Non-topological soliton

Quantum field theory has been developed to predict the scattering probability of elementary particles. However, in the mid 1970s it was found out{{according to whom|date=September 2016}} that this theory predicts one more class of stable compact objects: non-topological solitons (NTS). The NTS represents an unusual coherent state of matter, called also bulk matter. Models were suggested for the NTS to exist in forms of stars, quasars, the dark matter and nuclear matter.

A NTS configuration is the lowest energy solution of classical equations of motion possessing a spherical symmetry. Such a solution has been found for a rich variety of field Lagrangians. One can associate the conserved charge with global, local, Abelian and non-Abelian symmetry. It appears to be possible that the NTS configuration exists with both bosons as well as with fermions. In different models either one and the same field carries the charge and binds the NTS, or there are two different fields: charge carrier and binding field.

Image:grenrad.jpg

The spatial size of the NTS configuration may be elementary small or astronomically large, depending on the model fields and constants. The NTS size could increase with its energy until the gravitation complicates its behavior and finally causes the collapse. In some models, the NTS charge is bounded by the stability (or metastability) condition.

For a complex scalar field with the U(1) invariant Lagrange density{{cite journal | last=Coleman | first=Sidney | title=Q-balls | journal=Nuclear Physics B | publisher=Elsevier BV | volume=262 | issue=2 | year=1985 | issn=0550-3213 | doi=10.1016/0550-3213(85)90286-x | pages=263–283| bibcode=1985NuPhB.262..263C }}

: $\backslash mathcal\; L=|\backslash partial\_\backslash mu\; \backslash Phi|^2-U(|\backslash Phi|)\; \backslash ,$

the NTS is a ball with radius R filled with the field $\backslash Phi=(\backslash phi\_0/\backslash sqrt\{2\})e^\{i\backslash omega\; t\}$. Here $\backslash phi\_0$ is a constant inside the ball except for a thin surface coat where it sharply drops to the global U(1) symmetrical minimum of $U(|\backslash Phi|)$. The value $\backslash phi\_0$ is adjusted so that it minimises the energy of the configuration

: $E=\backslash Big(\; \backslash frac\{\backslash phi\_0^2\backslash omega^2\}\{2\}+U(\backslash frac\{\backslash phi\_0\}\{\backslash sqrt\{2\}\})\backslash Big)\; \backslash frac\{4\}\{3\}\backslash pi\; R^3.\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,(1)$

Since the U(1) symmetry gives the conserved current $j\_\backslash mu=-i(\backslash Phi^*\backslash partial\_\backslash mu\backslash Phi-\backslash partial\_\backslash mu\backslash Phi^*\backslash Phi),\; \backslash ,$

the ball possesses the conserved charge

: $Q=\backslash int\; j\_0\; d^3x\; =\; \backslash omega\backslash varphi\_0^2\backslash frac\{4\}\{3\}\backslash pi\; R^3.$

The minimization of the energy (1) with R gives

: $E=Q\backslash sqrt\{\backslash frac\{2U(\backslash varphi\_0/\backslash sqrt\{2\})\}\{\backslash phi\_0^2\}\}.\; \backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,(2)$

The charge conservation allows the decay of the ball into Q particles exactly. This decay is energetically unprofitable if the sum mass Qm exceed the energy (2). Therefore, for the NTS existence it is necessary to have

:

The thin wall approximation, which was used above, allows to omit the gradient term $E\_\backslash text\{surface\}$ in the expression for energy (1), since $E\_\backslash text\{surface\}\backslash sim\; U(\backslash varphi\_0/\backslash sqrt\{2\})R^2\backslash Delta\; R\backslash ll\; U(\backslash varphi\_0/\backslash sqrt\{2\})R^3$. This approximation is valid for $Q\backslash gg\; 1$ and is justified by the exact solution of the motion equation.

Image:Twof.jpg

The NTS configuration for a couple of interacting scalar fields{{cite journal | last1=Friedberg | first1=R. | last2=Lee | first2=T. D. | last3=Sirlin | first3=A. | title=Class of scalar-field soliton solutions in three space dimensions | journal=Physical Review D | publisher=American Physical Society (APS) | volume=13 | issue=10 | date=1976-05-15 | issn=0556-2821 | doi=10.1103/physrevd.13.2739 | pages=2739–2761| bibcode=1976PhRvD..13.2739F }} is sketched here.

