# Gravitational wave signals from long lasting binary–single black hole
encounters.

Elena Codazzo  Matteo Di Giovanni  Jan Harms

###### Abstract

In the dense regions of star clusters, close encounters with black holes (BHs)
can occur giving rise to a new class of gravitational-wave (GW) signals.
Binary-single encounters between three BHs are expected to dominate the rate
of signals from unbound systems in the frequency band of terrestrial GW
detectors. The encounter can describe a quasi-hyperbolic trajectory, which was
the focus of a recent study. In some cases, the encounter can take a more
complex form including one or two BH mergers as a result of the encounter,
repeating cycles of close proximity between the BHs, and the exchange of a BH
that is part of the binary. The variety of types of encounters leads to a
variety of GW signals emerging from these encounters. Using the ARWV N-body
code, we performed 42 numerical simulations, to explore various outcomes of
binary-single interaction, and we characterize the diverse GW signatures
produced during these encounters. Additionally, we evaluated the detectability
of these GW signals by injecting them into the simulated noise of the Einstein
Telescope and exploring different methods to detect the signals. Our findings
shed light on the complexities of these interactions and their potential
implications for GW astronomy.

###### keywords:

Close encounters, black holes, burst signals.

††journal: .

[a]Gran Sasso Science Institute (GSSI), I-67100 L‘Aquila, Italy [b]INFN,
Laboratori Nazionali del Gran Sasso, I-67100 Assergi, Italy [c]Dipartimento di
Fisica, Universita‘ di Roma La Sapienza, I-00185 Rome, Italy [d]INFN, Sezione
di Roma, I-00185 Rome, Italy

##  1 Introduction

After the first successful observation runs with the current-generation
gravitational-wave (GW) detectors Advanced Virgo [1] and Advanced LIGO [2],
the GW community is preparing to leap into the 3rd generation of GW detectors,
such as the proposed Einstein Telescope (ET) [3, 4] and Cosmic Explorer [5].
The improved sensitivity of these future ground-based detectors is expected to
increase the number of observed events from tens per year to hundreds of
thousands per year opening an enormous science case [6, 7]. Moreover, the
addition of the space-based detector LISA [8] together with pulsar-timing
arrays [9] and possibly with decihertz Moon-based GW detectors [10, 11] will
enable the scientific community to cover the GW spectrum from nanohertz to
kilohertz.

Taking all these elements into consideration, the accessible frequency band of
the GW spectrum will be significantly broadened, therefore prompting a series
of studies focused at the GW emission from non-canonical sources, i.e.,
different from compact object mergers, continuous waves and a stochastic
background, that are out of reach for current detectors.

Among these sources are hyperbolic encounters between compact objects, black
holes (BHs) in particular, which are enduring a growing interest in the
astrophysical and GW communities after the detection of GW190521 [12, 13], the
most massive BH merger detected so far. Given its masses and eccentricity of
the orbit, some follow-up studies [14, 15, 16] suggested that the merger could
have been the outcome of the dynamical interaction between two unbound solar-
mass BHs, therefore providing the first indirect detection of dynamical
interaction in a star cluster. Previously, the GW170817A binary black-hole
(BBH) merger had been found consistent with a hierarchical merger due to
dynamical interactions [17] as well, but the large uncertainties in the
reconstructed parameters damped its significance. Later, also the GW190425
[18] binary neutron star (BNS) merger, given its unusually high total mass,
led to the hypothesis, as one of the possible formation scenarios, of a
dynamical encounter in a cluster between a BNS and an isolated neutron star
that replaced the lightest component of the binary. Although this explanation
appears intriguing, it has been shown that the dynamical formation channel has
a negligible contribution to the BNS merger rate in the local universe,
therefore making the dynamical origin for GW190425 unlikely [19, 20].
Nevertheless, the binary pulsars B2127$+$11C in M15 [21] and J1807$-$2500 in
NGC 6544 [22] still provide evidence of neutron stars that experienced
dynamical interactions in clusters [23].

Dynamical interactions are not new to the GW community. Since the start of the
century, several theoretical studies have focused on the analytical [24, 25,
26, 27, 28, 29, 30, 31] and numerical [32, 33] tools to determine the GW
emission of a fly-by or of a dynamical capture. Recently, linking for the
first time the astrophysical properties about star population and cluster
dynamics [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59] with the GW emission of hyperbolic flybys,
[60] provided a first realistic estimate of the detectability of gravitational
radiation from binary-single hyperbolic flyby between BHs in star clusters and
the rate of these events.

As a follow up to [60], the aim of this work is to identify, within a set of
injections in simulated ET noise, which are the typical features of these
signals. This should enable us to determine, among other things, not only if
there are characteristics in common that could facilitate their recognition by
a data-analysis pipeline, but also whether a typical eccentric merger,
characterized by an inspiral phase, was preceded by a dynamical interaction or
represents a second-generation merger following an initial capture event.

The paper is organized as follows: in Section 2, an overview of the underlying
astrophysical assumptions of the systems considered in this study is provided;
Section 3 details the simulations carried out for this work and the
methodology used to simulate BBH-BH encounters and to characterize their GW
signals. The results are presented and discussed in Section 4. A short summary
along with final remarks conclude the paper in Section 5.

##  2 Astrophysical background

Nuclear star clusters (NSC) are one of the most promising locations for
dynamical encounters between compact objects [61, 62]. Since an extensive
overview of NSC and binary-single encounters has already been given in [60]
and other past works [63, 64, 65, 66, 67, 68, 69, 70, 71], here we will limit
ourselves to the highlights of their properties.

In star clusters, binary systems are classified as hard if their binding
energy is higher than the average kinetic energy of the stars in the cluster
[61]. BBHs in the core harden not only for GW emission but also through
binary–single encounters [61], namely three-body interactions in the form of a
fly-by or a resonant interaction between the BBH and an intruder object. In
the latter case, an unstable triple system is formed until one of the three
objects acquires enough speed to escape; typically, it is the least massive
component that most probably escapes [72]. In general, flybys are the most
common outcome of such interactions. Since, statistically, the velocity of the
intruder after the encounter is greater than the one with which it approached
the binary, as a consequence of the conservation of energy, the binary
tightens according to Heggie’s law, for which hard binaries tend to become
harder.

The semi-major axis a of the binary will decrease over time due to
binary–single encounters and due to GW emission [73] as follows [74]:

$\dfrac{\text{d}a}{\text{d}t}=-2\pi\xi\dfrac{G\rho_{\rm
c}}{\sigma}a^{2}-\frac{64}{5}\dfrac{G^{3}m_{1}m_{2}(m_{1}+m_{2})}{c^{5}a^{3}(1-e^{2})^{7/2}}f_{1}(e).$
(1)

where $m_{1}$ and $m_{2}$ are the primary and the secondary mass of the BBH, e
is its eccentricity, $m_{3}$ is the mass of the intruder, $\sigma$ is the
three-dimensional dispersion velocity, $\rho_{\rm c}$ is the local density of
stars, $\xi\approx$ 3 is a dimensionless hardening rate [75] and

$f_{1}(e)=\left(1+\frac{73}{24}e^{2}+\frac{37}{96}e^{4}\right).$ (2)

The contribution of dynamical hardening, proportional to $a^{2}$, dominates
over that due to the GW emission by the system, which instead takes over when
the semi-major axis is small since it is proportional to $a^{-3}$ [73].

Similarly, dynamical encounters and GW emission [73] cause the binary
eccentricity to change over time as [74]:

$\dfrac{\text{d}e}{\text{d}t}=2\pi\xi\kappa\dfrac{G\rho_{\rm
c}}{\sigma}a-\frac{304}{15}e\dfrac{G^{3}m_{1}m_{2}(m_{1}+m_{2})}{c^{5}a^{4}(1-e^{2})^{5/2}}f_{2}(e).$
(3)

with the $\kappa$ parameter defined as $\kappa\equiv\text{d}e/\text{d
ln}(1/a)$[76]; and

$f_{2}(e)=\left(1+\frac{121}{304}e^{2}\right).$ (4)

Using these properties, in [60], we conducted a study on the detectability of
gravitational signals from binary-single hyperbolic encounters. We used
stellar-mass black holes as input masses and considered hard binaries in an
environment that is the core of an NSC.

