EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN)

 CERN-EP-2022-050 LHCb-PAPER-2021-053 21 April 2022

Nuclear modification factor of neutral pions in the forward and backward
regions in $p$Pb collisions

LHCb collaboration

The nuclear modification factor of neutral pions is measured in proton-lead
collisions collected at a center-of-mass energy per nucleon of
$8.16\,\text{\,Te\kern-1.00006ptV}$ with the LHCb detector. The ${\pi}^{0}$
production cross section is measured differentially in transverse momentum
($p_{\mathrm{T}}$) for $1.5<p_{\mathrm{T}}<10.0\,\text{\,Ge\kern-1.00006ptV}$
and in center-of-mass pseudorapidity ($\eta_{\rm CM}$) regions $2.5<\eta_{\rm
CM}<3.5$ (forward) and $-4.0<\eta_{\rm CM}<-3.0$ (backward) defined relative
to the proton beam direction. The forward measurement shows a sizable
suppression of ${{\pi}^{0}}$ production, while the backward measurement shows
the first evidence of ${{\pi}^{0}}$ enhancement in proton-lead collisions at
the LHC. Together, these measurements provide precise constraints on models of
nuclear structure and particle production in high-energy nuclear collisions.

Submitted to Phys. Rev. Lett.

© 2024 CERN for the benefit of the LHCb collaboration. CC BY 4.0 licence.

Neutral pion production is an important probe of nuclear effects in heavy ion
collisions. In proton-lead ($p{\rm Pb}$) collisions at the Large Hadron
Collider (LHC), ${{\pi}^{0}}$ production is particularly sensitive to cold
nuclear matter (CNM) effects on the initial state of the bound nucleons in the
colliding nucleus. Pion production in $p{\rm Pb}$ collisions can be described
using the collinear factorization framework where CNM effects are encoded into
nuclear parton distribution functions (nPDFs) [1, 2, 3, 4] and the parton-
parton collision dynamics are described by perturbative quantum chromodynamics
(pQCD) [5]. These nPDFs are determined using fits to data and are poorly
constrained for partons with momentum fraction $x$ smaller than about
$10^{-4}$. At low $x$ and low momentum transfer $Q$, the parton number density
may become so large that parton saturation effects become sizable. Parton
saturation would lead to nonlinear evolution of parton densities and is often
described using the color glass condensate (CGC) effective field theory [6].
Measurements of ${{\pi}^{0}}$ production in $p{\rm Pb}$ collisions at forward
and backward rapidities with the LHCb detector can provide constraints on
nPDFs for $x$ between $10^{-6}$ and $10^{-1}$, potentially helping identify
the onset of parton saturation effects [7].

Measurements of angular correlations in small-collision systems at the LHC [8,
9, 10, 11, 12] and the Relativistic Heavy Ion Collider (RHIC) [13, 14] show
evidence of collective flow, which cannot be described using only collinear
factorization and nPDFs. In addition, the LHCb experiment recently measured
the nuclear modification factor of inclusive charged particles in $p{\rm Pb}$
collisions at a center-of-mass energy per nucleon of $\sqrt{s_{\rm
NN}}=5\,\text{\,Te\kern-1.00006ptV}$, observing a much larger enhancement at
backward pseudorapidities than predicted by nPDF calculations [15]. Similar
enhancements have been observed in charged-particle production at RHIC [16,
17, 18, 19] and have been attributed to the Cronin effect [20]. The Cronin
effect is often described as initial-state multiple scattering within the
nucleus [21], although radial collective flow and final-state recombination
could produce similar enhancements [22, 23]. However, measurements of
inclusive charged-particle and ${{\pi}^{0}}$ production at central rapidity at
the LHC show no evidence of a Cronin-like enhancement [24, 25, 26, 27],
potentially pointing to interplay between enhancing effects and low-$x$
effects such as parton saturation [28, 29].

Untangling the causes of the effects observed at RHIC and the LHCb experiment
requires measurements of the nuclear modification factor of identified
particles, such as neutral pions, over a broad rapidity range. Multiple
scattering in the nucleus would result in similar enhancements for all
particle species, while radial flow is expected to have a larger transverse
momentum ($p_{\mathrm{T}}$) hardening effect on higher mass particles included
in charged-particle measurements [30]. Alternatively, recombination is
expected to produce a particularly large baryon enhancement [23]. The PHENIX
collaboration recently reported enhancements of ${{\pi}^{0}}$ production in
small-collision systems, observing a system size dependence qualitatively
consistent with radial flow, although still quantitatively consistent with
nPDF predictions [31]. A study of ${{\pi}^{0}}$ production with the LHCb
detector will help differentiate between contributions from nuclear parton
density effects, initial state multiple scattering, and final-state collective
flow.

