††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Experimental probe of multi-mobility edges in quasiperiodic mosaic lattices

Jun Gao<EMAIL_ADDRESS>Department of Applied Physics, KTH Royal Institute of
Technology, Albanova University Centre, Roslagstullsbacken 21, 106 91
Stockholm, Sweden  Ivan M. Khaymovich<EMAIL_ADDRESS>Nordita,
Stockholm University and KTH Royal Institute of Technology, Hannes Alfvéns väg
12, SE-106 91 Stockholm, Sweden  Institute for Physics of Microstructures,
Russian Academy of Sciences, 603950 Nizhny Novgorod, GSP-105, Russia  Xiao-Wei
Wang  Center for Integrated Quantum Information Technologies (IQIT), School of
Physics and Astronomy and State Key Laboratory of Advanced Optical
Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai
200240, China  Ze-Sheng Xu  Department of Applied Physics, KTH Royal Institute
of Technology, Albanova University Centre, Roslagstullsbacken 21, 106 91
Stockholm, Sweden  Adrian Iovan  Department of Applied Physics, KTH Royal
Institute of Technology, Albanova University Centre, Roslagstullsbacken 21,
106 91 Stockholm, Sweden  Govind Krishna  Department of Applied Physics, KTH
Royal Institute of Technology, Albanova University Centre, Roslagstullsbacken
21, 106 91 Stockholm, Sweden  Alexander V. Balatsky  Nordita, Stockholm
University and KTH Royal Institute of Technology, Hannes Alfvéns väg 12,
SE-106 91 Stockholm, Sweden  Department of Physics, University of Connecticut,
Storrs, Connecticut 06269, USA  Val Zwiller  Department of Applied Physics,
KTH Royal Institute of Technology, Albanova University Centre,
Roslagstullsbacken 21, 106 91 Stockholm, Sweden  Ali W. Elshaari
<EMAIL_ADDRESS>Department of Applied Physics, KTH Royal Institute of
Technology, Albanova University Centre, Roslagstullsbacken 21, 106 91
Stockholm, Sweden

The mobility edge (ME) is a crucial concept in understanding localization
physics, marking the critical transition between extended and localized states
in the energy spectrum. Anderson localization scaling theory predicts the
absence of ME in lower dimensional systems. Hence, the search for exact MEs,
particularly for single particles in lower dimensions, has recently garnered
significant interest in both theoretical and experimental studies, resulting
in notable progress. However, several open questions remain, including the
possibility of a single system exhibiting multiple MEs and the continual
existence of extended states, even within the strong disorder domain. Here, we
provide experimental evidence to address these questions by utilizing a
quasiperiodic mosaic lattice with meticulously designed nanophotonic circuits.
Our observations demonstrate the coexistence of extended and localized states
in lattices with broken duality symmetry and varying modulation periods. By
single site injection and scanning the disorder level, we could approximately
probe the ME of the modulated lattice. These results corroborate recent
theoretical predictions, introduce a new avenue for investigating ME physics,
and offer inspiration for further exploration of ME physics in the quantum
regime using hybrid integrated photonic devices.

Disorder-induced localization, a phenomenon initially predicted by P. W.
Anderson in 1958 Anderson1958 , has been a prominent topic in condensed matter
physics disrev1 ; disrev2 . The scaling theory of localization scaling
revealed that in lower-dimensional disordered systems, all states become
localized, whereas in three-dimensional systems, localized and extended
eigenstates can coexist, resulting in the existence of a critical energy
$E_{c}$ known as the mobility edge (ME) Evers2008 . Notably, when a
quasiperiodic potential replaces random disorder, such as in the Aubry-André
(AA) model AAH1 ; AAH2 , a distinct picture emerges. This model suggests an
energy-independent critical metal-insulator transition at a self-dual point,
subsequently confirmed by experiments conducted both on photonic AAH3  and
atomic AAH4 ; AAH5  platforms. However, due to its self-dual symmetry, the AA
model does not possess a ME. The existence of MEs in one-dimensional (1D)
systems is primarily conjectured in more generalized models Prange1983 ;
Biddle2009 ; Biddle2010 ; Gopalakrishnan2017 ; Celardo2016 ; Deng2019 ;
Saha2019 ; Deng2018 ; Nosov2019 ; Nosov2019-2 ; Kutlin2020 ; Deng2022 ;
Das2022beta-ens ; Das2023beta-ens_IMK ; DasSarma1988 ; DasSarma1990 ;
Ganeshan2015 ; Li2017 ; Yao2019 ; Yin2020 ; Roy2018 ; Danieli2015 ; Ahmed2022
; Lee2022 ; Kim2022 , serving as a catalyst for numerous experimental
investigations based on ultracold atoms Ucold1 ; Ucold2 ; Ucold3 .

Recently, a significant advancement in the field of ME physics has emerged
with the introduction of Avila’s global theory Avila2015 , one of his Fields
Medal work. This novel theoretical framework has uncovered a distinct class of
exactly solvable 1D models, where quasiperiodic on-site potentials are
incorporated with certain periods Wang2020 . Referred to as mosaic lattices,
these models exhibit a range of compelling features. Notably, unlike previous
models employing random or other quasiperiodic disorders, the mosaic lattice
displays the remarkable property of hosting multiple MEs while breaking self-
duality symmetry. Moreover, regardless of the strength of the quasiperiodic
potential, extended states persist throughout the system –– a striking
departure from the previous findings.

Here, we conducted experimental implementation of quasiperiodic mosaic
lattices using integrated silicon nitride ($\text{Si}_{3}\text{N}_{4}$)
photonic circuits with complementary metal-oxide-semiconductor (CMOS)
compatible fabrication technology TPRev1 ; TPRev2 ; TPRev3 ; TPRev4 ; InteP ;
Elshaari2020 ; Moody2022 ; Chang2023 . By precisely engineering the on-site
potential of each lattice site and adjusting the in-between gaps, while
maintaining uniform hopping terms, we successfully achieved the desired
quasiperiodic modulation over a wide tuning range at room temperature. Through
single site excitation of the photonic mosaic lattice in the strong tuning
regime, we observe clear signatures of multiple MEs, which arise from the
energy-dependent coexistence of localized and extended states. The existence
of MEs is further confirmed by scanning the quasperiodic potential strength
and probing the average energy of the injected state. Our results showcase the
capacity of integrated photonics platforms to investigate ME physics in a
scalable and precise manner, with the potential to explore and uncover unique
quantum features depending on bosonic coalescence Lahini2010 ; Segev2013 ;
Crespi2013  in quasiperiodic lattice models.

Figure 1: Mobility edge in $\kappa=2$ quasiperiodic mosaic lattice. (a)
Schematic of the 1D quasiperiodic mosaic lattice lattice, exemplifying the
mobility edge phenomena in condensed matter physics. For the case of
$\kappa=2$, the energy of every second lattice site is modified in accordance
with equation 2, while the hopping constant $J$ between lattice sites is held
constant throughout. (b) Energy diagram illustrating the dependence of
eigenstate energies on quasiperiodic potential strength ($\lambda$). Each data
point signifies the energy of a specific eigenstate at a given potential
strength, with color coding employed to represent the energy-dependent fractal
dimension $D_{2}$. The 1D photonic lattice under investigation comprises 21
sites. (c) Real-space distribution of three distinct eigenstates at a disorder
strength $\lambda=5$: (i) the highest positive energy eigenstate, localized
predominantly at site 4; (ii) an extended state near zero energy; and (iii)
the highest negative energy eigenstate, localized primarily at site 8. The
system exhibits two mobility edges, as described by equation (5), where
eigenstates transition from localized to extended upon crossing the mobility
edge. (d) Sketch of the experimental setup. A 786nm laser is first prepared in
TE mode and then injected into the lattice via a lensed fiber. The figure
shows SEM image of the fabricated device. The input waveguide is split into
the mosaic lattice and a monitor waveguide. The intensity distribution in the
lattice is reflected by the grating couplers (red) and recorded through top
imaging.

The Hamiltonian of the quasiperiodic mosaic model can be described as

$H=J\sum_{j}\left(c_{j}^{\dagger}c_{j+1}+\text{ H.c.
}\right)+2\sum_{j}\lambda_{j}n_{j}.\\\ $ (1)

Here, $c_{j}^{\dagger}$ is the creation operator at site $j$, $J$ is the
nearest neighbor hopping term, and $\lambda_{j}$ represents the on-site
quasiperiodic potential modulation, which is given by the following formula,

$\lambda_{j}=\begin{cases}\lambda\cos[2\pi(\omega j+\theta)],&j=\kappa m,\\\
0,&\text{ otherwise }.\end{cases}$ (2)

$\theta$ is the phase offset during the modulation, $\omega$ is an irrational
number, for instance $(\sqrt{5}-1)/2$ in our case, and $\kappa$ is an integer
determining the mosaic modulation period. When $\kappa=1$, the lattice reduces
to the AA model with self-dual symmetry and transition at $\lambda=1$, whereas
when $\kappa\neq 1$, the duality symmetry of the lattices is broken. Fig. 1(a)
demonstrates a 1D quasiperiodic mosaic model with $\kappa=2$ modulation
period. In our simulation and experimental design, we set $\theta=0$ for the
convenience, and $J=0.015$ enabling wide tuning range of the potential
amplitude modulation $\lambda$.

