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.
Q-balls have recently attracted much attentions in cosmology  and astrophysics [2, 3, 4]. A Q-ball
 is a nontopological soliton  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
, Sine-Gordon models , parabolic-type models , confinement models [14, 15, 16, 17], two-field
models [6, 18], and flat models .
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 . 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  or gauge-mediated scenarios
, 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 . These type of Q-balls can be unstable to decay into baryons and the lightest super-
symmetric particle dark matter, such as neutralinos , gravitinos [25, 26] and axinos . 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 . 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 .
Scalar field potentials arising through gauge-mediated SUSY breaking  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 , Shoemaker and Kusenko recently explored
the minimum energy configuration for baryo-leptonic Q-balls, whose scalar field consists of both squarks
and sleptons . 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 , 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) , may end up contributing to
the energy density of dark matter [29, 34]; thus, the Q-balls can provide the baryon-to-photon ratio ,
i.e. nb/nγ ≃ (4.7 − 6.5) × 10−10 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 . 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  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
Q = ω
;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:
0dr rD−1, ΩD−1≡
generality, we can take positive values of ω and Q. By defining the effective potential Uωof a potential
dr, and D is the number of spatial dimensions. Without loss of
Uω≡ U −1
the Q-ball equation is
σ′′+D − 1
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 :
ω−≤ ω < ω+,
where we have defined the lower limit ω2
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
σ=0. The existence condition Eq. (4)
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  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
1 + K ln
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
• thin-wall Q-balls when ω ≃ ω−or equivalently σ0∼ σ+(ω),
• thick-wall Q-balls when ω ≃ ω+or equivalently σ0≃ σ−(ω).
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 . For absolutely stable Q-balls, the energy per unit charge is smaller than the rest
mass m for the field φ,
< 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
ǫω ≡ −Uω(σ+(ω)) =1
+(ω) − U(σ+(ω)),
ω≡ m2− ω2
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 . 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
→ EQ= ωQ + Sω. (10)
Assuming that γ is not a function of ω, we can compute the advertised characteristic slope,
ωQ= γ → EQ∝ Q1/γ
where we have used another Legendre relation ω =
in which we have fixed Sω. If a Q-ball is
classically stable, it satisfies
dω≤ 0 ⇔
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),
dQ≤ 0 in  so that
γ(ω) = 1 +
D − 2 + DU
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 .
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 , 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.
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
(2D − 1)/2(D − 1)
(D − 1)/(D − 2)
for S ≪ U,
for S ∼ U,
for S ≫ U.
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 .
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
The scalar potential we are considering at present is [21, 23]
U = Ugrav+ UNR=1
1 + K ln
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≡
ρω= M exp
?D − 1
ω≡ m2− ω2. Note that the constant M
where we set the integration constant as zero. Recall m2
has the same mass dimension, (D − 1)/2, as σ so that the only physical case is D = 3. The profile,
Eq. (A1), is an exact solution for Ugravwith the “core” radius RQ=?2/m2|K| , which is very large
σ′(0) = 0 = σ(∞) = σ′(∞) . In the extreme limit ω ≫ m, we obtain ρω → 0 for |K| ? O(1)
which implies σ0≡ σ(0) → 0. For large σ, the potential becomes asymptotically flat, tending towards
an infinite negative value. By adding the nonrenormalisable term UNR, the potential Ugravis lifted for
large σ in Eq. (15), then the full potential Ugrav+ UNRis bounded from below, see Sec. IIIA. We can
see the ansatz given in  corresponds to the case where ρω≃ M, which is valid only for |K| ≪ O(1)
and ω ≃ m, see Eq. (A3).
compared with m−1for small |K| ≪ O(1), and satisfies the boundary conditions for Q-balls, namely
APPENDIX B: THICK-WALL Q-BALL WITH A GAUSSIAN ANSATZ
In this appendix, we will investigate the thick-wall Q-ball in gravity-mediated models by introducing a
Gaussian ansatz and keeping all terms in Eq. (17) as opposed to the analysis in Sec.IIIC. By using this
profile we can perform the Gaussian integrations, and will obtain the generalised results of Eqs. (33, 34)
in Sec. IIIC. The test profile for the case, ω ? O(m), coincides with the solution σsolin Eq. (A1), which
implies that the nonrenormalisable term UNRin Eq. (15) is negligible.