This Letter presents a measurement of the nuclear modification factor of
${{\pi}^{0}}$ meson production in $p{\rm Pb}$ collisions at $\sqrt{s_{\rm
NN}}=8.16\,\text{\,Te\kern-1.00006ptV}$ with the LHCb detector. The nuclear
modification factor is given by

$R_{p{\rm Pb}}\equiv\frac{1}{A}\frac{{\rm d}\sigma_{p{\rm Pb}}/{\rm
d}p_{\mathrm{T}}}{{\rm d}\sigma_{pp}/{\rm d}p_{\mathrm{T}}},$ (1)

where $A=208$ is the atomic mass number of the lead nucleus and ${\rm
d}\sigma_{p{\rm Pb}}/{\rm d}p_{\mathrm{T}}$ and ${\rm d}\sigma_{pp}/{\rm
d}p_{\mathrm{T}}$ are the ${{\pi}^{0}}$ differential production cross sections
in $p{\rm Pb}$ and proton-proton ($pp$) collisions, respectively. The
${{\pi}^{0}}$ cross section is measured in $p{\rm Pb}$ collisions at
$\sqrt{s_{\rm NN}}=8.16\,\text{\,Te\kern-1.00006ptV}$ at both forward and
backward rapidities relative to the proton beam. The LHCb detector collected
only a small sample of unbiased $pp$ collisions at
$\sqrt{s}=8\,\text{\,Te\kern-1.00006ptV}$, hence a $pp$ reference cross
section is constructed by interpolating between measurements performed using
$pp$ collisions at $\sqrt{s}=5$ and $13\,\text{\,Te\kern-1.00006ptV}$. The
nuclear modification factor is measured for
$1.5<p_{\mathrm{T}}<10.0\,\text{\,Ge\kern-1.00006ptV}$ in the pseudorapidity
regions $2.5<\eta_{\rm CM}<3.5$ and $-4.0<\eta_{\rm CM}<-3.0$, where
$\eta_{\rm CM}$ is the pseudorapidity in the center-of-mass frame related to
the lab frame pseudorapidity $\eta_{\rm lab}$ by $\eta_{\rm CM}=\eta_{\rm
lab}-0.465$ in $p{\rm Pb}$ collisions and $\eta_{\rm CM}=\eta_{\rm lab}$ in
$pp$ collisions (natural units are used throughout this Letter).

The LHCb detector is a single-arm forward spectrometer covering the
pseudorapidity range $2<\eta_{\rm lab}<5$, described in detail in Refs. [32]
and [33]. The detector includes a high-precision tracking system consisting of
a silicon-strip vertex detector (VELO) surrounding the interaction region, a
large-area silicon-strip detector located upstream of a dipole magnet, and
three stations of silicon-strip detectors and straw drift tubes placed
downstream of the magnet. Photons and electrons are identified by a
calorimeter system consisting of scintillating-pad and preshower detectors, an
electromagnetic (ECAL) and a hadronic calorimeter. Particularly important for
this analysis is the ECAL, which consists of alternating layers of lead and
scintillator and has an energy ($E$) resolution of $13.5\%/\sqrt{E/{\rm
GeV}}\oplus 5.2\%\oplus(0.32\,\text{\,Ge\kern-1.00006ptV})/E$ [34]. Simulated
data samples are used to model the detector response to neutral pion
reconstruction. In the simulation, $p{\rm Pb}$ collisions are generated using
EPOS-LHC [35], while $pp$ collisions are generated using Pythia [36] with a
specific LHCb configuration [37]. Decays of unstable particles are described
by EvtGen [38] and final state radiation is generated using Photos [39]. The
interaction of the generated particles with the detector, and its response,
are implemented using the Geant4 toolkit [40, 41] as described in Ref. [42].

Proton-lead collisions are recorded in the forward configuration in which the
proton beam travels from the interaction region towards the spectrometer, and
in the backward configuration in which the lead ion beam travels towards the
spectrometer. The forward (backward) datasets correspond to an integrated
luminosity of $328\pm 9\,\,\upmu\text{b}^{-1}$ ($267\pm
7\,\,\upmu\text{b}^{-1}$). The forward configuration data are used to perform
measurements for positive $\eta_{\rm CM}$, while the backward sample is used
for negative $\eta_{\rm CM}$. All $p{\rm Pb}$ events must have at least one
reconstructed track in the VELO, while the $pp$ samples consist of unbiased
events.

Neutral pions are reconstructed using their decays to pairs of photons. To
ensure that both photons are in the detector acceptance, ${{\pi}^{0}}$
candidates must have $2.5<\eta_{\rm lab}<4.0$. Photons may reach the ECAL, or
they may convert to electron-positron pairs in the detector material. Photons
that reach the ECAL or that convert in the material downstream of the dipole
magnet are reconstructed as energy deposits (clusters) in the ECAL and are
referred to as ECAL photons. Photons that convert upstream of the dipole
magnet are referred to as converted photons. Only ${{\pi}^{0}}$ candidates
reconstructed from combinations of one ECAL photon and one converted photon
are considered. This reconstruction method provides better momentum resolution
than ECAL photon pairs alone and does not suffer from systematic effects due
to the small opening angle between ${{\pi}^{0}}$ decay photons. ECAL photon
clusters must contain at least two ECAL cells and must be far from the
extrapolated trajectories of all reconstructed tracks. Converted photons are
reconstructed from pairs of good-quality tracks with
$p>2\,\text{\,Ge\kern-1.00006ptV}$. To minimize the effects of energy loss via
bremsstrahlung radiation, only photons that convert downstream of the VELO are
considered. Converted photons are required to have a small invariant mass,
with a maximum that varies based on the position of the conversion vertex
along the beam axis. The reconstructed converted photon momentum is also
required to point to a reconstructed primary vertex. The ECAL photon must have
$p_{\mathrm{T}}>400\text{\,Me\kern-1.00006ptV}$ and the converted photon must
have $p_{\mathrm{T}}>500\text{\,Me\kern-1.00006ptV}$.

