Fixed point structure of supersymmetric O(N) theories

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1 Fixed point structure of supersymmetric O(N) theories Tobias Hellwig FS University Jena PAF TPI Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

2 Table of contents 1 Physical fundamentals 2 The case of large N[Heilmann 2012] Renormalized eld theory Eective eld theory 3 Corrections given by nite N analysis Spontaneously broken phase Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

3 Physical fundamentals Why O(N) theory Bardeen-Moshe-Bander phenomenon seen (Hartree-Fock method, gap equation) Exact solution is available Can derive exact critical exponents Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

4 Physical fundamentals Supersymmetry Linear O(N) model L ψ=0 = q 2 ρ U 2 (ρ)ρ, ρ(x) = 1 2 φ i(x)φ i (x) Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

5 Physical fundamentals Supersymmetry Linear O(N) model L ψ=0 = q 2 ρ U 2 (ρ)ρ, ρ(x) = 1 2 φ i(x)φ i (x) Looking for the following potential U bos (ρ) = (U (ρ)) 2 ρ 0 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

6 Physical fundamentals Supersymmetry Linear O(N) model L ψ=0 = q 2 ρ U 2 (ρ)ρ, ρ(x) = 1 2 φ i(x)φ i (x) Looking for the following potential U bos (ρ) = (U (ρ)) 2 ρ 0 Eective average action [ ( 1 1 Γ k = d 3 q Z (2π) 3 2 F 2 q 2 ρ 1 ) ψ/qψ Y (... ) +U (F φ 1 ψψ) 1 ] 2 2 U ψ i ψ j φ i φ j Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

7 Physical fundamentals Flow equation Moving in theory space from one energy scale k 1 to another k 2 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

8 Physical fundamentals Flow equation Moving in theory space from one energy scale k 1 to another k 2 Done by integrating out momenta shells Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

9 Physical fundamentals Flow equation Moving in theory space from one energy scale k 1 to another k 2 Done by integrating out momenta shells Flow equation t Γ k = 1 2 STr ( Γ (2) k + R k ) 1 t R k, t = log k Λ Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

10 Physical fundamentals Flow equation Moving in theory space from one energy scale k 1 to another k 2 Done by integrating out momenta shells Flow equation t Γ k = 1 2 STr ( Γ (2) k + R k ) 1 t R k, t = log k Λ Result t U k (N 1) = U f N ( U k ) 1 N (3U + 2ρU )f ( U + 2ρU ) k Large N Limit t U k = U f (U /k), f (x) = 1 x 2 (1 + x 2 ) 2 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

11 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) U bos (ρ) = U 2 (ρ)ρ = τ 2 ( ρ 3 2κρ 2 + κ 2 ρ ) Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

12 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) U bos (ρ) = U 2 (ρ)ρ = τ ( 2 ρ 3 2κρ 2 + κ 2 ρ ) Physical minimum of U bos (ρ) U (ρ) = 0 or ρ = 0 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

13 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) U bos (ρ) = U 2 (ρ)ρ = τ 2 ( ρ 3 2κρ 2 + κ 2 ρ ) Physical minimum of U bos (ρ) U (ρ) = 0 or ρ = 0 κ < 1 symmetric phase (SYM) κ > 1 phase of spontaneously broken O(N) symmetry (SSB) Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

14 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) U bos (ρ) = U 2 (ρ)ρ = τ 2 ( ρ 3 2κρ 2 + κ 2 ρ ) Physical minimum of U bos (ρ) U (ρ) = 0 or ρ = 0 κ < 1 symmetric phase (SYM) κ > 1 phase of spontaneously broken O(N) symmetry (SSB) Parametrisation τ = 1 c, κ 1 ρ 0 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

15 The case of large N[Heilmann 2012] Ways to look at our theory Renormalized eld theory Theory is valid for all energy scales Λ no cuto scale Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

16 The case of large N[Heilmann 2012] Ways to look at our theory Renormalized eld theory Theory is valid for all energy scales Λ no cuto scale Eective eld theory Eective theory of a high energy theory Cuto scale is nite (Λ < ) Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

