#### Electroweak monopoles and the electroweak phase transition

Eur. Phys. J. C
Electroweak monopoles and the electroweak phase transition
Suntharan Arunasalam 0
Archil Kobakhidze 0
0 ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University of Sydney , Sydney, NSW 2006 , Australia
We consider an isolated electroweak monopole solution within the Standard Model with a nonlinear BornInfeld extension of the hypercharge gauge field. Monopole (and dyon) solutions in such an extension are regular and their masses are predicted to be proportional to the Born-Infeld mass parameter. We argue that cosmological production of electroweak monopoles may delay the electroweak phase transition and make it more strongly first order for monopole masses M 9.3 · 103 TeV, while the nucleosynthesis constraints on the abundance of relic monopoles impose the bound M 2.3 · 104 TeV. The monopoles with a mass in this shallow range may be responsible for the dynamical generation of the matter-antimatter asymmetry during the electroweak phase transition.
1 Introduction
For a long time, there was a prevailing view that topologically
stable monopole solutions do not exist in the Standard Model
because the vacuum manifold SU (
2
) × U (
1
)Y /U (
1
)E M
allows no non-trivial second homotopy group. This has been
questioned in [1], where topological stable monopole (and
dyon) solutions, representing a non-trivial hybrid between
the U (
1
)E M Dirac monopole [2] and the non-Abelian ’t
Hooft–Polyakov monopole [3–5], have been found in [1].
While the SU (
2
) non-Abelian configuration is regular, the
U (
1
)E M configuration exhibits a point singularity at the
origin. As a result, the monopole mass is divergent. There is no
obvious problem with the energy of a classical configuration
being divergent as it may be regularized in a more complete
quantum theory. In fact, also some regularized monopole
solutions have been proposed [6–9] which indicate that
electroweak monopoles as light as ∼5–10 TeV may actually exist.
In this paper, we consider the Standard Model where the
standard kinetic term for the UY (
1
) hypercharge gauge boson
is a part of a nonlinear Born–Infeld Lagrangian. In this
theory we account for an extra mass parameter, √β, which
controls the nonlinearity of the hypercharge field. Similar
to the regularization of the divergent energy of a point-like
charge in the original Born–Infeld electrodynamics [10], we
find that the electroweak monopole gets also regularized
and it mass predicted to be ∝ √β. Interestingly enough,
the Born–Infeld extension considered here emerges as an
effective low energy theory of open strings attached to the
3branes [11]. In this picture, the parameter √β defines the
distance between 3-branes in the extra-dimensional bulk
spacetime.
The electroweak monopoles must be copiously produced
during the electroweak phase transition via the Kibble
mechanism [12,13].1 Furthermore, as we will argue in this
paper, cosmological production of electroweak monopoles
may delay the electroweak phase transition and make it
stronger first order. The physics behind this can be
heuristically explained as follows. Magnetic monopoles (and
antimonopoles) with symmetric vacuum configuration within the
monopole core are trapped in the region surrounded by the
domains with symmetry-breaking vacua with different
orientation of the Higgs field in the SU (
2
) × U (
1
)Y /U (
1
)E M
vacuum manifold. This costs in energy, leading to a higher
free energy in the broken phase relative to the case without
monopole production. In particular, we find that φc/ Tc
1 can be achieved without violating nucleosynthesis
constraints on relic monopole abundance. This may have
important implications for electroweak baryogenesis. Namely,
sphaleron mediated B + L-violating processes become
ineffective below the critical temperature, Tc, preventing the
washout of previously generated baryon asymmetry.
1 Strictly speaking the Kibble mechanism is applicable to global
monopole production. The refined mechanism in the case of gauge
theories is discussed in [14].
The rest of the paper is organized as follows. In Sect. 2,
we discuss the electroweak monopole solution within the
Born–Infeld hypercharge extension of the Standard Model.
Section 3 is devoted to a discussion of monopole production
and its impact on the electroweak phase transition. In Sect. 4,
we conclude.
2 Electroweak monopoles in the Born–Infeld
hypercharge model
Let us consider Standard Model extended by the
nonlinear Born–Infeld type hypercharge gauge field. The relevant
bosonic Lagrangian reads
1
− 4
Fμν F μν
1 1
1 + 2β2 Bμν Bμν − 16β4 (Bμν B˜ μν )2
where Dμ = ∂μ − i g2 τ a Aaμ − i g2 Bμ is the SU (
2
)L × U (
1
)Y
gauge covariant derivative with Aaμ and Bμ being SU (
2
)L
and U (
1
)Y gauge vector fields, respectively and H is the
electroweak doublet Higgs field. Fμaν (a = 1, 2, 3) denote the
SU (
2
)L gauge field strength tensors, and B˜ μν = 21 μναβ Bαβ
is the Hodge dual to the U (
1
)Y field strength tensor Bμν .
