Download Investigating the Structure of Electroweak Stars

Investigating the Structure of Electroweak
Stars
Benjamin Elder
with Glenn Starkman
Department of Physics, Case Western Reserve University
April 2011
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Summary
At extremely high temperatures a process called electroweak burning is expected to inter-convert quarks and (anti-)leptons. Experimental evidence
is lacking, since even the most powerful particle accelerators are unable to
create the necessary temperatures. Those conditions could exist in the core
of a new class of star. What would such a star look like? What would its
mass and radius be? An approximate static solution has been found which
demonstrates that such a star might exist , although it does not guarantee
stability. This research is concerned with improving the model for the structure of the star, and exploring the roles that various approximations and
boundary conditions play. It also takes steps towards more accurately accounting for particle masses and multiple generations of quarks and leptons.
This improved model may be used to investigate other areas of interest, such
as rate of burning and stability.
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Contents
1 Summary
2
2 Introduction
4
3 Structure
3.1 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Neutron Stars
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4.1 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 Electroweak Stars in the Low
5.1 The Electroweak Core . . .
5.2 Equation of State . . . . . .
5.3 Boundary Conditions . . . .
5.4 Results . . . . . . . . . . . .
Mass Approximation
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6 The Effects of Boundary Conditions
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7 Including Mass
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8 Future Work
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9 Acknowledgements
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2
Introduction
In all experimentally observed particle reactions, the number of quarks produced always exactly equals the number of antiquarks produced - a requirement known as baryon-number conservation. This gives rise to one of the
biggest open questions in cosmology today: how is it possible that the universe contains matter and hence a nonzero baryon number today? It is
possible that the universe simply started out with an excess of matter over
antimatter, although there are theoretical preferences for this not to be the
case. It is therefore likely that baryon number conservation is violated by
some process not yet confirmed experimentally.
The standard model of particle physics predicts that at temperatures on
the order of 100 GeV the electroweak phase transition occurs, which allows
baryon number to be violated. This temperature is exceedingly high, and
thus it is unlikely that this process occurred after the universe cooled below
100 GeV, which occurred about 10−10 seconds after the emergence of the
universe from its inflationary beginnings[1].
Although many of the standard model’s predictions have been experimentally confirmed to extraordinary accuracy, the existence of electroweak phase
transition is still uncertain because the unusually high temperature required
for this process currently precludes direct experimentation. The energy density associated with a temperature of T = 100 GeV is ρ = 108 GeV4 (in units
where c, ~, and kB are 1). This is a huge density – for comparison, the den-
4
sity of water is only about 10−32 GeV4 . The highest density achievable in a
laboratory today is approximately 108 MeV4 - 12 orders of magnitude lower
than what is needed for electroweak interactions to proceed [1]. As a result,
it is very unlikely that the required conditions could be artificially created in
the near future.
However, it has recently been proposed that such conditions could occur
naturally in the core of a star [2]. Such stars, called electroweak stars, would
be extremely dense, with core densities on the order of 108 GeV4 , and would
only be prevented from collapsing upon themselves by the huge amount of
energy released by the baryon number violating interactions going on in the
core. A static solution for such a star has been calculated, but it remains to
be seen whether such stars are stable. This project investigates the structure
(pressure and energy density as a function of radius) in more detail.
Why is the detailed structure interesting? First of all, it provides a more
accurate estimate of the mass and radius of the star, which are two of the
most direct observables of a star. Furthermore, for low density stars that can
be described classically, it can be shown that the transition between stable
and unstable is determined entirely by the equation of state inside the star,
so an analysis of the structure could help to illuminate electroweak stars’
stability. Finally, more precise knowledge of the mass and density inside the
star may be helpful when calculating the magnitude of quantities such as
time dilation near the core, which should be helpful in determining things
like the rate of burning and lifetime of the star.
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3
Structure
A star that is completely spherically symmetric and static (no convection or
rotation) may be completely described by its pressure and density profiles as
a function of radius. The temperature profile is important as well, but in the
present analysis temperature was set to zero for simplicity. Since electroweak
stars are extremely dense, a full general relativistic description of the matter
is necessary.