An example ${{\pi}^{0}}$ candidate invariant mass ($M(\gamma\gamma)$)
distribution in one interval of $p_{\mathrm{T}}$ and $\eta_{\rm CM}$ is shown
in Fig. 1. The ${{\pi}^{0}}$ yield is determined using a binned maximum
likelihood fit to the $M(\gamma\gamma)$ distribution. The ${{\pi}^{0}}$ signal
is modelled by a two-sided Crystal Ball function [43]. The parameters
describing the tails of the signal distribution are determined from
simulation, while the Gaussian mean and width are left to vary in the fit to
the data. The combinatorial background is modelled using charged particles as
proxies for neutral pions. Charged particles reconstructed in the tracking
system are given the ${{\pi}^{0}}$ mass, and their decays to two photons are
simulated. The simulated photons are combined with reconstructed ECAL photons
to form background candidates. The mass distribution of the proxy background
candidates accurately describes combinatorial backgrounds in simulation and is
used as a background model in the fit. An additional background component
arises when a converted photon is combined with its own bremsstrahlung
radiation to form a $\pi^{0}$ candidate, producing a peak at low mass. This
background is modelled by convolving the reconstructed converted photon mass
distribution with the sum of two positive half-normal distributions. The width
of the narrower half-normal distribution is left to vary in the fit, while the
larger width is fixed using simulation.

Figure 1: Example $M(\gamma\gamma)$ distribution in forward $p{\rm Pb}$
collisions at $\sqrt{s_{\rm NN}}=8.16\,\text{\,Te\kern-1.00006ptV}$ for
$2.5<\eta_{\rm CM}<3.5$ and
$2.0<p_{\mathrm{T}}({{\pi}^{0}})<2.2\,\text{\,Ge\kern-1.00006ptV}$. Fit
results are overlaid with the total fit shown as a solid line. The
combinatorial and bremsstrahlung backgrounds are shown as light and dark
shaded regions, respectively.

The ${{\pi}^{0}}$ yields are corrected for the effects of the detector
response using simulation and an iterative unfolding procedure. First,
correction factors are calculated for each $p_{\mathrm{T}}$ interval from
simulation and are used to correct the measured ${{\pi}^{0}}$ yields. The
corrected ${{\pi}^{0}}$ spectrum in data and the true ${{\pi}^{0}}$ spectrum
in simulation are then fit to Hagedorn functions [44]. The ratio of these fits
is used to weight the true ${{\pi}^{0}}$ $p_{\mathrm{T}}$ spectrum in
simulation. The procedure is then repeated with the weighted simulated data
sample. The procedure consistently converges after three iterations.

Differences in reconstruction efficiency between data and simulation are
measured and corrected for [45]. The ECAL photon reconstruction efficiency is
the product of the cluster reconstruction efficiency and the photon cluster
identification efficiency. The former is estimated with a tag-and-probe method
using converted photons. The tag electron must be matched to an ECAL cluster
and be identified as an electron. The cluster efficiency is then the fraction
of probe electron tracks with a matching ECAL cluster. The photon cluster
identification efficiency is evaluated by repeating the photon reconstruction
without any photon cluster identification requirements in a subset of events
in each data sample and measuring the fraction of ${{\pi}^{0}}$ mesons that
pass the requirements. The converted-photon efficiency is measured using the
ratio of the yields of ${{\pi}^{0}}$ mesons reconstructed as an ECAL photon
and a converted photon to those reconstructed as two ECAL photons. This ratio
is measured for $1.0<p_{\mathrm{T}}<1.5\,\text{\,Ge\kern-1.00006ptV}$ where
ECAL photons from ${{\pi}^{0}}$ decays are always reconstructed as two
separate clusters. The result is then combined with the measured ECAL photon
efficiency correction factors to calculate a converted photon efficiency
correction. The corrected efficiencies range from about $0.5\%$ at
$1.5\,\text{\,Ge\kern-1.00006ptV}$ to about $5\%$ at
$10\,\text{\,Ge\kern-1.00006ptV}$.

The ${{\pi}^{0}}$ cross section in $pp$ collisions at
$\sqrt{s}=8.16\,\text{\,Te\kern-1.00006ptV}$ is estimated by interpolating
between the measured $pp$ cross sections at
$\sqrt{s}=5\,\text{\,Te\kern-1.00006ptV}$ and
$\sqrt{s}=13\,\text{\,Te\kern-1.00006ptV}$. The interpolation is performed
independently in each $p_{\mathrm{T}}$ interval. The cross section is
interpolated using the functional form $\sigma(s)=as^{b}$. A linear
interpolation in $\sqrt{s}$ is also considered, as is a relative placement
interpolation method [46]. In the latter, placement factors are calculated
using simulation assuming a linear or power-law dependence of the cross
section on $\sqrt{s}$. The placement factors are then applied to data to
determine the interpolated cross section.

