# Quantum Biology

Alessandro Sergi<EMAIL_ADDRESS>School of Physics, University of KwaZulu-
Natal, Pietermaritzburg, Private Bag X01 Scottsville, 3209 Pietermaritzburg,
South Africa

###### Abstract

A critical assessment of the recent developments of molecular biology is
presented. The thesis that they do not lead to a conceptual understanding of
life and biological systems is defended. Maturana and Varela’s concept of
autopoiesis is briefly sketched and its logical circularity avoided by
postulating the existence of underlying living processes, entailing
amplification from the microscopic to the macroscopic scale, with increasing
complexity in the passage from one scale to the other. Following such a line
of thought, the currently accepted model of condensed matter, which is based
on electrostatics and short-ranged forces, is criticized. It is suggested that
the correct interpretation of quantum dispersion forces (van der Waals,
hydrogen bonding, and so on) as quantum coherence effects hints at the
necessity of including long-ranged forces (or mechanisms for them) in
condensed matter theories of biological processes. Some quantum effects in
biology are reviewed and quantum mechanics is acknowledged as conceptually
important to biology since without it most (if not all) of the biological
structures and signalling processes would not even exist. Moreover, it is
suggested that long-range quantum coherent dynamics, including electron
polarization, may be invoked to explain signal amplification process in
biological systems in general.

Published on line in:
Atti della Accademia Peloritana dei Pericolanti Vol. LXXXVII, C1C0901001
(2009).
DOI: 10.1478/C1C0901001

##  I Introduction

Biology offers to scientists the most complex systems to study in the
universe. Since scientists themselves are biological systems, such a study is
not just the most interesting that one can think of but effectively introduces
a circularity in the process of knowledge, as was noted by Maturana and Varela
tree : Life systems (the scientists) who try to know life systems (possibly
themselves); in other words, life that tries to know life. It is a platitude
to assert that studying biology from the point of view of physics (i.e., the
point of view of the fundamental laws of the universe) is very difficult st .
From a physicist’s perspective, there are universal laws (and, perhaps,
building blocks) of reality and one should be able to predict the emergence
and the characteristics of biological systems from these very fundamental
laws. Such a gigantic endeavour has not been successful to date.

In this contribution, the thesis that quantum mechanics is a powerful tool for
explaining the characteristics of biological systems will be defended and some
(speculative, at the moment) lines of research will be suggested. From a
certain point of view, the use of quantum mechanics in biology might seem
logical since it is the fundamental theory describing microscopic phenomena in
physics. Within a fully reductionist philosophy (which it is not invoked here)
chemistry, biochemistry, and biology would be only epiphenomena of the
fundamental microscopic laws of physics, i.e., they would be secondary
manifestations of the main microscopic reality with its laws. From another
point of view, it is very strange that one would invoke such a controversial
theory, as quantum mechanics actually is, in order to explain the most complex
phenomena in the universe. As a matter of fact, while everybody agrees on the
main technical points of quantum mechanics, almost nobody agrees on its
interpretation, which seems to depend oddly on the area of research the theory
is applied to. It is not entirely wrong to write that there are almost as many
interpretations of quantum mechanics as there are theoretical physicists: It
is sufficient to search in the contemporary literature of physics journals to
be convinced of this. The ideas underlying this contribution are that quantum
mechanics is the fundamental physical theory in the microscopic world, that
important concepts can arise from its application to biological systems, and
that biological systems, while requiring a multi-level approach, must also be
studied from a microscopic point of view (in order to unfold their universal
characteristics).

The main-stream scientific discipline that currently undertakes the endeavour
of a microscopic explanation of life is molecular biology. Its method is based
on enumerating and elucidating the role of molecules in the life process.
While such a work is of fundamental importance for medicine and applied
biochemistry, it does not seem to lead to a better understanding of the
universal properties of life itself. Such a critic opinion, which has been
defended among others by Kaneko kaneko , will be the starting point of this
contribution. However, while Kaneko invokes dynamical system theory kaneko ,
here quantum mechanics is considered necessary in order to unveil the
universal properties of living systems.

This contribution is organized as follows. In Section II the arguments of
Kaneko kaneko , trying to characterize life and to criticize molecular biology
(for failing to provide an understanding), are followed. In Section II.3 the
universal logic of Maturana and Varela, which is founded on the concept of
autopoiesis, is summarized. Autopoiesis is further analyzed and the more
fundamental concept of living process (based on amplification mechanisms) is
introduced. We feel that _living processes_ can be directly linked to quantum
phenomena and, as such, are more suited to physical modeling. The current
paradigm of condensed matter physics is illustrated and criticized in Section
III. The features of quantum mechanics, with a particular emphasis on those of
interest to biological processes, are sketched in the same Section. In Section
IV a certain number of biological phenomena, where quantum mechanics is
necessary, are reviewed. Van der Waals interactions and quantum mechanical
dispersion forces are presented (in a speculative way) as the main candidates
for the amplification processes necessary to living systems. Finally,
conclusions and perspectives are elucidated in Section V.

##  II What is life?

The questions that will be addressed in this Section are: “What kind of system
is life?” and “What does understanding life really means?”. The main thesis is
that such basic questions on life systems are not answered by the main-stream
approach of current biology, which enumerates molecules and genes.

As for understanding life as a process, the molecular paradigm embraced by
contemporary biology has a fundamental flaw: There is no particular molecule,
including DNA, whose presence by itself implies life. In his book kaneko ,
Kaneko presents the semi-serious example of an omelette, which possesses DNA
but which is clearly not alive. Moreover, until one does not gain a general
understanding of what life is, speculations about the possibility that the
molecules, used by living creatures on Earth, could not be the only thing
playing an important role for life are not entirely unreasonable. Hence, if
not the molecules, the specific conditions for life remain to be clarified.

One may attempt to compile a list of the characteristics of living systems.
These are the ability of reproduction, the potentiality to undergo evolution,
the existence of some kind of structure separating an individual’s body from
the external world, some kind of metabolic capacity through which an
individual body is maintained, the existence of some degree of autonomy, and
so on. Despite the various attempts at listing such characteristics, no list
has ever been completely satisfactory or agreed upon. However, there must be a
solution since in many cases human beings have an intuitive ability to
distinguish between living and non-living creatures (excluding limit cases
such as viruses and so on). Acknowledging that certain properties are common
to all living systems, a theoretical physicist would like to search for the
universal properties of living systems. In other words, one would like to
understand the universal logic of life (its logos underlying it as a process)
instead of understanding the specific functioning of a definite organism.
Indeed, since its birth biology has attempted to escape mere enumerationism
through Darwin’s theory of evolution: A universal logic based on the three
processes of variation, reproduction, and selection. Such an evolutionary
logic has been mimicked by computer scientists in order to devise the so-
called genetic algorithms. However, Darwin’s theory alone does not allow one
to determine what kind of properties (or functions) of organisms are possible
in general, nor does it allow one to determine whether any specific property
can be realized in practice. Hence, a universally applicable logic explaining
the emergence of the fundamental properties of living systems is yet to be
found.

###  II.1 Molecular biology

Physicists introduced a major trend in biology more than half a century ago.
It is worth mentioning Delbruck and collaborators, who were strongly
influenced by the lectures of Niels Bohr. Perhaps, the most far-reaching
speculation is due to the father of (quantum) wave mechanics, Schrödinger
himself, who suggested that an aperiodic solid could be the “storing device”
for biological information, in his famous book “What is life?” whatislife .
This steered the search leading to the discovery of the DNA molecule by Watson
and Crick.

Since the discovery of DNA, molecular biology has attempted to describe the
universal properties of the phenomena exhibited by living systems in terms of
molecules. The goal was, and still is, to trace down chemical processes from
the level of cells to that of the composing molecules, and to understand the
functioning of each molecule in biological processes (e.g., heredity,
metabolism, motility, and so on). The methodology of molecular biology can be
sketched in the following way. First, one has to consider a system at the
macroscopic level and identify the molecules and genes that are important in
some function under study. The role of each molecule must be clarified and the
interactions of such molecules with other molecules must be found. Then, one
has to devise how the macroscopic functions of the organism arise from the
cooperativity of the microscopic relevant molecules. In order to bring such a
program to completion, one has to cope with the enormous combinatorial
complexity that is due to the great number of different types of molecules
involved and, nevertheless, devise the network/circuit of chemical reactions
and back-reactions entailing life as a process. Hence, molecular biology
originally started as the pursuit of universality, rejecting the
“enumerationism” that preceded it. However, the present days witness a re-
emergence of the enumerative doctrine of the past, even if in different and
more subtle forms. Indeed, under the push of gigantic funding from medical
(and perhaps, army) research companies, molecular biology has now become an
enumerative science again. One can classify the genome project (the listing of
all the human genes), the proteome project (the listing of all proteins), and
the metabolome project (the listing of all the molecules involved in
metabolism) as mere enumerative science. Of course, such projects are of
utmost importance for practical reasons such as the health care of human
beings. The point is that such projects alone do not lead scientists one inch
further in the understanding of the universal logic of life.

Essentially, reductionism is the philosophy underlying such enumerative
projects. However, there are some hidden assumptions behind reductionism that
need to be brought to the foreground. Typically, reductionism is based on the
premise that the properties of individual elements change little in response
to their interaction with the other elements composing the whole. Here one
faces a first problem because interactions are often not small in biological
systems. Kaneko illustrates the example of proteins in the crowded environment
of a living cell, where they effectively constitute a gel kaneko . In some
cases, the distance between two neighbouring atoms in a single protein
molecule is greater than that between either of these atoms and the closest
atoms of other protein molecules. In reality, it is not unreasonable to
believe that a hard-core version of reductionism is bound to fail in the
search for the universal logic of life simply because living systems are not
machines. Typically, in living systems the behaviour of the parts/molecules
alone is different from that of the parts/molecules acting collectively. In
other words, the dynamics of the parts composing the whole is determined by
the whole; an example is given by the process of morphogenesis. In addition,
living systems display no fixed response to a specific stimulus: A given
stimulus can be associated to various possible responses (a mathematical
analogy is provided by many-valued functions). Technically, such a variety of
responses to a fixed stimulus is referred to as absence of stiffness. Such an
absence of stiffness can be further analyzed in terms of softness, or the
dependence of the response on the environmental conditions, and of
spontaneity, or the possibility of associating different outputs to the same
input, depending on the internal state of the living system and its
fluctuations. Softness and spontaneity, together with some form of memory,
give rise to (perhaps) the most striking feature of living systems: Autonomy
(flexibility and adaptability). In other words, living systems do not always
behave in strict accordance with the rules applied to them and, depending on
the situation, the rules they perceive will change (or the living systems are
able to change the rules they abide). Linking the autonomy of living systems
to mere molecular processes seems impossible to the present author. Other
universal properties of living systems that cannot be traced back to molecules
alone kaneko  are the stability, the irreversibility in the development
process (i.e., the loss of multipotency from embryonic stem cells to stem
cells, and to committed cells capable of reproducing their own kind only), and
the compatibility of the two faces of the reproduction: The ability to produce
nearly identical offspring and the capacity to generate variations leading to
diversity through evolution.

