Supersymmetric Q-balls and boson stars in (d + 1) dimensions

Supersymmetric Q-balls and boson stars in
(d + 1) dimensions
Jürgen Riedel
in collaboration with Betti Hartmann
Phys. Rev. D 87 (2013) 044003 [arXiv:1210.0096 [hep-th]]
School of Engineering and Science
Jacobs University Bremen, Germany
Models of Gravity
8TH IBERICOS MEETING
Granada, April 26th 2013
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions

Agenda
Q-balls and boson star soliton solutions
Set up model to support boson star solutions as well
as Q-ball solutions as limit without back reaction
Study solutions with gauge-mediated SUSY potential in
d + 1 dimensions
Possible candidates for dark matter in the early universe
Investigate stability conditions of solutions
Understand difference to standard Φ6-potential ansatz

Solitons in non-linear field theories
General properties of soliton solutions
localized, finite energy, stable, regular solutions of
non-linear equations
can be viewed as models of elementary particles
Examples
Topological solitons: Skyrme model of hadrons in high
energy physics one of first models and magnetic
monopoles, domain walls etc.
Non-topological solitons: Q-balls (flat space-time) and
boson stars (generalisation in curved space-time)

Non-topolocial solitons
Properties of topological solitons
Solutions possess the same boundary conditions at
inﬁnity as the physical vacuum state
Degenerate vacuum states do not necessarily exist
Require an additive conservation law, e.g. gauge
invariance under an arbitrary global phase
transformation
S. R. Coleman, Nucl. Phys. B 262 (1985), 263, R. Friedberg, T. D. Lee and A. Sirlin, Phys. Rev. D 13 (1976) 2739),
D. J. Kaup, Phys. Rev. 172 (1968), 1331, R. Friedberg, T. D. Lee and Y. Pang, Phys. Rev. D 35 (1987), 3658, P.
Jetzer, Phys. Rept. 220 (1992), 163, F. E. Schunck and E. Mielke, Class. Quant. Grav. 20 (2003) R31, F. E.
Schunck and E. Mielke, Phys. Lett. A 249 (1998), 389.

Model for Q-balls and boson stars in d + 1 dimensions
Action
S =
√
−gdd+1x R−2Λ
16πGd+1
+ Lm + 1
8πGd+1
dd x
√
−hK
negative cosmological constant Λ = −d(d − 1)/(2 2)
Matter Lagrangian
Lm = −∂MΦ∂MΦ∗ − U(|Φ|) , M = 0, 1, ...., d
Gauge mediated potential
USUSY(|Φ|) =
m2|Φ|2 if |Φ| ≤ ηsusy
m2η2
susy = const. if |Φ| > ηsusy
(1)
USUSY(|Φ|) = m2
η2
susy 1 − exp −
|Φ|2
η2
susy
(2)
A. Kusenko, Phys. Lett. B 404 (1997), 285; Phys. Lett. B 405 (1997), 108, L. Campanelli and M. Ruggieri,
Phys. Rev. D 77 (2008), 043504

Model for Q-balls and boson stars in d + 1 dimensions
Einstein-Klein-Gordon Equations are a coupled ODE
GMN + ΛgMN = 8πGd+1TMN M, N = 0, 1, .., d; (3)
−
∂U
∂|Φ|2
Φ = 0 (4)
Energy-momentum tensor
TMN = gMNL − 2
∂L
∂gMN
(5)
Locally conserved Noether current jM
and globally
conserved Noether charge Q
jM
= −
i
2
Φ∗
∂M
Φ − Φ∂M
Φ∗
jM
;M = 0; Q = − dd
x
√
−gj0
(6)

Ansatz Q-balls and boson stars for d + 1 dimensions
Metric in spherical Schwarzschild-like coordinates
ds2
= −A2
(r)N(r)dt2
+
1
N(r)
dr2
+ r2
dΩ2
d−1, (7)
where
N(r) = 1 −
2n(r)
rd−2
−
2Λ
(d − 1)d
r2
(8)
Stationary Ansatz for complex scalar ﬁeld
Φ(t, r) = eiωt
φ(r) (9)
Rescaling using dimensionless quantities
r →
r
m
, ω → mω, → /m, φ → ηsusyφ, n → n/md−2
(10)

Einstein equations in (d + 1) dimensions
Equations for the metric functions:
n = κ
rd−1
2
Nφ 2
+ U(φ) +
ω2φ2
A2N
, (11)
A = κr Aφ 2
+
ω2φ2
AN2
, (12)
κ = 8πGd+1η2
susy = 8π
η2
susy
Md−1
pl,d+1
(13)
Matter ﬁeld equation:
rd−1
ANφ = rd−1
A
1
2
∂U
∂φ
−
ω2φ
NA2
. (14)
Appropriate boundary conditions:
φ (0) = 0 , n(0) = 0 , A(∞) = 1

Expressions for Charge Q and Mass M
The explicit expression for the Noether charge
Q =
2πd/2
Γ(d/2)
∞
0
dr rd−1
ω
φ2
AN
(15)
Mass for κ = 0
M =
2πd/2
Γ(d/2)
∞
0
dr rd−1
Nφ 2
+
ω2φ2
N
+ U(φ) (16)
Mass for κ > 0
n(r 1) = M + n1r2∆+d
+ .... (17)
Y. Brihaye and B. Hartmann, Nonlinearity 21 (2008), 1937, D. Astefanesei and E. Radu, Phys. Lett. B 587
(2004) 7, D. Astefanesei and E. Radu, Nucl. Phys. B 665 (2003) 594 [gr-qc/0309131].

