SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time - Cancun talk 2012

IntroductionQ-balls

Boson starsAdS/CFT correspondence

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

Conclusion

SUSY Q-Balls and Boson Stars in Anti-deSitter space-time

Jürgen Riedelin Collaboration with Betti Hartmann, Jacobs University Bremen

School of Engineering and ScienceJacobs University Bremen, Germany

CANCUN TALK 2012Cancun, March 10th 2012

Jürgen Riedel & Betti Hartmann SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time

IntroductionQ-balls



Conclusion

Outline

1 Introduction

2 Q-balls

3 Boson stars

4 AdS/CFT correspondence

5 SUSY Q-balls in AdS background

6 SUSY boson stars in AdS background

7 Conclusion


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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∗φ̇− φφ̇∗)

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


IntroductionQ-balls



Conclusion

Prominent examples for topological solitons

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

Prominent examples for non-topological solitons

Q-ballsBoson stars


IntroductionQ-balls



Conclusion

Outline

1 Introduction

2 Q-balls

3 Boson stars




7 Conclusion


IntroductionQ-balls



Conclusion

The model

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

i

∫d3(Φ∗Φ̇− ΦΦ̇∗), 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 [|Φ̇|2 + |OΦ|2 + U(|Φ|)]under assumption that gµν is time-independent


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

Outline

1 Introduction

2 Q-balls

3 Boson stars




7 Conclusion


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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)


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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)


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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

)Jürgen Riedel & Betti Hartmann SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time

IntroductionQ-balls



Conclusion

Outline

1 Introduction

2 Q-balls

3 Boson stars




7 Conclusion


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

Outline

1 Introduction

2 Q-balls

3 Boson stars




7 Conclusion


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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

M0.5 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


IntroductionQ-balls



Conclusion


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)

Q0 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


IntroductionQ-balls



Conclusion


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

ωφ(

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


IntroductionQ-balls



Conclusion


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 200000.

010

0.01

50.

020

0.02

5

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


IntroductionQ-balls



Conclusion


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


IntroductionQ-balls



Conclusion

Outline

1 Introduction

2 Q-balls

3 Boson stars




7 Conclusion


IntroductionQ-balls



Conclusion

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

φ(r) = φ∆r∆, ∆ = −32 −

√94 + `2


IntroductionQ-balls



Conclusion

Calculating the mass

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

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


IntroductionQ-balls



Conclusion


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

Q0.2 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


IntroductionQ-balls



Conclusion


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

ωφ(

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


IntroductionQ-balls



Conclusion


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

Q0.2 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)


IntroductionQ-balls



Conclusion


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

ωφ(

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)


IntroductionQ-balls



Conclusion


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

Q0.2 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)


IntroductionQ-balls



Conclusion


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

ωφ(

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)


IntroductionQ-balls



Conclusion

Outline

1 Introduction

2 Q-balls

3 Boson stars




7 Conclusion


IntroductionQ-balls



Conclusion

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


IntroductionQ-balls



Conclusion

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]


IntroductionQ-balls



Conclusion

Outlook

Studying the condensate of boson stars in AdS withSUSY potentialInterpreting the condensate in the context of CFTStudying Q-balls and boson stars in AdS in (d+1)dimensionsStudying rotating boson stars in AdS with SUSYpotential


SUSY Q-Balls and Boson Stars in Anti-de Sitter space-time - Cancun talk 2012

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