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Transcript of 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
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 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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Prominent examples for topological solitons
Further examplesvortices, magnetic monopoles, domain walls, cosmicstrings, textures
Prominent examples for non-topological solitons
Q-ballsBoson stars
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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 Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
First results of the numerical analysis
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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]
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
The model for d + 1 dimensions
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
The model for d + 1 dimensions
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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)
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) 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
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) 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.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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).
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0)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
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)
= −0.1 (analytical)
= −0.5 (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).
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0)background
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.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0)background
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 .
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis Q-balls in Minkowski (Λ = 0) andAdS background (Λ < 0) background
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 .
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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).
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) background
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.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) background
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.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) background
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 .
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) 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 their chargeQ for different values of κ and d . Λ = 0.001
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
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 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.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions
Numerical analysis boson stars in Minkowski (Λ = 0)and AdS background (Λ < 0) background
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.
Jürgen Riedel & Betti Hartmann Supersymmetric Q-balls and boson stars in (d + 1) dimensions