Existence conditions for solutions
φ
V(φ)
−4 −2 0 2 4
−0.02−0.010.000.010.02
ω = 1.2
V = 0.0
φ
V
−5 0 5
−0.050.050.150.25
ω
= 0.02
= 0.05
= 0.7
= 0.9
= 1.2
Figure : Effective potential V(φ) = ω2φ2 − U(|Φ|).

Existence conditions for solutions
Condition 1:
V (0) < 0; Φ ≡ 0 local maximum ⇒ ω2 < ω2
max ≡ 1
2 U (0)
Condition 2:
ω2 > ω2
min ≡ minφ[2U(φ)/φ2] minimum over all φ
Consequences:
Restricted interval ω2
min < ω2 < ω2
max ;
U (0) > minφ[2U(φ)/φ2]
Q-balls are rotating in inner space with ω stabilized by
having a lower energy to charge ratio as the free
particles
For USUSY: ω2
max = 1 and ω2
min = 0

Q-balls in Minkowski background κ = 0 and Λ = 0
ω
Q
0.2 0.4 0.6 0.8 1.0
1e+001e+021e+041e+06
d
= 2
= 3
= 4
= 5
= 6
ω= 1.0
r
φ(r)
0 50 100 150 200
0.00.20.40.60.81.0
Profile
= Q min
= Q max at omega min
= Q max at omega max
Figure : Charge Q in dependence on the frequency omega for different values of d
φ(r >> 1) ∼
1
r
d−1
2
exp − 1 − ω2r + ... (18)

Q-balls in Minkowski background
Q
M
1e+00 1e+02 1e+04 1e+06
1e+001e+021e+041e+06
Λ
= 0.0 2d
= 0.0 3d
= 0.0 4d
= 0.0 5d
= 0.0 6d
= (M=Q)
20 40 60 100
204080
2d
200 300 400
200300450 3d
1500 2500 4000
15003000
4d
16000 19000 22000
1600020000
5d
140000 170000 200000
140000180000
6d
ω
M−Q
0.2 0.4 0.6 0.8 1.0
−200−1000100200
d
= 2
= 3
= 4
= 5
= 6
ω= 1.0
Figure : Mass M in dependence on their charge Q for different values of d in
Minkowski space-time

Boson stars in Minkowski background κ > 0 and Λ = 0
ω
M
0.2 0.4 0.6 0.8 1.0 1.2
10505005000
κ
= 0.005 5d
= 0.01 5d
= 0.005 4d
= 0.01 4d
= 0.005 3d
= 0.01 3d
= 0.005 2d
= 0.01 2d
ω= 1.0
0.95 0.98 1.01
50200500
3d
0.995 0.998 1.001
20006000
4d
0.95 0.98 1.01
20006000
5d
Figure : The value of the mass M of the boson stars in dependence on the frequency
ω and different values of d and κ.

Boson stars in Minkowski background
ω
M
0.9980 0.9985 0.9990 0.9995 1.0000
1e+011e+031e+051e+07
D
= 4.0d
= 4.5d
= 4.8d
= 5.0d
ω= 1.0
0.9990 0.9994 0.9998
5e+035e+05
5d
Figure : Mass M of the boson stars in asymptotically ﬂat space-time in dependence
on the frequency ω close to ωmax.

r
φ
φ(0)
0 200 400 600 800 1000
0.00.20.40.60.81.0
φ(0) & ω
= 2.190 & 0.9995 lower branch
= 1.880 & 0.9999 middle branch
= 0.001 & 0.9999 upper branch
0 5 10 15 20
0.000.100.20
Figure : Proﬁles of the scalar ﬁeld function φ(r)/φ(0) for the case where three
branches of solutions exist close to ωmax in d = 5. Here κ = 0.001.

Q
M
1e+01 1e+03 1e+05 1e+07
1e+011e+031e+051e+07
κ
= 0.001 5d
= 0.005 5d
= 0.001 4d
= 0.005 4d
= 0.001 3d
= 0.005 3d
= 0.001 3d
= 0.005 2d
ω= 1.0
10000 15000 20000 25000
200030005000
100000 150000 250000 400000
1e+045e+04
Figure : Mass M of the boson stars in asymptotically ﬂat space-time in dependence
on their charge Q for different values of κ and d.

Excited Q-balls
φ
V(φ)
−4 −2 0 2 4
−0.02−0.010.000.010.02
ω = 1.2
V = 0.0
r
φ
0 5 10 15 20
−0.10.10.20.30.40.5
k
= 0
= 1
= 2
φ = 0.0
Figure : Effective potential V(φ) and excited solutions.

Excited Q-balls
ω
Q
0.2 0.4 0.6 0.8 1.0
2e+031e+045e+042e+05
k
= 0
= 1
ω = 1.0
r
φ(r)
0 20 40 60 80 100
−0.20.00.20.40.60.81.0
Profile
= Q min
= Q max at omega min
= Q max at omega max
φ = 0
Figure : Charge Q in dependence on ω for d=4 for nodes k = 1, 2.

Excited boson stars
ω
Q
0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
2e+031e+045e+042e+05
k & κ
= 0 & 0.01
= 1 & 0.01
= 2 & 0.01
= 3 & 0.01
ω = 1.0
Figure : Charge Q over frequency ω for excited boson star solutions with nodes
k=0,1,2,3.

Supersymmetric Q-balls and boson stars in (d + 1) dimensions

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