Electroweak stars: how nature may capitalize on the standard model's ultimate fuel

0.3 MB·English

by  De-Chang Dai

Electroweak stars: how nature may capitalize on the standard model’s ultimate fuel
De-Chang Dai1, Arthur Lue2, Glenn Starkman2 and Dejan Stojkovic1
1Department of Physics, SUNY at Buffalo, Buffalo, NY 14260-1500 and
2CERCA, Department of Physics, Case Western Reserve University, Cleveland, OH 44106-7079
We study the possible existence of an electroweak star - a compact stellar-mass object whose
central core temperature is higher than the electroweak symmetry restoration temperature. We
found a solution to the Tolman-Oppenheimer-Volkoff equations describing such an object. The
parametersofsuchastararenotsubstantiallydifferentfromaneutronstar-itsmassisaround1.3
1 Solarmasseswhileitsradiusisaround8km. Whatisdifferentistheexistenceofasmallelectroweak
1 core. The source of energy in the core that can at least temporarily balance gravity are standard-
0 model non-perturbative baryon number (B) and lepton number (L) violating processes that allow
2 the chemical potential of B+L to relax to zero. The energy released at the core is enormous, but
gravitationalredshiftandtheenhancedneutrinointeractioncrosssectionattheseenergiesmakethe
n
energy release rate moderate at the surface of the star. The lifetime of this new quasi-equilibrium
a
can be more than ten million years. This is long enough to represent a new stage in the evolution
J
of a star if stellar evolution can take it there.
9
1 PACSnumbers: 04.50.Kd

]
h
INTRODUCTION processessuchasthose mediated by instantons. At tem-
p
- peratureswellbelowtheelectroweaksymmetry-breaking
p Thelaststageoftheevolutionofastarmassiveenough scale, T <100GeV,these non-perturbativeprocessesare
he to go through the supernova explosion (but not massive suppress∼edbythe extremelysmallfactore−8π/α (giving,
[ enough to become a black hole) is believed to be a neu- for example, a proton life-time of 10141yrs [3]). (Note,
tron star. A neutron star has no active energy source unless otherwise indicated we shall employ natural units
v3 which balances the force of gravity. Instead, it is sup- where ~ = c = kB ≡ 1.) However, above this scale,
0 ported by degeneracy pressure, which arises due to the baryon and lepton number violating processes are ex-
2 Pauliexclusionprinciple. However,thispressurecansup- pectedtobeessentiallyunsuppressed(althoughthecom-
05 pisokrntoownnlyansetuhteroTnolsmtaarns-Oligphpteenrhtehiamnear-bVooultko2ff.1lMim⊙it,.wNheicuh- beffineacttiiovnelyBc−onLverretmedaiinnstocolenpsteornves.d)I.nQthuiasrkpsroccaensst,hwenhibche
.
2 tronstarsheavierthanthislimiteventuallybecomeblack we call electroweak burning, huge amounts of energy can
1 holes. be released.
9
0 Recent studies point out the existence of a state be- Let us assume that in the core the temperature is
: tween the neutron star and black hole. It is called a above,oratleastverynear,theweakscale,andthemat-
v quark star. This state owes its existence to the QCD ter density is also comparable ρ > (100GeV)4. At these
i
X phase transitionin whichthe originalnuclearmatter be- densities,the matterisopaqueev∼entoneutrinosandthe
r comesquarkmatter. Thisprocesscanrelease1053 ergin energy released at the center cannot stream freely out
a about10−3 to10−2s[1,2]intheformofneutrinobursts. of the star. Nevertheless, the energy can be carried out,
While this a huge amount of energy, it is released in too mostly by neutrinos and anti-neutrinos. These can also
short a time to provide the pressure to prevent gravita- carryoutanyexcessanti-leptonnumbergeneratedinthe
tional collapse. After this energy release, a star contain- electroweak burning, and thus prevent the lepton num-
ing effectively only three quark flavors (u,d,s) can exist ber chemical potential from halting the consumption of
ina stableequilibriumwherethe pressureis providedby the baryon number. This mechanism can likely provide
the Pauli exclusion principle. However, at higher densi- a stable energy source which can counteract gravity for
ties where four or more quark flavors are present, quark a while. On a side track, we remark that studying the
matter cannot avoid gravitationalcollapse. electroweak star physics, in particular the consequences
Since in the gravitational collapse matter gets com- ofaverypowerfulenergysourcethatwassofarneglected,
pressed to ever increasing densities/temperatures, it is mayprovideimportantnewcluesaboutsupernovaexplo-
naturalto explorewhatcouldhappenatthe electroweak sions,butwereservefurtherdiscussiononthismatterto
phasetransition,thenext(andlast)withinthe standard a future paper (in preparation).