Following Avila’s profound global theory Avila2015 , it has been theoretically
proved Wang2020  that the mosaic model indeed manifests energy-dependent MEs
by computing the Lyapunov exponent (see Supplementary Information for
details), which can be described by the following expression:

$\displaystyle\left|\lambda a_{\kappa}\right|=J\quad\text{ for }\quad E$
$\displaystyle=$ $\displaystyle E_{c}\quad\text{, with}$ (3) $\displaystyle
a_{\kappa}=\frac{\sin(\kappa p)}{\sin p},\quad E$ $\displaystyle=$
$\displaystyle 2J\cos p$ (4)

the parametrization of the energy $E$ via the real (imaginary)-valued momentum
$p$ for $|E|\leq 2J$ ($|E|>2J$). A mosaic lattice with $\kappa$ modulation
period hosts $2(\kappa-1)$ MEs, which are distributed in energy spectrum
around $E=2J\cos(\pi m/\kappa)$, where $a_{\kappa}=0$ and the extended states
survive at arbitrarily strong potential $\lambda$, which is a new fundamental
feature of mosaic lattices (see Supplementary Information A.1). For the
simplest yet nontrivial case of $\kappa=2$, the two MEs are given by

$E_{c}=\pm{J}/{\lambda}.$ (5)

In Fig. 1(b) we show the calculated eigenvalues of a 21-site lattice versus
the modulation strength $\lambda$ based on our chosen parameters. To
characterize the MEs, we utilize the fractal dimension Evers2008 , defined
$D_{2}=-\lim_{L\rightarrow\infty}[\ln\langle\mathrm{IPR}\rangle/\ln L]$ via
the inverse participation ratio $\mathrm{IPR}=\sum_{j}|\psi_{i}(j)|^{4}$ of
eigenstates $\psi_{i}(j)$, averaged over the energies $E_{i}$, to distinguish
extended ($D_{2}\rightarrow 1$) and localized states ($D_{2}\rightarrow 0$).
The dashed blue lines represent the exact MEs for $\kappa=2$ mosaic lattice as
the transition between $D_{2}=0$ and $1$.

Figure 2: Experimental probe of localized and extended states in $\kappa=2$
mosaic lattice. (a) Real-space distribution of light intensity probed every
$200\mu$m along the lattice, with light injected at lattice site 4. The
observed confinement of light primarily at site 4 with minimal spreading to
neighboring waveguides indicate the strong localization due to overlap with
the localized highest positive energy eigenstate depicted in Fig.1(c). (b)
Similar measurement for light injected at waveguide 8, corresponding to the
highest negative energy localized eigenstate depicted in Fig.1(c). The
observed localization of light, consistent with the behavior described in
Fig.2(a), emphasizes the strong overlap with the localized states. (c) Real-
space distribution of light intensity for light injected at site 15,
exhibiting significant wave-packet spreading. Bar plot below shows the light
intensity on a logarithmic scale, measured after $1000\mu$m of light
propagation in the lattice. Unfilled bars represent simulation results and
colored bars indicate experimental measurements. (d) Inverse participation
ratio (IPR) calculated for the three wave packets, presented in Figs. 2(a),
(b), and (c). Single-site excitations, overlapping strongly with localized
states, maintain a high level IPR for varying propagation lengths, while the
one, overlapping with an extended state, as in Fig.2(c), exhibits IPR
reduction with the propagation distance. This behavior confirms the ME
presence for a specific potential strength $\lambda$.

In Fig. 1(c), we showcase three distinct eigenstates intensity distribution at
a disorder level of $\lambda=5$. We could see the spatial distributions of the
wave functions are exponentially localized at disordered sites $j=\kappa m$
(shown for the highest and lowest energy). For the extended eigenstate, the
particle tends to stay at the sites without potential modulation (see
Supplementary Information A.2), this also explains the survival of extended
states in the mosaic lattice at strong potential.

Experimentally, we implement the photonic quasiperiodic mosaic lattices based
on integrated $\text{Si}_{3}\text{N}_{4}$ photonics platform (see
Supplementary Information B for more fabrication details) silicon . A scanning
electron microscope (SEM) image of the nanophotonic device is presented in
Fig. 1(d). To design the desired on-site potential of each modulated site, we
control the width of each waveguide according to numerical vectorial mode
solver lumerical . We set 550 nm as the default width for a flexible tuning
range while maintaining single-mode profile operation, and the modulated sites
are designed to yield the potential modulation at a given modulation level of
$\lambda=5$, shown in Supplementary Information A.1 to be enough to form all
the MEs. The waveguide separation is carefully designed to keep the hopping
term uniform due to the asymmetric coupling of different waveguide widths (see
Supplementary Information C for the design methods). We choose the 4th, 8th
and 15th inputs to probe different regimes in the energy diagram. We
adiabatically expand the output array by a fan-out structure, and all the
output waveguides are coupled to grating couplers for the spatial intensity
measurement. An additional monitor waveguide is fabricated for the facet beam
profile imaging and polarization control as our previous designs Gao2022 ;
Xu2022 ; gao2023scalable . We also vary the propagation lengths in the samples
from 200 to 1000 µm with an interval of 200 µm to probe the light dynamics in
the lattices.

Figure 3: Experimental probe of $\kappa=3$ mosaic lattice of 34 sites. (left)
Eigenvalue evolution with increasing quasiperiodic disorder strength
$\lambda$. Each point represents the energy of an individual eigenstate, color
coded by the corresponding fractal dimension. As the disorder strength
increases, the system develops four MEs, separating three localization regimes
I, II, III, see labeled panels, with two extended eigenstates regimes in
between highlighted by a fractal dimension approaching unity. Propagation of a
single-site excitation probes different regimes of the energy spectrum at a
disorder strength of $\lambda=5$. The eigenstates distributions in real space
and their overlap with the corresponding choice of single site excitation are
given in the Supplementary Information E. I, II, III present the output
spatial distribution of the light intensity in a logarithmic scale after
propagating 1000 µm in the lattice, using single-site excitation at the 21st,
30th, and 34th waveguide, respectively. Unfilled bars represent simulation
results and colored bars indicate experimental measurements, with the color
code highlighting the light intensity. The experiment shows good agreement
with the simulation, where the light is exponentially localized in the
excitation lattice site. (bottom right) Experimental and numerical intensity
distribution after propagating 1000 µm in the lattice for a single-site
excitation at the 4th lattice site, overlapping with the extended eigenstates
in the energy spectrum. More spreading of light highlights delocalization
transition in the eigenstate spectrum, thus, exhibiting a ME. The inset shows
the measured experimental distribution in a linear scale.

The schematic of the experimental setup is shown in Fig. 1(d). The photonic
quasiperiodic mosaic lattice is probed using a coherent laser at a wavelength
of 786 nm, which is prepared with horizontal polarization (TE mode). The light
is coupled to the lattice through a lensed fiber mounted on a 6-axis nano-
positioning stage (Thorlabs NanoMax). The input waveguide is divided into two
paths: one serves as a monitor waveguide for polarization control, while the
other leads to the injection site. The output intensity is top-imaged using a
40X objective and directly measured by a charge-coupled device (CCD) camera
that records the reflected light from the grating couplers. An example of
image acquisition can be found in the Supplementary Information Fig. S3.

We first probe the two localized states in $\kappa=2$ photonic mosaic lattice.
Figs. 2(a) and (b) present the top images of light intensity distribution of
localized states every 200 µm along the propagation distance. The grey circles
mark the position of every site in the lattice. The figures clearly show the
strong spatial confinement of the injected light at site 4 and site 8. These
two excitation cases correspond to the highest and lowest energy eigenstates,
shown in Fig. 1(c). As a comparison, we also excite the 15th site to probe an
extended state in the lattice. Fig. 2(c) illustrates the light intensity
distribution in the lattice over a propagation distance of 1000 µm with both
linear (top) and logarithmic (bottom histogram) scale, confirming mostly odd-
site occupancy. To quantify the transport behavior, we calculate the IPR
values for spreading wave-packets in all three cases versus propagation
distance. The IPR values in Fig. 2(d) for the localized cases remain close to
1 as the propagation length increases, while the IPR value for the extended
state shows the expansion of the initial single-site wave packet (see
Supplementary Information A.3). Our measurements provide experimental evidence
of energy-dependent localization phase transition in the mosaic model.

Then, we investigate the $\kappa=3$ photonic mosaic lattice comprising 34
sites at the same modulation strength. The lattice contains four MEs given by
the expression of $E_{c}=\pm J\sqrt{1\pm 1/\lambda}$. The corresponding energy
diagram is depicted in Fig. 3, where the four MEs have divided the spectrum
into various regions. Here, we mark three distinct localized eigenstates as I,
II, and III (see Supplementary Information E for intensity distributions of
$\kappa=3$ eigenstates), and we probe the localized states with single site
excitation. After a propagation distance of 1000 µm, the evolution pattern
distributions are illustrated in Fig. 3. The experimental data demonstrates
remarkable agreement with the theoretical simulation results (shown as blank
bars in the figure), confirming the exponential localization in the excited
site. Moreover, we also inject light into the 4th waveguide, to probe the
extended state. This particular injection position is chosen since the site
has a large overlap with the two extended eigenstates, survived at such larger
potential (also shown in Supplementary Information E). Both linear and
logarithmic scale intensity distributions are exhibited in Fig. 3, and clearly
depict a marked contrast to the localized states in terms of light evolution
patterns. Our results prove the coexistence of energy-dependent localized and
extended states induced by multiple MEs in the $\kappa=3$ mosaic model.

Figure 4: Probing the mobility edge in $\kappa=2$ mosaic lattice. Panel (a)
illustrates the spatial distribution of the 14th eigenstate, which is found to
have a significant overlap with the 21st lattice site. This specific site is
chosen for the application of single site excitation to effectively probe the
system. The overlaid inset provides a detailed view of this eigenstate’s
relative location in the broader energy spectrum. It is noted that this
eigenstate resides in close proximity to the exact ME, as determined by the
theoretical model. The critical energy $E_{c}$ is given by
$E_{c}=\frac{J}{\lambda}$, wherein the ME occurs. A number of samples are
fabricated with varying degrees of disorder strength and then subsequently
probed for the presence of the 14th eigenstate near the ME. (b) The color map
presents the weight overlap analysis between the eigenstates and the output
intensity distribution, obtained experimentally following the single-site
excitation at the 21st waveguide and after the wave has propagated a distance
of 100 µm through the lattice structure. It is observed that the output
intensity distribution shows a considerable overlap with the 14th eigenstate,
positioned near the ME across a range of disorder strengths. (c) The final
plot shows the constructed energy level of the experimental excitation for
various disorder strengths, the measured energy conforms to the ME within the
system, thus providing further evidence for the transition between extended
and localized states in the mosaic lattice. The points represent the
experimentally measured data, while the line shows analytical ME, Eq. (5).

We further confirm the existence of the ME by scanning the quasiperiodic
modulation strength in a $\kappa=2$ mosaic lattice and investigating the
critical state near ME. As shown in Fig. 4(a), we first locate the 14th
eigenstate in the energy diagram. As we vary the modulation strength from
$\lambda=1.5$ to 5, this eigenstate consistently remains close to the upper
ME. Considering the spatial intensity distribution of the eigenstate, we
select lattice site 21 as the excitation site due to its larger overlap, and
analyze the weights of different eigenenergies by projecting the evolution
pattern over all 21 eigenstates. Notably, the weight analysis is time-
independent when calculating with state vectors. However in our case, since
the phase information is missing during the intensity measurement, we choose a
relatively small propagation distance as 100 µm to make the phase change
negligible, such experimental condition allows us to mimic a delta function
excitation. We measure five samples with different modulation potentials and
record all the intensity distributions $\\{I^{exp}\\}$. The weight
$\\{w_{i}\\}$ for each sample is given by
$w_{i}=|\langle\psi_{i}|\sqrt{I^{exp}}\rangle|^{2}$, where $|\psi_{i}\rangle$
represents the $i$-th eigenstate under the corresponding potential. The
calculated results in Fig. 4(b) clearly show the excitation state maintains a
dominant overlap with the 14th eigenstate while scanning the disorder level.
Based on the weight analysis, we estimate the energy of the excitation state
$E_{ex}=\sum_{i}w_{i}E_{i}$ with different modulation amplitudes. These values
are depicted in the energy spectrum shown in Figure 4(c), which follow the
trend of the $\kappa=2$ ME as the strength of the quasiperiodic potential
increases.

In conclusion, we have experimentally implemented a novel class of
quasiperiodic mosaic lattices, offering a crucial advancement in the quest for
understanding ME physics. Leveraging integrated photonics platforms, we could
design and realize the mosaic lattices in a scalable and flexible fashion with
fast prototype. We could effectively probe the intricate behaviour of
coexistence of localized and extended states in the mosaic model, revealing
crucial insights into the energy-dependent localization transition. Our work
demonstrates that quasiperiodic mosaic systems indeed exhibit richer physics
than well appreciated random disorder models, substantially extending our
understanding of the mechanisms driving phase transitions in disordered
systems. Our findings, therefore, underscore a significant advancement in the
field, with the potential to catalyze new research directions in quantum
physics, materials science, and beyond.

## Acknowledgments

A.W.E acknowledges support Knut and Alice Wallenberg (KAW) Foundation through
the Wallenberg Centre for Quantum Technology (WACQT), Swedish Research Council
(VR) Starting Grant (Ref: 2016-03905), and Vinnova quantum kick-start project
2021. V.Z. acknowledges support from the KAW and VR. Work at Nordita was
supported by European Research Council under the European Union Seventh
Framework ERS-2018-SYG HERO, KAW 2019.0068 and the University of Connecticut.

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# Scattering cross-section in gravitating cosmic string spacetimes

Marcos Silva
Departamento de Física, Universidade Federal de Pernambuco,
Av. Prof. Moraes Rego, 1235, Recife - PE - 50670-901, Brazil
<EMAIL_ADDRESS>Azadeh Mohammadi
Departamento de Física, Universidade Federal de Pernambuco,
Av. Prof. Moraes Rego, 1235, Recife - PE - 50670-901, Brazil
<EMAIL_ADDRESS>
###### Abstract

We propose a modification of the partial wave approach to deal with the
relativistic quantum scattering of bosonic and fermionic particles in a class
of models concerning gravitating cosmic string spacetimes. These spacetimes
are characterized by the Minkowski line element at the center of the vortex,
non-vanishing curvature at a finite distance from the center but with a
conical structure far from the core. We find the correction in the partial
wave expansion and the phase shift. Consequently, we show the explicit form of
the scattering amplitude and the correction to the differential cross-section
for a massive scalar field. We also implement our formalism in a toy model
mimicking this class of gravitating cosmic string spacetime. Moreover, we
discuss the procedure to apply this formalism to a massive Dirac field.

##  1 Introduction

In 1976, Kibble showed that cosmological phase transitions leading to distinct
non-zero vacuum states of the Higgs field should, in principle, cause the
formation of topological defects, namely domain walls, monopoles, and cosmic
strings [1]. Among these defects, cosmic strings are the most promising ones.
Ever since, many cosmic strings solutions have been found in models with
spontaneous symmetry breaking, including Nielsen-Olesen vortex in the Abelian-
Higgs model in 1973 [2], Semilocal strings by Vachaspati and Achucarro in 1991
[3], and electroweak strings in the Weinberg-Salam model by Vachaspati in 1993
[4]. For a thorough review of the field theoretical and cosmological aspects
of various types of cosmic string solutions, see [5, 6].

The gravitational aspects of cosmic strings with both wire and non-wire
approximation have been extensively studied considering different matter
models [7, 8, 9, 10, 11, 12, 13, 14, 15]. In these models, in order to find
the metric for a gravitating cosmic string, one needs to couple a Higgs
potential with a stable topological solution to gravity. In most cases, the
metric is found numerically. Although there are various possibilities for the
Higgs model leading to such solutions, one of their common features is a
conical spacetime far from the vortex core. Conical structures _per se_ may
create many exciting features in the dynamics of fields. For example, they can
affect the vacuum expectation values of the physical observables. An ideal
cosmic string can induce vacuum polarization as well as bosonic and fermionic
charge and current densities (see for example [16, 17] and references
therein). Not surprisingly, the scattering problem shall present rich features
as well. In [18] it was shown how the scattering of two massive point
particles is related to the scattering of one particle on a cone, and in [19]
the authors analyzed the classical and quantum non-relativistic scattering of
a particle in a conical space. One of the main results in [19] is that the
conical geometry creates singularities in the scattering amplitude and
consequently in the scattered wavefunction if one follows the canonical
procedure in quantum mechanics. The authors solved this issue by modifying the
asymptotic incident wave. Motivated by this approach, [20] showed that the
evolution of a wave packet in the conical spacetime of an ideal cosmic string
is highly dependent on the many possible boundary conditions of the scalar
field at the vertex of the cone. In [21], the authors generalized the
procedure proposed in [19] to the relativistic scattering of a charged scalar
field by an ideal cosmic string considering non-minimal coupling to gravity,
and in [22] they analyzed the scattering of scalar particles in the same
spacetime but considering minimal and non-minimal coupling to vector and
scalar fields. In the present work, we propose a new formalism dealing with
the scattering problem of bosonic and fermionic fields in the general
spacetime of a gravitating cosmic string, which possesses a conical structure
far from the core. In this formalism, in order to find the scattering
amplitude and differential cross-section, we have also modified the asymptotic
incident wave, although with a different approach from the one in [19].

The structure of the paper is as follows. In section 2, we study the main
aspects of a gravitating cosmic string spacetime, while in section 3, we
derive the formula for the scattering amplitude of a massive scalar field and
discuss novel aspects of our approach. We also give a simple example, in
section 4, to show the utility of the new approach. In section 5, we comment
on how this approach is employed in the fermionic case. Finally, in section 6,
we present our conclusions. Throughout the paper we use natural units
($\hbar=c=1$).