Our findings indicate that the rate of these encounters within NSCs, up to
redshift z = 3.5, falls within the range of
$[0.006-0.345]\,$yr${}^{-1}\,$Gpc-3. For the lower and upper limits, we
considered core radii of 0.1 pc and 1 pc, respectively. We observed that the
majority of these encounters produced GW signals in the frequency band of
LISA, but no signals were in the ground-based interferometer frequency band.
However, we discovered a correlation between the characteristic frequency of
the signals with the binary’s semi-major axis and the impact parameter with
which the intruder approaches. In this study, we tuned these parameters to
obtain signals suitable for analysis within the frequency band of ET. All the
parameters considered are aligned with the presence of a binary and an
intruder in an environment resembling the core of an NSC, which we extensively
examined in our previous work [60].

##  3 Method and simulations

To simulate binary-single encounters, we use the N-body simulation code ARWV
[77, 78, 79]. ARWV makes use of the algorithmic regularization chain method to
integrate the equations of motion [80] [81]. In this way the round-off errors
are reduced, making the regularization algorithm more efficient, especially
for close interactions. ARWV incorporates a post-Newtonian treatment up to
order 2.5 to correct the equations of motion in case of strong gravitational
interactions [82]. These simulations provide the positions of all three bodies
at each time step, allowing for computation of the strength of the
gravitational signal emitted by their configuration.

The initial conditions are chosen to represent realistic encounters within the
core of a NSC, as already described by [60]. Furthermore, [60] found that the
frequency of the GW signal increases as both the binary’s semi-major axis and
the impact parameter of the intruder decrease. Therefore, after the generation
of the initial conditions, we select only the systems with impact parameter
and semi-major axis eligible to generate a GW signal within the ET frequency
band and sensitivity. Finally, we use the Hilbert-Huang Transform analysis
[83] to reveal distinctive features of the signals.

###  3.1 Initial conditions

The simulation of a binary-single encounter requires the use of a set of 15
parameters to define the initial conditions of the simulation [62, 84, 60]
including the spins of the BHs. Among these parameters are the angles $\phi$,
$\psi$, $\theta$ that define the geometry of the encounter, the impact
parameter $b$, the velocities of the BHs and the orbital parameters of the
binary. A detailed description of these parameters is beyond the scope of this
paper, but can be found in [62, 84, 60]. All these parameters are sampled
according to the distributions described in [60]. Moreover, from the initial
conditions generated for [60], we select an event with initial conditions
eligible to generate a GW signal detectable by ET (Table 1).

Using this event as a reference, we decided to divide the simulations into two
categories: spinning BHs and non-spinning BHs. In the former case, the spins
are randomly sampled according to a Maxwellian distribution with a root mean
square of 0.1. For each category, we redefine the reference event as e0 (Table
1) for spinless simulations and e5 for simulations with spinning BHs. The
initial conditions within the two categories are exactly the same, with the
only difference being the spin values. At this point, we decide to gradually
move from e0 and e5 by changing only one parameter at a time while keeping all
the others fixed in each simulation. The goal of this procedure is to assess
whether the spin has an impact on the outcome of the encounters or not and to
show how small and predictable changes in the initial conditions can lead to
unpredictable outputs that dramatically affect the outcome and the
detectability of each event.

In particular, we decide to keep the spins and the geometry of the encounter
fixed for all simulations and to change only the following parameters:

  * 1.

eccentricity: e $\in$ [0, 0.5, 1, 1.5, 2] $\times 10^{-5}$;

  * 2.

relative velocity between binary and intruder: v $\in$ [30, 40, 50, 60, 70]
km/s;

  * 3.

impact parameter: b $\in$ [3, 4, 5, 6, 7] $\times 10^{-3}$ AU;

  * 4.

intruder mass: m${}_{3}\in$ [5, 7, 10, 13, 15] M$\odot$;

  * 5.

mass ratio of the binary: q $\in$ [0.3, 0.5, 0.7, 0.83, 1].

In total, we ran 42 simulations, equally divided between the spinning and non-
spinning BHs cases.

Table 1: Initial values of the parameters for the baseline event e0. From top to bottom: the two masses of the binary, the mass of the intruder, the semi-major axis and the eccentricity of the binary, relative velocity and impact parameter of the intruder approaching the binary, and the geometrical parameters (fase, $\psi$, $\phi$, and $\theta$) fixed for all the simulations. All three BHs spinless. Parameters | Initial Values
---|---
m1 [M⊙] | 30.0
m2 [M⊙] | 25.0
m3 [M⊙] | 10.0
a [AU] | 8e-05
e | 0.0
v [km/s] | 50.0
b [AU] | 0.005
fase | 2.8
$\psi$ | 3.0
$\phi$ | 3.0
$\theta$ | 1.5
spins | 0.0

###  3.2 Characterization of the signal

Through the output of ARWV, we track the evolution of each BBH-BH encounter.
Figures 1 and 2 show the time evolution of the mutual distances among the
three bodies for non-spinning and spinning BHs respectively, for different
initial conditions. The baseline events e0 and e5 are also depicted once for
each row for comparison and the outcome of the simulations will be discussed
in Sec. 4.

Figure 1: Mutual distance among the three bodies as a function of time for
simulations with initial BH spins set to zero. Each plot in the 5x5 grid
displays a unique simulation. Each row represents the variation of a single
parameter in the initial conditions, in particular eccentricity e $\in$ [0,
0.5, 1, 1.5, 2] $\times 10^{-5}$ (first row); relative velocity between binary
and intruder v $\in$ [30, 40, 50, 60, 70] km/s (second row); impact parameter
b $\in$ [3, 4, 5, 6, 7] $\times 10^{-3}$ AU (third row); intruder mass m$3\in$
[5, 7, 10, 13, 15] M$\odot$ (fourth row); mass ratio q $\in$ [0.3, 0.5, 0.7,
0.83, 1] (bottom row). The title of each panel highlights the occurrence of
merging bodies, their identity in parentheses, along with the time of the
merger. The event e0 serves as baseline for all events and is shown multiple
times for comparison. Figure 2: Same as Figure 1, but in all simulations the
BHs have initial spins. The reference event is denoted as e5.

To characterize the GW emission of each event, we compute the wave amplitude
as [85]:

$h^{TT}_{jk}=\dfrac{2G}{c^{4}R}\ddot{Q}^{\rm TT}_{jk},$ (5)

where $j,\,k=$1,2 and 3, R is the luminosity distance of the source from us
and $\ddot{Q}^{\rm TT}_{jk}$ is the second time derivative of the transverse-
traceless part of the quadrupole moment. We set our sources at a distance of
230$\operatorname{Mpc}$. From now on, we consider the wave propagating along
the $z-$direction and, of the full GW tensor, we choose the $h_{11}$ component
only, i.e. the plus polarization h+, as GW amplitude of our signals.

In order to investigate the influence of the third body on the signal, we
define a parameter, referred to as closeness, that provides a measure of their
spatial distance as:

$C=\text{log}\frac{\text{d}_{min}}{\text{d}_{max}},$ (6)

where dmin and dmax are the minimum and the maximum values of the three mutual
distances and $C<0$ by definition. Figure 3 presents the time evolution of h+
during 1200 s long simulations, unless they are interrupted before.
Interruptions occur when, due to consecutive mergers, only one of the three
bodies is left in the simulation. The closeness parameter is visualized using
a color scale with magenta identifying $C=0$, which specifies that the number
of bodies in the system decreased from three to two due to a merger.

Figure 3: Left. The plots illustrate eight time-series, each representing the
signal of an entire simulation - denoted on the top left. Each simulation
lasts 1200 seconds unless two mergers occur, leading to an early termination.
The color scale corresponds to the closeness parameter, with the magenta color
denoting a value of zero, indicating the presence of only two remaining bodies
due to the merger of two out of three bodies. Right. The plots display zoomed-
in views of the corresponding plots on the left, focusing on encounters and
mergers, providing detailed insights into the evolution of the signal and
closeness parameter during critical phases of the simulation.

####  3.2.1 Hilbert-Huang Transform

We use the Hilbert Huang Transform (HHT) [83] to characterize the signals,
utilizing the EMD Python Package [86]. HHT is a signal processing technique
designed for the analysis of non-linear and non-stationary data. This method
involves two steps. First, it employs empirical mode decomposition (EMD) to
decompose complex signals into intrinsic mode functions (IMF) and residuals,
effectively segregating the time domain data into various frequency scales.
These IMFs, when aggregated with the residual, reproduce the original time
series. Notably, this sifting process is performed adaptively, how the modes
are biased depend on the data towards locally dominant frequencies.
Subsequently, the HHT method applies the Hilbert Transform to each IMF,
facilitating the computation of instantaneous frequency (IF) and amplitude
(IA) for each IMF with respect to time. This capability allows for the
monitoring of temporal fluctuations in the frequency and amplitude of signal
components.

