obiasT Hellwig - uni-jena.de€¦ · ableT of contents 1 Physical fundamentals 2 The case of large...

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

Physical fundamentals

Why O(N) theory

Bardeen-Moshe-Bander phenomenon seen (Hartree-Fock method, gapequation)

Exact solution is available

Can derive exact critical exponents

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

∫d3q

(2π)3

2F 2 − q2ρ− 1

2ψ̄/qψ

4Y (. . . )

+U ′(Fφ− 1

2ψ̄ψ)− 1

2U ′′ψiψjφ

Supersymmetry

Linear O(N) model

L|ψ=0 = −q2ρ− U ′2(ρ)ρ, ρ(x) =1

2φi (x)φi (x)

Ubos(ρ) = (U ′(ρ))2ρ ≥ 0

∫d3q

(2π)3

2F 2 − q2ρ− 1

2ψ̄/qψ

4Y (. . . )

+U ′(Fφ− 1

2ψ̄ψ)− 1

2U ′′ψiψjφ

Supersymmetry

Linear O(N) model

L|ψ=0 = −q2ρ− U ′2(ρ)ρ, ρ(x) =1

2φi (x)φi (x)

Ubos(ρ) = (U ′(ρ))2ρ ≥ 0

∫d3q

(2π)3

2F 2 − q2ρ− 1

2ψ̄/qψ

4Y (. . . )

+U ′(Fφ− 1

2ψ̄ψ)− 1

2U ′′ψiψjφ

Flow equation

Moving in theory space from one energy scale k1 to another k2

Done by integrating out momenta shells

Flow equation

∂tΓk =1

(Γ(2)k + Rk

)−1∂tRk , t = log

Result

∂tU′

k= −(N − 1)

NU ′′f

(U ′

)− 1

N(3U ′′ + 2ρU ′′′)f

(U ′ + 2ρU ′′

)Large N Limit

∂tU′

k= −U ′′f (U ′/k), f (x) =

1− x2

(1 + x2)2

Flow equation

∂tΓk =1

(Γ(2)k + Rk

Result

∂tU′

k= −(N − 1)

NU ′′f

(U ′

)− 1

N(3U ′′ + 2ρU ′′′)f

(U ′ + 2ρU ′′

)Large N Limit

∂tU′

k= −U ′′f (U ′/k), f (x) =

1− x2

(1 + x2)2

Flow equation

∂tΓk =1

(Γ(2)k + Rk

Result

∂tU′

k= −(N − 1)

NU ′′f

(U ′

)− 1

N(3U ′′ + 2ρU ′′′)f

(U ′ + 2ρU ′′

)Large N Limit

∂tU′

k= −U ′′f (U ′/k), f (x) =

1− x2

(1 + x2)2

Flow equation

∂tΓk =1

(Γ(2)k + Rk

Result

∂tU′

k= −(N − 1)

NU ′′f

(U ′

)− 1

N(3U ′′ + 2ρU ′′′)f

(U ′ + 2ρU ′′

)Large N Limit

∂tU′

k= −U ′′f (U ′/k), f (x) =

1− x2

(1 + x2)2

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

)Physical minimum of Ubos(ρ)

⇔ U ′(ρ) = 0 or ρ = 0

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 <∞)

Ways to look at our theory

Theory is valid for all energy scalesΛ→∞ no cuto� scale

E�ective theory of a high energy theoryCuto� scale is �nite (Λ <∞)

The case of large N[Heilmann 2012] Renormalized �eld theory

Solution of the RG �ow

ρ− ρ0k = cU ′(ρ) + kH

(U ′

), ρ0k = k + ρ0

Masses in di�erent regimes of c and ρ0 at k = 0,[Moshe 2003],[Bardeen 1985]

Solution of the RG �ow

ρ− ρ0k = cU ′(ρ) + kH

(U ′

), ρ0k = k + ρ0

Masses in di�erent regimes of c and ρ0 at k = 0,[Moshe 2003],[Bardeen 1985]

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=2c=cM

c=Πc=cL

X-100 - 5 -1 - 0.5 0.5 1 5 100

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=2c=cM

c=Πc=cL

X-100 - 5 -1 - 0.5 0.5 1 5 100

The case of large N[Heilmann 2012] E�ective �eld theory

Solution of the �ow equation

ρ− ρ0k = cU ′(ρ) + kH

(U ′

)− ΛH

(U ′

), ρ0k = k + ρ0

Masses in di�erent regimes of c and ρ0 at k = 0 again

Solution of the �ow equation

ρ− ρ0k = cU ′(ρ) + kH

(U ′

)− ΛH

(U ′

), ρ0k = k + ρ0

Masses in di�erent regimes of c and ρ0 at k = 0 again

Structure of our solution in di�erent regimes of c and ρ0 derived fromRG �ow

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)

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)

c = π ⇒ |u| ρ→0−→ ∞u*

-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

c = π ⇒ |u| ρ→0−→ ∞u*

-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

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

(√N + 17

N − 1− 1

), i = 0, 1, . . .

Does not exist for all ρ/k > 0

Critical exponents:

θi = (1− i)− (i + 1)i

(√N + 17

N − 1− 1

), i = 0, 1, . . .

Critical exponents:

θi = (1− i)− (i + 1)i

(√N + 17

N − 1− 1

), i = 0, 1, . . .

Critical exponents:

θi = (1− i)− (i + 1)i

(√N + 17

N − 1− 1

), i = 0, 1, . . .

Ρ � k

u'N=1000N®¥

2 4 6 8 10

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

[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

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

|u(0)| > 1 is the remaining case

0.2 0.4 0.6 0.8 1.0Ρ

numerical solution of �xed pointequation

� � �

� � ��

� � � ��

8 10 12 14 16

� N = ¥

� N =10 3

� N =10 2

� N = 50

� N = 20

� N =10

running of critical exponents derivedwith a power series ansatz

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