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 Riedelin collaboration with Betti Hartmann, Jacobs University Bremen

School of Engineering and ScienceJacobs 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 solutionslocalized, finite energy, stable, regular solutions ofnon-linear equationscan be viewed as models of elementary particles

ExamplesTopological solitons: Skyrme model of hadrons in highenergy physics one of first models and magneticmonopolesNon-topological solitons: Q-balls (flat space-time) andboson stars (generalisation in curved space-time)


Topolocial solitons

Properties of topological solitonsBoundary conditions at spatial infinity are topologicaldifferent from that of the vacuum stateDegenerated vacua states at spatial infinitycannot be continuously deformed to a single vacuum

Example in one dimension: L = 12 (∂µφ)2 − λ

4

(φ2 − m2

λ

)broken symmetry φ→ −φ with two degenerate vacua atφ = ±m/

√λ

N.S. Manton and P.M. Sutcliffe Topological solitons, Cambridge University Press, 2004


Non-topolocial solitons

Properties of topological solitonsSolutions possess the same boundary conditions atinfinity as the physical vacuum stateDegenerate vacuum states do not necessarily existRequire an additive conservation law, e.g. gaugeinvariance under an arbitrary global phasetransformation

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 theoremDerrick’s theorem puts restrictions to localized solitonsolutions in more than one spatial dimensionNo (stable) stationary point of energy exists with respectto λ for a scalar with purely potential interactions

Around Derrick’s Theoremif one includes appropriate gauge fields, gravitationalfields or higher derivatives in field Lagrangianif one considers solutions which are periodic in time



Model in one dimension)With complex scalar fieldΦ(x, t) : L = ∂µΦ∂µΦ∗ − U(|Φ|), U(|Φ|) minimum at Φ = 0Lagrangian is invariant under transformationφ(x)→ eiαφ(x)

Give rise to Noether charge Q = 1i

∫dx3φ∗φ− φφ∗)

Solution that minimizes the energy for fixed Q:Φ(x, t) = φ(x)eiωt

Solutions have been constructed in (3 + 1)-dimensionalmodels 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 < ω2max ≡ U

′′(0)

Condition 2:ω2 > ω2

min ≡ minφ[2U(φ)/φ2] minimum over all φConsequences:

Restricted interval ω2min < ω2 < ω2

max ;U′′

(0) > minφ[2U(φ)/φ2]

Q-balls are rotating in inner space with ω stabilized byhaving a lower energy to charge ratio as the freeparticles


Existence conditions for Q-balls

φ

V(φ

)

−4 −2 0 2 4

−0.0

2−

0.0

10.0

00.0

10.0

2

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

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)

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 = − i2

(Φ∗∂MΦ− Φ∂MΦ∗

)with jM;M = 0 . (6)

Globally conserved Noether charge Q

Q = −∫

ddx√−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 + 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)


Einstein equations in (d + 1) dimensions

Equations for the metric functions:

n′ = κrd−1

2

(Nφ′2 + U(φ) +

ω2φ2

A2N

), (12)

A′ = κr(

Aφ′2 +ω2φ2

AN2

), (13)

κ = 8πGd+1η2susy = 8π

η2susy

Md−1pl,d+1

(14)

Matter field equation:(rd−1ANφ′

)′= rd−1A

(12∂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−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 κ 6= 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

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)


Q-balls in Minkowski and AdS 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

ω

Q

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 (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

−0.0

50.0

50.1

50.2

5

ω

= 0.02

= 0.05

= 0.7

= 0.9

= 1.2

φ

V

−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).


Q-balls in Minkowski 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 + 1

ωm

ax

3 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).

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+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-balls in AdS 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.


Excited Q-balls

φ

V(φ

)

−4 −2 0 2 4

−0.0

2−

0.0

10.0

00.0

10.0

2

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

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

Q

M

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 .


Boson stars in Minkowski 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 .


Boson stars in AdS 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.


Boson stars in AdS 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


supersymmetric q-balls and boson stars in (d + 1) dimensions - jena talk mar 1st 2013

Science

fixed q

boson stars ind

noether charge q

sutcliffe topological

finite energy

spatial infinitycannot

cambridge university

field lagrangianif