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 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)



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 = − i2

(Φ∗∂MΦ− Φ∂MΦ∗

)with jM;M = 0 . (6)

Globally conserved Noether charge Q

Q = −∫

ddx√−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 + 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)


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


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)


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)


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



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



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.



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).



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)



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).



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

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 .



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 .


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).



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.



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.



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 .



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



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.



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.
