}(u,\mu)-[\ldots]\phi^{\prime}_{\pi\sigma}(u,\mu)+[\ldots]\phi_{\pi\sigma}(u,\mu)\Big{]}\Big{\\}}\,.$
(4)

The first term in (4) corresponds to the twist-2 part, while the twist-3 part
is proportional to the chiral condensate
$\mu_{\pi}=m_{\pi}^{2}/(m_{u}+m_{d})\cong 2$ GeV (at the factorization scale
$\mu_{F}=2$ GeV). This parameter is large and although the twist-3 cross-
section for pion electroproduction is suppressed by $\mu_{\pi}^{2}/Q^{2}$ as
compared to twist-2 cross section, for the range of $Q^{2}$ accessible in
current experiments the suppression factor is of order unity222 Twist-3
effects can also be generated by twist-3 GPDs. However, these are expected to
be small and therefore neglected.. The 3-body $\pi\rightarrow q\bar{q}g$
projector contributes to the amplitudes corresponding to Fig. 3b,

${\cal P}_{3}^{P}\sim
f_{3\pi}(\mu)\gamma_{5}[\ldots]\,\phi_{3\pi}(u_{1},u_{2},u_{g},\mu)\,.$ (5)

The helicity non-flip amplitudes are generated by twist-2, while the helicity
flip ones are of twist-3 origin.

In addition to twist-2 DA $\phi_{\pi}$ there are two 2-body twist-3 DAs,
$\phi_{\pi p}$ and $\phi_{\pi\sigma}$, and 3-body twist-3 DA $\phi_{3\pi}$.
Twist-3 DAs are connected by equations of motion (EOMs). By EOMs and DA
symmetry properties, it is possible to express the twist-3 subprocess
amplitudes in terms of only two twist-3 DAs, and combine 2- and 3-body
contributions. Applying EOMs also results in an inhomogeneous linear first-
order differential equation, which can be used to determine $\phi_{\pi p}$
(and $\phi_{\pi\sigma}$) from a known 3-body DA $\phi_{3\pi}$ [34]333It is
important to note that the same gauge must be used consistently for the
constituent gluon in the $q\bar{q}g$ projector and EOMs. .

In meson electroproduction, both transverse and longitudinal photons
contribute to twist-2 subprocess amplitudes. As expected, the longitudinal
contribution vanishes in the photoproduction limit , while in the DVMP limit
only longitudinal photons contribute. The general structure of the twist-3
contributions for both transverse and longitudinal photons reads

$\displaystyle{\cal H}^{P,tw3}$ $\displaystyle=$ $\displaystyle{\cal
H}^{P,tw3,q\bar{q}}+{\cal H}^{P,tw3,q\bar{q}g}$ (6) $\displaystyle=$
$\displaystyle\big{(}{\cal H}^{P,\phi_{\pi p}}+\underbrace{{\cal
H}^{P,\phi_{\pi 2}^{EOM}}\big{)}+\big{(}{\cal H}^{P,q\bar{q}g,C_{F}}}+{\cal
H}^{P,q\bar{q}g,C_{G}}\big{)}$ $\displaystyle=$ $\displaystyle{\cal
H}^{P,\phi_{\pi p}}\;\;+\quad\qquad{\cal
H}^{P,\phi_{3\pi},C_{F}}\qquad\quad+{\cal H}^{P,\phi_{3\pi},C_{G}}\,,$

where ${\cal H}^{P,tw3,q\bar{q}}$ is the twist-3 2-body contribution
proportional to the $C_{F}$ color factor, and ${\cal H}^{P,tw3,q\bar{q}g}$ is
the twist-3 3-body contribution with $C_{F}$ and $C_{G}$ proportional parts.
The $C_{G}$ part is gauge invariant, whereas for $C_{F}$ contributions, only
the sum of 2- and 3-body parts is gauge invariant with respect to the choice
of photon or virtual gluon gauge. EOMs are used to obtain this sum, as well as
the complete twist-3 contribution expressed through only two twist-3 DAs,
$\phi_{3\pi}$ and $\phi_{\pi p}$. The twist-3 subprocess amplitude for
longitudinal photons vanishes both for photoproduction and DVMP. One finds
that for photoproduction ${\cal H}^{P,\phi_{\pi p}}=0$ [19]. For DVMP, ${\cal
H}^{P,\phi_{\pi 2}^{EOM}}=0$, and while no end-point singularities are present
for $t\neq 0$, they must be considered in the limit $t\to 0$ since for ${\cal
H}^{P,\phi_{\pi p}}\sim\int_{0}^{1}\frac{du}{u}\phi_{Pp}(u)$. In [14] the
modified hard-scattering picture has been used to regularize the 2-body
contributions. With the complete twist-3 contribution now available [21] the
analysis in modified and collinear picture is underway.

In [19] the cross-section for $\pi^{0}$ photoproduction has been fitted to
[17] data. The results are displayed in Fig. 4.

Figure 4:  The cross section for $\pi^{0}$ photoproduction with twist-3
contributions. Solid (dotted) curve: full (twist-2) result. Dashed curve: full
result with fixed renormalization and factorization scale. Data taken CLAS
[17] (open circles) and from SLAC [15] ($s=10.3$ GeV2) (Ref. [21]).

The twist-2 prediction lies well below the data. However, by including the
twist-3 contributions one obtains reasonable agreement with the experiment.
Twist-3 is more important in the backward hemisphere ($\theta$ is c.m.s.
scattering angle). In [21], the analysis was extended to $\pi^{+}$ and
$\pi^{-}$, using only a few available experimental data [15, 16]. In [20],
$\eta$ (preliminary GlueX data) and $\eta^{\prime}$ photoproduction was
studied. A similar behavior in photoproduction cross-sections was observed,
except for $\eta^{\prime}$, where the twist-2 contribution was significant,
offering the possibility of determining the 2-gluon twist-2 DA.

For pion electroproduction there are four partial cross sections. In [21] the
theoretical predictions were given and the importance of the measurement was
stressed. Different combinations of form factors make it possible to extract
transversity GPDs ($F^{q}_{T}$), which have a large $-t$ behavior that is
important for parton tomography.

Figure 5: Results for the helicity correlation parameters $A_{LL}$ and
$K_{LL}$ for $\pi^{+}$, $\pi^{-}$ and $\eta^{\prime}$ photoproduction (Refs.
[21, 20]).

In meson photoproduction, spin-dependent observables such as the correlations
of the helicities of the photon and either the incoming or outgoing nucleon,
i.e., $A_{LL}$ and $K_{LL}$, offer additional insight that is less sensitive
to particular parameters. It can be shown that $A_{LL}^{P,tw2}=K_{LL}^{P,tw2}$
and $A_{LL}^{P,tw3}=-K_{LL}^{P,tw3}$, indicating that the measurement of
$A_{LL}$ and $K_{LL}$ offers a characteristic signature for the dominance of
twist-2 or twist-3, similar to the role that the comparison of $\sigma_{T}$
and $\sigma_{L}$ has in DVMP. From Fig. 5, it is clear that our numerical
results suggest the dominance of twist-3 for large $\theta$, while twist-2
increases in the forward direction.

