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User: As a text script expert, please help me to write a short text script with the topic "2 Large Logarithms from Inflationary QFT The point of this section is to describe the various large logarithms which appear in loop corrections on de Sitter background, both to matter from inflationary gravitons and to gravity from inflationary matter.ν ≡ ρν µν ≡ Because the spatial plane wave mode functions for dynamical gravitons are identical to those of the massless, minimally coupled scalar [25], the 3 propagator for which breaks de Sitter invariance [29], one would think that the graviton propagator must also break de Sitter invariance.Three key insights seem to resolve the controversy in favor of de Sitter breaking: • The linearization instability of de Sitter precludes the addition of any de Sitter invariant gauge fixing term to the action [39], although one canstill impose de Sitter invariant gaugesas strong operator equations.LGF −2 − µ ν µ µρ,σ − 2 ρσ,µ − µρ σ h i (12) In this gauge the graviton and ghost propagators are, i[ ∆ ](x;x) = [ TI ] i∆ (x;x) , (13) µν ρσ ′ µν ρσ × I ′µν ≡ µν µ ν One of the things that makes the propagator in this gauge so simple is that the tensor factors are constant, [ TA] = 2η η 2 η η , [ TB] = 4δ0 η δ0 , (15) µν ρσ µ(ρ σ)ν − D 3 µν ρσ µν ρσ − (µ ν)(ρ σ) − [ TC] = 2 [(D 3)δ0 δ0 +η ][(D 3)δ0 δ0 +η ] .(16) µν ρσ (D 2)(D 3) − µ ν µν − ρ σ ρσ − − 4 The A propagator is expression (5); the B and C propagators are, i∆ B = Γ( 4D π2 D− 21) [aa′∆x1 2]D 2−1 + 1Γ 6( πD 2 D 2) [aa′∆H x22 ]D 2−2 − Dk −2 +Γ(D 2+1) H4aa′∆x2 k H2aa∆x2 +... , (17) 128πD 2 [aa′∆x2]D 2−2 − 2D ′ i∆ C = Γ( 4D π2 D− 21) [aa′∆x1 2]D 2−1 + (D 2− 13 6) πΓ( D 2D 2−1) [aa′∆H x22 ]D 2−2 + (D −2)k (D −3) +(D 2 1− 284 π)Γ(73) L −2 µ ν − − − Its leading logarithms can be captured by replacing the Heisenberg equation for the (dimensionally regulated) full field φ(x), with the Langevin equation for the UV finite (hence D = 4) stochastic random variable ϕ(x) [83], ¨ ˙ 2 φ+(D 1)Hφ φ = V (φ) 3H[ϕ˙ ϕ˙ ] = V (ϕ) .One first acts the inverse kinetic operator of the full field equation to the reach the Yang-Feldman equation, with free field φ (x), 0 φ(x) = φ (x) d4x g(x)iθ(t t)[φ (x),φ (x)]V φ(x) .(89) µν − 2 µν µν µ ν − 2 µν ρ σ n o 17 Because the propagator of the graviton field h has the same logarithmic µν tail as the nonlinear sigma model scalars of the previous subsection, the differentiated scalar φ in equation (89) needs to be evaluated in the presence of a constant h background.µν µν µν A key insight is that a metric g = a2g , with constant g , corresponds µν µν µν to de Seitter with a different cosmological constant [28,98].µν µ ν ν µ− µν ⇒ σµν − µ σν − ν σµ (cid:16) (cid:17) (90) e e e This means that relation (83) for the coincident scalar propagator becomes, hΩ |∂ µφ(x)∂ νφ(x) |Ω ieg −→ −3(H 322 πeg0 20)2 g µν .For de Sitter the result would be just −4π − 3 GH4, and explicit computations show that making the 1-loop graviton − sel4 fπ -energy conserved requiresethat precisely this term must be canceled by a combination of the Eddington (R2) counterterm (which makes only a fi- nite contribution to the 1-point function for de Sitter) plus an extra, finite renormalization of the cosmological constant [81].(93) This agrees with (92) because the general Ricci scalar, R = 12g00H2 +6Ha 1gαβΓ0 +a 2R , (94) − − αβ − coincides with 12g00H2 feor constant g .e e e µν − e e 18 3.4 Pure Gravity It is desirable to express the Lagrangian of gravity (10) in terms of the conformally rescaled fields [45], = aD 2 ggαβgρσgµν 1h h 1h h LEinstein − − 2 αρ,µ νσ,β − 2 αβ,ρ σµ,ν p n +1h h ee1he eh +(D 2)aD 1H ggρσgµνh h .where replaced by H. The various tensor factors are, e [ TA] = 2γ γe 2 γ γ , [ TB] = 4u γ u ,(106) µν ρσ µ(ρ σ)ν − D 3 µν ρσ µν ρσ − (µ ν)(ρ σ) − [ µνTe ρC σ] =(107) − − Theerequired coincidence limits are, ∂ i[ ∆ ](x;x) = aHku [ TA] , (108) α µν ρσ ′ (cid:12)x′=x − α µν ρσ (cid:12) e ee e ∂ ∂ i[ ∆ ](x;x)(cid:12) = α β′ µν ρσ ′ (cid:12)x′=x e (cid:12) (cid:12) e kHe2 g (D 1)[ TA]+[ TB] 2 [ TC] ,(109) D αβ − − µν ρσ µν ρσ − D 2 µν ρσ n − o e e e ∂ i[ ∆ ](x;x) = aHku γ , (110) α µ ρ ′ α µρ (cid:12)x′=x − ∂ ∂ i[ ∆e ](x;x)(cid:12) (cid:12) = e kHee2e g (D 1)γ +u u .