The Lagrange density

: $\backslash mathcal\; L=|\backslash partial\_\backslash mu\backslash Phi|^2+\backslash frac\{1\}\{2\}(\backslash partial\_\backslash mu\backslash sigma)^2-g|\backslash Phi|^2\backslash sigma^2-\backslash frac\{\backslash lambda\}\{4\}(\backslash sigma^2-\backslash sigma\_0^2)^2$

is invariant under U(1) transformation of the complex scalar field $\backslash Phi.$ Let this field depends on time and coordinate simply as $\backslash Phi(\backslash vec\; \{r\},t)\; =(\backslash phi(r)/\backslash sqrt\{2\})e^\{i\backslash omega\; t\}$. It carries the conserved charge $Q=4\backslash pi\backslash omega\backslash int\backslash limits\_\{0\}^\{\backslash infty\}\backslash phi^2(r)r^2dr$. In order to check that the energy of the configuration is smaller than Qm, one should either to calculate this energy numerically or to use the variational method. For trial functions

$\backslash phi(r)=\backslash varphi\_0(\backslash sin\backslash omega\; r/r)$ and $\backslash sigma(r)=0$ for r < R,

: $\backslash phi(r)=0\backslash text\{\; and\; \}\backslash sigma(r)=\backslash sigma\_0\backslash Big(\; 1-\backslash exp(-(r-R)/l)\backslash Big)\backslash text\{\; for\; \}r\; >\; R,\; \backslash ,$

the energy in the large Q limit is approximately equal to

$E\backslash approx\; \backslash frac\{\backslash pi\; Q\}\{R\}+\backslash frac\{\backslash pi\}\{3\}\backslash lambda\backslash sigma\_0^4\; R^3$.

The minimization with R gives the upper estimation $E\_\{up\}=\backslash frac\{4\}\{3\}\backslash pi\backslash lambda^\{1/4\}Q^\{3/4\}\backslash sigma\_0$

for the energy of the exact solution of motion equations

$\backslash omega^2\backslash phi+\backslash vec\{\backslash partial^2\}\backslash phi-g\backslash sigma^2\backslash phi=0$ and $\backslash vec\{\backslash partial^2\}\backslash sigma-g\backslash phi^2\backslash sigma-\backslash lambda\backslash sigma(\backslash sigma^2-\backslash sigma\_0^2)=0$.

It is indeed smaller than $Qg\backslash sigma\_0$ for Q exceeding the crucial charge

: $Q\_\backslash min\; =\{\backslash Big(\backslash frac\{4\backslash pi\}\{3g\}\backslash Big)\}^4\backslash lambda.$

If instead of boson, fermions carry the conserved charge, an NTS also exists. At this time one could take

: $\backslash mathcal\; L=\backslash sum\_\{k=1\}^N\; \backslash left(\backslash frac\{i\}\{2\}\backslash overleftrightarrow\{\backslash overline\{\backslash Psi\}\_k\backslash gamma^\backslash mu\backslash partial\_\backslash mu\backslash Psi\_k\}-(m+g\backslash sigma)\backslash overline\{\backslash Psi\}\_k\backslash Psi\_k\; \backslash right)\; -\; U(\backslash sigma)+\backslash frac\{1\}\{2\}(\backslash partial\_\backslash mu\backslash sigma)^2.$

\,\,\,\,\,(3)

N is the number of fermion species in the theory. Q can't exceed N due to the Pauli exclusive principle if the fermions are in the coherent state. This time the NTS energy E is bound by

: $E\_\backslash text\{up\}=\backslash frac\{4\}\{3\}\backslash pi\backslash sqrt\{2\}U^\{1/4\}\; \backslash left(-\backslash frac\{m\}\{g\}\backslash right)N^\{3/4\},$

See Friedberg/Lee.{{cite journal | last1=Friedberg | first1=R. | last2=Lee | first2=T. D. | title=Fermion-field nontopological solitons | journal=Physical Review D | publisher=American Physical Society (APS) | volume=15 | issue=6 | date=1977-03-15 | issn=0556-2821 | doi=10.1103/physrevd.15.1694 | pages=1694–1711| bibcode=1977PhRvD..15.1694F }}

The condition

Image:fisstab.jpg

The condition of stability against the fission looks as follows:

:

It signifies that $E(Q\_1)+E(Q\_2)>E(Q\_1\; +\; Q\_2)$. This condition is satisfied for the NTS in examples 2.2 and 2.3. The NTS in example 2.1, called also Q-ball, is stable against the fission as well, even though the energy (2) does not satisfy (4): one has to recollect the omitted gradient surface energy and to add it to the Q-ball energy (1). Perturbatively, $E\_\backslash text\{surface\}\backslash propto\; R^2(Q)\backslash propto\; Q^\{2/3\}$. Thus

:

Another job $E\_\backslash text\{surface\}$ does, is to set $Q\_\backslash min$ for the thin-wall description of Q-ball: for small Q the surface becomes thicker, $E\_\backslash text\{surface\}$ grows and kills the energy gain $Q(m-\backslash sqrt\{U(|\backslash Phi|)/|\backslash Phi|^2\}\backslash Big\; |\_\backslash min)$. However the formalism for the thick-wall approximation has been developed by KusenkoA. Kusenko, CERN-Th/97-69, hep-th/9704073 who says that for small Q, NTS also exists.