##  4 Summary and outlook

Twist-2 NLO contributions to DVMP amplitudes are available and need to be
compared with experimental data. The preliminary comparison of vector meson
production to data seems satisfactory, but NLO corrections are significant,
and the first DIS, DVCS, and DV$V_{L}$P fits have been performed. For
pseudoscalar meson production, twist-3 contributions dominate, and a complete
analysis of 2- and 3-body twist-3 contributions is ongoing. The available
twist-2 NLO contributions should also be tested. It is important to note that
the choice of meson distribution amplitude significantly affects the DVMP
predictions. In WA photoproduction of $\pi$ mesons, the twist-2 analysis falls
short by an order of magnitude. The complete twist-3 contribution has been
included, and it was found that the meson’s twist-3 contributions dominate for
$\pi$s and $\eta$. Future experimental goals include the clear separation of
longitudinally and transversely polarized photon contributions.

Acknowledgements This publication is supported by the Croatian Science
Foundation project IP-2019-04-9709, by the EU Horizon 2020 research and
innovation programme, STRONG-2020 project, grant agreement No 824093.

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# Understanding Revision Behavior in Adaptive Writing Support Systems for
Education

Luca Mouchel

Thiemo Wambsganss

Paola Mejia-Domenzain

EPFL<EMAIL_ADDRESS>EPFL<EMAIL_ADDRESS>EPFL
<EMAIL_ADDRESS>Tanja Käser

EPFL<EMAIL_ADDRESS>
###### Abstract

Revision behavior in adaptive writing support systems is an important and
relatively new area of research that can improve the design and effectiveness
of these tools, and promote students’ self-regulated learning (SRL).
Understanding how these tools are used is key to improving them to better
support learners in their writing and learning processes. In this paper, we
present a novel pipeline with insights into the revision behavior of students
at scale. We leverage a data set of two groups using an adaptive writing
support tool in an educational setting. With our novel pipeline, we show that
the tool was effective in promoting revision among the learners. Depending on
the writing feedback, we were able to analyze different strategies of learners
when revising their texts, we found that users of the exemplary case improved
over time and that females tend to be more efficient. Our research contributes
a pipeline for measuring SRL behaviors at scale in writing tasks (i.e.,
engagement or revision behavior) and informs the design of future adaptive
writing support systems for education, with the goal of enhancing their
effectiveness in supporting student writing. The source code is available at
https://github.com/lucamouchel/
Understanding-Revision-Behavior.

###### keywords:

Revision Behavior, Writing Support Systems, ML-based adaptive feedback, Self-
Regulated Learning

##  1 Introduction

Intelligent writing support tools (e.g., Grammarly, WordTune or Quilbot) offer
new ways for learners to receive feedback and thus revise their texts [19].
These and other writing support systems bear the potential to provide learners
with needed adaptive feedback on their writing exercises when educators are
not present, (e.g., on grammatical mistakes [32], argumentation [26], empathy
[29], or general persuasive writing [27]). They can help students in their
self-regulated learning (SRL) process [9, 36], to organize their thoughts and
ideas, reflect on their learnings, or simply receive feedback on frequently
occurring grammar or argumentation mistakes. From an educational perspective,
it is important to understand how these tools are used by learners in
educational settings and how they improve the effectiveness of educational
scenarios [7, 18]. Present research is largely focused on designing and
building writing support systems [8, 20]. However, there are not many insights
into the effects of the usage of these tools and their impact on students’ SRL
processes [4, 25], which is why we contribute a novel pipeline analyzing and
visualizing revision behavior to better understand how we can design, develop
and improve existing systems to better support students. Techniques from the
field of data mining are a solution to understanding revision behavior and
explaining SRL. One such technique is Keystroke Logging (KL). KL allows us to
use educational data mining to analyze user behavior in writing tasks [16, 35,
23]111Tools such as InputLog [16] or ETS [35] are examples of KL programs.. In
this study, we model, inspect, and analyze quantitative data in learners’
writing interactions through KL by developing a novel pipeline. We use a
keystroke log from an experiment, where users were divided into two groups.
The first one was given adaptive feedback and the second one was not. A
detailed description of our dataset and the experiment demographics and
procedure are available in Section 3. To the best of our knowledge, no
publicly available pipeline exists that focuses on processing the keystroke
behavior of learners and helps analyze SRL characteristics such as engagement,
revision, or visualize the learning path. We intend to first identify and
visualize the differences between these two groups in their revision process
and compare different user profiles and measure their engagement over time. We
use an exemplary data set to build this pipeline and apply data mining in
order to gain insights into the underlying process of this writing activity.

##  2 Background

### Research on Automatic Data Mining for Writing Behaviour

Research in writing process analysis can be traced to the 1970s [10, 24].
However, only more recently have studies been focusing theoretically on
behavioral and cognitive processes of writing [14, 17]. In fact, Flower and
Hayes [11] laid the groundwork for research on the psychology of writing. They
propose that the act of writing is propelled by goals, which are created by
the writer and grow in number as the writing progresses. Today, writing
support tools need to support this cognitive process as it emphasizes writers’
intentions, rather than their actions [12]. It is important to understand what
these tools help with, and how we may design new ones [12]. While prior works
on text revision [8, 15, 20, 28] have proposed machine collaborative writing
interfaces, they focus on collecting human-machine interaction data to better
train neural models, rather than understanding the underlying processes of
text revision. Several studies in the past have used KL as a technique to
study revision [16, 23, 35] in different settings and some of the aims were to
understand and evaluate keystroke log features in a writing task context.
However, until now, KL has been scarcely used in the classroom [25]. One issue
with keystroke loggers is their invasive nature. KL raises several ethical
issues, most notably privacy violation [25], but in this study, participants
gave their consent for the collection of their data, all the while preserving
their privacy. Previous research has suggested that writing time and number of
keystrokes, which are indicative of general writing fluency and effort, are
related to writing quality [1, 33]. Another feature of interest is pause
times, [34] found that under a certain timed-writing test condition, shorter
pauses are preferred as that indicates an adequate understanding of the task
requirements, more familiarity with the writing topic, and better task
planning [23].

### Self-Regulated Learning

To analyze revision behavior, we rely on the lens of self-regulated learning
(SRL). SRL refers to the pro-active process that learners engage in to
optimize their learning outcome [36]. According to Zimmerman’s model of SRL
[36], there are three major phases: forethought, performance and self-
reflection. The forethought phase includes task analysis, such as goal setting
and strategic planning and self-motivational beliefs. The performance phase
includes self-control processes, such as task and attention-focusing
strategies. The self-reflection phase includes processes involving self-
judgment and self-reaction [31]. SRL is essential in the context of studying
revision behavior in writing support systems as it allows writers to take an
active role in identifying and addressing their own writing weaknesses, rather
than simply relying on the writing support system to automatically detect and
correct errors. This can lead to a deeper understanding of the writing
process.