model. In the standard model, both baryon and lep- In this paper, we study the inner structure of such a
ton number U(1) global symmetries are accidental, con- star and calculate its lifetime. We find that this new
served perturbatively, but violated by non-perturbative phase can last up to 1015s, which is long enough to give
2
a new name to this type of stars - electroweak stars. In pathsincrease,however,theyarestilltooshortforparti-
ordertofindoutwhetherelectroweakstarsareverycom- clesto streamfreelyoutofthe star. Eventually,however
mon or very exotic objects, much more detailed analysis we reach the star’s neutrino release shell (i.e. neutri-
isrequired. Futureworkwillconsiderquestionsofstabil- nosphere), where the neutrino (and anti-neutrino) mean
ity,ofevolution–ifandwhenthe electroweakstararises free path is long enoughfor these particlesto flowfreely.
in the late-phases of evolution of ordinary stars – of the [This is well within the radius at which the same is true
structure of the outer cooler layers, and of observational of photons.] The neutrinos and anti-neutrinos carry off
signatures of these objects. notjustmostoftheenergyproducedinthe coreburning
region (the rest is carried off by photons), but also the
leptonnumber, thus permitting the burning to continue.
DENSITY, PRESSURE, AND PARTICLE Sincethefuelispredominantlybaryonsnotanti-baryons,
NUMBER DENSITY INSIDE THE resulting in the production of an excess of anti-leptons
ELECTROWEAK STAR overleptons,therewillbeagreaterfluxofanti-neutrinos
than neutrinos from the star. Moreover,the electroweak
Weseparatethecoreoftheelectroweakstarintothree processes(unlikeinversebetadecaysinnormalstars)cre-
regions,as showninFig. 1. The centralregionis hotter ates equal numbers of anti-neutrinos of all three genera-
and denser than the electroweak symmetry restoration tions (see Eq. 1). Thus the excess is likely to be equal in
temperature (T > 100GeV). This region is very dense all generations. Such a signal, taking into account neu-
(ρ > 108GeV4),∼but small (several cm). We find that trino oscillations, might be an indication that the elec-
the totalmass inside this regionis 5 10−6M⊙. Since troweakprocessesaretakingplaceatthecoreofthestar.
∼ ×
the lepton and baryon number are not conserved in this However,theexcessislikelytobeverysmall,since(asin
region, but B L is, baryons are freely transformed to a supernova), the emission is expected to be dominated
−
anti-leptonsso longas itis thermodynamically favorable bythermalemissionfromthe neutrinosphere(whichcar-
to do so. A SU(2)-preserving instanton interaction can ries no excess), not by direct anti-neutrino production
convert 9 quarks into 3 anti-leptons [4], for example: in the core. More likely, a detailed analysis of the elec-
troweak star will show that the neutrinospheres for the
udd css tbb ν¯ ,ν¯ ,ν¯ (1) threegenerationsofneutrinosandanti-neutrinosaresep-
e µ τ
→
aratedduetothedifferingmassesandhenceabundances
and all B L and electromagnetic charge conserving and momentum distributions as a function of radius of
−
variations involving quarks, anti-quarks, neutrinos, an- the quarks and charged leptons of the associated gener-
tineutrinos and charged leptons. At these densities and ations. Therefore the fluxes and spectra of the different
temperatures each particle carries about 100GeV of en- neutrino andanti-neutrino species should be predictably
ergy, so this process can release about 300GeV per neu- (and hopefully observably) different.
trino. Thetypicalenergyoftheproducedneutrinoisnot The size of a quark star is about 10km, while its mass
the only relevant quantity here, we also need the rate at is of the order of M⊙. A small electroweak core cannot
which the energy is released. The sphaleron baryogene- significantly change the region outside the core. We use
sis/baryodestructionrate(abovetheelectroweaksymme- the so-called“bagmodel”[5]asanapproximatesolution
tryrestorationtemperature)isproportionaltoT4. How- of the region outside the core. The pressure and energy
ever,wewillshowlaterthatthe energyreleaseratefrom density are [6]
the core is limited by the quark supply rate (dictated
by the gravitational collapse) rather than the maximal P = p B (2)
i N
−
electroweak burning rate.
i
X
The energy in the form of neutrinos flows out of the
ǫ = ǫ +B .