As for quantum correction, it also diminishes the binding energy per charge $m-E(Q)/Q$ for small NTS, making them unstable. The small NTS are especially important for the fermion case, since it is naturally to expect rather small number of fermions species N in (3), and consequently, Q. For Q=2 the quantum correction decreases the binding energy by 23%.{{cite journal | last1=Friedberg | first1=R. | last2=Lee | first2=T. D. | title=Fermion-field nontopological solitons. II. Models for hadrons | journal=Physical Review D | publisher=American Physical Society (APS) | volume=16 | issue=4 | date=1977-08-15 | issn=0556-2821 | doi=10.1103/physrevd.16.1096 | pages=1096–1118| bibcode=1977PhRvD..16.1096F | url=https://digital.library.unt.edu/ark:/67531/metadc1409614/ }}

For Q=1 a calculation based on the path integral method has been carried out by Baacke.{{cite journal | last=Baacke | first=Jürgen | title=The effective action of a spin 1/2 field in the background of a nontopological soliton | journal=Zeitschrift für Physik C | publisher=Springer Nature | volume=47 | issue=4 | date=1990 | doi=10.1007/BF01552327 | pages=619–624| bibcode=1990ZPhyC..47..619B | s2cid=120656769 }}

The quantum energy has been derived as a time derivative of the one-loop fermion effective action

: $S\_\{\backslash text\{eff\}\}=-i\backslash ln\; \backslash det\backslash Big(\backslash frac\{i\backslash gamma^\{\backslash mu\}\{\backslash partial\}\_\{\backslash mu\}-\; g\backslash sigma(x)\}\{i\backslash gamma^\{\backslash mu\}\{\backslash partial\}\_\{\backslash mu\}-g\backslash sigma\_0\}\backslash Big).$

This calculation gives the loop energy of the order of binding energy.

In order to find the quantum correction following the canonical method of quantization, one has to solve the Schrödinger equation for the Hamiltonian built with quantum expansion of field functions. For the boson field NTS it reads

: $\backslash Phi=1/\backslash sqrt\{2\}(\backslash phi\_\{cl\}(\backslash vec\{r\}-\backslash vec\{R(t)\})+\backslash sum\_n\; q\_n(t)\backslash beta(\backslash vec\{r\}))e^\{i\; \backslash zeta(t)\},\; \backslash sigma=\backslash sigma\_\{cl\}(\backslash vec\{r\}-\backslash vec\{R(t)\})+\backslash sum\_n\; q\_n(t)\backslash alpha(\backslash vec\{r\}).$

Here $\backslash phi\_\{cl\}\backslash ,$ and $\backslash sigma\_\{cl\}$ are the solutions of the classical equation of motion, $\backslash vec\{R(t)\}$ represents motion of the mass center, $\backslash zeta(t)$ is the over-all phase, $q\_\{n\},\; n=5,6,\backslash dots,\backslash infty$ are the vibration coordinates, by analogy with the oscillator decomposition of photon field

: $A^\backslash mu=\; \backslash sum\_k\; a\_k^\backslash mu(t)e^\{i\backslash vec\; k\backslash vec\; r\}\; +c.c.$

For this calculation the smallness of four-interaction constant is essential, since the Hamiltonian is taken in the lowest order of that constant. The quantum decreasing of the binding energy increases the minimal charge $Q\_\backslash min$ making the NTS metastable between old and new values of this charge.

Image:Maxch.jpg

NTSs in some models become unstable as Q exceeds some stable charge $Q\_\backslash max:E(Q>Q\_\backslash max)>Qm$. For example, NTS with fermions carrying a gauge charge{{cite journal | last1=Lee | first1=Kimyeong | last2=Stein-Schabes | first2=Jaime A. | last3=Watkins | first3=Richard | last4=Widrow | first4=Lawrence M. | title=GaugedQballs | journal=Physical Review D | publisher=American Physical Society (APS) | volume=39 | issue=6 | date=1989-03-15 | issn=0556-2821 | doi=10.1103/physrevd.39.1665 | pages=1665–1673| pmid=9959828 | bibcode=1989PhRvD..39.1665L }} has $E(Q)\backslash propto\; (C\_1Q^\{4/3\}+C\_2Q^2)^\{2/3\}$ exceeding Qm for Q large enough as well as for small one. Besides, the gauged NTS probably is unstable against a classical decay without conservation of its charge due to complicated vacuum structure of the theory.G. G. Petriashvilly, Yad. Phys. 50 (1989) 573.