##  3 Method

To investigate revision behavior in the writing process, we propose a pipeline
for the automatic analysis of the SRL behavior of users during a writing task.
Our work follows the Knowledge Discovery in Databases process by following the
methodology in Fig. 1.

Overview of our pipeline and methodology, following the KDD process

Figure 1: Overview of our pipeline and methodology, following the KDD process

### Demographics, Procedure & Dataset Description

With approval of the ethical board of our university, we collected data from a
writing experiment which consisted of 73 users divided into two groups, as
illustrated in Table 1.

Table 1: Demographics of the participants per group from the exemplary data set |  With Adaptive Feedback (G1) |  Without Adaptive Feedback (G2)
---|---|---
No. Participants |  34 |  39
Age Mean |  26.8 |  26.3
Age Std |  3.3 |  2.8
% Female |  43 |  51
% Male |  51 |  46
% Other |  6 |  3

The two groups of users were tasked with writing three cooking recipes. Both
groups were given a sample recipe as reference. The first group (G1) received
adaptive feedback from the platform when they submitted their texts. The
second group (G2) did not receive any feedback. Once they submitted their
recipes to the system, users in G1 had the option to reset and start a new
recipe or revise their texts based on the feedback. The same protocol was
followed for G2, but they did not receive feedback. Here are several examples
of the feedback the platform provided users: ’List each ingredient
separately.’, ’Enumerate the steps.’, ’How can your recipe be more specific?’,
’Use stir, mix, or beat instead of “add” to be more specific.’ or ’Indicate
whether the meat, poultry, or seafood is boned, skinned, or otherwise
prepared.’

With regards to the dataset, the entries of the log data we collected
consisted of user ids, event dates, the keystroke logs as a JSON file and the
final version of the text submitted at that particular date. An example of an
entry is as follows: 2023-01-01, 12:00:00, user1, [{’time’: 1, ’character’:
’a’}, ... }], "a) Cook ...".

### Qualitative Perception

Following the experiment, users were tasked with answering follow-up questions
and we identified eight different topics regarding the reported revisions,
including, adding missing ingredients, improving the clarity, not making any
changes and others. To do this, we used BERTTopic [13], a topic modeling
technique that clusters sentence embeddings generated by Sentence-BERT [22],
to perform qualitative analysis of participants’ open responses about recipe
revisions: (What did you edit (add, remove or change) from the original text
(the recipe you wrote)?). We split the sentences into clusters based on their
relevance, assigned names to each cluster, and computed the probability of
each sentence belonging to a cluster. We grouped the sentences by participant
to obtain the set of topics associated with their entire text answer. For
example, if a participant’s answer consisted of sentences with assigned topics
$A$, $B$, and $C$, the set of topics associated with their answer would be
$Z=\\{A,B,C\\}$.

### Data Processing

Given that the logs consist of the users’ first attempts at writing one of the
three recipes and their respective revision phases, it is important to
separate them in order to focus only on the revision steps. We define sessions
for a user as all the data collected from them for one recipe. To separate
sessions, we use cosine distance to detect where the session ends and where
the next one starts. One advantage of using cosine distance for text
comparison is that it is relatively insensitive to the length of the strings.
In contrast, other measures of distance such as Euclidean distance are
sensitive to the length of the vectors and can be affected by the presence of
common words that do not contribute significantly to the meaning of the
strings. To map sentences to 50-dimensional vectors, we use a GloVe model
[21], which is already trained on Wikipedia. First, we map each word to their
embeddings and then compute the sum of the vectors component-wise. Formally,
each text submitted $t$ has a set of words $\mathcal{W}=\\{w_{i}\;|\;1\leq
i\leq N_{t}\\}$, where $N_{t}$ is the number of words for text $t$. Then, we
map each $w_{i}$ to their embeddings $\tilde{w}_{i}$ which are 50-dimensional
vectors. Now let $\tilde{t}$ be the embedding of the text $t$, then
$\tilde{t}=\sum_{i=1}^{N_{t}}\tilde{w}_{i}$. This allows us to capture each
word of the text and this way, we can collect the set of all text embeddings
in the dataset
$\mathcal{T}=\\{\tilde{t}_{1},\tilde{t}_{2},...\\}=\left\\{\sum_{i=1}^{N_{1}}\tilde{w}_{i},\sum_{i=1}^{N_{2}}\tilde{w}_{i},...\right\\}$.
We use $\mathcal{T}$ to run the recursive algorithm described in Appendix C on
the recipes submitted and compute the cosine distance between the text
embeddings of $t_{k}$, $t_{k+1}$, $...$, starting at $k=0$, until we find
$n>k$

$1-\frac{\langle\tilde{t}_{k},\;\tilde{t}_{n}\rangle}{\lVert\tilde{t}_{k}\rVert\cdot\lVert\tilde{t}_{n}\rVert}<0.995$

When we do, we define $n$ as the index of a new recipe in our data. Then, we
repeat the process by starting at $t_{n}$ and comparing $\tilde{t}_{n}$ with
the text embeddings $\tilde{t}_{n+1},\tilde{t}_{n+2},...$ to find the next
index. This way, we collect the indices of new recipes in our dataset so that
we can focus on the revision between these indices.

Moreover, to apply process mining techniques, we built event logs from the
writing task. For each group, we collect the activities for each user, by
looking at when they submit the first, second and third recipes and all the
revision steps in between.

### Feature Extraction

Different aspects of SRL have been researched extensively [18]. In a meta-
analysis on online education, [6] found significant associations with academic
achievement for five sub-scales of SRL: effort regulation (persistence in
learning), time management (ability to plan study time), metacognition
(awareness and control of thoughts), critical thinking (ability to carefully
examine material), and help-seeking (obtaining assistance if needed)222The
nature of our log data does not allow to represent help-seeking.. Based on
these findings, we use the following dimensions to represent student behavior:
effort regulation (Number of Revisions, Number of Edits, Time Spent Revising),
time management (Time Spent Revising, Pause Times), metacognition (Efficiency,
Pause Time), and critical thinking (DIRatio). A detailed description of these
feature variables can be found in Appendix A, Table 3.

### Building the Learning Path

Understanding revision behavior implies understanding the underlying process
in the writing task (e.g., how long do users in a group take to revise on
average or how many users revise). In order to understand this better, process
mining, especially process discovery [5], can help us model and visualize the
writing process for users in a group and design a learning path when using
adaptive writing support systems [30]. In this study, we use Directly-Follows
Graphs (DFGs) [3], which represent activities and their relationships333Other
data structures like Petri Nets could also be used. Petri Nets are commonly
used to apply process mining [3].. This is useful for the field of SRL as it
provides a way to visualize and analyze the steps involved in a process,
especially revision. A formal definition of DFGs can be found in Appendix B.

##  4 Results

### Revision Strategies

With this study, we find that users in different groups revise their texts
differently. Recall that G1 is given adaptive feedback and G2 is not. By
providing insightful feedback on what a user can change in their writing,
users tend to have more revision steps with fewer edits at each step. However,
users not receiving feedback follow the opposite trend, they have fewer
revision steps, with a larger number of revisions at each step. This
phenomenon is visible in Fig. 2. In fact, for the first and second texts
(Appendix D, Tables 4 and 5), we find $p$-values < 0.05 for Number of
Revisions using $t\text{-tests}$, which indicates a significant difference in
the number of revisions. This is also underlined by the mean number of
revisions and edits. On average, users in G1 tend to revise their texts more
often, with fewer edits at each step (Appendix D, Tables 4, 5 and 6).

bubble plots sorted by the number of revisions

Figure 2: Bubble plot for the first recipe sorted by the number of revisions.
The bubbles correspond to the total number of edits (insertions and deletions)
for a user at each revision step

From the directly-follows graphs (Fig. 3), we see that users spend
approximately the same amount of time writing recipes and the same amount of
time revising at the first revision step. However, we see that users in G2
revise much longer when having consecutive revision sessions (6 min on
average) compared to G1 (56 s)(Fig. 3). This confirms that users in G1 have
shorter revision sessions, whereas users in G2 have longer revision steps.

DFGs for the first and second group

Figure 3: Overview of the SRL behavior of students revising their texts as
directly-followed graphs for G1 (left) and G2 (right) automatically calculated
and drawn by our pipeline

### Engagement

From the second recipe onwards, we find that users revise less often, perform
fewer edits, spend less time revising and type faster (Fig. 4). This stems
from users being less engaged in the task at hand. In fact, users spend 67%
less time revising in G2 (Fig. 4) (from 264 seconds on average for the first
recipe to 86.5 seconds for the third one (Tables 4 and 6)) and 64% less in G1
(from 224 seconds to 81.2). In G2, users perform 74% fewer edits between the
first and last recipe (Fig. 4). Users in G2 performed on average 222 edits
when revising for the first recipe and only 57 for the third one (Tables 4 and
6). The decrease in pause time for the two groups also declines over time
(0.822 to 0.553 seconds on average for G1 and 0.646 to 0.525 seconds for G2),
even though participants in G2 consistently maintain a smaller average pause
time when revising. This is one interpretation of the results and Fig. 4,
another one would be to consider users are improving in this task. On average,
pause time for G1 decreased by 32.7% and 18.7% for G2 (Fig. 4). Shorter pause
times indicate better understanding of the task requirements and better task
planning [34]. This is coherent with the participants’ reported changes. As
seen in Fig. 5, we found that participants from G2 increasingly reported
making no changes to their recipes (36% for the third recipe). In contrast,
participants in G1 continued reporting making changes based on the received
adaptive feedback. Nevertheless, there was also an increase in the
participants in G1 that did not edit the recipe, one participant noted I
didn’t edit as much this time as I remembered to add them the first time
around.

user engagement and feature evolution on 4 feature variables over the entirety
of the writing experiment

Figure 4: Visualizing user engagement and feature evolution on 4 feature
variables over the entirety of the writing experiment

Percentage of participants that stated that they made no changes when editing
their recipes

Figure 5: Percentage of participants that stated that they made no changes
when editing their recipes in the survey following the experiment

### Gender Comparison

Research has often found that males tend to be more impaired at composing text
in comparison with women. The study in [35] found that female students
performed better than male students on a number of levels. Females had higher
scores, revised more, and were more efficient: they revised more per unit of
time, exhibiting greater writing fluency. In this study, we found that there
is a clear distinction in the writing capabilities between males and females.
Like [35], we find females are more efficient in this writing task. They tend
to have higher efficiency scores (Fig. 6) and we find $p=0.0038$ when
comparing efficiency scores in G2 (Table 2), which demonstrates the disparity
in efficiency distribution between the two groups. Curiously, males revise
less often when receiving feedback (as seen on the x-axis by the number of
times revised, (Fig. 6)444Some outliers were removed (e.g., users who spent
over 10’000 seconds revising or users who have very low efficiency scores (few
edits over a long period of time)).). On the contrary, when users do not
receive feedback, females revise once at most (because index 0 is not a
revision phase, Fig. 6). This also reinforces women’s abilities in their
writing, suggesting they feel less need for revision if they do not receive
feedback on what they can improve. Regarding the Delete-Insert ratio
(DIRatio), although we find there is no statistical difference (Table 2), we
find that males in G2 generally have higher scores, especially in G2. Having
higher DIRatio scores means users delete a larger portion of their texts (over
15% for several male users in G2, Fig. 6). Looking back at SRL, especially on
the critical thinking aspect [6], which is defined as the ability to examine
material, we can see males are more self-critical and delete a larger portion
of their texts compared to females when they do not receive feedback.

Table 2: $p$-values for Efficiency and DIRatio features comparing males and females in each group; *p<0.05, **p<0.01, ***p<0.001 |  With Adaptive Feedback (G1) |  Without Adaptive Feedback (G2)
---|---|---
Efficiency |  $p=0.215$ |  $p=0.0038$**
DIRatio |  $p=0.387$ |  $p=0.088$

gender comparison over 2 features

Figure 6: Overview of 2 SRL features from our pipeline comparing males and
females

##  5 Discussion & Conclusion

With this research, we contribute to the field of understanding the use of
intelligent writing systems by learners. We do this by gaining insights into
their SRL by inspecting revision behavior. From the log data we collected, we
built and modelled a pipeline to analyze and visualize user behavior in the
revision phases of the writing task, by observing different features extracted
from the revisions of G1 (with adaptive feedback) and G2 (without adaptive
feedback)(Figs. 2, 4 and 6). Our analysis revealed that learners in different
groups revise using different strategies. Learners who were equipped with
adaptive feedback revised more often, with fewer edits at each revision step
and users without adaptive feedback followed the opposite trend. This suggests
that the support provided by the system may influence revision behavior and
how it is used. Additionally, we found users seemed to be improving in the
writing task as demonstrated by the post-survey and the data, even though they
seem to be less engaged from the evolution of the feature variables in Fig. 4.
Finally, we concluded females were more efficient than males in this
experiment, by having higher efficiency scores. While there has been research
on the effectiveness of such systems in improving writing skills, there is a
limited understanding of how users revise their writing when using these
tools. To evaluate users’ SRL, it is crucial to have a better understanding of
how they self-regulate, especially in writing activities, in order to provide
them with the correct tools to improve their writing skills and understand the
underlying writing process [4, 31].

Regarding future directions, one can focus on clustering revision data in
order to gain further insights into the revision behavior in a writing task.
We have already done this, by identifying eight reported revisions, including
adding more details, changing the structure, improving the clarity or not
making any changes. Nevertheless, we focus on not making any changes, but
analyzing other revision reports could help shed light on more differences
between the groups. As such, clustering could be used for each group to
identify the differences between the two groups or between learners in the
same group, to see how users revise when receiving feedback or not.

