compat=1.1.0 ampmtime ampmtime

# Dynamics of Confined Monopoles
and Similarities with Confined Quarks

Gia Dvali  Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,
Theresienstraße 37, 80333 München, Germany  Max-Planck-Institut für Physik,
Föhringer Ring 6, 80805 München, Germany  Juan Valbuena-Bermudez
<EMAIL_ADDRESS>Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,
Theresienstraße 37, 80333 München, Germany  Max-Planck-Institut für Physik,
Föhringer Ring 6, 80805 München, Germany  Michael Zantedeschi
<EMAIL_ADDRESS>Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,
Theresienstraße 37, 80333 München, Germany  Max-Planck-Institut für Physik,
Föhringer Ring 6, 80805 München, Germany

(, )

###### Abstract

In this work, we study the annihilation of a pair of ‘t Hooft-Polyakov
monopoles due to confinement by a string. We analyze the regime in which the
scales of monopoles and strings are comparable. We compute the spectrum of the
emitted gravitational waves and find it to agree with the previously
calculated point-like case for wavelengths longer than the system width and
before the collision. However, we observe that in a head-on collision,
monopoles are never re-created. Correspondingly, not even once the string
oscillates. Instead, the system decays into waves of Higgs and gauge fields.
We explain this phenomenon by the loss of coherence in the annihilation
process. Due to this, the entropy suppression makes the recreation of a
monopole pair highly improbable. We argue that in a similar regime, analogous
behaviour is expected for the heavy quarks connected by a QCD string. There
too, instead of re-stretching a long string after the first collapse, the
system hadronizes and decays in a high multiplicity of mesons and glueballs.
We discuss the implications of our results.

##  I Introduction

It is well known Langacker and Pi (1980); Lazarides  _et al._ (1982); Vilenkin
(1982); Vilenkin and Shellard (2000) that monopoles carrying the opposite
magnetic charges under a $U(1)$ gauge symmetry become connected by a string
when $U(1)$ is Higgsed. The string represents a magnetic flux tube of Nielsen-
Olesen type Nielsen and Olesen (1973). As long as the mass of the monopole is
larger than the scale of the string tension, energy per unit length $\mu$, the
breakup of the string via nucleation of monopole pairs is exponentially
unlikely Vilenkin (1982).

This system shares some similarity with a quark-anti-quark pair connected by a
QCD string in a confining gauge theory. The QCD string represents a flux tube
of the color electric field. The string tension, $\mu=\Lambda^{2}$, is set by
the QCD scale, $\Lambda$. As long as all quarks in the theory are heavier than
$\Lambda$, the probability of breaking the string by a pair creation is
exponentially small.

Due to this analogy, studying monopoles connected by a magnetic string can
serve as a useful test-laboratory for understanding certain features of
confined heavy quarks.

The above systems have a number of interesting applications in particle
physics and cosmology.

For example, recently a novel mechanism for producing the primordial black
holes was proposed in Dvali  _et al._ (2021). Therein, quark pairs, produced
and diluted in the inflationary era, are confined in the late Universe. Upon
horizon re-entry, they collapse and form black holes (BHs) due to the large
amount of energy stored in the flux tubes connecting them. Given the constant
acceleration of quarks sourced by the string, gravitational waves (GWs) of
frequency comparable to the inverse of the horizon size are produced.
Analogous considerations could be applied to the case of confined monopoles
Matsuda (2009).

Previous calculations of the radiated GW spectrum were performed by Martin and
Vilenkin Martin and Vilenkin (1997) in the point-like approximation, in which
the size of monopoles as well as the width of the string are set to zero. In
this limit, they obtained the following emitted power for a large range of
frequencies

$P_{\rm n}\sim\frac{\Lambda^{4}}{M_{\rm p}^{2}}\frac{1}{n},$ (1)

$n$ being the frequency number, $M_{\rm p}$ the planck mass, and $\Lambda$ the
confining scale. This relation was derived considering a pair of monopoles
connected by a string and is expected to be valid in the case of confined
quarks too. Such sources could explain the recent hints of stochastic GW
background obtained from pulsar timing arrays Arzoumanian  _et al._ (2020);
Antoniadis  _et al._ (2022). Moreover, given the flatness of the resulting
energy density across several orders of frequency Leblond  _et al._ (2009), in
the future, it will be possible to cross-check with other gravitational wave
detectors sensitive to the lengths shorter than the pulsar timing arrays.

Another scenario for which our study is relevant is the Langacker-Pi mechanism
Langacker and Pi (1980). In an attempt to solve the monopole abundance
problem, the theory ensures a temporary (thermal) window in which the $U(1)$
group associated to the monopole charge is broken, leading to their
confinement.

At lower temperatures the $U(1)$ symmetry is restored again. This mechanism
can be achieved by adequately choosing the spectrum and couplings of the
theory Weinberg (1974). The system therefore has a finite window of
opportunity for getting rid of monopoles. If monopoles connected by string can
oscillate for too long, this window of opportunity is insufficient for solving
the monopole problem.

The goal of the present work is to analyze the dynamics of a monopole/anti-
monopole pair in the confined phase in detail. In order to do so, we
considered a $SU(2)$ gauge theory and chose a simple scalar sector capable of
achieving the above-mentioned configuration via spontaneous symmetry breaking:
a scalar field in the adjoint representation, and a complex scalar doublet.
The former breaks (Higgses) the gauge group $SU(2)$ to $U(1)$, therefore
admitting t’Hooft-Polyakov monopoles as a solution ’t Hooft (1974); Polyakov
(1975). The latter breaks the residual $U(1)$ gauge group leading to the
confinement of the associated “magnetic flux”. Our study covers the regime in
which the monopole size and the string width are comparable. The similarities
with confined quarks are established in the analogous regime.

It turns out that the point-like limit approximates very well the part of the
classical dynamics in which the monopole separation is much larger then the
characteristic width of the system. However, beyond this regime we observe
some new features.

Naively, it is expected that a collapsing straight string performs several
oscillations. That is, one would think that after shrinking, the end points
(monopoles) scatter and fly apart stretching a long string again. In this way,
the string would contract and expand with certain periodicity, as some sort of
a rubber band.

However, we observe that in head-on collision the outcome is very different.
After the first shrinkage the string never recovers. Instead, the entire
energy is converted into the waves of Higgs and gauge particles. These waves
can also be thought of as large number of overlapping short strings.

We explain this phenomenon and argue that in analogous kinematic regime the
similar effect takes place in case of confined quarks. In this particular
regime, in both cases, the outcome can be understood as the result of the
entropy suppression for production of a highly coherent state in a collision
process Dvali (2021). Due to this, instead of stretching a long string, the
system prefers to produce many particles (short strings) which have a much
higher entropy. In case of QCD, the collapse of a long string results into a
high multiplicity of glueballs (closed strings) and mesons (open strings).

We also point out that inability of monopole and antimonopole to going through
each other, falls in the same category as the suppression of the passage of a
magnetic monopole through a domain wall, studied in Dvali  _et al._ (1998). In
that example, the domain wall provides a support base for unwinding the
monopole, similar to the role of the antimonopole in the present case. The
recreation of the monopole state on the other side of the wall is unlikely due
to the insufficiency of the microstate entropy of the monopole for overcoming
the exponential suppression of the corresponding multi-particle amplitude
Dvali (2021). This leads to the “erasure” of monopoles by domain walls. In
Dvali  _et al._ (1998), this mechanism was used to solve the cosmological
monopole problem in grand unified theories. However, the phenomenon of erasure
is of broader fundamental interest. In particular, this is indicated by the
similarities between the erasure processes of confined quarks and confined
monopoles discussed in the present paper.

It emerges that in the studied regime, the processes of the collapse of the
confined pairs in both theories are governed by the same universal effect: the
exponential suppression of production of a high occupation number (coherent)
state, albeit of insufficient entropy Dvali (2021)

The GW spectrum produced by confined monopoles is appropriately captured by
the point-like result for scales larger than the monopole width. As expected,
we observe non-negligible corrections to the power spectrum for scales
comparable to the monopole radius, where the emitted radiation is boosted,
therefore providing corrections to the GWs emission produced by the
confinement dynamics.

We expect that our results have implications for the collapse of the generic
bounded strings such as the string-theoretic strings bounded by $D$-branes
Copeland  _et al._ (2004); Dvali and Vilenkin (2004).

The paper is organized as follows. First we discuss the system of confined
monopoles and study it numerically. Next, we explain the underlying physics
that is shared by confined quarks and monopoles. We then study emission of
gravitational waves. Finally we discuss sphalerons and give outlook and
conclusions.

