Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Properties of bound states from the FRG and the DSE-BSE approaches Jordi Par´ ıs L´ opez Advisors: R. Alkofer and H. Sanchis-Alepuz Karl-Franzens-Universit¨ at Graz, Austria Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 1 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Content Motivation Brief introduction to the FRG FRG techniques Results in the Quark-Meson model Preliminary comparison with the DSE-BSE approach Summary and Outlook Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 2 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Motivation Observable properties of hadrons are difficult to extract from QCD’s degrees of freedom: Need theoretical assumptions: Bound states and QCD (at hadronic energies) are not perturbative. Many approaches and models are built to solve these problems. Lattice QCD. Functional methods. Dyson-Schwinger equations (DSE) . Bethe-Salpeter equations (BSE) . Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 3 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Both equations need to be truncated to be numerically solved. Rainbow-Ladder truncation: Bethe-Salpeter equation: Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 4 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Results for the pion and other ground state mesons are well understood. However: The solution relies strongly on the truncation. For more complex systems: Other terms appear in the DSE. Rainbow-Ladder truncation is not good enough. New technical issues appear. Work in a different approach: Use of the Functional Renormalization Group (FRG) to find properties of mesons. The FRG approach is consistent with BSE. Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 5 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Introduction to the FRG Generating Functional in Euclidean space as starting point: � Z [ J ] = e W [ J ] = D ψ e − S [ ψ ]+ � x J ψ Effective Action Γ[ φ ] from W [ J ] Legendre transformation: � d Γ[ φ ] � � � e − Γ[ φ ] = D ψ exp − S [ φ + ψ ] + d φ ψ x δψ ≡ J , φ ≡ δ W [ J ] with δ Γ = � ψ � J . Γ[ φ ] expressed as sum of 1PI diagrams. δ J Introduction of scale k and regulator ∆ S k : Γ k [ φ ] = Γ[ φ ] − ∆ S k [ φ ] Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 6 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Initial and final conditions are fixed in theory space: The choice of the regulator is not unique. Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 7 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary � With the choice of quadratic regulators ∆ S k [ φ ] = p φ R k φ , the properties of the scale-dependent effective action lead to the 1-loop integral-differential equation: ∂ t Γ k = 1 � � − 1 � � Γ (2) 2 Tr ∂ t R k + R k k Wetterich’s Flow Equation with � k � ∂ t = k d t = ln Λ UV dk The properties of this exact flow equation are very convenient for physical calculations since it is an Euclidean 1-loop integral-differential equation. Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 8 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Wetterich’s equation can be solved applying vertex expansion: ∞ 1 � Γ ( n ) � Γ k [ φ ] = k ( p 1 , ... p n ) φ ( p 1 ) ...φ ( p n ) n ! p 1 ... p n n =0 Applying n − derivatives and averaging the fields one obtains the flow of the momentum dependent vertex functions. Remark: a truncation/approximation is needed. Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 9 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Dynamical Hadronization Macroscopic QCD degrees of freedom are mesons and baryons. Introduced in the effective action through 4-Fermi Hubbard-Stratonovich transf.: Problem : 4-Fermi interaction flow non-zero, H-S transfomation must be applied in every RG-step = ⇒ Solved by Dynamical Hadrnization. Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 10 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Introduction of scale dependent bosonic field: ∂ t φ k ( p ) = ∂ t A k ( p )( ¯ ψτψ )( p ) + ∂ t B k ( p ) φ k ( p ) with ∂ t A k and ∂ t B k defined such that 4-Fermi flow is cancelled: This generalizes Hubbard-Stratonovich transf. for every RG-step. Green’s functions computed with meson exchange diagrams. Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 11 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Analytical Continuation Search for bound state properties through real-time Green’s functions: analytical continuation must be performed. Since the inverse propagator of 2-point function is proportional to ( p 2 + M 2 ) in Euclidean Space, goal is to continue to purely imaginary p 0 . Extrapolation by fitting Euclidean momenta p 2 data to a parametrized function and evaluating it at Minkowski momenta − p 2 . Direct calculation using the properties of the regulators. Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 12 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Analytical continuation by extrapolation: Pad´ e approximant: m � c i x i i =0 R ( m n ) ( x ) = n � d j x j 1 + j =1 Schlessinger point method for a set of M data points: F ( x 1 ) C ( x ) = z 1 ( x − x 1 ) 1 + z 2( x − x 2) 1+ . . . zM − 1( x − xM − 1) Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 13 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Analytical continuation by direct calculation: Complex momenta accessed by direct calculation, not extrapolation. Already achieved using 3D-regulators [R-A Tripolt et al, arXiv:1311.0630v2]. Successfully performed with a 4D modified regulator for zero temperature O ( N ) model [J. M. Pawlowski and N. Strodthoff, arXiv:1508.01160v3]. � p 2 + ∆ m 2 � � ∆Γ (2) � r ( p 2 ) = k ( p 2 ) | φ = φ 0 + ∆ m 2 r R k ;∆ m 2 r r k 2 Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 14 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Modified regulator moves poles out the integrating region = ⇒ physical implications induced. [J. M. Pawlowski and N. Strodthoff, arXiv:1508.01160v3] Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 15 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Preliminary results in the Quark-Meson model Why using QM as starting point? A.Cyrol et al, arXiv:1605.01856 Low-energy effective theory with gluons decoupled. Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 16 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Free massless quark effective action + 4-Fermi interaction: d 4 p � (2 π ) 4 Z k ,ψ ¯ Γ k [ ¯ Γ 4 − int [ ¯ p ψ A ψ A ψ, ψ ] = a i / + ψ, ψ ] a k Applying Hubbard-Stratonovich transformation introducing φ = ( σ,� π ) fields: � � ¯ � Z ψ, k ¯ ψ A p ψ A � Γ k ψ, ψ, σ,� π = a i / a + p + 1 � Z k ,φ p 2 + m 2 � � σ 2 + π z π z � − c σ + k ,φ 2 � � � σ 2 δ ab + i γ 5 ( τ z ) ab π z � h k ¯ ψ A ψ A + a b q Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 17 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Results using the LPA’+DH Wave-function renormalizations Z k , i � = 1 and non-kinetic bosonic terms rewritten in the same O(N) potential: V ( n ) ∞ � n ! ( ρ − ρ 0 ) n k V k ( ρ ) = n =0 σ 2 + � with ρ = 1 � π 2 � and ρ 0 scale independent expansion point. 2 For non-zero renormalization wavefunctions the definition of the renormalized couplings/fields is needed: 1 ˆ ρ = Z k ,φ ρ ψ = Z k ,ψ ψ 2 h k h k = 1 Z k ,ψ Z 2 k ,φ Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 18 / 32
Motivation Introduction to the FRG FRG techniques Results DSE-FRG comparison Summary Consequences of rewriting effective action: Renormalized potential V ( n ) ∞ � n ! ( ρ − ρ 0 ) n k V k ( ρ ) = n =0 = V ( n ) With V ( n ) k ,ρ and ρ 0 = Z k ,φ ρ 0 = ⇒ running expansion point. k Z n k Terms proportional to (momentum dependent) anomalous dimensions η i appear explicitly in the flow equation: η k , i = − ∂ t Z k , i Z k , i Jordi Par´ ıs L´ opez Properties of bound states from FRG and DSE-BSE 19 / 32
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