Internal workshop jub talk jan 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

INTERNAL WORKSHOP JUB TALK

Bremen, Jan 19th 2013

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

IntroductionQ-balls

Boson starsAdS/CFT correspondence

SUSY Q-balls in AdS backgroundSUSY boson stars in AdS background

Summary results in 4 dimensionsQ-Balls and boson stars in d + 1 dimensions

Numerical results in d + 1 dimensions

Outline

1 Introduction2 Q-balls in 3+1 dimensions3 Boson stars in 3+1 dimensions4 AdS/CFT correspondence5 SUSY Q-balls in AdS5 background6 SUSY boson stars in AdS background7 Summary results in 4 dimensions8 Q-Balls and boson stars in d + 1 dimensions9 Numerical results 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 particlesdimension

Examples and restrictionsSkyrme model of hadrons in high energy physics one offirst modelsDerrick’s theorem puts restrictions to localized solitonsolutions in more than one spatial dimension


Solitons in non-linear field theories

Derrick’s non-existence theoremProof proceeds by contradictionSuppose a solitonic solution φ0(~x) existsDeformations φλ(λ~x)=φ0(~x), where λ is dilation parameterNo (stable) stationary point of energy exists with respect toλ for a scalar with purely potential interactions.

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


Topolocial solitons

PropertiesBoundary 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


Non-topolocial solitons

Classical example in one dimensionWith 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


Prominent examples for topological solitons

Further examplesvortices, magnetic monopoles, domain walls, cosmicstrings, textures

Prominent examples for non-topological solitons

Q-ballsBoson stars


The model

Lagrangian L =∂µΦ∂µΦ∗ − U(|Φ|); the signature of themetric is (+,-,-,-)Noether current j = i(Φ∗Φ− ΦΦ∗) symmetry under U(1)Conserved Noether charge Q = 1

i

∫d3(Φ∗Φ− ΦΦ∗), with

Φ := Φ(t , r) we have dQdt = 0

Ansatz for solution Φ(x, t) = φ(x)eiωt

Energy-momentum tensorTµν = ∂µΦ∂νΦ∗ + ∂νΦ∂µΦ∗ − gµνLTotal Energy E =

∫d3xT 0

0 =∫

d3x [|Φ|2 + |OΦ|2 + U(|Φ|)]under assumption that gµν is time-independent


Existence conditions of Q-balls

Condition 1

V′′

(0) < 0; Φ ≡ 0 local maximum⇒ ω2 < ω2max ≡ U

′′(0)

Condition 2

ω2 > ω2min ≡ minφ[2U(φ)/φ2] minimum over all φ

Consequences

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 free particles


Thin wall approximation of Q-balls

If the Q-ball is getting large enough, surface effects canbe ignored: thin wall limit.

Minimum of total energy ωmin = Emin = 2U(φ0)φ2 , for φ0 > 0

The energy and charge is proportional to the volumewhich is similarly found in ordinary matter→ Q = ωφ2VTherefore Q-balls in this limit are called Q-matter and havevery large charge, i.e. volumeSuitable potential U = m2|Φ|2 − a|Φ|4 + b|Φ|6, with a andb are constants


Rotating Q-balls

The Ansatz Φ = φ(r , θ)eiωt+inϕ, where n is an integerNon-linear field equation:dU(φ)

dφ =(∂2φ∂r2 + 2

r∂φ∂r + 1

r2∂2φ∂θ2 + cosθ

r2sinθ∂φ∂θ −

n2φr2sinθ + ω2φ

)Charge Q = 4πω

∫∞0 drr2 ∫ π

0 dθsinθφ2

Uniqueness of the scalar field under a completerotation Φ(ϕ) = Φ(ϕ+ 2π) requires n to be an integer


Rotating Q-balls

ConsequencesThe angular momentum J is quantized:J =

∫T0φd3x = nQ: n = rotational quantum number

One requires that φ→0 for r →0 or r →∞φ(r)|r=0 = 0 is a direct consequence of the term n2φ2

r2sin2θ


Boson stars

Action ansatz: S =∫ √−gd4x

( R16πG + Lm

)Matter Lagrangian Lm = −1

2∂µΦ∂µΦ∗ − U(|Φ|); thesignature of the metric is (-,+,+,+)Variation with respect to the scalar field

1√−g∂µ (

√−g∂µΦ) = ∂U

∂|Φ|2 Φ

Metric ansatzds2 = −f (r)dt2 + l(r)

f (r)

(dr2 + r2dθ2 + r2sin2θdφ2)

Conserved current jµ = i√−ggµν(Φ∗∂νΦ− Φ∂νΦ∗)

Noether charge Q =∫

dx3j0 associated to the globalU(1) transformation


Boson star models

Simplest model U = m2|Φ|2 (by Kemp, 1986)