##  II Setup

We will work with a $\text{SU}(2)$ gauged field theory that contains a scalar
field in the adjoint representation, $\varphi^{a}$ (a = 1, 2, 3), a scalar
field in the fundamental representation, $\psi$, and gauge fields,
$W^{a}_{\mu}$. The Lagrangian of the system is given by

$\mathcal{L}=\frac{1}{2}D_{\mu}\varphi^{a}D^{\mu}\varphi^{a}+(D_{\mu}\psi)^{\dagger}D_{\mu}\psi-\frac{1}{4}{W^{a}}_{\mu\nu}{W^{a}}^{\mu\nu}-V(\varphi,\psi)$
(2)

where summation over repeated $\text{SU}(2)$ indices is understood, and the
field strengths for the gauge field is

$W_{\mu\nu}^{a}=\partial_{\mu}W^{a}_{\mu}-\partial_{\nu}W^{a}_{\mu}+g\epsilon^{abc}W^{b}_{\mu}W^{c}_{\nu}.$
(3)

The covariant derivatives are defined as

$\displaystyle D_{\mu}\varphi^{a}$
$\displaystyle=\partial_{\mu}\varphi^{a}+g\epsilon^{abc}W^{b}_{\mu}\varphi^{c},$
(4) $\displaystyle D_{\mu}\psi$ $\displaystyle=\partial_{\mu}\psi-
ig\frac{\sigma^{a}}{2}W^{a}_{\mu}\psi,$ (5)

and the potential is given by Kibble and Vachaspati (2015)

$V(\varphi,\psi)=\frac{\lambda}{4}(\varphi^{a}\varphi^{a}-\eta^{2})^{2}+\frac{\tilde{\lambda}}{2}(\psi^{\dagger}\psi-v^{2})^{2}+c\,\psi^{\dagger}\sigma^{a}\psi\varphi^{a}.$
(6)

As the first stage of symmetry breaking, we give vacuum expectation value to
the adjoint field while keeping $\psi=0$. The system admits ’t Hooft-Polyakov
monopoles ’t Hooft (1978); Polyakov (1975). As $\psi$ acquires vacuum
expectation value, the $SU(2)$ gauge symmetry is Higgsed to $U(1)$, and the
magnetic flux of a monopole is trapped into a tube which can end on an anti-
monopole. In this way, monopoles become confined. The dynamics of such
configuration are the main focus of this work.

Figure 1: A sketch of the initial monopole/anti-monopole initial
configuration.

The initial configuration utilized in this work is sketched in Fig. 1. The
monopoles are aligned along the $z-$axis at a distance $d$. In the figure
$\theta$ and $\overline{\theta}$ denote the respective monopole and
antimonopole position azimuthal angle. In the approximation when the distance
$d$ is much longer than the monopole size the string configuration can be
derived as follows. For $\overline{\theta}=0$, a monopole should be recovered
$\psi\propto(\cos{\theta/2},\sin{\theta/2}e^{i\phi})^{t}$ Nambu (1974, 1977),
while for $\theta=\pi$ an antimonopole should be obtained
$\psi\propto(\sin\overline{\theta}/2,\cos\overline{\theta}/2e^{i\phi})^{t}$,
$\phi$ being the polar angle. Therefore the string configuration is given by
Vachaspati and Field (1994); Vachaspati (2016); Saurabh and Vachaspati (2017)

$\psi\propto\begin{pmatrix}\sin(\theta/2)\sin(\bar{\theta}/2)e^{i\gamma}+\cos(\theta/2)\cos(\bar{\theta}/2)\\\
\sin(\theta/2)\cos(\bar{\theta}/2)e^{i\phi}-\cos(\theta/2)\sin(\bar{\theta}/2)e^{i(\phi-\gamma)}\end{pmatrix}$
(7)

with $\gamma$ accounting for the possibility of twisting the antimonopole
w.r.t. the monopole. In fact, for $\theta=\pi$, $\psi$ corresponds to the
above mentioned antimonopole under the shift $\phi\rightarrow\phi+\gamma$.
Above the configuration, for $\theta=\overline{\theta}=0$, $\psi=(v,0)^{t}$,
while below $\theta=\overline{\theta}=\pi$, $\psi=(ve^{i\gamma},0)^{t}$.
Finally between the two monopoles, $\theta=\pi$ and $\overline{\theta}=0$,
$\psi\propto(e^{i\phi},0)^{t}$ corresponding to the unit winding string.

Asymptotically the string is proportional to the positive eigenvector of the
third Pauli matrix. Since we choose $c<0$ in the last term of the potential
(6), the adjoint field direction $\hat{\varphi}^{a}$ of the monopoles can be
built asymptotically as Nambu (1974, 1977)

$\hat{\varphi}^{a}=\frac{1}{v^{2}}\psi^{\dagger}\tau^{a}\psi,\quad a=1,2,3$
(8)

where $\tau^{a}$ denotes the three Pauli matrices (see Fig. 2).

In the next section we analyze the monopole/anti-monopole configuration after
the first phase transition, ignoring the doublet. Although this was already
explored by Vachaspati and Saurabh Vachaspati (2016); Saurabh and Vachaspati
(2017), it serves as a useful exercise before turning to the symmetry broken
phase.

##  III Monopole/anti-monopole system

The explicit equations of motion can be found in Appendix A. From now, we work
in energy units of $\eta^{-1}$ and set the gauge coupling $g=1$. Thus
$\lambda$ is the parameter in the theory that controls the mass and size of
the monopoles.

For a (spherically symmetric) monopole field configurations, we use the
following ansatz

$\varphi^{a}=h(r)\hat{r}^{a},\quad
W^{a}_{i}=\frac{(1-k(r))}{r}\epsilon^{aij}\hat{r}^{j}$ (9)

where $r$ is the radial coordinate and $\hat{r}^{a}=r^{a}/|\vec{r}|$. Under
ansatz (9), the equations of motion become:

$\displaystyle h^{\prime\prime}(r)+\frac{2}{r}h^{\prime}(r)$
$\displaystyle=\frac{2}{r^{2}}k(r)^{2}h(r)-\lambda(h(r)^{2}-1)h(r),$ (10)
$\displaystyle k^{\prime\prime}(r)$
$\displaystyle=\frac{1}{r^{2}}(k(r)^{2}-1)k(r)+h(r)^{2}k(r),$ (11)

with asymptotic conditions

$\displaystyle h(r)\xrightarrow{r\rightarrow 0}0,\ $ $\displaystyle\
k(r)\xrightarrow{r\rightarrow 0}1,$ (12) $\displaystyle
h(r)\xrightarrow{r\rightarrow\infty}1,\ $ $\displaystyle\
k(r)\xrightarrow{r\rightarrow\infty}0.$ (13)

The above equations were solved numerically in order to obtain the monopoles
profile.

Finally the ansatz for the initial adjoint field configuration is given by

$\varphi^{a}=h(r_{m})h(\bar{r}_{m})\hat{\varphi}^{a},$ (14)

with $r_{m}$ ($\bar{r}_{m}$) denoting the monopole (anti-monopole) coordinate
center and $\hat{\varphi}^{a}$ is defined in (8).

Figure 2: Example of field configuration $\hat{\varphi}^{a}$ for twist
$\gamma=0$ according to Eq. (14) in the $yz$ plane. The circles denote the
position of the monopole and antimonopole.

An example of such configuration is shown in Fig. 2.

The stationary ansatz for the gauge fields considered by Vachaspati and
Saurabh Vachaspati (2016); Saurabh and Vachaspati (2017), follows from the
requirement that the covariant derivative of the Higgs isovector vanish at
spatial infinity, $D_{\mu}\hat{\varphi}|_{r\rightarrow\infty}=0$. This gives

$W^{a}_{\mu}=-(1-k(r_{m}))(1-k(\bar{r}_{m}))\epsilon^{abc}\hat{\varphi}^{b}\partial_{\mu}\hat{\varphi}^{c}.$
(15)

As expected, and verified in Saurabh and Vachaspati (2017), the monopole/anti-
monopole are attracted to each other due to a “magnetic” Coulomb-like
interaction (for distances much bigger than the monopole size). Moreover, the
potential energy is also affected by the initial system twist parametrized by
$\gamma$ Taubes (1982a) \- such a correction, however, is exponentially
suppressed at large distances. The verification of these properties served as
a valuable check of the numerics presented in this work.

##  IV Monopole/Anti-monopole connected by strings

We solve field equations (24)-(27), with an initial configuration of the
monopole/anti-monopole given by (14) and (15). Before starting the dynamical
evolution a numerical relaxation of the configuration was performed (c.f.
Saurabh and Vachaspati (2017)).

Asymptotically $\varphi^{a}=\eta\delta^{a3}$, implying that $\psi$, due to the
last interaction term in (6), needs to be proportional to the positive
$\sigma^{3}$ eigenvector $\langle|\psi|\rangle\propto(1,0)^{t}$, as correctly
implied by ansatz (7)111we used $c<0$ in our numerical simulations.