In conclusion, our research on revision behavior in adaptive writing support
systems has shed light on how users in different groups approach revision. The
development of a pipeline to study this topic has allowed us to collect and
analyze data on user writing and revision activity, leading to the discovery
of important patterns and trends. Overall, our study has made a significant
contribution to the field by providing a deeper understanding of revision
behavior.

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##  Appendix A Features

Table 3 gives a detailed description of the features we study in this paper,
extracted in our pipeline.

Table 3: Overview of feature variables automatically calculated through our pipeline to measure SRL behavior of users in their writing exercises based on keystroke logs Feature Variables |  Description
---|---
Number Of Revisions |  For each user, we count the amount of times they revise each time they write a recipe (i.e., when they submitted then re-edited their texts). This gives a sense of the effort put into the revision phase of the writing task.
Number of edits |  The total number of insertions and deletions during a revision step. Insertions are counted as any characters that are typed including whitespaces, and deletions are counted as the number of times the user presses any of the Backspace or Delete buttons.
Time Spent Revising in seconds |  We compute the average time users spend revising for each group, for each recipe. This allows us to compare the two groups and to estimate the effort put in by both groups.
Delete-Insert Ratio (DIRatio) |  The average deletions over insertions ratio, which approximately captures the extent of editing and revision of any kind [35].
Efficiency |  Estimated by the number of insertions per second, which indicates a general writing speed. This feature is arguably an indicator of writing fluency [35].
Pause Time during Revision in seconds |  For each user, we collect the inter-key time interval and compute the mean of these intervals. This captures the average lag time between two adjacent keystroke actions [35]. This feature captures the effort and persistence level of users.

##  Appendix B Directly-Follows Graphs

In process mining, a Directly-Follows Graph is a directed graph that
represents the sequence of activities in a process based on event logs.
Formally, given a set of activities $\mathcal{A}$, an event log $\mathcal{L}$
is a multiset of traces $t\in\mathcal{L}$, where $t$ is a sequence of
activities $t=(a_{1},a_{2},\;...,a_{n})$, with $a_{i}\in\mathcal{A}$ $1\leq
i\leq n$ [2]. Given the event log $\mathcal{L}$, the DFG $\mathcal{G}$ is a
directed graph such that $\mathcal{G}=(V,E)$. $V$ is the set of activities in
$\mathcal{L}$: $V=\\{a\in\mathcal{A}\;|\;\exists t\in\mathcal{L}\land a\in
t\\}$. $E$ is defined as $E=\\{(u,v)\in V\times{V}\;|\;\exists
t=(a_{1},a_{2},...,a_{n}),\;t\in\mathcal{L}\;\land\;a_{i}=u\;\land\;a_{i+1}\\}$
[2].

##  Appendix C Separating writing sessions

Algorithm 1 describes our implementation of session separation. Each
participant wrote a first version of their recipes, then revised it, before
starting the next recipes. To focus on the revision sessions, we needed to
implement a function which captures the indices in the dataset where
participants started their recipes. First, we preprocess the submitted text by
removing noisy characters, such as punctuation and return the list of
sanitized words. Then we use the GloVe model to convert the words to vectors
and return one 50-dimensional vector which is the sum of each word embedding.
Then we recursively find the indices of new recipes using cosine distance.
However, the algorithm is 91% accurate: this is because sometimes users
submitted random strings or the revisions led to the algorithm detecting
another recipe. We adjusted the missing indices by hand by looking at the
dataset.

Algorithm 1 Separating writing sessions using cosine distance

function separateSessions

$model\leftarrow\textsc{GloVeModel}$

function getVector($text$)

$p\leftarrow\textsc{preProcess}(text)$ $\triangleright$ splits and sanitizes
$text$

$arr\leftarrow$ Initialize an empty list

for $word\in p$ do

add $model[word]$ to $arr$ if $word\in model$

end for

return $\textsc{np.sum}(arr,axis=0)$ $\triangleright$ uses numpy

end function

function computeIndices($startIndex,accumulator$)

$recipes\leftarrow$ retrieve $recipes$ from the dataset

$size\leftarrow$ the total number of $recipes$

if $startIndex\geq size-1$ then

return $accumulator$

end if

$vec\leftarrow\textsc{getVector}(recipes[startIndex])$

for $n\leftarrow startIndex$ to $size$ do

$d\leftarrow 1-\textsc{cosineDist}(vec,\textsc{getVector}(recipes[n]))$

if $d<0.995$ then

add $n$ to the $accumulator$

return computeIndices($n$, $accumulator$)

end if

end for

end function

return computeIndices($0$, empty $accumulator$)

end function

##  Appendix D Results

### Detailed Results

Tables 4, 5 and 6 report the mean, standard deviation of different feature
variables for both groups, as well as $p$-values.

Table 4: Overview of SRL features from our pipeline for the first written text between students receiving adaptive feedback (G1) and no feedback (G2) based on our data set | With Adaptive Feedback (G1) | Without Adaptive Feedback (G2) |
---|---|---|---
Feature Variables |  Mean | Std |  Mean | Std |  $p$-values
Number of Revisions |  1.882 | 2.166 |  0.846 | 0.735 | 0.0071
Number of Edits |  75.734 | 95.114 |  222.73 | 338.574 | 0.24
Time Spent Revising (sec) |  224.48 | 237.37 |  264.01 | 530.47 | 0.694
Pause Time in Revision (sec) |  0.822 | 0.225 |  0.646 | 0.08 | 0.339
Table 5: Overview of SRL features from our pipeline for the second written text between students receiving adaptive feedback (G1) and no feedback (G2) based on our data set | With Adaptive Feedback (G1) | Without Adaptive Feedback (G2) |
---|---|---|---
Feature Variables |  Mean | Std |  Mean | Std |  $p$-values
Number of Revisions |  1.147 | 1.033 |  0.692 | 0.722 | 0.033
Number of Edits |  95.051 | 192.92 |  57.52 | 61.31 | 0.26
Time Spent Revising (sec) |  121.19 | 148 |  70.15 | 120.9 | 0.11
Pause Time in Revision (sec) |  0.69 | 0.365 |  0.421 | 0.2 | 0.089
Table 6: Overview of SRL features from our pipeline for the third written text between students receiving adaptive feedback (G1) and no feedback (G2) based on our data set | With Adaptive Feedback (G1) | Without Adaptive Feedback (G2) |
---|---|---|---
Feature Variables |  Mean | Std |  Mean | Std |  $p$-values
Number of Revisions |  1.12 | 1.05 |  0.737 | 0.676 | 0.11
Number of Edits |  87.61 | 270.75 |  57.7 | 71.7 | 0.45
Time Spent Revising (sec) |  81.2 | 115.8 |  86.5 | 231.9 | 0.905
Pause Time in Revision (sec) |  0.553 | 0.212 |  0.525 | 0.296 | 0.85


