Slowly rotating Q-balls

TL;DR

This paper explores the existence of long-lived, metastable rotating Q-balls with small angular momentum across various scalar theories, extending understanding of their potential cosmological roles.

Contribution

It demonstrates the possibility of metastable rotating Q-balls with small angular momentum in all scalar theories supporting non-rotating Q-balls, broadening the scope of their theoretical and cosmological significance.

Findings

Metastable rotating Q-balls can exist with small angular momentum.

These solutions are relevant for scalar theories supporting non-rotating Q-balls.

Implications for cosmological models involving boson stars and Q-balls.

Abstract

Q-balls are non-topological solitons arising in scalar field theories. Solutions for rotating Q-balls (and the related boson stars) have been shown to exist when the angular momentum is equal to an integer multiple of the Q-ball charge $Q$. Here we consider the possibility of classically long-lived metastable rotating Q-balls with small angular momentum, even for large charge, for all scalar theories that support non-rotating Q-balls. This is relevant for rotating extensions of Q-balls and related solitons such as boson stars as it impacts their cosmological phenomenology.

Equations

$$f(r,\theta ,\varphi )=g(r,\theta )\exp (iN\varphi ),$$

$$\mathcal{L}={\left|{\partial }_{\mu }\Phi \right|}^2-U(|\Phi {|}^2),$$

$$0\le \sqrt{\frac{2}{{\varphi }_0^2}U\left(\frac{{\varphi }_0^2}{2}\right)}\equiv {\omega }_0<m_{\varphi }.$$

$$\Phi (t,\vec{x})={\Phi }_0(r=|\vec{x}|)e^{-i\omega t}\equiv \frac{{\varphi }_0}{\sqrt{2}}f(r)e^{-i\omega t},$$

$$-{\omega }^2{\Phi }_0-{\nabla }^2{\Phi }_0+\frac{\partial U(|{\Phi }_0{|}^2)}{\partial {\Phi }_0^{\ast }}=0$$

$$\overline{r}$$

$$\overline{\omega }$$

$$Q$$

$$E$$

$$\text{d}E=\omega \text{d}Q,$$

$$L_z=-\int {\text{d}}^{3}x\left(\dot{\Phi }{\partial }_{\varphi }{\Phi }^{\ast }+{\dot{\Phi }}^{\ast }{\partial }_{\varphi }\Phi \right)$$

$$\Phi (t,r,\theta ,\varphi )=f(r,\theta )e^{iN\varphi }{e}^{-i\omega t}$$

$$L_z=iN\int {\text{d}}^{3}x\left({\Phi }^{\ast }\dot{\Phi }-{\dot{\Phi }}^{\ast }\Phi \right)=NQ.$$

$$\Phi =\frac{{\varphi }_0}{\sqrt{2}}\left[f(r)e^{-i\omega t}+\sum _{L,M}{h}_{LM}(r,t)Y_{LM}(\theta ,\varphi )\right],$$

$$E_{\omega ,\mu }$$

$$+\mu \left[L_z+\int {\text{d}}^{3}x({\dot{\Phi }}^{\ast }{\partial }_{\varphi }\Phi +\dot{\Phi }{\partial }_{\varphi }{\Phi }^{\ast })\right]$$

$$=\int {\text{d}}^{3}x[|\dot{\Phi }+i\omega \Phi +\mu {\partial }_{\Phi }\Phi {|}^2+|\nabla \Phi {|}^2$$

$$+U-{\left|i\omega \Phi +\mu {\partial }_{\varphi }\Phi \right|}^2].$$

$$\dot{\Phi }+i\omega \Phi +\mu {\partial }_{\varphi }\Phi =0,$$

$$h_{LM}(r,t)=h_{LM}(r)e^{-i(\omega +M\mu )t}.$$

$$\text{d}E=\omega \text{d}Q+\mu \text{d}{L}_z.$$

$$\frac{\text{d}{L}_x}{\text{d}t}=-\mu L_y,$$

$${\Phi }^{\ast }=\frac{{\varphi }_0}{\sqrt{2}}\left[f{e}^{i\omega t}+\sum _{L,M}{h}_{L,-M}^{\ast }(-1{)}^M{Y}_{LM}{e}^{i(\omega -M\mu )t}\right]$$

$$h_{LM}^{\pm }\equiv h_{LM}\pm (-1{)}^M{h}_{L,-M}^{\ast },$$

$$h_{LM}^{\pm \ast }=\pm (-1{)}^M{h}_{L,-M}^{\pm }.$$

$$0$$

$$+\left({\overline{\omega }}^2+M^2{\overline{\mu }}^2\right)h_{LM}^{+}+2\overline{\omega }M\overline{\mu }{h}_{LM}^{-}$$

$$-h_{LM}^{+}{\varphi }_0^2{\frac{\partial \overline{U}}{\partial (\Phi {\Phi }^{\ast })}|}_{\Phi ={\Phi }_0}-h_{LM}^{+}{f}^2{\varphi }_0^4{\frac{{\partial }^2\overline{U}}{\partial (\Phi {\Phi }^{\ast }{)}^2}|}_{\Phi ={\Phi }_0}$$

$$0$$

$$+\left({\overline{\omega }}^2+M^2{\overline{\mu }}^2\right)h_{LM}^{-}+2\overline{\omega }M\overline{\mu }{h}_{LM}^{+}$$

$$-h_{LM}^{-}{\varphi }_0^2{\frac{\partial \overline{U}}{\partial (\Phi {\Phi }^{\ast })}|}_{\Phi ={\Phi }_0}.$$

$$h_{LM}^{+}$$

$$h_{LM}^{-}$$

$$0$$

$$-h_{LM}^{+(0)}{\left[{\varphi }_0^2\frac{\partial \overline{U}}{\partial (\Phi {\Phi }^{\ast })}+f^2{\varphi }_0^4\frac{{\partial }^2\overline{U}}{\partial (\Phi {\Phi }^{\ast }{)}^2}\right]}_{\Phi ={\Phi }_0}.$$

$$0$$

Full text

††preprint: UCI-TR-2022-28

I Introduction

In a seminal paper Coleman:1985ki , Coleman proved that nontopological solitons called Q-balls exist in scalar field theories where the potential satisfies certain conditions. This was extended to supersymmetric models Kusenko:1997zq ; Enqvist:1997si , and Q-balls were shown to be good dark matter candidates in both supersymmetric Kusenko:1997si ; Kusenko:1997vp ; Kusenko:2001vu and nonsupersymmetric Krylov:2013qe ; Ponton:2019hux ; Bai:2019ogh ; Bai:2021mzu theories. Q-balls can be produced by the collapse of an Affleck-Dine condensate Multamaki:2002hv ; Tsumagari:2009na , or by the formation of miniclusters in the early universe which subsequently collapse Griest:1989cb ; Griest:1989bq ; Griest:1989bq ; Hiramatsu:2010dx ; Bai:2022kxq .

In these discussions, it is usually assumed that miniclusters with angular momentum can still collapse into Q-balls. This is a normal assumption for collapse to black holes or disks, for which any angular momentum is allowed. However, for rotating Q-balls Volkov:2002aj ; Campanelli:2009su ; Arodz:2009ye ; Shnir:2011gr ; Loiko:2018mhb (and the closely related boson stars Silveira:1995dh ; Kleihaus:2005me ; Kleihaus:2007vk ; Kleihaus:2011sx ; Liebling:2012fv ; Davidson:2016uok ; Herdeiro:2019mbz ; Collodel:2019ohy ; Delgado:2020udb ; Kling:2020xjj ; Gervalle:2022fze ; Siemonsen:2023hko ) the dominant paradigm is that the angular momentum must be an integer multiple of the charge $Q$. If this is the case, then the formation of large Q-balls would be drastically different from the usual collapse, since angular momentum would need to be shed in a very precise manner.

This would in fact be a puzzling scenario, since one would expect that a large classical object like a Q-ball could be given a small angular velocity, and hence a small angular momentum. Classically, at least, these are continuous quantities, and one should be able to make these arbitrarily small. Naturally, the angular momentum is quantized in the quantum theory, but even so, one would expect that it should be possible to place a small number of particles in a state of nonzero angular momentum, so that the angular momentum of the Q-ball does not scale with the total charge.