Proper boson stars U = m2|Φ|2 − λ|Φ|4/2(by Colpi, Sharpio and Wasserman, 1986)

Sine-Gordon boson starU = αm2

[sin(π/2

[β√|Φ|2 − 1

]+ 1]

Cosh-Gordon boson star U = αm2[cosh(β

√|Φ|2 − 1

]Liouville boson star U = αm2 [exp(β2|Φ|2)− 1

](Schunk and Torres, 2000)


Self-interacting boson stars models

Model U = m2|Φ|2 − a|Φ|4 + b|Φ|6, with a and b areconstants (Mielke and Scherzer, 1981)

Soliton stars U = m2|Φ|2(1− |Φ|2/Φ2

0)2

(Friedberg, Lee and Pang, 1986)

Represented in the limit of flat space− time, by Q -ballsas non-topological solitonsHowever, terms of |Φ|6 or higher-order terms implies thatthe scalar part of the theory is not re-normalizable


Charged Boson stars

System of complex scalar fields coupled to aU(1) gauge field with quartic self-interactionThe metric ansatzds2 = gµνdxµdxν = −A2Ndt2 + dr2

N + r2 (dθ2 + sin2θdφ2),with N = 1− 2m(r)

r andSolution ansatz: Φ = φ(r)eiωt , Aµdxµ = A0(r)dtA gauge coupling constant e does increase themaximum mass M and bf conserved charge QUsing a V-shaped scalar potential(Kleihaus, Kunz, Lammerzahl, and List, 2009)


Rotating Boson stars

The metric ansatzds2 = −f (r , θ)dt2 + l(r,θ)

f (r,θ)

[g(r , θ)(dr2 + r2dθ2) + r2sin2θ

(dφ− χ(r,θ)

r dt)2

]Stationary spherically symmetric ansatzΦ(t , r , θ, ϕ) = φ(r , θ)eiωt+inϕ

Uniqueness of the scalar field under a completerotation Φ(ϕ) = Φ(ϕ+ 2π) requires n to be an integer (, i.e.

n = 0,±1,±2, ...)

Conserved scalar chargeQ = −4πω

∫∞0

∫ π0√−g 1

f

(1 + n

ωχr

)φ2drdθ


Rotating Boson stars continued

Total angular momentum J = −∫

T 0ϕ

√−gdrdϕdθ

With T 0ϕ = nj0, since ∂Φ

∂φ = i nΦ one finds: J = nQSolution is axially symmetric (for n 6= 0 )This means that a rotating boson star is bf proportional tothe conserved Noether chargeIf n = 0, it follows that a spherically symmetric bosonstar has angular momentum J = 0Rotating boson stars were intensively studied in 4dimensions (Kleihaus et al) as well in 5 dimensions (Hartmannet al) with U(|Φ|) = λ

(|Φ|6 − a|Φ|4 + b|Φ|2

)


AdS/CFT correspondence

Important result from StringTheory (Maldacena, 1997):A theory of classical gravity in (d + 1)-dimensionalasymptotically Anti-de Sitter (AdS) space-time is dual to astrongly-coupled, scale-invariant theory (CFT) living onthe d-dimensional boundary of AdSAn important example: Type IIB string theory in AdS5× S5dual to 4-dimensional N = 4 supersymmetric Yang-MillstheoryOne can use classical gravity theory, i.e. weakly-coupled,to study strongly coupled quantum field theories


Holographic conductor/ superconductor

Taken from arxiv: 0808.1115

Boundary of SAdS ≡ AdS

Dual theory“lives” here

r → ∞

r

x,yr=r

h horizon

Temperature represented bya black hole

Chemical potentialrepresented by a chargedblack hole

Condensate represented bya non-trivial field outside theblack hole horizon if T < Tc

⇒ One needs an electricallycharged plane-symmetrichairy black hole


The model

Action ansatz:S =

∫dx4√−g

(R + 6

`2 − 14 FµνFµν − |DµΦ|2 −m2|Φ2|

)Metric with r = rh event horizon (AdS for r →∞) +negative cosmological constant Λ = −3/`2

ds2 = −g(r)f (r)dt2 +dr2

f (r)+ r2(dx2 + dy2)

Ansatz: Φ = Φ(r), At = At (r)

Presence of the U(1) gauge symmetry allows to gaugeaway the phase of the scalar field and make it real


Holographic insulator/ superconductor

double Wick rotation (t → iχ, x → it) of SAdS with rh → r0

ds2 = dr2

f (r) + f (r)dχ2 + r2(−dt2 + dy2

)with f (r) = r2

`2

(1− r3

0r3

)It is important that χ is periodic with period τχ = 4π`2

3r0

Scalar field in the background of such a soliton has astrictly positive and discrete spectrum (Witten, 1998)

