supersymmetric q-balls and boson stars in (d + 1) dimensions - mexico city talk 2012
DESCRIPTION
Supersymmetric Q-balls and boson stars in (d + 1) dimensionsTRANSCRIPT
Supersymmetric Q-balls and boson stars in(d + 1) dimensions
Jürgen Riedelin collaboration with Betti Hartmann, Jacobs University Bremen
School of Engineering and ScienceJacobs 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
ActionS =
∫ √−gdd+1x
(R−2Λ
16πGd+1+ Lm
)+ 1
8πGd+1
∫ddx√−hK
negative cosmological constant Λ = −d(d − 1)/(2`2)
Matter LagrangianLm = −∂MΦ∂MΦ∗ − U(|Φ|) , M = 0,1, ....,dGauge mediated potential
USUSY(|Φ|) =
m2|Φ|2 if |Φ| ≤ ηsusy
m2η2susy = const . if |Φ| > ηsusy
(1)
U(|Φ|) = m2η2susy
(1− exp
(− |Φ|
2
η2susy
))(2)
(Campanelli and Ruggieri)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
The model for d + 1 dimensions
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
The model for d + 1 dimensions
Locally conserved Noether current jM , M = 0,1, ..,d
jM = − i2
(Φ∗∂MΦ− Φ∂MΦ∗
)with jM;M = 0 . (6)
Globally conserved Noether charge Q
Q = −∫
ddx√−gj0 . (7)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
The model Ansatz for d + 1 dimensions
Metric in spherical Schwarzschild-like coordinates
ds2 = −A2(r)N(r)dt2 +1
N(r)dr2 + r2dΩ2
d−1, (8)
whereN(r) = 1− 2n(r)
rd−2 −2Λ
(d − 1)dr2 (9)
Stationary Ansatz for complex scalar field
Φ(t , r) = eiωtφ(r) (10)
Rescaling using dimensionless quantities
r → rm, ω → mω, `→ `/m, φ→ ηsusyφ,n→ n/md−2 (11)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Coupled system of non-linear ordinary differential
Einstein equations read
n′ = κrd−1
2
(Nφ′2 + U(φ) +
ω2φ2
A2N
), (12)
A′ = κr(
Aφ′2 +ω2φ2
AN2
), (13)
(rd−1ANφ′
)′= rd−1A
(12∂U∂φ− ω2φ
NA2
). (14)
κ = 8πGd+1η2susy = 8π
η2susy
Md−1pl,d+1
(15)
φ′(0) = 0 , n(0) = 0 ,A(∞) = 1
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Expressions for Charge Q and Mass M
The explicit expression for the Noether charge
Q =2πd/2
Γ(d/2)
∞∫0
dr rd−1ωφ2
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 6= 0
n(r 1) = M + n1r2∆+d + .... (18)
(Radu et al)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Expressions for Charge Q and Mass M
The scalar field function falls of exponentially for Λ = 0
φ(r >> 1) ∼ 1
rd−1
2
exp(−√
1− ω2r)
+ ... (19)
The scalar field function falls of power-law for Λ < 0
φ(r >> 1) =φ∆
r∆, ∆ =
d2±√
d2
4+ `2 . (20)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) background
ω
M
0.4 0.6 0.8 1.0 1.2 1.41e+
00
1e+
02
1e+
04
1e+
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 1.41e+
00
1e+
02
1e+
04
1e+
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 valuesof d in Minkowski space-time
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) background
Q
M
1e+00 1e+02 1e+04 1e+061e
+0
01
e+
02
1e
+0
41
e+
06
Λ
= 0.0 2d
= 0.0 3d
= 0.0 4d
= 0.0 5d
= 0.0 6d
= (M=Q)
20 40 60 100
20
40
80 2d
200 300 400
200
300
450
3d
1500 2500 4000
1500
3000
4d
16000 19000 22000
16000
20000 5d
140000 170000 200000
140000
180000
6d
Figure : Mass M of the Q-balls in dependence on their charge Q for different valuesof d in Minkowski space-time
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) background
Q
M
1e+00 1e+02 1e+04 1e+06 1e+081e
+0
01
e+
02
1e
+0
41
e+
06
1e
+0
8
Λ
= −0.1 2d
= −0.1 3d
= −0.1 4d
= −0.1 5d
= −0.1 6d
= (M=Q)
1500 2500 4000
1500
3000
2d
1500 2500 4000
1500
3000
3d
1500 2500 4000
1500
3000
4d
1500 2500 4000
1500
3000
5d
1500 2500 4000
1500
3000
6d
Figure : Mass M in dependence on Q for d = 2, 3, 4, 5, 6 and Λ = −0.1.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) background
φ
V
−5 0 5
−0.0
50.0
50.1
50.2
5
ω
= 0.02
= 0.05
= 0.7
= 0.9
= 1.2
φV
−5 0 5
01
23
4
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 backgroundfor fixed r = 10,Λ = −0.1 and different values of ω (left),for fixed r = 10, ω = 0.3 anddifferent values of Λ (right).