###  II.2 The information paradigm

Since this is the computer age, it is not strange to witness another more
subtle reductionist approach to the logic of life by means of the the
information (computer) paradigm. Such an attempt is subtle because, in
principle, it promises to explain even the autonomy of living systems.
Typically, within an algorithmic approach, the process of development
exhibited by living creatures would be represented as some kind of computer
program (a logical expression in the form of a chain of if-then statements).
For example, if $X$ denotes the concentration of a species of molecules and
$Y$ the concentration of another species, a fundamental bio-chemical process
could be understood by the code:

${\rm If}~{}X~{}\rangle~{}X^{0}~{}{\rm then~{}express}~{}Y\;,$

where $X^{0}$ is a concentration threshold. One serious problem that such an
algorithmic approach must face is that, for continuous values of the
concentration $X$, the fluctuations of $X$ can cause big errors and one should
then explain the algorithmic robustness (stability) of such “living”
computers. The problem of stability is not trivial even when one considers
discrete values of $X$. In fact, life processes involve an enormous variety of
molecules. But, the number of molecules of a given type is often small. This
means that the fluctuations of $X$ are always big and one is in the presence
of a very large chemical noise. An example is given by the process of
development of a multicellular organism kaneko : It seems miraculous that
systems with such a variety of molecules, participating in such a large number
of processes, can result again and again in an almost identical macroscopic
pattern. The situation is analogous to that of a person attempting to stack
many irregularly shaped blocks into a particular form during an intense
earthquake kaneko . A possible solution to the stability problem might be
given by the existence of negative feedback processes in living systems (i.e.,
by some form of error-correction). However, there must also be positive
feedback (amplification) in living systems. Indeed, in the following, the
thesis that the amplification processes themselves are a key to the
understanding of life will be proposed. It is not clear to the author how
error-correction schemes might work in the presence of amplification. However,
for the sake of scientific fairness, one should acknowledge that the above
arguments alone are not enough to rule out completely the information
paradigm. Perhaps, it is the author’s dislike of such a paradigm that leads
him to reject it. Typically, if the universe is some kind of supercomputer and
the phenomena we observe are just the results of its calculations, then
everything should be computable by the computer-universe. However, according
to the laws we know, many non-linear problems show an extreme sensitivity to
the initial conditions, and thus require an infinite precision representation
of the initial information for an exact solution. There are even phenomena
which are not computable in a finite time with a finite memory. In classical
mechanics the three-body problem is not exactly solvable, in quantum mechanics
the two-body problem (e.g., electron dynamics in the helium atom) is not
exactly solvable, and in quantum field theory the zero-body problem (i.e., the
vacuum) is not exactly solvable. Scientists are able to calculate realistic
processes only at the expense of tremendous simplifications and
approximations. The author is not able to imagine what kind of computer-
universe would be able to solve all these problems in the time they take to
occur in reality.

###  II.3 Autopoiesis: The universal logic of Maturana and Varela

Molecular biology is not the only attempt to provide a universal logic for
life. In recent times, Maturana and Varela proposed a theory of life that is
not based on molecules and that tries to explain the general features of
living systems, including those features that are not observed but which are
nevertheless possible, in abstract terms tree . These authors state that
understanding a living system means understanding the network of relations
that must occur so that it can exist as a unit. The set of such relations is
called “organization”. The particular and concrete realization of the
organization of a living system (molecules, network of specific chemical
reactions, and so on) is called “structure”. From this distinction, it follows
that the same organization can, in principle, be realized through various
structures.

These authors make another very important distinction. In their analysis tree
it emerges that living systems are closed (e.g., isolated) from the point of
view of their organization. In other words, whatever the logic of life may be,
it must specify the living system as an independent unit, knowing and reacting
to nothing else than its internal state. However, Maturana and Varela also
clarified that, on a different level, viz., the level of thermodynamics,
living systems are open systems: They are not isolated but (from the point of
view of their structure) interact continuously with their own environment. At
this stage, Maturana and Varela explain the process of life in terms of an
unavoidable circularity: It is a peculiar feature of living systems that the
product of their organization is themselves. In other words, their
organization is such that it maintains their structure which, in turn,
implements in practice their organization. Such an unavoidable circularity is
called autopoiesis, and Maturana and Varela proceeds with the identification
of living systems with autopoietic units.

Clearly, the logic of autopoiesis is a universal logic, based on the
circularity of the abstract level of the organization maintaining itself by
means of the concrete (and physical) level of the structure of living systems.
A platonic philosopher might, perhaps, use the word “form” in place of
organization, hence clarifying from the very begining that Maturana and
Varela’s approach is not entirely reductionist. Nevertheless, reductionism is
to be found at the level of the structure, the concrete physical realization
of the living system which must be invoked in order to maintain the
organization itself. Such a circularity is fascinating and problematic at the
same time since, in general, physics and human logic do not like circularity.
It seems difficult to devise concrete mathematical models implementing such
general ideas.

###  II.4 Living processes

One way to escape the circularity of autopoiesis would be to decompose it in
terms of more fundamental processes. Indeed, it appears to the present author
that autopoiesis is necessary to explain the persistence in time of a given
living system: The system can stay alive because its actions mantain its
existence. The following question naturally arises: Would it be possible to
speak of transient living systems? In other words, if living systems would not
produce themselves they would disappear almost instantaneously; however, it is
altogether tempting to call “life” such an ephemeral existence. I propose to
consider living processes (although the meaning of this concept is at the
moment unclear) as the fundamental building block of life. Such living
processes would be something more fundamental than autopoiesis for
characterizing life. From such a point of view, autopoiesis would explain the
stable existence of a living system over an extended time interval. it is
intuitive that when autopoiesis stops, a living system dies. Hence, an
autopoietic unit would then be defined as a network of self-sustaining living
processes.

Although the living process has been defined as the fundamental (transient)
building block of life, its characteristics have been left as yet mysterious.
The working hypothesis that is introduced here as a postulate (to be verified
by further analysis) is that the living process is an amplification process,
from the microscopic to the macroscopic scale, which builds up complexity in
structure and organization. Of course, the autopoiesis of Maturana and Varela
would require a feedback process from the macro to the micro scale. However,
according to the above discussion, such a feedback mechanism would be required
for mantaining life, not for life itself. The introduction of the concept of
living process breaks the circularity of autopoiesis. The identification of
the living process with a particular type of amplification process,
transferring information from the microscopic level to the macroscopic one,
which also leads to an increase of complexity in organization, permits, at
least in principle, to devise structural (i.e., physical or mathematical)
models. At this stage, it is also necessary to explain what we mean when
saying that complexity increases in going from smaller to larger scales.
Complexity is the number of constraints (or laws) that the process must
fulfill as the scale increases. The nature of such constraints can be static
or dynamic, and one thus considers a fixed structure or to the time evolution
of the system. It is not difficult to understand that an increasing number of
constraints leads to the generation of forms: What is form if not something
that is specified by boundaries and constraints? Therefore, one can also
define the living process as an amplification process that builds up forms.

The above reflection is, at the moment of writing, only a working hypothesis
and here the author is not going to provide the reader with any specific
mathematical or computer model of a living process. Perhaps, the best that can
be presently done is leaving the reader with a metaphoric image that tries to
convey the idea of amplification building complexity. Hence, one can imagine a
lightning flash which, from the shape of a flux-tube, enlarges
(amplification), not unlike the delta of a river, in order to build up an
intricate tree (augmented complexity) of smaller lightning flashes. Keeping
the poetic spirit of the above example, one can draw the main conclusion of
this Section and state that biological systems, far from being mere machines,
are matter that dances (i.e., matter that moves with an incredible level of
coordination among its constituents). Theoretical physicists are left with the
question whether the standard theories of condensed matter physics can explain
biological systems.

##  III The current paradigm of condensed matter physics

The physical description of condensed matter systems is currently dominated by
electrostatics. The most sophisticated, and state-of-the-art, molecular
dynamics simulations of protein molecules in water, see Refs. sergi-rome ;
pellicane  as an example, are based on semi-phenomenological force fields,
describing interactions arising from fixed electric charges, Lennard-Jones and
harmonic potentials plus bond constraints that mimic covalent bonding. In
practice, charge shielding causes the existence of short-ranged forces in such
models, which effectively treat matter as an erector set (or meccano). Even
first-principles theories, such as the electron Density Functional Theory
gross , are currently based on electrostatics alone. As a result, they
describe hydrogen bonding, van der Waals, and charge polarization effects only
with difficulty and, on the whole, with unsatisfactory results. Clearly, such
a condensed matter paradigm tries to build long-ranged correlations from
statistical fluctuations of short-ranged interactions.

We surmise that such a paradigm might be flawed at a fundamental level. For
example, it is clear that both the structure and the function of biological
macromolecules largely depend on hydrogen bonding as well as on hydrophobic
and hydrophilic interactions. These are determined by dispersion or van der
Waals forces, precisely those interactions that are not properly described by
the current paradigm. Such dispersion forces depend on the temperature and on
the molecular environment (i.e., they are not additive) milonni ; vanderwaals
. They lead to the existence of long-ranged networks of structural and
dynamical correlations. Van der Waals forces, also known as induced-dipole-
induced-dipole forces, arise from a highly correlated motion of the electronic
clouds of otherwise neutral atoms. Such a correlated motion, which in quantum
mechanical terms is called coherent, takes place even at room temperature and
in densely packed matter. Poetically, one could say that such forces in matter
arise from “dancing” electronic clouds. The explanation of such a dance is
provided by quantum mechanics milonni ; vanderwaals ; ballentine .