17 The case of large N[Heilmann 2012] Renormalized eld theory Renormalized eld theory Solution of the RG ow ρ ρ 0k = cu (ρ) + kh ( U k ), ρ 0k = k + ρ 0 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

18 The case of large N[Heilmann 2012] Renormalized eld theory Renormalized eld theory Solution of the RG ow ρ ρ 0k = cu (ρ) + kh ( U k ), ρ 0k = k + ρ 0 Masses in dierent regimes of c and ρ 0 at k = 0,[Moshe 2003], [Bardeen 1985] Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

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 [ ρ 0 /k, ] Information about negative eld amplitudes X may be important for solution Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

20 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 [ ρ 0 /k, ] Information about negative eld amplitudes X may be important for solution Masses in dierent regimes of c and ρ 0 (c L 3.07, c M 3.18) 100 u X I c= IV c=0 c=c M II III 5 c=π 100 c=cl c=2 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

21 The case of large N[Heilmann 2012] Eective eld theory Eective eld theory Solution of the ow equation ( ) ρ ρ 0k = cu U (ρ) + kh ΛH k ( U Λ ), ρ 0k = k + ρ 0 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

22 The case of large N[Heilmann 2012] Eective eld theory Eective eld theory Solution of the ow equation ( ) ρ ρ 0k = cu U (ρ) + kh ΛH k ( U Masses in dierent regimes of c and ρ 0 at k = 0 again Λ ), ρ 0k = k + ρ 0 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

23 The case of large N[Heilmann 2012] Eective eld theory Eective eld theory Structure of our solution in dierent regimes of c and ρ 0 derived from RG ow Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

24 The case of large N[Heilmann 2012] Eective eld theory Phase transition Second order phase transition between SSB and SYM exact critical exponents θ i = i 1, i = 0, 1,... Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

25 The case of large N[Heilmann 2012] Eective eld theory Phase transition Second order phase transition between SSB and SYM exact critical exponents θ i = 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 / 19

26 The case of large N[Heilmann 2012] Eective eld theory BMB c = π u ρ 0 u c=π Ρ Resulting mass M = (ku(0)) 2 does not have to vanish for k = 0 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

27 The case of large N[Heilmann 2012] Eective eld theory BMB c = π u ρ 0 u c=π Ρ Resulting mass M = (ku(0)) 2 does not have to vanish for k = 0 No scale invariance goldstone boson (dilaton) and goldstone fermion (dilatino) Critical exponent is given to ν BMB = = θ 0 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

28 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 exponents for innite N Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

29 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 exponents for innite N One solution independent of the order of truncation in power series c = 2(N ) strong coupling regime Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

30 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 exponents for innite N One solution independent of the order of truncation in power series c = 2(N ) strong coupling regime Critical exponents: (i + 1)i θ i = (1 i) 6 ( ) N + 17 N 1 1, i = 0, 1,... Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

31 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 exponents for innite N One solution independent of the order of truncation in power series c = 2(N ) strong coupling regime Critical exponents: (i + 1)i θ i = (1 i) 6 Does not exist for all ρ/k > 0 ( ) N + 17 N 1 1, i = 0, 1,... Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

32 Corrections given by nite N analysis Spontaneously broken phase 4 3 u' N=1000 N Ρ 2 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

33 Corrections given by nite N analysis Spontaneously broken phase Summary Exact Solution for N Derived masses of the model for dierent 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 using LPA Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

34 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: v1 [hep-th] 27. Aug 2012 [Moshe 2003] Moshe Moshe, Jean Zinn-Justin. Quantum eld theory in the large N limit: A review. Phys. Rept., 385:69-228,2003. [Bardeen 1985] William A. Bardeen, Kyoshi Higashijima, Moshe Moshe. Spontaneous Breaking of Scale Invariance in a Supersymmetric Model. Nucl. Phys.,B250:437, 1985 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

35 Corrections given by nite N analysis Symmetric phase Solution existing for all ρ > 0 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

36 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 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

37 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 Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19

38 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 / 19

39 Corrections given by nite N analysis Symmetric phase u(0) > 1 is the remaining case N 10 N 20 N 50 N N N numerical solution of xed point equation running of critical exponents derived with a power series ansatz Tobias Hellwig (TPI FSU Jena) Fixed point of O(N) theories / 19