Above the pair, $\psi$ ansatz minimizes the interaction energy. As it crosses
the monopole, the configuration becomes singular at the south pole and its
phase is flipped by $2\pi$. It follows that between the two monopoles, along
the axial axes, $\psi\propto(0,e^{i\phi})^{t}$, $\phi$ being the polar angle.
By construction the associated winding number is one. Finally, as it reaches
the anti-monopole, the string flips by $2\pi$ again and it becomes
proportional to $\psi\propto(e^{i\gamma},0)^{t}$. While minimization of scalar
potential is independent on the value of $\gamma$, the same is not true for
both the kinetic terms and the gauge sector. In fact, $\gamma\neq 0$ generates
a repulsive interaction Saurabh and Vachaspati (2017) between the quark/anti-
quark pair which can be easily shown to be maximal for $\gamma=\pi$. Unless
otherwise stated, from now on we will focus on the case $\gamma=0$.

As the “magnetic” field of the pair is confined into tubes, the monopoles
accelerate towards each other, turning rapidly, relativistic and annihilate.

Figure 3: Evolution of the magnetic field norm. The picture is sliced at $y=0$
for better readability.

In Fig. 3, snapshots of the time evolution of the “magnetic” field norm are
shown. The string formation is observed at early times. For another visual
representation of the collapse dynamics see the following link (the sphaleron
dynamics described below is also shown there).

Effectively, the system behaves as a type II superconductor and the magnetic
field is neutralized by the longitudinal massive photon component which is
nothing but the eaten up Goldstone boson. Due to the compactness of the group,
as expected, no residual magnetic field is observed. The confining string,
being related to the breaking of the $U(1)$ subgroup, carries the same
magnetic flux as the original monopoles and is therefore able to fully
neutralize the magnetic field.

An example of string formation is shown in Fig. 7, where the initial phase of
$\psi$ was randomly chosen at each lattice point. The emergence of the winding
further justifies the chosen initial conditions mentioned at the beginning of
this section, which is adopted from here on wards.

Figure 4: Time evolution of the position of the cores of the monopole and
anti-monopole. The dots are numerical results while the dashed line is the
point-like solution (17). Here $a=\mu/m_{m}=0.16\eta$.

The dynamics is well approximated by the point-like study of Martin and
Vilenkin Martin and Vilenkin (1997). The system they considered has action

$I=-m_{m}\int ds_{1}-m_{m}\int ds_{2}-\Lambda^{2}\int dS,$ (16)

where the first two terms correspond to the monopole/anti-monopole worldline,
while the last term describes the string world-sheet for Nambu actionNambu
(1977). The equation of motion obtained by varying action (16) admits a
particular solution of an exactly straight string with the monopole/anti-
monopole pair accelerating towards its center. Their trajectories are given by
Martin and Vilenkin (1997):

$x(t)=\pm\frac{{\rm{sgn}}(t)}{a}(\gamma_{0}-\sqrt{1+(\gamma_{0}v_{0}-a|t|)^{2}})$
(17)

where $t_{0}=-\gamma_{0}v_{0}/a$, $a=\Lambda^{2}/m_{m}$, $v_{0}$ and
$\gamma_{0}=(1-v_{0}^{2})^{-1/2}$ are respectively the maximum velocity and
Lorentz factor of the monopoles, reached at $t=0$.
As shown in Fig. 4, the trajectory followed by the centers of the monopoles
(black dots) are nicely fitted by solution (17) (dashed line). As expected,
the monopoles, due to the constant acceleration, rapidly become relativistic.
However, differently from the point-like solution, they annihilate upon
reaching approximately zero distance. In contrast, in solution (17), they
simply pass through each other and oscillate.

##  V Similarities between confined quarks and monopoles

Our numerical analysis shows that after the first direct collision, the
monopole and antimonopole always annihilate and decay into waves, without
further oscillations. At first glance, this looks counterintuitive, since a
priori there is no reason for monopole and antimonopole not to pass through
each other stretching the string of the opposite polarity.

This phenomenon can be explained by the loss of coherence, or equivalently, by
the entropy suppression. The basic point is that, once the monopole and
antimonopole come on top of each other, the system looses coherence due to the
emission of waves. A further re-creation of monopole-antimonopole pair
connected by a string represents a process of a transition from a highly
energetic localized source into a coherent state of many soft quanta. Very
general arguments Dvali (2021) indicate that such a process is exponentially
suppressed, unless the microstate entropy of the coherent state is close to
saturating a certain upper bound set by unitarity.

In the present case, the microstate degeneracy of the coherent state
describing the pair of confined monopoles is not even close to this value.

Due to this, it is insufficient for matching the phase space occupied by the
waves of Higgs and gauge bosons. Correspondingly, rather than creating a
highly coherent configuration of monopoles confined by a long string, with
much higher probability, the system chooses to decay into those waves.

Interestingly, the reasoning of Dvali (2021) indicates that, in the analogous
regime, a similar behaviour is expected for the quarks confined by a long QCD
string.

In order to display the arguments in the language applicable to both systems,
let us consider a $SU(N_{c})$ gauge theory with a heavy quark transforming in
a fundamental representation. The term “heavy” implies that the mass of the
quark, $m_{q}$, is higher than the the QCD scale, $\Lambda$. No light quarks
are assumed to exist in the theory. The spin of the quark is not important for
our analysis.

The scale $\Lambda$ sets the boundary between the two descriptions. At
distances shorter than the QCD length, $\Lambda^{-1}$, the viable degrees of
freedom are gluons and quarks. At distances larger than $\Lambda^{-1}$, the
theory confines and the physical degrees of freedom are colorless composites
such as glueballs and mesons.

The phenomenon of confinement is often described as a dual version of the
Meissner effect Nambu (1974); Mandelstam (1976); ’t Hooft (1978). The essence
of this description is that the gluon “electric” field gets trapped in a flux
tube when sources are separated by a distance $d\gg\Lambda^{-1}$. This is
analogous to the behaviour of the magnetic field of monopoles in the
$U(1)$-Higgs phase. In the present discussion, the role of the sources will be
played by a heavy quark-anti-quark pair. When separated by a distance
$d\gg\Lambda^{-1}$, the pair is connected by a QCD string. The thickness of
the string is set by the QCD length $\Lambda^{-1}$. The tension of the string
is given by the QCD scale, $\mu\sim\Lambda^{2}$.

Thus, a quark-anti-quark pair connected by a QCD string is strikingly similar
to a monopole-anti-monopole connected by a magnetic flux tube. It is a long
standing question of how much of QCD physics is captured by this analogy.

In the processes in which the string stays longer than its thickness, the
similarity between the two systems is not surprising. Naively, this similarity
is not expected to extend to the processes in which the string shrinks to the
size of its thickness, since physics governing the two systems at such
distances are very different:

  * •

In the case of confined monopoles, the gauge theory is in the Higgs phase. The
degrees of freedom are gauge and Higgs bosons. The monopoles, as well as the
string connecting them, represent the solitonic objects.

  * •

On the other hand, in QCD the quarks are fundamental particles rather than
solitons. Also, the color string connecting them is not describable as a
solitonic solution of the classical equations of motion.

Despite these differences, we can find out that the two systems do exhibit
similarities even in certain processes controlled by physics below and around
the scale $\Lambda^{-1}$.

This similarity concerns the observation in our numerical analysis that in a
head-on collision the re-creation of a long string never takes place. This
behaviour is expected to be shared by the confined quarks in the similar
kinematic regime. Below we give supporting arguments. Our discussion will be
mostly qualitative. In order to keep close to the parameters of our numerical
analysis, we shall assume that $m_{q}\sim\Lambda$. Of course, we must assume
that quark masses are somewhat larger than $\Lambda$ in order for the long
string to be sufficiently stable against the quantum break-up via pair
nucleation. The system then is characterised by the two scales: $\Lambda$ and
the initial separation of quarks, $d$.

A useful physical way for understanding the suppression is via describing the
string-formation process in an effective theory of glueballs and mesons. As
said, these represent the correct physical degrees of freedom at distances
larger than the QCD length. In our parameter regime, the characteristic size
of an unexcited glueball or a meson is $\Lambda^{-1}$, and their masses are
$\sim\Lambda$.

Qualitatively, one can think of glueballs and mesons as of closed and open
strings of sizes $\Lambda^{-1}$. We must however remember that such strings
cannot be described as classical solitons. For this, it suffices to notice
that the Compton wavelengths of glueballs and mesons are $\sim\Lambda^{-1}$.
We shall assume that the initial energy is dominated by the energy of the
string, and that the contributions from the quark masses are subdominant. That
is, $d\gg 1/m_{q}\sim\Lambda^{-1}$. This implies that in the moment of the
collision the quarks are ultra-relativistic.

The string that is shrunk to the size of its width $\Lambda^{-1}$, represents
a blob of energy $\sim d\Lambda^{2}$. The energy per volume of quark Compton
wavelength, is much larger than the masses of quarks. Correspondingly, quarks
are effectively massless. Of course, the energy density is also much larger
than the masses of unexcited mesons and glueballs. Correspondingly, the shrunk
string is free to produce mesons and glueballs of high multiplicity, $n\sim
d\Lambda$.