What the Hilbert transform actually does is give an analytical signal, z(t),
from real data, f(t), to calculate instantaneous properties. The analytical
signal will be:

$z(t)=f(t)+iH\\{f(t)\\}=A(t)e^{i\theta(t)}.$ (7)

$H\\{f(t)\\}$ is the Hilbert transform of $f(t)$ :

$H\\{f(t)\\}=\frac{1}{\pi}\mathcal{P}\int_{-\infty}^{\infty}\frac{f\left(\tau\right)}{t-\tau}d\tau,$
(8)

where $\mathcal{P}$ indicates the Cauchy principal value. $A(t)$ and
$\theta(t)$ are the amplitude and the phase of the signal defined as:

$\displaystyle A(t)$ $\displaystyle=\sqrt{f^{2}(t)+H\\{f(t)\\}^{2}},$ (9)
$\displaystyle\theta(t)$ $\displaystyle=\tan^{-1}[H\\{f(t)\\}/f(t)].$

The IF and IA to compute the Hilbert spectrum are respectively:

$\displaystyle\omega(t)$ $\displaystyle=\frac{d\theta(t)}{dt},$ (10)
$\displaystyle e$ $\displaystyle=|A(t)|^{2}.$

###  3.3 Signal injections in ET noise

The signals obtained from our simulations are injected into 104 s long data
segments of ET noise sampled at 4096 Hz. ET noise is simulated by generating
white Gaussian noise colored with the ET design sensitivity curve [87]. We use
the Welch to obtain the amplitude spectral density (ASD) of the noise. In
doing so, we divide the data into segments of 103 s in length, with an overlap
of 0.5. The ASD of the noise is shown in Figure 4 together with the same noise
in the time domain. The injections are done through the gwpy [88, 89] and
pycbc [90] Python packages.

We use the ASD of the signals, $S_{h}$, and the ASD of the noise, $S_{n}$, to
calculate the signal-to-noise ratio (SNR) as follows:

$SNR=\sqrt{4\cdot\sum_{k=1}^{N}\left(\frac{S_{h}(f_{k})^{2}}{S_{n}(f_{k})^{2}}\right)}$
(11)

The SNR is related to the area between the characteristic strain $h_{c}$ and
the characteristic noise strain $h_{n}$, computed as:

$\displaystyle h_{c}$ $\displaystyle=f\sqrt{TS_{h}},$ (12) $\displaystyle
h_{n}$ $\displaystyle=\sqrt{fS_{n}},$

where T is the segment length. The plots of the characteristic strain of all
the simulated signals are shown in Figures 8 and 9 of the Appendix.

The time series of the signals injected into the noise were tapered and
bandpassed between 2 Hz and 30 Hz to compute the spectrograms of these data,
lasting up to 1200 s, containing the signals.

Figure 4: Left side: amplitude spectral density of the ET simulated noise in
green compared to its sensitivity curve [87]. Right side: 104 s of ET
simulated noise in the time domain.

##  4 Results

In the first instance, we discuss the outcome of the simulated encounters,
where the components of the binaries are identified as bodies 1 and 2 whereas
the intruder is body 3. Just considering the two baseline events e0 and e5,
identical except for the spins, the distinction between the non-spinning
(Figure 1) and spinning case (Figure 2) significantly alters the outcome of
the two events. In the former case, as can be better seen from the time series
in Figure 3, the third body completes two close orbits around the binary,
which remains gravitationally bound even after the encounter, resulting in an
increase of its eccentricity. In contrast, in the latter case, the third body
interacts multiple times with the binary and leads it to a merger. The newly
formed BH then forms a new binary system with the third body, remaining
gravitationally bound for the rest of the simulation, with an eccentricity of
approximately 0.8. In events e2 and e4 (Figure 1 and 3), after the initial
encounter, the system remains bound for several orbits until the merger of
bodies 1 and 3 occurs. In e2, the resulting BH and body 2 move away from each
other, while in e4, a new binary with high eccentricity forms. The event e21
(Figure 1 and 3) exhibits a particularly intriguing behavior. Following the
encounter, the third body remains gravitationally bound to the binary, which,
due to the interaction, enters the inspiral phase. A close examination of the
time series reveals that the third body undergoes several close fly-bys during
the inspiral phase, remaining bound to the new BH in a binary system with an
eccentricity of 0.7. In e25 and e26 (Figure 2 and 3), the simulations are
interrupted shortly after, 800 s and 1000 s respectively, due to a second
merger. While we expect similar mergers to occur when a second-generation
binary is formed, we are not able to notice them in our simulations since this
could happen after the time limit of 1200 s. What is important to note here is
that only the second merger occurs through the spiraling of the two bodies,
while in both cases, the first merger is a rapid plunge resulting from the
close interaction.

To understand the features of the stronger events, we characterize the high
SNR regime in the ideal case in which the SNR is calculated using the
characteristic strain $h_{c}$ of the simulated signal over the noise strain
$h_{n}$ of simulated ET noise. After that, we inject the signal in the ET
noise to have an insight on what to expect in the real case and whether the
insights obtained from our simulations can be useful to analyze the real
signal or not.

The top row of Figure 5 shows the SNR of the simulated signals, whereas the
bottom row shows the corresponding number of mergers per event. The mergers
are categorized as follows:

  * 1.

no mergers: 0 M;

  * 2.

one merger resulting from a plunge or capture event: 1 M;

  * 3.

one merger following an inspiral phase between two bodies: 1 M/I;

  * 4.

two mergers, with at least the second one involving an inspiral phase: 2 M/I.

It becomes apparent that the presence of an inspiral leads to a higher SNR, as
clearly seen by comparing the heatmap showing the number of mergers with the
SNR heatmap in Figure 5. Of particular interest are the cases with 0 M (e.g.,
e3) or 1 M (e.g., e5, e20, e27, e35), where unexpectedly high SNRs for the
type of event are observed. This is due to the presence of an inspiral that
has been cut by the end of the simulation because the merger time was longer
than the simulation time. The striking observation from Figure 5 is that
spinning systems invariably lead to at least one merger event, unlike their
non spinning counterparts.

Figure 5: Top. Heatmaps showing the SNRs for Figures 1 (left) and 2 (right).
Bottom. Heatmaps illustrate the number of mergers occurring in the
simulations. The arrangement of the 5x5 grid corresponds to Figures 1 (left)
and 2 (right). The two heatmaps exhibit discrete color variations representing
different scenarios: zero mergers (0 M), one merger resulting from plunge or
capture events (1 M), one merger following an inspiral between two bodies (1
M/I), and two mergers, with at least the second involving an inspiral phase (2
M/I). Notably, the presence of the inspiral phase significantly increases the
SNR, as evident when comparing the cells in the heatmaps with 1 M/I and 2 M/I
to the corresponding SNR heatmap cells. The presence of an inspiral phase,
even if interrupted by the end of the simulation, leads to an increase in the
SNR, even in cases with zero mergers (e.g., e3) or with one merger (e.g., e5,
e20, e27, e35). Figure 6: HHT spectrum of simulations e0, e3 and e21.
Encounters cause rapid changes in frequency.

We then compute the HHT of the simulated signal across time to explore changes
in patterns of the instantaneous frequency. Figure 6 shows the HHT of a
segment of the simulations e0, e3 and e21. As we can see for e0, from
$T=500\,s$ the IMF containing the highest frequencies shows sharp peaks. In
the higher IMFs, IMF 1 and IMF 2, the trace of the two encounters, happening
at 500 s and 515 s respectively (Figure 3 and 6), is particularly visible.
Before the encounter, the binary with $e\sim 0$ keeps a steady value of IMF 1;
after the first encounter, IMF 1 changes by showing sharp frequency peaks;
after the second encounter the peaks change again. This is also visible by the
rapid increase of IMF 2. All in all, the close encounters between the intruder
and the binary, which drastically modify the semi-major axis and the
eccentricity of the binary, show sudden frequency changes in the HHT. A very
similar pattern can be observed in the plot referring to event e3, with a
first steady frequency trace, followed by sudden peaks reaching up to 10 Hz
and by a further change after the second encounter. A slightly different
pattern can be observed for e21. During this resonant encounter, we have
multiple fly-bys - at 485 s, 490 s, 505 s, 525 s and 545 s - due to the
formation of a triple system. The inspiraling motion triggered after the
encounter at 505 s translates to a more fuzzy frequency pattern. The following
encounters are still visible both in the time series and in the HHT: in
particular, we can see that, after the inspiral starts at 505 s, another
encounter at 525 s produces a clearly visible change in the IMF trend.

Figure 7: Time series and spectrograms of four simulations: e3, e14, e24 and
e36. The top panel of each event contains the h+ as a function of time for the
entire simulation. The spectrograms of the signals injected in the noise -
tapered and bandpassed - are normalized by the median of the frequency bins.

