Introduction to the Exact Renormalization Group Informal Seminar Bertram Klein, GSI literature: • J. Berges, N. Tetradis, and C. Wetterich [hep-ph/0005122]. • lectures H. Gies, UB Heidelberg. • D. F. Litim, J. M. Pawlowski [hep-th/0202188]. [Wetterich (1993), Wegner/Houghton (1973), Polchinski (1984)]. 1
Outline • motivation • exact RG • scale-dependent effective action • one-loop flow equations for effective action • hierarchy of flow equations for n -point functions • truncations • connection to perturbative loop expansion • O ( N )-model in a derivative expansion: flow equations 2
Motivation • need to cover physics across different scales • microscopic theory → macroscopic (effective) theory • “bridge the gap” between microscopic theory and effective macroscopic description (in terms of effective/thermodynamic potentials, . . . ) • loose the irrelevant details of the microscopic theory ⇒ How do we decide what is relevant and what is not? • important rˆ ole of fluctuations: long-range in the vicinity of a critical point ⇒ How do we treat long-range flucutations? • universality : certain behavior in the vicinity of a critical point independent from the details of the theory (e.g. critical exponents) • often additional complications: need to go from one set of degrees of freedom (at the microscopic level) to a different set (at the macroscopic level) here: we want to use an average effective action in the macroscopic description 3
Exact RG Flows � What do we mean by “exact” renormalization group flows? • derived from first principles • connects (any given) initial action (classical action) with full quantum effective action ⇒ exact flow reproduces standard perturbation theory • flow in “theory space”: trajectory is scheme-dependent, but end point is not • truncations project “true” flow onto truncated action S [ φ ] [fig. nach H. Gies] Γ[ φ ] 4
Goal: A scale-dependent effective action Our goal is an “averaged effective action” Γ k [ φ ] which is . . . • . . . a generalization of the effective action which includes only fluctuations with q 2 � k 2 1 • . . . a “coarse-grained” effective action, averaged over volumes ∼ k d (i.e. quantum flucutations on smaller scales are integrated out!) • . . . for large k ( → small length scales) very similar to the microscopic action S [ φ ] (since long-range correlations do not yet play a rˆ ole) • . . . for small k ( → large length scales) includes long-range effects (long-range correlations, critical behavior, . . . ) • . . . and which can be derived from the generating functional. How does this look in practice? ⇒ we look at derivation of such an effective action starting from the generating functional for n -point correlation functions 5
Derivation of the scale-dependent effective action [1] • scalar theory, fields χ a , a = 1 , . . . , N , d Euclidean dimensions • start from the generating functional of the n -point correlation functions (path integral rep.) � � � � Z [ J ] = Dχ exp − S [ χ ] + Jχ x • define a scale-dependent generating functional by inserting a cutoff term � � � � Z k [ J ] = Dχ exp − S [ χ ] + Jχ − ∆ S k [ χ ] x • define scale-dependent generating functional W k [ J ] for the connected Greens functions by Z k [ J ] = exp [ W k [ J ]] • cutoff term for a scalar theory: � 1 ∆ S k [ χ ] = χ ∗ ( q ) R k ( q ) χ ( q ) 2 q [cutoff term quadratic in the fields ensures that a one-loop equation can be exact (Litim)] 6
Intermezzo: Properties of the cutoff function • required properties for R k ( q ): 1. R k ( q ) → 0 for k → 0 at fixed q (so that W k → 0 [ J ] = W [ J ] and thus Γ k → 0 [ φ ] = Γ[ φ ]) 2. R k ( q ) → ∞ ( divergent ) for k → Λ ( k → ∞ for Λ → ∞ ) (so that Γ k → Λ [ φ ] = Γ Λ [ φ ] = S [ φ ]) 3. R k ( q ) > 0 for q 2 → 0 (e.g. R k ( q ) → k 2 for q 2 → 0) (must be an IR regulator, after all!) • examples for popular cutoff functions: 1. without finite UV cutoff Λ → ∞ 1 q 2 R k ( q ) = � � q 2 exp − 1 k 2 2. with a finite UV cutoff Λ 1 q 2 R k ( q ) = � � � � q 2 q 2 exp − exp k 2 Λ 2 7