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) background
Λ
ωm
ax
−0.10 −0.15 −0.20 −0.25 −0.30 −0.35 −0.40 −0.45
1.2
1.4
1.6
1.8
2.0
φ(0) = 0
= 2d
= 4d
= 6d
= 8d
= 10d
= 2d (analytical)
= 4d (analytical)
= 6d (analytical)
= 8d (analytical)
= 10d (analytical)
−0.1010 −0.1014 −0.1018
1.2
65
1.2
75
1.2
85
6d8d
d + 1ω
ma
x3 4 5 6 7 8 9 10
1.0
1.2
1.4
1.6
1.8
2.0
Λ
= −0.01
= −0.1
= −0.5
= −0.01 (analytical)
= −0.1 (analytical)
= −0.5 (analytical)
3.0 3.2 3.4
1.3
21.3
41.3
6
Λ = −0.1
Figure : The value of ωmax = ∆/` in dependence on Λ (left) and in dependence on d(right).
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) background
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.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) background
ω
M
0.5 1.0 1.5 2.0
110
100
1000
10000
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+051e+
01
1e+
02
1e+
03
1e+
04
1e+
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 thecharge Q (right) in AdS space-time for different values of d and number of nodes k .
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) background
φ(0)
<O
>1 ∆
0 5 10 15 20
0.0
00
.05
0.1
00
.15
0.2
0Λ
= −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 onφ(0) for different values of Λ and d .
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
10
50
50
05
00
0
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
50
200
500
3d
0.995 0.998 1.001
2000
6000
4d
0.95 0.98 1.01
2000
6000
5d
Figure : The value of the mass M of the boson stars in dependence on the frequencyω for Λ = 0 and different values of d and κ. The small subfigures show the behaviourof M, respectively at the approach of ωmax for d = 3, 4, 5 (from left to right).
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) background
ω
M
0.9980 0.9985 0.9990 0.9995 1.00001e
+0
11
e+
03
1e
+0
51
e+
07 D
= 4.0d
= 4.5d
= 4.8d
= 5.0d
ω= 1.0
0.9990 0.9994 0.9998
5e+
03
5e+
05
5d
Figure : Mass M of the boson stars in asymptotically flat space-time in dependenceon the frequency ω close to ωmax.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) background
r
φ
φ(0
)
0 200 400 600 800 1000
0.0
0.2
0.4
0.6
0.8
1.0
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
0.0
00.1
00.2
0
Figure : Profiles of the scalar field function φ(r)/φ(0) for the case where threebranches of solutions exist close to ωmax in d = 5. Here κ = 0.001.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) background
Q
M
1e+01 1e+03 1e+05 1e+071e
+0
11
e+
03
1e
+0
51
e+
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
2000
3000
5000
100000 150000 250000 400000
1e+
04
5e+
04
Figure : Mass M of the boson stars in asymptotically flat space-time in dependenceon their charge Q for different values of κ and d .
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) background
Q
M
1 10 100 1000 10000
11
01
00
10
00
10
00
0
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
500
600
800
1000
Figure : Mass M of the boson stars in AdS space-time in dependence on theircharge Q for different values of κ and d . Λ = 0.001
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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 1.4
110
100
1000
10000
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
110
100
1000
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 independence on the frequency ω in asymptotically flat space-time (Λ = 0) andasymptotically AdS space-time (Λ = −0.1) for different values of d and κ.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) background
φ(0)
<O
>1 ∆
0 1 2 3 4 5 6 7
0.0
00.0
50.1
00.1
50.2
0
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
0.0
00.0
50.1
00.1
5
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 onφ(0) (left) and in dependence on M (right) for different values of κ and d with Λ = −0.1.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions