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Fixed point structure of supersymmetric O(N) theories
Tobias Hellwig
FS University Jena PAF TPI
24.09.2012
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 1 / 19
Table of contents
1 Physical fundamentals
2 The case of large N[Heilmann 2012]Renormalized �eld theoryE�ective �eld theory
3 Corrections given by �nite N analysisSpontaneously broken phase
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 2 / 19
Physical fundamentals
Why O(N) theory
Bardeen-Moshe-Bander phenomenon seen (Hartree-Fock method, gapequation)
Exact solution is available
Can derive exact critical exponents
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 3 / 19
Physical fundamentals
Supersymmetry
Linear O(N) model
L|ψ=0 = −q2ρ− U ′2(ρ)ρ, ρ(x) =1
2φi (x)φi (x)
Looking for the following potential
Ubos(ρ) = (U ′(ρ))2ρ ≥ 0
E�ective average action
Γk =
∫d3q
1
(2π)3
[Z
(1
2F 2 − q2ρ− 1
2ψ̄/qψ
)+
1
4Y (. . . )
+U ′(Fφ− 1
2ψ̄ψ)− 1
2U ′′ψiψjφ
iφj]
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 4 / 19
Physical fundamentals
Supersymmetry
Linear O(N) model
L|ψ=0 = −q2ρ− U ′2(ρ)ρ, ρ(x) =1
2φi (x)φi (x)
Looking for the following potential
Ubos(ρ) = (U ′(ρ))2ρ ≥ 0
E�ective average action
Γk =
∫d3q
1
(2π)3
[Z
(1
2F 2 − q2ρ− 1
2ψ̄/qψ
)+
1
4Y (. . . )
+U ′(Fφ− 1
2ψ̄ψ)− 1
2U ′′ψiψjφ
iφj]
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 4 / 19
Physical fundamentals
Supersymmetry
Linear O(N) model
L|ψ=0 = −q2ρ− U ′2(ρ)ρ, ρ(x) =1
2φi (x)φi (x)
Looking for the following potential
Ubos(ρ) = (U ′(ρ))2ρ ≥ 0
E�ective average action
Γk =
∫d3q
1
(2π)3
[Z
(1
2F 2 − q2ρ− 1
2ψ̄/qψ
)+
1
4Y (. . . )
+U ′(Fφ− 1
2ψ̄ψ)− 1
2U ′′ψiψjφ
iφj]
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 4 / 19
Physical fundamentals
Flow equation
Moving in theory space from one energy scale k1 to another k2
Done by integrating out momenta shells
Flow equation
∂tΓk =1
2STr
(Γ(2)k + Rk
)−1∂tRk , t = log
k
Λ
Result
∂tU′
k= −(N − 1)
NU ′′f
(U ′
k
)− 1
N(3U ′′ + 2ρU ′′′)f
(U ′ + 2ρU ′′
k
)Large N Limit
∂tU′
k= −U ′′f (U ′/k), f (x) =
1− x2
(1 + x2)2
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 5 / 19
Physical fundamentals
Flow equation
Moving in theory space from one energy scale k1 to another k2
Done by integrating out momenta shells
Flow equation
∂tΓk =1
2STr
(Γ(2)k + Rk
)−1∂tRk , t = log
k
Λ
Result
∂tU′
k= −(N − 1)
NU ′′f
(U ′
k
)− 1
N(3U ′′ + 2ρU ′′′)f
(U ′ + 2ρU ′′
k
)Large N Limit
∂tU′
k= −U ′′f (U ′/k), f (x) =
1− x2
(1 + x2)2
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 5 / 19
Physical fundamentals
Flow equation
Moving in theory space from one energy scale k1 to another k2
Done by integrating out momenta shells
Flow equation
∂tΓk =1
2STr
(Γ(2)k + Rk
)−1∂tRk , t = log
k
Λ
Result
∂tU′
k= −(N − 1)
NU ′′f
(U ′
k
)− 1
N(3U ′′ + 2ρU ′′′)f
(U ′ + 2ρU ′′
k
)Large N Limit
∂tU′
k= −U ′′f (U ′/k), f (x) =
1− x2
(1 + x2)2
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 5 / 19
Physical fundamentals
Flow equation
Moving in theory space from one energy scale k1 to another k2
Done by integrating out momenta shells
Flow equation
∂tΓk =1
2STr
(Γ(2)k + Rk
)−1∂tRk , t = log
k
Λ
Result
∂tU′
k= −(N − 1)
NU ′′f
(U ′
k
)− 1
N(3U ′′ + 2ρU ′′′)f
(U ′ + 2ρU ′′
k
)Large N Limit
∂tU′
k= −U ′′f (U ′/k), f (x) =
1− x2
(1 + x2)2
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 5 / 19
The case of large N[Heilmann 2012]
Solutions of the �ow equation
Solution of the PDE is known
Initial condition at t = 0, k = Λ:
U ′(ρ) = τ(ρ− κ)⇒ Ubos(ρ) = U ′2(ρ)ρ = τ2(ρ3 − 2κρ2 + κ2ρ
)
Physical minimum of Ubos(ρ)
⇔ U ′(ρ) = 0 or ρ = 0
κ < 1 symmetric phase (SYM) κ > 1 phase of spontaneously brokenO(N) symmetry (SSB)
Parametrisation τ = 1c, κ− 1 ∝ ρ0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 6 / 19
The case of large N[Heilmann 2012]
Solutions of the �ow equation
Solution of the PDE is known
Initial condition at t = 0, k = Λ:
U ′(ρ) = τ(ρ− κ)⇒ Ubos(ρ) = U ′2(ρ)ρ = τ2(ρ3 − 2κρ2 + κ2ρ
)Physical minimum of Ubos(ρ)
⇔ U ′(ρ) = 0 or ρ = 0
κ < 1 symmetric phase (SYM) κ > 1 phase of spontaneously brokenO(N) symmetry (SSB)
Parametrisation τ = 1c, κ− 1 ∝ ρ0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 6 / 19
The case of large N[Heilmann 2012]
Solutions of the �ow equation
Solution of the PDE is known
Initial condition at t = 0, k = Λ:
U ′(ρ) = τ(ρ− κ)⇒ Ubos(ρ) = U ′2(ρ)ρ = τ2(ρ3 − 2κρ2 + κ2ρ
)Physical minimum of Ubos(ρ)
⇔ U ′(ρ) = 0 or ρ = 0
κ < 1 symmetric phase (SYM) κ > 1 phase of spontaneously brokenO(N) symmetry (SSB)
Parametrisation τ = 1c, κ− 1 ∝ ρ0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 6 / 19
The case of large N[Heilmann 2012]
Solutions of the �ow equation
Solution of the PDE is known
Initial condition at t = 0, k = Λ:
U ′(ρ) = τ(ρ− κ)⇒ Ubos(ρ) = U ′2(ρ)ρ = τ2(ρ3 − 2κρ2 + κ2ρ
)Physical minimum of Ubos(ρ)
⇔ U ′(ρ) = 0 or ρ = 0
κ < 1 symmetric phase (SYM) κ > 1 phase of spontaneously brokenO(N) symmetry (SSB)
Parametrisation τ = 1c, κ− 1 ∝ ρ0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 6 / 19
The case of large N[Heilmann 2012]
Ways to look at our theory
Renormalized �eld theory
Theory is valid for all energy scalesΛ→∞ no cuto� scale
E�ective �eld theory
E�ective theory of a high energy theoryCuto� scale is �nite (Λ <∞)
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 7 / 19
The case of large N[Heilmann 2012]
Ways to look at our theory
Renormalized �eld theory
Theory is valid for all energy scalesΛ→∞ no cuto� scale
E�ective �eld theory
E�ective theory of a high energy theoryCuto� scale is �nite (Λ <∞)
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 7 / 19
The case of large N[Heilmann 2012] Renormalized �eld theory
Renormalized �eld theory
Solution of the RG �ow
ρ− ρ0k = cU ′(ρ) + kH
(U ′
k
), ρ0k = k + ρ0
Masses in di�erent regimes of c and ρ0 at k = 0,[Moshe 2003],[Bardeen 1985]
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 8 / 19
The case of large N[Heilmann 2012] Renormalized �eld theory
Renormalized �eld theory
Solution of the RG �ow
ρ− ρ0k = cU ′(ρ) + kH
(U ′
k
), ρ0k = k + ρ0
Masses in di�erent regimes of c and ρ0 at k = 0,[Moshe 2003],[Bardeen 1985]
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 8 / 19
The case of large N[Heilmann 2012] Renormalized �eld theory
Renormalized �eld theory
Dimensionless solution u = U ′/k of the RG �ow
u(ρ/k) = u∗(X ), X ∈ [−ρ0/k ,∞]
Information about negative �eld amplitudes X may be important forsolution
Masses in di�erent regimes of c and ρ0 (cL ≈ 3.07, cM ≈ 3.18)
c=0
c=2c=cM
c=Πc=cL
c=50I
IIIII
IV
u*
X-100 - 5 -1 - 0.5 0.5 1 5 100
-100
- 5
-1
- 0.5
0.5
1
5
100
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 9 / 19
The case of large N[Heilmann 2012] Renormalized �eld theory
Renormalized �eld theory
Dimensionless solution u = U ′/k of the RG �ow
u(ρ/k) = u∗(X ), X ∈ [−ρ0/k ,∞]
Information about negative �eld amplitudes X may be important forsolution
Masses in di�erent regimes of c and ρ0 (cL ≈ 3.07, cM ≈ 3.18)
c=0
c=2c=cM
c=Πc=cL
c=50I
IIIII
IV
u*
X-100 - 5 -1 - 0.5 0.5 1 5 100
-100
- 5
-1
- 0.5
0.5
1
5
100
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 9 / 19
The case of large N[Heilmann 2012] E�ective �eld theory
E�ective �eld theory
Solution of the �ow equation
ρ− ρ0k = cU ′(ρ) + kH
(U ′
k
)− ΛH
(U ′
Λ
), ρ0k = k + ρ0
Masses in di�erent regimes of c and ρ0 at k = 0 again
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 10 / 19
The case of large N[Heilmann 2012] E�ective �eld theory
E�ective �eld theory
Solution of the �ow equation
ρ− ρ0k = cU ′(ρ) + kH
(U ′
k
)− ΛH
(U ′
Λ
), ρ0k = k + ρ0
Masses in di�erent regimes of c and ρ0 at k = 0 again
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 10 / 19
The case of large N[Heilmann 2012] E�ective �eld theory
E�ective �eld theory
Structure of our solution in di�erent regimes of c and ρ0 derived fromRG �ow
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 11 / 19
The case of large N[Heilmann 2012] E�ective �eld theory
Phase transition
Second order phase transition between SSB and SYM
exact critical exponents
θi = i − 1, i = 0, 1, . . .
Case c = π additional new phenomenon
Bardeen-Moshe-Bander phenomenon (BMB)
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 12 / 19
The case of large N[Heilmann 2012] E�ective �eld theory
Phase transition
Second order phase transition between SSB and SYM
exact critical exponents
θi = i − 1, i = 0, 1, . . .
Case c = π additional new phenomenon
Bardeen-Moshe-Bander phenomenon (BMB)
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 12 / 19
The case of large N[Heilmann 2012] E�ective �eld theory
BMB
c = π ⇒ |u| ρ→0−→ ∞u*
Ρ
c=Π
-100 - 5 -1 - 0.5 0.5 1 5 100
-100
- 5
-1
- 0.5
0.5
1
5
100
Resulting mass M = (ku(0))2 does not have to vanish for k = 0
No scale invariance ⇒ goldstone boson (dilaton) and goldstonefermion (dilatino)
Critical exponent is given to
νBMB =1
26= 1 = θ0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 13 / 19
The case of large N[Heilmann 2012] E�ective �eld theory
BMB
c = π ⇒ |u| ρ→0−→ ∞u*
Ρ
c=Π
-100 - 5 -1 - 0.5 0.5 1 5 100
-100
- 5
-1
- 0.5
0.5
1
5
100
Resulting mass M = (ku(0))2 does not have to vanish for k = 0
No scale invariance ⇒ goldstone boson (dilaton) and goldstonefermion (dilatino)
Critical exponent is given to
νBMB =1
26= 1 = θ0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 13 / 19
Corrections given by �nite N analysis Spontaneously broken phase
Linear polynomial approximation in the SSB
Two �xed point solutions with correct limit of their critical exponentsfor in�nite N
One solution independent of the order of truncation in power seriesc = 2(N →∞) ⇒ strong coupling regime
Critical exponents:
θi = (1− i)− (i + 1)i
6
(√N + 17
N − 1− 1
), i = 0, 1, . . .