To recap, the notation we have adopted in Eq. (17) is ˜ σ = σ/M, ˜ ω = ω/m, β2is defined in Eq. (16)
and we are considering the case of n > 2. To begin with we introduce a Gaussian ansatz inspired by
Eq. (A1) for the potential Eq. (17)
˜ σ(r) = λωexp(−κ2
where ˜ σ0≡ ˜ σ(0) = λω= finite, and λω, κωwill be functions of ω implicitly. λωshould not be confused
with the coupling constant λ in Eq. (15). Both λωand κωcan be determined by extremising the Euclidean
action Sω; hence the actual free parameter here will be only ω. It is crucial to note that λω cannot be
infinite in the thick-wall limit since we know that λωis finite and tending to 0. If the nonrenormalisable
term UNRis negligible, we can expect λω ∼ ρω/M ∼ ˜ σ−(ω) and κ2
implies that the “core” radius RQof the thick-wall Q-ball is RQ∼
limit ω ≫ m, we shall also confirm λω→ 0, which means ˜ σ0→ 0.
By substituting Eq. (B1) into Eq. (1) with the potential Eq. (17), we obtain Q and Sω using the
following Gaussian integrations: ΩD−1
Q = M2πD/2ωλ2
Sω = M2πD/2κ−D
[A(κω, λω) + B(ω, λω) + C(λω)],
ω∼ |K|m2due to Eq. (A1), which
?2/m2|K|. For the extreme thick-wall
?D/2for real k where ΩD−1 ≡
whereA(κω, λω) ≡
B(ω, λω) ≡
, C(λω) = m2β2λn
Notice that A(κω, λω) comes from the gradient term and the logarithmic term in Sω and depends on
both κωand λω. Similarly, B(ω, λω) is given by the quadratic term in the potential Eq. (17) and depends
both on λω and explicitly on ω, whereas C(λω) arises simply from the nonrenormalisable term in the
potential. An alternative (but in this case more complicated) approach to obtain Q would be the use of
Legendre transformations in Eq. (10).
By extremising Sωin terms of the two free parameters κωand λω:
= 0, (B6)
A + B + C =λ2
,A + B +nC
which implies that
m2= |K| − (n − 2)β2λn−2
where we have eliminated the A + B terms in the two expressions of Eq. (B7). Using Eq. (B8) and the
second expression of Eq. (B7), we obtain the relations between ω and λω
m2= 1 + |K|(D − 1 − 2lnλω) +2(n + D) − nD
1 − 2|K|lnλω
1 + |K|(D − 1 − 2lnλω) for |K| ∼ O(1),
for |K| < O(1),
|K|m2< 0, (B11)
where we have differentiated Eq. (B9) with respect to ω to obtain Eq. (B11) and have defined F as
F ≡ 1 − (n − 2)2(n+D)−nD
that both κωand λωare functions of ω; however, these are not solvable in closed forms unless the particular
limits, which were introduced in Sec.IIIC, are taken, as we will now show. Comparing Eqs. (B9, B11) with
Eqs. (28, 29), we can see an extra contribution of O(|K|) in Eq. (B9), which is not present in Eq. (28).
This difference of (D − 1)|K| arises because in calculating Eq. (B9) we have used λω, whereas we have
used ˜ σ−(ω) in obtaining Eq. (28), and although related they are not precisely the same. In the extreme
thick-wall limit ω ≫ m, and from Eq. (B9) this implies λω→ 0+(recall from Eq. (B1) that λω has to
remain finite). Considering the nonrenormalisable term in Eq. (B9), the fact that β2? |K| ? O(1) and
λω→ 0+with n > 2, implies that this term is subdominant and can be ignored. As long as λω< O(1),
then F ∼ 1 and the second relation of Eq. (B11) follows, which implies that λωis a monotically decreasing
function in terms of ω. The limit λω∼ O(1) corresponds to ω ? O(m), see Eq. (B9). We will call this
the “moderate limit” and represent it by ’∼’. The other case, ω ≫ m (or equivalently λω≪ O(1)), we
shall call the “extreme limit” and represent it by ’→’. Depending on the logarithmic strength of |K|, we
can obtain Eq. (B10), which leads to the approximated expressions for λωand can also obtain κωfrom
?D/2= 1 +2(n+D)−nD
ω− m2|K|?. Equations (B8, B9) imply
˜ σ−(ω) for |K| < O(1)
for |K| ∼ O(1)
m2∼ |K| → |K| for |K| ? O(1),
where κωis independent of ω in both the “moderate” and “extreme” limits.