Thus, system has to decide whether to produce a single long string or to
hadronize into $n$ short strings, in form of the glueballs and mesons. The
outcome is decided by the entropy of the final state. This entropy, in case of
a long string, is much less than the entropy of a generic state with high
multiplicity of mesons and glueballs.

This can be understood from the fact that in an effective theory of glueballs
and mesons, a long QCD string represents a coherent state. For
$d\gg\Lambda^{-1}$, such a state can be viewed as approximately-classical. The
mean occupation number of its constituents can be estimated as

$n\sim d\Lambda\,.$ (18)

Their characteristic de Broglie wavelengths are given by $\sim d$. Now, it is
intuitively clear that from all possible microstates of $n$ mesons and
glueballs, only a small fraction represents a highly coherent state of a long
string. Correspondingly, the phase space for string-formation is tiny.

For better estimate, let us follow the scattering process more closely. When
the initial string of size $d$ collapses to the size $\Lambda^{-1}$, it can be
viewed as a highly excited state of a meson. In this language a further
formation of a long string, represents a transition process from a highly
energetic quantum meson into a coherent state of $n$ soft quanta. According to
Dvali (2021), the probability of such a transition process is suppressed by
the following universal factor,

$\sigma\,\lesssim\,{\rm e}^{-n+S}\,.$ (19)

The first term ${\rm e}^{-n}$ encodes an exponential suppression
characteristic of the transitions from a single particle state into a state of
high occupation number $n$. The second factor, ${\rm e}^{S}$, comes from the
microstate entropy $S$ of the final $n$-particle state. Not surprisingly, the
entropy enhances the transition probability as it counts the number of the
available final states.

In the present case, the final macrostate we are looking for is a QCD string
of length $d$. For such a string, the degenerate microstates are given by all
possible distinct orientations. Their number is $\sim(d\Lambda)^{2}$ and the
corresponding microstate entropy is given by,

$S\sim\ln(d^{2}\Lambda^{2})\,.$ (20)

Taking into account (18), we obtain the following estimate for the suppression
factor

$\sigma\,\lesssim\,{\rm e}^{-d\Lambda+2\ln(d\Lambda)}\,.$ (21)

This exponentially small number explains why in head-on collisions the system
prefers to hadronize in a high multiplicity of short strings, rather than to
produce a single long one.

In other words, due to insufficient entropy, a long QCD string cannot saturate
the transition process from a highly energetic meson state. As argued in Dvali
(2021), for an unsuppressed production of an $n$-particle coherent state in a
transition process of the type, $1\rightarrow n$, its microstate entropy must
be $S\simeq n$. This is clear from the equation (19). The states with such a
high entropy were referred to as “saturons”. The expression (20) shows that
the entropy of the long string is much less than the saturation entropy. This
is the main reason for the suppression. Since the production of a long string
is exponentially suppressed, the system fills the remaining phase space by a
hadronization into a high multiplicity of mesons and glueballs.

Physics that governs dynamics in the similar kinematic process with monopoles
is analogous. After the collapse of a long string, the system forms a blob
with very high energy density. It is entropically preferred to dissipate this
energy in high multiplicity of Higgs and gauge bosons rather then in a highly
coherent state of monopoles connected by a long string.

Thus, the suppression of the monopoles-passage through each other, as well as
the inability of self-recreation of a long QCD string after its fist collapse,
represent the phenomena belonging to the same universality class of effects
taking place in high energy collisions. Their unifying feature is the
exponential suppression of production of coherent states with insufficient
microstate entropy Dvali (2021).

Another phenomenon belonging to the same universality class is the inability
of a monopole to go through a domain wall, discussed in Dvali  _et al._
(1998). In this theory, the domain wall represents a two-dimensional sheet in
which the Higgs field vanishes and the nonabelian (grand unification) symmetry
is restored. Due to this, when a monopole meets the wall, it is no longer
subject to the topological obstruction that keeps the entire magnetic charge
in one point. Correspondingly, monopole unwinds and the magnetic charge is
spread along the wall. 222 Of course, within specific grand unified models,
there can exist monopoles that are not affected topologically by a given
domain wall and thereby can pass through without spreading the magnetic charge
Brush  _et al._ (2015). These are not relevant for the present discussion.

In Dvali  _et al._ (1998) this effect was called the “erasure” of topological
defects. It was argued that, as a bonus, this mechanism can substantially
reduce the cosmological abundance of monopoles in grand unified theories. In
such theories the phase transition that forms monopoles, at the same time,
forms the unstable domain walls. The walls sweep away monopoles and then
disappear. A somewhat analogous effect of erasure was observed in
$2+1$-dimensions in interactions between skyrmions and domain walls
Kudryavtsev  _et al._ (1998), and, more recently, between vortices and domain
walls Dvali and Valbuena-Bermúdez .

Although it is both energetically and topologically permitted for the monopole
(or a skyrmion) to go through the wall and materialize on the other side, the
process is highly unlikely. This is also confirmed by more recent numerical
analysis of the monopole-wall system Bachmaier  _et al._ . Again, the reason
is the loss of coherence and the entropy suppression. In the example of Dvali
_et al._ (1998), the domain wall provides a support for unwinding the magnetic
charge in the same way as the antimonopole does in the present case. The loss
of coherence due to induced waves makes the recreation of monopole
configuration improbable 333In this respect, it is interesting to compare the
situation with scattering of solitons in $1+1$ dimensional theory, which are
known to be able to go through each other Rajaraman (1982). There are at least
two factors that make difference with the monopole case. First is
dimensionality which restricts the phase space for the loss of coherence.
Unlike monopoles, in case of kinks there is no transverse direction available
for emitting the waves. Correspondingly, the precursors that could potentially
take away coherence can only travel in the same directions as kinks and cannot
escape efficiently. The second factor is confinement. It will be interesting
to study the system within our parameter space. We thank Tanmay Vachaspasti
for raising this question..

From the point of view of Ref. Dvali (2021), all the considered cases fall in
the universal category of the processes in which the microstate entropy of the
final coherent state is not sufficient for compensating the exponential
suppression (19). As discussed, the same conclusion applies to a long QCD
string viewed as a coherent state in effective theory of mesons and glueballs.

This said, one can certainly imagine the parameter regimes in which the long
string can be re-created with less suppression. For example, let us assume
$m_{q}\gg\Lambda$ and the energy of the initial string $=\,3m_{q}$. In such a
case, the quark-antiquark pair moving apart after the first collision, can
stretch a string of the length $d\sim m_{q}/\Lambda^{2}$. This is because the
system does not possess the energy required for creating an additional quark-
antiquark pair, necessary for the string breakup. Of course, even in this
regime, quarks can annihilate into gluons which then will hadronize, without
recreating a long string. However, the annihilation cross-section is
perturbative since the scattering length is much shorter than $\Lambda^{-1}$.

Also, in case of monopoles, behaviour can be very different in the regime in
which the monopole size is much smaller than the width of the string. In this
situation, for non-zero impact parameter, the monopole and antimonopole can
miss each other and the string can oscillate, while shrinking to the size of
its width. One may expect that the efficiency of annihilation requires that
the impact parameter is less than the size of the monopole core. In this case,
the annihilation cross-section presumably will be suppressed by a geometric
factor444We thank Alex Vilenkin for commenting.. However, the understanding of
this regime requires a separate study.

##  VI Gravitational Waves

Figure 5: Angularly integrated power emitted $P_{\rm n}$ according to (22).The
amplitude was normalized w.r.t. to the lowest frequency comparable to the
initial distance between monopole and antimonopole, up to frequencies
comparable to the monopoles size $P_{\rm n}\propto n^{-1}$ (shaded region).

The radiated power at frequency $\omega_{n}=2\pi n/T$ ($T$ being the collapse
time)555For practical purposes we chose $T=2d\eta^{-1}$, $d$ being the lattice
distance between monopoles., per unit solid angle in the direction k,
$|\textbf{k}|=\omega_{n}$ can be computed as Weinberg (1972)

$\begin{split}P&=\sum_{n}P_{n}=\sum_{n}\int d\Omega\frac{dP_{n}}{d\Omega},\\\
\frac{dP_{n}}{d\Omega}&=\frac{G\,\omega_{n}^{2}}{\pi}\left(T_{\mu\nu}^{*}(\omega_{n},\textbf{k})T^{\mu\nu}(\omega_{n},\textbf{k})-\frac{1}{2}|T_{\mu}^{\mu}(\omega_{n},\textbf{k})|^{2}\right),\end{split}$
(22)

where the energy-momentum tensor in momentum space is given by

$T^{\mu\nu}(\omega_{n},\textbf{k})=\frac{1}{T}\int_{0}^{T}dt\,e^{i\omega_{n}t}\int
d^{3}\textbf{x}e^{-i\textbf{k}\cdot\textbf{x}}T^{\mu\nu}(t,\textbf{x}).$ (23)

The radiated spectrum obtained from our simulation is shown in Fig. 5, where
the dependence of $(P_{n}\cdot n)$ vs $n$ is shown over logarithmic intervals.
We normalised the plot w.r.t $P_{2}$, corresponding to a wavelength comparable
to half of the monopoles initial distance. Direct comparison with the point-
like study of Martin and Vilenkin Martin and Vilenkin (1997) shows interesting
salient features.

As expected, for low frequencies, $n\lesssim 7$ (red shaded area in the plot),
corresponding to an emitted radiation when the monopoles are still far from
each other, our result is well approximated by the point-like study. In fact,
a spectrum behaviour $P_{n}\propto n^{-1}$ is observed similar to the point-
like case Martin and Vilenkin (1997). However, the spectrum is different when
the monopoles annihilate. In fact, as shown in Fig. 5 the monopole
annihilation boosts the radiation spectrum for frequencies comparable to the
monopoles and string width. Even though it was not possible to find an exact
scaling of the spectrum in this region, this behaviour is different from the
point-like case where no enhancement is observed.

The corrections to the spectrum of gravitational waves coming from the finite
widths of the string and monopoles, can be of observational interest in cases
when this length-scale is macroscopic. For example, a string of the tension
$\mu=(10^{16}{\rm GeV})^{2}$ can have a width of order km, provided the gauge
coupling of the theory is $g\sim 10^{-35}$. Existence of such super-weak gauge
interaction in some hidden sector of the theory is fully consistent with all
known laws of physics. Of course, the parameters in our simulation could not
be taken with such extreme values. However, the dynamic of the string can be
significantly affected due to the difference of Higgs and magnetic cores as in
the regimes discussed in Adelberger  _et al._ (2007). Correspondingly,
applications of our results for such parameter ranges, can only be viewed as
indicative.

##  VII Sphaleron

Figure 6: Evolution of the magnetic and scalar cores of the monopole/anti-
monopole pair for initial twist $\gamma=\pi$.

So far we have been focused on the untwisted configuration i.e., $\gamma=0$.
This corresponds to a minimum of the string energy. While it is obvious from
(7) that rotating $\gamma$ by $2\pi$ gives back the same configuration, it is
straightforward to see how the energy varies with $\gamma$ (at fixed distance
$d$) and find that $\gamma=\pi$ is an energy maximum.

The role of the twist $\gamma$ has a deep topological meaning. There exists a
static bound state configuration known as “sphaleron” – see, eg., Manton and
Sutcliffe (2004) for a review and Klinkhamer and Manton (1984); Manton (1983);
Brihaye and Kunz (1994) for solutions in the electroweak sector. This bounded
configuration interpolates between vacua of different Chern-Simons number. It
corresponds to the maximum of the non-contractible loop interpolating between
two such minima and has Chern-Simons number $1/2$ Manton (1983).

It was argued in Taubes (1982b) that such unstable solutions can be understood
as a monopole and a rotated antimonopole. For the maximal twist $\gamma=\pi$,
the Chern-Simons number of (15) corresponds to that of a sphaleron as it can
be easily checked.

Refs. Saurabh and Vachaspati (2017); Kleihaus and Kunz (2000) have numerically
verified the behaviour of this object in the unconfined case. They observed
that the twist $\gamma=\pi$ prevents the monopole-antimonopole annihilation.
To our knowledge, the behaviour of this configuration in the confined phase
has not been studied so far.

In Fig. 6, the position of the two monopoles core centers is shown as a
function of time. As it can be seen, at initial time the two magnetic cores
start to accelerate towards each other due to the flux tubes connecting them,
dragging the scalar core along and the system becomes relativistic.

The dynamics is analogous to the untwisted one, c.f. Fig. 4, up to the
collapse moment. While in the untwisted case the two monopoles annihilate
right away, in the twisted case they do not. Instead, the magnetic cores repel
each other, slow down and bounce back.

From there on the two monopoles have a bouncing behaviour, and dissipate
energy up to settling to a constant distance where the confining energy is
balanced by the repulsive twist energy. Eventually, the pair fully
annihilates. Concomitantly a deviation from axial symmetry is observed. Given
the axial symmetry of the initial configuration, and the fact that the
dynamics should preserve such symmetry, we believe this to be a numerical
artefact.

In Fig. 6, $\Lambda\simeq\eta$ was used. For lower value of this parameter the
bouncing sizes, as well as distance of the bound state were observed to be
larger. For a different initial distance, 2-dimensional slices of the dynamics
can be found at the following link.

For $\gamma\neq\pi$, no bounce is observed, as the system can always untwist
in the direction of favourable minimum, thereby annihilating right away.

##  VIII Conclusion and outlook

In this article we numerically studied the confinement of a $SU(2)$
monopole/anti-monopole pair. Initially the pair is in a dipolar configuration
as in Fig. 2. The confinement is instantaneously imposed by setting an extra
complex doublet field in its Higgs-phase, therefore breaking the residual
$U(1)$ and giving mass to the previously massless “photon”.

As shown in Fig. 4, very quickly the monopoles become relativistic due to
confinement. Taking initial random phases for the confining scalar field, it
is possible to see the dynamical emergence of the tube as shown in Fig. 7.

The GW spectrum is also computed and is compared with the point-like study of
Martin and Vilenkin Martin and Vilenkin (1997). Before the collision and for
the wavelengths longer than the relevant widths of the system (i.e., monopoles
and string width) we find perfect agreement. Namely, we confirm a power
spectrum $P_{n}\propto n^{-1}$, $n$ denoting the frequency number. As
previously mentioned, this type of spectrum can lead to a flat density of GWs
$\Omega_{GW}$ across several orders of the frequency, therefore making it
interesting from a phenomenological perspective in view of future refinement
of the stochastic GW background hints obtained from pulsar timing arrays
Arzoumanian  _et al._ (2020); Antoniadis  _et al._ (2022).

However, in our parameter regime, in which the scales of string and monopoles
are of the same order, we observe the following differences from the point-
like case.

Firstly, an enhancement of the GW spectrum for wavelengths comparable to the
monopoles and string size as shown in Fig. 5, therefore providing finite-width
corrections to the point-like case Martin and Vilenkin (1997). This can be of
phenomenological interest for the strings of a macroscopic width. Notice that
such strings can have a high tension provided the gauge coupling is
sufficiently weak and can produce intense gravitational waves.

Secondly, in our numerical analysis we have observed an interesting effect.
After the first collision, the monopoles annihilate without any further
oscillations. We gave a qualitative explanation to this observation in terms
of a general phenomenon of suppressed production of low-entropy coherent
states in a collision process Dvali (2021). In the present case, the loss of
coherence in the monopole-anti-monopole collision, makes the recreation of a
monopole-string system unlikely due to an insufficient entropy of such a
state.

Applying the general reasoning of Dvali (2021), we argued that in the similar
parameter regime this behaviour must be shared by heavy quarks confined by a
long QCD string. There too, after the collapse of a straight long string,
instead of re-stretching it in an oscillatory mode, the system prefers to
directly decay into high multiplicity of mesons and glueballs. This similarity
extends the connection between the confined quarks and monopoles to the domain
of processes controlled by the short-distance physics.

Aknowledgements: We thank Tanmay Vachaspati and Alex Vilenkin for useful
discussions and valuable comments.
This work was supported in part by the Humboldt Foundation under Humboldt
Professorship Award, by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC-2111 -
390814868, and Germany’s Excellence Strategy under Excellence Cluster Origins.