Derivation of the scale-dependent effective action [2] • exchange dependence on the source J for dependence on expectation value δW k [ J ] δJ ( x ) → φ ( x ) = φ a φ ( x ) = k [ J ( x )] • employ a modified Legendre transformation and define the scale dependent effective action as � Γ k [ φ ] = − W k [ J ] + J ( x ) φ ( x ) − ∆ S k [ φ ] � �� � x ( ∗ ) ( ∗ ): cutoff term depends on expectation value φ : crucial for connection to the “bare” (classical) action S [ φ ] at the UV scale, and to quench only fluctuations around the expectation value! • Variation condition on the action/equation of motion for φ ( x ) � � δ Γ k [ φ ] δW k [ J ] δJ ( y ) δJ ( y ) + J ( x ) − δ ∆ S k [ φ ] = − δφ ( x ) + δφ ( x ) φ ( y ) δφ ( x ) δJ ( y ) δφ ( x ) y y � �� � =0 δ = J ( x ) − δφ ( x )∆ S k [ φ ] = J ( x ) − ( R k φ )( x ) 8
Derivation of Flow equation [1] Flow equation : It describes the change of the scale-dependent effective action at scale k with a change of the RG scale, and thus how the effective actions on different scales are connected. to derive the flow equation we need • modified Legendre transform • scale-dependent generating functional of the connected Greens functions • take the derivative with regard to the scale of the modified Legendre transformation (introduce t = log ( k/ Λ) ⇒ ∂ t = k∂ k ): � � δW k [ J ] ∂ t Γ k [ φ ] = − ∂ t W k [ J ] − ∂ t J + φ ( x )( ∂ t J ) − ∂ t ∆ S k [ φ ] = − ∂ t W k [ J ] − ∂ t ∆ S k [ φ ] δJ ( x ) x x � �� � = φ ( x ) � �� � =0 • derivative of the cutoff term (remember that φ is the independent variable in Γ k [ φ ]) � � 1 φ ∗ ( q ) R k ( q ) φ ( q ) = 1 φ ∗ ( q )( ∂ t R k ( q )) φ ( q ) ∂ t ∆ S k [ φ ] = ∂ t 2 2 q q 9
Derivation of Flow equation [2] • we need the scale derivative of W k [ J ] • first express the derivative in terms of exp( W k [ J ]) ∂ t W k [ J ] = exp( − W k [ J ]) exp( W k [ J ]) ∂ t W k [ J ] = exp( − W k [ J ]) ( ∂ t W k [ J ]) exp( W k [ J ]) � �� � =1 = exp( − W k [ J ]) ( ∂ t exp( W k [ J ])) • now go back to the path integral representation: scale dependence appears only in cutoff term � � � � ∂ t W k [ J ] = exp( − W k [ J ]) ∂ t Dχ exp − S [ χ ] + Jχ − ∆ S k [ χ ] x � � � � = exp( − W k [ J ]) Dχ ( − ∂ t ∆ S k [ χ ]) exp − S [ χ ] + Jχ − ∆ S k [ χ ] x � � � � � � � − 1 χ ∗ ( q )( ∂ t R k ( q )) χ ( q ) = exp( − W k [ J ]) Dχ exp − S [ χ ] + Jχ − ∆ S k [ χ ] 2 q x � � � � � − 1 Dχ χ ∗ ( q ) χ ( q ) = ( ∂ t R k ( q )) exp( − W k [ J ]) � exp − S [ χ ] + Jχ − ∆ S k [ χ ] 2 � �� q x 10
Derivation of Flow equation [3] Express this in terms of the connected Greens functions: δ 2 δ 2 W k [ J ] δJ ( q ) δJ ∗ ( q ) + δW k [ J ] δW k [ J ] exp( − W k [ J ]) δJ ( q ) δJ ∗ ( q ) exp( W k [ J ]) = δJ ∗ ( q ) δJ ( q ) � χ ∗ ( q ) χ ( q ) � k, connected + φ ∗ ( q ) φ ( q ) = = G k ( q, q ) + φ ∗ ( q ) φ ( q ) ⇒ we find for the flow of W k [ J ] � − 1 ( ∂ t R k ( q )) ( G k ( q, q ) + φ ∗ ( q ) φ ( q )) ∂ t W k [ J ] = 2 q � � − 1 ( ∂ t R k ( q )) G k ( q, q ) − 1 φ ∗ ( q )( ∂ t R k ( q )) φ ( q ) = 2 2 q q � − 1 = ( ∂ t R k ( q )) G k ( q, q ) − ∂ t ∆ S k [ φ ] 2 q • insert this into the flow equation for Γ k . . . 11
Derivation of Flow equation [4] . . . result for ∂ t W k [ J ] into flow equation: ∂ t Γ k [ φ ] = − ∂ t W k [ J ] − ∂ t S k [ φ ] � 1 = ( ∂ t R k ( q )) G k ( q, q ) + ∂ t ∆ S k [ φ ] − ∂ t ∆ S k [ φ ] 2 q The result for the flow equation for the effective action is � 1 ∂ t Γ k [ φ ] = ( ∂ t R k ( q )) G k ( q, q ) 2 q • This should now be expressed as a functional differential equation for the effective action. 12
Inversion of scale-dependent propagator [1] What remains to do in order to obtain a (functional) differential equation for the scale-dependent effective action is to express G ( p, q ) in terms of this effective action δ 2 W k [ J ] φ ( q ) = δW k [ J ] G ( p, q ) = δJ ∗ ( p ) δJ ( q ) , δJ ∗ ( q ) use variation condition on effective action (from modified Legendre transformation) δ Γ k [ φ ] J ∗ ( q ) − φ ∗ ( q ) R k ( q ) = δφ ( q ) second variation with respect to φ ∗ ( q ′ ) δ 2 Γ k [ φ ] δ 2 Γ k [ φ ] δJ ∗ ( q ) δφ ( q ′ ) − R k ( q ) δ ( q − q ′ ) ⇒ δJ ∗ ( q ) δφ ∗ ( q ′ ) δφ ( q ) + R k ( q ) δ ( q − q ′ ) . = δφ ( q ′ ) = δφ ∗ ( q ′ ) δφ ( q ) Now start from an identity to show that this is the inverse of G ( q ′ , q ) 13
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