For any spherically symmetric and static distribution of matter in space,
the Einstein field equations may be solved to yield a pair of differential equations [4]:
dM
= 4πρr2
dr
(1)
dP
P +ρ
=−
(4πP r3 + M )
dr
r(r − 2M )
(2)
P is the pressure, ρ is the energy density of the matter, and M is the total
enclosed mass contained in a spherical volume of radius r. These differential
equations describe how the enclosed mass and pressure vary from the center
to the boundary of the star. These equations are quite general, providing
a basic relativistic model of any static, spherically symmetric collection of
matter. As such, they are often referred to as the structure equations of the
star.
There is currently no way known to integrate these equations analytically.
It is possible to proceed numerically by setting boundary conditions on M , ρ,
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and P near the core of the star and then integrating outward to the surface.
The choice of boundary conditions will be discussed separately for each type
of star.
Once boundary conditions have been set, the structure equations may
be integrated with an appropriate equation of state linking P and ρ. The
integration proceeds until P and ρ drop to 0, which is the boundary of the
star (it will be shown later that they both vanish at the same point). The
radius at which they vanish is the radius of the star, so P (rstar ) = ρ(rstar ) = 0
and M (rstar ) = Mstar is the total mass of the star.
3.1
Equation of State
As previously mentioned, an equation of state is required in order to solve
the structure equations. A general equation of state (for fermions only) may
be found via a relativistic “particle in a box” analysis. For simplicity, it
is assumed here that temperature is zero. This assumption has no physical motive, and is done only to be able to proceed analytically. Since this
equation of state ignores burning and temperature, it describes only the degeneracy pressure of the fermions, and no other outward forces are included
(e.g. pressure from burning in the core, rotation, convection, etc.)
With these assumptions, the partial pressures and energy densities associated with each fermion are:
pi =
ti
Ki
(sinh ti − 8 sinh + 3ti )
3
2
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(3)
ρi = Ki (sinh ti − ti )
(4)
where Ki is a constant, and ti is a dimensionless parameter that is large for
high pressures and densities, and as it drops to zero both pressure and energy
density vanish. Ki and ti are:
Ki =
πm4i
4(2π)3
ti = 4arcsinh
kf,i
m
(5)
(6)
The partial pressures and energy densities may be combined as
P =
X
pi − B
(7)
ρ=
X
ρi + B
(8)
where B is the bag energy density, and is chosen as B = (145 MeV)4 . [2]
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Neutron Stars
Before turning to electroweak stars it is helpful to first gain some intuition by
looking at neutron stars, which are relatively simple because they are dominated by one species of fermion (neutrons), with only small concentrations
of protons and electrons.
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4.1
Equation of State
Only neutrons are being included, so the pressure and energy density are described entirely by tn , the dimensionless parameter associated with neutrons.
It is then simple to substitute the equations for P and ρ (equations (3) and
(4)) into the structure equations, so that the new structure equations are:
dM
= 4πKn (sinh tn − tn )ρr2
dr
(9)
sinh tn − 2 sinh t2n
dtn
−4
4
tn
= 2
( πKn r3 (sinh tn −8 sinh +3tn )+M )G
tn
2GM
dr
2
r (1 − r ) cosh tn − 4 cosh 2 + 3 3
(10)
These equations may now be integrated numerically with an appropriate
choice of boundary conditions.
4.2
Boundary Conditions
The integration starts in the center of the star, so the enclosed mass M at
a radius close to 0 is 0. The central density is chosen to be in the range
where neutrons are beginning to “touch,” that is, approximately 1 neutron
per fm3 . This corresponds to a central density of approximately 10−2 GeV4 ,
or tn (0) ≈ 4.
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Figure 1: M, ρ, and P of a neutron star.
4.3
Results
The results from integrating the structure equations for a central density of
ρ(0) = 10−2 GeV4 are plotted in Figure 1. (For all of the calculations in this
paper, an adaptive fourth-order Runge-Kutta solver was used).
Qualitatively, pressure and energy density start out large near the core
and drop off monotonically to zero. The enclosed mass M starts out low,
and rises slowly at the beginning, even though ρ is large, because r is still
small. When r and ρ are both relatively large, M grows rapidly, and then its
growth slows in the upper crust where ρ approaches zero. Note also that, in
this example, the radius and mass are 7.2 km and 0.68 M , respectively.