##  2 Conical structure

One of the simplest conical structures is an ideal cosmic string, or
equivalently a "confined" vortex represented by the line element

$ds^{2}=dt^{2}-dr^{2}-b^{2}r^{2}d\varphi^{2}-dz^{2},$ (1)

with $b=1-4\mu$, $\mu$ being the string’s mass per unit length. It is a
locally flat and cylindrically symmetric spacetime. Defining $b\varphi$ as the
angular coordinate, one can see that it does not cover the whole $2\pi$ for
$b<1$, which clearly gives origin to the conical structure. However, the
conical structure also appears when one deals with an "open" or extended
vortex, in which the energy density is spread over space. In this case, the
most general form of the line element respecting the cylindrical symmetry is
as follows

$ds^{2}=N^{2}(r)dt^{2}-dr^{2}-L^{2}(r)d\varphi^{2}-N^{2}(r)dz^{2}.$ (2)

It presents a flat spacetime in two regions although, in general, with
different parametrizations. When $r\rightarrow 0$, the metric coefficients are

$\begin{gathered}N(r)\rightarrow 1\\\ L(r)\rightarrow r,\end{gathered}$ (3)

and when $r\rightarrow\infty$ we have

$\begin{gathered}N(r)\rightarrow a\\\ L(r)\rightarrow br+c,\end{gathered}$ (4)

where $a$, $b$ and $c$ are constants in spacetime. However, they may depend on
the parameters of the model for the vortex. In both limits, $r\rightarrow 0$
and $r\rightarrow\infty$, the Ricci tensor and consequently the curvature are
zero, representing a flat spacetime. Though, in between, $0<r<\infty$, the
metric can be arbitrarily complicated and, in general, presents non-vanishing
curvature. Concrete examples of this behavior can be found in [13] for a
gravitating cosmic string originating from the Abelian-Higgs model, and in
[15] for a non-abelian extension with two bosonic sectors, taking the line
element (2).

##  3 Scalar field scattering

One of the most relevant physically observable quantities, especially in
particle physics, is the cross-section of particle scattering. In this
section, our goal is to find the phase shift and the corresponding
differential cross-section of a scalar field in a gravitating cosmic string
spacetime background. Let us start with the Klein-Gordon equation non-
minimally coupled with gravity

$(\Box+M^{2}+\xi R)\Phi=0,$ (5)

where $R$ is the Ricci scalar, $M$ is the mass, and $\xi$ is the non-minimal
coupling. Replacing the d’Alembertian operator by
$\Box=\frac{1}{\sqrt{-g}}\partial_{\mu}{(\sqrt{-g}g^{\mu\nu}\partial_{\nu}{\Phi})}$,
we get

$\Bigg{\\{}\frac{1}{N^{2}L}\Bigg{[}L\partial_{t}^{2}-\Big{(}2NN^{\prime}L+N^{2}L^{\prime}\Big{)}\partial_{r}-N^{2}L\partial_{r}^{2}-\frac{N^{2}}{L}\partial_{\varphi}^{2}-L\partial_{z}^{2}\Bigg{]}+M^{2}+\xi
R\Bigg{\\}}\Phi=0,$ (6)

where the prime denotes the derivative with respect to $r$. Taking into
account the cylindrical symmetry and energy conservation, we substitute the
following ansatz

$\Phi=e^{\mp iEt}e^{ikz}\sum_{m=-\infty}^{\infty}a_{m}R_{m}(r)e^{im\varphi},$
(7)

into the differential eq. (6) and define $\lambda^{2}\equiv E^{2}-M^{2}-k^{2}$
which result in

$R_{m}^{\prime\prime}(r)+\left(\frac{2N^{\prime}(r)}{N(r)}+\frac{L^{\prime}(r)}{L(r)}\right)R_{m}^{\prime}(r)+\left[\frac{\lambda^{2}}{N^{2}(r)}-M^{2}\left(1-\frac{1}{N^{2}(r)}\right)-\frac{m^{2}}{L^{2}(r)}+\xi
R\right]R_{m}(r)=0.$ (8)

The summation over $m$ and the parameter $a_{m}$ reflects the fact that for a
linear equation, the sum of the solutions with any arbitrary constant
prefactor is still a solution.

The solution to eq. (8) near the origin is a Bessel function in the form
$R_{m}(r\rightarrow 0)\propto J_{m}(\lambda r)$, where we absorb the
proportionality constant in $a_{m}$. If we impose the solution near the origin
to be a plane wave in the x-direction, the coefficients are automatically
determined, $a_{m}=i^{m}$. The solution in the limit $r\rightarrow\infty$,
where the metric components converge to eq. (4), is
$R_{m}(r\rightarrow\infty)=b_{m}J_{m^{\prime}}(\lambda^{\prime}w)+c_{m}Y_{m^{\prime}}(\lambda^{\prime}w)$.
In this solution, $Y_{m}$ is the Neumann function of order $m$, $w=r+c/b$,
$m^{\prime}=m/b$ and
${\lambda^{\prime}}^{2}=\lambda^{2}/a^{2}-M^{2}(1-1/a^{2})$. Taking the
assymptotic form of both Bessel and Neumann functions together with
$b_{m}=C_{m}\cos d_{m}$ and $c_{m}=-C_{m}\sin d_{m}$, leads to

$R_{m}(r\rightarrow\infty)=C_{m}\sqrt{\frac{2}{\pi\lambda^{\prime}r}}\cos\left(\lambda^{\prime}r+\beta_{m^{\prime}}\right),$
(9)

where
$\beta_{m^{\prime}}=\frac{\lambda^{\prime}c}{b}-\alpha_{m^{\prime}}+d_{m}(\lambda)$,
$\alpha_{m}=\frac{\pi}{2}(m+1/2)$ while $C_{m}(\lambda)$ and $d_{m}(\lambda)$
are model-dependent constants to be determined, usually numerically. It is
important to notice that the phase shift of the m-th mode is given by
$\delta_{m}(\lambda)=\beta_{m^{\prime}}+\alpha_{m}=\frac{\lambda^{\prime}c}{b}+\frac{m\pi}{2}\left(1-\frac{1}{b}\right)+d_{m}(\lambda)$.
Also, note that if the flat spacetime for $r\rightarrow 0$ and r
$\rightarrow\infty$ is with the same parameterization, i.e. $c=0$ and $b=1$,
the phase shift becomes equal to $d_{m}(\lambda)$ determined by the form of
the potential in $0<r<\infty$. Moreover, the non-minimal coupling $\xi$
becomes irrelevant in the general form of the solutions since the curvature
vanishes in both limits, $r\rightarrow 0$ and $r\rightarrow\infty$. Actually,
one could add any interaction to the region $0<r<\infty$ without changing the
general form of the solution at infinity. For instance, suppose we add an
interaction between the scalar test field and the gauge field of the vortex.
Depending on the vortex field configuration, local coupling with the gauge
field may affect the asymptotically defined constants $C_{m}$ and $d_{m}$. The
phase-shift formula does not change by the gauge interaction, but its values
are certainly modified by the factor $d_{m}$, which, like $C_{m}$, store
information on the local interaction between the origin and infinity.
Different models, like semilocal or Alice strings, shall present distinct non-
minimal local interactions. The amount of this effect is hard to estimate
without doing an explicit example, which, if one wants to tackle realistic
gravitating vortices, requires numerical simulations. However, in the next
section we give a simplified example mimiching this class of gravitating
cosmic string background.

Usually, in the partial wave approach, we rewrite the cosine in eq. (9) with
complex exponentials, which gives

$\Phi_{solution}=\frac{1}{\sqrt{2\pi\lambda^{\prime}r}}\left[e^{-i\lambda^{\prime}r}\left(\sum_{m}C_{m}i^{m}e^{im\varphi}e^{-i\beta_{m^{\prime}}}\right)+e^{i\lambda^{\prime}r}\left(\sum_{m}C_{m}i^{m}e^{im\varphi}e^{i\beta_{m^{\prime}}}\right)\right].$
(10)

For the asymptotic ansatz we take

$\Phi_{ansatz}=f(\varphi)\frac{e^{i\lambda^{\prime}r}}{\sqrt{r}}+\sum_{m=-\infty}^{\infty}A_{m}i^{m}J_{m^{\prime}}(\lambda^{\prime}w)e^{im\varphi}=f(\varphi)\frac{e^{i\lambda^{\prime}r}}{\sqrt{r}}+(e^{i\lambda^{\prime}r\cos\varphi})_{mod}$
(11)

One of the reasons for writing the above equation in this form is that when
$A_{m}=1$, $a=1$, $b=1$ and $c=0$ we recover the usual asymptotic ansatz known
in quantum mechanics. In contrast to the standard approach, here $A_{m}$ is
left to be determined based on the form of the solution at infinity, i.e.
after the scattering.

