Supersymmetric Q-balls and boson stars in (d + 1) dimensions - Mexico city talk 2012

Supersymmetric Q-balls and boson stars in
(d + 1) dimensions
Jürgen Riedel
in collaboration with Betti Hartmann, Jacobs University Bremen
School of Engineering and Science
Jacobs University Bremen, Germany
MEXICO CITY, OCT 16TH, 2012
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions

The model for 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)
U(|Φ|) = m2
η2
susy 1 − exp −
|Φ|2
η2
susy
(2)
(Campanelli and Ruggieri)

Einstein Equations are a coupled ODE
GMN + ΛgMN = 8πGd+1TMN , M, N = 0, 1, .., d (3)
Energy-momentum tensor
TMN = gMNL − 2
∂L
∂gMN
(4)
Klein-Gordon equation
−
∂U
∂|Φ|2
Φ = 0 . (5)

Locally conserved Noether current jM, M = 0, 1, .., d
jM
= −
i
2
Φ∗
∂M
Φ − Φ∂M
Φ∗
with jM
;M = 0 . (6)
Globally conserved Noether charge Q
Q = − dd
x
√
−gj0
. (7)

The model Ansatz for d + 1 dimensions
Metric in spherical Schwarzschild-like coordinates
ds2
= −A2
(r)N(r)dt2
+
1
N(r)
dr2
+ r2
dΩ2
d−1, (8)
where
N(r) = 1 −
2n(r)
rd−2
−
2Λ
(d − 1)d
r2
(9)
Stationary Ansatz for complex scalar ﬁeld
Φ(t, r) = eiωt
φ(r) (10)
Rescaling using dimensionless quantities
r →
r
m
, ω → mω, → /m, φ → ηsusyφ, n → n/md−2
(11)

Coupled system of non-linear ordinary differential
Einstein equations read
n = κ
rd−1
2
Nφ 2
+ U(φ) +
ω2φ2
A2N
, (12)
A = κr Aφ 2
+
ω2φ2
AN2
, (13)
rd−1
ANφ = rd−1
A
1
2
∂U
∂φ
−
ω2φ
NA2
. (14)
κ = 8πGd+1η2
susy = 8π
η2
susy
Md−1
pl,d+1
(15)
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
(16)
Mass for κ = 0
M =
2πd/2
Γ(d/2)
∞
0
dr rd−1
Nφ 2
+
ω2φ2
N
+ U(φ) (17)
Mass for kappa = 0
n(r 1) = M + n1r2∆+d
+ .... (18)
(Radu et al)

Expressions for Charge Q and Mass M
The scalar ﬁeld function falls of exponentially for Λ = 0
φ(r >> 1) ∼
1
r
d−1
2
exp − 1 − ω2r + ... (19)
The scalar ﬁeld function falls of power-law for Λ < 0
φ(r >> 1) =
φ∆
r∆
, ∆ =
d
2
±
d2
4
+ 2 . (20)

Numerical analysis Q-balls in Minkowski (Λ = 0) and
AdS background (Λ < 0) background
ω
M
0.4 0.6 0.8 1.0 1.2 1.4
1e+001e+021e+041e+06
Λ
= 0.0 2d
= 0.0 3d
= 0.0 4d
= 0.0 5d
= 0.0 6d
= −0.1 2d
= −0.1 3d
= −0.1 4d
= −0.1 5d
= −0.1 6d
ω= 1.0
ωQ
0.2 0.4 0.6 0.8 1.0 1.2 1.4
1e+001e+021e+041e+06
Λ
= 0.0 2d
= 0.0 3d
= 0.0 4d
= 0.0 5d
= 0.0 6d
= −0.1 2d
= −0.1 3d
= −0.1 4d
= −0.1 5d
= −0.1 6d
ω= 1.0
Figure : Mass M of the Q-balls in dependence on their charge Q for different values
of d in Minkowski space-time

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
Figure : Mass M of the Q-balls in dependence on their charge Q for different values
of d in Minkowski space-time

Q
M
1e+00 1e+02 1e+04 1e+06 1e+08
1e+001e+021e+041e+061e+08
Λ
= −0.1 2d
= −0.1 3d
= −0.1 4d
= −0.1 5d
= −0.1 6d
= (M=Q)
1500 2500 4000
15003000 2d
1500 2500 4000
15003000
3d
1500 2500 4000
15003000
4d
1500 2500 4000
15003000
5d
1500 2500 4000
15003000
6d
Figure : Mass M in dependence on Q for d = 2, 3, 4, 5, 6 and Λ = −0.1.