Finally, we proceed to inject the signals in simulated ET noise. The plots in
Figure 7 show the spectrograms, each 1200 s long, of e3, e14, e24 and e36. The
data, sampled at 4096 Hz, are tapered and bandpassed between 2 Hz and 30 Hz
and then apply the Welch method using 16 s long segments with an 8 s overlap
to compute the spectra. Each frequency bin is normalized by its median to
highlight the time variation. There is a notable excess of power during the
first encounter, as shown in the examples in Figure 7, a signature evident in
all 42 simulations. Due to the encounters, the binary systems exhibit high
eccentricity, a distinctive feature visible in the spectrograms of e3 and e14
with multiple tails during the inspiral. In simulation e3, there are three
encounters of interest. The first two are very close to each other, at around
the 500 s, whereas the third is just before 700 s. The third encounter leads
to the inspiral of the binary system. Despite the simulation ending before the
actual merger, the phase preceding it is clearly visible in the spectrogram.
Simulation e14 exhibits a similar behavior, where the initial two encounters,
characterized by an excess of power, promptly trigger the merger of the binary
system, perfectly recognizable in the time-frequency plot. In simulation e24,
the initial interaction is a plunge. The intruder immediately merges with the
less massive component of the binary system. However, the binary system itself
remains bound with high eccentricity, approximately 0.9. The formation of this
more massive and high eccentric binary is evident in the spectrogram, where
after the first excess of power, unstable peaks manifest before the system
gradually start to circularize. Looking at the spectrogram of e36, it is clear
that there is a gap between an initial burst (at 485 s) and the subsequent
merger (starting at 550 s) with the inspiral part that follows, showing how
another scenario - the burst - triggered the merger. Specifically, this burst
that we can observe in the spectrogram refers to the close encounter and to
the subsequent plunge. Although the specifics of the phenomenon underlying the
burst cannot be discerned from the spectrograms alone, observing similar
behavior in a significant percentage of our simulations suggests that the
bursts preceding the mergers have an astrophysical nature and are indicative
of a dynamic environment.

##  5 Conclusions

We simulated 42 single-binary encounters with the N-body code ARWV starting
from a common set of initial conditions. For each simulation, we varied only
one of these parameters, in order to show how sensitive the simulations are
even to a small change in initial conditions. We then inspected the
gravitational signals of these events, which involve mergers with their
typical inspiral phase, as well as bursts due to encounters or plunges. We
employed the HHT, a method used to analyze data exhibiting non-linear and non-
stationary patterns. It decomposes signals into their individual components,
enabling the analysis of their time-evolving frequency and amplitude
characteristics. By applying this method, we were able to see clearly how
encounters, after modifying some of the geometric parameters of the binary
(semi-major axis and eccentricity), produce sudden and visible frequency
changes. Finally, after the injection of our signals in the simulated ET
noise, we were still able to observe these frequency changes in the
spectrograms of the events.

Our findings suggest that a highly eccentric merger could be preceded by
encounters - or even by previous mergers (plunges) - that act as boosters for
the merger. In our experiments we considered very tight initial binaries, thus
obtaining quite short simulations, lasting even less than 20 minutes; in
reality, this behavior could also vary in a wider temporal range. By
inspecting the spectrograms of the signals injected in simulated ET noise, we
showed that the main features of these signals are distinguishable also
without any prior knowledge on the nature of the event under examination.

For a future development of this work, we foresee the improvement of the
simulation of spinning bodies. In fact, although ARWV includes the spins, it
does not provide enough information about their orientation. This means that
we are limited to the specific case in which the spins are assumed to be
aligned and that we do not know how an encounter affects the orientation of
the spins in a binary system.

The meaningfulness of this development resides in the fact that, as we have
already extensively discussed, dynamical encounters lead to some typical
signatures like eccentric orbits, unusual mass ratios and misaligned spins.
Therefore, the study of how dynamical interactions affect the spins of a
binary system would be of great help to support follow-up studies of future
detections, numerical waveform studies and detection pipeline developments.
This will become paramount when, in future detectors, we will be able to
detect also the continuous GW signal from the coalescence of compact objects
from which we will be able to infer relevant information about the orbital
parameters of the system.

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##  6 Appendix

In this appendix, we report the Table (2) containing the initial values of the
parameters and the plots of the characteristic strains of all the simulations.

In Table 2 are shown only the parameters that were modified in each
simulation. As explained in Sec 3.1 the value that we varied from e0 or e5 is
one between: eccentricity (e), relative velocity (v), impact parameter (b),
intruder mass (m3) and mass ratio of the binary (q=m2/m1). In the simulations
that include spins, the spins of the three BHs are taken from a Maxwellian
distribution with a root mean square of 0.1 and do not change across the
different simulations.

Figures 8 and 9 illustrate the characteristic strain of the gravitational
signal from simulations in Figure 1 and 2 respectively. The presence of a
spike at 0.6 Hz is attributed to the accumulated signal from the initial
binary before the encounter.

Table 2: Initial parameter values for all conducted simulations. Only values that may differ from simulation e0 are listed. event | m1 [M⊙] | m2 [M⊙] | m3 [M⊙] | e | v [km/s] | b [AU] | spin
---|---|---|---|---|---|---|---
e0 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.005 | 0.0
e1 | 30.0 | 25.0 | 10.0 | 5e-06 | 50.0 | 0.005 | 0.0
e2 | 30.0 | 25.0 | 10.0 | 1e-05 | 50.0 | 0.005 | 0.0
e3 | 30.0 | 25.0 | 10.0 | 1.5e-05 | 50.0 | 0.005 | 0.0
e4 | 30.0 | 25.0 | 10.0 | 2e-05 | 50.0 | 0.005 | 0.0
e5 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.005 |  $\neq$ 0
e6 | 30.0 | 25.0 | 10.0 | 5e-06 | 50.0 | 0.005 |  $\neq$ 0
e7 | 30.0 | 25.0 | 10.0 | 1e-05 | 50.0 | 0.005 |  $\neq$ 0
e8 | 30.0 | 25.0 | 10.0 | 1.5e-05 | 50.0 | 0.005 |  $\neq$ 0
e9 | 30.0 | 25.0 | 10.0 | 2e-05 | 50.0 | 0.005 |  $\neq$ 0
e10 | 30.0 | 25.0 | 10.0 | 0.0 | 30.0 | 0.005 | 0.0
e11 | 30.0 | 25.0 | 10.0 | 0.0 | 40.0 | 0.005 | 0.0
e12 | 30.0 | 25.0 | 10.0 | 0.0 | 60.0 | 0.005 | 0.0
e13 | 30.0 | 25.0 | 10.0 | 0.0 | 70.0 | 0.005 | 0.0
e14 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.003 | 0.0
e15 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.004 | 0.0
e16 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.006 | 0.0
e17 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.007 | 0.0
e18 | 30.0 | 25.0 | 5.0 | 0.0 | 50.0 | 0.005 | 0.0
e19 | 30.0 | 25.0 | 7.0 | 0.0 | 50.0 | 0.005 | 0.0
e20 | 30.0 | 25.0 | 13.0 | 0.0 | 50.0 | 0.005 | 0.0
e21 | 30.0 | 25.0 | 15.0 | 0.0 | 50.0 | 0.005 | 0.0
e22 | 30.0 | 9.0 | 10.0 | 0.0 | 50.0 | 0.005 | 0.0
e23 | 34.0 | 17.0 | 10.0 | 0.0 | 50.0 | 0.005 | 0.0
e24 | 30.0 | 21.0 | 10.0 | 0.0 | 50.0 | 0.005 | 0.0
e25 | 27.0 | 27.0 | 10.0 | 0.0 | 50.0 | 0.005 | 0.0
e26 | 30.0 | 25.0 | 10.0 | 0.0 | 30.0 | 0.005 |  $\neq$ 0
e27 | 30.0 | 25.0 | 10.0 | 0.0 | 40.0 | 0.005 |  $\neq$ 0
e28 | 30.0 | 25.0 | 10.0 | 0.0 | 60.0 | 0.005 |  $\neq$ 0
e29 | 30.0 | 25.0 | 10.0 | 0.0 | 70.0 | 0.005 |  $\neq$ 0
e30 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.003 |  $\neq$ 0
e31 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.004 |  $\neq$ 0
e32 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.006 |  $\neq$ 0
e33 | 30.0 | 25.0 | 10.0 | 0.0 | 50.0 | 0.007 |  $\neq$ 0
e34 | 30.0 | 25.0 | 5.0 | 0.0 | 50.0 | 0.005 |  $\neq$ 0
e35 | 30.0 | 25.0 | 7.0 | 0.0 | 50.0 | 0.005 |  $\neq$ 0
e36 | 30.0 | 25.0 | 13.0 | 0.0 | 50.0 | 0.005 |  $\neq$ 0
e37 | 30.0 | 25.0 | 15.0 | 0.0 | 50.0 | 0.005 |  $\neq$ 0
e38 | 30.0 | 9.0 | 10.0 | 0.0 | 50.0 | 0.005 |  $\neq$ 0
e39 | 34.0 | 17.0 | 10.0 | 0.0 | 50.0 | 0.005 |  $\neq$ 0
e40 | 30.0 | 21.0 | 10.0 | 0.0 | 50.0 | 0.005 |  $\neq$ 0
e41 | 27.0 | 27.0 | 10.0 | 0.0 | 50.0 | 0.005 |  $\neq$ 0
Figure 8: characteristic strain (brown) of the gravitational signal from
simulations in Figure 1 and the strain amplitude of the ET noise (green). At
the top of each plot, the simulation label is displayed. Furthermore, the SNR
of each event is provided in the top-left corner of the corresponding plot.
Figure 9: Similar to Figure 8, but displaying the characteristic strain of the
signals from Figure 2.


††thanks: now at Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA††thanks: now at University of Michigan, Ann Arbor, MI 48109, USA††thanks:
Now at University College London, London WC1E 6BT, UK††thanks: now at Tufts
University, Medford, MA 02155, USA

The MINERvA Collaboration

#  Measurement of final-state correlations in neutrino muon-proton mesonless
production on hydrocarbon at $\langle E_{\nu}\rangle=3$ GeV

X.-G. Lu  Oxford University, Department of Physics, Oxford, United Kingdom  M.
Betancourt  Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
USA  T. Walton  Fermi National Accelerator Laboratory, Batavia, Illinois
60510, USA  F. Akbar  AMU Campus, Aligarh, Uttar Pradesh 202001, India  L.
Aliaga  Department of Physics, College of William & Mary, Williamsburg,
Virginia 23187, USA  Sección Física, Departamento de Ciencias, Pontificia
Universidad Católica del Perú, Apartado 1761, Lima, Perú  O. Altinok  Physics
Department, Tufts University, Medford, Massachusetts 02155, USA  D.A. Andrade
Campus León y Campus Guanajuato, Universidad de Guanajuato, Lascurain de
Retana No. 5, Colonia Centro, Guanajuato 36000, Guanajuato México.  M.
Ascencio  Sección Física, Departamento de Ciencias, Pontificia Universidad
Católica del Perú, Apartado 1761, Lima, Perú  L. Bellantoni  Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA  A. Bercellie  University
of Rochester, Rochester, New York 14627 USA  A. Bodek  University of
Rochester, Rochester, New York 14627 USA  A. Bravar  University of Geneva,
1211 Geneva 4, Switzerland  H. Budd  University of Rochester, Rochester, New
York 14627 USA  T. Cai  University of Rochester, Rochester, New York 14627 USA
M.F. Carneiro  Department of Physics, Oregon State University, Corvallis,
Oregon 97331, USA  J. Chaves  Department of Physics and Astronomy, University
of Pennsylvania, Philadelphia, PA 19104  D. Coplowe  Oxford University,
Department of Physics, Oxford, United Kingdom  H. da Motta  Centro Brasileiro
de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, Rio de
Janeiro, 22290-180, Brazil  S.A. Dytman  Department of Physics and Astronomy,
University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA  G.A. Díaz
University of Rochester, Rochester, New York 14627 USA  Sección Física,
Departamento de Ciencias, Pontificia Universidad Católica del Perú, Apartado
1761, Lima, Perú  J. Felix  Campus León y Campus Guanajuato, Universidad de
Guanajuato, Lascurain de Retana No. 5, Colonia Centro, Guanajuato 36000,
Guanajuato México.  L. Fields  Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA  Northwestern University, Evanston, Illinois 60208  R.
Fine  University of Rochester, Rochester, New York 14627 USA  A.M. Gago
Sección Física, Departamento de Ciencias, Pontificia Universidad Católica del
Perú, Apartado 1761, Lima, Perú  R.Galindo  Departamento de Física,
Universidad Técnica Federico Santa María, Avenida España 1680 Casilla 110-V,
Valparaíso, Chile  H. Gallagher  Physics Department, Tufts University,
Medford, Massachusetts 02155, USA  A. Ghosh  Departamento de Física,
Universidad Técnica Federico Santa María, Avenida España 1680 Casilla 110-V,
Valparaíso, Chile  Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier
Sigaud 150, Urca, Rio de Janeiro, Rio de Janeiro, 22290-180, Brazil  R. Gran
Department of Physics, University of Minnesota – Duluth, Duluth, Minnesota
55812, USA  D.A. Harris  Fermi National Accelerator Laboratory, Batavia,
Illinois 60510, USA  S. Henry  University of Rochester, Rochester, New York
14627 USA  S. Jena  Knowledge city, Sector 81, SAS Nagar, Manauli PO 140306
D.Jena  Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
J. Kleykamp  University of Rochester, Rochester, New York 14627 USA  M.
Kordosky  Department of Physics, College of William & Mary, Williamsburg,
Virginia 23187, USA  T. Le  Physics Department, Tufts University, Medford,
Massachusetts 02155, USA  Rutgers, The State University of New Jersey,
Piscataway, New Jersey 08854, USA  E. Maher  Massachusetts College of Liberal
Arts, 375 Church Street, North Adams, MA 01247  S. Manly  University of
Rochester, Rochester, New York 14627 USA  W.A. Mann  Physics Department, Tufts
University, Medford, Massachusetts 02155, USA  C.M. Marshall  University of
Rochester, Rochester, New York 14627 USA  K.S. McFarland  University of
Rochester, Rochester, New York 14627 USA  Fermi National Accelerator
Laboratory, Batavia, Illinois 60510, USA  A.M. McGowan  University of
Rochester, Rochester, New York 14627 USA  B. Messerly  Department of Physics
and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
J. Miller  Departamento de Física, Universidad Técnica Federico Santa María,
Avenida España 1680 Casilla 110-V, Valparaíso, Chile  A. Mislivec  University
of Rochester, Rochester, New York 14627 USA  J.G. Morfín  Fermi National
Accelerator Laboratory, Batavia, Illinois 60510, USA  J. Mousseau  University
of Florida, Department of Physics, Gainesville, FL 32611  D. Naples
Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh,
Pennsylvania 15260, USA  J.K. Nelson  Department of Physics, College of
William & Mary, Williamsburg, Virginia 23187, USA  C. Nguyen  University of
Florida, Department of Physics, Gainesville, FL 32611  A. Norrick  Department
of Physics, College of William & Mary, Williamsburg, Virginia 23187, USA
Nuruzzaman  Rutgers, The State University of New Jersey, Piscataway, New
Jersey 08854, USA  Departamento de Física, Universidad Técnica Federico Santa
María, Avenida España 1680 Casilla 110-V, Valparaíso, Chile  A. Olivier
University of Rochester, Rochester, New York 14627 USA  V. Paolone  Department
of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania
15260, USA  C.E. Patrick  Northwestern University, Evanston, Illinois 60208
G.N. Perdue  Fermi National Accelerator Laboratory, Batavia, Illinois 60510,
USA  University of Rochester, Rochester, New York 14627 USA  M.A. Ramírez
Campus León y Campus Guanajuato, Universidad de Guanajuato, Lascurain de
Retana No. 5, Colonia Centro, Guanajuato 36000, Guanajuato México.  R.D.
Ransome  Rutgers, The State University of New Jersey, Piscataway, New Jersey
08854, USA  L. Ren  Department of Physics and Astronomy, University of
Pittsburgh, Pittsburgh, Pennsylvania 15260, USA  D. Rimal  University of
Florida, Department of Physics, Gainesville, FL 32611  P.A. Rodrigues
University of Mississippi, Oxford, Mississippi 38677, USA  University of
Rochester, Rochester, New York 14627 USA  D. Ruterbories  University of
Rochester, Rochester, New York 14627 USA  H. Schellman  Department of Physics,
Oregon State University, Corvallis, Oregon 97331, USA  Northwestern
University, Evanston, Illinois 60208  J.T. Sobczyk  University of Wroclaw,
plac Uniwersytecki 1, 50-137 Wrocław, Poland  C.J. Solano Salinas  Universidad
Nacional de Ingeniería, Apartado 31139, Lima, Perú  H. Su  Department of
Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania
15260, USA  M. Sultana  University of Rochester, Rochester, New York 14627 USA
E. Valencia  Department of Physics, College of William & Mary, Williamsburg,
Virginia 23187, USA  Campus León y Campus Guanajuato, Universidad de
Guanajuato, Lascurain de Retana No. 5, Colonia Centro, Guanajuato 36000,
Guanajuato México.  D. Wark  Oxford University, Department of Physics, Oxford,
United Kingdom  A. Weber  Oxford University, Department of Physics, Oxford,
United Kingdom  J. Wolcott  University of Rochester, Rochester, New York 14627
USA  M.Wospakrik  University of Florida, Department of Physics, Gainesville,
FL 32611  B. Yaeggy  Departamento de Física, Universidad Técnica Federico
Santa María, Avenida España 1680 Casilla 110-V, Valparaíso, Chile