There exists an energy gap which allows theinterpretation of this soliton as the gravity dual of aninsulatorAdding a chemical potential µ to the model reduces theenergy gap


The e = 0 limit

In the case of vanishing gauge coupling constant e:

The scalar field decouples from gauge fieldOne cannot use gauge to make scalar field realThe simplest ansatz for complex scalar field:φ(r) = φeiωt

This leads to Q-balls and boson stars solutions


The model for G = 0

SUSY potential U(|Φ|) = m2η2susy

(1− exp

(−|Φ|2/η2

susy))

Metric ds2 = −N(r)dt2 + 1N(r)dr2 + r2

(dθ2 + sin2 θdϕ2

)with N(r) = 1 + r2

`2and ` =

√−3/Λ

Using Φ(t , r) = eiωtφ(r), rescaling

Equation of motion φ′′ = −2r φ′ − N′

N φ′ − ω2

N2φ+ φ exp(−φ2)N

Power law for symptotic fall-off for Λ < 0:

φ(r) = φ∆r∆, ∆ = −32 −

√94 + `2

Charge and mass Q = 8π∫∞

0 φr2dr andM = 4π

∫∞0

[ω2φ2 + φ′2 + U(φ)

]r2dr


First results of the numerical analysis

ω

M

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

5010

020

050

010

0020

00

Mass over Omega

Λ= 0= −0.01= −0.02= −0.025

ω

M

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

5010

020

050

010

0020

00

Charge over Omega

Λ= 0= −0.01= −0.02= −0.025

Figure: Properties of SUSY Q-balls in AdS background mass M (left) and charge Q(right) versus frequency ω for various values of Λ



φ(0)

M

0 2 4 6 8 10

110

100

1000

1000

0

Mass over Phi(0)

Λ= 0= −0.01= −0.02= −0.025

φ(0)

Q

0 2 4 6 8 10

110

100

1000

1000

0

Charge over Phi(0)

Λ= 0= −0.5= −0.−1= −5

Figure: Properties of SUSY Q-balls in AdS background mass M (left) and charge Q(right) versus scalar field function at the origin φ(0) for various values of Λ



M

Q

200 500 1000 2000 5000 10000 20000200

500

2000

5000

2000

050

000

Charge over Mass

Λ= 0= −0.01= −0.02= −0.025

ω

φ(0)

0.2 0.4 0.6 0.8 1.0 1.2

02

46

810

Phi(0) over Omega

Λ= 0= −0.01= −0.02= −0.025

Figure: Properties of SUSY Q-balls in AdS background mass M versus charge Q(left) and the scalar field function at the origin φ(0) versus frequency ω (right) forvarious values of Λ



M

Con

dens

ate

0 5000 10000 15000

0.01

00.

015

0.02

00.

025

Condensate over Mass

Λ= −0.03= −0.04= −0.05= −0.075

Q

Con

dens

ate

0 5000 10000 15000 20000

0.01

00.

015

0.02

00.

025

Condensate over Charge

Λ= −0.03= −0.04= −0.05= −0.075

Figure: Condensate O1∆ over Mass M (left) and charge Q (right) for various values of

Λ



φ(0)

Con

dens

ate

0 2 4 6 8 10

0.01

00.

015

0.02

00.

025

Condensate over Phi(0)

Λ= −0.03= −0.04= −0.05= −0.075

Figure: Condensate O1∆ as function of the scalar field at φ(0) for various values of Λ


SUSY potential U(|Φ|) = m2η2susy

(1− exp

(−|Φ|2/η2

susy))

The coupling constant κ is given with κ = 8πGη2susy

Metricds2 = −A2(r)N(r)dt2 + 1

N(r)dr2 + r2 (dθ2 + sin2θdϕ2) with

N(r) = 1− 2n(r)r − Λ

3 r2 and ` =√−3/Λ

Using Φ(t , r) = eiωtφ(r) and rescalingEquations of motionn′ = κ

2 r2(

N(φ′)2 + ω2φ2

A2N + 1− exp(−φ2))

,

A′ = κr(ω2φ2

AN2 + Aφ′)

and(r2ANφ′

)′= −ω2r2

AN + r2Aφexp(−φ2)


φ(r) = φ∆r∆, ∆ = −32 −

√94 + `2


Calculating the mass


φ(r) = φ∆r∆, ∆ = −32 −

√94 + `2

The mass in the limit r 1 and κ > 0 isn(r 1) = M + n1φ

2∆r2∆+3 + ... with n1 = −Λ∆2+3

6(2∆+3)

For the case κ = 0 the Mass M is with n(r) ≡ 0, A(r) ≡ 1:M =

∫d3xT00 = 4π

∫∞0

[ω2φ2 + N2(φ′)2 + NU(φ)