The systematic uncertainties are summarized in Table 1. The largest sources of
systematic uncertainty come from the interpolation between $pp$ cross-sections
and the ${{\pi}^{0}}$ fit model. The maximum variation of interpolation
results using the different $pp$ interpolation methods is taken as the
interpolation uncertainty. The fit model uncertainty is estimated by varying
the default fit parameters by their uncertainties from the simulation. The
variance of the resulting yields is taken as a systematic uncertainty. The fit
model uncertainty also includes a contribution from the background model,
which is estimated by repeating the fit using a polynomial background and
taking the difference relative to the default result as a systematic
uncertainty. An additional contribution to the fit model uncertainty is
assigned to the yield extraction method itself by using an alternative method,
where a background-only fit is performed in the mass region
$[0,50]\cup[250,400]\,\text{\,Me\kern-1.00006ptV}$. The background is then
subtracted, and the background-subtracted distribution is integrated over
$[50,250]\,\text{\,Me\kern-1.00006ptV}$ to estimate the ${{\pi}^{0}}$ yield.
The difference between the alternative and default yield extraction methods is
taken as a systematic uncertainty.

Smaller systematic uncertainties come from unfolding, the luminosity estimate,
and the efficiency correction factors. The unfolding uncertainty is estimated
by using a closure test in simulation. The unfolded yields agree with the true
yields in simulation to within about $1\%$ in most $p_{\mathrm{T}}$ intervals.
An additional unfolding uncertainty arises from differences in ${{\pi}^{0}}$
$p_{\mathrm{T}}$ resolution in data and simulation. This difference is
estimated to be less than $10\%$ by comparing the fitted widths of the
${{\pi}^{0}}$ peaks in data and simulation. The resolution is varied in the
unfolding by $\pm 10\%$, resulting in a systematic uncertainty of less than
$1\%$ in every $p_{\mathrm{T}}$ interval. The efficiency correction
uncertainty arises from the finite size of the simulated data samples and
results in a global uncertainty of about $1\mbox{--}2\%$. The luminosity has
been measured in $pp$ collisions with a precision of $2\%$ and in $p{\rm Pb}$
collisions with a precision of $2.6\%$ in the forward configuration and
$2.5\%$ in the backward configuration. The luminosity uncertainty is $50\%$
correlated between datasets. The differential cross sections have an
additional $4\%$ uncertainty due to uncertainties in the detector material
budget. This uncertainty is fully correlated between datasets and cancels in
the nuclear modification factor.

Table 1: Relative systematic and statistical uncertainties in ${\rm d}\sigma/{\rm d}p_{\mathrm{T}}$ and $R_{p{\rm Pb}}$ in percent. The ranges correspond to the minimum and maximum values of the associated uncertainties across all $p_{\mathrm{T}}$ intervals and both $\eta_{\rm CM}$ regions. The ${\rm d}\sigma/{\rm d}p_{\mathrm{T}}$ ranges cover the uncertainties for each of the $p{\rm Pb}$ and $pp$ samples. All sources of systematic uncertainty are fully correlated across $p_{\mathrm{T}}$ intervals. Source |  | ${\rm d}\sigma/{\rm d}p_{\mathrm{T}}\,[\%]$ |  | $R_{p{\rm Pb}}\,[\%]$
---|---|---|---|---
Fit model |  | 2.0 | – | 12.6 |  | 0.9 | – | 15.8
Unfolding |  | 0.3 | – | 6.4 |  | 0.4 | – | 6.4
Interpolation |  |  | $-$ |  |  | 0.9 | – | 4.5
Material |  |  | 4.0 |  |  |  | $-$ |
Efficiency |  | 1.3 | – | 1.9 |  | 1.9 | – | 2.1
Luminosity |  | 2.0 | – | 2.6 |  | 2.2 | – | 2.3
Total systematic |  | 5.4 | – | 15.0 |  | 4.3 | – | 17.4
Statistical |  | 1.0 | – | 9.6 |  | 1.4 | – | 9.1

The fully corrected ${{\pi}^{0}}$ differential cross sections and nuclear
modification factor are shown in Fig. 2 and Fig. 3, respectively. The nuclear
modification factor shows a Cronin-like enhancement at backward pseudorapidity
and a strong suppression at forward pseudorapidity. These measurements are
compared to next-to-leading order pQCD calculations [47] using the EPPS16 [2]
and nCTEQ15 [3] nPDF sets and the DSS14 ${{\pi}^{0}}$ fragmentation functions
[48]. The nPDFs have been reweighted to incorporate LHCb $D^{0}$ production
data [49, 50, 51], resulting in considerably smaller uncertainties than
calculations using the default EPPS16 and nCTEQ15 nPDFs [51]. The measurement
uncertainties in the forward region are much smaller than the nPDF
uncertainties, indicating that this measurement can provide powerful
constraints on nPDFs at low $x$. In addition, the forward results present
tension with the CGC calculation [52]. The enhancement in the backward
direction between $2$ and $4~{}\text{\,Ge\kern-1.00006ptV}$ is larger than
predicted by the pQCD calculation, suggesting that effects not described by
nPDFs contribute to the enhancement.