###### Abstract

Final-state kinematic imbalances are measured in mesonless production of
$\nu_{\mu}+\textrm{A}\to\mu^{-}+\textrm{p}+\textrm{X}$ in the MINERvA tracker.
Initial- and final-state nuclear effects are probed using the direction of the
$\mu^{-}$-p transverse momentum imbalance and the initial-state momentum of
the struck neutron. Differential cross sections are compared to predictions
based on current approaches to medium modeling. These models under-predict the
cross section at intermediate intranuclear momentum transfers that generally
exceed the Fermi momenta. As neutrino interaction models need to correctly
incorporate the effect of the nucleus in order to predict neutrino energy
resolution in oscillation experiments, this result points to a region of phase
space where additional cross section strength is needed in current models, and
demonstrates a new technique that would be suitable for use in fine grained
liquid argon detectors where the effect of the nucleus may be even larger.

An accurate understanding of nuclear medium modifications to neutrino-nucleon
interactions is required for reliable measurements of fundamental neutrino
properties. The distributions of final-state observables reflect complicated
and intertwined effects from nucleon and nuclear dynamics, and the
interpretation of single-particle kinematics is thereby obscured Nieves:2011yp
. These underlying dynamics can influence neutrino energy reconstruction in
oscillation experiments Coloma:2013rqa ; Abe:2017vif . Certain categories of
nuclear effects, however, can be separated by variables Lu:2015tcr ;
Furmanski:2016wqo  designed to elicit final-state correlations that are absent
for neutrino interactions on free nucleons, but are in-play in neutrino-
nucleus scattering. This Letter reports the measurements with such variables
for the purpose of constraining nuclear effects in neutrino interactions.

In charged-current neutrino-nucleus scattering there is an imbalance,
$\delta\vec{p}$, between the initial neutrino momentum and the sum of final-
state lepton and hadron momenta as a result of nuclear effects. This imbalance
is the sum of Fermi motion (FM) and intranuclear momentum transfer (IMT),
which is the sum of all other effects including nucleon correlations
Singh:1992dc ; Gil:1997bm ; Nieves:2004wx ; Nieves:2005rq ; Martini:2009uj ;
Benhar:1994hw ; Delorme:1985ps ; Marteau:1999jp ; Martini:2010ex ;
Nieves:2011pp ; Nieves:2011yp ; Martini:2011wp  and final-state interactions
(FSI):

$\displaystyle\delta\vec{p}=\vec{p}_{\textrm{FM}}-\vec{p}_{\textrm{IMT}}.$ (1)

In $\nu_{\mu}$ charged-current quasielastic (QE) interactions,
$\nu_{\mu}+\textrm{n}\to\mu^{-}+\textrm{p}$, momentum is transferred from the
leptonic current to the target neutron. However if the neutron is correlated
with other nucleons, the momentum transfer is shared among the correlated
partners. In both cases, final-state interactions occur as particles from the
primary interaction propagate through the nucleus exchanging energy, momentum
and charge with the nuclear environment. Primary particles can be absorbed
during this propagation; baryonic resonances (RES) can be produced in the
primary interaction and the resulting products can undergo FSI and be
absorbed. These non-QE processes give the same observable final state as QE
scattering.

This study focuses on the QE-like process

$\displaystyle\nu_{\mu}+\textrm{A}\to\mu^{-}+\textrm{p}+\textrm{X},$ (2)

where X is a final-state hadronic system consisting of the nuclear remnant
with possible additional protons but without pions that indicate RES or other
processes. In Eq. (2), the incident neutrino energy, $E_{\nu}$, is unknown,
but the dependence of $\delta\vec{p}$ on $E_{\nu}$ can be removed. This can be
done as follows:

Firstly, decompose $\delta\vec{p}$ into longitudinal and transverse components
with respect to the neutrino direction,

$\displaystyle\delta\vec{p}$ $\displaystyle\equiv(\delta
p_{\textrm{L}},~{}\delta\vec{p}_{\textrm{T}}),$ (3) $\displaystyle E_{\nu}$
$\displaystyle=p_{\textrm{L}}^{\,\mu}+p_{\textrm{L}}^{\,\textrm{p}}-\delta
p_{\textrm{L}},$ (4) $\displaystyle\vec{0}$
$\displaystyle=\vec{p}_{\textrm{T}}^{\,\mu}+\vec{p}_{\textrm{T}}^{\,\textrm{p}}-\delta\vec{p}_{\textrm{T}},$
(5)

where $\vec{p}^{\,\mu}$ and $\vec{p}^{\,\textrm{p}}$ are the muon and proton
momenta, respectively. The direction of the transverse momentum imbalance
$\delta\vec{p}_{\textrm{T}}$ (see schematic definition in Fig. 1),

$\displaystyle\delta\alpha_{\textrm{T}}$
$\displaystyle\equiv\arccos\frac{-\vec{p}_{\textrm{T}}^{\,\mu}\cdot\delta\vec{p}_{\textrm{T}}}{p_{\textrm{T}}^{\,\mu}\delta
p_{\textrm{T}}},$ (6)

is uniformly distributed in the absence of IMT because of the isotropic nature
of Fermi motion. This variable is thus sensitive to IMT Lu:2015tcr . Because
$\left|\vec{p}_{\textrm{T}}^{\,\textrm{p}}\right|>\left|\vec{p}_{\textrm{T}}^{\,\mu}\right|$
for $\delta\alpha_{\textrm{T}}<90^{\circ}$, accelerating FSI can be
distinguished from decelerating FSI using $\delta\alpha_{\textrm{T}}$. Recent
measurements of $\delta p_{\textrm{T}}$ and $\delta\alpha_{\textrm{T}}$ on
hydrocarbon at beam energy around 600 MeV by the T2K Collaboration can be
found in Ref. Abe:2018pwo .

Figure 1: Schematic definition of the transverse kinematics Lu:2015tcr .

Secondly, under the assumption that X is just the remnant nucleus,
$\textrm{A}^{\prime}$, then $\delta p$ gives the magnitude of its recoil
momentum, and

$\displaystyle\delta p_{\textrm{L}}$
$\displaystyle=\frac{1}{2}R-\frac{m_{\textrm{A}^{\prime}}^{2}+\delta
p_{\textrm{T}}^{2}}{2R},$ (7) $\displaystyle R$ $\displaystyle\equiv
m_{\textrm{A}}+p_{\textrm{L}}^{\,\mu}+p_{\textrm{L}}^{\,\textrm{p}}-E^{\mu}-E^{\textrm{p}},$
(8)

where $m_{\textrm{A}^{(\prime)}}$, $E^{\mu\,(\textrm{p})}$ are the nuclear
target (remnant) mass, and the muon (proton) energy, respectively
Furmanski:2016wqo . In the limit of zero IMT (that is, pure QE), the recoiling
momentum of $\textrm{A}^{\prime}$ balances the initial neutron momentum and

$\displaystyle p_{\textrm{n}}$ $\displaystyle=\delta p,$ (9)

which can be estimated using the relation
$m_{\textrm{A}^{\prime}}=m_{\textrm{A}}-m_{\textrm{n}}+b$, where
$m_{\textrm{n}}$ is the neutron mass, and $b=+27.13~{}\textrm{MeV}$ for carbon
obtained from the probabilistic model for excitation energy Furmanski:2016wqo
.

This Letter presents the measurement of $\delta\alpha_{\textrm{T}}$ and
$p_{\textrm{n}}$ in $\nu_{\mu}$ induced production on polystyrene by the
MINERvA experiment. The signal is defined as an event with no pions, one muon
and at least one proton satisfying

$\displaystyle 1.5~{}\textrm{GeV}/c<$ $\displaystyle
p_{\mu}<10~{}\textrm{GeV}/c,~{}\theta_{\mu}<20^{\circ},$ (10) $\displaystyle
0.45~{}\textrm{GeV}/c<$ $\displaystyle
p_{\textrm{p}}<1.2~{}\textrm{GeV}/c,~{}\theta_{\textrm{p}}<70^{\circ},$ (11)

where $p_{\mu}$ and $\theta_{\mu}$ ($p_{\textrm{p}}$ and
$\theta_{\textrm{p}}$) are the muon (proton) momentum and polar angle with
respect to the neutrino direction, respectively, when exiting the nucleus.
Nuclear effects in terms of Fermi motion and IMT are measured and compared to
model predictions.

The MINERvA experiment is in the NuMI beam line Adamson:2015dkw  at Fermilab.
The detector is described in detail elsewhere Aliaga:2013uqz . The tracker is
constructed of hexagonal planes which are approximately perpendicular to the
incoming neutrino beam and made from triangular scintillator strips.
Scintillator strips in adjacent planes are rotated by 60∘ with respect to each
other, permitting three-dimensional track reconstruction which is efficient up
to 70∘ from the detector axis. The scintillator is embedded in polystyrene,
containing the carbon target nuclei. The MINOS Near Detector is two meters
downstream of the MINERvA detector and serves as a magnetized muon
spectrometer Michael:2008bc . The data used in this analysis corresponds to
3.28$\times 10^{20}$ protons on target (POT) delivered from 2010 to 2012; the
integrated $\nu_{\mu}$ flux prediction ($2.88\times
10^{-8}/\textrm{cm}^{2}/\textrm{POT}$) is from Aliaga:2016oaz .

Neutrino interactions are simulated with genie 2.8.4 Andreopoulos:2009rq  in
both a nominal form, with and without FSI, and also with a MINERvA ‘tune’,
(mnvgenie-v1). The nuclear initial state is modeled as relativistic Fermi gas
Bodek:1980ar . Quasielastic LlewellynSmith:1971uhs , RES, and deep inelastic
scattering (DIS) kinematics are modeled with a dipole axial form factor using
$M^{\textrm{QE}}_{\textrm{A}}=0.99~{}\textrm{GeV}/c^{2}$, the Rein-Sehgal
model Rein:1980wg , and the 2003 Bodek-Yang model Bodek:2002ps , respectively.
pythia6 Sjostrand:2006za  and models based on Koba-Nielsen-Olesen scaling
Koba:1972ng  are used to describe hadronization. The $hA$ option of genie was
used to model FSI Dytman:2011zz . The general performance of the genie FSI
treatment for pions has been demonstrated in data versus simulation
comparisons published by MINERvA Eberly:2014mra ; Aliaga:2015wva ;
McGivern:2016bwh ; Altinok:2017xua ; Betancourt:2017uso .

mnvgenie-v1 includes two-particle-two-hole (2p2h) excitations of the nucleus
as formulated in the Valencia model Nieves:2011yp ; Sobczyk:2012ms ;
Gran:2013kda ; Schwehr:2016pvn . The interaction strength with 2p2h has been
tuned to MINERvA inclusive scattering data Rodrigues:2015hik , resulting in a
significant enhancement relative to the Valencia model in a restricted region
of energy-momentum transfer. mnvgenie-v1 also includes a modification to the
nonresonant pion production as constrained by deuterium data Wilkinson:2014yfa
, and collective excitations of the nucleus for the QE channel. The latter are
approximated as a superposition of one-particle-one-hole (1p1h) excitations
and calculated with the Random Phase Approximation Nieves:2004wx . Because the
affected events contribute little to the sample, the effects of nonresonant
pion production and RPA in this analysis are negligible.

In the nominal genie configuration, FSI are further categorized as follows:

  1. 1.

Noninteracting proton FSI \- FSI without pion absorption in which the proton
propagates as a free particle.

  2. 2.

Accelerating proton FSI \- FSI without pion absorption in which the proton
energy increases as a result of FSI.

  3. 3.

Decelerating proton FSI \- FSI without pion absorption in which the proton
energy decreases as a result of FSI.

  4. 4.

Pion FSI \- FSI in which a pion is absorbed.

In addition to genie, data is also compared to the predictions of nuwro
Golan:2012wx . The initial state is modeled either as a local Fermi gas or
with a Spectral Function Benhar:1994hw . FSI are treated as intranuclear
cascades of hadronic interactions Pandharipande:1992zz  incorporating the Oset
model Salcedo:1987md , and 2p2h excitations are from the Valencia model
Nieves:2011yp ; Sobczyk:2012ms .

Events with at least two reconstructed tracks in the MINERvA tracker
satisfying Eqs. (10)-(11) are selected. The muon candidate track must match a
track in the MINOS Near Detector, necessitating Eq. (10). The two tracks are
combined to determine $p_{\mu}$ and $\theta_{\mu}$, with resolutions of
$\sim$8% at 5 $\textrm{GeV}/c$ and $\sim$0.6 degrees, respectively. The proton
candidate is distinguished from possible mesons with specific energy loss
($\textrm{d}E/\textrm{d}x$). Its momentum $p_{\textrm{p}}$ is determined
according to the $\textrm{d}E/\textrm{d}x$ profile along the track
Walton:2014dka ; Walton:2014esl . If two or more protons satisfy Eq. (11)
(less than 1% of all selected events), the most energetic one is taken as the
proton candidate.

To improve $p_{\textrm{p}}$ resolution, additional selection requirements to
obtain elastically scattered and contained (ESC) protons Lu:2016mjf  are
introduced. When a proton is not contained in the tracker or undergoes
inelastic scattering it has a deteriorated momentum estimate. ESC protons are
selected by requiring large $\textrm{d}E/\textrm{d}x$ near the track
endpoints. This reduces the spread of the reconstructed $p_{\textrm{p}}$ to
about 60% of its previous measurement Walton:2014esl , resulting in a
resolution of $\sim$2% at 1 $\textrm{GeV}/c$, at the cost of a reduction of
statistics to about 40% of the initial proton sample.

The efficiency (including acceptance effects) of event selection is estimated
to be 8.6% and the purity is 78%. The predicted background contributions
mainly come from RES where the pion from baryon decay exits the nucleus but is
not identified (13.4%) and DIS (5.4%). A data-driven method Walton:2014esl  is
used to determine backgrounds. Sidebands are determined in the plane of
unattached visible energy away from the interaction vertex vs. transferred
four-momentum squared, $Q^{2}$. Backgrounds from the genie simulation are
rescaled so that the sidebands describe the data (See Fig. 2) and are then
extrapolated into the signal region. The background-subtracted distributions
are then unfolded DAgostini:1994fjx  with four iterations, where the number of
iterations is chosen to balance between the bias and fluctuation of the
unfolded distributions. After a subsequent efficiency correction, event
distributions are normalized by the product of the number of target nucleons
($3.11\times 10^{30}$), POT, and $\nu_{\mu}$ flux, to obtain the flux-averaged
differential cross sections.

Figure 2: Reconstructed data distribution of $p_{\textrm{n}}$ in a sideband
region, compared to full mnvgenie-v1+geant4 simulation after background
tuning. The QE-like signal and the background are decomposed into interaction
modes.

Systematic uncertainties are evaluated for all observables. As an example, the
cross section uncertainty in $\delta\alpha_{\textrm{T}}$ is summarized here.
Besides statistical uncertainty (5-7%), uncertainties arising from the NuMI
flux prediction (6%), genie modeling (6-9%) and detector response (6-16%) are
accounted for. For the latter two, parameters in the physics and detector
models are varied within uncertainties and the resulting cross section
variations are the assigned systematic uncertainties Walton:2014esl . genie
model uncertainties predominantly arise from modeling 2p2h, while the
transverse projection of the muon and proton kinematics and the ESC proton
selection have significant contributions from detector systematics. The total
uncertainty is 20% at $\delta\alpha_{\textrm{T}}=0^{\circ}$ and 180∘, and
decreases to about 13% at 90∘.

Figure 3: Differential cross section in $\theta_{\textrm{p}}$ compared to
genie predictions (nominal genie version: 2.8.4). Outer error bars represent
the combined statistical and systematic uncertainties, while the inner ones
statistical only. The white space between the lines for nominal genie and
mnvgenie-v1 is mostly from the tuned 2p2h component in the latter.  Figure 4:
Differential cross sections in $p_{\textrm{n}}$, compared to genie
predictions. The _no-FSI_ prediction maximum is $11.5$ (see legend) in unit of
$10^{-39}$ cm${}^{2}/\textrm{GeV}/c/\textrm{nucleon}$.  Figure 5: Differential
cross sections in $\delta\alpha_{\textrm{T}}$, compared to genie predictions.
The peak in accelerating proton FSI on the right side is from events where
$\vec{p}_{\textrm{T}}^{\,\textrm{p}}$ has a large angular separation from
$-\vec{p}_{\textrm{T}}^{\,\mu}$ despite
$\left|\vec{p}_{\textrm{T}}^{\,\textrm{p}}\right|>\left|\vec{p}_{\textrm{T}}^{\,\mu}\right|$.