i N
central core, and eventually out of the star. However,
i
in the central region,the mean free paths of all particles X
are short compared to the size of the star. The energy Here, P is the total pressure, p is the partial pressure
i
must therefore diffuse out of the central core. Similarly, of particle i, B is the bag energy density, ǫ is the to-
N
the non-perturbative B-violating processes, while reduc- tal energy density, and ǫ is the energy density of par-
i
ing B, reduce L. Initially B L > 0 in stars, because ticle i. The subscript i runs over all types of particles
−
neutrinos escape, and thus there are more neutrons plus presentinthestar(howeverhereforsimplicityweinclude
protons than there are electrons. Thus, the B-violating only fermions, i.e. quarks, neutrinos and electrons). We
processeswouldcauseananti-leptonnumbertobuildup. choose the specific value B1/4 =145MeV.The condition
N
This must either stop the burning, or be carried out of for the electric charge neutrality is
the core by a flux of anti-leptons. Outside the central
core,baryonandleptonnumberareconserved. Theden-
q n =0. (3)
i i
sity falls with increasing radius, and particle mean free
i
X
3
SOLUTION TO THE
TOLMAN-OPPENHEIMER-VOLKOFF
EQUATIONS DESCRIBING AN ELECTROWEAK
STAR
Inthis sectionwe setup andnumericallysolvethe sys-
tem of equations that describe the structure of an elec-
troweak star.
Outside the central region I, baryon and lepton num-
bers are conserved. We assume for simplicity that the
chemicalpotentialsoftheleft-andright-handedfermions
are the same, and that all fermions of the same charge
(e.g. u, c and t quarks) have the same chemical poten-
tial. An antiparticle has the chemical potential of the
opposite sign. The interaction d u+e+ν¯ gives one
e
→
relationship
µ =µ +µ µ . (6)
d u e ν
−
Eq. (1) implies another relation inside the electroweak
core:
µ +2µ +µ =0, (7)
u d ν
FIG.1: Thestructureofanelectroweakstar. Weseparatethe
core of the star into three regions: I. central core where the We assume µ = 0 outside the electroweak core, so that
ν
electroweak symmetry is restored, II. region where the high neutrinos or anti-neutrinos can propagate away. We
energyneutrinosaretrapped,III.regionfrom whichhighen-
set the boundary conditions at the center that pres-
ergyneutrinoscanescape. Neutrinoswillbeemittedbetween
sure and energy density are (100GeV)4, and then cal-
the regions II. and III.Photons are emitted at thesurface of
culatethe structure ofthe staraccordingto the Tolman-
thestar.
Oppenheimer-Volkoff equations
dP (ǫ+P)(M +4πPr3)
= , (8)
Here, qi is the charge of the particle i. The pressure, dr − Mp2r2(1− M2Mp2r)
energy density, and number density of particles can be
dM
well approximated from Fermi-Dirac and Bose-Einstein = 4πǫr2. (9)
dr
gas distribution:
gtt = (1− M22MRstar )e−R0P(r) P2d+Pǫ (10)
p surface
g ∞ 1
pi = 6πi2 dE(E2−m2i)3/2fi(E) (4) grr = 1 2M(r) (11)
g Zm∞i − Mp2r
ǫi = 2πi2 dE(E2−m2i)1/2E2fi(E) Rsurfaceisradiusofthestar,Mstar isthemassofthestar,
Zmi while M is the Planck mass. Since we are dealing with
∞ p
g
n = i dE(E2 m2)1/2Ef (E). very dense objects, general relativistic effects described
i 2π2 Zmi − i i bythemetriccomponentsgtt andgrr willbeveryimpor-
tant.
If we supply an equation of state ǫ(P), the Tolman-
Here, g is the number of degrees of freedom of particle
i Oppenheimer-Volkoff equations could be solved directly.
i (e.g. g = 2 for leptons and g = 6 for quarks), m is
i However,wedonotwanttoassumeanequationofstate,
the mass of the particle, µ is the chemical potential of
i and instead we try to solve the system of coupled equa-
the particle and T is the temperature. The distribution
tions we already set-up.