The ${{\pi}^{0}}$ nuclear modification factor is also compared to the charged-
particle nuclear modification factor measured by the LHCb experiment in $p{\rm
Pb}$ collisions at $\sqrt{s_{\rm NN}}=5\,\text{\,Te\kern-1.00006ptV}$ [15].
The forward ${{\pi}^{0}}$ measurement agrees with the charged-particle
measurement, and the enhancement at backward $\eta_{\rm CM}$ is smaller than
that seen for charged particles at the LHCb experiment. Because the charged-
particle measurement includes heavier mesons and baryons, this ordering could
indicate a mass-dependent enhancement consistent with radial flow or baryon
enhancement from final state recombination [23]. Studies of protons and
heavier unflavored mesons, such as $\eta$ and $\eta^{\prime}$ mesons, could
help differentiate between these explanations.

Figure 2:  Measured ${{\pi}^{0}}$ differential cross sections versus
$p_{\mathrm{T}}$ in the (top) backward and (bottom) forward $\eta_{\rm CM}$
regions. Statistical uncertainties are shown by error bars, while systematic
uncertainties are shown by boxes. The $pp$ cross sections are scaled by the
atomic mass of the lead ion, $A=208$. Figure 3:  Measured ${{\pi}^{0}}$
nuclear modification factor in the (left) backward and (right) forward
$\eta_{\rm CM}$ regions. Error bars show the statistical uncertainty, while
the open boxes show the $p_{\mathrm{T}}$-dependent systematic uncertainties.
The solid gray boxes show the overall normalization uncertainties from the
luminosity estimate and efficiency correction factors. The results are
compared to (top) theoretical predictions [47, 49, 52] and (bottom) to
charged-particle data from Ref. [15]. The hatched regions show the nPDF
uncertainties of the pQCD calculations. The vertical error bars on the
charged-particle results show the combined systematic and statistical
uncertainties.

In conclusion, the ${{\pi}^{0}}$ nuclear modification factor is measured at
forward and backward rapidities in $p{\rm Pb}$ collisions at $\sqrt{s_{\rm
NN}}=8.16\,\text{\,Te\kern-1.00006ptV}$. The measured nuclear modification
factor has a total uncertainty of less than $6\%$ in most $p_{\mathrm{T}}$
intervals, which will provide strong constraints on models of nuclear
structure and particle production in heavy ion collisions. In particular, the
forward measurement is sensitive to nPDFs for $x$ as low as $10^{-6}$ and can
provide useful constraints in future nPDF analyses. Furthermore, the backward
measurement shows the first evidence of enhanced ${{\pi}^{0}}$ production in
proton-ion collisions at the LHC. These measurements will help constrain
nuclear parton densities and models of collectivity in small-collision
systems, as well as explain the origin of Cronin enhancement at the LHC.

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Supplemental Material for LHCb-PAPER-2021-053

Nuclear modification factor of neutral pions in the forward and backward
regions in $p{\rm Pb}$ collisions

## Additional Comparisons to Theory

Fig. S1 shows comparisons of the measured ${{\pi}^{0}}$ nuclear modification
factor to NLO pQCD predictions calculated using the nominal EPPS16 [2] and
nCTEQ15 [3] nPDF sets. These nPDF analyses do not incorporate LHCb $D^{0}$
data [50], which provide strong constraints at low $x$. As a result,
calculations using these nPDF sets have much larger uncertainties than those
incorporating LHCb measurements.

Figure S1: Measured $\pi^{0}$ nuclear modification factor in the (left)
backward and (right) forward $\eta_{\rm CM}$ regions. The error bars show the
statistical uncertainties, while the open boxes show the
$p_{\mathrm{T}}$-dependent systematic uncertainties. The solid black boxes
show the overall normalization uncertainties from the luminosity estimate and
efficiency correction factors. The measurements are compared to NLO pQCD
predictions [47] calculated using the EPPS16 [2] and nCTEQ15 [3] nPDF sets.


# A shadow of the repulsive Rutherford scattering and Hamilton vector

D A Shatilov1 and Z K Silagadze1,2 1 Novosibirsk State University, Novosibirsk
630090, Russia<EMAIL_ADDRESS>2 Budker Institute of Nuclear Physics,
Novosibirsk 630 090, Russia<EMAIL_ADDRESS>
###### Abstract

The fact that repulsive Rutherford scattering casts a paraboloidal shadow is
rarely exploited in introductory mechanics textbooks. Another rarely used
construction in such textbooks is the Hamilton vector, a cousin of the more
famous Laplace-Runge-Lenz vector. We will show how the latter (Hamilton’s
vector) can be used to explain and clarify the former (paraboloidal shadow).