Muon momentum and polar angle distributions are satisfactorily described by
mnvgenie-v1 (Supplemental Material 1 Fig. 1), as is the proton momentum
distribution (Supplemental Material 1 Fig. 2). There is a significant over-
prediction beyond 1-$\sigma$ total uncertainty at $\theta_{\textrm{p}}\gtrsim
60^{\circ}$ (Fig. 3). Of the four FSI categories, noninteracting proton and
accelerating proton FSI are the ones that appear only in this high angle
region. Away from it, at $\theta_{\textrm{p}}$ below $40^{\circ}$ where pion
FSI and 2p2h contributions are large, the mnvgenie-v1 prediction describes the
data very well.

For genie without FSI and nominal genie with noninteracting proton, the pure
QE assumption of Eqs. (7)-(9) is valid. The corresponding predictions show a
Fermi-gas peak at $p_{\textrm{n}}\lesssim 0.25~{}\textrm{GeV}/c$ and the
additional Bodek-Ritchie tail Bodek:1980ar  (Fig. 4). As only Fermi motion is
considered for such pure QE events, the distribution of
$\delta\alpha_{\textrm{T}}$ is flat to first order; a secondary effect arises
from the cross section dependence on the center-of-mass energy of the
interacting neutrino-neutron system (Fig. 5).

With IMT, $\delta p$, and therefore $p_{\textrm{n}}$ [Eq. (9)], is (the
modulus of) the convolution of Fermi motion and IMT [Eq. (1)]. As Fermi motion
alone has a small effect on $\delta\alpha_{\textrm{T}}$, the effect of IMT is
revealed by the non-flatness of the $\delta\alpha_{\textrm{T}}$ distribution.
Following the definition Eq. (6) (see also Fig. 1), it can be seen that when
$\left|\vec{p}_{\textrm{T}}^{\,\textrm{p}}\right|<\left|\vec{p}_{\textrm{T}}^{\,\mu}\right|$,
$\delta\alpha_{\textrm{T}}$ becomes larger than $90^{\circ}$. Since in the
plane transverse to the neutrino direction the muon and proton momenta are
back-to-back for a free nucleon target (no nuclear effects),
$\left|\vec{p}_{\textrm{T}}^{\,\textrm{p}}\right|<\left|\vec{p}_{\textrm{T}}^{\,\mu}\right|$
means that the proton is (transversely) decelerated by nuclear effects. As a
result, in Fig. 5, the nominal genie prediction with decelerating proton FSI
does not contribute greatly to the transverse forward boosting region
$\delta\alpha_{\textrm{T}}\lesssim 90^{\circ}$ Lu:2015tcr  where accelerating
proton FSI are the dominating IMT. Such accelerating FSI are responsible for
the QE peak distortion beyond 5-$\sigma$ total uncertainty at the lowest
$p_{\textrm{n}}$ (Fig. 4). Of this distortion, the largest contribution comes
from the elastic interaction of the genie hA FSI model Lu:2015tcr  which does
not occur in other models, such as genie’s hN FSI model Dytman:2011zz . This
elastic interaction was designed to resemble hadron-nucleus elastic scattering
where the scattered particle could gain energy at the expense of a decelerated
recoil nucleus. Turning this effect off can provide a better shape agreement
with data.

Non-exclusive IMT—pion FSI and 2p2h—are well separated from the Fermi motion
prediction of the $p_{\textrm{n}}$ QE peak (Fig. 4). At $p_{\textrm{n}}$ below
$0.25~{}\textrm{GeV}/c$, data points constrain the modeling of Fermi motion;
at $p_{\textrm{n}}\gtrsim 0.4~{}\textrm{GeV}/c$, where pion FSI and 2p2h
effects are large, mnvgenie-v1’s prediction follows the data. In the
transition region, $p_{\textrm{n}}\sim 0.3~{}\textrm{GeV}/c$, mnvgenie-v1
shows a clear deficit beyond 1-$\sigma$ total uncertainty. With pion
absorption and 2p2h events, the measured proton carries a fraction of the
total momentum transfer given to multiple particles. Therefore, these
reactions behave similarly as decelerating proton FSI in both $p_{\textrm{n}}$
and $\delta\alpha_{\textrm{T}}$. The overall mnvgenie-v1 prediction describes
the $\delta\alpha_{\textrm{T}}$ distribution well (Fig. 5).

Comparison to the predictions of nuwro are shown in Fig. 6. Here, nuwro’s
Spectral Function model works better than its local Fermi gas model. Like
mnvgenie-v1, nuwro with Spectral Function is lacking strength at the
transition region between Fermi motion and non-exclusive IMT (RES and 2p2h),
also with a deficit significantly beyond 1-$\sigma$ total uncertainty. Except
for this, the pion production and 2p2h treatment in nuwro provide a good
description for IMT. Furthermore, the predictions with Spectral Function for
single particle kinematics are all within 1-$\sigma$ total uncertainties
(Supplemental Material 1 Figs. 3 and 4).

Figure 6: Differential cross sections in $p_{\textrm{n}}$ (_left_) and
$\delta\alpha_{\textrm{T}}$ (_right_), compared to nuwro predictions with
local Fermi gas (LFG) or Spectral Function (SF).

nuwro’s Spectral Function option does introduce a high-momentum component for
the initial nucleon motion, more than the Bodek-Ritchie addition gives to
genie. The Spectral Function enhancement with respect to local Fermi gas
appears in the transition region $p_{\textrm{n}}\sim 0.3~{}\textrm{GeV}/c$.
This is also a kinematic region where the 2p2h tuning process of mnvgenie-v1
increases the cross section predictions up to $0.4\times
10^{-39}~{}\textrm{cm}^{2}/\textrm{GeV}/c/\textrm{nucleon}$. So introduction
of such a tuning process to nuwro with Spectral Function, or a Spectral
Function-based mnvgenie-v1 might produce a model that agrees better with the
data.

In conclusion, the cross sections of QE-like production in terms of
$p_{\textrm{n}}$ and $\delta\alpha_{\textrm{T}}$, whereby the target Fermi
motion and the intranuclear momentum transfer are separated, have been
presented. MINERvA’s tuned implementation of 2p2h processes gives rate and
shape corrections that enable genie to accurately describe the data. Both
mnvgenie-v1 and nuwro with Spectral Function provide good descriptions of the
single particle kinematics and reasonable predictions for $p_{\textrm{n}}$ and
$\delta\alpha_{\textrm{T}}$. However, both generators fail in the transition
region between Fermi motion and IMT dominated regions. Combination of a
mnvgenie-v1-like 2p2h tune and Spectral Function might correct this.

This technique relies on the fine-grained tracking capability of MINERvA to
identify and precisely measure ESC protons. This technique will also be used
in experiments with homogeneous trackers such as liquid argon time projection
chambers Antonello:2015lea ; Acciarri:2015uup .

Because the variables $p_{\textrm{n}}$ and $\delta\alpha_{\textrm{T}}$ have
sensitivity to initial- and final-state nuclear effects, their measurements on
other nuclear targets from MINERvA and from liquid argon experiments should
reveal individual A-dependence for Fermi motion and IMT such as FSI and 2p2h,
and thus provide a detailed modeling of the nuclear effects that will be
valuable for future neutrino oscillation experiments.

This work was supported by the Fermi National Accelerator Laboratory under US
Department of Energy contract No. DE-AC02-07CH11359 which included the MINERvA
construction project. Construction support was also granted by the United
States National Science Foundation under Award PHY-0619727 and by the
University of Rochester. Support for participating scientists was provided by
NSF and DOE (USA), by CAPES and CNPq (Brazil), by CoNaCyT (Mexico), by
Proyecto Basal FB 0821, CONICYT PIA ACT1413, Fondecyt 3170845 and 11130133
(Chile), by DGI-PUCP and UDI/VRI-IGI-UNI (Peru), by the Latin American Center
for Physics (CLAF), by Science and Technology Facilities Council (UK), and by
NCN Opus Grant No. 2016/21/B/ST2/01092 (Poland). We thank the MINOS
Collaboration for use of its near detector data. We acknowledge the dedicated
work of the Fermilab staff responsible for the operation and maintenance of
the beam line and detector and the Fermilab Computing Division for support of
data processing.

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