Using Eqs. (B2, B3) and Eq. (B7), we obtain the characteristic slope in both the “moderate” and
ωQ= 1 +κ2
2ω2∼ 1 +m2|K|
→ 1. (B13)
In order to show their classical stability, we shall differentiate Q with respect to ω using Eqs. (B8, B9)
and Eq. (B11):
= 1 −
1 −D(n − 2)
ω+(n − 2)ω2
→ 1 > 0,
∼ 1 −
= 1 −
∼ 1 −m2|K|
where we have taken the “moderate limit” and “extreme limit” and used κ2
ω∼ m2|K|, F = 1 +
ω− m2|K|?∼ 1. The classical stability condition Eq. (B14) is consistent with Eq. (B15),
and is consistent with Eq. (12). This is different from the result we obtained for the polynomial poten-
tials [see Eq. (74) in ], because in that case the Gaussian ansatz does not give the exact solution
unlike here in Eq. (B1) where it does become the exact solution Eq. (A1) in both limits. The results,
Eqs. (B13, B14) and Eq. (B15), in both the “moderate” and “extreme” limits recover the key results,
Eqs. (33, 34), and are independent of D; hence, the thick-wall Q-balls for all D have similar properties.
We can also see the small additional effects arising from the nonrenormalisable term in Eqs. (B14, B15).
Let us summarise the important results we found in this appendix. By introducing a Gaussian test
profile Eq. (B1) inspired by the exact solution Eq. (A1) for Ugrav, we computed the Euclidean action
Sω and the charge Q using Gaussian integrations. Then, we extremised Sω in terms of λω and κω in
Eq. (B6), which gave the relations of both λωand κωas a function of ω. By introducing two limits called
“moderate limit” and “extreme limit”, we confirmed that the ansatz, Eq. (B1), approaches Eq. (A1)
in the “moderate limit”. We established that the results Eqs. (B13, B14) and Eq. (B15) recovered the
previous results in Eqs. (33, 34) which are obtained simply by reparameterising in Sωand extracting the
explicit ω-dependence from the integral in Sω with U = Ugrav where the nonrenormalisable term was
neglected at the beginning of the analysis by applying L’Hˆ opital rules.
In addition, we would like to emphasise the main differences between our work and other earlier analyses
in the literature [30, 49]. The analytical framework adopted in  is valid only for |K| = 1, D = 3, n = 4.
Our work has shown that this can be generalised to arbitrary integer values of D and n(> 2) under the
conditions β2? |K| ? O(1), and that the thick-wall Q-ball can be classically stable. In Sec.IIIC, we also
found that the thick-wall Q-ball may be absolutely stable under certain additional conditions, Eq. (35).
Furthermore, Enqvist and McDonald in  analytically obtained the same “core” size of thick-wall Q-
balls, although they obtained a slightly different value for EQ/Q (see their Eq. (112)). The reason for
this is because their ansatz assumed λω ≃ 1 in Eq. (B1) by simply neglecting the nonrenormalisable
term, which implies that the third term of B(ω, λω) and term C(λω) in Eq. (B5) are absent. Hence,
their analysis is valid for |K| ≪ O(1) and ω ≃ m, see Eq. (A3). We, however, have kept all the terms in
Eq. (17) and used a more general ansatz, which can be applied for |K| ? O(1) and ω ? O(m) with the
restricted coupling constant of the nonrenormalisable term β2? |K|. In summary, in this appendix we
have extensively investigated analytically both the absolute and classical stability of Q-balls in Eq. (B13)
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