In this letter, we revisit these issues. We point out that the reason the angular momentum is found to scale with the charge is that in the literature, a specific ansatz is chosen for the rotating Q-ball profile in which it depends on the azimuthal angle $\phi$ only through an overall factor of $\exp(iN\phi)$, where $N$ is an integer. That is, the spatial profile is assumed to have the form Volkov:2002aj

[TABLE]

which leads to the angular momentum scaling with $Q$.

Here we show that the restriction (1) is too severe. We find that Q-balls can be constructed with a more general dependence on $\phi$. In these solutions, the angular momentum need not scale with the charge, and, classically, can be made arbitrarily small.

After reviewing nonrotating Q-balls in the following section, and setting up notation, we present our new ansatz for rotating Q-balls in Sec. III. These involve a new parameter $\mu$, which we show to correspond to the angular velocity of the Q-ball. For small angular velocities (Sec. IV), we find an analytical solution to the equations of motion of the Q-balls to leading order in $\mu$.

We find the energy of this leading order solution is lower than that of a soliton whose angular momentum is carried by external particles (Sec. V). This implies that our ansatz is stable to dissociation, and that there must exist a Q-ball solution with nonzero angular momentum. We close with a discussion of our results in Sec. VI.

II Non-rotating Q-balls

In theories of a complex scalar field $\Phi(t,\vec{x})$ with $U(1)$-invariant Lagrangian

[TABLE]

solitonic solutions called Q-balls exist if the function $U(|\Phi|^{2})/|\Phi|^{2}$ has a minimum at $|\Phi|=\phi_{0}/\sqrt{2}$ with $0<\phi_{0}<\infty$, and

[TABLE]

The Q-ball solutions are found by making the spherically-symmetric ansatz

[TABLE]

and solving the equation of motion

[TABLE]

for the radial profile $\Phi_{0}(r)$. This equation can be shown to have a solution when $\omega_{0}<\omega<m_{\phi}$ Coleman:1985ki . It is often preferable to use dimensionless variables

[TABLE]

for analytic and numeric methods to construct these and related solitons Heeck:2020bau ; Heeck:2021zvk ; Heeck:2021gam ; Heeck:2021bce ; Almumin:2021gax . The Q-ball charge $Q$ and energy $E$ are given by the integrals

[TABLE]

For fields that satisfy the equations of motion (5) we have

[TABLE]

which indicates that $\omega$ acts like a chemical potential Nugaev:2019vru . Q-balls that are stable against dissociation and fission have an increasing $Q$ for decreasing $\omega$, with $Q\to\infty$ for $\omega\to\omega_{0}$ Coleman:1985ki ; PaccettiCorreia:2001wtt .

III Rotating Q-balls

Clearly, rotating Q-balls cannot be spherically symmetric, or else the angular momentum $\vec{L}$ would be zero. In particular, we have for the $z$ component of $\vec{L}$

[TABLE]

in spherical coordinates. For a nonzero $L_{z}$, the field $\Phi$ must therefore depend on the azimuthal angle $\phi$.

In constructing rotating solutions, typically studies have considered a profile of the form Volkov:2002aj ; Campanelli:2009su ; Arodz:2009ye ; Shnir:2011gr ; Loiko:2018mhb ; Silveira:1995dh ; Kleihaus:2005me ; Kleihaus:2007vk ; Kleihaus:2011sx ; Liebling:2012fv ; Davidson:2016uok ; Herdeiro:2019mbz ; Collodel:2019ohy ; Delgado:2020udb ; Gervalle:2022fze

[TABLE]

with integer $N$, which leads to

[TABLE]

Numerical solutions for $f(r,\theta)$ have been found, providing evidence for the existence of rotating Q-balls whose angular momentum scales with the charge.

However, there is no compelling reason for the $\phi$ dependence to take the simple form of (11). To allow for the existence of Q-balls with $|L_{z}|\ll Q$, we consider a more general ansatz where the profile contains different components with different $\phi$ dependence:

[TABLE]

where the $Y_{LM}$ are the usual spherical harmonics; an $L=0$ term can be absorbed into $f(r)$ so the sum begins with $L=1$. As these ansatze are continuously connected to the non-rotating solution $\Phi_{0}e^{-i\omega t}$, one expects that the angular momentum can be made arbitrarily small, unlike in the ansatz (11). For example if the $h_{LM}$ are small, this could correspond to the introduction of a few rotating particles to a nonrotating Q-ball.

To motivate a suitable ansatz for the time dependence, we note that the ground states of rotating Q-balls should have the lowest energy with a fixed charge $Q$ and angular momentum $L_{z}$. These are found by introducing two Lagrange multipliers $\omega$ and $\mu$, and minimizing the functional

[TABLE]

Minimizing the first term leads to

[TABLE]

which ensures that charge, angular momentum, and energy are time independent. Equation (17) also implies that

[TABLE]

For fields that satisfy the equations of motion, we find that Eq. (9) is generalized to

[TABLE]

Since $\mu$ is conjugate to $L_{z}$, it should be related to the angular velocity of the soliton about the $z$-axis. This is also seen in the form of our ansatz (18), which depends on the combination $\phi-\mu t$. Finally, it can be shown that

[TABLE]

which correspond to the the Euler equations of a rotating rigid body in the absence of external torques, when $\mu$ is taken to be the angular velocity about the $z$-axis.

IV A solution at small $L_{z}$

As we construct solutions to the ansatz (13), in the limit of small $L_{z}$, we use the spherical harmonic convention $Y_{L,M}^{\ast}=(-1)^{M}Y_{L,-M}\,.$ This allows us to write

[TABLE]

as a sum over the same spherical harmonics as in (18). The time dependence is absent from the equations of motion and the conserved quantities, making it useful to define the fields

[TABLE]

where

[TABLE]

In general, the $h^{\pm}_{LM}$ of different $(L,M)$ are all coupled since the equations of motion are nonlinear. However, for small perturbations $h_{LM}\to\varepsilon h_{LM}$ with $\varepsilon\ll 1$, the leading order functions decouple:

[TABLE]

For each $(L,M)$, these are two real coupled differential equations. To obtain solutions that correspond to small angular momentum, we further expand each $h^{\pm}_{LM}$ in a power series in $\overline{\mu}$. We note that $\overline{\mu}\to-\overline{\mu}$ leaves the equations of motion invariant if combined with $h^{\pm}_{LM}\to\pm h^{\pm}_{LM}$. The expansion therefore has the form

[TABLE]

We set all $h_{L0}=0$ since these modes do not contribute to $L_{z}$. To zeroth order in $\overline{\mu}$, the equation is

[TABLE]

To find a solution, we take the $\overline{r}$ derivative of equation (5) to find the new equation

[TABLE]

We therefore find that there is an exact solution of Eq. (28) for $L=1$:

[TABLE]

To first order in $\bar{\mu}$, we then find an equation for $h^{-}_{11}$

[TABLE]

One can verify, using equation (5), that this is solved by

[TABLE]

Thus for $L=1$ we have found a solution up to order $\overline{\mu}$

[TABLE]

We illustrate these profiles in Fig. 1 for a sextic potential.

We can iterate this procedure to higher orders in $\overline{\mu}$ to find solutions to arbitrary accuracy, at least when $\overline{\mu}\ll 1$. The angular momentum is positive for $\overline{\mu}>0$ and to leading order is

[TABLE]

with $L_{x,y}=0$. This shows we have a perturbative solution for a Q-ball with small $|L_{z}|$ independent of $Q$.

V Nonperturbative stability argument

In this section, we further argue for the existence of (slowly) rotating Q-balls. The idea is to compare the energies of (i) a specific localized ansatz for a rotating Q-ball, and (ii) the energy of a solution with a nonrotating Q-ball and some particles at infinity which carry the angular momentum. If the localized solution has a lower energy than the delocalized solution, then there must exist a stable localized solution for a Q-ball with nonzero angular momentum.

In fact, in a classical universe, a rotating Q-ball can reduce its angular momentum by emitting an arbitrarily small charge, which can carry an arbitrarily large angular momentum if it moves far away from the Q-ball with a small velocity. If the charge can be made arbitrarily small, the binding energy is also small, and this emission is energetically favorable. However, quantum mechanically this process is eventually unphysical since there is a minimal possible charge corresponding to the emission of a single particle. This must be taken into account in the stability analysis.

The delocalized configuration is the sum of the profile for the nonrotating Q-ball and the fields of the particles at infinity. As reviewed above, the nonrotating Q-ball has a form $\Phi_{0}=\phi_{0}f(r)e^{-i\omega t}/\sqrt{2}$. The corresponding charge and energy Heeck:2020bau are

[TABLE]

The first term in the energy scales like the volume of the Q-ball, while the second scales like the surface area and so is typically subleading.

The angular momentum is carried by the delocalized particles. Since these particles are far away, even a small velocity can produce a finite angular momentum. We can therefore minimize the energy by taking these particles to be essentially at rest. Their wave functions go as $e^{-im_{\phi}t}$, which means that the energy of these particles is $E=m_{\phi}\Delta Q$, where $\Delta Q\geq 1$ is the charge of the delocalized particles. The total charge is $Q_{0}+\Delta Q$, and the total energy is $E_{0}+m_{\phi}\Delta Q$.

The energy of the free particles could in some cases be reduced by forming separate Q-balls. As an extreme example, consider two Q-balls of equal charge $\tfrac{1}{2}(Q_{0}+\Delta Q)$ with a small angular momentum about their center of mass. From the volume terms the energy goes like

[TABLE]

where $\omega^{\prime}>\omega$ because for stable Q-balls the charge is a monotonically decreasing function of $\omega$. This means that $\Delta E\geq\omega^{\prime}\Delta Q$.

For the localized solution we use an ansatz motivated by the previous section, and take the configuration to be

[TABLE]

where

[TABLE]

which leads to $L_{x}=L_{y}=0$ and

[TABLE]

We find that the charge is shifted from $Q_{0}$ by a positive amount

[TABLE]

and the energy is shifted from the nonrotating value by

[TABLE]

The energy of the localized solution is hence smaller than the delocalized solutions as long as $\omega<m_{\phi}$ (which is always true for a Q-ball) or $\omega<\omega^{\prime}$ (which is guaranteed for stable Q-balls), and if $\epsilon,\overline{\mu}$ are sufficiently small. This proves the existence of stable localized Q-ball solutions with small angular momentum.

VI Conclusion

We have analyzed Q-balls with nonzero angular momentum and found that there exist solutions where the angular momentum does not scale as an integer multiple of the charge. This agrees with expectations about large classical objects but contrasts with rotating Q-ball solutions found in the literature. Consequently, this vastly enlarges the possibilities for rotating Q-balls.

This is not simply a formal question; there are important phenomenological consequences. Specifically, the existence of this new class of Q-ball solutions is essential for the validity of standard calculations of the Q-ball relic density; without these solutions, only Q-balls with specific values of the angular momentum would be produced, and the relic density would presumably be much smaller.

There are still many questions regarding rotating Q-balls. We focused on small angular velocities in order to prove the existence of rotating Q-balls in this limit. It would be interesting to have a full characterization of these solutions for arbitrary angular momentum, perhaps using the methods of Kling:2017mif ; Kling:2017hjm . Extending these solutions to boson stars, oscillons, and other solitons in the literature would also be very interesting. We hope to return to these questions in future work.

Acknowledgements

We are grateful to Ruth Gregory, Eric Hirschmann, and Nicolás Yunes for helpful discussions. The work of Y.A. and A.R. is supported by the National Science Foundation Grant No. PHY-1915005. The research of Y.A. was supported by Kuwait University. J.H. is supported in part by the National Science Foundation under Grant No. PHY-2210428. C.B.V. is supported in part by the National Science Foundation under Grant No. PHY-2210067 .