# Exclusive decay study of the 16.62 MeV (2-, T=1) resonance in 12C

M. Kuhlwein  K. Lytje  H. O. U. Fynbo<EMAIL_ADDRESS>A. Gad  E. Jensen  O.
S. Kirsebom  M. Munch  J. Refsgaard  K. Riisager  Department of Physics and
Astronomy, Aarhus University, DK-8000 Aarhus, Denmark  Department of Astronomy
and Physics, Saint Mary’s University, Halifax, Nova Scotia, B3H 3C3 Canada,
and TRIUMF, 4004 Wesbrook Mall, Vancouver BC, V6T 2A3 Canada  Institute for
Big Data Analytics, Dalhousie University, Halifax, NS B3H 4R2, Canada

###### Abstract

The 3$\alpha$ decay of the 16.62 MeV (2-, T=1) resonance in 12C has been
studied for nearly a century starting with one of the first nuclear reaction
studies at the Cavendish Laboratory in the 1930s. In the hitherto latest study
published a decade ago a model based on earlier work from the 1960s was found
to give a good account of a set of inclusive data. This model describes the
decay as an l=3 $\alpha$-particle populating the 2+ state of 8Be. Here we
provide new exclusive data on the 3$\alpha$ decay of the 16.62 MeV resonance,
and demonstrate that the decay is best described by a model with predominantly
l=1 emission with an admixture of l=3.

###### keywords:

Collective levels , breakup and momentum distributions , multifragment
emission and correlations

iint TXFiint

##  1 Introduction

The 16.62 MeV (2-, T=1) resonance in 12C is positioned 663 keV above the
proton threshold in 12C. With a width of 300 keV the state is prolifically
produced by impinging $<$1 MeV protons on 11B, which was one of the first
nuclear reactions to be studied at the Cavendish Laboratory [1, 2, 3].

There are at least three different interests in the p+11B reaction at the
energies relevant for the population of the 2- resonance. Firstly, roughly 50%
of the decay leads to the 3$\alpha$-final state (with the rest being back to
p+11B ), and the mechanism of this three-body decay has been in focus since
the early work at the Cavendish Laboratory. Secondly, the 11B(p,3$\alpha$)
reaction is considered a candidate for fusion energy generators [4], a
possibility which is still being pursued in research [5] and by several
commercial companies. Finally, the reaction is considered as a potential way
to enhance proton cancer therapy efficiency by adding a concentration of boron
to the tumor volume, which will then produce high-LET $\alpha$ particles when
protons in the Bragg peak enter the tumor and interact with the boron, e.g.
[6].

Here we present a new complete kinematics measurement of the 3$\alpha$-decay
of the 16.62 MeV resonance and demonstrate how conflicting conclusions on its
decay mechanism from previous studies throughout almost a century are a
consequence of previous analyses being based primarily on inclusive data.

##  2 Review of previous work

With unnatural parity s-wave partial waves cannot contribute to the
3$\alpha$-decay of the 16.62 MeV resonance. The initial work at the Cavendish
Laboratory in the 1930s established that the 3$\alpha$-decay is strongly
influenced by the broad 2+ state in 8Be as illustrated in Fig. 1. Within this
picture the first $\alpha$ particle can have orbital angular momentum with
either l=1 or l=3 relative to 8Be.

$p+{}^{11}$B$15.96$$E_{p}=$675\text{\,}\mathrm{keV}$$$2^{-}$12C$16.57$$\alpha_{1}$$\cancel{\alpha_{0}}$$2^{+}$$10.39$$0^{+}$$\alpha+{}^{8}$Be$7.36$$\alpha+\alpha$$\alpha+\alpha+\alpha$$7.27$

Fig. 1:  Population by p+11B of the $16.62\text{\,}\mathrm{MeV}$ state in
${}^{12}\mathrm{C}$ and the subsequent decay via ${}^{8}\mathrm{Be}$.
Influence of the ground state in ${}^{8}\mathrm{Be}$ is forbidden by
conservation of angular momentum and parity. All energies are measured in MeV
from the ground state of ${}^{12}\mathrm{C}$.

Until the end of the 1960s, the problem was only visited intermittently. Then,
in the period 1967-1972 at least five different groups published detailed work
on the problem. Some of this work was spurred by MacDonald, who in 1965
pointed to the importance of symmetry effects related to the boson character
of the three $\alpha$ particles in the final state, which should lead to
characteristic patterns in the Dalitz plot of the decay [7].

In 1967 the first high quality data came from Kamke and Krug [8] with a
detailed analysis and interpretation given by Schäfer in 1970 [9], who
confirmed the presence of symmetry effects and concluded that a strong l=3
component is needed to understand the data. In 1969 Quebert and Marquez in an
independent study also identified the symmetry effect and pointed to a
dominant l=3 contribution [10]. Next, Giorni et al. published data in 1968
[11] and interpretation in 1970 [12] where the data were fitted with an
adjustable coherent sum of l=1 and l=3 contributions leading to 83% branching
ratio for l=1 (no uncertainty given). In 1970 Cockburn et al. also provided
high quality data and a remarkably illuminating analysis and discussion of the
data [13]. Cockburn et al. were able to make a fit to the full Dalitz plot
including a coherent sum of l=1 and l=3 contributions, which resulted in a
ratio of reduced widths of ${\gamma_{3}}^{2}/{\gamma_{1}}^{2}$=0.60-0.65.
Since the l=3 wave is damped by the angular momentum barrier, this could be in
rough agreement with the ratio deduced by Giorni [12]. By analysing regions in
the Dalitz plot with the largest discrepancies between theory and data,
Cockburn et al. could identify the effect of rescattering between primary
(emitted from 12C) and secondary (emitted from 8Be) $\alpha$ particles, and
also examined if the properties of the 8Be 2+ resonance were affected by the
presence of the third $\alpha$ particle. The last result from this period was
published by Treado et al. [14]. In this work a ratio of 10(3) between the
branching ratios to the l=3 and l=1 components in favour of l=1 was deduced
from an analysis of angular correlations between primary and secondary
$\alpha$ particles. This indicates a stronger dominance of l=1 over l=3
relative to the findings of Cockburn et al. and Giorni et al.

Somewhat later two scans of the reaction 11B(p,3$\alpha$) were published with
the aim of deducing the astrophysical reaction rate [15, 16]. In these studies
the treatment of the 3$\alpha$ breakup of 12C states populated in the reaction
was somewhat crude, e.g. neglecting symmetrisation effects, and including only
the l=1 contribution.

In 2011 Stave et al. [17] published a new measurement of the 3$\alpha$-decay
of the 16.62 MeV state. They refer to [16] for providing the modern view of
the decay of the 2- state emphasizing the choice of l=1. Stave et al. then use
a comparison of their new data to a model neglecting symmetry effects to
conclude that the primary $\alpha$ particle is emitted with l=3 and the
conclusions from [16] on the breakup mechanism of the 2- state therefore is
erroneous.