##  1 Introduction

Recently, in an interesting paper [1], a shadow of the repulsive Rutherford
scattering was considered in great detail in fixed-target and center-of-mass
frames. It was noted that this shadow, in addition to playing an important
role in low-energy ion scattering spectroscopy, is of great educational value.

Despite its educational attractiveness, this simple fact concerning repulsive
Coulomb orbits was seemingly forgotten. After the boundary parabola equation
was derived for the envelope of a family of scattered orbits in [2], the
problem did not attract attention and disappeared into oblivion, as witnessed
by the accidental rediscovery of parabolic shadow by an undergraduate student
in computer simulations [3].

In this article, we will show how the Hamilton vector can be used to obtain
the paraboloidal shape of the shadow in a simple and transparent way. We hope
that the publication of [1] will stimulate interest in this neither too
trivial nor too difficult problem, and help it to finally find its way into
introductory textbooks on classical mechanics.

##  2 Hamilton vector

The Hamilton vector is an extra constant of motion in the Kepler problem, like
the well-known Laplace-Runge-Lenz vector (in fact, the two are simply related)
[4]. This “lost sparkling diamond of introductory level mechanics” [5] was
well known in the past, but then almost forgotten (see [4, 6, 7, 8] and
references therein).

A great virtue of the Hamilton vector is that it can be introduced in a simple
and natural way [9]. Let

${\bf e}_{r}=\cos{\phi}\,{\bf i}+\sin{\phi}\,{\bf j},\;\;{\bf
e}_{\phi}=-\sin{\phi}\,{\bf i}+\cos{\phi}\,{\bf j},$ (1)

be time-dependent polar-system unit vectors. Then (the dot denotes time-
derivative)

$\dot{\bf e}_{r}=\dot{\phi}\,{\bf e}_{\phi},\;\;\dot{\bf
e}_{\phi}=-\dot{\phi}\,{\bf e}_{r}.$ (2)

Using the second equation in (2), we can write Newton’s second law for the
Coulomb/Kepler problem in the form (in fixed-target frame)

$\dot{\bf V}=\frac{\alpha}{\mu r^{2}}\,\dot{\bf e}_{r}=-\frac{\alpha}{\mu
r^{2}\dot{\phi}}\,\dot{\bf
e}_{\phi}=\frac{\mathrm{d}}{\mathrm{d}t}\left(-\frac{\alpha}{L_{z}}\,{\bf
e}_{\phi}\right),$ (3)

where $\alpha=\frac{Z_{p}Z_{t}e^{2}}{4\pi\epsilon_{0}}$, $Z_{p}$ and $Z_{t}$
being the projectile and target charges, respectively, in units of the
elementary charge $e$ and ${\bf V}$ is the the target-relative projectile
velocity. At the last step we have used that $L_{z}=\mu r^{2}\dot{\phi}$,
$z$-component of the total angular momentum of the system, is conserved in the
central field (like any other components of the angular momentum). It is
important to note that, similar to the traditional formulation of the two-body
problem, a reduced mass of the system is introduced

$\mu=\frac{m_{p}m_{t}}{m_{p}+m_{t}},$ (4)

where $m_{p}$ and $m_{t}$ are the projectile and target masses in the
specified order. Equation (3) indicates that the following vector (the
Hamilton vector) is conserved in the Coulomb/Kepler field:

${\bf H}={\bf V}+\frac{\alpha}{L_{z}}\,{\bf e}_{\phi}.$ (5)

The physical meaning of the Hamilton vector is especially clear in the case of
an attractive potential, when $\alpha<0$. Then

${\bf H}={\bf V}-{\bf V_{c}},$ (6)

where ${\bf V_{c}}=\frac{|\alpha|}{L_{z}}\,{\bf e}_{\phi}$ is the velocity
that the particle would have if it moved in a circular orbit around the center
of the field with the same $L_{z}$ component of the angular momentum.
Therefore, the Hamilton vector is a kind of velocity defect (deficit) that is
conserved during motion in the Coulomb/Kepler field.

Note that the better known Laplace-Runge-Lenz vector ${\bf A}$ is related to
the Hamilton vector:

${\bf A}={\bf H}\times{\bf L}={\bf V}\times{\bf L}+\alpha\,{\bf e}_{r}.$ (7)

##  3 The shape of the shadow

Let us consider some projectile trajectory (in the fixed-target frame)
launched at $x=-\infty$ and with an initial velocity $\bf{V_{\infty}}$
directed along the x axis. Selected point $B$ at this trajectory has the polar
angle $\phi$. When point $A$ on this trajectory moves towards negative
infinity on the $x$-axis, the corresponding unit vector ${\bf e}_{\phi}$
approaches negative direction of the $y$ axis, and its $x$-projection tends to
zero (see figure 1).

Figure 1: Projectile trajectory along which ${\bf H}(A)={\bf H}(B)$ for any
choice of $A$ and $B$.

