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arXiv:0905.0125v3 [hep-th] 4 Aug 2009

Q-balls in flat potentials

Edmund J. Copeland∗and Mitsuo I. Tsumagari†

School of Physics and Astronomy, University of Nottingham,

University Park, Nottingham NG7 2RD, UK

We study the classical and absolute stability of Q-balls in scalar field theories with flat potentials

arising in both gravity-mediated and gauge-mediated models. We show that the associated Q-matter

formed in gravity-mediated potentials can be stable against decay into their own free particles as

long as the coupling constant of the nonrenormalisable term is small, and that all of the possible

three-dimensional Q-ball configurations are classically stable against linear fluctuations. Three-

dimensional gauge-mediated Q-balls can be absolutely stable in the “thin-wall-like” limit, but are

completely unstable in the “thick-wall” limit.

I. INTRODUCTION

Q-balls have recently attracted much attentions in cosmology [1] and astrophysics [2, 3, 4]. A Q-ball

[5] is a nontopological soliton [6] whose stability is ensured by the existence of a continuous global charge

Q (for a review see [7, 8, 9, 10, 11] and references therein), and a number of scalar field theory models

have been proposed to support the existence of nontopological solitons. They include polynomial models

[5], Sine-Gordon models [12], parabolic-type models [13], confinement models [14, 15, 16, 17], two-field

models [6, 18], and flat models [1].

From a phenomenological point of view, the most interesting examples are probably the supersymmetric

Q-balls arising within the framework of the Minimal Supersymmetric Standard Model (MSSM), which

naturally contains a number of gauge invariant flat directions. Many of the flat directions can carry baryon

(B) or/and lepton (L) number which is/are essential for Affleck-Dine (AD) baryogenesis [19]. Following

the AD mechanism, a complex scalar (AD) field acquires a large field value during a period of cosmic

inflation and tends to form a homogeneous condensate, the AD condensate. In the presence of a negative

pressure [20, 21], the condensate is unstable against spatial fluctuations so that it develops into nonlinear

inhomogeneous lumps, namely Q-balls. The stationary properties and cosmological consequences of the

Q-balls depend on how the Supersymmetry (SUSY) is broken in the hidden sector, transmitting to the

observable sector through so-called messengers. In the gravity-mediated [22] or gauge-mediated scenarios

[1], the messengers correspond respectively either to supergravity fields or to some heavy particles charged

under the gauge group of the standard model.

Q-balls can exist in scalar field potentials where SUSY is broken through effects in the supergravity

hidden sector [23]. These type of Q-balls can be unstable to decay into baryons and the lightest super-

symmetric particle dark matter, such as neutralinos [24], gravitinos [25, 26] and axinos [27]. Recently,

McDonald has argued that enhanced Q-ball decay in AD baryogenesis models can explain the observed

positron and electron excesses detected by PAMELA, ATIC and PPB-BETS [28]. By imposing an upper

bound on the reheating temperature of the Universe after inflation, this mode of decay through Q-balls

has been used to explain why the observed baryonic (Ωb) and dark matter (ΩDM) energy densities are

so similar [29, 30], i.e. ΩDM/Ωb= 5.65 ± 0.58 [31].

Scalar field potentials arising through gauge-mediated SUSY breaking [22] tend to be extremely flat.

Using one of the MSSM flat directions, namely the QdL direction (where Q and d correspond to squark

fields and L to a slepton field), which has a nonzero value of B − L and therefore does not spoil AD

baryogenesis via the sphaleron processes that violate B+L [30], Shoemaker and Kusenko recently explored

the minimum energy configuration for baryo-leptonic Q-balls, whose scalar field consists of both squarks

and sleptons [32]. It had been assumed to that point that the lowest energy state of the scalar field

corresponds to being exactly the flat direction; however in [32], the authors showed that the lowest energy

state lies slightly away from the flat directions, and that the relic Q-balls, which are stable against decay

into both protons/neutrons (baryons) and neutrinos/electrons (leptons) [33], may end up contributing to

the energy density of dark matter [29, 34]; thus, the Q-balls can provide the baryon-to-photon ratio [34],

∗ed.copeland@nottingham.ac.uk

†ppxmt@nottingham.ac.uk

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i.e. nb/nγ ≃ (4.7 − 6.5) × 10−10[35] where nband nγ are respectively the baryon and photon number

densities in the Universe.

In this paper we examine analytically and numerically the classical and absolute stability of Q-balls

using flat potentials in the two specific models mentioned above. In order to study the possible existence

of lower-dimensional Q-balls embedded in 3+1 dimensions, we will work in arbitrary spatial dimensions

D; although of course the D = 3 case is of more phenomenological interest. Previous work [21, 30, 36]

on the gravity-mediated potential has used either a steplike or Gaussian ansatz to study the analytical

properties of the thin and thick-wall Q-balls. Introducing more physically motivated ans¨ atze, we will

show that the thin-wall Q-balls can be quantum mechanically stable against decay into their own free

particle quanta, that both thin and thick-wall Q-ball solutions obtained are classically stable against

linear fluctuations, and confirm that a Gaussian ansatz is a physically reasonable one for the thick-wall

Q-ball. The one-dimensional Q-balls in the thin-wall limit are excluded from our analytical framework.

The literature on Q-balls with gauge-mediated potentials has tended to use a test profile in approximately

flat potentials. We will present an exact profile for a generalised gauge-mediated flat potential, and show

that we naturally recover results previously published in [22, 30, 34].

The rest of this paper is organised as follows. In Sec. II we briefly review the important Q-ball

properties that were established in [37]. Section III provides a detailed analyses for gravity-mediated

potentials, and in Sec. IV we investigate the case of a generalised gauge-mediated potential. We confirm

the validity of our analytical approximations with complete numerical Q-ball solutions in Sec. V before

summarising in Sec. VI. Two appendices are included. In Appendix A, we obtain an exact solution

for the case of a logarithmic potential, and in Appendix B, we confirm that the adoption of a Gaussian

ansatz is appropriate for the thick-wall Q-ball found in the gravity-mediated potentials.

II.THE BASICS

Here, we review the basic properties of Q-balls as described in [37] and introduce a powerful technique

that enables us to find the charge Q and energy EQof the Q-ball as well as the condition for its stability,

and characteristic slope γ(ω) ≡ EQ/ωQ where ω is defined through the Q-ball ansatz, which is given by

decomposing a complex scalar field φ into φ = σ(r)eiωt. σ is a real scalar field, r is a radial coordinate,

and therefore ω is a rotational frequency in the U(1) internal space. By scaling the radius r of the Q-ball

ansatz, which minimises EQ, we can find the characteristic slopes in terms of the ratio between the surface

energy S and the potential energy U of the Q-ball. When the characteristic slope, γ, is independent of

ω, we obtain the relation: EQ∝ Q1/γ. In general the charge, energy and Euclidean action Sωare given

by

Q = ω

?

VD

σ2;Sω=

?

VD

?1

2σ′2+ Uω

?

;EQ= ωQ + Sω,(1)

where our metric is ds2= −dt2+ hijdxidxj, the determinant of the spherically symmetric spatial

metric hij is defined by h ≡ det(hij), and we have used the following notation:

ΩD−1

?∞

U(σ)

?

VD≡

?dDx√h =

0dr rD−1, ΩD−1≡

generality, we can take positive values of ω and Q. By defining the effective potential Uωof a potential

2πD/2

Γ(D/2), σ′≡dσ

dr, and D is the number of spatial dimensions. Without loss of

Uω≡ U −1

2ω2σ2,(2)

the Q-ball equation is

σ′′+D − 1

r

σ′=dUω

dσ,

(3)

where σ(r) is a monotonically decreasing function in terms of r. Given a potential U(σ), which has a

global minimum at σ = 0, it is possible to show that Q-balls exist within the restricted range of ω [5]:

ω−≤ ω < ω+,

−≡

(4)

where we have defined the lower limit ω2

2U

σ2

??

σ+(ω−)≥ 0, σ+(ω) is the nonzero value of σ where

+≡

Uω(σ+(ω)) is minimised (see Fig. 1), and the upper limit ω2

d2U

dσ2

???

σ=0. The existence condition Eq. (4)

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restricts the allowed form of the potential U, which implies that the potential should grow less quickly

than the quadratic term (i.e. mass term) for small values of σ. The case ω−= 0 corresponds to degenerate

vacua potentials (DVPs), whilst ω−?= 0 has nondegenerate vacua (NDVPs). In [37] we examined the

case of polynomial potentials and restricted ourselves to the case of ω2

the potentials. In this paper we extend our analysis allowing us to investigate the case ω2

since the potentials include one-loop corrections to the bare mass m. Here, the potential which we will

consider in the gravity-mediated models, is U = Ugrav+ UNRwhere UNRis a nonrenormalisable term

(to be discussed below), and

+= m2where m is a bare mass in

+≫ m2, needed

Ugrav≡1

2m2σ2

?

1 + K ln

?σ2

M2

??

.(5)

Here, K is a constant factor arising from the one-loop correction and M is the renormalisation scale.

To proceed with analytical arguments, we consider the two limiting values of ω or σ0≡ σ(r = 0) which

describe

?

• thin-wall Q-balls when ω ≃ ω−or equivalently σ0∼ σ+(ω),

• thick-wall Q-balls when ω ≃ ω+or equivalently σ0≃ σ−(ω).

(6)

Note, this limit doe not imply that a thick-wall Q-ball has to have a large thickness that is comparable to

the size of the core size. For the extreme thin-wall limit, ω = ω−, thin-wall Q-balls satisfyEQ

In particular, Coleman demonstrated that a steplike profile for Q-balls, which generally exist for ω−?= 0,

satisfies γ = 1, which implies that the charge Q and energy EQare proportional to the volume, and he

called this Q-matter [5]. For absolutely stable Q-balls, the energy per unit charge is smaller than the rest

mass m for the field φ,

Q= γ(ω−)ω−.

EQ

Q

< m.(7)

Thus the Q-ball satisfying Eq. (7) is stable against free-particle decays because the Q-ball energy EQis

less than a collection of Q free-particles of total energy Efree= mQ. If the Q-ball has decay channels

into other fundamental scalar particles that have the lowest mass mmin, we need to replace m by mmin

in the absolute stability condition Eq. (7). In the opposite limit ω ≃ ω+, the Q-ball energy approaches

the free particle energy, EQ→ mQ. For later convenience, we define two positive definite quantities, ǫω

and mωby

ǫω ≡ −Uω(σ+(ω)) =1

1

2

m2

2ω2σ2

+(ω) − U(σ+(ω)),

≃

?ω2− ω2

−

?σ2

+, (8)

ω≡ m2− ω2

(9)

which can be infinitesimally small for either thin- or thick-wall limits. By assuming σ+(ω) ≃ σ+(ω−) ≡ σ+

in the thin-wall limit, we immediately obtain the second line in Eq. (8). Notice that this assumption was

implicitly imposed in our previous thin-wall analysis [37]. While this is fine for the gravity-mediated case,

with Gauge mediated potentials which are extremely flat, this implicit assumption cannot hold because

σ+(ω) does not exist. Therefore we will not use the variable ǫω for the case of the Gauge mediated

potentials. Notice that the variable m2

ωcannot be infinitesimally small when we consider the gravity-

mediated case: ω2

Q-ball parameters is the Legendre relation [37, 38]. For example the energy follows from

+?∼ m2. A powerful tool we can make use of when calculating some of the physical

Sω→ Q = −dSω

dω

????

EQ

→ EQ= ωQ + Sω.(10)

Assuming that γ is not a function of ω, we can compute the advertised characteristic slope,

EQ

ωQ= γ → EQ∝ Q1/γ

(11)

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where we have used another Legendre relation ω =

dEQ

dQ

???

Sω

in which we have fixed Sω. If a Q-ball is

classically stable, it satisfies

ω

Q

dQ

dω≤ 0 ⇔

d

dω

?EQ

Q

?

= −Sω

Q2

dQ

dω≥ 0. (12)

These classical stability conditions are equivalent to the fission condition, i.e.

the charge Q for classically stable Q-balls is a decreasing function in terms of ω. By scaling a Q-ball

solution with respect to the radius r, we also obtain the virial relation DU = −(D − 2)S + DωQ/2 and

the characteristic slope γ(ω),

dω

dQ≤ 0 in [37] so that

γ(ω) = 1 +

?

D − 2 + DU

S

?−1

(13)

once the ratio S/U is given where S ≡?

for D ≥ 2, see Eq. (1), whilst Sωis positive for D = 1 only when U ≥ S. It implies that we have to be

careful to use the second relation of Eq. (12) for D = 1 to evaluate the classical stability condition as

we saw in the case of using the Gaussian ansatz, which is valid for D = 1 for polynomial potentials [37].

Our key results for D ≥ 2 are

The first case in Eq. (14) corresponds to the extreme case of thin and thick-wall Q-balls. Furthermore,

in [37], we saw that for the extreme thin-wall Q-balls in DVPs, then there was a virialisation between

S and U, which corresponds to the second case in Eq. (14). At present it is not known what kind of

Q-ball potentials correspond to the third case; therefore, we will not be considering that case in the rest

of our paper. Notice that in the case S ≫ U for D = 2, we obtain the characteristic slope γ ≫ 1 from

Eq. (13). Similarly, for D = 1, the characteristic slopes are obtained, i.e. γ ≃ 1, ≫ 1, ≃ 0, respectively

for S ≪ U, S ∼ U, S ≫ U. We will use these 1D analytic results to interpret numerical results of

one-dimensional Q-balls in the thin-wall limit.

To end this section we note a nice duality that appears in Eqs. (13, 14) between the two cases S ∼ U

and S ≫ U. In particular, for S ∼ U in D dimensions, the same result for γ is obtained (to leading order)

in 2 × D dimensions when S ≫ U.

VD

1

2σ′2and U ≡?

VDU are the surface and potential energies,

respectively. For D ≥ 2, we can see γ(ω) ≥ 1 because S, U ≥ 0, which implies that Sωis positive definite

γ ≃

1

(2D − 1)/2(D − 1)

(D − 1)/(D − 2)

for S ≪ U,

for S ∼ U,

for S ≫ U.

(14)

III.GRAVITY-MEDIATED POTENTIALS

The MSSM consists of a number of flat directions where SUSY is not broken. Those flat directions

are, however, lifted by gauge, gravity, and/or nonrenormalisable interactions. In what follows the gravity

interaction is included perturbatively via the one-loop corrections for the bare mass m in Eq. (5) and

the nonrenormalisable interactions (UNR), which are suppressed by high energy scales such as the grand

unified theory scale MU ∼ 1016GeV or Planck scale mpl ∼ 1018GeV. Here, m is of order of SUSY

breaking scale which could be the gravitino mass ∼ m3/2, evaluated at the renormalisation scale M [23].

We note that, following the majority of work in this field, we will ignore A-term contributions ( U(1)

violation terms), thermal effects [39, 40] which come from the interactions between the AD field and

the decay products of the inflaton, and the Hubble-induced terms which gives a negative mass-squared

contribution during inflation. It is possible that their inclusion could well change the results of the

following analysis.

The scalar potential we are considering at present is [21, 23]

U = Ugrav+ UNR=1

2m2σ2

?

1 + K ln

?σ2

M2

??

+

|λ|2

mn−4

pl

σn

(15)

where we used Eq. (5), K is a factor for the gaugino correction, which depends on the flat directions, and

M is the renormalisation scale. Also λ is a dimensionless coupling constant, and UNR≡

|λ|2

mn−4

pl

σn, where

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n > 2. If the MSSM flat directions include a large top quark, K can be positive and then Q-balls do not

exist. For flat directions that do not have a large top quark component, we typical find K ≃ −[0.01−0.1]

[21, 41]. The power n of the nonrenormalisable term depends on the flat directions we are choosing along

which we maintain R parity. As examples of the directions involving squarks, the ucdcdcdirection has

n = 6, whilst the ucucdcecdirection requires n = 10. A complete list of the MSSM flat directions can

be found in Table 1 of [42]. Since the potential in Eq. (15) for K < 0 could satisfy the Q-ball existence

condition in Eq. (4), where ω+≫ m, Q-balls naturally exist.

In the rest of this paper, we will focus on potentials of the form of Eq. (15) for general D(≥ 1) and

ω and n(> 2) so that M and mpl have the same mass dimension, (D − 1)/2, as σ. It means that the

parameters M and mplare only physical for D = 3. For several cases of n and D, the term UNRcan be

renormalisable, but we will generally call it the nonrenormalisable term for the future convenience. The

readers should note that the potential Eq. (15) has been derived only with N = 1 supergravity in D = 3;

therefore, the potential form could well be changed in other dimensions. Furthermore, the logarithmic

correction breaks down for small σ and the curvature of Eq. (15) at σ = 0 is finite due to the gaugino

mass, which affects our thick-wall analysis and their dynamics. However, we concentrate our analysis

on this potential form for arbitrary D, n and any values of σ for two main reasons. The first is that it

contains a number of general semiclassical features expected of all the potentials, and the second is that

it offers the opportunity to consider the lower-dimensional Q-balls embedded in D = 3.

In Appendix A, we obtain the exact solution of Eq. (3) with the potential U = Ugrav; however, exact

solutions of the general potential U in Eq. (15) are fully nonlinear and can be obtained only numerically.

Therefore, we will analytically examine the approximate solutions in both the thin and thick-wall limits.

Before doing so, we shall begin by imposing a restriction on λ in Eq. (15) in order to obtain stable

Q-matter in NDVPs. With the further restrictions on λ and |K|, we can proceed with our analytical

arguments, and we will finally obtain the asymptotic Q-ball profile for large r which will be used in the

numerical section, Sec. V.

A. The existence of absolutely stable Q-matter

As we have seen, the first restriction on the gravity-mediated potential Eq. (15) which will allow for

the existence of a Q-ball solution Eq. (4) is K < 0. However, given values for m, mpl, M, n, and K in

Eq. (15), we need to restrict the allowed values of the parameter, λ in the potential in order to ensure we

obtain absolutely stable Q-matter. Notice that Q-matter exists in NDVPs, whilst the extreme thin-wall

Q-balls in DVPs, which will not be Q-matter as it will turn out, may exist with the lowest possible limit

of λ.

By using the definitions of ω−and σ+, namely, ω2

−≡

2U

σ2

??

σ+and

dUω−

dσ

???

σ+= 0, we shall find the range

of values of λ for which absolutely stable Q-matter solutions exist. Moreover, we will obtain the curvature

µ, which is proportional to |K|, of the effective potential Uωat σ+.

The effective potential for Eq. (15) can be rewritten in terms of new dimensionless variables ˜ σ =

σ/M, ˜ ω = ω/m, and

β2=|λ|2Mn−2

mn−4

pl

m2> 0,(16)

as

U˜ ω=1

2M2m2˜ σ2?1 − ˜ ω2− 2|K|ln ˜ σ?+ M2m2β2˜ σn.

−≡2U

(17)

After some simple algebra and introducing ˜ ω2

˜ σ2|˜ σ+and

dU˜ ω−

d˜ σ

???

˜ σ+= 0, we obtain

˜ σ+=

?

|K|

(n − 2)β2

?

1

n−2

,˜ ω2

−=

1

n − 2

?

n − 2 + 2|K| − 2|K|ln

?

|K|

(n − 2)β2

??

. (18)

Notice that ˜ ω2

thin-wall Q-balls do exist and are absolutely stable as we will see. In NDVPs, Q-matter solutions exist

and are absolutely stable when 0 < ˜ ω2

−< 1, see Eq. (7). Combining these facts and using the second

−= 0 corresponds to DVPs where Q-matter solutions do not exist [37], whilst the extreme