The fixed point structure of the 3d O(N) model and its non-trivial UV fixed point

Transcription

1 The fixed point structure of the 3d O(N) model and its non-trivial UV fixed point

2 Contents Introduction FRG O(N) model in the large N limit Flow equation Solving the flow equation Local flow Exact solution Universality & fixed points Eigenperturbation, critical exponents Bardeen-Moshe-Bander phenomenon Conclusion

3 Functional Formalism INTRODUCTION Euclidean QFT n-point function can be produced from the generating functional The effective action (by Legendre trf.) Q-EOM: Describes the dynamics of the VEV + quantum fluc. included

4 FRG INTRODUCTION Wilsonian idea: instead PT, we integrate out momentum shell by momentum shell Effective average action: How? k

5 FRG INTRODUCTION Wilsonian idea: instead PT, we integrate out momentum shell by momentum shell Effective average action: How? k Momentum-dependent mass term

6 FRG INTRODUCTION Wilsonian idea: instead PT, we integrate out momentum shell by momentum shell Effective average action: How? k Momentum-dependent mass term

7 FRG INTRODUCTION We apply the same routine for our new theory The average effective action The VEV Q-EOM The scale dependence of the avarage effective action: the flow eq. q

8 FRG INTRODUCTION We apply the same routine for our new theory The average effective action The VEV Q-EOM The scale dependence of the avarage effective action: the flow eq. q We can choose regulator One-loop structure PT expansion can be recovered full propagator

9 FRG INTRODUCTION

10 Flow Equation To solve the RG flow: an ansatz for the effective action is needed Slowly varying fields INTRODUCTION Derivative Expansion/ Local Potential Approximation (LPA) The O(N) sym. avr. eff. action

11 Flow Equation To solve the RG flow: an ansatz for the effective action is needed Slowly varying fields INTRODUCTION Derivative Expansion/ Local Potential Approximation (LPA) The O(N) sym. avr. eff. action Plugging into the flow

12 Flow Equation To solve the RG flow: an ansatz for the effective action is needed Slowly varying fields INTRODUCTION Derivative Expansion/ Local Potential Approximation (LPA) The O(N) sym. avr. eff. action Plugging into the flow Flow of effective potential

13 Flow Equation INTRODUCTION Using the optimized regulator: the loop integral is analytic Taking the large N-limit (the universality class of the ideal Bose gas) The flow for the dimensionless effective potential Dimensionless quantities

14 Flow Equation INTRODUCTION Using the optimized regulator: the loop integral is analytic Taking the large N-limit (the universality class of the ideal Bose gas) The flow for the dimensionless effective potential Dimensionless quantities A PDE for the derivative(!) of the effective potential. We need to solve this.

15 Local Flow SOLVING THE F.E. Expanding the potential in terms of polynomial couplings Plug it in our equation for the effective potential

16 Local Flow SOLVING THE F.E. Expanding the potential in terms of polynomial couplings (decoupled) The b-functions (exactly marginal)

17 Local Flow SOLVING THE F.E. Expanding the potential in terms of polynomial couplings (decoupled) Running The b-functions (exactly marginal)

18 Local Flow SOLVING THE F.E. Expanding the potential in terms of polynomial couplings (decoupled) Running Fixed Points The b-functions (exactly marginal) Tricritical (UV attractive) Wilson-Fisher (IR attractive) Constants from initial value

19 Global Flow SOLVING THE F.E. The flow equation can be solved ANALITICALLY in the large N

20 Global Flow SOLVING THE F.E. The flow equation can be solved ANALITICALLY in the large N analytic continuation * * Here G-s on the RHS are depending on the initial conditions

21 Global Flow SOLVING THE F.E. Fixed points are described by scaling solutions 0

22 Global Flow SOLVING THE F.E. Fixed points are described by scaling solutions 0 it turns out:

23 Global Flow SOLVING THE F.E. Four branches describe the solution epending on the sign of c and u Issues 1. Branch continuation 2. Turning points 0 1

24 Global Flow SOLVING THE F.E. 1. Branch continuation 0 1 Continuation from positive to negative branch is not smooth enough