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Figure 2: Dependence of Mstar and Rstar on central density ρ(0).
These equations were then integrated for a range of central densities to
find the mass of the star corresponding to each central density. The results
are plotted in Figure 2.
These results are nearly identitical to those presented in [4], the original
paper on neutron stars. Note that the choice of central density has a drastic
effect on the star’s mass. It is possible to show that stars with ρ(0) . 10−2
GeV4 are stable, and that stars with central densities greater than this are
unstable.
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5
Electroweak Stars in the Low Mass Approximation
Electroweak stars differ from neutron stars in several respects: first, they
are assumed to contain all three generations of particles, instead of just neutrons. Second, they have a small core that is dense enough to allow burning
via electroweak processes. Third, they should have a nonzero temperature
and have a luminosity due to the burning in the core. In this paper, the
temperature in the area outside the electroweak core is assumed to be zero
so that the degenerate equation of state (equations (3) and (4)) applies. Furthermore, pressure from burning is ignored so that the structure equations,
(1) and (2), apply as well. At certain points, it is also assumed that only
a single generation of fermions are present, which simplifies the equation of
state considerably.
5.1
The Electroweak Core
The core is assumed to be a sphere of uniform density. The minimum density
to enable electroweak burning is ρ = 108 GeV4 . [2]
The maximum radius of a sphere of constant density before it becomes a
black hole, rcritical , can be calculated from the equation for the Schwarzschild
radius: rcritical = 2GM . This implies
r
rcritical =
12
3
8πGρ
(11)
For a density of ρ = 108 GeV4 , rcritical ≈ 8 cm. The radius of the electroweak
core rcore must be less than rcritical , and for the calculations in this paper was
set to rcore = 15 rcritical . In a later section it will be shown that the mass and
radius of the star are not sensitive to how exactly rcore is set.
5.2
Equation of State
In the case of electroweak stars, it is necessary to account for at least an entire
generation of fermions (e.g. up quark, down quark, electron, and electron
neutrino). This gives an equation of state that involves at least four ti (in the
same example, it would be tu , td , te , and tνe ). This introduces extra variables,
and so additional equations relating the variables must be introduced in order
to proceed. In [2], these assumptions are given as:
µd = µu + µe − µν
(12)
µν = 0
(13)
Σqi ni = 0
(14)
where µi , qi , and ni represent the chemical potential, electric charge, and
number density of a particle i. Equation 12 comes from weak interaction
equilibrium (d → u+e+ ν¯e ). Equation 13 allows neutrinos to escape the star,
a necessary condition if the star is to continue burning. The last equation
is a statement of local charge neutrality: given any volume contained in the
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star, this statement requires that volume to contain a net charge of zero.
When more than one generation of particles is being considered, all present
fermions of identical charge are taken to have the same chemical potential
(for example, µu = µc = µt ) because of quark mixing in the standard model.
It can be shown that number density ni is related to the Fermi momentum
kf,i by the equation
ni =
gi 3
k
6π 2 f,i
(15)
where gi = 6 for quarks and gi = 2 for leptons. The equation E 2 = m2 + k 2
implies a relation between between chemical potential and fermi momentum:
kf,i =
q
µ2i − m2i
(16)
Number density cannot be less than zero, so if µi < mi then kf,i = ni = ti =
0. These relations allow the charge neutrality equation to be written in terms
of chemical potentials. Assuming that only the first generation of fermions
is present, the charge neutrality equation (equation (14)) can be rewritten as
3
3
3
0 = 4(µ2u − m2u ) 2 − 2(µ2d − m2d ) 2 − 2(µ2e − m2e ) 2
(17)
(An identical equation exists if only third generation fermions are present.
Simply replace µu with µt , md with mb , etc.) If the chemical potentials are all
assumed to be much greater than their respective masses, then this equation
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becomes
0 = 4µ3u − 2µ3d − 2µ3e
(18)
This is what is meant by the “low mass approximation.”