functions are
Because of the properties of Fermi-Dirac statistics,
fermions can have large energies at T = 0. In that case
1 theenergyofneutrinosisnotproportionaltothetemper-
fi(E)= 1 e(E−µi)/T (5) ature,insteaditisproportionaltothechemicalpotential
± µ , as can be seen from Eq. (5). We will use this fact
ν
to eliminate T from the equations and simplify our cal-
for bosons and fermions. culations. The exact treatment would require keeping
4
T as a variable (beside the energy density, pressure and
nceunmtebrerTd=en1s0it0yG),eVse,ttainndgfithnedibnogutnhdeatreymcpoenrdaittuiorne partotfihlee ensity 1020 energy density (MeV4)
d
Tǫ((rr))a.nSdeMtti(nrg),Tbu=tm0awyicllhannogtesing(nri)fi.cTanhtilsychchaannggeeinPn((rr)), mber 1018 pnruemssbuerre d (eMnesVity4) (MeV3)
willseeminglyaffecttheneutrinomeanfreepath,butone y, nu 101146
can explicitly verify that this does not happen since the sit 10
n
mean free path in Eq. (13) will effectively depend only de 1012
y
on the energy density ǫ = n E . Thus, the structure of g 10
i i i er 10
the star will not change much if we set T = 0. [Note en 8
that the properties of Bose-Einstein statistics are differ- ure, 106
ent and one cannot set T = 0 when including photons ss 10
e
and weak gauge bosons into consideration.] pr 104
-4 -3 -2 -1 0 1
Thus, we have four independent equations (3), (6), 10 10 10 10 10 10
radius (km)
(7), and (8), with four independent variables µ , µ ,
u d
µ and µ . The chemical potentials µ , µ , µ and µ
e ν u d e ν
FIG. 2: The pressure, energy density and particle number
thenuniquelyspecifyP(r),ǫ(r),n (r)andM(r)through
i density change with the radius of the star. The total radius
equations (2), (4), (5), (8) and (9).
ofthestarisabout 8.2km (radiuswherepressureandenergy
The numerical results are plotted in Fig. 2. The total density drop to zero), while the mass is about 1.3M⊙ (mass
size of the star is 8.2km (radius where pressure and en- within that radius).
ergy density drop to zero) and its mass is 1.3M⊙ (mass
within that radius). The size of the central electroweak
core is severalcm (this is the distance at which the den-
21
sity falls below (100GeV)4. Within this central region, 10
thepressureandenergydensitygradientsareverysmall, ) 1019 Neutron star
so pressure and energy density are practically constant 4MeV 1017 EPloelcyttrroowpieca k star
tthhaertet.heWeequcaatnioanlsooffisntadtethdeoedsepneontdcehnacnegeǫ(mPu)cahndacrfionsds sity ( 1015 model n=1.5
n
the star and is approximately ǫ=3P. In the absence of y de 1013
theactiveenergysource,suchanequationofstatewould erg 1011
n
signalinstability. However,electroweakstarshaveavery e
9
10
powerful energy source which can balance gravity. Also,
7
this equation of state will tip toward a non-relativistic 10
-5 -4 -3 -2 -1 0 1
10 10 10 10 10 10 10
equation of state once weak gauge bosons (and interac-
radius (km)
tions among the particles) are included into calculation,
since they are not ultra-relativistic at these energies. FIG. 3: Comparison between the energy density profiles of
an electroweak star, neutron star (a star made from neu-
For comparison, in Fig. 3 we plot the energy density
trons only) and polytropic star (with the polytropic index
profiles of an electroweak star, a neutron star (a star
1.5). Forall threeplotsthecentraldensity is3×1020MeV4.
made from neutrons only) and a polytropic star (with
The neutron and electroweak stars are almost indistinguish-
thepolytropicindex1.5),where,toemphasizethediffer- ableatsmall radiibecauseparticle masses arenotimportant
ence(andsimilarity)inthestructures,wesetthecentral inthatregion,buttheybecomedistinguishableatlargeradii.
density for all three to 3 1020MeV4. (Of course, no
×
stable neutronstar can have this high a centraldensity.)
The neutron and electroweak stars are almost indistin- tweentheburningandtherefuelingisachieved. Ifso,the
guishable at small radii because particle masses are not star enters a quasi-stable state that can last for a while.
important in that region, but they become distinguish- The question of dynamical stability will be addressed in
able at large radii. Furthermore, we are currently treat- future work.
ingthequarksintheelectroweakstarasadegenerategas
of non-interacting particles. Adding in the interactions,
whichislikelytoimprovethestabilityoftheelectroweak LOCATION OF THE NEUTRINOSPHERE
star, is the subject of a future work.