Let us write the conservation law of the Hamilton vector in the $x$ and
$y$-projections for points $A$ and $B$ in the limit, when the point $A$
corresponds to $x=-\infty$:

$V_{\infty}=V_{x}-\frac{\alpha}{L_{z}}\sin{\phi},\;\;-\frac{\alpha}{L_{z}}=V_{y}+\frac{\alpha}{L_{z}}\cos{\phi},$
(8)

where ${\bf V}=(V_{x},V_{y})$ is the projectile velocity at point $B$. Then

$\displaystyle
V^{2}=\left(V_{\infty}+\frac{\alpha}{L_{z}}\sin{\phi}\right)^{2}+\frac{\alpha^{2}}{L_{z}^{2}}(1+\cos{\phi})^{2}=$
$\displaystyle
V_{\infty}^{2}+2\,\frac{\alpha^{2}}{L_{z}^{2}}+2\,\frac{\alpha}{L_{z}}\,V_{\infty}\sin{\phi}+2\,\frac{\alpha^{2}}{L_{z}^{2}}\,\cos{\phi}.$
(9)

On the other hand, $V^{2}$ can be determined from the energy conservation law

$\frac{\mu V_{\infty}^{2}}{2}=\frac{\mu V^{2}}{2}+\frac{\alpha}{r},$ (10)

which gives

$V^{2}=V_{\infty}^{2}-2\,\frac{\alpha}{\mu r},$ (11)

where $r$ is the target-relative distance at point $B$. Then, substituting
(11) into the l.h.s. of (9), we get after a simple algebra

$\frac{1}{\mu r}=-\alpha\sigma^{2}(1+\cos{\phi})+\sigma V_{\infty}\sin{\phi},$
(12)

where $\sigma=\frac{1}{|L_{z}|}$ (note that $L_{z}<0$, because
$\dot{\phi}<0$). This formula describes every possible (for different initial
parameters) projectile trajectory, which is a hyperbola in polar coordinates.

The rationale in finding shadow shape from this point can be followed as in
[1]. Namely, for a given polar angle $\phi$, the shadow boundary is determined
by the minimum value of $r$. Therefore, we must choose $\sigma_{m}$ (inverse
angular momentum) in such a way that

$\left.\frac{\mathrm{d}}{\mathrm{d}\sigma}\left[-\alpha\sigma^{2}(1+\cos{\phi})+\sigma
V_{\infty}\sin{\phi}\right]\right|_{\sigma=\sigma_{m}}=0.$ (13)

This gives

$\sigma_{m}=\frac{V_{\infty}\sin{\phi}}{2\alpha(1+\cos{\phi})}.$ (14)

To calculate the minimum value of $r$ for a given $\phi$, we substitute (14)
into (12) and get

$\frac{1}{\mu
r_{min}}=\frac{V_{\infty}^{2}\sin^{2}{\phi}}{4\alpha(1+\cos{\phi})}=\frac{V_{\infty}^{2}}{4\alpha}(1-\cos{\phi}).$
(15)

Therefore, the boundary of the shadow is given by the equation

$\frac{1}{r_{min}}=\frac{\mu V_{\infty}^{2}}{4\alpha}(1-\cos{\phi}),$ (16)

which is the equation of the parabola in polar coordinates.

Note that finding the envelope of trajectories from the equation (12) can be
put in a general envelope determination context by treating this formula as a
condition $F(r,\phi,\sigma)=0$ and then applying the general rule for the
envelope $\frac{\partial}{\partial\sigma}F(r,\phi,\sigma)=0$ (more on this in
the concluding remarks).

So far, we have implicitly assumed that all possible projectile trajectories
are confined to the $x-y$ plane. In the case of three-dimensional flow, due to
the axial symmetry of the problem, the shape of the shadow is a paraboloid
obtained by rotating the parabola (16) around its axis of symmetry.

Introducing the cylindrical coordinates $z=r_{min}\cos{\phi}$,
$\rho=r_{min}\sin{\phi}$, equation (16) can be rewritten in the form

$\frac{4\alpha}{\mu V_{\infty}^{2}}=\sqrt{\rho^{2}+z^{2}}-z.$ (17)

Solving for $z$, we obtain the following equation for the shadow paraboloid in
cylindrical coordinates:

$z=\frac{\rho^{2}}{8\xi}-2\xi,$ (18)

where

$\xi=\frac{\alpha}{mV_{\infty}^{2}}.$ (19)

##  4 Concluding remarks

As we can see, using the Hamilton vector allows a simple and transparent way
to get the main result of [1], the equation (18) for the shape of the shadow.
In fact, we have been using this problem for some time as an exercise in an
introductory mechanics course at Novosibirsk State University [10, 11]. In
[11], two more solutions to the problem are given.

The first solution is based on the fact that the shadow boundary is the
envelope of the projectile trajectories. A one-parameter family of the
projectile trajectories is given by the equation (for details, see [11]. Note
that in [11] the flow falls from the right).