Equations (12), (13), and (18) make it possible to parametrize all of the
chemical potentials in terms of one of them, i.e. we have µu (µe ) and µd (µe ),
and hence pressure and density as a function of a single chemical potential
(P (µe ) and ρ(µe )). Although P and ρ can be parametrized by any of the
chemical potentials, µe was chosen because the electron has the lowest mass
and its chemical potential was therefore expected to go to zero last of all the
particles.
With P (µe ) and ρ(µe ) in hand it is in principle possible to insert them into
the structure equations, transforming
dP
dr
into
dP dµe
dµe dr
in a manner analogous
to what was done with neutron stars. However, equation (18) changes once
µu → 0, making this process difficult. Instead, the structure equations were
left as is, and the conversion from P to ρ was calculated as follows:
• Given a pressure P , we want to find the associated ρ using the equation
of state, but have no knowledge of the chemical potentials.
• Use a Regula-Falsi routine to guess at the µe that gives this pressure
P.
• Once the correct µe has been found, use it to calculate the chemical
potentials of the other species, the partial energy densities, and finally
the total energy density at this µe . This is the desired ρ.
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This algorithm provides ρ(P ) and, mutatis mutandis, P (ρ).
5.3
Boundary Conditions
The boundary conditions for electroweak stars are slightly more complicated
than for neutron stars. Because the electroweak core, the integration starts
not at the center, but instead starts at the surface of the core (approximately
r = 2 cm for ρ(0) = 108 GeV4 ). Since the integration does not start at the
center, the initial enclosed mass is not zero, but is instead the mass of the
electroweak core. The mass of the electroweak core is calculated as simply
3
.
M (rcore ) = 34 πρ(0)rcore
The boundary condition ρ is set to be the same as the density as the
electroweak core, so ρ(rcore ) = ρ(0).
5.4
Results
The solution for the one generation model using first-generation particles is
plotted in Figure 3.
The radius of the star is similar to that of the neutron star, although it
has about twice the mass. Unlike the neutron star case, the enclosed mass
M (r) rises roughly linearly, instead of undergoing a period of rapid growth
in the middle. As with neutron stars, there appears to be a de facto equation
of state of ρ ≈ 3P . This star is extremely dense. The Schwarzschild radius
for 1.5 M is approximately 4.5 km, so the star is not too far from becoming
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Figure 3: M, ρ, and P of an electroweak star, using only first generation
particles, in the low mass approximation.
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Figure 4: M, ρ, and P of an electroweak star, using only third generation
particles, in the low mass approximation.
a black hole.
A one generation model using third-generation particles is plotted in Figure 4. In this case, the star is much smaller, which is not surprising because
third generation particles are much more massive than their one generation
counterparts, so their chemical potentials start out lower. Due to the high
particle masses, the low mass approximation is no longer valid, and so the
mass and radius of the star will probably change significantly when particle
mass is properly accounted for.
A model including all three generations of fermions is plotted in Figure 5.
This star is even smaller than the third generation case because it includes
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Figure 5: M, ρ, and P of an electroweak star, using all three generations of
particles, in the low mass approximation.
first and generation fermions in addition to third generation particles. As
with the third generation model, the actual mass and radius will probably
be quite different once particle mass is correctly included.
6
The Effects of Boundary Conditions
In the previous section, the boundary conditions were chosen arbitrarily:
ρ(0) = 108 GeV4 was merely the minimum density to allow electroweak
burning in the core, and the ratio
rcore
rcritical
=
1
5
was chosen simply to ensure
that the core was not a black hole.
How sensitive are the mass and radius of the star on these initial condi19
Figure 6: Mass and radius of an electroweak star as a function of the central
density. For these calculations, rcore = 15 rcritical .
tions? If the star’s core density is changed, does the mass change drastically,
as was observed for neutron stars? What about the radius of the electroweak
core compared to its associated Schwarzschild radius rcritical ?
The effects of these parameters on total mass and radius are plotted in
Figures 6 and 7. In both of these cases, these boundary conditions have little
to no effect of the mass or radius of the star in this range.
