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Introduction to the Exact Renormalization Group

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Introduction to the Exact Renormalization Group 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 r^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" Γk[φ] which is . • . a generalization of the effective action which includes only fluctuations with q2 & k2 1 • . a \coarse-grained" effective action, averaged over volumes ∼ kd (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 r^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 χa; 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 Z Zx • define a scale-dependent generating functional by inserting a cutoff term Zk[J] = Dχ exp −S[χ] + Jχ − ∆Sk[χ] Z Zx • define scale-dependent generating functional Wk[J] for the connected Greens functions by Zk[J] = exp [Wk[J]] • cutoff term for a scalar theory: 1 ∗ ∆S [χ] = χ (q)R (q)χ(q) k 2 k Zq [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 Rk(q): 1. Rk(q) ! 0 for k ! 0 at fixed q (so that Wk!0[J] = W [J] and thus Γk!0[φ] = Γ[φ]) 2. Rk(q) ! 1 (divergent) for k ! Λ (k ! 1 for Λ ! 1) (so that Γk!Λ[φ] = ΓΛ[φ] = S[φ]) 2 2 2 3. Rk(q) > 0 for q ! 0 (e.g. Rk(q) ! k for q ! 0) (must be an IR regulator, after all!) • examples for popular cutoff functions: 1. without finite UV cutoff Λ ! 1 2 1 Rk(q) = q q2 exp k2 − 1 2. with a finite UV cutoff Λ 2 1 Rk(q) = q q2 q2 exp k2 − exp Λ2 7 Derivation of the scale-dependent effective action [2] • exchange dependence on the source J for dependence on expectation value δW [J] φ(x) = k ! φ(x) = φa[J(x)] δJ(x) k • employ a modified Legendre transformation and define the scale dependent effective action as Γk[φ] = −Wk[J] + J(x)φ(x) − ∆Sk[φ] x Z (∗) (∗): cutoff term depends on expectation value φ: crucial for connection| {z } 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) δΓ [φ] δW [J] δJ(y) δJ(y) δ∆S [φ] k = − k + φ(y) +J(x) − k δφ(x) δJ(y) δφ(x) δφ(x) δφ(x) Zy Zy =0 δ = J| (x) − ∆S{z[φ] = J(x) − (R}φ)(x) δφ(x) k k 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 [J] @ Γ [φ] = −@ W [J] − k @ J + φ(x)(@ J) −@ ∆S [φ] = −@ W [J] − @ ∆S [φ] t k t k δJ(x) t t t k t k t k Zx Zx =φ(x) | {z } =0 • derivative of the cutoff term| (remember{zthat φ is the indep} endent variable in Γk[φ]) 1 ∗ 1 ∗ @ ∆S [φ] = @ φ (q)R (q)φ(q) = φ (q)(@ R (q))φ(q) t k t 2 k 2 t k Zq Zq 9 Derivation of Flow equation [2] • we need the scale derivative of Wk[J] • first express the derivative in terms of exp(Wk[J]) @tWk[J] = exp(−Wk[J]) exp(Wk[J]) @tWk[J] = exp(−Wk[J]) (@tWk[J]) exp(Wk[J]) =1 = exp| (−Wk[J{z]) (@t exp(W}k[J])) • now go back to the path integral representation: scale dependence appears only in cutoff term @tWk[J] = exp(−Wk[J]) @t Dχ exp −S[χ] + Jχ − ∆Sk[χ] Z Zx = exp(−Wk[J]) Dχ(−@t∆Sk[χ]) exp −S[χ] + Jχ − ∆Sk[χ] Z Zx 1 ∗ = exp(−W [J]) Dχ − χ (q)(@ R (q))χ(q) exp −S[χ] + Jχ − ∆S [χ] k 2 t k k Z Zq Zx 1 ∗ = − (@ R (q)) exp(−W [J]) Dχ χ (q)χ(q) exp −S[χ] + Jχ − ∆S [χ] 2 t k k k Zq Z Zx | {z } 10 Derivation of Flow equation [3] Express this in terms of the connected Greens functions: δ2 δ2W [J] δW [J] δW [J] exp(−W [J]) exp(W [J]) = k + k k k δJ(q)δJ ∗(q) k δJ(q)δJ ∗(q) δJ ∗(q) δJ(q) ∗ ∗ = hχ (q)χ(q)ik;connected + φ (q)φ(q) ∗ = Gk(q; q) + φ (q)φ(q) ) we find for the flow of Wk[J] 1 ∗ @ W [J] = − (@ R (q)) (G (q; q) + φ (q)φ(q)) t k 2 t k k Zq 1 1 ∗ = − (@ R (q))G (q; q) − φ (q)(@ R (q))φ(q) 2 t k k 2 t k Zq Zq 1 = − (@ R (q))G (q; q) − @ ∆S [φ] 2 t k k t k Zq • insert this into the flow equation for Γk . 11 Derivation of Flow equation [4] . result for @tWk[J] into flow equation: @tΓk[φ] = −@tWk[J] − @tSk[φ] 1 = (@ R (q))G (q; q) + @ ∆S [φ] − @ ∆S [φ] 2 t k k t k t k Zq The result for the flow equation for the effective action is 1 @ Γ [φ] = (@ R (q)) G (q; q) t k 2 t k k Zq • 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 δ2W [J] δW [J] G(p; q) = k ; φ(q) = k δJ ∗(p)δJ(q) δJ ∗(q) use variation condition on effective action (from modified Legendre transformation) δΓ [φ] ∗ ∗ k = J (q) − φ (q)R (q) δφ(q) k second variation with respect to φ∗(q0) 2 ∗ ∗ 2 δ Γ [φ] δJ (q) 0 δJ (q) δ Γ [φ] 0 k = − R (q) δ(q − q ) ) = k + R (q) δ(q − q ): δφ∗(q0)δφ(q) δφ(q0) k δφ(q0) δφ∗(q0)δφ(q) k Now start from an identity to show that this is the inverse of G(q0; q) 13 Inversion of scale-dependent propagator [2] • start from the identity δ ∗ 0 φ (q) = δ(q − q) δφ∗(q0) 2 ∗ 00 δ δWk[J] δ Wk[J] δJ (q ) = ∗ 0 = ∗ 00 ∗ 0 δφ (q ) δJ(q) 00 δJ (q ) δJ(q) δφ (q ) Zq • use expression for δJ ∗/δφ∗ established above 2 2 0 δ Wk[J] δ Γk[φ] 0 00 =) δ(q − q ) = ∗ 00 ∗ 0 00 + Rk(q) δ(q − q ) 00 δJ (q ) δJ(q) δφ (q ) δφ(q ) Zq ) the scale dependent inverse propagator is given by − 2 1 0 δ Γ [φ] 0 G(q; q ) = k + R (q) δ(q − q ) δφ∗(q) δφ(q0) k (result is as expected, but it is necessary to establish the particular form of the scale dependence of the propagator) 14 Result for the Flow equation − 1 δ2Γ [φ] 1 @ Γ [φ] = (@ R (q)) k + R (q) t k 2 t k δφ∗(q) δφ(q) k Zq Graphical representation (insertion stands for the derivative of the cutoff function @tRk) Γ 1 k 2 The line represents the full propagator (which includes the complete field dependence).
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