Thenon-perturbativeelectroweakinteractionsprovide Neutrinos near the centralregionhave very high ener-
the source of energy that keeps the star from collapsing. gies (but below the weak scale) and the local density is
As the baryons are burned, gravitationalcollapse brings extremely high. Their mean free path in this region is
more material to the central part of the core feeding the thus very short, and they cannot stream freely from the
fire. We assume that eventually a quasi-equilibrium be- star. However,as the local density falls with radius,and
5
the neutrinos’ average energy decreases, their mean free
path increases. We define the neutrino escape radius (as 0
10
a function of neutrino energy) as the distance from the
center at which a neutrino with that energy has a mean
freepathgreaterthanthethicknessoftheoverlyingmat-
tiserthaendsectaonftshuecrhefsoprheeersiccaapleshtehlelssftoarr.nTeuhterinneoustorifneonseprhgeirees hift 10-1
s
d
characteristic of the local temperature. e
r
Now we study the neutrino mean free path in dense
-2
10
media as presented in [8]. We assume that the cross
section of neutrino scattering is [9]
σi G2Fs G2FEνEi. (12) 10-3 -4 -3 -2 -1 0 1
∼ π ∼ π 10 10 10 10 10 10
radius (km)
Here, G 1.166 10−5GeV−2 is Fermi constant, s is
F
∼ ×
thecenterofmassenergy,E istheenergyoftheparticle
i FIG.4: Theredshiftfactorpgtt(r)vs. radiusofthestar. The
i,whileE istheneutrinoenergy. Themeanfreepathis
ν redshiftnearthecenterismuchlargerthanatthesurface. A
1 particle with the original energy of 100GeV near the center
= σini (13) carriesawayonly165MeVasitleavesthesurface. Opacityof
λ
i thedense medium will further reducethis energy.
X
Unlike any other active star, particles propagating
through the electroweak stars suffer large gravitational propagating from the center to the surface. This energy
redshift. Thus Eν is not a constant, but changes as loss will in turn further change the neutrino mean free
path. Properly calculated neutrino energy (plotted in
g (r )
E (r)= tt 0 E (r ). (14) Fig. 7) will imply that the neutrino mean free path at
ν ν 0
p√gtt(r) the neutrinosphere is λ 0.1km.
∼
With the redshift taken into account, the neutrinos
The redshift ratio is shown in Fig 4. Obviously, the red-
have roughly the same energy as the background they
shift effect near the center is much larger than at the
are propagatingin, which is true in generalfor the other
surface. Since inside the starǫ 3P,the redshift canbe
∼ types of particles. If a particle has the energy density ǫ,
estimated from Eq. (10). It is
itsenergycanbeestimatedas ǫ1/4 P1/4. Comparing
∼ ∼
P(r) 1/4 thiswithEq.(15),theenergyofaneutrinodecreaseswith
E E (r ) (15) the distance the same way as the background. Thus,
ν ν 0
∼ P(r )
(cid:18) 0 (cid:19) the neutrinos coming from the center and arriving to a
The pressure is about (100GeV)4 near the center, and certainpointinsidethestarhaveroughlythesameenergy
about(100MeV)4nearthesurface(beforethebagenergy as the neutrinos that were emitted by thermal processes
at that point.
changes the pressure a lot). Therefore, a particle retains
only a fraction 10−3 of its original energy. Though in-
stanton processes release huge amounts of energy at the
TRANSPORT OF ENERGY THROUGH THE
center, the star can emit much more moderate amounts.
STAR, ENERGY RELEASE RATE, AND THE
The result is shown in Fig. 5. A particle with the origi-
LIFETIME OF AN ELECTROWEAK STAR
nalenergyof100GeVnearthe centerindeedcarriesonly
165MeVasitleavesthe surfacedue to theredshift. This
We would like now to estimate the (neutrino) lumi-
energywillbe further reduceddue to opacityofthe high
nosity of the electroweak star. A naive luminosity (i.e.
density medium that neutrinos are traveling through.
energyreleaserate)canbeobtainedfromtheblack-body
Wenowusethisenergycorrectedfortheredshifttocal-
radiation at the stars core
culate the mean free path on the surface. From Eq. (12)
and Eq. (13) the mean free path on the surface is
σ ǫ πR2 (17)
Lν ≤Lbb ∼ bb ν core
1 G2ǫ
F Eν(Rsurface) (16) This places an upper bound of about 3 1041MeV2 =
λ ∼ π ×
7 1056erg/s. At this rate it would take less than a
Eq. (16) yields λ 1012MeV−1 10−4km. At the core, se×cond to release 1M⊙. Fortunately, Eq. (17) is a severe
Eq.(16)yields λ ∼10−14m. An∼eutrino withthis energy over-estimate of the luminosity. It does not take general
∼
cannotleavethestarfreely,andmustloseitsenergywhile relativityintoaccount,nordoesitallowforthefactthat
6
released at the surface do not crucially depend on pre-
5 cise details of the electroweak processed at the core, i.e.
10
the star is capable of burning much more quarks than it
actually does.
We now calculate the neutrino energy release rate at
V) 4
e 10 infinitymoreaccuratelytakingintoaccounttherelativis-
M
y ( tic transport of energy through the star. Gravitational
g
r redshiftandtimedelayalongwiththeenhancedneutrino
e
En 3 interaction cross section at these energies will affect the
10
energy transport through the star and make the energy
releaserate moderate at the surface of the star. Imagine
two surfaces inside the star whose radii are one λ apart.
102 The outward flux of neutrinos will be partially canceled
-5 -4 -3 -2 -1 0 1
10 10 10 10 10 10 10 bythebackwardfluxduetoneutrinocollisionswithother
radius (km) particles, and we are interested only in the net outward
flux. The relativistic transport of neutrino energy can
FIG. 5: Neutrino energy as a function of the distance from then be described by
thecenterwhereonlyeffectsofgravitationalredshiftaretaken
intoaccount(noenergylossduetointeractions). Theenergy d(S(r)g (r))
4πr2λ tt = L. (20)
on the surface is about 1000 times smaller than the original
g (r)dr −
energy at the center. − rr
p
Here S(r) dE(r)/(dtdA) is the energy flux at the ra-
≡
dius r, A is the area, while L dE/dt is the energy
the luminosity depends not just on the temperature and ≡
releaserate at infinity (which is about the same order as
densityattheneutrino-spherebutalsoontheirgradients,
the intensity at the neutrinosphere). The factor g (r)
since those determine the net outward flux of energy. tt
describes both the redshift and time delay. The factor
First we will estimate the energyrelease rate when we
g (r) takes into account that the space-time inside the
includeredshifteffectsincalculatingtheratiodE/dt. An rr
star is not flat. The energy flux S(r) can be found from
upper bound on the energy release rate can be obtained
the energy density ǫ given by (4) knowing the energy
fromthefreefalltimeoftheincomingquarkshellintothe ν
(i.e. chemical potential) of neutrinos E . The metric
electroweak-burningcore. Thisdeterminesthemaximum ν
coefficients are given by Eq. (10) and (11). It is then
rate at which the electroweak core burning can be fed.
straightforward to calculate L, given all the other pa-
The free fall acceleration over a short distance given by
rameters.
the mean free path λ is roughly given by a = M /r2 ,
ew ew Theneutrinomeanfreepathisnotaconstantthrough-
wherer referstoaradiusonemeanfreepath(λ)away
ew
out the star,but changesfrompoint to point. We there-
fromtheelectroweakcore(relativisticcorrectionswillnot
fore set a practical definition of the location of the neu-
change the result significantly). Therefore, the energy
trinosphere as R with
release rate can be no more than nsph
Rstar dr
(cid:18)ddEt (cid:19)max ∼4πre2ws2λMre2(wrew)ǫ(rew)gtt(rew), (18) ZRnsph λ(r) =1, (21)
where λ(r) is given by Eq. (16). This definition requires
This is just 1027MeV2, which is already much lower
thataneutrino,atagivenradiusrandagivenenergyE ,
than the earlier upper limits obtained from Eq. (17). ν
interacts only once with other particles before it leaves
We compare this rate with the electroweak baryogene-
the star.
sis/baryodestructioninteractionrate,which is abovethe
With these definitions, we cancalculate intensity L as
electroweak phase transition proportional to T4 [10]
the function of the radius of the neutrinosphere R .
nsph
ddEtw ∼4πre2w×20α5ewT4 (19) wFrhoimchtihnetcuornndigtiiovnes(2th1e),rweequfiinreddtheenenreguytrflinuoxeSn(err)gyfoErνa,
givenR . TheresultisplottedinFig.6. Largerradius
nsph
This rate is about 1034MeV2, which is much larger than R implies lower energy density, which means that
nsph
the rate in (18). Therefore we can reasonably assume higher energy neutrinos can escape. The energy release
that as quarks reach the electroweak core, they are con- rate therefore increases with the radius of the shell and
vertedintoneutrinosinstantaneously. Otherwise,thein- it is maximal at the surface of the star. However, there
falling matter would pile up and form a black hole. This itexceedsthe limitfromthe quarkfreefall. Thisimplies
result also implies that the star’s structure and energy that the neutrino release shell must be inside the star.