Quantum mechanics is widely believed to be the fundamental theory underling
the phenomenological reality. Although the majority of physicists agrees on
its mathematical formulations, its interpretation is highly controversial.
However, on some points there is a wide consensus. For example, there is
almost no dispute on the issue that quantum mechanics has some form of non-
locality built inside bell . The most advanced mathematical formulation of
quantum mechanics takes the form of a field theory zee . Notwithstanding
infinities, field phenomenology is perhaps more soundly funded than
conventionalism or the spooky attitude arising from “particle” interpretations
(see, for example, the discussion of the Einstein-Podolski-Rosen paradox from
the point of view of field theory in Ref. preparata ). In simple terms, the
field is “something” that is extended in space and time by its very definition
and, from a conceptual point of view, this aspect can be accorded more easily
with the non-locality of quantum mechanics, which appears rather puzzling when
interpreted in terms of localized particles. What is important, both for
physics and for the present discussion, is that the quantum phenomenology
(therein including discreteness, diffraction, and coherence ballentine ) does
not substantially raise any dispute.

Discreteness in quantum mechanics is usually associated with the appearance of
stationary energy levels separated by “quanta” of energy: Transitions between
these levels can only take place through the transfer of the required amount
of energy. Such a discreteness arises from the boundary conditions imposed on
the wave function (or functional) and is considered to be well understood. The
existence of discrete values for the magnetic moments, charges, and masses of
fundamental particles is less clearly understood but is easily embedded in the
current formalism of quantum mechanics.

Quantum diffraction takes its name from an analogy with the wave propagation
of light. Ensembles of microscopic particles exhibit wave-like motion arising
from the correlation and the spatial and time memory of single events in an
ensemble: One single particle is able to influence the “whole” so that the
entire ensemble appears to “move” like a wave, thus also displaying
interference effects.

Coherence also takes its name from an analogy with the wave motion of light:
It immediately brings to mind the condition of phase stability that is
necessary to observe interference and, thus, diffraction. In quantum mechanics
one would consider the phase stability of the wave function (or functional in
field theory). However, this would be confined within a formulation of quantum
mechanics in terms of wave functions. Quantum mechanics can also be formulated
in terms of path integrals, charge and mass densities, distributions in phase
space and so on nine . Hence, one needs a definition of quantum coherence less
bound to the mathematical formulation of quantum mechanics itself. Here it is
proposed to define quantum coherence as the property underlying the typical
and highly correlated motion which takes place in an unperturbed, isolated
quantum system. In condensed matter physics, striking examples of quantum
coherent motion are provided by superfluids and superconductive materials zee
. In superfluids the atomic motion is so highly correlated that friction
disappears and the fluid moves as a whole without dissipation. In many
respects, a superconductive material can be considered as a charged superfluid
of paired electrons (Cooper’s pairs) moving in a frictionless way in the
lattice of positively charged ions (making up the solid material). Both
phenomena take place at very low temperatures. Thermal fluctuations are
incoherent by their very essence and destroy coherent quantum fluctuations
with a surprisingly high efficiency: This is the phenomenon of decoherence
decoherence  that is displayed by open quantum systems petruccione . Hence, in
real systems some kind of shielding from thermal fluctuations is necessary in
order to observe quantum coherent motions. It is worth remembering that in
both cases of superfluidity and superconductivity, physicists do not possess a
widely agreed microscopic dynamical explanation. In superfluids, a microscopic
picture of rotons, which are the typical many-body excitations of such
systems, has not yet emerged. In other words, although one can approximately
calculate the roton energy spectrum, it is not yet known what a roton is on
the microscopic scale: In other words, nobody actually knows what the rotonic
motion of atoms in a superfluid is. The situation is somewhat better for
superconductivity, where, at least, Cooper’s pairs have been postulated.
However, there is no first-principle explanation of the formation of Cooper’s
pairs. Typically, the celebrated Bardeen-Cooper-Schrieffer theory of
superconductivity ballentine ; zee  works only after assuming the phenomenon
of electron pairing. In analogy with mechanics, one can call such theories
kinematical, since they describe the time evolution of the system without
considering the microscopic causes of the motion. This state of affairs should
be contrasted with that of dynamical theories which try to explain the time
evolution of the system under study starting from the underlying microscopic
causes (in analogy with dynamics, the branch of mechanics which explains
motion in relation to forces).

As for its applications in condensed matter systems, quantum mechanics might
provide a synthesis of reductionism and holism. The big fight of the $19^{\rm
th}$ century, between Mach and Ostwald on one side (the champions of
thermodynamics and of the holistic vision of matter) and Maxwell and Boltzmann
(the champions of atomism) on the other, has been resolved in favor of the
latter: The regularity of the chemical laws is nowadays commonly seen as the
realization of the ancient atomistic dream of Democritus and Epicurus.
However, the importance of the specification of the boundary conditions in
quantum mechanics, arising from its non-local or global (holistic) features,
renders it conceptually similar to thermodynamics, where the ensemble must be
specified in terms of the macroscopic conserved quantities.

The main conclusion of this Section is that condensed matter systems can be
dynamically understood only by resorting to quantum mechanics and to the
coherent motion of the electrons, which gives rise to chemical bonding:
Quantum mechanical electronic clouds are matter that dances, the poet would
say. The main question that is left to theoretical physicists is the
following: Condensed matter requires the coherent motion of electrons while
biological systems seem to require a highly correlated motion of heavy atoms;
is quantum mechanics necessary to understand biological systems as dancing
matter? In other words, may coherent electronic motion be the cause of
correlated heavy atomic motion?

##  IV Quantum phenomena in biology

From a certain point of view, it should not be a surprise that quantum
mechanics is relevant to biology. After all, quantum mechanics is definitely
relevant to chemistry. However, there is the serious possibility that all
quantum effects in biology are, in practice, trivial nontrivial . By this it
is meant that, although necessary to explain the details of a given biological
phenomenon, quantum mechanics does not need to be understood mathematically by
a biologist who wants to study life processes. Up to a certain extent, such a
position is also held in the present contribution and phantasmal concepts such
as “quantum consciousness” penrose  are not even discussed. Because of
decoherence decoherence , it appears highly implausible to the present author
that extended coherent states of heavy atoms, such as those of superfluids and
superconductors, may exist at room temperature inside a biological system
(such as a cell). Is this the end of the story? Since electronic coherence
seems to be fundamental for condensed matter systems (in practice all
electromagnetic forces can be seen as a manifestation of quantum coherent
behaviour collective ), there is still the possibility that such an electronic
coherence is really fundamental biological systems. Such a thesis is here
defended and a working hypothesis is also proposed.

There are some quantum phenomena in biology that definitely need quantum
mechanics for their very existence and whose detailed description is
challenging for the theoretical physicist. However, once their existence is
postulated, they can be used by the biologist without almost any reference to
quantum mechanics itself (in the jargon used in this contribution, one can say
that such phenomena are kinematically described by the biologist). This
occurrence is not very dissimilar from the understanding of the stability of
matter: Without quantum mechanical laws atoms could not exist. However, in a
kinematical way, one can postulate the stability of atoms, disregarding its
cause, and study, for example, the physics of noble gases (at temperatures far
from the absolute zero). On a second thought, perhaps it is not intellectually
fair to classify effects like these as trivial. Typically, the fact that some
fundamental concept can be used as a “black box”, within a more approximate
level of description, should not be used to deem the concept itself as
trivial. Therefore, in disagreement with the definition of triviality adopted
in Ref. nontrivial , here it is defended the thesis that when quantum
mechanics is necessary for the existence of a given biological phenomenon,
that phenomenon is nontrivial on a quantum mechanical basis, even if
biologists may choose to describe it kinematically as a black box. Electric
charge and exciton transfer processes in proteins and photoactive complexes
are examples of such pseudo-trivial quantum effects. The exciton is a many-
body excitation of an interacting system: A charge is excited from its ground
state and undergoes a transition to the excited state leaving a hole (a
missing charge) in the ground state; afterwards, because of many-body
interactions, the charge and the hole move in a coherent way thus giving life
to the exciton. Without quantum mechanics, one would not have ground and
excited energy states and would not have the coherent motion of the charge and
the hole. In biological systems, there are cases in which the motion of the
exciton can be approximated by classical mechanics (and there are cases when
this is not possible) but without quantum mechanics the exciton itself would
not exist. Single charge transfers are intrinsically quantum mechanical only
when tunneling through energy barriers takes place. As a matter of fact, some
charge transfer processes can be modeled classically. However, quantum
coherence could be fundamental also in non-tunneling transfers in order to
establish the right degree of correlation with the environment rearrangement
before, during, and after the charge transfer. An example could be provided by
the process of molecular recognition taking place in odor sensing by human
beings. Brookes and coworkers odor  proposed a quantum model to explain odor
selectivity. According to this model, molecules are recognized not only in
terms of their shape (allowing them to dock at the right recognition site) but
also in terms of their phonon frequencies: The molecular phonons provide the
necessary energy to realize an inelastic electron transfer process which takes
place at the recognition site. Such a mechanism would explain why molecules
with the same shape may have different odors and why molecules with different
shape may have the same odor. If this is true, quantum mechanics would be
fundamental even in odor sensing. Of course, a biologist could just assume the
existence of phonons in molecules and their coupling to charges in proteins to
kinematically explain the “mechanics” of odor sensing. But the possibility of
such a mechanism would come from quantum effects anyway and this would be
conceptually very important for the understanding of life.

There are other types of phenomena in biology where quantum coherence is also
fundamental in the kinematics itself. One such phenomenon is the wave-like
motion of massive molecules. In a different context, the possibility that an
ensemble of massive molecules could behave as waves and display diffraction
has been experimentally proven by Zeilinger and coworkers zeilinger . They
have shown that slow beams of fullerene molecules (${\rm C}_{60}$) can give
rise to diffraction effects as much as lighter particles (e.g., electrons and
neutrons) do. The key to understand such a phenomenon is that the de Broglie
wavelength $\lambda$ of any object is inversely proportional to its momentum
$p$: $\lambda=h/p$, where $h$ is the celebrated Plank’s constant. Now, the
momentum is equal to the product of the mass times the velocity of the object
($p=mv$). Therefore, even if $m$ is large, as is the case of a fullerene
molecule, a small velocity $v$ of translation of the centre of mass can be
associated with an appreciable de Broglie wavelength: Indeed, Zeilinger has
been able to measure it. A second crucial step, necessary for the experimental
measurement of $\lambda$ in fullerenes, is that decoherence decoherence
(which is a universal and fast process, taking place on the scales of
femtosecons) must be inhibited in some way, otherwise the wave nature of the
beam would persist for too short times with no observable effects. In ${\rm
C}_{60}$ it turns out that the centre of mass, displaying the wave properties,
is effectively decoupled from the relative motion of the other sixty atoms in
the molecule: The sixty carbon atoms constitute the environment of the centre
of mass of the molecule and without a coupling to the environment (or in the
presence of an important energy gap) there is no decoherence. Can all this be
relevant to biological systems? In biological systems one finds long and heavy
carbon chains. They typically constitute the backbone of any protein. The
motion of such long chains can be represented collectively in terms of modes,
viz., they can be expressed in Fourier frequencies describing the motion of
all the atoms in the chain globally. The dispersion relation links the smaller
frequencies to the motion of the greater number of atoms in the chain, so that
one can say that the massive modes of the chain are slow. As a result they can
have an appreciable de Broglie wavelength. If such slow massive modes become
very weakly coupled to the environment (i.e., the water molecules and/or other
proteins around) they can exhibit interference and diffraction effects in
their motion. This has been been theoretically proven by Tuckerman in a study
of an inter-molecular proton transfer in Malonaldehyde. Such a study was
carried out by means of a very sophisticated first-principle simulation
technique which combines a path integral representation of the heavy carbon
atoms of the molecule and a density functional representation of the valence
electrons tuck . By means of this technique Tuckerman could alternatively
represent the motion of the heavy carbon atoms by means of quantum mechanics
and classical mechanics while always representing the transferring proton
quantum mechanically. Upon calculating the free energy barriers of the
transfer process in different cases, he was able to show the importance of the
quantum motion of the heavy atoms at $T=300$ K for a quantitatively correct
description of the phenomenon.

Another quantum effect in biological systems has been experimentally revealed
quite recently. It has to do with the exciton propagation in photosynthentic
systems such as the Fenna-Matthews-Olson complex photo-1  and in
photosynthetic proteins photo-2 . The charge propagation in such systems
experimentally appears to display a wave-like character even at relatively
high temperatures, of the order of $100\sim 200$ K. It has been suggested that
the protein environment might, in some way, shield the coherence of the
exciton transfer process but the detailed mechanism is not yet understood.

The last example that will be discussed here is similar to the coherent
dynamics in photosynthesis: We refer to the coherent dynamics in chromophore
molecules and to the subsequent process of vision. Such an example will also
be used to introduce and discuss a general characteristic of biological
processes and to propose the working hypothesis pointing to the necessity of
quantum mechanical coherence in living processes (which has already been
discussed in Section II.4). Chromophores are small molecules which are tightly
binded in protein pockets. Examples are given by the p-coumaric acid, the
chromophore of the Photoactive Yellow Protein sergi , and by the retinal in
bacteriorhodopsin duane . Chromophores inside a protein can catch light. After
catching light, the energy is transformed into small atomic rearrangements of
the atoms of the chromophore: Typically, a double bond is twisted and two
hydrogen atoms make a transition from the trans configuration (staying on
opposite sites of the double bond) to the cis configuration (staying on the
same site of the double bond) or vice versa, depending if one speaks of the
p-coumaric acid or retinal, respectively. It has been experimentally
demonstrated in the case of the retinal that the chromophore dynamics inside
the protein is a coherent process duane . Hence, also in this case, one finds
that the protein allows, in some way, the coherent dynamics of extended and
massive systems to take place even at room temperature. An aspect that is not
yet understood is that, while the chromophores freely flip between the trans
and cis configurations in the vacuum, inside the protein this only happens
through a photocycle composed of various steps (of which the trans-cis
transition is only the first stage). The work reported in sergi  tried to
elucidate this point in the case of the p-coumaric acid in the Photoactive
Yellow Protein, but no final conclusion could be drawn. Proton pumping from
the chromophore to the protein is also involved. What is of interest to us is
that there is currently no real explanation for the signalling state of the
whole protein after light catching. In other words, the microscopic dynamics,
photon absorption plus atomic motion of the small chromophore molecule, must
be amplified at the level of the protein in order to cause the macroscopic
signalling state (or, at least, its first stage). Therefore, in this
phenomenon one encounters the amplification process that has been postulated
to be the key for living processes in Section II.4. Indeed, such an
amplification step could have been discussed for all the previous examples.
The question in all cases would be: What is the physical mechanism that gives
rise to the amplification process?

The hypothesis that is proposed here is that the amplification process could
be explained by a long-range, coherent polarization dynamics of electronic
degrees of freedom. These are in fact shielded from decoherence because the
interaction with the thermal environment may only take place by overcoming a
significant energy gap: In other words, the energy associated with thermal
fluctuations at room temperature is usually much less than the energy
necessary for a transition to the first electronic excited state. If this were
not the case, van der Waals forces and the hydrogen bonding, for example,
would not be present: The incoherent transitions of different atoms to and
from the excited state (caused by thermal fluctuations) would destroy the
coherence needed by dispersion forces. Moreover, if the dynamics of the
electron clouds is coherent, considering that the forces on the nuclei arise
from the electrons, can one really think of the dynamics of the nuclei as
incoherent? Indeed, there are theoretical reasons that suggest that when
classical degrees of freedom interact consistently with quantum ones, they
also acquire some quantum features kapral . In such a case, atomic
correlations of biological functions in thermally disordered and crowded
environments could also be interpreted in term of quantum electron coherence.
The hypothesis that quantum coherent electron dynamics/polarization could
explain in general the amplification mechanism in biological process is bold
and, at the moment, not substantiated by scientific evidence. As already noted
before, at this stage, it should be interpreted just as a working hypothesis
that needs to be tested.

##  V Conclusions and perspectives

Following Kaneko’s book, a critical assessment of the usefulness of the
current trends in molecular biology has been presented. The criticism is that
the elucidation of molecules does not lead to an understanding of life as a
process. The general idea of Maturana and Varela’s autopoiesis, explaining
living systems, has been briefly sketched. Its intrinsic circularity has been
superseded by postulating the existence of a living process, proceeding from
the microscopic scale to the macroscopic scale and building complexity. Both
the microscopic amplification and the increasing complexity have been assumed
to be of fundamental importance for living processes. The amplification
process, in particular, seems easier to be modeled in a mathematical way than
autopoiesis. The concepts behind condensed matter theory and quantum mechanics
have been reviewed, emphasizing that van der Waals interactions and chemical
bonding require, in general, quantum electronic coherence even at room
temperature. Hence, meccano-like (electrostatically founded) condensed matter
theory has been deemed inadequate for the understanding of biological
phenomena. Some quantum effects in biology have also been discussed and the
suggestion that signal amplification may be explained in terms of coherent
quantum electron dynamics has been proposed: Coherent charge distribution
dynamics might be a key to the understanding of biological matter.

Does all this mean that another paradigm of condensed matter theory is needed
in order to understand biological matter? Here, the affirmative answer has
been defended and it has been proposed that long-ranged interactions and
correlations must be included from the start in the theories of biological
processes. This conclusion leads immediately to some working perspectives. One
could try to devise phenomenological computer models that include/postulate
long-ranged correlations in the dynamics and then use them to mimic biological
processes. On a more fundamental level, such correlated states should arise
from first-principle theories like quantum electrodynamics. Hence, one can
embark onto the very ambitious process of finding novel (perhaps, non-
perturbative) solutions of the ground (or first excited) states of quantum
electrodynamics in densely packed (condensed) matter systems.

One last question needs to be clearly answered here. Should a biologist study
quantum mechanics and learn all about Hilbert spaces and linear Hermitian
operators? This is not necessary for describing biological process in a
kinematical way (i.e., disregarding their causes) since quantum mechanics can
be used in many instances as a black-box theory. Nevertheless, one should bear
in mind that most biological structures and signalling process would not even
exist without quantum mechanics. This may not be important for the practice of
biology but is certainly fundamental for understanding its conceptual basis.

## Acknowledgments

This work has been funded through a competitive research grant of the
University of KwaZulu Natal. The travel to the University of Messina (Italy)
has been funded by the National Research Foundation (NRF) of South Africa
through a Knowledge and Interchange (KIC) grant.

Useful discussions with Prof. Giacomo Tripodi and Prof. Owen de Lange (who
also carefully read the manuscript) are acknowledged. I am particularly
indebted to Prof. Paolo Giaquinta, who did not only critically review many of
the ideas here reported (thus helping me shaping them up) but, out of
friendship, also took upon himself the burden of a very meticolous (and for me
very precious) editing of the manuscript.

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# A two-dimensional soliton system in the Maxwell-Chern-Simons
gauge model

A.Yu. Loginov<EMAIL_ADDRESS>V.V. Gauzshtein  Tomsk Polytechnic University,
634050 Tomsk, Russia

###### Abstract

The $(2+1)$-dimensional Maxwell-Chern-Simons gauge model consisting of two
complex scalar fields interacting through a common Abelian gauge field is
considered. It is shown that the model has a solution that describes a soliton
system consisting of vortex and Q-ball constituents. This two-dimensional
soliton system possesses a quantized magnetic flux and a quantized electric
charge. Moreover, the soliton system has a nonzero angular momentum.
Properties of this vortex-Q-ball system are investigated by analytical and
numerical methods. It is found that the system combines properties of a vortex
and a Q-ball.

###### keywords:

vortex , flux quantization , Q-ball , Noether charge , Chern-Simons term

††journal: Physics Letters B

##  1 Introduction

Topological solitons of $(2+1)$-dimensional field models play an important
role in various areas of field theory, physics of condensed state, cosmology,
and hydrodynamics. Among them, we should first mention vortices of the
effective theory of superconductivity [1], vortices of the $(2+1)$-dimensional
Abelian Higgs model [2], and lumps of the $(2+1)$-dimensional nonlinear $O(3)$
sigma model [3].

In contrast to the $(3+1)$-dimensional case, electrically charged solitons do
not exist in the $(2+1)$-dimensional Maxwell electrodynamics for a fairly
straightforward reason: the electric field goes like $1/r$, so the electric
field’s energy diverges logarithmically. In $(2+1)$ dimensions, however, the
dynamics of gauge field may be governed not only by the Maxwell term but also
by the Chern-Simons term [4, 5, 6]. In the presence of the Chern-Simons term,
a gauge field becomes topologically massive, thus making possible the
existence of two-dimensional electrically charged solitons. These solitons
exist both in the Maxwell-Chern-Simons models [7, 8, 9] and in the Chern-
Simons models [10, 11, 12, 13, 14] and can be both topological and
nontopological. Topological solitons are electrically charged vortices,
whereas nontopological ones are two-dimensional electrically charged spinning
(possessing an angular momentum) Q-balls. The numerical research of such a
two-dimensional Q-ball has been performed in [15]. The three-dimensional
counterparts of these Q-balls have been described in [16, 17]. More recently,
the influence of the Chern-Simons term on electrically charged and spinning
solitons of several $(2+1)$-dimensional Abelian gauge models has been studied
in [18].

In this Letter a two-dimensional soliton system in the Maxwell-Chern-Simons
gauge model is considered. As well as in the Maxwell gauge model [19], the
soliton system consists of a vortex and a Q-ball interacting through a common
Abelian gauge field. This vortex-Q-ball system possesses a radial electric
field, carries a quantized magnetic flux, and has a nonzero angular momentum,
but in contrast to [19], it also has a quantized electric charge. It is shown
that the vortex-Q-ball system combines properties of topological and
nontopological solitons.

##  2 Lagrangian and field equations of the model

The Lagrangian density of the model is

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\mu}{4}\epsilon^{\rho\sigma\tau}F_{\rho\sigma}A_{\tau}$
(1)
$\displaystyle+\left(D_{\mu}\phi\right)^{\ast}D^{\mu}\phi-V\left(\left|\phi\right|\right)$
$\displaystyle+\left(D_{\mu}\chi\right)^{\ast}D^{\mu}\chi-U\left(\left|\chi\right|\right),$

where the complex scalar fields $\phi$ and $\chi$ are minimally coupled to the
Abelian gauge field $A_{\mu}$ through the covariant derivatives:

$D_{\mu}\phi=\partial_{\mu}\phi+ieA_{\mu}\phi,\quad
D_{\mu}\chi=\partial_{\mu}\chi+iqA_{\mu}\chi.$ (2)

The self-interaction potentials used in this paper are the same as those used
in [19]:

$\displaystyle V\left(\left|\phi\right|\right)$
$\displaystyle=\frac{\lambda}{2}\left(\phi^{\ast}\phi-v^{2}\right)^{2}\\!,$
$\displaystyle U\left(\left|\chi\right|\right)$
$\displaystyle=m^{2}\chi^{\ast}\chi-g\left(\chi^{\ast}\chi\right)^{2}+h\left(\chi^{\ast}\chi\right)^{3}\\!.$
(3)

In Eq. (3), $\lambda$, $g$, and $h$ are the positive self-interaction
constants, $m$ is the mass of the scalar $\chi$-particle, and $v$ is the
vacuum average of the complex scalar field $\phi$. We suppose that the
parameters $m$, $g$, and $h$ satisfy the condition

$\frac{g^{2}}{4m^{2}}<h<\frac{g^{2}}{3m^{2}}.$ (4)

In this case, the potential $U\left(\left|\chi\right|\right)$ has the two
minima: the global minimum at $\chi=0$ and a local one at some nonzero
$\left|\chi\right|$.

The model’s action $S=\int\mathcal{L}d^{3}x$ is invariant under the local
gauge transformations:

$\displaystyle\phi\left(x\right)$ $\displaystyle\rightarrow$
$\displaystyle\phi^{\prime}\left(x\right)=\exp\left(-ie\Lambda\left(x\right)\right)\phi\left(x\right),$
$\displaystyle\chi\left(x\right)$ $\displaystyle\rightarrow$
$\displaystyle\chi^{\prime}\left(x\right)=\exp\left(-iq\Lambda\left(x\right)\right)\chi\left(x\right),$
$\displaystyle A_{\mu}\left(x\right)$ $\displaystyle\rightarrow$
$\displaystyle
A_{\mu}^{\prime}\left(x\right)=A_{\mu}\left(x\right)+\partial_{\mu}\Lambda\left(x\right)$
(5)

if the local gauge parameter $\Lambda\left(x\right)$ decreases rapidly at
infinity. Because of neutrality of the Abelian gauge field $A_{\mu}$, the
Lagrangian density (1) is also invariant under the two independent global
gauge transformations:

$\displaystyle\phi\left(x\right)$ $\displaystyle\rightarrow$
$\displaystyle\phi^{\prime}\left(x\right)=\exp\left(-i\alpha\right)\phi\left(x\right),$
$\displaystyle\chi\left(x\right)$ $\displaystyle\rightarrow$
$\displaystyle\chi^{\prime}\left(x\right)=\exp\left(-i\beta\right)\chi\left(x\right).$
(6)

This invariance leads to the two Noether currents:

$\displaystyle j_{\phi}^{\mu}$ $\displaystyle=$ $\displaystyle
i\left[\phi^{\ast}D^{\mu}\phi-\left(D^{\mu}\phi\right)^{\ast}\phi\right],$
$\displaystyle j_{\chi}^{\mu}$ $\displaystyle=$ $\displaystyle
i\left[\chi^{\ast}D^{\mu}\chi-\left(D^{\mu}\chi\right)^{\ast}\chi\right].$ (7)

Under the discrete transformations $C$, $P$, and $T$, the Chern-Simons term
$\mathcal{L}_{C\\!S}=\mu\epsilon^{\rho\sigma\tau}F_{\rho\sigma}A_{\tau}/4$
behaves as follows:

$\mathcal{L}_{\mathrm{CS}}^{\left(C\right)}=\mathcal{L}_{\mathrm{CS}},\;\mathcal{L}_{\mathrm{CS}}^{\left(P\right)}=-\mathcal{L}_{\mathrm{CS}},\;\mathcal{L}_{\mathrm{CS}}^{\left(T\right)}=-\mathcal{L}_{\mathrm{CS}}.$
(8)

It follows from Eq. (8) that the Chern-Simons term breaks the $P$, $CP$, and
$T$-invariance of the model’s Lagrangian.

The field equations of the model have the form:

$\displaystyle\partial_{\mu}F^{\mu\nu}+\mu\left.\\!\right.^{\ast}\\!F^{\nu}=j^{\nu},$
(9) $\displaystyle
D_{\mu}D^{\mu}\phi+\lambda\left(\phi^{\ast}\phi-v^{2}\right)\phi=0,$ (10)
$\displaystyle
D_{\mu}D^{\mu}\chi+\left(m^{2}-2g\left(\chi^{\ast}\chi\right)+3h\left(\chi^{\ast}\chi\right)^{2}\right)\chi=0,$
(11)

where the dual field strength
$\left.\\!\right.^{\ast}\\!F^{\nu}=\epsilon^{\nu\alpha\beta}F_{\alpha\beta}/2$,
and the electromagnetic current $j^{\nu}$ is expressed in terms of the Noether
currents:

$j^{\nu}=ej_{\phi}^{\nu}+qj_{\chi}^{\nu}.$ (12)

Integrating the left and right hand sides of Eq. (9) with the index $\nu=0$
over the spatial plane, we obtain an important relation between the electric
charge and the magnetic flux:

$Q=eQ_{\phi}+qQ_{\chi}=-\mu\Phi,$ (13)

where $Q_{\phi}=\int j_{\phi}^{0}d^{2}x$ and $Q_{\chi}=\int
j_{\chi}^{0}d^{2}x$ are the conserved Noether charges.

The symmetric energy-momentum tensor of the model and the corresponding
expression for the energy density are written as:

$\displaystyle T_{\mu\nu}=$ $\displaystyle-
F_{\mu\lambda}F_{\nu}^{\;\lambda}+\frac{1}{4}g_{\mu\nu}F_{\lambda\rho}F^{\lambda\rho}$
$\displaystyle+\left(D_{\mu}\phi\right)^{\ast}D_{\nu}\phi+\left(D_{\nu}\phi\right)^{\ast}D_{\mu}\phi$
$\displaystyle-
g_{\mu\nu}\left(\left(D_{\mu}\phi\right)^{\ast}D^{\mu}\phi-V\left(\left|\phi\right|\right)\right)$
$\displaystyle+\left(D_{\mu}\chi\right)^{\ast}D_{\nu}\chi+\left(D_{\nu}\chi\right)^{\ast}D_{\mu}\chi$
$\displaystyle-
g_{\mu\nu}\left(\left(D_{\mu}\chi\right)^{\ast}D^{\mu}\chi-U\left(\left|\chi\right|\right)\right),$
(14) $\displaystyle T_{00}=$
$\displaystyle\frac{1}{2}E_{i}E_{i}+\frac{1}{2}B^{2}$ (15)
$\displaystyle+\left(D_{0}\phi\right)^{\ast}D_{0}\phi+\left(D_{i}\phi\right)^{\ast}D_{i}\phi+V\left(\left|\phi\right|\right)$
$\displaystyle+\left(D_{0}\chi\right)^{\ast}D_{0}\chi+\left(D_{i}\chi\right)^{\ast}D_{i}\chi+U\left(\left|\chi\right|\right),$

where $E_{i}=F_{0i}$ are the components of electric field strength and
$B=-F_{12}$ is the magnetic field strength.

In this paper we adopt the following gauge condition: $\partial_{0}\phi=0$.
Using the analogy of Q-ball, we find a soliton solution of model (1) that
minimizes the energy $E=\int T_{00}d^{2}x=H=\int\mathcal{H}d^{2}x$
($\mathcal{H}$ is the density of the Hamiltonian $H$) at a fixed value of the
Noether charge $Q_{\chi}=\int j_{\chi}^{0}d^{2}x$. In this case, the soliton
solution is an unconditional extremum of the functional

$F=\int\mathcal{H}d^{2}x-\omega\int j_{\chi}^{0}d^{2}x=H-\omega Q_{\chi},$
(16)

where $\omega$ is the Lagrange multiplier. The Noether charge $Q_{\chi}$ is
written in terms of the canonically conjugated fields $\chi$, $\chi^{\ast}$,
$\pi_{\chi}=\partial\mathcal{L}/\partial\left(\partial_{0}\chi\right)=\left(D_{0}\chi\right)^{\ast}$,
and
$\pi_{\chi^{\ast}}=\partial\mathcal{L}/\partial\left(\partial_{0}\chi^{\ast}\right)=D_{0}\chi$
as follows:

$Q_{\chi}=-i\int\left(\chi\pi_{\chi}-\chi^{\ast}\pi_{\chi^{\ast}}\right)d^{2}x.$
(17)

From Eq. (17), we obtain the variation of the Noether charge $Q_{\chi}$ in
terms of the canonically conjugate fields:

$\delta
Q_{\chi}=-i\int\left(\chi\delta\pi_{\chi}+\pi_{\chi}\delta\chi-\text{c.c.}\right)d^{2}x.$
(18)

At the same time, the first variation of the functional $F$ vanishes for the
soliton solution:

$\delta F=\delta H-\omega\delta Q_{\chi}=0.$ (19)

From Eqs. (18) and (19), we obtain the following Hamilton field equations:

$\partial_{0}\chi=\frac{\delta
H}{\delta\pi_{\chi}}=-i\omega\chi,\quad\partial_{0}\chi^{\ast}=\frac{\delta
H}{\delta\pi_{\chi^{\ast}}}=i\omega\chi^{\ast},$ (20)

while time derivatives of the other model’s fields are equal to zero. We see
that the scalar field $\chi$ has the time dependence of Q-ball type:

$\chi\left(x\right)=\chi\left(\mathbf{x}\right)\exp\left(-i\omega t\right),$
(21)

whereas the other model’s fields do not depend on time for the adopted gauge
condition $\partial_{0}\phi=0$. Extremum condition (19) leads to the important
relation for the soliton solution:

$\frac{dE}{dQ_{\chi}}=\omega,$ (22)

where it is understood that the Lagrange multiplier $\omega$ is some function
of the Noether charge $Q_{\chi}$.

##  3 The ansatz and some properties of the solution

To solve field equations (9) – (11), we apply the following ansatz:

$\displaystyle A^{\mu}\left(x\right)$
$\displaystyle=\left(\frac{A_{0}\left(r\right)}{er},\frac{1}{er}\epsilon_{ij}n_{j}A\left(r\right)\right),$
$\displaystyle\phi\left(x\right)$
$\displaystyle=v\exp\left(iN\theta\right)F\left(r\right),$
$\displaystyle\chi\left(x\right)$
$\displaystyle=\sigma\left(r\right)\exp\left(-i\omega t\right),$ (23)

where $\epsilon_{ij}$ are the components of the two-dimensional antisymmetric
tensor $\left(\epsilon_{12}=1\right)$ and $n_{j}$ are those of the two-
dimensional radial unit vector
$\mathbf{n}=\left(\cos\left(\theta\right)\\!,\,\sin\left(\theta\right)\right)$.
We suppose that the function $\sigma\left(r\right)$ is real, so ansatz (23)
completely fixes the model’s gauge.

It can be shown that ansatz (23) is consistent with field equations (9) –
(11), so we get the system of second order nonlinear differential equations
for the ansatz functions:

$\displaystyle
A_{0}^{\prime\prime}(r)-\frac{A_{0}^{\prime}(r)}{r}+\frac{A_{0}(r)}{r^{2}}-\mu
A^{\prime}(r)$
$\displaystyle-2\left(e^{2}v^{2}F\left(r\right)^{2}+q^{2}\sigma\left(r\right)^{2}\right)$
$\displaystyle\times A_{0}(r)+2eq\omega r\sigma\left(r\right)^{2}=0,$ (24)
$\displaystyle A^{\prime\prime}(r)-\frac{A^{\prime}(r)}{r}-\mu
A_{0}^{\prime}(r)$
$\displaystyle-2e^{2}v^{2}\left(N+A(r)\right)F\left(r\right)^{2}$
$\displaystyle-2q^{2}\sigma\left(r\right)^{2}A\left(r\right)+\mu\frac{A_{0}(r)}{r}=0,$
(25) $\displaystyle
F^{\prime\prime}(r)+\frac{F^{\prime}(r)}{r}-\frac{F(r)}{r^{2}}$
$\displaystyle\times\left((N+A(r))^{2}-A_{0}(r)^{2}\right)$
$\displaystyle+\lambda v^{2}\left(1-F(r)^{2}\right)F(r)=0,$ (26)
$\displaystyle\sigma^{\prime\prime}(r)+\frac{\sigma^{\prime}(r)}{r}+\sigma\left(r\right)$
$\displaystyle\times\left(\left(\omega-\frac{q}{e}\frac{A_{0}\left(r\right)}{r}\right)^{2}-\frac{q^{2}}{e^{2}}\frac{A\left(r\right)^{2}}{r^{2}}\right)$
(27)
$\displaystyle-\left(m^{2}-2g\sigma\left(r\right)^{2}+3h\sigma\left(r\right)^{4}\right)\sigma\left(r\right)=0.$

The expression for the energy density $\mathcal{E}=T_{00}$ can also be written
in terms of the ansatz functions:

$\displaystyle\mathcal{E}=$
$\displaystyle\,\frac{1}{2}\frac{A^{\prime}{}^{2}}{e^{2}r^{2}}+\frac{1}{2}\left(\left(\frac{A_{0}}{er}\right)^{\prime}\right)^{2}+v^{2}F^{\prime}{}^{2}$
$\displaystyle+\frac{\left((N+A)^{2}+A_{0}{}^{2}\right)}{r^{2}}v^{2}F^{2}$
$\displaystyle+\frac{\lambda}{2}v^{4}\left(F^{2}-1\right)^{2}+\sigma^{\prime}{}^{2}$
$\displaystyle+\left(\omega-q\frac{A_{0}}{er}\right)^{2}\sigma^{2}+\frac{q^{2}}{e^{2}}\frac{A^{2}}{r^{2}}\sigma^{2}$
$\displaystyle+m^{2}\sigma^{2}-g\sigma^{4}+h\sigma^{6}.$ (28)

The regularity of the soliton solution at $r=0$ and the finiteness of the
soliton’s energy
$E=2\pi\int\nolimits_{0}^{\infty}\mathcal{E}\left(r\right)rdr$ lead us to the
following boundary conditions for the ansatz functions:

$\displaystyle A_{0}(0)$ $\displaystyle=0,\quad
A_{0}(r)\underset{r\rightarrow\infty}{\longrightarrow}0,$ $\displaystyle A(0)$
$\displaystyle=0,\quad\,A(r)\underset{r\rightarrow\infty}{\longrightarrow}-N,$
$\displaystyle F(0)$
$\displaystyle=0,\quad\,F(r)\underset{r\rightarrow\infty}{\longrightarrow}1,$
$\displaystyle\sigma^{\prime}(0)$
$\displaystyle=0,\quad\,\sigma(r)\underset{r\rightarrow\infty}{\longrightarrow}0.$
(29)

From the boundary conditions for $A(r)$ it follows that the magnetic flux of
the vortex-Q-ball system is quantized:

$\Phi=2\pi\int_{0}^{\infty}B\left(r\right)rdr=\frac{2\pi}{e}N,$ (30)

where $B(r)=-A^{\prime}(r)/(er)$ is the magnetic field strength. Moreover,
from Eqs. (13) and (30) it follows that the total electric charge of the
vortex-Q-ball system is also quantized:

$Q=-\frac{2\pi}{e}\mu N.$ (31)

In terms of the ansatz functions, the $C$-transformation is written as

$\displaystyle\omega\rightarrow-\omega,\;\ N\rightarrow-N,\;A_{0}\rightarrow-
A_{0},\;$ $\displaystyle A\rightarrow-A,\;\ F\rightarrow
F,\;\sigma\rightarrow\sigma.$ (32)

It is easily shown that Eqs. (24) – (27) are invariant under transformation
(32) as well as energy density (28). According to Eq. (8), $C$-transformation
(32) is the only discrete one that leaves Eqs. (24) – (27) invariant. All
other discrete transformations ($P$, $CP$, and $T$) do not leave Eqs. (24) –
(27) invariant. We see that transformation (32) changes the sign of $\omega$,
but at the same time, it also changes the sign of the soliton’s winding
number. From this it follows that the energy of a soliton with a given fixed
$N$ is not invariant under the change of sign of the phase frequency:
$E\left(-\omega\right)\neq E\left(\omega\right)$. Similarly, it can be shown
that $Q_{\phi,\chi}\left(-\omega\right)\neq-Q_{\phi,\chi}\left(\omega\right)$,
so the absolute values of the Noether charges are also not invariant.