φ
V
−5 0 5
−0.050.050.150.25
ω
= 0.02
= 0.05
= 0.7
= 0.9
= 1.2
φ
V
−5 0 5
01234
Λ
= 0.0
= −0.01
= −0.05
= −0.1
= −0.5
Figure : Effective potential V(φ) = ω2φ2 − U(|Φ|) for Q-balls in an AdS background
for ﬁxed r = 10,Λ = −0.1 and different values of ω (left),for ﬁxed r = 10, ω = 0.3 and
different values of Λ (right).

Λ
ωmax
−0.10 −0.15 −0.20 −0.25 −0.30 −0.35 −0.40 −0.45
1.21.41.61.82.0
φ(0) = 0
= 2d
= 4d
= 6d
= 8d
= 10d
= 2d (analytical)
= 4d (analytical)
= 6d (analytical)
= 8d (analytical)
= 10d (analytical)
−0.1010 −0.1014 −0.1018
1.2651.2751.285
6d
8d
d + 1
ωmax
3 4 5 6 7 8 9 10
1.01.21.41.61.82.0
Λ
= −0.01
= −0.1
= −0.5
= −0.01 (analytical)
3.0 3.2 3.4
1.321.341.36
Λ = −0.1
Figure : The value of ωmax = ∆/ in dependence on Λ (left) and in dependence on d
(right).

r
φ
0 5 10 15 20
−0.10.10.20.30.40.5
k
= 0
= 1
= 2
φ = 0.0
Figure : Proﬁle of the scalar ﬁeld function φ(r) for Q-balls with k = 0, 1, 2 nodes,
respectively.

ω
M
0.5 1.0 1.5 2.0
110100100010000
Λ & k
= −0.1 & 0 4d
= −0.1 & 1 4d
= −0.1 & 2 4d
= −0.1 & 0 3d
= −0.1 & 1 3d
= −0.1 & 2 3d
QM
1e+01 1e+02 1e+03 1e+04 1e+05
1e+011e+021e+031e+041e+05
Λ & k
= −0.1 & 0 4d
= −0.1 & 1 4d
= −0.1 & 2 4d
= −0.1 & 0 3d
= −0.1 & 1 3d
= −0.1 & 2 3d
Figure : Mass M of the Q-balls in dependence on ω (left) and in dependence on the
charge Q (right) in AdS space-time for different values of d and number of nodes k.

φ(0)
<O>
1
∆
0 5 10 15 20
0.000.050.100.150.20
Λ
= −0.1 2d
= −0.1 3d
= −0.1 4d
= −0.1 5d
= −0.1 6d
= −0.1 7d
= −0.5 2d
= −0.5 3d
= −0.5 4d
= −0.5 5d
= −0.5 6d
= −0.5 7d
Figure : Expectation value of the dual operator on the AdS boundary < O >1/∆
corresponding to the value of the condensate of scalar glueballs in dependence on
φ(0) for different values of Λ and d.

Numerical analysis boson stars in Minkowski (Λ = 0)
and AdS background (Λ < 0) background
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 for Λ = 0 and different values of d and κ. The small subﬁgures show the behaviour
of M, respectively at the approach of ωmax for d = 3, 4, 5 (from left to right).

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.

Q
M
1 10 100 1000 10000
110100100010000
κ
= 0.01 6d
= 0.005 6d
= 0.01 5d
= 0.005 5d
= 0.01 4d
= 0.005 4d
= 0.01 3d
= 0.005 3d
= 0.01 2d
= 0.005 2d
ω= 1.0
1000 1500 2000 2500
5006008001000
Figure : Mass M of the boson stars in AdS space-time in dependence on their
charge Q for different values of κ and d. Λ = 0.001

ω
M
0.2 0.4 0.6 0.8 1.0 1.2 1.4
110100100010000
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
ω
Q
0.2 0.4 0.6 0.8 1.0 1.2 1.4
1101001000
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
Figure : The value of the mass M (left) and the charge Q (right) of the boson stars in
dependence on the frequency ω in asymptotically ﬂat space-time (Λ = 0) and
asymptotically AdS space-time (Λ = −0.1) for different values of d and κ.

φ(0)
<O>
1
∆
0 1 2 3 4 5 6 7
0.000.050.100.150.20
κ
= 0.005 5d
= 0.01 5d
= 0.005 4d
= 0.01 4d
= 0.005 3d
= 0.01 3d
M
<O>
1
∆
0 500 1000 1500 2000 2500
0.000.050.100.15
κ
= 0.005 5d
= 0.01 5d
= 0.005 4d
= 0.01 4d
= 0.005 3d
= 0.01 3d
Figure : Expectation value of the dual operator on the AdS boundary < O >1/∆
corresponding to the value of the condensate of scalar glueballs in dependence on
φ(0) (left) and in dependence on M (right) for different values of κ and d with Λ = −0.1.

Supersymmetric Q-balls and boson stars in (d + 1) dimensions - Mexico city talk 2012

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