7
25
10
20 V) 104
10 e
M
2)V 1015 o ( 102
e n
(y M 1010 utri 100
Intensit 110005 y of ne 10-2
g -4
10-5 ner 10
10-10 E 10-6
-15 -8
10 -4 -3 -2 -1 0 10 -4 -3 -2 -1 0
10 10 10 10 10 10 10 10 10 10
radius (km) radius (km)
FIG. 6: The energy release rate vs. neutrino release radius: FIG. 7: The neutrino energy at the surface of the central
The energy release rate increases with the radius of the neu- electroweak core vs. the radius of the neutrino release shell
trino release shell (i.e. neutrinosphere). The maximum en- (i.e. neutrinosphere) Rnsph. From Fig. 6 we see that the
ergy release rate is on the surface, but there it exceeds the energy release rate increases with the Rnsph. Higher energy
limit from the quark shell free fall into the electroweak core release rate needs the source of higher energy. The energy
(whichminimizesthedenominatorin(17)). Thisimpliesthat of neutrinos produced at the core therefore increases with
theneutrino release shell must be inside thestar. the radius of release shell. However, the energy is already
larger than 500GeV at 8.2km. The star cannot support this
amount of energy implying that its radius must be smaller
At the neutrinosphere, R , the following must be than that. In fact, 300GeV neutrinos produced by baryon
nsph
true, 4πr2S(r = R )g (r = R ) = L, since there numberviolating processes at the core imply that the radius
nsph tt nsph
of the neutrinorelease shell is 8.1km.
is no backward flux of neutrinos (all the neutrinos that
reachR leavethestarfreely). Fromtherewecanfind
nsph
the neutrino energy, E , at the central electroweak core
ν
needed to support a certain intensity at the surface of
the star. Fig. 7 shows the results. Higher energy release
Many decades have elapsed since modeling of the core
rate needs the source of higher energy. If the neutrino
collapsesupernovamechanismbeganindetail. However,
releaseradiusis8.1km,thentheoriginalneutrinoenergy
the exact mechanism remains elusive so far. It appears
must be about 300GeV,which is equalto the energyper
that the explosion of supernova is a multiscale and mul-
neutrinothattypicalelectroweakprocessescanreleaseat
tiphysics event, where gravity, neutrino transport, fluid
the center. We therefore conclude that the radius of the
instabilities,rotation,magneticfields etcareallveryim-
neutrino release shell can be at most 8.1km. If the neu-
portant ingredients. The existing models failed to pro-
trino escape radius is about this size, the energy release
duce the supernova explosion. It will be of high impor-
rate is about 1024MeV2. Integrating this, we find that it
tance to check if the electroweakphysics can help super-
takes about 1015sec (10 million years) to release energy
novaeexplodebyprovidingaverypowerfulenergysource
equivalent of 1M⊙. This is the minimal life-time of the that was neglected so far in the studies of this problem.
electroweak star, provided that the electroweak burning
The expected final outcome of the stability analysis of
does not stop before all the quark fuel is spent.
the electroweak stars sequence will perhaps yield a sta-
In our calculations we neglected the fact that some bility curve. The possible outcomes are that the whole
fraction of energy is carriedaway by photons, since they electroweak sequence is stable or unstable. However, a
have shorter mean free path. Further, though the net genericoutcomewithdifferentregionsinparameterspace
leptonnumber is conservedinthis region,the number of correspondingto stable andunstable starsdepending on
neutrinos and antineutrinos are not conserved. We also the initial conditions is more likely. In unstable regions,
ignoredtheeffectsofenergytransportduetoconvection. theelectroweakstarswithacertaininitialmassandcen-
Finally, neutrino energy was calculated from the chemi- tral density could be unstable to explosions. These ex-
cal potential change, instead of the temperature change. plosions could be in the form of gamma ray bursts or
However, we assume that these effects will not change perhaps supernova explosions. To find out the answer, a
the order of magnitude estimate. very detailed modeling of the electroweak star is needed
On a completely different side, it is not difficult to takingallthenecessaryingredientslikegravity(withfull
imagine that the electroweak physics could be responsi- GReffects),neutrinotransport,fluidinstabilitiesetcinto
ble for some of the Type II supernova explosions. account.
8
CONCLUSIONS be expected from the selfsimilar solutions (homologous
collapse,PPNandothercommonlyusedapproximations
failinthisregime). Inparticular,theFig. 6in[13]shows
We studied the possibility of the existence of a new
that, with appropriate initial conditions (say those of a
phase in the stelar evolution. After all of the thermonu-
neutron star), the electroweak densities can be reached
clear fuel is spent, and possibly after the supernova ex-
before the ”gravothermal catastrophe”, i.e. black hole
plosion, but before the remaining mass crosses its own
formation. Oncethatstageisreached,electroweakburn-
Schwarzschildradius,thetemperatureofthecentralcore
ingwillgetignitedandthe collapsewillbe stoppedfor a
may surpass the electroweak symmetry restoration tem-
while. This certainly does not imply that the whole star
perature. Atthispoint,non-perturbativebaryon-number
is at the electroweak density, in contrary, it is not much
violating interactions in the ordinary SU(2) U(1) elec-
× differentfrom the ordinaryneutronstar,as the TOVso-
troweakstandardmodelbecomeunsuppressed. Thecon-
lution demonstrates.
sequentrelaxationofthestellarbaryonnumberchemical
One intriguing possibility is that the authors of [14]
potential, i.e. the conversion of baryons to anti-leptons,
and [15] are correct that gravitational collapse of a stel-
mayprovideasourceofpressurewhichcanbalancegrav-
lar mass does not result in a black hole but in an object
ity. We constructed a solution to TOV equation whose
of very high density and temperature toward its center.
central pressure is non-singular (and thus not a black
Then one would not have to worry about the collaps-
hole). We showed that the in-falling matter gets con-
ing object crossing its own Schwarzschild radius before
verted into neutrinos at the rate much faster than the
reaching electroweak densities.
free-fall rate, which indicates that the mater burns be-
Finally, electroweakphase transitionmay be triggered
forecrossingitsownSchwarzschildradius(assumingthat
insideaneutronstarbyamechanismreminiscentofsono-
the core is not within the Schwarzschild radius itself).
luminescence, where non-linear waves (e.g. star-quakes)
Our analysis shows that lifetime of this new phase can
may concentrate huge energy within some small volume
be as long as 10 million years which we propose to call
[16].
an electroweak star. We emphasize that the electroweak
We emphasize further that electroweak stars do not
star relies only on the Standard Model physics for its
have to be formed exclusively in stellar matter collapse.
existence.
Primordial black holes are formed in large density fluc-
Electroweakstarswouldbeanexcitingadditiontothe tuations in the early universe, with masses ranging from
diverse menagerie of astrophysical bodies that the uni- onePlanckmasstoafewsolarmasses,whichisthemass
verse provides (see also recent proposals in [11]). Never- within the horizon at the QCD phase transition (phase
theless, considerable work remains to be done before we transitions usually produce fluctuations large enough to
can claim with confidence that such objects will form in
cause the collapse of the whole causally connected re-
the natural process of stellar evolution, or that they will gion). Electroweak densities are reached very easily in
indeed burn steadily for an extended period. Similarly, such violent processes where usual homologous approx-
assessing the visibility of these fascinating new objects
imation clearly fails. In that case, however, the elec-
requires a careful modeling of their outer structure to troweak stars should be very long lived in order to be of
determine the neutrino/antineutrino [12] and photon lu- astrophysical significance. One more caveat is of course
minosity and spectrum. thatsofarnoprimordialblackholeshavebeenobserved.
While we constructed a non-singular solution to TOV Studying the details of the the electroweak star
equations which describes the electroweak star, the cru- physics, may give some clues for understanding of the
cial next step would be to verify that a full time de- supernova explosion mechanism and phenomenology on
pendent evolution will take the collapsing system to this the side track. Also, the electroweak processes give an
configuration. This is not unlikely to happen since the effectivemechanismoferasingbaryonnumberfromacol-
global parameters of the electroweak star are not much lapsing star before it becomes a black hole, which may
different from those of the neutron star. The radius is be interesting from the point of view of the information
8.2km and the mass is 1.3 Solar masses. What is differ- loss paradox (see discussion in [17]).
entisthecentralcorewhichisatorabovetheelectroweak SeeingthegravitationalcollapseasaninverseBigBang
density. In gravitational collapse, the collapsing object process, one might ask what happens when the next
is not crossing its own Schwarzschild radius at once as phase transition is hit, i.e the Grand Unifying Theory
a whole. The simulations show that the central density (GUT) phase transition. Our preliminary estimates in-
is much higher than the rest of the star (since the outer dicate that the core of such an object should be micro-
layers push the inner layers stronger and stronger). It scopic. It is very unlikely that this stage will be reached
appears that it is not difficult to achieve central density before the object crosses its own Schwarzschildradius.
of the order of the electroweak phase transition. For ex- DS acknowledges the financial support from NSF,
ample,in[13],the authorsfindthatdensitiesachievedin grantnumberPHY-0914893. GDSandALthanksCERN
gravitational collapse are much larger than what would for its hospitality. GDS is supported by a grant from
9
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