]r2dr

The charge Q is given for all values of κ as:Q = 8π

∫∞0

ωr2

AN dr



ω

M

0.2 0.4 0.6 0.8 1.0

1050

500

5000

Mass over Omega

κ= 0.0= 0.001= 0.01= 0.05= 0.1

ω

Q

0.2 0.4 0.6 0.8 1.0

1050

500

5000

Charge over Omega

κ= 0.0= 0.001= 0.01= 0.05= 0.1

Figure: Properties of SUSY boson stars in AdS background mass M (left) andcharge Q (right) versus frequency ω for various values of κ and fixed Λ = 0.0



φ(0)

Q

0 2 4 6 8 10

1050

500

5000

Charge over Phi(0)

κ= 0.0= 0.001= 0.01= 0.05= 0.1

ω

φ(0)

0.2 0.4 0.6 0.8 1.0

05

1015

Phi(0) over Omega

κ= 0.0= 0.001= 0.01= 0.05= 0.1

Figure: Properties of SUSY boson stars in AdS background charge Q versus φ(0)

(left) and φ(0) versus frequency ω (right) for various values of κ and fixed Λ = 0.0



ω

Q

0.2 0.4 0.6 0.8 1.0

1050

500

5000

Charge over Omega

κ= 0.0= 0.001= 0.01= 0.075= 0.1

ω

Q

0.2 0.4 0.6 0.8 1.0

1050

500

5000

Charge over Omega

κ= 0.0= 0.001= 0.01= 0.075= 0.1

Figure: Properties of SUSY boson stars in AdS background charge Q versusfrequency ω for various values of κ and fixed Λ = −0.001 (left) and fixed Λ = −0.01(right)



ω

φ(0)

0.2 0.4 0.6 0.8 1.0

05

1015

20

Phi(0) over Omega

κ= 0.0= 0.001= 0.01= 0.075= 0.1

ω

φ(0)

0.2 0.4 0.6 0.8 1.0

05

1015

20

Phi(0) over Omega

κ= 0.0= 0.001= 0.01= 0.075= 0.1

Figure: Properties of SUSY boson star in AdS background φ(0) versus frequency ωfor various values of κ and fixed Λ = −0.001 (left) and fixed Λ = −0.01 (right)



ω

Q

0.2 0.4 0.6 0.8 1.0 1.2 1.4

1050

500

5000

Charge over Omega

Λ= 0.0= −0.001= −0.01= −0.05= −0.1

ω

Q

0.2 0.4 0.6 0.8 1.0 1.2 1.4

1050

500

5000

Charge over Omega

Λ= 0.0= −0.001= −0.01= −0.05= −0.1

Figure: Properties of SUSY boson stars in AdS background charge Q versusfrequency ω for various values of Λ and fixed κ = 0.0 (left) and fixed κ = 0.01 (right)



ω

φ(0)

0.2 0.4 0.6 0.8 1.0 1.2 1.4

02

46

810

Phi(0) over Omega

Λ= 0.0= −0.001= −0.01= −0.05= −0.1

ω

φ(0)

0.2 0.4 0.6 0.8 1.0 1.2 1.4

02

46

810

Phi(0) over Omega

Λ= 0.0= −0.001= −0.01= −0.05= −0.1

Figure: Properties of SUSY boson star in AdS background φ(0) versus frequency ωfor various values of Λ and fixed κ = 0.0 (left) and fixed κ = 0.01 (right)


Summary of first Results

Shift of ωmax for Q-balls and boson stars to higher valuesfor increasingly negative values of Λ, i.e.ωmax →∞ for Λ→ −∞The minimum value of the frequency for Q-balls isωmin = 0 for all Λ

The minimum value of the frequency for boson starsωmin increases for increasingly negative values of Λ

The curves mass M over frequency ω and charge Qversus ω for Q-balls and boson stars show

M → 0 for ω → ωmaxQ → 0 for ω → ωmax


Summary of first Results continued

For boson stars the cosmological constant Λ ’kills’ thelocal maximum of the charge Q and Mass M near ωmax ,similarly as large values of κ

The curve of the condensate for Q-balls, i.e. O1∆ as a

function of the scalar field φ(0), has qualitatively thesame shape as in Horowitz and Way, JHEP 1011:011, 2010[arXiv:1007.3714v2]


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 κ 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.


Numerical analysis Q-balls in Minkowski (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).



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

6d

8d

d + 1ω

ma

x

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).



r

φ

0 5 10 15 20

−0

.10

.10

.20

.30

.40

.5

k

= 0

= 1

= 2

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 2dω= 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 2dω= 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 their chargeQ 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 2dω= 1.0

ωQ

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 2dω= 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.


Technology

Internal workshop jub talk jan 2013