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Figure 7: Mass and radius of an electroweak star as a function the fraction
rcore
. For these calculations, ρ(0) = 108 GeV4 .
rcritical
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7
Including Mass
To properly acount for particle mass in a one generation model, it would in
general be necessary to solve equation (17) for µu (or, in the third generation
case, µt ) exactly. No way was found to do this. Instead, it was assumed that
the low mass approximation holds for all particles except one.
For the third generation case, the mass of the top quark was left in because
it is the most massive of the third generation particles. Then equation (17)
becomes:
3
0 = 4(µ2t − m2u ) 2 − 2µ3b − 2µ3τ
(19)
Ideally, this would allow one to solve for µt (µτ ), as before. Equation (19) is
a polynomial of order 6 in µt , making it difficult to solve, especially because
the first term must be set to zero for µt ≤ mt . However, the equation is only
a polynomial of order 3 in µτ , so it was solved instead for µτ (µt ). With this
solution, one can build a table of µτ values calculated at their corresponding
values for µt . With this table, it is straightforward calculate µt (µτ ) via
interpolation.
The relationship between µτ and µt is plotted in Figure 8. The important
feature of this graph is that it shows µτ → mτ = 1.7 GeV before µt → mt =
171 GeV. The low mass approximation breaks down for µi ≈ mi , and so
evidently if only one particle mass is to be included at first, it should be the
mass of the tau.
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Figure 8: Relationship between µτ and µt where only the mass of the top
quark has been included.
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Figure 9: Relationship between µτ and µt where only the mass of the tau
lepton has been included.
Keeping only the mass of the tau instead, equation (17) becomes:
3
0 = 4µ3t − 2µ3b − 2(µ2τ − m2τ ) 2
(20)
This equation is a polynomial of order 3 in µt , so it is straightforward to
solve for µt (µτ ). The results are plotted in Figure 9.
This time, µt → mt before µτ → mτ , which is the opposite of what
occurred in the first approximation. This implies that neither is a good
approximation, and so it is likely necessary to solve equation (17) in its
entirety without removing masses.
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8
Future Work
Much is still unknown about electroweak stars. The logical next step is an
exact treatment of the particle masses, which would involve solving equation
(17) exactly, or finding a better approximation. However, there are many
other aspects to be explored, such as including nonzero temperature outside
the electroweak core and the inclusion of bosons. Questions of burning are
also of interest, and were discussed in [2] – exactly how much energy is
released in the core? How do neutrinos propagate out of the star? What is
the lifetime of the star?
Finally, there are still many questions regarding formation and evolution.
It is still unknown whether natural processes can form electroweak stars in
nature, and if they can be created, what is the mechanism that does so?
Once formed, what happens to them? Do they eventually explode, or do
they collapse to become black holes?
Collaborators are currently investigating many of these questions, including how electroweak stars might evolve from other types of stars, and of observables that could facilitate detection. If electroweak stars turn out to have
stable equilibria and are physically permitted to form, then observation of
these stars could provide an invaluable window into electroweak physics that
is otherwise completely unobservable for the time being.
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9
Acknowledgements
I would like to thank De-Chang Dai and Dejan Stojkovic for their thoughts
and advice on this topic. I would also like to thank Ken Singer and Rolfe
Petschek for their helpful thoughts and discussions, as well as their input at
various stages. Finally, I would like to thank my advisor, Glenn Starkman,
for his constant help and guidance on this project.
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References
[1] G. Starkman, private communication.
[2] De-Chang Dai, Arthur Lue, Glenn Starkman, Dejan Stojkovic. Electroweak stars: how nature may capitalize on the standard model’s ultimate
fuel. arXiv: 0912.0520 (2010)
[3] R.C. Tolman. Relativity, Thermodynamics, and Cosmology. Oxford Press
(1934)
[4] J.R. Oppenheimer and G.M. Volkoff. On Massive Neutron Cores. Physical
Review, Vol. 55, p. 374 (1939)
[5] S. Weinberg. Gravitation and Cosmology. John Wiley and Sons, Inc.
(1972)
[6] S. Chandrasekhar. The highly collapsed configurations of a stellar mass.
Monthly Notices of the Royal Astronomical Society, Vol. 95, p. 207 (1935)
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