All experiments referred to in this section used similar detector setups
consisting of one or two small detectors with one at a fixed positions and the
other free to rotate around the target position. In contrast, the present
study uses a compact array of DSSDs allowing for a complete-kinematics
measurement, as detailed later.

##  3 3$\alpha$ breakup model

We use the R-matrix based description and parametrisation of the 3$\alpha$
decay from [18]. This parametrisation has also been used to describe the decay
of other states in 12C: e.g. 0+ (7.65 MeV) [19], 1+ (12.71 MeV) [20], and 2+
(16.11 MeV) [21]. This description takes into account the symmetry stemming
from the Boson character of the $\alpha$-particles as pointed out by MacDonald
[7]. The description is also very similar to those used by [10, 9, 12, 13].

In this description the decay amplitude is given by

$\displaystyle f_{c}^{m_{a}}$ $\displaystyle(123)=\sum_{m_{b}}\langle
J_{b}l_{1}m_{b}(m_{a}-m_{b})|J_{a}m_{a}\rangle$
$\displaystyle\times\bigl{[}i^{l_{1}}Y_{l_{1}}^{m_{a}-m_{b}}(\Omega_{1})\bigr{]}\bigl{[}i^{l_{2}}Y_{l_{2}}^{m_{b}}(\Omega_{23})\bigr{]}$
$\displaystyle\times\gamma_{c}\bigl{(}2P_{l_{1}}/\rho_{1}\bigr{)}^{\frac{1}{2}}\exp[i(\omega_{l_{1}}-\phi_{l_{1}})\bigr{]}F_{c}(E_{23}),$
(1)

where $F_{c}(E_{23})$ is a factor describing the resonant strength of the
intermediate system. In R-matrix theory the single-level approximation is

$\displaystyle
F_{c}(E_{23})=\frac{\gamma_{\lambda_{b}l_{2}}\bigl{(}2P_{l_{2}}/\rho_{23}\bigr{)}^{\frac{1}{2}}\exp[i(\omega_{l_{2}}-\phi_{l_{2}})\bigr{]}}{E_{\lambda_{b}}-E_{23}-\bigl{[}S_{l_{2}}-B{l_{2}}+iP_{l_{2}}\bigr{]}\gamma_{\lambda_{b}l_{2}}^{2}}.$
(2)

To obtain the total decay weight the expression is symmetrised by permutation
of the indices of the $\alpha$ particles:

$\displaystyle
W=\sum_{m_{a}}\Bigl{\lvert}\sum_{c}\Bigl{\\{}f_{c}^{m_{a}}(123)+f_{c}^{m_{a}}(231)+f_{c}^{m_{a}}(312)\Bigr{\\}}\Bigr{\rvert}^{2}.$
(3)

The symbols appearing in eqs. (3)–(3) are defined in 1.

Table 1: Explanation of the parameters appearing in eqs. (3)–(3). $J_{a},m_{a}$ |  Angular momentum quantum numbers of 12C.
---|---
$J_{b},m_{b}$ |  Same for 8Be.
$l_{1},l_{2}$ |  Orbital angular momentum in 12C$\rightarrow$8Be$+\alpha$ and 8Be$\rightarrow$2$\alpha$
$\lambda_{b}$ |  The level populated in 8Be. Implicitly specifies $J_{b}$ and $l_{2}$.
$c$ |  Decay channel specifying $\\{l_{1},\lambda_{b}\\}$.
$\gamma_{c}$ |  Reduced width amplitude for decay of 12C through channel $c$.
$\gamma_{\lambda_{b}l_{2}}$ |  Same for decay of 8Be.
$\Omega_{1}$ |  Direction of the first emitted $\alpha$ in the rest frame of 12C.
$\Omega_{23}$ |  Direction of the second emitted $\alpha$ in the rest frame of 8Be.
$E_{23}$ |  Relative energy between $\alpha_{2}$ and $\alpha_{3}$
$\rho_{1}$ |  $=k_{1}a_{1}$, where $k_{1}$ is the wave number and $a_{1}$ is the channel radius for the primary breakup channel.
$\rho_{23}$ |  Same for the secondary breakup channel.
$P_{l_{1}},P_{l_{2}}$ |  Penetrability for the primary and secondary breakup channels.
$\omega_{l_{1}},\omega_{l_{2}}$ |  Coulomb phase shifts.
$\phi_{l_{1}},\phi_{l_{2}}$ |  Hard-sphere phase shifts.
$E_{\lambda_{b}}$ |  Level energy of $\lambda_{b}$ in the intermediate system.
$S_{l_{2}},B_{l_{2}}$ |  Shift function and boundary condition for the secondary breakup channel.

Contributions from $l_{1}=1$ and $l_{1}=3$ can be added coherently. We will
also allow an additional adjustable phase $\delta$ between the $l_{1}=1$ and
$l_{1}=3$ amplitudes.

$\left\lvert
f\right\rvert^{2}=\sum_{m_{a}}\left\lvert\sqrt{k}f_{\ell_{1}=1}+\sqrt{1-k}\cdot
e^{i\delta}f_{\ell_{1}=3}\right\rvert^{2}$ (4)

The question that will be adressed in the following is then: is the 3$\alpha$
breakup of the 16.62 MeV resonance well described by this model, and if so, is
the breakup dominated by $l_{1}=1$, $l_{1}=3$, or a mixture of the two? We
anticipate that the model cannot provide a perfect description of the data
since it ignores any interaction between the $\alpha$-particle emitted in the
process 12C$\rightarrow$8Be$+\alpha$ and those from 8Be$\rightarrow$2$\alpha$
[13, 20].

##  4 Experiment

The experiment was conducted at the $5\text{\,}\mathrm{MeV}$ Van de Graaff
accelerator at Aarhus University. Other results from the same experimental
campaign have been published elsewhere [22, 23].

The 16.62 MeV resonance was populated via
${}^{11}\mathrm{B}$(p${}^{12}\mathrm{C}$), as depicted in Fig. 1. The
accelerator provided a $1361(2)\text{\,}\mathrm{keV}$ beam of $H_{2}^{+}$,
which was guided through a sequence of slits and steerers into the reaction
chamber where it impacted a $\sim$120\text{\,}\mathrm{nm}$$
${}^{11}\mathrm{B}$ target on a carbon backing produced in house. The
measurement time was $43.1$ hours with an integrated beam current of
$55(3)\text{\,}\mathrm{\SIUnitSymbolMicro C}$.

The detection setup consisted of four double-sided silicon-strip detectors
(DSSSD’s). Two of the detectors were square
($$5\text{\,}\mathrm{cm}$\times$5\text{\,}\mathrm{cm}$$ with 16 rectangular
strips on each side) and two were annular (inner radius
$11\text{\,}\mathrm{mm}$ and outer radius is $35\text{\,}\mathrm{mm}$ with 32
radial strips (spokes) on the front side and 24 circular strips (rings) on the
back), all from Micron Semiconductor Ltd.