Does not exist for all ρ/k > 0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 14 / 19
Corrections given by �nite N analysis Spontaneously broken phase
Linear polynomial approximation in the SSB
Two �xed point solutions with correct limit of their critical exponentsfor in�nite N
One solution independent of the order of truncation in power seriesc = 2(N →∞) ⇒ strong coupling regime
Critical exponents:
θi = (1− i)− (i + 1)i
6
(√N + 17
N − 1− 1
), i = 0, 1, . . .
Does not exist for all ρ/k > 0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 14 / 19
Corrections given by �nite N analysis Spontaneously broken phase
Linear polynomial approximation in the SSB
Two �xed point solutions with correct limit of their critical exponentsfor in�nite N
One solution independent of the order of truncation in power seriesc = 2(N →∞) ⇒ strong coupling regime
Critical exponents:
θi = (1− i)− (i + 1)i
6
(√N + 17
N − 1− 1
), i = 0, 1, . . .
Does not exist for all ρ/k > 0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 14 / 19
Corrections given by �nite N analysis Spontaneously broken phase
Linear polynomial approximation in the SSB
Two �xed point solutions with correct limit of their critical exponentsfor in�nite N
One solution independent of the order of truncation in power seriesc = 2(N →∞) ⇒ strong coupling regime
Critical exponents:
θi = (1− i)− (i + 1)i
6
(√N + 17
N − 1− 1
), i = 0, 1, . . .
Does not exist for all ρ/k > 0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 14 / 19
Corrections given by �nite N analysis Spontaneously broken phase
Ρ � k
u'N=1000N®¥
2 4 6 8 10
- 2
-1
1
2
3
4
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 15 / 19
Corrections given by �nite N analysis Spontaneously broken phase
Summary
Exact Solution for N →∞Derived masses of the model for di�erent phases in the large N limit
Same result as Hartree-Fock method by looking at gap equation
Improved our knowledge from H-F by looking at RG �ow
Could derive the approximately realised strong coupling constant usingLPA
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 16 / 19
Corrections given by �nite N analysis Spontaneously broken phase
[Heilmann 2012] Heilmann, M; Litim, D.F.; Synatschke-Czerwonka, F.;Wipf, A.: �Phases of supersymmetric O(N) theories� (Artikel),arxiv:1208.5389v1 [hep-th] 27. Aug 2012
[Moshe 2003] Moshe Moshe, Jean Zinn-Justin. Quantum �eld theory inthe large N limit: A review. Phys. Rept., 385:69-228,2003.
[Bardeen 1985] William A. Bardeen, Kyoshi Higashijima, MosheMoshe. Spontaneous Breaking of Scale Invariance in a SupersymmetricModel. Nucl. Phys.,B250:437, 1985
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 17 / 19
Corrections given by �nite N analysis Symmetric phase
Solution existing for all ρ > 0
solely one of u(0) and u′(0) can be chosen
Solutions with |u(0)|>1 do exist for all ρ/k > 0
Solutions with |u(0)|<1 cease to exist at some value ≈ ρ/k > 0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 18 / 19
Corrections given by �nite N analysis Symmetric phase
Solution existing for all ρ > 0
solely one of u(0) and u′(0) can be chosen
Solutions with |u(0)|>1 do exist for all ρ/k > 0
Solutions with |u(0)|<1 cease to exist at some value ≈ ρ/k > 0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 18 / 19
Corrections given by �nite N analysis Symmetric phase
Solution existing for all ρ > 0
solely one of u(0) and u′(0) can be chosen
Solutions with |u(0)|>1 do exist for all ρ/k > 0
Solutions with |u(0)|<1 cease to exist at some value ≈ ρ/k > 0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 18 / 19
Corrections given by �nite N analysis Symmetric phase
Solution existing for all ρ > 0
solely one of u(0) and u′(0) can be chosen
Solutions with |u(0)|>1 do exist for all ρ/k > 0
Solutions with |u(0)|<1 cease to exist at some value ≈ ρ/k > 0
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 18 / 19
Corrections given by �nite N analysis Symmetric phase
|u(0)| > 1 is the remaining case
0.2 0.4 0.6 0.8 1.0Ρ
5
10
15
20
25
30
u
numerical solution of �xed pointequation
� � �
�
�
� � �
�
�
� � ��
�
� � ��
�
� � � ��� � � � �
8 10 12 14 16
-10
- 5
5
� N = ¥
� N =10 3
� N =10 2
� N = 50
� N = 20
� N =10
running of critical exponents derivedwith a power series ansatz
Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories 24.09.2012 19 / 19