From Eqs. (24) – (27) and boundary conditions (29) it follows that at $r=0$,
the power expansion of the soliton solution has the form:

$\displaystyle A_{0}\left(r\right)$
$\displaystyle=a_{1}r+\frac{a_{3}}{3!}r^{3}+O\left(r^{5}\right),$
$\displaystyle A\left(r\right)$
$\displaystyle=\frac{b_{2}}{2!}r^{2}+\frac{b_{4}}{4!}r^{4}+O\left(r^{6}\right),$
$\displaystyle F\left(r\right)$
$\displaystyle=\frac{c_{\left|N\right|}}{\left|N\right|!}r^{\left|N\right|}\\!+\\!\frac{c_{\left|N\right|+2}}{\left(\left|N\right|\\!+\\!2\right)!}r^{\left|N\right|+2}\\!+\\!O\\!\left(\\!r^{\left|N\right|+4}\\!\right)\\!,$
$\displaystyle\sigma\left(r\right)$
$\displaystyle=d_{0}+\frac{d_{2}}{2!}r^{2}+O\left(r^{4}\right).$ (33)

In Eq. (33), the expressions of the next-to-leading coefficients $a_{3}$,
$b_{4}$, $c_{\left|N\right|+2}$, and $d_{2}$ are

$\displaystyle a_{3}=$
$\displaystyle\,3qd_{0}^{2}\left(a_{1}q-e\omega\right)+\frac{3}{2}\mu b_{2},$
$\displaystyle b_{4}=$
$\displaystyle\,3\left(q^{2}b_{2}d_{0}^{2}+2Ne^{2}v^{2}c_{\left|N\right|}^{2}\delta_{1,\left|N\right|}\right)+\mu
a_{3},$ $\displaystyle c_{\left|N\right|+2}=$
$\displaystyle-\frac{c_{\left|N\right|}}{4}\left(\left|N\right|+2\right)\left(a_{1}^{2}+\left|N\right|\left|b_{2}\right|+\lambda
v^{2}\right),$ $\displaystyle d_{2}=$
$\displaystyle\frac{d_{0}}{2}\left[d_{0}^{2}\left(3d_{0}^{2}h-2g\right)\right.$
$\displaystyle\left.+e^{-2}\left(qa_{1}+e\left(m-\omega\right)\right)\right.$
$\displaystyle\left.\times\left(-qa_{1}+e\left(m+\omega\right)\right)\right],$
(34)

where $\delta_{1,\left|N\right|}$ is the Kronecker symbol. Linearization of
Eqs. (24) – (27) at large $r$ and use of corresponding boundary conditions
(29) lead us to the asymptotic form of the solution as $r\rightarrow\infty$:

$\displaystyle A_{0}\left(r\right)$ $\displaystyle\sim
a_{\infty}\sqrt{m_{A}r}\exp\left(-m_{A}r\right),$ $\displaystyle
A\left(r\right)$
$\displaystyle\sim-N+a_{\infty}\sqrt{m_{A}r}\exp\left(-m_{A}r\right),$
$\displaystyle F\left(r\right)$ $\displaystyle\sim
1+c_{\infty}\frac{\exp\left(-m_{\phi}r\right)}{\sqrt{m_{\phi}r}},$
$\displaystyle\sigma\left(r\right)$ $\displaystyle\sim
d_{\infty}\frac{\exp\left(-\sqrt{m^{2}-\omega^{2}}r\right)}{\sqrt{mr}},$ (35)

where

$m_{A}=\sqrt{2e^{2}v^{2}+\frac{\mu^{2}}{4}}-\frac{\mu}{2}$ (36)

is the mass of the gauge boson and $m_{\phi}=\sqrt{2\lambda}v$ is the mass of
the scalar $\phi$-particle.

For symmetric energy-momentum tensor (14), the angular momentum tensor has the
form

$J^{\lambda\mu\nu}=x^{\mu}T^{\lambda\nu}-x^{\nu}T^{\lambda\mu}.$ (37)

Use of Eqs. (14), (23), and (37) leads us to the angular momentum’s density
expressed in terms of the ansatz functions:

$\displaystyle\mathcal{J}$
$\displaystyle=\frac{1}{2}\epsilon_{ij}J^{0ij}=-rBE_{r}+2\frac{q}{e}A\left(\omega-q\frac{A_{0}}{er}\right)\sigma^{2}$
$\displaystyle-2\frac{A_{0}\left(N+A\right)}{r}v^{2}F^{2}.$ (38)

In Eq. (38),
$E_{r}(r)=-\left(A_{0}\left(r\right)/\left(er\right)\right)^{\prime}$ is the
radial electric field strength. Integrating the term
$-rBE_{r}=-e^{-2}A^{\prime}\left(A_{0}/r\right)^{\prime}$ by parts, taking
into account boundary conditions (29), and using Eq. (24) to eliminate
$A_{0}^{\prime\prime}$, we obtain the expression for the soliton’s angular
momentum $J=2\pi\int_{0}^{\infty}\mathcal{J}\left(r\right)rdr$:

$J=-4\pi
Nv^{2}\int_{0}^{\infty}A_{0}\left(r\right)F^{2}(r)dr+\pi\frac{\mu}{e^{2}}N^{2}.$
(39)

At the same time, Eqs. (7) and (23) lead us to the following expression of the
Noether charge $Q_{\phi}$:

$Q_{\phi}=-4\pi v^{2}\int_{0}^{\infty}A_{0}\left(r\right)F^{2}(r)dr.$ (40)

From Eqs. (13), (31), (39), and (40) it follows that for the vortex-Q-ball
system the important relation holds between the angular momentum $J$ and the
Noether charges $Q_{\phi}$ and $Q_{\chi}$:

$\displaystyle
J=NQ_{\phi}+\pi\frac{\mu}{e^{2}}N^{2}=-N\frac{q}{e}Q_{\chi}-\pi\frac{\mu}{e^{2}}N^{2}.$
(41)

We see that the angular momentum depends linearly on the Noether charges of
the scalar fields.

Figure 1:  The numerical solution $m^{-1/2}A_{0}(r)/(er)$ (dotted), $A(r)$
(dashed), $F(r)$ (solid), and $m^{-1/2}\sigma(r)$ (dash-dotted) for the
vortex-Q-ball system. The model’s parameters are $e=q=0.3\,m^{1/2}$,
$\mu=0.5\,m$, $\lambda=0.335\,m$, $v=1.221\,m^{1/2}$, $g=1.0\,m$, $h=0.26$,
and $N=1$. The phase frequency $\omega=0.38\,m$.

Any solution of field equations (9) – (11) is an extremum of the action
$S=\int\mathcal{L}d^{2}xdt$. It is readily seen, however, that the Lagrangian
density (1) does not depend on time if the field configurations are those of
ansatz (23). It follows that any solution of system (24) – (27) is an extremum
of the Lagrangian $L=\int\mathcal{L}d^{2}x$. Let $A_{0}(r)$, $A(r)$, $F(r)$,
and $\sigma(r)$ be a solution of system (24) – (27) satisfying boundary
conditions (29). After the scale transformation of the solution’s argument
$r\rightarrow\lambda r$, the Lagrangian $L$ becomes a function of the scale
parameter $\lambda$. The function $L\left(\lambda\right)$ has an extremum at
$\lambda=1$, so its derivative with respect to $\lambda$ vanishes at this
point: $\left.dL/d\lambda\right|_{\lambda=1}=0$. From this equation it follows
that the virial relation holds for the vortex-Q-ball system:

$2\left(E^{\left(E\right)}-E^{\left(H\right)}+E^{\left(P\right)}\right)+L^{\left(C\\!S\right)}-\omega
Q_{\chi}=0,$ (42)

where

$E^{\left(E\right)}=\frac{1}{2}\int
E_{i}E_{i}d^{2}x=\pi\int_{0}^{\infty}\left(\left(\frac{A_{0}}{er}\right)^{\prime}\right)^{2}rdr$
(43)

is the electric field’s energy,

$E^{\left(B\right)}=\frac{1}{2}\int
B^{2}d^{2}x=\pi\int_{0}^{\infty}\frac{A^{\prime}{}^{2}}{e^{2}r}dr$ (44)

is the magnetic field’s energy,

$E^{\left(P\right)}=2\pi\int_{0}^{\infty}\left[V\left(\left|\phi\right|\right)+U\left(\left|\chi\right|\right)\right]rdr$
(45)

is the potential part of the soliton’s energy, and

$L^{\left(C\\!S\right)}=\frac{\mu}{4}\int\epsilon^{\rho\sigma\tau}F_{\rho\sigma}A_{\tau}d^{2}x$
(46)

is the Chern-Simons part of the model’s Lagrangian.

##  4 Numerical results

Figure 2:  The numerical solution $m^{-1/2}A_{0}(r)/(er)$ (dotted), $A(r)$
(dashed), $F(r)$ (solid), and $m^{-1/2}\sigma(r)$ (dash-dotted) for the system
of vortex and Q-ball that do not interact with each other. The model’s
parameters are the same as in Fig. 1.

Now we present some numerical results concerning the vortex-Q-ball system. For
numerical calculations, we use the natural units $c=1$, $\hbar=1$. In
addition, the mass $m$ of scalar $\chi$-particle is used as the energy unit.
After that, the model is completely determined by the seven parameters: $e$,
$q$, $\mu$, $\lambda$, $v$, $g$, and $h$. We chose the following values of
these parameters: $e=q=0.3\,m^{1/2}$, $\mu=0.5\,m$, $\lambda=0.335\,m$,
$v=1.221\,m^{1/2}$, $g=1.0\,m$, and $h=0.26$, where the parameters’ dimensions
correspond to the $(2+1)$-dimensional case. The correctness of the numerical
solution were checked by use of Eqs. (13), (22), (41), and (42).

In Fig. 1, we can see the dimensionless zero component $m^{-1/2}A_{0}(r)/(er)$
of the gauge potential along with the dimensionless ansatz functions $A(r)$,
$F(r)$, and $m^{-1/2}\sigma(r)$. The vortex part of the soliton system is in
the topological sector with $N=1$, the phase frequency $\omega$ is equal to
$0.38\,m$. Figure 2 presents the numerical solution for the case $q=0$,
whereas the other model’s parameters remain the same as in Fig. 1. This
corresponds to superimposed gauged vortex and non-gauged Q-ball that do not
interact with each other. From Figs. 1 and 2, we can conclude that the gauge
interaction between the vortex and Q-ball components leads to significant
changes in the shapes of the ansatz functions $A_{0}(r)$, $A(r)$, and
$\sigma(r)$, while the shape of $F(r)$ does not change significantly.