The signals from the detectors were analysed in a conventional scheme of
preamplifiers, amplifiers, ADCs and TDCs.

##  5 Data analysis and discussion

In the off-line analysis events of interest were selected by imposing energy
and momentum conservation of three detected particles, and that the individual
energies were larger than 250 keV. Finally, events corresponding to population
of the ground state in 8Be, which cannot originate from the 2- state, were
identified from the relative energy of pairs of $\alpha$-particles, and
removed from further analysis. Of the order of 6 $\times$ 105 3$\alpha$-events
were collected in the experiment.

In the lower panel in fig.2 we display the 3$\alpha$ events selected in this
way in a Dalitz plot with axes $x=\sqrt{3}(E_{2}-E_{3})/\sum_{i}E_{i}$ and
$y=(2E_{1}-E_{2}-E_{3})/\sum_{i}E_{i}$, where $E_{i}$ is the energy of the
$i$’th alpha particle in the center-of-mass frame.

Fig. 2: Dalitz plots of 3$\alpha$ coincidence data and comparison to models
with $l_{1}$=1, $l_{1}$=3 and the mixture of the two. The Dalitz plot
coordinates are $x=\sqrt{3}(E_{2}-E_{3})/\sum_{i}E_{i}$ and
$y=(2E_{1}-E_{2}-E_{3})/\sum_{i}E_{i}$.

A qualitative understanding of the measured distribution of data in the Dalitz
plot comes from the general description given in [24]: The events are confined
to the inside of a circle due to momentum conservation. As first pointed out
by [7], the centre and the rim of the circle are excluded due to symmetry
requirements characteristic for a 2- state. The influence of resonances
between two of the three $\alpha$-particles is seen as lines corresponding to
fixed values of the relative energy of two $\alpha$-particles. The width of
these lines are determined by the width of the $\alpha\alpha$ resonance. Due
to the presence of three identical particles this leads to the appearance of
the triangular region of enhanced intensity in fig. 2. The intensity
distribution along the lines of this triangle is determined by angular
correlations. Interference effects are strongest where the lines overlap.
Note, interference effects were ignored in several studies referred to in
section 2, e.g. [15, 16, 17].

To facilitate a quantitative understanding of the data and to elucidate the
veracity of the different conclusions on the 3$\alpha$-decay of the 2-
resonance outlined in section 2 we explore if the data is well described by
the model from section 3, and if so, if the breakup is dominated by $l_{1}=1$,
$l_{1}=3$, or a mixture of the two. The upper three panels in fig.2 show the
model prediction for $l_{1}=1$, $l_{1}=3$, and our best fit to a mixture of
the two, respectively. In table 2 we give the result of fits of the models to
the data. For completeness we also give the result of a fit to Eq. 4 with
$\delta$=0. Obviously, the best model is obtained by allowing an admixture of
$l_{1}=1$ and $l_{1}=3$ \- this is true whether or not the extra phase
$\delta$ is included.

The fitted values of the admixture are in broad agreement with the ratios
found by [12, 13]. The two models with only $l_{1}$=1 or $l_{1}$=3 give
similar quality fits with a slight preference for the former. This preference
is somewhat surprising based on visual inspection of the Dalitz plots in fig.
2.

To understand the difference between $l_{1}$=1 and $l_{1}$=3 we show in fig. 3
the projected $\alpha$-spectra from the Dalitz-plots in fig. 2.

Fig. 3: Single $\alpha$-spectra from the same data shown in fig. 2 and
comparison to models with $l_{1}$=1, $l_{1}$=3 and a mixture of the two. Data
is black, and model is orange.

There are primarily two places where $l_{1}$ influences the decay amplitude
given in Eq. 3: In the penetrability $P_{l_{1}}$ and in the angular part
through Clebsh-Gordon coefficients and spherical harmonic functions. The
effect of the penetrability can be seen on the high energy side of the energy
distribution where the larger barrier from $l_{1}$=3 appears to push the main
peak a little up in energy. The difference in the low-energy part of the
spectra is due to the difference in angular correlations for $l_{1}$=1,3; the
latter effect is discussed in more detail in [20, 21]. The angular part also
accounts for the difference in the size of the hole in the centre of the
Dalitz-plot, which corresponds to events with equal energies of the three
$\alpha$-particles. This corresponds to individual energies around 3 MeV.

Allowing for an admixture of the two angular momenta gives the best fit
because all of these features can be better reproduced. Allowing the 8Be 2+
resonance energy to move by more than 100keV does not change the conclusion
that $l_{1}$=1 dominates when the two values are both allowed to contribute.

Note, the energy spectra displayed in fig. 3 are very similar to those in [10]
and to some extent also those in [17]. To further elucidate the conclusions
from previous studies we have also compared our data to a model neglecting
interference effects, which is obtained by using only one of the three
permutations in Eq. 3. This model is equivalent to those used by [15, 16, 17].
With this model, $l_{1}$=3 gives the best reproduction of our
$\alpha$-spectrum in agreement with the conclusion of [17]. It therefore
appears that the conclusion of the dominance of $l_{1}$=3 is based on using
too simple a model that only allows for one value of $l_{1}$ and neglects
interference, and basing the conclusion primarily on inclusive data.

$\ell_{1}=1$ [%] |  $\ell_{1}=3$ [%] | $\chi^{2}$ | $\delta$
---|---|---|---
$100$ | $0$ | 22705 | n/a
$0$ | $100$ | 22808 | n/a
85(1) | 15(1) | 15360 | $0$
76(5) | 25(5) | 12297 | $67(1)\%\cdot 2\pi$
Table 2:  Fit results and $\chi^{2}$ values for the tested models.
Uncertainties are statistical only. The first two columns give the percentage
of $\ell_{1}=1$ and $\ell_{1}=3$. The number of degrees of freedom for the
$\chi^{2}$-distribution is $\sim 5350$.

As seen from the values of $\chi^{2}$ in table 2, even our best model deviates
significantly from the data. Those differences are presumable caused by final
state interactions as discussed in some detail [25, 13, 26, 20]. This aspect
will not pursued further here.

##  6 Conclusion

In summary, we have collected exclusive data from the 3$\alpha$-decay of the
16.62 MeV (2-, T=1) resonance in 12C and compared our measurements with
R-matrix based sequential models. We found that the model with admixture of
$l_{1}$=1 and 3 describes the data quite well. We also demonstrate that the
conflicting conclusions from previous measurements spanning almost a century
may be partly explained by neglecting the possible admixture or two angular
momenta, neglecting interference effects or basing conclusions only on
inclusive data.

It remains to be examined what the consequences are of our results on e.g. the
rate of the 11B(p,3$\alpha$) and therefore on scenarios where this rate plays
a role including astrophysics and possibly fusion reactors.

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