$F(r,\phi,\phi_{0})=\frac{\alpha}{\mu
V_{\infty}^{2}}\,\frac{\tan^{2}{\phi_{0}}}{r}+1+\frac{\cos{(\phi-\phi_{0})}}{\cos{\phi_{0}}}=0,$
(20)

where $\phi_{0}$ is the polar angle of the symmetry axis of the hyperbolic
trajectory. Then it is known from mathematics (for inquisitive readers: there
are some subtleties in the mathematical definition of an envelope, see [12,
13, 14]) that the envelope equation is obtained by excluding the parameter
$\phi_{0}$ from the system

$\displaystyle F(r,\phi,\phi_{0})=0,$
$\displaystyle\frac{\partial}{\partial\phi_{0}}F(r,\phi,\phi_{0})=0.$ (21)

In this way we can obtain the envelope equation

$\frac{1}{r_{min}}=\frac{\mu V_{\infty}^{2}}{4\alpha}(1-\cos{\phi}).$ (22)

The sign in front of $\cos{\phi}$ can change assuming that the flow falls from
the right, for example, as it is done in [11]. In the present article (as in
[1]) it was assumed that the flow falls from the left.

In the second solution method, considered in [11], we fix the polar angle
$\phi$ in the family (20) and choose the parameter $\phi_{0}$ so that the
corresponding trajectory minimizes $r$. This is the same idea that was used in
[1] (and in this article), just the parameterizations in the one-parameter
family of projectile trajectories are different. Therefore, we emphasize that
the most original contribution relative to [1] is not so much about the
procedure for obtaining the shadow itself, but rather obtaining trajectories,
which is facilitated by the use of the Hamilton vector.

The method used in this article can be extended to the interesting case when
the infalling flux is originated from an emitter at a finite distance from the
repulsive Coulomb potential centre. Such a modification of the problem of the
shape of the shadow in repulsive Rutherford scattering is also of considerable
pedagogical interest. It represents a physical situation that actually occurs
in atomic physics, and allows students to become familiar with such
interesting phenomena as the glory and rainbow effects in scattering processes
[15].

Consider projectile trajectories all starting from some point $A$ on the
$x$-axis at a distance $R$ from the center of the field (see figure 2). We
assume that the particles are released from the emitter with the same in
magnitude initial velocity $V_{0}$.

Figure 2: The definition of kinematic variables for the projectile trajectory
launched from the point $A$ on the $x$-axis.

If the initial velocity ${\bf V}_{0}$ makes an angle $\theta$ with the
$x$-axis, the conservation equations (8) of the Hamilton vector components
will be modified in the following way:

$V_{0}\cos{\theta}=V_{x}-\frac{\alpha}{L_{z}}\sin{\phi},\;\;V_{0}\sin{\theta}-\frac{\alpha}{L_{z}}=V_{y}+\frac{\alpha}{L_{z}}\cos{\phi}.$
(23)

Then

$V^{2}={V_{0}}^{2}+2\,\frac{\alpha}{L_{z}}V_{0}\left[\,\sin{(\phi-\theta)}-\sin{\theta}\,\right]+2\left(\frac{\alpha}{L_{z}}\right)^{2}\left(1+\cos{\phi}\right),$
(24)

and using the energy conservation law

$\frac{\mu V_{0}^{2}}{2}+\frac{\alpha}{R}=\frac{\mu
V^{2}}{2}+\frac{\alpha}{r},$ (25)

we derive the following relation

$\displaystyle\frac{1}{\mu r}=-\alpha\sigma^{2}(1+\cos{\phi})+\sigma
V_{0}\sin{(\phi-\theta)}=$
$\displaystyle-\alpha\sigma^{2}(1+\cos{\phi})+V_{0}\sin{\phi}\,\sqrt{\sigma^{2}-\frac{1}{\mu^{2}R^{2}V_{0}^{2}}}-\,\frac{\cos{\phi}}{\mu
R},$ (26)

where we have taken into account that $\frac{1}{\sigma}=|L_{z}|=\mu
V_{0}R\sin{\theta}$.

The equation (26) indicates that the minimum $r$ for a given $\phi$
corresponds to $\sigma_{m}$ such that

$\sqrt{\sigma_{m}^{2}-\frac{1}{\mu^{2}R^{2}V_{0}^{2}}}=\frac{V_{0}\sin{\phi}}{2\alpha\,(1+\cos{\phi})}.$
(27)

Substituting $\sigma_{m}$ from (27) into (26), we finally obtain for the
shadow boundary the expression

$\frac{1}{r_{min}}=\frac{\mu
V_{\infty}^{2}}{4\alpha}(1-A)\left(1-\frac{1+A}{1-A}\,\cos{\phi}\right),$ (28)

where we have introduced a convenient dimensionless parameter [15, 16]

$A=\frac{V_{\infty}^{2}}{V_{0}^{2}}-1=\frac{2\alpha}{\mu V_{0}^{2}R}.$ (29)

Since $A>0$ equation (28) corresponds to a hyperbola with vertex at the
emission point and focus at the force center [15]. As the emission point moves
towards infinity, $A\to 0$ and the hyperbola (28) degenerates into the
parabola (16).

For an attractive potential, the envelope of Kepler ellipses, all starting
from the same point in space, can be obtained in a similar way [16].

## Acknowledgments

The work is supported by the Ministry of Education and Science of the Russian
Federation.

## References

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