Parameter β is a new Born–Infeld parameter of dimension
mass2. It controls the nonlinearity of the hypercharge field
and, as we will see shortly, provides ultraviolet regularization
of the electroweak monopole mass. In the limit β → ∞, we
recover the Standard Model theory.
The above Lagrangian leads to the following set of field
equations of motion:
Dμ(Dμ H ) = λ
∂μ − i g τ a Aa
2 μ
μ2
H † H −
λ
F μiν = i g2
H,
H †τ i (Dν H )−(Dν H )†τ i H ,
∂μ ⎣
⎡
g
= i 2
1
− 4β2
Bαβ B˜ αβ B˜ μν
⎤
1 + 2β12 Bαβ Bαβ − 161β4 (Bαβ B˜ αβ )2 ⎦
H †(Dν H ) − (Dν H )† H .
(
2
)
(
3
)
(
4
)
Here, (r, θ , ϕ) are the usual spherical polar co-ordinates
and rˆ is the unit vector in the radial direction. Now, as done
in [1], consider the following ansatz:
1
H = √ ρξ, ρ = ρ(r ), ξ = i
2
sin(θ /2)e−iϕ
− cos(θ /2)
1 1
Aμ = g A(r )∂μtrˆ + g ( f (r ) − 1)rˆ × ∂μrˆ,
1 1
Bμ = − g B(r )∂μt − g (1 − cos θ )∂μϕ.
In particular, the functions, A(r ) and B(r ) represent dyon
solutions of this model. For A(r ) = B(r ) = 0, one obtains
pure magnetic monopole, which is also the lightest object and
thus we concentrate on this solution. For A(r ) = B(r ) = 0,
Eq. (
4
) is trivially satisfied and Eqs. (
2
) and (
3
) yield
2 f 2
ρ¨ + r ρ˙ − 2r 2 ρ = λ
ρ2 μ2
2 − λ
ρ
f¨ −
f 2 − 1
r 2
f = g42 ρ2 f.
The following boundary conditions can be chosen for these
equations:
(
1
)
f (0) = 1, ρ(0) = 0, f (∞) = 0, ρ(∞) = ρ0 =
2μ2
λ
(
5
)
(
6
)
(
7
)
.
(
8
)
(
9
)
(
10
)
(
11
)
Under these boundary conditions, it can be seen that near the
origin
f ≈ 1 + α1r 2, ρ ≈ β1r δ
with δ = (−1 + √3)/2 and asymptotically,
−gρ0
2
f ≈ f1 exp
r , ρ ≈ ρ0 + ρ1
exp(−√2μr )
r
.
The energy of this monopole is given by
E = E0 + E1,
0
∞
E0 =
E1 = 4π
⎡
drβ2 ⎣
0
∞
dr
h2Y ⎤
(4πr 2)2 + β2 − 4πr 2⎦ ,
1 ( f 2 − 1)2
g2 2r 2
1
+ 2 (r ρ˙)2
1 2
+ g2 f˙ +
λr 2
8
ρ2 − ρ02 2
+ 41 f 2ρ2 ,
where hY = 4gπ is the hypermagnetic charge of the
monopole, g being the hypercharge gauge coupling.
Here, E0 is the term corresponding to Born–Infeld
hypercharge term and E1 is due to the remainder of the Lagrangian.
In the usual standard model, E1 is finite due to the above
boundary conditions and asymptotics and E0 is infinite.
However, due to the Born–Infeld modification, E0 is also made
finite. E1 has been calculated by [7] to be roughly 4 TeV. As
discussed below, the mass of the monopoles that provide a
significant impact on the electroweak phase transition must
be at least of order 104 TeV. Hence, E0 must dominate this
mass and hence, it is assumed that E ≈ E0. This term can
be calculated exactly using elliptic integrals as [15]:
π 3/2
E ≈ 3 (
3
)2
4
βh3Y
where we have used g = 0.357. Thus the monopole mass
is proportional to the Born–Infeld mass parameter, √β. One
can verify that the magnetic charge of this monopole solution
4π .
is h = e
In the perturbative expansion of the Born–Infeld
Lagrangian, which is valid for low hypercharge field strength,
|Bμν | < β, the lowest order Born–Infeld correction appears
as operators of mass dimension 8. They involve only
hypercharge field and are suppressed by a factor ∝ β−2. The best
bound on the Born–Infeld mass parameter can be inferred
from the PVLAS measurements of nonlinearity in light
propagation [16] (see also [17]):
β
5.0 · 10−4 GeV.
This is clearly a very weak constraint compared to
constraints from direct searches of massive monopoles [18],
which in our case implies √β 15.1 GeV. In contrast, the
monopole mass is regularized in [7,8] by non-renormalizable
operators with mass dimension n > 8 + 2√3 operator which
also involve the Higgs field. These operators are significantly
constrained by LHC data on the Higgs-to-2γ decay [9].
3 Monopole production and electroweak phase
transition
Consider the one loop high temperature effective potential:
V (φ, T ) = D(T 2 − T02)φ2 − E T φ3 − 41 λT φ4
where the parameters are defined by
1
D = 8v2 (2m2W + m2Z + 2mt2),
1
E = 4π v3 (2m3W + m3Z ),
(
13
)
(14)
1
T0 = 2D (μ2 − 4Bv2)
with ln a f = 1.14, ln ab = 3.91, m H = 125 GeV,
mW = 80.2 GeV, m Z = 91.2 GeV, mt = 173 GeV and
v = 246 GeV.
Let φc(T ) be the value of the Higgs field at the second
minimum. If monopoles are not produced, the Gibbs free
energy of the unbroken and broken phases are simply the
value of the potential at φ = 0 and φ = φc(T ) and the critical
temperature, Tc is defined as the temperature at which these
are equal. In order to avoid the sphaleron washout constraint,
one requires that (e.g. see [19] and the references therein).
(16)
φc(Tc)
Tc
1.
With the standard model parameters, this ratio is roughly
0.17, implying that sphaleron mediated processes will
washout any pre-existing matter–antimatter asymmetry in
the universe. This conclusion is altered significantly once
the electroweak phase transition is supplemented by the
production of the electroweak monopoles.
The details of the production of monopoles have been
discussed in [13], which we closely follow. Although the
electroweak monopoles are typically heavier that the
critical temperature, M >> Tc, they are produced out of
equilibrium during the phase transition when the Higgs field
becomes frozen in the broken phase [13]. During the
electroweak phase transition, there is a finite distance over which,
the field is correlated with itself. At distances larger than
the correlation length, the Higgs field may point in
different directions in the manifold of degenerate vacua.
Following Kibble [12] and Preskill [13], one can argue that a
certain density of monopoles is to be expected on this account
alone. Monopoles (like vortices in a superconductor) can
be thought of as a measure of the disorder remaining in
the system, where symmetric (normal) regions trapped by
flux quantization in the broken (superconducting) ground
state. Hence, production of monopoles will drive the
surrounding plasma out of equilibrium. The equilibrium will be
eventually restored once the monopole/antimonopole
density will drop due to the monopole–antimonopole
annihilation.
The production of monopoles during the electroweak
phase transition offers a qualitatively new picture of
baryogenesis at the electroweak scale. In addition to sphalerons,
there is additional source of anomalous B + L violation
through matter-monopole scattering. These scatterings are
known to be unsuppressed [20], and thus are potentially
rapid to contribute to the generation of baryon
asymmetry at around Tc. Once the equilibrium is achieved,
however, below Tc, sphaleron and monopole mediated
processes must become irrelevant in the broken phase. The
sphalerons are ineffective if condition (16) is satisfied,
while matter-monopole scatterings must decouple once the
monopole density becomes low enough due to monopole–
antimonopole annihilation. The latter process is also
important to not upset the standard Big Bang nucleosynthesis.
In what follows we will concern ourselves with washout
issues, postponing the full discussion of baryogenesis to
future work.
3.1 Circumventing the sphaleron induced washout
of baryon asymmetry
Since monopoles are assumed to be heavier than the
critical temperature Tc, they can be treated as nonrelativistic
point-like particles. Monopole–antimonopole interactions,
as well as interactions of monopoles with charged particles
of plasma, are due to the long-range electromagnetic forces.
The monopole–antimonopole annihilation cross section can
be approximated by σM M¯ = dc2, where dc = h2/4π T is the
Coulomb capture distance and h = 4π/e is the monopole
charge. Similarly, a relativistic charged particle with a charge
q will scatter off the monopole (antimonopole) with the
cross section σq M = (qh/4π )2T −2. The initial density of
isolated monopoles (and antimonopoles) can be estimated
as n0 = dc−3 [13]. Below the capture distance monopole–
antimonopole pair would form a unstable bound state and
decay subsequently through annihilation.
The production of monopoles cost in energy, so the free
energy in the broken phase becomes
Gb = V (φc(T )) + n0 M.
(17)
Equating, now, the free energies in the symmetric and broken
phases, it becomes clear that the electroweak phase
transition happens at lower critical temperatures Tc, and,
therefore, sphalerons may start to satisfy the non-washout
condition (16). As seen from Fig. 1, this indeed takes place when
monopoles are sufficiently heavy, M > 9.3 · 103 TeV.
Unfortunately, such heavy monopoles are not able to be produced
at the LHC or at other future planned colliders.
3.2 Monopole–antimonopole annihilation and monopole
washout constraints
The monopoles (antimonopoles) can drift towards
antimonopoles (monopoles) through the scatterings on charged
particles of plasma. Each of such scattering rate can be
estimated as qi σqi M nqi = i (hqi /4π )2T −2nqi =
(3/4π 2)ζ (
3
)T i (hqi /4π )2, where ζ (
3
) ≈ 1.20 and the
sum goes over relativistic charged particles (we included only
fermions) which are at thermal equilibrium at temperature T .
After ∼ M/ T such occurrences, a monopole will scatter at
large angle and drift towards the antimonopole. Hence, the
monopole/antimonopole mean free path is given by [13]
λ ≈
M
qi σqi M nqi T
T
M
1/2
1
= B
M
T 3
1/2
where B = (3/4π 2)ζ (
3
) i (hqi /4π )2. As long as the mean
free path (18) is smaller than the Coulomb capture
distance dc, monopole–antimonopole pairs can annihilate as
described. However, as the universe expands and cools down,
λ grows faster than dc, and below the temperature
T f =
4π
h2
where λ ≈ dc, the monopole–antimonopole annihilation
rate becomes negligible. Solving the Boltzmann equation for
monopole/antimonopole number density and evaluating it at
T f one obtains2 [13]:
where C = 0.6N −1/2, N is the number of relativistic degrees
of freedom and MP is the Planck mass. Below T f this number
density simply dilutes as T −3 due to the expansion of the
universe. The monopole/antimonopole number density is
constrained by the standard Big Bang nucleosynthesis. Namely,
at T = 1 MeV the monopole/antimonopole density should
be such that
n/ T 3 = n f / T f3
(1MeV)
M
.
Plugging in the numbers B ≈ 3 and C ≈ 0.06 and imposing
the obvious requirement T f < Tc, we obtain from (21) the
following upper bound on the monopole mass:
M < 2.28 · 104 TeV.
Hence, it is seen that the monopoles required to suppress
sphalerons in the broken phase can still satisfy the
nucleosynthesis constraints.
As has been mentioned above, quark/lepton scatterings
off monopoles and antimonopoles also lead to unsuppressed
anomalous B + L violation [20]. These processes must also
decouple once the equilibrium is achieved, otherwise they
will washout the asymmetry. Hence, we demand the rate of
such processes at T f , σqi M n f , is less than the expansion rate
of the universe, H (T f ) = T f2/C MP , which implies
(19)
(20)
(21)
4 Conclusion
In this paper we have postulated the existence of electroweak
monopoles regularized within the Born–Infeld hypercharge
extension of the Standard Model. Such monopoles (and
antimonopoles) must be copiously produced during the
electroweak phase transition and can drive local
nonequilibrium in plasma. The production of the monopoles cost in
energy, thus postponing electroweak phase transition. We
have shown that if the monopole mass, defined through
the Born–Infeld mass parameter, is within a narrow range
0.9 · 104 TeV < M < 2.3 · 104 TeV, sphaleron mediated
processes can be made ineffective, thus preventing washout
of the previously generated matter–antimatter asymmetry,
while still satisfying the nucleosynthesis constraints. We have
also verified that anomalous B + L violation processes due
to the quark/lepton–monopole scatterings, while being active
during the phase transition at ∼ Tc, become suppressed
in the broken phase (T < Tc) due to the efficient enough
monopole–antimonopole annihilation.
If the electroweak phase transition is indeed
accompanied by the production of the electroweak monopoles of
mass, M ∼ 104 TeV, a new mechanism for the electroweak
baryogenesis can be realized. Namely, out-of-equilibrium
quark/lepton scatterings off monopoles may generate
nonzero B + L number due to the anomaly. The issue of CP
violation, which left outside this paper, must be considered
carefully. We plan to study this mechanism in more detail in
future work.
Acknowledgements We would like to thank Tsutomu Yanagida for
useful discussions. This work was partially supported by the Australian
Research Council.
Note added After the submission of this paper, a new, more stringent
constraint on the lower bound of the Born–Infeld parameter, √β, has
been obtained in [21] by considering light-by-light scattering at the
LHC. However, the monopole masses considered in this paper, M ∼
104 TeV, are unaffected.
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Funded by SCOAP3.
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4π BT f
α3
,
where α ≈ 1/137 is the fine structure constant. Taking
into account Eq. (19), one immediately sees that the above
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2 Here, for the sake of simplicity, we ignore the potential imbalance
between the monopole and antimonopole number densities due to CP
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