Supersymmetric Q-balls and boson stars in (d + 1) dimensions - Jena Talk Mar 1st 2013

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
February 10, 2014
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions

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
Non-topological solitons: Q-balls (flat space-time) and
boson stars (generalisation in curved space-time)

Topolocial solitons
Properties of topological solitons
Boundary conditions at spatial inﬁnity are topological
different from that of the vacuum state
Degenerated vacua states at spatial inﬁnity
cannot be continuously deformed to a single vacuum
Example in one dimension: L = 1
2 (∂µφ)2
− λ
4 φ2 − m2
λ
broken symmetry φ → −φ with two degenerate vacua at
φ = ±m/
√
λ
N.S. Manton and P.M. Sutcliffe Topological solitons, Cambridge University Press, 2004

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.

Derrick’s non-existence theorem
Derrick’s theorem puts restrictions to localized soliton
solutions in more than one spatial dimension
No (stable) stationary point of energy exists with respect
to λ for a scalar with purely potential interactions
Around Derrick’s Theorem
if one includes appropriate gauge fields, gravitational
fields or higher derivatives in field Lagrangian
if one considers solutions which are periodic in time

Model in one dimension)
With complex scalar ﬁeld
Φ(x, t) : L = ∂µΦ∂µΦ∗ − U(|Φ|), U(|Φ|) minimum at Φ = 0
Lagrangian is invariant under transformation
φ(x) → eiαφ(x)
Give rise to Noether charge Q = 1
i dx3φ∗ ˙φ − φ ˙φ∗)
Solution that minimizes the energy for ﬁxed Q:
Φ(x, t) = φ(x)eiωt
Solutions have been constructed in (3 + 1)-dimensional
models with non-normalizable Φ6-potential
M.S. Volkov and E. Wöhnert, Phys. Rev. D 66 (2002), 085003, B. Kleihaus, J. Kunz and M. List, Phys. Rev. D 72
(2005), 064002, B. Kleihaus, J. Kunz, M. List and I. Schaffer, Phys. Rev. D 77 (2008), 064025.

Existence conditions for Q-balls
Condition 1:
V (0) < 0; Φ ≡ 0 local maximum ⇒ ω2 < ω2
max ≡ 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

Existence conditions for Q-balls
φ
V(φ)
−4 −2 0 2 4
−0.02−0.010.000.010.02
ω = 1.2
V = 0.0
Figure : Effective potential V(φ) = ω2φ2 − U(|Φ|).

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)
U(|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

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)

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, (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)

Einstein equations in (d + 1) dimensions
Equations for the metric functions:
n = κ
rd−1
2
Nφ 2
+ U(φ) +
ω2φ2
A2N
, (12)
A = κr Aφ 2
+
ω2φ2
AN2
, (13)
κ = 8πGd+1η2
susy = 8π
η2
susy
Md−1
pl,d+1
(14)
Matter ﬁeld equation:
rd−1
ANφ = rd−1
A
1
2
∂U
∂φ
−
ω2φ
NA2
. (15)
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
(16)
Mass for κ = 0
M =
2πd/2
Γ(d/2)
∞
0
dr rd−1
Nφ 2
+
ω2φ2
N
+ U(φ) (17)
Mass for κ = 0
n(r 1) = M + n1r2∆+d
+ .... (18)
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].

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)

Q-balls in Minkowski and AdS 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
ω
Q
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 (Λ = 0) (left) and AdS (Λ < 0) (right)
B. Hartmann and J. Riedel, Phys. Rev. D 86 (2012) 104008 [arXiv:1204.6239 [hep-th]], B. Hartmann and J. Riedel,
Phys. Rev. D 87 (2013) 044003 [arXiv:1210.0096 [hep-th]]

Q-balls in Minkowski and AdS background
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).

Q-balls in Minkowski background
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).
E. Radu and B. Subagyo, Spinning scalar solitons in anti-de Sitter spacetime, arXiv:1207.3715 [gr-qc]

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

Q-balls in AdS background
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.

Excited Q-balls
φ
V(φ)
−4 −2 0 2 4
−0.02−0.010.000.010.02
ω = 1.2
V = 0.0
Figure : Effective potential V(φ) and excited solutions.

Excited Q-balls
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.

Excited Q-balls
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
Q
M
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.

Boson stars in Minkowski 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.

Boson stars in AdS background
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.

Boson stars in AdS background
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

Supersymmetric Q-balls and boson stars in (d + 1) dimensions - Jena Talk Mar 1st 2013

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