Dynamics of Conned Monopoles and Similarities with Conned Quarks Gia Dvali Juan Valbuena-Bermudezand Michael Zantedeschiy

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Dynamics of Conﬁned Monopoles

and Similarities with Conﬁned Quarks

Gia Dvali, Juan Valbuena-Bermudez,∗and Michael Zantedeschi†

Arnold Sommerfeld Center, Ludwig-Maximilians-Universit¨at,

Theresienstraße 37, 80333 M¨unchen, Germany and

Max-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, Germany

(Dated: Friday 28th October, 2022, 12:09am)

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 [1–4] that monopoles carrying the op-

posite 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 [5]. As long as the mass of the monopole is larger

than the scale of the string tension, energy per unit length

µ, the breakup of the string via nucleation of monopole

pairs is exponentially unlikely [3].

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, µ= Λ2, is set

by the QCD scale, Λ. As long as all quarks in the theory

are heavier than Λ, 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 applica-

tions in particle physics and cosmology.

For example, recently a novel mechanism for producing

the primordial black holes was proposed in [6]. 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 pro-

∗juanv@mpp.mpg.de

†michaelz@mpp.mpg.de

duced. Analogous considerations could be applied to the

case of conﬁned monopoles [7].

Previous calculations of the radiated GW spectrum were

performed by Martin and Vilenkin [8] 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

Pn∼Λ4

n,(1)

nbeing the frequency number, Mpthe planck mass, and

Λ the conﬁning scale. This relation was derived consider-

ing a pair of monopoles connected by a string and is ex-

pected 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 [9,10].

Moreover, given the ﬂatness of the resulting energy den-

sity across several orders of frequency [11], 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 [1]. In an attempt to solve the

monopole abundance problem, the theory ensures a tem-

porary (thermal) window in which the U(1) group asso-

ciated 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 [12].

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 opportu-

nity is insuﬃcient for solving the monopole problem.

The goal of the present work is to analyze the dynamics

arXiv:2210.14947v1 [hep-th] 26 Oct 2022

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 sponta-

neous symmetry breaking: a scalar ﬁeld in the adjoint

representation, and a complex scalar doublet. The for-

mer breaks (Higgses) the gauge group SU (2) to U(1),

therefore admitting t’Hooft-Polyakov monopoles as a so-

lution [13,14]. The latter breaks the residual U(1) gauge

group leading to the conﬁnement of the associated “mag-

netic 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 character-

istic 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 peri-

odicity, as some sort of a rubber band.

However, we observe that in head-on collision the out-

come 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 analo-

gous 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 [15]. 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 re-

sults into a high multiplicity of glueballs (closed strings)

and mesons (open strings).

We also point out that inability of monopole and anti-

monopole 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 [16]. 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 [15]. This

leads to the “erasure” of monopoles by domain walls.

In [16], this mechanism was used to solve the cosmolog-

ical monopole problem in grand uniﬁed theories. How-

ever, the phenomenon of erasure is of broader fundamen-

tal interest. In particular, this is indicated by the simi-

larities 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 [15]

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 ob-

serve non-negligible corrections to the power spectrum

for scales comparable to the monopole radius, where the

emitted radiation is boosted, therefore providing correc-

tions to the GWs emission produced by the conﬁnement

dynamics.

We expect that our results have implications for the col-

lapse of the generic bounded strings such as the string-

theoretic strings bounded by D-branes [17,18].

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 SU(2) gauged ﬁeld theory that con-

tains a scalar ﬁeld in the adjoint representation, ϕa(a =

1, 2, 3), a scalar ﬁeld in the fundamental representation,

ψ, and gauge ﬁelds, Wa

µ. The Lagrangian of the system

is given by

L=1

2DµϕaDµϕa+(Dµψ)†Dµψ−1

4Waµν Waµν −V(ϕ, ψ)

(2)

where summation over repeated SU(2) indices is under-

stood, and the ﬁeld strengths for the gauge ﬁeld is

µν =∂µWa

µ−∂νWa

µ+gabcWb

µWc

ν.(3)

The covariant derivatives are deﬁned as

Dµϕa=∂µϕa+gabcWb

µϕc,(4)

Dµψ=∂µψ−ig σa

2Wa

µψ, (5)

and the potential is given by [19]

V(ϕ, ψ) = λ

4(ϕaϕa−η2)2+˜

2(ψ†ψ−v2)2+c ψ†σaψϕa.

(6)

As the ﬁrst stage of symmetry breaking, we give vac-

uum expectation value to the adjoint ﬁeld while keep-

ing ψ= 0. The system admits ’t Hooft-Polyakov

monopoles [14,20]. As ψacquires vacuum expectation

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

initial conﬁguration.

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 con-

ﬁguration are the main focus of this work. 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 θand θdenote the respective monopole

and antimonopole position azimuthal angle. In the ap-

proximation when the distance dis much longer than

the monopole size the string conﬁguration can be derived

as follows. For θ= 0, a monopole should be recovered

ψ∝(cos θ/2,sin θ/2eiφ)t[21,22], while for θ=πan an-

timonopole should be obtained ψ∝(sin θ/2,cos θ/2eiφ)t,

φbeing the polar angle. Therefore the string conﬁgura-

tion is given by [23–25]

ψ∝sin(θ/2) sin(¯

θ/2)eiγ + cos(θ/2) cos(¯

θ/2)

sin(θ/2) cos(¯

θ/2)eiφ −cos(θ/2) sin(¯

θ/2)ei(φ−γ)

(7)

with γaccounting for the possibility of twisting the anti-

monopole w.r.t. the monopole. In fact, for θ=π,ψcor-

responds to the above mentioned antimonopole under the

shift φ→φ+γ. Above the conﬁguration, for θ=θ= 0,

ψ= (v, 0)t, while below θ=θ=π,ψ= (veiγ ,0)t. Fi-

nally between the two monopoles, θ=πand θ= 0,

ψ∝(eiφ,0)tcorresponding 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 ˆϕaof the monopoles can be built asymptotically

as [21,22]

ˆϕa=1

v2ψ†τaψ, a = 1,2,3 (8)

where τadenotes 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 [24,25], 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 Ap-

pendix A. From now, we work in energy units of η−1and

set the gauge coupling g= 1. Thus λis the parame-

ter in the theory that controls the mass and size of the

monopoles.

For a (spherically symmetric) monopole ﬁeld conﬁgura-

tions, we use the following ansatz

ϕa=h(r)ˆra, W a

i=(1 −k(r))

raij ˆrj(9)

where ris the radial coordinate and ˆra=ra/|~r|. Under

ansatz (9), the equations of motion become:

h00(r) + 2

rh0(r) = 2

r2k(r)2h(r)−λ(h(r)2−1)h(r),(10)

k00(r) = 1

r2(k(r)2−1)k(r) + h(r)2k(r),(11)

with asymptotic conditions

h(r)r→0

−−−→ 0, k(r)r→0

−−−→ 1,(12)

h(r)r→∞

−−−→ 1, k(r)r→∞

−−−→ 0.(13)

The above equations were solved numerically in order to

obtain the monopoles proﬁle.

Finally the ansatz for the initial adjoint ﬁeld conﬁgura-

tion is given by

ϕa=h(rm)h(¯rm) ˆϕa,(14)

with rm(¯rm) denoting the monopole (anti-monopole) co-

ordinate center and ˆϕais deﬁned in (8). An example of

such conﬁguration is shown in Fig. 2.

The stationary ansatz for the gauge ﬁelds considered

by Vachaspati and Saurabh [24,25], follows from the

requirement that the covariant derivative of the Higgs

isovector vanish at spatial inﬁnity, Dµˆϕ|r→∞ = 0. This

gives

µ=−(1 −k(rm))(1 −k(¯rm))abc ˆϕb∂µˆϕc.(15)

As expected, and veriﬁed in [25], 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 γ[26] - such a correction, however, is exponentially

suppressed at large distances. The veriﬁcation of these

properties served as a valuable check of the numerics pre-

sented in this work.

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DynamicsofConnedMonopolesandSimilaritieswithConnedQuarksGiaDvali,JuanValbuena-Bermudez,andMichaelZantedeschiyArnoldSommerfeldCenter,Ludwig-Maximilians-Universitat,Theresienstrae37,80333Munchen,GermanyandMax-Planck-InstitutfurPhysik,FohringerRing6,80805Munchen,Germany(Dated:Friday28thOctober...

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Dynamics of Conned Monopoles and Similarities with Conned Quarks Gia Dvali Juan Valbuena-Bermudezand Michael Zantedeschiy

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