Figure 3:  The dimensionless versions of the electric field strength
$\tilde{E}_{r}(r)=m^{-3/2}E_{r}(r)$ (solid), the magnetic field strength
$\tilde{B}(r)=m^{-3/2}B(r)$ (dashed), the scaled energy density
$0.5\,\tilde{\mathcal{E}}(r)=0.5\,m^{-3}\mathcal{E}(r)$ (dash-dotted), the
electric charge density $\tilde{j}_{0}(r)=m^{-5/2}j_{0}(r)$ (dash-dot-dotted),
and the scaled angular momentum’s density
$0.5\,\tilde{\mathcal{J}}(r)=0.5\,m^{-2}\mathcal{J}(r)$ (dotted),
corresponding to the solution in Fig. 1.

Figure 3 shows the dimensionless versions of the electric field strength
$\tilde{E}_{r}(r)=m^{-3/2}E_{r}(r)$, the magnetic field strength
$\tilde{B}(r)=m^{-3/2}B(r)$, the scaled energy density
$0.5\,\tilde{\mathcal{E}}(r)=0.5\,m^{-3}\mathcal{E}(r)$, the electric charge
density $\tilde{j}_{0}(r)=m^{-5/2}j_{0}(r)$, and the scaled angular momentum’s
density $0.5\,\tilde{\mathcal{J}}(r)=0.5\,m^{-2}\mathcal{J}(r)$ that
correspond to the soliton solution in Fig. 1. We see that just as in [19], the
vortex-Q-ball system consists of three parts: the central transition region,
the inner region, and the external transition region. We also see that the
densities of the energy and the angular momentum are approximately constant in
the inner region, while the electric and magnetic field strengths are close to
zero there.

In Fig. 4, we can see the dimensionless soliton energy $\tilde{E}=m^{-1}E$ as
a function of the dimensionless phase frequency $\tilde{\omega}=m^{-1}\omega$.
The function $\tilde{E}\left(\tilde{\omega}\right)$ is presented in the range
from the minimum values of $\left|\tilde{\omega}\right|$ that we have reached
by numerical methods to its maximum value of $1$. The most important feature
of Fig. 4 is that the soliton’s energy is not invariant under the change of
sign of the phase frequency:
$\tilde{E}\left(-\tilde{\omega}\right)\neq\tilde{E}\left(\tilde{\omega}\right)$.
This fact is a direct consequence of the $T$-invariance breaking, which is
caused by the Chern-Simons term in the Lagrangian (1). From Fig. 4 it follows
that the soliton’s energy $E$ tends to infinity as
$\left|\tilde{\omega}\right|$ tends to its minimum values (thin-wall regime).
In the thin-wall regime, the spatial size of the soliton’s inner region
increases indefinitely, so the main contribution to the soliton’s energy and
angular momentum comes from this region.

Figure 4:  The dimensionless soliton energy $\tilde{E}=m^{-1}E$ as a function
of the dimensionless phase frequency $\tilde{\omega}=m^{-1}\omega$. The
model’s parameters are the same as in Fig. 1.

As $\tilde{\omega}\rightarrow 1$, the vortex-Q-ball system goes into the
thick-wall regime. As well as in the thin-wall regime, the soliton’s Noether
charge $Q_{\chi}$ and energy $E$ tend to infinity in the thick-wall regime. It
was found numerically that $Q_{\chi}(\tilde{\omega})$ and
$\tilde{E}(\tilde{\omega})$ have the following behaviour as
$\tilde{\omega}\rightarrow 1$:

$\displaystyle Q_{\chi}\underset{\tilde{\omega}\rightarrow
1}{\longrightarrow}B+A\left(1-\tilde{\omega}\right)^{-\frac{1}{2}},$
$\displaystyle\tilde{E}\underset{\tilde{\omega}\rightarrow
1}{\longrightarrow}C+A\left(2-\tilde{\omega}\right)\left(1-\tilde{\omega}\right)^{-\frac{1}{2}},$
(47)

where $A$, $B$, and $C$ are positive constants. From Eq. (47) it follows that
the behaviour of $Q_{\chi}\left(\tilde{\omega}\right)$ and
$\tilde{E}\left(\tilde{\omega}\right)$ in the thick-wall regime is in
agreement with Eq. (22). Such behaviour of $Q_{\chi}(\tilde{\omega})$ and
$\tilde{E}(\tilde{\omega})$ in a neighborhood of the maximum value
$\tilde{\omega}=1$ is very different from that of the two-dimensional non-
gauged Q-ball [20]. It is also quite different from the behaviour of the
vortex-Q-ball system [19] in the Maxwell gauge model. However, the behaviour
of $Q_{\chi}(\tilde{\omega})$ and $\tilde{E}(\tilde{\omega})$ in a
neighborhood of $\tilde{\omega}=1$ is similar to that of the usual three-
dimensional Q-ball [20].

In Fig. 5, we can see the dependences of the dimensionless soliton energy
$\tilde{E}$ and the absolute value of Noether charge $Q_{\chi}$ (which is
negative for $\tilde{\omega}<0$) on the absolute value of $\tilde{\omega}$ in
a neighborhood of $\left|\tilde{\omega}\right|=1$. From Fig. 5 it follows that
the behaviour of the vortex-Q-ball system in the neighborhoods of
$\tilde{\omega}=-1$ and $\tilde{\omega}=1$ is completely different. Indeed,
its behaviour near $\tilde{\omega}=1$ corresponds to thick-wall regime (47).
At the same time, its behaviour near $\tilde{\omega}=-1$ is rather unusual.
Firstly, there is no thick-wall regime here. Secondly, the Q-ball component of
the vortex-Q-ball system disappears at $\tilde{\omega}=-0.93$. For
$\tilde{\omega}\in\left(-1,-0.93\right)$, we have only the single Maxwell-
Chern-Simons vortex without any Q-ball component.

Figure 5:  The dimensionless soliton energy $\tilde{E}=m^{-1}E$ (solid for
$\tilde{\omega}>0$ and dash-dotted for $\tilde{\omega}<0$) and the absolute
value of Noether charge $Q_{\chi}$ (dashed for $\tilde{\omega}>0$ and dotted
for $\tilde{\omega}<0$) as functions of the absolute value of dimensionless
phase frequency $\left|\tilde{\omega}\right|$ in a neighborhood of
$\left|\tilde{\omega}\right|=1$.

Figure 6 shows the dimensionless energy $\tilde{E}$ as a function of the
absolute value of Noether charge $Q_{\chi}$ for the both signs of
$\tilde{\omega}$. It also shows the similar dependence
$\tilde{E}\left(\left|Q_{\chi}\right|\right)$ for the two-dimensional non-
gauged Q-ball with the same parameters $m$, $g$, and $h$ as for the vortex-Q-
ball system. In addition, the straight line $\tilde{E}=\left|Q_{\chi}\right|$
is also shown in Fig. 6. We can see that the curve
$\tilde{E}(\left|Q_{\chi}\right|)$ corresponding to the two-dimensional Q-ball
is tangent to the straight line $\tilde{E}=\left|Q_{\chi}\right|$ at some
nonzero $\left|Q_{\chi}\right|$ as it should be [20]. In contrast to this, the
vortex-Q-ball system is described by the two curves, which correspond to the
both signs of the phase frequency $\tilde{\omega}$. The curve
$\tilde{E}(Q_{\chi})$ corresponding to the positive $\tilde{\omega}$ is
similar to that of three-dimensional Q-ball. In particular, it has the cusp
and consists of two branches. As $Q_{\chi}\rightarrow\infty$, the lower branch
goes into the thin-wall regime, while the upper one goes into the thick-wall
regime. At the same time, the curve $\tilde{E}(-Q_{\chi})$ corresponding to
the negative $\tilde{\omega}$ has no cusp and consists of only one branch. The
curve starts at $Q_{\chi}=0$ and goes into the thin-wall regime as
$Q_{\chi}\rightarrow-\infty$. From Fig. 6, we can conclude that in the thin-
wall regime the Q-ball component of the the vortex-Q-ball system is stable to
the decay into the massive scalar $\chi$-particles.

Figure 6:  The vortex-Q-ball system’s dimensionless energy $\tilde{E}$ as a
function of the absolute value of Noether charge $Q_{\chi}$ for
$\tilde{\omega}>0$ (solid) and for $\tilde{\omega}<0$ (dash-dotted), and that
for the two-dimensional non-gauged Q-ball (dash-dot-dotted) with the same
parameters $m$, $g$, and $h$ as for the vortex-Q-ball system. The dashed line
is the straight line $\tilde{E}=\left|Q_{\chi}\right|$.

##  5 Conclusions

In the present paper, we have researched the soliton system consisting of a
vortex and a Q-ball that interact with each other through a common Abelian
gauge field. Like a vortex, this two-dimensional soliton system has quantized
magnetic flux (30). Due to the Chern-Simons term in the Lagrangian, the
quantized magnetic flux leads to quantized electric charge (31) of the soliton
system and, as a consequence, to a nonzero radial electric field. As a result,
the soliton system possesses nonzero angular momentum (41) that depends
linearly on the Noether charges of the scalar fields. Owing to the Chern-
Simons term, the energy of the vortex-Q-ball system is not invariant under the
sign reversal of the phase frequency $\omega$. This in turn leads to the
significant change of the dependence $E(Q_{\chi})$ in comparison with the
vortex-Q-ball system [19] and with the two-dimensional non-gauged Q-ball [20].
The vortex-Q-ball system combines properties of both nontopological (Eq. (22))
and topological (boundary condition (29) for $A(r)$ and, as a consequence,
magnetic flux quantization (30)) solitons.

Finally, let us point out a possible application of the results obtained in
[19] and in the present paper. A vortex-Q-ball string may arise when a cosmic
string passes through a charged scalar condensate. Such a condensate could
exist in the early universe; electrically charged boson stars [21], if they
exist, also consist of such a condensate. A part of the condensate may be
carried away by the passing cosmic vortex string, with the result that the
vortex-Q-ball string arises. In this case, the gauge interaction between
vortex and Q-ball components of the vortex-Q-ball string leads to significant
changes of their properties.

## Acknowledgments

The research is carried out at Tomsk Polytechnic University within the
framework of Tomsk Polytechnic University Competitiveness Enhancement Program
grant.

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