25 Global Flow SOLVING THE F.E. 1. Branch continuation 0 1 Continuation from positive to negative branch is not smooth enough

26 Global Flow SOLVING THE F.E. 2. Turning points In this regime we have to choose either the dashed or the full line: strong coupling 0 1

27 The structure THE FP STRUCTURE The proper definition of the function: We consider only 0 1 The treshold value for the turning point

28 Global VS Local THE FP STRUCTURE Thus if we tune the VEV to its critical value we can distinguish different type of fixed point solutions Gauss Wilson-Fisher Tricritical BMB 0 1 NEW

29 Parameter space THE FP STRUCTURE Fixed point solutions Topology of the parameter space Critical surfaces G t BMB

30 Parameter space THE FP STRUCTURE Fixed point solutions Topology of the parameter space Critical surfaces G t BMB

31 Parameter space THE FP STRUCTURE Fixed point solutions Topology of the parameter space Critical surfaces G t BMB

32 Parameter space THE FP STRUCTURE Fixed point solutions Topology of the parameter space Critical surfaces G t BMB

33 Parameter space THE FP STRUCTURE Fixed point solutions Topology of the parameter space Critical surfaces G t BMB

34 The Potential THE FP STRUCTURE Integrating u respect to BMB G

35 Conclusion * Fixed point solutions: - From local flow - From global flow Non-perturbative solution to a 3d, O(N) symmetric quantum field theory theory in the large N limit A fixed point is found which was invisible for the local flow LITERATURE [1] Bardeen-Moshe-Bander, Fixed Point and the Ultraviolet Triviality of (Φ 2)_3^ /PhysRevLett [2] D. F. Litim, Optimisation of the exact renormal-isation group, Phys. Lett. B 486(2000) 92, [hep-th/ ] [3] Edouard Marchais, PhD Thesis [4] M. Heilmann, D. F. Litim, F. Synatschke- Czerwonka and A. Wipf1, Critical behavior of supersymmetric O(N) models in the large-n limit, /physrevd [5] Daniel F. Litim, Marianne C. Mastaler, Franziska Synatschke-Czerwonka, Andreas Wipf,Critical behavior of supersymmetric O(N) models in the large-n limit [6] H. Gies, Introduction to the functional RG and applications to gauge theories, arxiv:hepph/ v1 * Escape point

36 Eigenperturbation THE FP STRUCTURE The idea: perturbing around the scaling solution Inserting it into the flow equation we obtain the fluctuation equation Solving it by separation of variables gives: The eigenperturbation equation reads: ANALITICITY CONDITION: the perturbation must be analytic Restriction on Remark:

37 Critical Exponents THE FP STRUCTURE Wilson-Fisher Tricritical (mean-field)

38 Bardeen-Moshe-Bander ph. THE FP STRUCTURE The BMB fixed point solution has a singularity at 0 demanding analyticity is useless BMB fixed point Arbitrarily large mass Breaking of scale invariance!

39 Conclusion Non-perturbative solution to a 3d, O(N) symmetric quantum field theory theory in the large N limit Study of the fixed point solutions and phase transitions (WF, Tricrit., BMB) Critical exponents recovered BMB: UV fixed point with breaking of the scale invariance LITERATURE [1] Bardeen-Moshe-Bander, Fixed Point and the Ultraviolet Triviality of (Φ 2)_3^ /PhysRevLett [2] D. F. Litim, Optimisation of the exact renormal-isation group, Phys. Lett. B 486(2000) 92, [hep-th/ ] [3] Edouard Marchais, PhD Thesis [4] M. Heilmann, D. F. Litim, F. Synatschke- Czerwonka and A. Wipf1, Critical behavior of supersymmetric O(N) models in the large-n limit, /physrevd [5] Daniel F. Litim, Marianne C. Mastaler, Franziska Synatschke-Czerwonka, Andreas Wipf,Critical behavior of supersymmetric O(N) models in the large-n limit [6] H. Gies, Introduction to the functional RG and applications to gauge theories, arxiv:hepph/ v1

40 THANK YOU FOR YOUR ATTENTION. Supported by: European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP A/ 2-11/ National Excellence Program