Now, let us express the asymptotic form of the ansatz (11) with plane waves,
precisely as we did for the actual solution (10)

$\displaystyle\Phi_{ansatz}$
$\displaystyle=\frac{1}{\sqrt{2\pi\lambda^{\prime}r}}\left[e^{-i\lambda^{\prime}r}\left(\sum_{m=-\infty}^{\infty}A_{m}i^{m}e^{-i\frac{\lambda^{\prime}c}{b}}e^{i\alpha_{m^{\prime}}}e^{im\varphi}\right)+\right.$
$\displaystyle\left.+e^{i\lambda^{\prime}r}\left(\sqrt{2\pi\lambda^{\prime}}f(\varphi)+\sum_{m=-\infty}^{\infty}A_{m}i^{m}e^{i\frac{\lambda^{\prime}c}{b}}e^{-i\alpha_{m^{\prime}}}e^{im\varphi}\right)\right].$
(12)

Comparing the coefficients of $e^{-i\lambda^{\prime}r}$, we obtain

$A_{m}e^{-i(\beta_{m^{\prime}}-d_{m})}=C_{m}e^{-i\beta_{m^{\prime}}}\rightarrow
A_{m}=C_{m}e^{-id_{m}(\lambda)}.$ (13)

Comparing the coefficients of $e^{i\lambda^{\prime}r}$ and considering eq.
(13), results in

$f(\varphi)=\frac{1}{\sqrt{2\pi
i\lambda^{\prime}}}\sum_{m=-\infty}^{\infty}D_{m}\left[e^{2id_{m}(\lambda)}-1\right]e^{im(\varphi-\delta\varphi)}=\sum_{m=-\infty}^{\infty}f_{m}(\varphi),$
(14)

where

$D_{m}\equiv C_{m}e^{i\left(\frac{\lambda^{\prime}c}{b}-d_{m}\right)},$ (15)

and $\delta\varphi=\frac{\pi}{2}\left(\frac{1}{b}-1\right)\geq 0$ knowing
$b\leq 1$. The solution (14) has the extra factors $D_{m}$ and $\delta\varphi$
when compared to the result in quantum mechanics. However, in the limit
$a\rightarrow 1$, $b\rightarrow 1$ and $c\rightarrow 0$ or equivalently,
$\lambda^{\prime}\rightarrow\lambda$, $m^{\prime}\rightarrow m$ and
$w\rightarrow r$ where one recovers the flat spacetime metric with the same
parametrization in both limits, $r\rightarrow 0$ and $r\rightarrow\infty$, the
extra factors dissappear and the results match.

Now suppose $A_{m}=1$, as it is in the standard quantum mechanics partial-wave
approach. In this situation, the scattering amplitude is given by

$f(\varphi)=\frac{e^{i\lambda^{\prime}c/b}}{\sqrt{2\pi
i\lambda^{\prime}}}\left(\sum_{m}e^{2id_{m}}e^{im(\varphi-\delta\varphi)}-\delta(\varphi-\delta\varphi)\right).$
(16)

It means that the conical structure produces a delta contribution to the
scattering amplitude in the standard QM approach. This divergence was also
found in [19]. In our approach, with $A_{m}$ free to be determined by the
field solution at infinity, the scattering amplitude becomes

$f(\varphi)=\frac{e^{i\lambda^{\prime}c/b}}{\sqrt{2\pi
i\lambda^{\prime}}}\left(\sum_{m}C_{m}e^{idm}-\sum_{m}C_{m}e^{-id_{m}}e^{im(\varphi-\delta\varphi)}\right),$
(17)

which does not have any delta contribution due to the nontrivial mode
dependent constants $C_{m}$ and $d_{m}$. Therefore, our formalism avoids the
delta-function divergence.

Let us pause and connect our approach with the one in [19] for the particle
scattering in a conical geometry. In their scenario, the whole space is
conical and analytically determined. Because of a non-vanishing deficit angle,
the scattering amplitude $f(\varphi)$ is singular, taking the standard partial
wave expansion ansatz in quantum mechanics. They circumvented this problem by
modifying the second term in (11) to match the solution determined at
$r\rightarrow\infty$. Our reasoning is similar. Due to the conical structure
at $r\rightarrow\infty$, we need to leave an extra degree of freedom to be
fixed with the asymptotic solution after the scattering. In contrast with the
result in [19], one needs to find the parameters in the extra factor $D_{m}$
numerically since our formalism deals with a class of spacetimes typically too
complex to have a metric in a closed-form due to the matter-gravity
interaction. Similar to [19], here, the optical theorem is not satisfied,
although there is no problem with the unitarity. The probability is conserved
since the probability current is divergenceless. From this, we see that in
this class of spacetimes, the optical theorem is no longer suitable to
determine particle conservation.

Also, one can see that, due to conical structure, the angular variable in (14)
has a deficit. In fact it is interesting to notice that $\delta\varphi$ is
related to the angular difference between geodesics in the ideal string
background [23], $\Delta\varphi=\frac{8\pi\mu}{b}$, $b=1-4\mu$, by the
following relation

$\delta\varphi=\frac{1}{4}\Delta\varphi.$ (18)

The appearance of the deficit in the angular part of the scattering amplitude
originates from the spacetime’s conical structure with a deficit angle equal
to $\delta=2\pi(1-L^{\prime}(\infty))$ [15].

Now, let us take a closer look at the phase shift. As we have shown before,
the phase shift of the scalar field scattering in a gravitating cosmic string
spacetime is given by

$\delta_{m}(\lambda)=\beta_{m^{\prime}}+\alpha_{m}=\lambda^{\prime}\frac{c}{b}+\frac{m\pi}{2}\left(1-\frac{1}{b}\right)+d_{m}(\lambda),$
(19)

in which the second term matches the result found in [19] but with the
addition of the first term accounting for the rescaling of the radial
coordinate, and the last one accounting for the nontrivial spacetime
configuration. As we mentioned before, the last term depends on the
gravitating cosmic string model under study. An exciting feature of this
formula is that the first term is already in the form of an approximation for
the low-energy scattering. In this situation, the isotropic mode, $m=0$, is
the only one to significantly contribute to the scattering amplitude since for
other modes, a portion of the incident flux is going to be in the
$\hat{\varphi}$-direction [24]. In quantum mechanics, we define the scattering
length, $l_{sc}$, by

$l_{sc}=\lim_{\lambda^{\prime}\rightarrow
0}\left|\frac{\delta_{0}(\lambda)}{\lambda^{\prime}}\right|.$ (20)

For instance, when considering the potential of a hard sphere of radius
$R_{0}$, the scattering length is equal to $l_{sc}=R_{0}$ as it should.
Following (20), the scattering length of the gravitating cosmic string can be
estimated as

$l_{sc}=\frac{c}{b}+\frac{d(d_{0})}{d\lambda^{\prime}}(\lambda^{\prime}=0),$
(21)

since for $\lambda^{\prime}=0$ there is no scattering, resulting in
$d_{0}(\lambda^{\prime}=0)=0$.

Now, knowing how to find the scattering amplitude, $f(\varphi)$, we can return
to the computation of the differential cross-section. As the incoming and
outgoing momenta, $\lambda$ and $\lambda^{\prime}$, are not in general the
same, the differential cross-section is given by [25]

$\frac{d\sigma}{d\varphi}=\frac{\lambda^{\prime}}{\lambda}|f(\varphi)|^{2},$
(22)

ignoring the z-axis due to the symmetry of the system in this direction.
Clearly, if one includes the $z$-direction, the total cross-section diverges.
By sustituting eq. (14) into eq. (22) and then integrating (22) in $\varphi$,
we obtain

$\sigma=\frac{4}{\lambda}\sum_{m=-\infty}^{\infty}|C_{m}|^{2}\sin^{2}(d_{m}).$
(23)

This is exactly the known result in quantum mechanics with the extra term
$|C_{m}|^{2}$. This term tends to 1 in the limit where one recovers the flat
spacetime with the same parametrization when $r\rightarrow 0$ and
$r\rightarrow\infty$.

##  4 Toy model

To show the utility of the new approach, we construct a simplified analytical
model similar to a gravitating cosmic string background where it is possible
to calculate the factors $C_{m}$ and $d_{m}$ explicitly. The metric is given
by

$\displaystyle\begin{aligned} r<r_{0}&:\quad N(r)=1,\quad L(r)=r\\\
r>r_{0}&:\quad N(r)=a,\quad L(r)=br+c,\end{aligned}$ (24)

which represents an empty cylinder of radius $r_{0}$ with all its energy
concentrated on the zero-thickness walls. The spacetime is flat both inside
and outside the cylinder, but the geometry outside is conical while the inside
is Minkowskian. Imposing continuity of $L(r)$ at $r=r_{0}$ yields

$r_{0}=\frac{c}{1-b},$ (25)

which suggests that the asymptotic conical geometry encodes information about
the cylinder size. Here we use the approximation $r_{0}\gg
c/b=\mathcal{O}(1)$, which means $b$ is close to 1. Also, for simplicity, we
set $a=1$ to avoid any delta function in the field equation. Later, when
tackling a more realistic model, we relax this condition.

The solutions to the field equations are

$\displaystyle\begin{aligned}
r<r_{0}&:\phi=e^{-iEt}e^{ikz}\sum_{m}i^{m}J_{m}(\lambda r)e^{im\varphi}\\\
r>r_{0}&:\phi=e^{-iEt}e^{ikz}\sum_{m}i^{m}C_{m}J_{m^{\prime}}(\lambda
w)e^{im\varphi}.\end{aligned}$ (26)

If we take the asymptotic form of the solution outside ($r_{0}\gg c/b$) when
imposing the continuity of $\phi$ at $r=r_{0}$, we get

$C_{m}\sqrt{\frac{2}{\pi\lambda
r_{0}}}\cos(\lambda^{\prime}r_{0}+\beta_{m^{\prime}})=J_{m}(\lambda r_{0}).$
(27)

Now imposing the continuity of the derivative $d\phi/dr$ on the border of the
tube results in

$\frac{\lambda}{2}\Delta J_{m}(\lambda r_{0})=C_{m}\sqrt{\frac{2}{\pi\lambda
r_{o}}}\cos(\lambda
r_{0}+\beta_{m^{\prime}})\left[\frac{1}{2r_{0}}+\lambda\tan(\lambda
r_{0}+\beta_{m^{\prime}})\right],$ (28)

where $\Delta J_{m}(x)=J_{m+1}(x)-J_{m-1}(x)$. Substituting (27) in (28), we
obtain

$\tan(\lambda r_{0}+\beta_{m^{\prime}})=\frac{1}{2}\left(\frac{\Delta
J_{m}(\lambda r_{0})}{J_{m}(\lambda r_{0})}-\frac{1}{\lambda r_{0}}\right),$
(29)

which is equivalent to

$d_{m}(\lambda)=\alpha_{m^{\prime}}-\lambda\left(r_{0}+\frac{c}{b}\right)+\tan^{-1}\left[\frac{1}{2}\left(\frac{\lambda}{\lambda}\frac{\Delta
J_{m}(\lambda r_{0})}{J_{m}(\lambda r_{0})}-\frac{1}{\lambda
r_{0}}\right)\right].$ (30)

Inserting (29) in (27) gives the amplitude of the scattered field

$C_{m}=\sqrt{\frac{\pi\lambda r_{0}}{2}}J_{m}(\lambda
r_{0})\left\\{1+\frac{1}{4}\left[\frac{\Delta J_{m}(\lambda
r_{0})}{J_{m}(\lambda r_{0})}-\frac{1}{\lambda
r_{0}}\right]^{2}\right\\}^{1/2}.$ (31)

It is worth mentioning that this toy model has its limitations. The metric
(24) has a curvature proportional to a delta function at $r=r_{0}$, so this is
still a singular spacetime. Computing the cross-section we observe that its
convergence is extremely slow. Therefore, in what follows, we develop a smooth
version of the metric (24). Notice that the metric functions (24) can be
expressed using the Heaviside step function, $\Theta(x)$, in the following
form

$\displaystyle\begin{aligned} N(r)&=\Theta(r_{0}-r)+a\Theta(r-r_{0}),\\\
L(r)&=r\Theta(r_{0}-r)+(br+c)\Theta(r-r_{0}),\end{aligned}$ (32)

which presents no transition region between Minkowski and the conical
spacetime. We can create a smooth transition using an analytical approximation
of the Heaviside function $\Theta(x)$, as follows

$H(x)=\frac{1}{2}\left(1+\tanh(px)\right),\quad
H(x)\xrightarrow{p\to\infty}\Theta(x),$ (33)

which gives

$\displaystyle\begin{aligned}
N(r)&=\frac{1}{2}\left\\{(a+1)+(a-1)\tanh\left[p(r-r_{0})\right]\right\\}\\\
L(r)&=\frac{1}{2}\left\\{\left((b+1)r+c\right)+\left((b-1)r+c\right)\tanh\left[p(r-r_{0})\right]\right\\},\end{aligned}$
(34)

when combined with (32). One can see that $r_{0}$ can still characterize the
vortex size in this toy model since the transition between the metric
components occurs _around_ $r_{0}$. In fact, since this model is designed to
mimic a class of gravitating cosmic string solutions, one can take (34) as an
approximation to realistic111Here, we use the term ”realistic” for the metrics
which are solutions of the Einstein field equations. scenarios. In Figure 1,
we plot the metric functions $N(r)$ and $L(r)$ compared with the Minkowski
case.

Figure 1: Metric functions (34) using $a=0.98,b=0.64,c=0.39$ and $p=3$. Dashed
lines show the Minkowski counterparts.

One can substitute (34) in (8) and solve the equation of motion numerically to
extract the mode-dependent constants $C_{m}$ and $d_{m}$. The solid curve in
Fig. 2 shows the scattering cross-section of a scalar field interacting with
the background spacetime (32). Now, let us add an extra interaction in the
aforementioned scenario. Consider the scalar test field interacting with the
gauge field generating the vortex. We take the Nielsen-Olesen vortex [2] with
the gauge field given by

$A^{\varphi}=\frac{n}{er}\alpha(r)\hat{\varphi},$ (35)

where $e$ is the coupling constant between the gauge field and the scalar
field of the vortex, $n$ is the winding number and $\alpha(r)$ satisfy
$\alpha(r\to 0)\to 0$ and $\alpha(r\to\infty)\to 1$. The Nielsen-Olesen
solution only depends on $n$ and $\beta=(m_{s}/m_{v})^{2}$, where $m_{s}$
($m_{v}$) is the mass of the scalar (gauge) boson [26]. In [15], the authors
solved the Maxwell-Einstein equations for this vortex taking $n=1$ and
$\beta=0.5$ and found conical parameters approximately around
$a=0.98,b=0.64,c=0.39$. We take the same conical and gauge field parameters,
together with $p=3.0$. In Figure 2, one can observe how the gauge field
coupling affects the total scattering cross-section of the scalar test field.

Figure 2: The solid (dashed) line shows the total scattering cross-section of
scalar field with $M=1.0$ in the absence (presence) of the gauge field.

It is easy to notice that for large momentum (small wavelength), the
scattering cross-section of the scalar field is much less pronounced compared
with the low momentum (large wavelength) case for both scenarios, with and
without the gauge field interaction, as expected. Only low momentum particles
are affected significantly by the non-trivial background metric. However, in
the presence of the gauge field interaction, the field is also scattered by
the magnetic field of the vortex, leading to a larger scattering cross-
section. For large enough momenta, the interaction with the local potentials
becomes less relevant, resulting in a small scattering cross-section. As the
momentum tends to infinity, the cross-section should vanish.

##  5 Fermion field scattering

In this section, we will show that with a simple adjustment, one can find the
scattering differential cross-section for a Dirac particle interacting with a
gravitating cosmic string background. In this case, we treat each component of
the spinorial field as a scalar field and find its associated differential
cross-section. The total differential cross-section for the scattering of a
fermionic particle is the average value of all components [27].

The Dirac equation in a curved spacetime is in the form

$(\gamma^{A}e^{\mu}\,_{A}\partial_{\mu}-\gamma^{A}\Gamma_{A}+iM_{f})\Psi=0,$
(36)

where $\Gamma_{A}$ and $M_{f}$ are the spin connection and the fermion mass,
respectively. The Greek indices are lowered and raised by the spacetime metric
$g_{\mu\nu}=diag(N^{2},-1,-L^{2},-N^{2})$ and the latin indices by
$\eta_{AB}=diag(1,-1,-1,-1)$. When $r\rightarrow 0$ the positive- and
negative-energy fermionic mode functions (36) are given by [28]

$\psi^{\pm}_{j}(t,r\rightarrow 0,\varphi,z)\propto e^{\mp
iEt}e^{ikz}e^{ij\varphi}\begin{pmatrix}J_{\beta_{j}}(\lambda
r)e^{-i\varphi/2}\\\ sJ_{\beta_{j}+\epsilon_{j}}(\lambda r)e^{i\varphi/2}\\\
\pm\frac{k-is\epsilon_{j}\lambda}{E\pm M}J_{\beta_{j}}(\lambda
r)e^{-i\varphi/2}\\\ \mp\frac{k-is\epsilon_{j}\lambda}{E\pm
M}J_{\beta_{j}+\epsilon_{j}}(\lambda r)e^{i\varphi/2}\\\ \end{pmatrix},$ (37)

where $j=\pm 1/2,\pm 3/2,...$, $\epsilon_{j}=sgn(j)$, $s=\pm 1$ and
$\beta_{j}=|j|-\epsilon_{j}/2$. The proportionality constant is to be defined
based on the chosen initial condition, just as we did with the scalar field.
Now, the solution, when $r\rightarrow\infty$, has a similar form

$\psi^{\pm}_{j}(t,r\rightarrow\infty,\varphi,z)=C_{j}e^{\mp
iEt}e^{ikz}e^{ij\varphi}\sqrt{\frac{2}{\pi\lambda^{\prime}r}}\begin{pmatrix}\cos(\lambda^{\prime}w-\alpha_{\beta_{j^{\prime}}}+d_{j}^{0}(\lambda))e^{-i\varphi/2}\\\
s\cos(\lambda^{\prime}w-\alpha_{\beta_{j^{\prime}}+\epsilon_{j^{\prime}}}+d_{j}^{1}(\lambda))e^{i\varphi/2}\\\
\pm\frac{k-is\epsilon_{j^{\prime}}\lambda^{\prime}}{E\pm
M^{\prime}}\cos(\lambda^{\prime}w-\alpha_{\beta_{j^{\prime}}}+d_{j}^{2}(\lambda))e^{-i\varphi/2}\\\
\mp\frac{k-is\epsilon_{j^{\prime}}\lambda^{\prime}}{E\pm
M^{\prime}}\cos(\lambda^{\prime}w-\alpha_{\beta_{j^{\prime}}+\epsilon_{j^{\prime}}}+d_{j}^{3}(\lambda))e^{i\varphi/2}\\\
\end{pmatrix},$ (38)

where $j^{\prime}=j/b$, $w=r+c/b$, $M^{\prime}=aM$ and
${\lambda^{\prime}}^{2}=(E/a)^{2}-(k/a)^{2}-M^{2}$. Notice that again, we have
to determine the constants $C_{j}(\lambda)$ and $d^{i}_{j}(\lambda)$ based on
the specific gravitating cosmic string model, and hence we are going to have
four different phase shifts, depending on how each component of the fermionic
field is affected by the non-trivial spacetime configuration.

Now, in order to apply our formalism to the fermionic field, one needs to
treat each component as a scalar field and find its associated scattering
amplitude $f^{i}(\varphi)$. The differential cross-section is then given by
the average of the differential cross-sections of all components as follows

$\frac{d\sigma}{d\varphi}=\frac{1}{4}\frac{\lambda^{\prime}}{\lambda}\sum_{i=0}^{3}|f^{i}(\varphi)|^{2}.$
(39)

##  6 Conclusion

In this work, we have developed a new approach to relativistic quantum
scattering in a class of models regarding gravitating cosmic string
spacetimes. Our approach is a modification of the usual partial wave
expansion, which was motivated by the formalism in [19] for a particle
scattering in a conical geometry. In the first part, we considered the Klein-
Gordon equation non-minimally coupled with gravity. The background metric
comes from a gravitating cosmic string spacetime in the form of an extended
vortex. This class of spacetimes is cylindrically symmetric with the general
line element given in eq. (2) where the metric components have the forms (3)
and (4) when $r\rightarrow 0$ and $r\rightarrow\infty$, respectively. These
forms reflect the fact that the background spacetime close to and far from the
vortex center is flat and generally with different parametrization. However,
in between, $0<r<\infty$, the curvature of the spacetime is non-zero in
general and can have a complicated form. The solution to the Klein-Gordon
equation is a linear combination of the Bessel functions, parametrized with
mode number $m$, at the center of the vortex. Far from the vortex, the
solution is a linear combination of Bessel and Neumann functions but with the
momentum eigenvalue corrected to account for the conical structure. We have
found the resulting phase shift of the scalar field interacting with the
background spacetime by taking the asymptotic forms of the above special
functions for $r\rightarrow 0$ and $r\rightarrow\infty$. We have shown that,
besides the contribution from the conical structure of the spacetime, which is
the same as the case of a pure conical geometry studied in [19], the phase
shift includes a contribution originating from the complex spacetime geometry
in $0<r<\infty$. The different spacetime parametrization resulting in
different momenta, $\lambda$ and $\lambda^{\prime}$, also affects the phase
shift.

It has already been shown that in conical spaces, the scattering problem is
nontrivial. As shown initially in [19], the partial wave approach needs to be
modified in these scenarios. Here we have shown how to modify it in a
gravitating cosmic string spacetime. We left a free parameter in the
unscattered wave and then fixed it by the field solution at
$r\rightarrow\infty$. This procedure is shown to remove singularities off the
scattering amplitude and yields a modified form of the partial wave expansion,
with an extra factor compared with the canonical form in quantum mechanics.
However, as expected, one recovers the canonical form when the flat spacetime
has the same parametrization for $r\rightarrow 0$ and $r\rightarrow\infty$. In
addition, we presented a toy model in order to calculate the factors $C_{m}$
and $d_{m}$ analytically.

We have also discussed how to implement other interactions besides gravity
into the formalism. The effect of any new local interaction enters our
approach via its effect on the amplitude, $C_{m}$, and phase, $d_{m}$, of the
field at infinity. For instance, the interaction with the gauge field that
creates the vortex will undoubtedly affect the cross-section, but the amount
of change is model-dependent. We have added the interaction of the field with
the Nielsen-Olesen vortex gauge field in the toy model and discussed the
interplay between the spacetime curvature and the gauge field configuration in
the scattering cross-section.

Finally, we have shown that the new partial wave approach is also valid for
the Dirac field in the same spacetime background, applying our scalar field
scattering formalism and the consequent scattering differential cross-section
for each component. This way, taking the average value gives the total
differential cross-section for a fermionic field.

## Acknowledgments

AM acknowledges the financial support from the Brazilian agencies CAPES and
CNPq, Grant Number 309368/2020-0, besides the financial support from
Universidade Federal de Pernambuco Edital Qualis A. MS also thanks the
Brazilian agency CNPq for the financial support.

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The gravitational energy-momentum for the super-energy Bel-Robinson tensor

Lau Loi So

###### Abstract

Describing the gravitational energy-momentum, the super-energy Bel-Robinson
tensor is the best candidate. In the past, people seems only explore the
lowest order: the electric part $E_{ab}$ and magnetic part $B_{ab}$ for the
Riemann tensor. These two components are related with the static case,
however, for the energy transfer situation, one may need to consider the time
varying $\dot{E}_{ab}$ and $\dot{B}_{ab}$. Here we use
$(\dot{E}_{ab},\dot{B}_{ab}$) to study the energy-momentum for the Bel-
Robinson tensor in a small sphere limit. Meanwhile, our result illustrates how
the gravitational field carries the 4-momentum including this extra
information.

##  1 Introduction

The super-energy Bel-Robinson tensor is the most elegant candidate to describe
the gravitational energy-momentum in general relativity [1]. The well known
physical and mathematical properties are: energy positivity, completely
symmetric, trace free and divergence free. Bel and Robinson demonstrated the
gravitational energy-momentum tensor since 1958 [2, 3, 4, 5]. In the
beginning, describing the gravitational energy using the Bel-Robinson tensor
had the dimensional embarrassment, but this difficulty can be solved by the
technique of a small sphere limit [6, 7].

Regarding the energy-momentum description of the Bel-Robinson tensor, people
seems only examine the lowest order: the electric part $E_{ab}$ and magnetic
part $B_{ab}$ for the Riemann curvature tensor. These two components are
related with the static case. However, if we consider the energy transferring
phenomenon such as the tidal heating for the Juipter-Io system or black hole
[8, 9, 10], one may need to consider the extra piece of information, the time
varying components $\dot{E}_{ab}$ and $\dot{B}_{ab}$. Here we use
$(\dot{E}_{ab},\dot{B}_{ab})$ to study the energy-momentum for the Bel-
Robinson tensor, under the limit of a small sphere region. In addition, our
result elucidates how the gravitational field carries the 4-momentum using
this extra piece of information.

##  2 Technical background

Throughout this work, we use the same spacetime signature and notation in MTW
[11], including the geometrical units $G=c=1$, where $G$ and $c$ are the
Newtonian constant and the speed of light. The Greek letters denote the
spacetime and Latin letters refer to spatial. In vacuum, the Bel-Robinson
tensor can be defined as follows

$\displaystyle
B_{\alpha\beta\mu\nu}:=R_{\alpha\lambda\mu\sigma}R_{\beta}{}^{\lambda}{}_{\nu}{}^{\sigma}+R_{\alpha\lambda\nu\sigma}R_{\beta}{}^{\lambda}{}_{\mu}{}^{\sigma}-\frac{1}{8}g_{\alpha\beta}g_{\mu\nu}R^{2}_{\lambda\sigma\rho\tau},$
(1)

where
$R^{2}_{\lambda\sigma\rho\tau}=R_{\lambda\sigma\rho\tau}R^{\lambda\sigma\rho\tau}$.
In order to extract the energy-momentum, one can use the analog of the
“electric” $E_{ab}$ and “magnetic” $B_{ab}$ parts of the Weyl tensor [12],

$\displaystyle E_{ab}:=C_{0a0b},\quad B_{ab}:=\ast{C_{0a0b}},$ (2)

where $C_{\alpha\beta\mu\nu}$ is the Weyl conformal tensor and
$\ast{C_{\alpha\beta\mu\nu}}$ is its dual,

$\ast{C_{\alpha\beta\mu\nu}}=\frac{1}{2}\epsilon_{\alpha\beta\lambda\sigma}C^{\lambda\sigma}{}_{\mu\nu}.$
(3)

(Here $\epsilon_{\alpha\beta\mu\nu}=\epsilon_{[\alpha\beta\mu\nu]}$ with
$\epsilon_{0123}=\sqrt{-g}$ is the totally anti-symmetric Levi-Civita tensor,
see [11], in particular Eq. 8.10 and Ex. 8.3.) In vacuum using the Riemann
tensor

$E_{ab}=R_{0a0b},\quad B_{ab}=*R_{0a0b}.$ (4)

Certain commonly occurring quadratic combinations of the Riemann tensor
components in terms of the electric $E_{ab}$ and magnetic $B_{ab}$ parts in
vacuum are

$R_{0a0b}R_{0}{}^{a}{}_{0}{}^{b}=E^{2}_{ab},\quad
R_{0abc}R_{0}{}^{abc}=2B^{2}_{ab},\quad R_{abcd}R^{abcd}=4E^{2}_{ab},$ (5)

where $E^{2}_{ab}:=E_{ab}E^{ab}$ and similarly for $B^{2}_{ab}$. In
particular, the Riemann squared tensor can then be written

$R^{2}_{\lambda\sigma\rho\tau}=8(E^{2}_{ab}-B^{2}_{ab}).$ (6)

Within a weak field the metric tensor can be decomposed as
$g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$, and its inverse is
$g^{\alpha\beta}=\eta^{\alpha\beta}-h^{\alpha\beta}$. Here we only consider
the lowest order (see (8) at [9]), the metric components can be written as