## References

  * Langacker and Pi (1980) Paul Langacker and So-Young Pi, “Magnetic Monopoles in Grand Unified Theories,” Phys. Rev. Lett. 45, 1 (1980).
  * Lazarides  _et al._ (1982) George Lazarides, Q. Shafi, and T. F. Walsh, “Cosmic Strings and Domains in Unified Theories,” Nucl. Phys. B 195, 157–172 (1982).
  * Vilenkin (1982) A. Vilenkin, “COSMOLOGICAL EVOLUTION OF MONOPOLES CONNECTED BY STRINGS,” Nucl. Phys. B 196, 240–258 (1982).
  * Vilenkin and Shellard (2000) A. Vilenkin and E. P. S. Shellard,  _Cosmic Strings and Other Topological Defects_ (Cambridge University Press, 2000).
  * Nielsen and Olesen (1973) Holger Bech Nielsen and P. Olesen, “Vortex Line Models for Dual Strings,” Nucl. Phys. B 61, 45–61 (1973).
  * Dvali  _et al._ (2021) Gia Dvali, Florian Kühnel, and Michael Zantedeschi, “Primordial Black Holes from Confinement,” (2021) arXiv:2108.09471 [hep-ph] .
  * Matsuda (2009) Tomohiro Matsuda, “Primordial black holes from monopoles connected by strings,” Astropart. Phys. 30, 333–337 (2009), arXiv:hep-ph/0509061 .
  * Martin and Vilenkin (1997) Xavier Martin and Alexander Vilenkin, “Gravitational radiation from monopoles connected by strings,” Phys. Rev. D 55, 6054–6060 (1997), arXiv:gr-qc/9612008 .
  * Arzoumanian  _et al._ (2020) Zaven Arzoumanian _et al._ (NANOGrav), “The NANOGrav 12.5 yr Data Set: Search for an Isotropic Stochastic Gravitational-wave Background,” Astrophys. J. Lett. 905, L34 (2020), arXiv:2009.04496 [astro-ph.HE] .
  * Antoniadis  _et al._ (2022) J. Antoniadis _et al._ , “The International Pulsar Timing Array second data release: Search for an isotropic Gravitational Wave Background,” Mon. Not. Roy. Astron. Soc. 510 (2022), 10.1093/mnras/stab3418, arXiv:2201.03980 [astro-ph.HE] .
  * Leblond  _et al._ (2009) Louis Leblond, Benjamin Shlaer, and Xavier Siemens, “Gravitational Waves from Broken Cosmic Strings: The Bursts and the Beads,” Phys. Rev. D 79, 123519 (2009), arXiv:0903.4686 [astro-ph.CO] .
  * Weinberg (1974) Steven Weinberg, “Gauge and Global Symmetries at High Temperature,” Phys. Rev. D 9, 3357–3378 (1974).
  * ’t Hooft (1974) G. ’t Hooft, “Magnetic Monopoles in Unified Theories,” in  _17th International Conference on High-Energy Physics_ (1974) pp. III.84–85.
  * Polyakov (1975) Alexander M. Polyakov, “Compact Gauge Fields and the Infrared Catastrophe,” Phys. Lett. B 59, 82–84 (1975).
  * Dvali (2021) Gia Dvali, “Entropy Bound and Unitarity of Scattering Amplitudes,” JHEP 03, 126 (2021), arXiv:2003.05546 [hep-th] .
  * Dvali  _et al._ (1998) G. R. Dvali, Hong Liu, and Tanmay Vachaspati, “Sweeping away the monopole problem,” Phys. Rev. Lett. 80, 2281–2284 (1998), arXiv:hep-ph/9710301 .
  * Copeland  _et al._ (2004) Edmund J. Copeland, Robert C. Myers, and Joseph Polchinski, “Cosmic F and D strings,” JHEP 06, 013 (2004), arXiv:hep-th/0312067 .
  * Dvali and Vilenkin (2004) Gia Dvali and Alexander Vilenkin, “Formation and evolution of cosmic D strings,” JCAP 03, 010 (2004), arXiv:hep-th/0312007 .
  * Kibble and Vachaspati (2015) T. W. B. Kibble and Tanmay Vachaspati, “Monopoles on strings,” J. Phys. G 42, 094002 (2015), arXiv:1506.02022 [astro-ph.CO] .
  * ’t Hooft (1978) Gerard ’t Hooft, “On the Phase Transition Towards Permanent Quark Confinement,” Nucl. Phys. B 138, 1–25 (1978).
  * Nambu (1974) Yoichiro Nambu, “Strings, Monopoles and Gauge Fields,” Phys. Rev. D 10, 4262 (1974).
  * Nambu (1977) Yoichiro Nambu, “String-Like Configurations in the Weinberg-Salam Theory,” Nucl. Phys. B 130, 505 (1977).
  * Vachaspati and Field (1994) Tanmay Vachaspati and George B. Field, “Electroweak string configurations with baryon number,” Phys. Rev. Lett. 73, 373–376 (1994), arXiv:hep-ph/9401220 .
  * Vachaspati (2016) Tanmay Vachaspati, “Monopole-Antimonopole Scattering,” Phys. Rev. D 93, 045008 (2016), arXiv:1511.05095 [hep-th] .
  * Saurabh and Vachaspati (2017) Ayush Saurabh and Tanmay Vachaspati, “Monopole-antimonopole Interaction Potential,” Phys. Rev. D 96, 103536 (2017), arXiv:1705.03091 [hep-th] .
  * Taubes (1982a) Clifford Henry Taubes, “The Existence of a Nonminimal Solution to the SU(2) Yang-Mills Higgs Equations on R**3,” Commun. Math. Phys. 86, 257 (1982a).
  * Mandelstam (1976) S. Mandelstam, “Vortices and Quark Confinement in Nonabelian Gauge Theories,” Phys. Rept. 23, 245–249 (1976).
  * Brush  _et al._ (2015) Micah Brush, Levon Pogosian, and Tanmay Vachaspati, “Magnetic monopole—domain wall collisions,” Phys. Rev. D 92, 045008 (2015), arXiv:1505.08170 [hep-th] .
  * Kudryavtsev  _et al._ (1998) Alexander Evgenyevich Kudryavtsev, B. M. A. G. Piette, and W. J. Zakrzewski, “Skyrmions and domain walls in (2+1)-dimensions,” Nonlinearity 11, 783–795 (1998), arXiv:hep-th/9709187 .
  * (30) G. Dvali and J.. Valbuena-Bermúdez, “Erasure of vortices,” in preparation .
  * (31) M. Bachmaier, G. Dvali, and J.. Valbuena-Bermúdez, “Radiation emission during the erasure of magnetic monopoles,” in preparation .
  * Rajaraman (1982) R. Rajaraman,  _Solitons and Instantons. An Introduction to Solitons and Instantons in Quantum Field Theory_ (1982).
  * Weinberg (1972) Steven Weinberg,  _Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity_ (John Wiley and Sons, New York, 1972).
  * Adelberger  _et al._ (2007) Eric Adelberger, Gia Dvali, and Andrei Gruzinov, “Photon mass bound destroyed by vortices,” Phys. Rev. Lett. 98, 010402 (2007), arXiv:hep-ph/0306245 .
  * Manton and Sutcliffe (2004) N. S. Manton and P. Sutcliffe,  _Topological solitons_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2004).
  * Klinkhamer and Manton (1984) Frans R. Klinkhamer and N. S. Manton, “A Saddle Point Solution in the Weinberg-Salam Theory,” Phys. Rev. D 30, 2212 (1984).
  * Manton (1983) N. S. Manton, “Topology in the Weinberg-Salam Theory,” Phys. Rev. D 28, 2019 (1983).
  * Brihaye and Kunz (1994) Yves Brihaye and Jutta Kunz, “On axially symmetric solutions in the electroweak theory,” Phys. Rev. D 50, 4175–4182 (1994), arXiv:hep-ph/9403392 .
  * Taubes (1982b) Clifford Henry Taubes, “The Existence of a Nonminimal Solution to the SU(2) Yang-Mills Higgs Equations on R**3,” Commun. Math. Phys. 86, 257 (1982b).
  * Kleihaus and Kunz (2000) Burkhard Kleihaus and Jutta Kunz, “A Monopole - anti-monopole solution of the SU(2) Yang-Mills-Higgs model,” Phys. Rev. D 61, 025003 (2000), arXiv:hep-th/9909037 .
  * Israeli and Orszag (1981) Moshe Israeli and Steven A. Orszag, “Approximation of radiation boundary conditions,” Journal of Computational Physics 41, 115–135 (1981).

##  Appendix A Fields equation

###  A.1 Field equations

The field equations can be written as

$\displaystyle\partial_{t}^{2}\varphi^{a}$
$\displaystyle=\nabla^{2}\varphi^{a}-g\epsilon^{abc}\partial_{i}\phi^{b}W_{i}^{c}-g\epsilon^{abc}(D_{i}\varphi)^{b}W_{i}^{c}-\lambda(\varphi^{b}\varphi^{b}-\eta^{2})\varphi^{a}-g\epsilon^{abc}\varphi^{b}\Gamma^{c}-c\psi^{\dagger}\sigma^{a}\psi,$
(24) $\displaystyle\partial_{t}^{2}\psi^{\alpha}$
$\displaystyle=\nabla^{2}\psi^{\alpha}-\frac{g^{2}}{4}W_{i}^{a}W_{i}^{a}\psi^{\alpha}-i\frac{g}{2}\Gamma^{a}(\sigma^{a}\psi)^{\alpha}-igW^{a}_{i}(\sigma^{a}\partial_{i}\psi)^{\alpha}-c\varphi^{a}\sigma^{a}\psi^{\alpha}-\tilde{\lambda}(\psi^{\dagger}\psi-v^{2})\psi^{\alpha},$
(25) $\displaystyle\partial_{t}W^{a}_{0i}$
$\displaystyle=\nabla^{2}W_{i}^{a}+g\epsilon^{abc}W_{j}^{b}\partial_{j}W_{i}^{c}-g\epsilon^{abc}W^{b}_{j}W^{c}_{ij}-D_{i}\Gamma^{a}-g\epsilon^{abc}\varphi^{b}(D_{i}\varphi)^{c}-\frac{1}{2}g^{2}\psi^{\dagger}\psi
W_{i}^{a}-ig\psi^{\dagger}\sigma^{a}\partial_{i}\psi,$ (26)
$\displaystyle\partial_{t}\Gamma^{a}$
$\displaystyle=\partial_{i}W_{0i}^{a}-g_{p}^{2}\left[\partial_{i}(W_{0i}^{a})+g\epsilon^{abc}W^{b}_{i}W^{c}_{0i}+g\epsilon^{abc}\varphi^{b}(D_{t}\varphi)^{c}+ig\psi^{\dagger}\sigma^{a}\partial_{t}\psi\right],$
(27)

where we are using the temporal gauge, $W^{a}_{0}=0$,
$\Gamma^{a}=\partial_{i}W^{a}_{i}$ are introduced as new variables, and
$g^{2}_{p}=1.5$ is a numerical parameter that we can choose to ensure
numerical stability. The equations were evolved with a Cranck-Nicolson
leapfrog algorithm combined with absorbing boundary conditions Israeli and
Orszag (1981).