Generally, the NTS charge is limited by the gravitational collapse:

.

If one adds to the Q-ball Lagrange density an interaction with massless fermion $\backslash Psi$

: $\backslash mathcal\; L\_\{\backslash Psi\}+\backslash mathcal\; L\_\{\backslash Psi\backslash Phi\}=\backslash Psi^+(i\backslash partial\_0+i\backslash vec\{\backslash partial\}\backslash vec\{\backslash sigma\})\backslash Psi-ig\backslash Phi\backslash Psi^+\backslash sigma\_2\backslash Psi^*+$

ig\Phi^*\Psi^T\sigma_2\Psi

which is also U(1) invariant assuming the global charge for boson twice as for fermion, Q-ball once created begins to emit its charge with $\backslash Psi$-pairs, predominantly from its surface. The evaporation rate per unit area{{cite journal | last1=Cohen | first1=Andrew | last2=Coleman | first2=Sidney | last3=Georgi | first3=Howard | last4=Manohar | first4=Aneesh | title=The evaporation of Q-balls | journal=Nuclear Physics B | publisher=Elsevier BV | volume=272 | issue=2 | year=1986 | issn=0550-3213 | doi=10.1016/0550-3213(86)90004-0 | pages=301–321| bibcode=1986NuPhB.272..301C }} {{nowrap begin}}.{{nowrap end}}

The ball of trapped right-handed Majorana neutrinos in $SU(2)\_L\; SU(2)\_R$ symmetric electroweak theory loses its charge (the number of trapped particles) through the neutrino-antineutrino annihilation by emitting photons from the whole volume.{{cite journal | last=Holdom | first=Bob | title=Cosmic balls of trapped neutrinos | journal=Physical Review D | publisher=American Physical Society (APS) | volume=36 | issue=4 | date=1987-08-15 | issn=0556-2821 | doi=10.1103/physrevd.36.1000 | pages=1000–1006| pmid=9958263 | bibcode=1987PhRvD..36.1000H }}A. D. Dolgov, O. Yu. Markin, Sov. Phys. JETP 71 (1990) 207.

The third example for a NTS metastable due to particle emission is the gauged non-Abelian NTS. The massive (outside the NTS) member of fermionic multiplet decays into a massless one and a gauged boson also massless in the NTS. Then the massless fermion carries away the charge since it does not interact at all with the Higgs field.

Three last examples represent a class for NTS metastable due to emission of particles which do not participate in the NTS construction. One more similar example: because of the Dirac mass term $m(\backslash overline\backslash Psi\_R\backslash Psi\_L\; +\; \backslash overline\backslash Psi\_L\backslash Psi\_R)$ , right-handed neutrinos convert to left-handed ones. That happens at the surface of neutrino ball mentioned above. Left-handed neutrinos are very heavy inside the ball and they are massless outside it. So they go away carrying the energy and diminishing the number of particles inside. This "leakage" appears to be much slower than the annihilation onto photons.A. E. Everett, Phys. Rev. D 10 (1974) 3126.

Image:grlim.jpg

As the charge Q grows and E(Q) the order of $M^2\_\{Pl\}R(Q)$, the gravitation becomes important for NTS. A proper name for such an object is a star. A boson-field Q-star looks like a big Q-ball. The way gravity changes E(Q) dependence{{cite journal | last=Lynn | first=Bryan W. | title=Q-stars | journal=Nuclear Physics B | publisher=Elsevier BV | volume=321 | issue=2 | year=1989 | issn=0550-3213 | doi=10.1016/0550-3213(89)90352-0 | pages=465–480| bibcode=1989NuPhB.321..465L | doi-access=free }} is sketched here. It is the gravity what makes for Q-star — stabilize it against the fission.

Q-star with fermions has been described by Bahcall/Selipsky.{{cite journal | last1=Bahcall | first1=Safi | last2=Lynn | first2=Bryan W. | last3=Selipsky | first3=Stephen B. | title=Fermion Q-stars | journal=Nuclear Physics B | publisher=Elsevier BV | volume=325 | issue=3 | year=1989 | issn=0550-3213 | doi=10.1016/0550-3213(89)90498-7 | pages=606–618| bibcode=1989NuPhB.325..606B | doi-access=free }} Similar the NTS of Friedberg & Lee, the fermion field carrying a global conserved charge, interacts with a real scalar field.

: $\backslash mathcal\; L=\backslash frac\{i\}\{2\}\backslash overleftrightarrow\{\backslash overline\backslash Psi\backslash gamma^\backslash mu\backslash partial\_\backslash mu\backslash Psi\}\; -\; m(\backslash sigma)\backslash overline\backslash Psi\backslash Psi-U(\backslash sigma)+\backslash frac\{1\}\{2\}(\backslash partial\_\backslash mu\backslash sigma)^2.$

The $\backslash sigma$ inside Q-star moves from a global maximum of the potential changing the mass of fermions and making them bound.

But this time Q is not the number of different fermion species but it is the large number of one and the same kind particles in the Fermi gas state. Then for the fermion field description one has to use $k\_F(r)$ instead of $\backslash Psi(r)$ and the condition of pressure equilibrium instead of the Dirac equation for $\backslash Psi(r)$. Another unknown function is the scalar field $\backslash sigma(r)$ profile which obeys the following motion equation : $\{\backslash vec\backslash partial\}^2\backslash sigma\; -\; (dm(\backslash sigma)/d\backslash sigma)\backslash langle\backslash overline\backslash Psi\backslash Psi\backslash rangle\; -\; dU(\backslash sigma)/d\backslash sigma=0$. Here$\backslash langle\backslash overline\backslash Psi\backslash Psi\backslash rangle$ is the scalar density of fermions, averaged on statistical ensemble:

:

Fermi energy of the fermion gas $\backslash varepsilon\_F=(k^2\_F+m^2)^\{1/2\}$.

Neglecting the derivatives of $\backslash sigma(r)$ for large Q, that equation together with the pressure equilibrium equation $P\_f-U=0$, constitute a simple system which gives $k\_F$ and $\backslash sigma$ inside the NTS. They are constant since we have neglected the derivatives. The fermion pressure

:

For example, if $m(\backslash sigma)=g\backslash sigma$ and $U(\backslash sigma)=\backslash lambda/4(\backslash sigma^2-\backslash sigma^2\_0)^2$, then $\backslash sigma=0$ and $k\_F=\backslash varepsilon\_F=(3\backslash pi^2\backslash lambda)^\{1/4\}\backslash sigma\_0$. That means fermions appear to be massless in the NTS. Then the full fermion energy $E\_f=3P\_f$. For an NTS with the volume $V$ and the charge $Q=V(k^3\_F/3\backslash pi^2)$, its energy is proportional to the charge: $E(Q)=(\; U+E\_f)V=\backslash varepsilon\_FQ$.

The described above fermion Q-star has been considered as a model for neutron starS. Bahcall, B. W. Lynn, S. B. Selipsky, Preprint SU-ITP-866 (1989).{{cite journal | last1=Bahcall | first1=Safi | last2=Lynn | first2=Bryan W | last3=Selipsky | first3=Stephen B | title=Are neutron stars Q-stars? | journal=Nuclear Physics B | publisher=Elsevier BV | volume=331 | issue=1 | year=1990 | issn=0550-3213 | doi=10.1016/0550-3213(90)90018-9 | pages=67–79| bibcode=1990NuPhB.331...67B | doi-access=free }} in the effective hadron field theory.

If the scalar field potential $U(\backslash sigma)$ has two degenerate or almost degenerate minima, one of them have to be the real (true) minimum in which we happen to leave. Inside NTS $\backslash sigma$ occupies another one. In such a model non-zero vacuum energy appears only at the NTS surface, not in its volume. This allows for the NTS to be very big without falling in gravitational collapse.

That is the case in the left-right symmetric electroweak theory. For a scale of symmetry breaking about 1 TeV, $\backslash nu$-ball of trapped right-handed massless neutrino might have the mass (energy) about 10⁸ solar masses and was considered as a possible model for quasar.{{cite journal | last1=Dolgov | first1=A. D. | last2=Markin | first2=O. Y. | title=Left-Right Symmetric Electroweak Theory and the Central Engine of Quasars | journal=Progress of Theoretical Physics | publisher=Oxford University Press (OUP) | volume=85 | issue=5 | date=1991-05-01 | issn=0033-068X | doi=10.1143/ptp/85.5.1091 | pages=1091–1104| bibcode=1991PThPh..85.1091D |doi-access=free}}

For the degenerate potential $U(\backslash sigma)=\backslash mu^2\backslash sigma^2(1-\backslash sigma/\backslash sigma\_0)^2/2$

both boson{{cite journal | last1=Friedberg | first1=R. | last2=Lee | first2=T. D. | last3=Pang | first3=Y. | title=Mini-soliton stars | journal=Physical Review D | publisher=American Physical Society (APS) | volume=35 | issue=12 | date=1987-06-15 | issn=0556-2821 | doi=10.1103/physrevd.35.3640 | pages=3640–3657| pmid=9957625 | bibcode=1987PhRvD..35.3640F }} and fermion{{cite journal | last1=Lee | first1=T. D. | last2=Pang | first2=Y. | title=Fermion soliton stars and black holes | journal=Physical Review D | publisher=American Physical Society (APS) | volume=35 | issue=12 | date=1987-06-15 | issn=0556-2821 | doi=10.1103/physrevd.35.3678 | pages=3678–3694| pmid=9957627 | bibcode=1987PhRvD..35.3678L }} soliton stars were investigated.

A complex scalar field could alone form the state of gravitational equilibrium possessing the astronomically large conserved number of particles.{{cite journal | last1=Friedberg | first1=R. | last2=Lee | first2=T. D. | last3=Pang | first3=Y. | title=Scalar soliton stars and black holes | journal=Physical Review D | publisher=American Physical Society (APS) | volume=35 | issue=12 | date=1987-06-15 | issn=0556-2821 | doi=10.1103/physrevd.35.3658 | pages=3658–3677| pmid=9957626 | bibcode=1987PhRvD..35.3658F }}J. J. Van der Bij, M. Gleiser, Preprint FERMILAB-Pub-87/41-A. Such objects are called minisoliton stars because of their microscopic size.

Could a system of the Higgs field and some fermion field of the Standard model be in the state of Friedberg & Lee NTS ? That is more possible for a heavy fermion field: for a such one the energy gain would be the most because it does lose its large mass in the NTS interior, were the Yukawa term $g\backslash sigma\backslash overline\backslash Psi\backslash Psi$ vanishes due to $\backslash sigma\backslash simeq0$. The more so if the vacuum energy in the NTS interior $U(0)=\backslash lambda\backslash sigma^4\_0/4$ is large, that would mean the large Higgs mass $m\_H=\backslash sigma\_0\backslash sqrt\{2\backslash lambda\}$. The large fermion mass implies strong Yukawa coupling $g$.

Calculation showsS. Dimopoulos, B. W. Lynn, S. Selipsky, N. Tetradis, Preprint CERN-TH.5761/90. that the NTS solution is energetically favored over a plane wave (free particle) only if $g>2.3$ for even very small $m\_H$. For

$m\_H$ =350 GeV (this is the point were $\backslash lambda=1$ for experimentally known $\backslash sigma\_0=$250 GeV) the coupling $g$ must be more than five.

The next question is whether or not multi-fermion NTS like a fermion Q-star is stable in the Standard model. If we restrict ourself by one fermion species, then the NTS has god the gauge charge. One can estimate the energy of gauged NTS as follows:

: $E=(4\backslash pi)^\{1/3\}(3/4)^\{5/3\}Q^\{4/3\}/R+\backslash beta\; e^2Q^2/R+(4\backslash pi\; R^3/3)\backslash lambda\backslash sigma^4\_0/4.$

Here $R$ and $Q$ are its radius and charge, the first term is the kinetic energy of the fermi-gas, the second is the Coulomb energy, $\backslash beta$ takes into account the charge distribution inside the NTS and the latest one gives the volume vacuum energy. Minimization with $R$ gives the NTS energy as a function of its charge:

: $E(Q)=4\backslash sigma\_0(\backslash pi\backslash lambda)^\{1/4\}/3\backslash left((4\backslash pi)^\{1/3\}(3/4)^\{5/3\}Q^\{4/3\}+\backslash beta\; e^2Q^2\backslash right)^\{3/4\}.$

An NTS is stable if $E(Q)$ is smaller than the sum of masses for $Q$ particles at infinite distance each from other. That is case for some $Q$, but such a $E(Q)$ dependence allows the fission for any $Q$.

Why could not quarks be bound in a hadron like in NTS. Friedberg and Lee investigated such a possibility. They assumed quarks getting huge masses from their interaction with a scalar field $\backslash sigma$. Thus free quarks are heavy and escape from detection. The NTS built with quarks and $\backslash sigma$ fields demonstrate static properties of hadrons with 15% accuracy. That model demands SU(3) symmetry additional to the color one in order to preserve the later unbroken so that QCD gluons get large masses by SU(3) symmetry breaking outside hadrons and also avoid detection.

Nuclei have been considered as NTS's in the effective theory of strong interaction which is easier to deal with than QCD.D. A. Hochron, Preprint CERN-TH-5991/91.

The way NTS's could be born by depends on whether or not the Universe carries a net charge. If it does not then NTS could be formed from random fluctuations of the charge. Those fluctuations grow up, disturb the vacuum and create NTS configurations.

If the net charge is present, i.e. charge asymmetry exists with a parameter $\backslash eta=(|\; n\_\backslash Phi-n\_\{\backslash overline\backslash Phi\}|)/\; (n\_\backslash Phi+n\_\{\backslash overline\backslash Phi\})$, NTS could be simply born as the space became divided onto finite regions of true and false vacuum during the phase transition in the early Universe. Those occupied by the NTS (false) vacuum are almost ready NTSs. The scenario of the region formation depends on the phase transition order.

Image:fopht.jpg

If the first order phase transition occurs, then nucleating bubbles of true vacuum grow and percolate, shrinking regions filled with the false vacuum. The later are preferable for charged particles to live in due to their smaller masses, so those regions become

NTSs.{{cite journal |last1=Jung |first1=Sunghoon |last2=Hong |first2=Jeong-Pyong |last3=Xie |first3=Ke-Pan |title=Fermi-ball dark matter from a first-order phase transition |journal=Physical Review D |date=2020 |issn=2470-0029 |volume=102 |issue=7 |page=075028 |doi=10.1103/PhysRevD.102.075028 | arxiv=2008.04430 |bibcode=2020PhRvD.102g5028H |doi-access=free }}

Image:thopht.jpg

In case of the second order phase transition as temperature drops below the crucial value $T\_c$ the space consist of interconnecting regions of both false and true vacua with characteristic size $\backslash xi\backslash propto\; T^\{-1\}\_c$. This interconnection "freezes out" as its rate becomes smaller than the expansion rate of the Universe at Ginzburg temperature $T\_G$, then the regions of two vacua percolate.

But if the false vacuum energy is large enough, $\backslash Lambda>0.8U\_M$ on the plot, the false vacuum forms finite clusters (NTS's) surrounded by the percolated true vacuum.{{cite journal | last1=Gelmini | first1=Graciela B. | last2=Gleiser | first2=Marcelo | last3=Kolb | first3=Edward W. | title=Cosmology of biased discrete symmetry breaking | journal=Physical Review D | publisher=American Physical Society (APS) | volume=39 | issue=6 | date=1989-03-15 | issn=0556-2821 | doi=10.1103/physrevd.39.1558 | pages=1558–1566| pmid=9959816 | bibcode=1989PhRvD..39.1558G | hdl=2060/19890007156 | hdl-access=free }}

The trapped charge stabilizes clusters against collapse.

Image:potbds.jpg

In the second scenario of the NTS formation the number of born $Q$-charged NTS's per unit volume is simply the number density of clusters holding $Q$ particles. Their number density is given{{cite journal | last=Stauffer | first=D. | title=Scaling theory of percolation clusters | journal=Physics Reports | publisher=Elsevier BV | volume=54 | issue=1 | year=1979 | issn=0370-1573 | doi=10.1016/0370-1573(79)90060-7 | pages=1–74| bibcode=1979PhR....54....1S | s2cid=41441907 }}

by $n(r)=\; br^\{-3/2\}\backslash exp\{(-cr)\}/V\_\backslash xi$, here b and c are constants of the order of unit, $r\backslash simeq\; (L/2\backslash xi)^3$ is the number of correlation volumes $V\_\backslash xi$ in a cluster of size $L$. The number of particle in a cluster is

$Q(r)\backslash simeq\; rV\_\backslash xi\backslash eta\; n\_Q$, here $n\_Q$ is the charge density in the universe at Ginzburg temperature. Thus big clusters are born very rarely and if the minimum stable charge $Q\_\backslash min$ is present, then overwhelming majority of born NTS carries $Q\_\backslash min$.

For the following Lagrange density with biased discrete symmetry{{cite journal | last1=Frieman | first1=Joshua A. | last2=Gelmini | first2=Graciela B. | last3=Gleiser | first3=Marcelo | last4=Kolb | first4=Edward W. | title=Primordial Origin of Nontopological Solitons | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=60 | issue=21 | date=1988-05-23 | issn=0031-9007 | doi=10.1103/physrevlett.60.2101 | pages=2101–2104| pmid=10038260 | bibcode=1988PhRvL..60.2101F }}

: $\backslash mathcal\; L=|\backslash partial\_\backslash mu\backslash Phi|^2+(\backslash partial\_\backslash mu\backslash sigma)^2/2-$

\lambda_1(\sigma^2-\sigma_0^2)^2/8-\lambda_2(\sigma-\sigma_0)^3\sigma_0/3-

h|\Phi|^2(\sigma-\sigma_0)^2-g|\Phi|^4-\Lambda

with

: $\backslash lambda\_1/\backslash lambda\_2\; =0.15,\backslash ,$

: $\backslash Lambda=0.6\; \backslash lambda\_1\backslash sigma\_0^4$ and $\backslash xi=1/\backslash lambda\_1T\_G,$

it appears to be $Q\_\backslash min\; =\; 18\backslash lambda\_1/h^2$ and

: $n(Q\_\backslash min)\backslash propto(\backslash eta/\backslash lambda\_1Q\_\backslash min)^\{3/2\}\backslash exp\{(-c\; Q\_\backslash min\backslash lambda\_1^3/\backslash eta)\}.$

The net charge could be also placed in the complex scalar field condensate $\backslash langle\; \backslash Phi\backslash rangle$ instead of free particles. This condensate could consist of spatially homogeneous $\backslash Phi=f(t)\backslash exp\{(i\backslash theta\; (t))\}/\backslash sqrt\; 2$ and

$|\backslash Phi|^2=f^2(t)/2$ provides its potential to be at minimum as the universe cools down and the temperature correction changes the form of the potential. Such a model was treated to explain the baryon asymmetry.S. Dodelson, L. M. Widrow, Phys. Rev. Lett. 41(1990)340.

If the field potential allows Q-ball to exist, then they could be born from this condensate as the charge volume density $j\_0=f^2\backslash partial\_0\backslash theta(t)$ drops in course of the universe expansion and becomes equal to Q-balls charge density.K. M. Benson, L. M. Widrow, Preprint HUTP-90/A054.

As follows from the equation of motion for $\backslash theta(t)$, this density $j\_0$ changes with the expansion as the minus third power of scale factor $a(t)$ for the expanding space-time with the differential length element $ds^2=dt^2-a^2(t)(dx^2+dy^2+dz^2)$.

Breaking the condensate onto Q-balls appears to be favorable over further dilution of the homogeneous charge density by expansion. The total charge in a comoving volume $Q=\backslash int\; j\_0a^3d^3x$ stays fixed of course.

The condensation of $\backslash Phi$ could occur at high temperature of the universe, due to the negative temperature correction to its mass: $m(T)=m^2-\backslash lambda\; ^2/2$ which provides with minimum its potential $U(|\backslash Phi|)=m^2|\backslash Phi|^2-\backslash lambda|\backslash Phi|^4/2+\backslash gamma|\backslash Phi|^6$. Here the last term is induced by the interaction $h\backslash chi^2|\backslash Phi|^2$ with additional field $\backslash chi$ that has to be introduced in order to satisfy the Q-ball existence condition

Once formed, the NTSs undergo complicated evolution, losing and acquiring the charge by interaction with each other and surrounding particles. Depending on theory parameters, they could either disappear at all or get statistical equilibrium and "freeze out" at some temperature of the universe, or be born "frozen out" if their interaction is slower than expansion rate at $T\_G$. In the first and the second cases, their up-to-date abundance (if any) has nothing to do with that at the moment of formation.{{cite journal | last1=Griest | first1=Kim | last2=Kolb | first2=Edward W. | last3=Massarotti | first3=Alessandro | title=Statistical fluctuations as the origin of nontopological solitons | journal=Physical Review D | publisher=American Physical Society (APS) | volume=40 | issue=10 | date=1989-11-15 | issn=0556-2821 | doi=10.1103/physrevd.40.3529 | pages=3529–3532| pmid=10011724 | bibcode=1989PhRvD..40.3529G | hdl=2060/19890016309 | s2cid=36256647 | hdl-access=free }}{{cite journal | last1=Frieman | first1=Joshua A. | last2=Olinto | first2=Angela V. | last3=Gleiser | first3=Marcelo | last4=Alcock | first4=Charles | title=Cosmic evolution of nontopological solitons | journal=Physical Review D | publisher=American Physical Society (APS) | volume=40 | issue=10 | date=1989-11-15 | issn=0556-2821 | doi=10.1103/physrevd.40.3241 | pages=3241–3251| pmid=10011692 | bibcode=1989PhRvD..40.3241F }}

Since an NTS is a composite object, it has to demonstrate properties different from those of a single particle, e.g. evaporation emission, excitation levels, scattering form-factor. Cosmic observations of such phenomena could provide the unique information about the physics beyond the ability of accelerators.

• Fermi ball
• Topological defect

{{reflist|33em}}

Category:Quantum field theory

Category:Solitons