$\displaystyle h^{00}$ $\displaystyle=$
$\displaystyle\frac{3}{r^{5}}I_{ab}x^{a}x^{b}-E_{ab}x^{a}x^{b},$ (7)
$\displaystyle h^{0j}$ $\displaystyle=$
$\displaystyle\frac{4}{r^{5}}\,\epsilon^{j}{}_{pq}J^{p}{}_{l}\,x^{q}x^{l}+\frac{2}{3}\epsilon^{j}{}_{pq}B^{p}{}_{l}x^{q}x^{l}+\frac{2}{r^{3}}\dot{I}^{j}{}_{a}\,x^{a}+\frac{10}{21}\dot{E}_{ab}x^{a}x^{b}x^{j}-\frac{4}{21}\dot{E}^{j}{}_{a}x^{a}r^{2},$
(8) $\displaystyle h^{ij}$ $\displaystyle=$
$\displaystyle\eta^{ij}h^{00}+\bar{h}^{ij},$ (9)

where

$\displaystyle\bar{h}^{ij}=\frac{8}{3r^{3}}\epsilon_{pq}{}^{(i}\dot{J}^{j)p}x^{q}+\frac{5}{21}x^{(i}\epsilon^{j)}{}_{pq}\dot{B}^{q}{}_{l}x^{p}x^{l}-\frac{1}{21}r^{2}\epsilon_{pq}{}^{(i}\dot{B}^{j)\,q}x^{p}.$
(10)

Zhang used $\bar{h}^{\alpha\beta}$ for the manipulation while we prefer using
$h^{\alpha\beta}$, the transformation is as follows

$\displaystyle\bar{h}^{\alpha\beta}=h^{\alpha\beta}-\frac{1}{2}\eta^{\alpha\beta}h.$
(11)

The corresponding first order harmonic gauge is
$\partial_{\beta}\bar{h}^{\alpha\beta}=0$. Moreover, we will substitute the
mass quadrupole moment $I_{ij}$ and current quadrupole moment $J_{ij}$ as
determined by Poisson [10]:

$\displaystyle
I^{ij}=\frac{32}{45}M^{6}\dot{E}^{ij},\quad{}J^{ij}=\frac{8}{15}M^{6}\dot{B}^{ij},$
(12)

where $M$ is the mass of the black hole. The value of the tidal heating is
something like
$I^{ij}\dot{E}_{ij}+J^{ij}\dot{B}_{ij}\simeq{}\dot{E}^{2}_{ij}+\dot{B}^{2}_{ij}$,
where $\dot{E}^{2}_{ij}$ means $\dot{E}^{ij}\dot{E}_{ij}$, likewise for
$\dot{B}^{2}_{ij}$. As the tidal heating is a kind of energy dissipation,
obviously that both $\dot{E}^{2}_{ab}$ and $\dot{B}^{2}_{ab}$ indicated the
physical interaction for the energy transfer.

##  3 The energy-momentum for the Bel-Robinson tensor

Referring to (1), the usual gravitational energy density presentation for the
Bel-Robinson tensor is

$\displaystyle B_{0000}$ $\displaystyle=$ $\displaystyle
2R_{0a0b}R_{0}{}^{a}{}_{0}{}^{b}-\frac{1}{8}g_{00}g_{00}R^{2}_{\lambda\sigma\rho\tau}$
(13) $\displaystyle=$ $\displaystyle E^{2}_{ab}+B^{2}_{ab}.$

This basic result can be obtained from (5) to (6) or from (7) to (9). As
mentioned before, this presentation had the dimensional problem, the
appropriate method is the small sphere limit. Here we use this small sphere
technique, including $(\dot{E}_{ab},\dot{B}_{ab})$ and ignored the higher
order, the energy is

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int_{V}B_{00mn}\,x^{m}x^{n}\,d^{3}x$ (17)
$\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int_{V}\left[2(R_{0amb}R_{0}{}^{a}{}_{n}{}^{b}-R_{0a0m}R_{0}{}^{a}{}_{0n})-\frac{1}{8}g_{00}g_{mn}R^{2}_{\lambda\sigma\rho\tau}\right]x^{m}x^{n}\,d^{3}x$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int_{V}\left[\begin{array}[]{ccc}(E^{2}_{ab}+B^{2}_{ab})\,r^{2}-2(E^{c}{}_{c\,q}E_{c\,q}+B^{c}{}_{p}B_{c\,q})\,x^{p}x^{q}\\\
+\frac{2}{9}(\dot{E}_{c\,d}\dot{E}_{p\,q}\,x^{c}x^{d}x^{p}x^{q}-4\dot{E}^{c}{}_{p}\dot{E}_{c\,q}\,x^{p}x^{q}\,r^{2}+2\dot{E}^{2}_{ab}\,r^{4})\\\
+\frac{2}{9}(\dot{B}_{c\,d}\dot{B}_{p\,q}\,x^{c}x^{d}x^{p}x^{q}-4\dot{B}^{c}{}_{p}\dot{B}_{c\,q}\,x^{p}x^{q}\,r^{2}+2\dot{B}^{2}_{ab}\,r^{4})\\\
\end{array}\right]d^{3}x$ $\displaystyle=$
$\displaystyle\frac{1}{60}(E^{2}_{ab}+B^{2}_{ab})\,r^{5}+\frac{2}{315}(\dot{E}^{2}_{ab}+\dot{B}^{2}_{ab})\,r^{7}.$
(18)

Note that the sign of this energy guarantees the positivity. Moreover, that
extra piece quantity $(\dot{E}^{2}_{ab}+\dot{B}^{2}_{ab})\,r^{7}$ illustrates
how the gravitational field carries the energy.

Likewise, we demonstrate the momentum for this Bel-Robinson tensor. Recall the
usual presentation for the momentum density

$\displaystyle
B_{000i}=2R_{0a0b}R_{0}{}^{a}{}_{i}{}^{b}=2\epsilon_{iab}E^{ac}B^{b}{}_{c}.$
(19)

Again, this result can be computed from (5) to (6) or from (7) to (9). Note
that the dimensional problem still exist. Using the method of a small sphere
limit, including $(\dot{E}_{ab},\dot{B}_{ab})$ and omitted the higher order,
the momentum is

$\displaystyle P_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int_{V}B_{0imn}\,x^{m}x^{n}d^{3}x$ (26)
$\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int_{V}2(R_{0amb}R_{i}{}^{a}{}_{n}{}^{b}-R_{0a0m}R_{i}{}^{a}{}_{0n})\,x^{m}x^{n}d^{3}x$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int_{V}\left[\begin{array}[]{ccc}2\epsilon_{c\,p\,q}(E^{c}{}_{i}B^{p}{}_{d}\,x^{q}x^{d}-E_{c\,d}B^{p\,d}\,x^{q}x_{i}+\frac{2}{3}E^{c}{}_{d}B^{p}{}_{i}\,x^{q}x^{d})\\\
+\frac{2}{3}\epsilon_{i\,p\,q}(E_{c\,d}B^{p\,c}\,x^{q}x^{d}+E^{p}{}_{c}B^{q}{}_{d}\,x^{c}x^{d})\\\
\end{array}\right]d^{3}x$
$\displaystyle+\frac{1}{16\pi}\int_{V}\frac{1}{9}\left[\begin{array}[]{ccc}4\epsilon_{c\,p\,q}\dot{E}^{c}{}_{d}\dot{B}^{p}{}_{l}\,x^{q}x^{l}x^{d}x_{i}\\\
+\epsilon_{c\,p\,q}(\dot{E}^{c}{}_{i}\dot{B}^{p}{}_{d}\,x^{q}x^{d}+2\dot{E}^{c\,d}\dot{B}^{p}{}_{i}\,x^{q}x^{d}-5\dot{E}^{c\,d}\dot{B}^{p}{}_{d}\,x^{q}x_{i})\,r^{2}\\\
+\epsilon_{i\,p\,q}(3\dot{E}^{p}{}_{c}\dot{B}^{q}{}_{d}\,x^{c}x^{d}-\dot{E}_{c\,d}\dot{B}^{p}{}_{c}\,x^{q}x^{d}-3\dot{E}^{p\,c}\dot{B}^{q}{}_{c}\,r^{2})\,r^{2}\end{array}\right]d^{3}x$
$\displaystyle=$
$\displaystyle-\left[\frac{1}{30}\epsilon_{iab}E^{ac}B^{b}{}_{c}\,r^{5}+\frac{4-\delta}{315}\epsilon_{iab}\dot{E}^{ac}\dot{B}^{b}{}_{c}\,r^{7}\right],$
(27)

where $\delta=\frac{1}{6}$. In addition, that extra piece quantity
$\epsilon_{iab}\dot{E}^{ac}\dot{B}^{b}{}_{c}\,r^{7}$ explains how the
gravitational field carries the momentum.

After obtained the extra piece information of the energy-momentum for the Bel-
Robinson tensor, we emphasis that it satisfies the future pointing and non-
spacelike property.

##  4 Conclusion

It is known that the super-energy Bel-Robinson tensor is the best candidate to
describe the gravitational energy-momentum in general relativity. According to
the phenomenon of the tidal heating, the Jupiter-Io system or the black hole,
we observed that the gravitational energy should be related with the time
varying components of the electric part $\dot{E}_{ab}$ and magnetic part
$\dot{B}_{ab}$ for the Riemann curvature tensor. Here we used the Bel-Robinson
tensor to demonstrate the gravitational energy-momentum, including these extra
components $(\dot{E}_{ab},\dot{B}_{ab})$, we find that it gives a desirable
result under a small sphere limit, i.e., the 4-momentum are future pointing
and non-spacelike. Moreover, our result also illustrates that how the
gravitational field carries the energy-momentum through the empty spacetime.

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