Figure 7: Phase of the confining scalar field in the xy plane at the middle of
the string height.


# Charged Particle Multiplicity and Pseudorapidity Density Measurements in pp
collisions with ALICE at the LHC

INFN Bologna & CERN
E-mail

###### Abstract:

These proceedings describe the charged-particle pseudorapidity densities and
multiplicity distributions measured by the ALICE detector in pp collisions at
$\sqrt{s}=0.9$ and 7 TeV in specific phase space regions. The pseudorapidity
range $|\eta|<0.8$, together with $p_{\rm T}$ cuts at 0.15, 0.5 and 1
$\mathrm{GeV}/c$ is considered. The classes of events considered are those
having at least one charged particle in the kinematical ranges just described.
The results obtained by ALICE are compared to Monte Carlo predictions.

##  1 Introduction

The ALICE results on charged-particle pseudorapidity density
($\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$) and multiplicity distributions in
pp collisions at $\sqrt{s}=0.9$ and 7 TeV presented in this document derive
from an analysis carried out with an event and track selection specially
chosen to make comparison with Monte Carlo calculations to allow for a better
Monte Carlo tuning. Tracks reconstructed as coming from primary
particles111The ensemble of primary charged particles includes those produced
in the collision and their decay products, excluding weak decays from strange
particles. in the Inner Tracking System (ITS) and in the Time Projection
Chamber (TPC) of ALICE have been used, and kinematical phase space regions
defined in $\eta$ ($|\eta|<0.8$) and $p_{\rm T}$ ($p_{\rm T}$ $>$ $p_{\rm
T,cut}$, with $p_{\rm T,cut}$ = 0.15, 0.5 and 1.0 $\mathrm{GeV}/c$) have been
considered. The first $p_{\rm T,cut}$ corresponds to the $p_{\rm T}$cutoff at
which the ALICE global tracking efficiency (i.e. including both ITS and TPC)
reaches $\sim 50\%$ and stays approximately constant ($\sim 70\\--75\%$) for
higher $p_{\rm T}$ [1], while the higher $p_{\rm T,cut}$ values allow a
comparison with measurements performed by ATLAS and CMS. The pseudorapidity
density and multiplicity distributions are measured for all charged particles
in the given $p_{\rm T}-\eta$ region for those events which have at least one
charged particle. This introduces a so-called “hadron level definition” of the
event class considered, named INEL$>0_{|\eta|<0.8,p_{\rm T}>p_{\rm T,cut}}$
hereafter.

##  2 The ALICE experiment and the data samples

The ALICE experiment consists of a set of different detectors placed in a
solenoidal magnetic field of 0.5 T (the central barrel) plus other detectors
outside. Details about the various subsystems can be found in [3]. For the
analysis presented herein, tracks reconstructed in the ALICE central barrel by
the ITS and TPC detectors were used, while the triggering and event selection
relied on both the ITS and VZERO detectors.

The data samples used consisted of Minimum Bias pp events collected in 2009
and 2010. The Minimum Bias trigger was defined as a signal in either one of
the two ALICE VZERO hodoscopes, or in the ITS pixel detector (one out of
three). A coincidence with the signals from the two beam pick-up counters
(BPTX) was also required to select the events and remove the background. In
such conditions, about $110000$ (collected in 2009) and $2.2\times 10^{6}$
events (collected in 2010) were used for the charged-particle pseudorapidity
density analysis at $\sqrt{s}=0.9$ and 7 TeV respectively. The multiplicity
distributions were obtained from approximately $2.9\times 10^{6}$ and
$2.7\times 10^{6}$ events (all collected in 2010) at $\sqrt{s}=0.9$ and 7 TeV
respectively.

In addition to requiring the Minimum Bias trigger in the collision and the
reconstruction of the primary vertex, a preselection of the events aimed at
reducing the beam background was applied based on the information from the
VZERO detector and on the correlation between the number of hits and the so-
called tracklets222A tracklet is built combining a pair of hits in the two
innermost ITS layers. found in the the two innermost ITS layers, corresponding
to the Silicon Pixel Detector (SPD, see also [4]). A selection on the vertex
was also applied, requiring it to be obtained either from the tracks
reconstructed from the TPC and the ITS detectors, or, in case this was not
available, from the tracklets found in the SPD detector (see also [4]), using
the SPD information. Moreover, only events for which the vertex position along
the $z$ coordinate ($vtx_{z}$) was such that $|vtx_{z}|<10$ cm were accepted.
Finally, only those events with at least one reconstructed track in the
kinematical region defined by the pseudorapidity interval $|\eta|<0.8$ and
$p_{\rm T}>p_{\rm T,cut}$ ($p_{\rm T,cut}=0.15,0.5,1$ $\mathrm{GeV}/c$) were
considered. For the 2010 data, an additional event selection criterion was
used in order to reduce the contribution from pile-up events, removing those
identified as coming from pile-up based on the SPD information. This sample
was then corrected back to the $\rm{INEL}>0_{|\eta|<0.8,p_{\rm T}>p_{\rm
T,cut}}$ “hadron level definition” described in Sec. 1.

##  3 Analysis strategy

The tracks used in the analysis are those reconstructed by the ALICE central
global tracking [5], which is based on the Kalman filter technique [5, 6].
Track selection criteria (cuts) have been applied in order to maximize the
tracking efficiency and minimize the contamination from secondaries and fake
tracks, as described in [2]. The raw
$\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ and multiplicity distributions
obtained from data were corrected using PYTHIA Monte Carlo simulations as
described in Sec. 3.1 and Sec. 3.2. The GEANT3 particle transport package was
used together with a detailed description of the geometry of the experiment,
and of the detector and electronics response. Moreover, the simulation was set
to reproduce the conditions of the LHC beam and of the detectors (in terms of
vertex position, calibration, alignment and response) at the time the data
under study were collected. The underestimate in the Monte Carlo simulations
of the event strangeness content with respect to that found in the data was
also taken into account during the correction phase. Further information about
the analysis strategy and the corrections applied can be found in [2].

###  3.1 Corrections for the charged-particle pseudorapidity density analysis

The charged-particle pseudorapidity distribution is given by the expression

$\frac{1}{N_{\rm{ev}}}\frac{{\rm{d}}N_{\rm{ch}}}{\rm{d}\eta}.$ (1)

where $N_{\rm{ev}}$ corresponds to the total number of events that belong to
the INEL$>0_{|\eta|<0.8,p_{\rm T}>p_{\rm T,cut}}$ class. The corrections
applied to the raw $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ distribution are
of three types: a track-to-particle correction is needed, in order to take
into account the difference between the measured tracks and the true charged
primary particles coming from acceptance effects, detector and reconstruction
efficiency; a second correction is applied, to account for the fact that
events without a reconstructed vertex are not considered; finally, the bias
due to the INEL$>0_{|\eta|<0.8,p_{\rm T}>p_{\rm T,cut}}$ event selection used
is considered.

###  3.2 Corrections for multiplicity distribution analysis

For the multiplicity analysis, the correction procedure is twofold. First, an
unfolding technique is applied in order to account for the fact that due to
the efficiency, acceptance and detector effects, the measured multiplicity
spectrum is distorted from the true one. In addition, vertex reconstruction
and event selection efficiency need to be taken into consideration, in a
similar way to the $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ analysis.

The unfolding procedure used for the analysis presented here is described in
[2]. It is based on a $\chi^{2}$-minimization approach, where the $\chi^{2}$
function used to evaluate the “guessed” unfolded spectrum $U$, can be written
as:

$\hat{\chi}^{2}(U)=\sum_{m}\left(\frac{M_{m}-\sum_{t}R_{mt}U_{t}}{e_{m}}\right)^{2}.$
(2)

Here, $M_{m}$ is the measured distribution at true multiplicity $t$ with error
$e_{m}$, and $R_{mt}$ is the response matrix element for measured multiplicity
$m$ and true multiplicity $t$ which encodes the probability that an event with
true multiplicity $t$ is measured with multiplicity $m$. Because this
minimization suffers from oscillations in the unfolded spectrum, a constraint
$P(U)$ was added to the $\chi^{2}$ function, favouring a certain shape in the
unfolded distribution [7], following the same approach as described and
discussed in [8]. The constraint $P(U)$ is called a $regularization$ $term$,
and the new function to be minimized becomes:

$\chi^{2}(U)=\hat{\chi}^{2}(U)+\beta P(U).$

As written in the formula, $P(U)$ depends only on the unfolded spectrum $U$.
$\beta$ is the weight of the regularization term.

For the unfolding procedure, a parameterization of the response matrix was
used, in order to avoid statistics issues at high multiplicities and in the
tails of the distributions for a given fixed true multiplicity. More details
on the choice of the regularization function and on the parameterization can
be found in [2].

###  3.3 Systematic Uncertainties

Various sources of systematic uncertainties have been taken into account, most
of which are common to the two analyses. They include:

  * •

track quality cuts variation;

  * •

tracking efficiency;

  * •

material budget;

  * •

particle species relative fraction;

  * •

event type (Single, Double, Non-Single Diffractive) relative composition;

  * •

pile-up;

  * •

Monte Carlo generator dependence of the corrections;

Moreover, for the multiplicity distribution analysis only, the following
sources of systematic uncertainties have been studied:

  * •

choice of the regularization function and weight;

  * •

bias introduced by the regularization [9];

  * •

unfolding dependence on the $\langle p_{\rm T}\rangle$ as a function of
multiplicity.

The complete description of the evaluation of the systematic uncertainties is
discussed in [2].

##  4 Results

Figure 1 shows the final charged particle
$\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ for the
$\rm{INEL}>0_{|\eta|<0.8,p_{\rm T}>p_{\rm T,cut}}$ classes of events. Here,
for the sake of brevity, only the results for the $p_{\rm T,cut}=0.15$ GeV/$c$
at $\sqrt{s}=0.9$ (left panel) and $p_{\rm T,cut}=0.5$ GeV/$c$ at 7 TeV
(right) are shown. The complete set of results is presented in [2].
Predictions from Monte Carlo generators are superimposed on the distributions.
They are indicated as follows:

  * •

PYTHIA-6

    * –

Atlas CSC (tune 306 [10]);

    * –

D6T (tune 109 [11]);

    * –

A (tune 100 [12]);

    * –

Perugia-0 (tune 320 [13]);

    * –

Perugia-2011 (tune 350 [14]).

  * •

PYTHIA-8

    * –

Pythia8 (tune 1 [15]);

    * –

Pythia8 (tune 4C) [16]);

  * •

PHOJET ([17]);

  * •

EPOS LHC ([18]).

The bottom panels of the figures show the ratio between the data and the Monte
Carlo predictions.

|
---|---
Figure 1: $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ versus $\eta$ obtained at
$\sqrt{s}=0.9$ (left) $p_{\rm T}>0.15$ $\mathrm{GeV}/c$ (left) and
$\sqrt{s}=7$ (left) $p_{\rm T}>0.5$ $\mathrm{GeV}/c$ for $|\eta|<0.8$
normalized to the $\rm{INEL}>0_{|\eta|<0.8,p_{\rm T}>p_{\rm T,cut}}$ event
class. The predictions from different Monte Carlo generators are also shown.
The grey bands represent the systematic uncertainties on the data. Bottom
panels: data to Monte Carlo prediction ratios for the different generators
considered. Here, the grey bands represent the total (statistical +
systematic) uncertainty on the data.

The charged particle multiplicity distribution results are shown in Fig. 2 for
the same cases as in Fig. 1. For the sake of visibility, the Monte Carlo
comparison (which was carried out with the same generators and tunes used for
Fig. 1) is shown for both figures in two different panels, as explained in the
legends.

For both analyses, the results show that, in general, a universal trend in
terms of comparison between the ALICE data and Monte Carlo calculations cannot
be identified. At different centre-of-mass energies and with different values
of $p_{\rm T,cut}$, the different models describe the data differently, and
one tune that gives reasonable comparison to data in one case fails in the
others. Moreover, for the multiplicity distributions, the level of
agreement/disagreement varies significantly as a function of multiplicity.

|
---|---
|
Figure 2: Multiplicity distributions for the analysis at $\sqrt{s}=0.9$ TeV,
$p_{\rm T,cut}=0.15$ GeV/$c$ (top) and at $\sqrt{s}=7$ TeV, $p_{\rm
T,cut}=0.5$ GeV/$c$ (bottom). In the left and right panels, the data are
compared with different Monte Carlo expectations, as indicated in the key. In
the upper panels, data are shown with both statistical (black line) and
systematic (grey band) uncertainties. The grey bands in the lower panels,
where the data/Monte Carlo ratios are presented, correspond to the total
uncertainty on the final results.

##  5 Conclusions

The charged particle pseudorapidity density and multiplicity distributions
measured by ALICE at $\sqrt{s}=0.9$ and 7 TeV with charged tracks
reconstructed in the ITS and TPC detectors have been presented. A $p_{\rm
T,cut}$ (with $p_{\rm T,cut}$ = 0.15, 0.5, 1.0 GeV/$c$) was used in order to
characterize the class of events to be considered for the analysis, namely the
$\rm{INEL}>0_{|\eta|<0.8,p_{\rm T}>p_{\rm T,cut}}$ class, defined requiring at
least one charged particle with $p_{t}>p_{\rm T,cut}$ in $|\eta|<0.8$. While
the lowest $p_{\rm T,cut}$ allows the most inclusive measurement for ALICE
with global tracks, the 0.5 and 1.0 GeV$/c$ cuts were chosen together with the
other LHC collaborations (ATLAS, CMS) to allow for the comparison with their
results (not shown here). The results were compared to different Monte Carlo
models, showing that the selected Monte Carlo generators do not reproduce the
measurements at both centre-of-mass energies and for all choices of $p_{\rm
T,cut}$.

## References

  * [1] K. Aamodt et al. [ALICE Collaboration], Phys. Lett. B 693 (2010) 53 [arXiv:1007.0719 [hep-ex]].
  * [2] ALICE Collaboration, ALICE-PUBLIC-2013-001.
  * [3] F. Carminati et al., ALICE Collaboration, Physics Performance Report Vol. I, CERN/LHCC 2003-049 and J. Phys. G30 1517 (2003); B. Alessandro et al., ALICE Collaboration, Physics Performance Report Vol. II, CERN/LHCC 2005-030 and J. Phys. G32 1295 (2006); K. Aamodt et al., ALICE Collaboration, JINST 3 (2008) S08002.
  * [4] K. Aamodt et al. [ALICE Collaboration], Eur. Phys. J. C 65 (2010) 111, arXiv:0911.5430 [hep-ex].
  * [5] B. Alessandro et al. [ALICE Collaboration], J. Phys. G: Nucl. Part. Phys. 32 (2006) 1295.
  * [6] P. Billoir, Nucl. Instrum. Meth. A 225 (1984) 352.
  * [7] V. Blobel in 8th CERN School of Comp., CSC 84, Aiguablava, Spain, 9 22 Sep. 1984, CERN-85-09, 88 (1985).
  * [8] K. Aamodt et al. [ALICE Collaboration], Eur. Phys. J. C 68 (2010) 89 [arXiv:1004.3034 [hep-ex]].
  * [9] G. Cowan, in Advanced statistical techniques in particle physics, Proceedings, Conference, Durham, UK, March 18-22, 2002, published in Conf. Proc. C 0203181 (2002) 248.
  * [10] A. Moraes [ATLAS Collaboration], ATLAS Note ATL-COM-PHYS-2009-119 (2009). ATLAS CSC (306) tune.
  * [11] M. G. Albrow et al. (Tev4LHC QCD Working Group), arXiv:hep-ph/0610012 (2006). D6T (109) tune.
  * [12] R. Field, Min-Bias and the Underlying Event at the Tevatron and the LHC, Fermilab ME/MC Tuning Workshop, Fermilab, Oct. 4, 2002.
  * [13] P. Z. Skands, in Multi-Parton Interaction Workshop, Perugia,Italy, 28 31 Oct. 2008, arXiv:0905.3418 [hep-ph] (2009).
  * [14] P. Z. Skands, Phys. Rev. D 82 (2010) 074018 [arXiv:1005.3457 [hep-ph]].
  * [15] T. Sjöstrand, S. Mrenna, P. Z. Skands, arXiv:0710.3820, CERN-LCGAPP-2007-04, LU TP 07-28, FERMILAB-PUB-07-512-CD-T (2007).
  * [16] R. Corke and T. Sjöstrand, JHEP 1103 (2011) 032 [arXiv:1011.1759 [hep-ph]].
  * [17] R. Engel, J. Ranft, S. Roesler, Phys. Rev. D 52, 1459 (1995).
  * [18] K. Werner, F.-M. Liu and T. Pierog, Phys. Rev. C 74 (2006) 044902 [hep-ph/0506232]; T. Pierog, I. .Karpenko, J. M. Katzy, E. Yatsenko and K. Werner, arXiv:1306.0121 [hep-ph]; K. Werner, F. -M. Liu and T. Pierog, Phys. Rev. C 74 (2006) 044902 [hep-ph/0506232].