(121) µν ≡ µν (µ ν) µ ν µν αβ α The first two terms of (116) represent the action of the gauge-fixed kinetic operator on the graviton field, ∂ a2 ggαβ[1gµρgνσ 1gµνgρσ]κh +2H2a4 gu(µgν)(ρuσ)κh α − 2 −4 ρσ,β − ρσ n p o p e −e → [ 21 egµρ egνσ −e 41gµ eνgρσ]D AA ρσ +eu(µgν)ρDeBB ρe+uµuνD CC.ij The double logarithmic corrections (70) that were found in the old gauge [15] could derive from a contribution to the stochastic Langevin equation which does seem to be present in (128), Ha 1uα∂ (A a) κ2H4A3 +... (129) − α − ∼ However, checking the numerical coefficient would require a dimensionally e e regulated and fully renormalized computation of the graviton loop contri- bution to i[µνΣρσ](x;x) in the new gauge (98).− µ − ρσ − ρν − µσ νρ − ρσ µν p hp i p (165) e e e e e e e ee e e e e These should probably be integrated out the same way that differentiated gravitons were integrated out of the pure gravitational field equations in section 3.4.When this is done, and the undifferentiated gravitons are set to zero, the result is, δS = ∂ [ ggµν∂ φ] 1(D 2)φR g ∂2φ κ2H4a2 φ+... (166) δφe µ − ν − 4 D− 1 − −→ − 8π2 p − p e ee e e However, one we oe uld need an extra factore of +κ2H4a2 φ in order to explain 12π2 the graviton loop corrections (42) and (43).4.3.3 Einstein-Dirac Renormalizing single graviton loop corrections to the fermion self-energy re- quires three counterterms (53), ∆ α ψ(iγµ∂ a 1)3aψ +α D(D 1)H2ψiγµ∂ ψ Dirac 1 µ − 2 µ(178) −→ 4πar × 4.4 Graviton Corrections to Gravity The flat space limit of the simplest gauge [45,46] implies that renormalizing the 1-loop graviton contribution to the graviton self-energy must have at least the same two invariant counterterms (28) as matter loop contributions, ∆ = c R2√ g +c CαβγδC √ g .However, loop corrections which involve graviton propagators do depend on the scheme by which the graviton gauge is fixed and this has led to a debate about the reality of the large logarithms these corrections induce [104–114].32 5.1 How It Works on Flat Space Consider graviton corrections to a massless, minimally coupled scalar on flat space background (g η +κh ) in the 2-parameter family of covariant µν µν µν ≡ gauge fixing functions, = 1 ηµνF F , F = ηρσ h βh .(182) LGF −2α µ ν µ µρ,σ − 2 ρσ,µ (cid:16) (cid:17) In the Schwinger-Keldysh formalism the renormalized self-mass is [116], M2(x;x) = (α,β) κ2∂8 θ(∆t ∆r) ln[µ2(∆t2 ∆r2)] 1 ,(183) 0 ′ −C0 × 128π3 − − − n (cid:16) (cid:17)o C0(α,β) = +3 4 − 3 4 ×α − 23 × β1 2 + 43 × (( βα − 23 )) 2 , (184) − − where ∆t t t and ∆r ~x ~x .Matter loop corrections to gravity are completely independent of the graviton gauge, and the fact that they induce the same sorts of large loga- rithms as are found from graviton loop corrections in a fixed gauge, is strong evidence that graviton-induced logarithms are real.For example, de Sitter (ǫ = 0) gravitons change the Coulomb potential [12] and the electric field strength of plane wave photons [13] to, Φ(t,r) = Q 1+ 2~G + 2~GH2 ln[a(t)Hr]+... , (2) 4πε0a(t)r n 3πc3a2(t)r2 πc5 c o F (t,~x) = Ftree(t,~x) 1+ 2~GH2 ln[a(t)]+... .µν ≡ µν µ ν One of the things that makes the propagator in this gauge so simple is that the tensor factors are constant, [ TA] = 2η η 2 η η , [ TB] = 4δ0 η δ0 , (15) µν ρσ µ(ρ σ)ν − D 3 µν ρσ µν ρσ − (µ ν)(ρ σ) − [ TC] = 2 [(D 3)δ0 δ0 +η ][(D 3)δ0 δ0 +η ] .(25) 4π −aHr a h i The tree order solution also approaches a constant at late times [61], φtree = HK ln(Hr) 1 +... HK ln(Hr) .The graviton self-energy can be used to quantum-correct the linearized Einstein equation, µνρσκh (x) d4x [µνΣρσ](x;x)κh (x) = κ2 Tµν(x) , (29) L ρσ −Z ′ ′ ρσ ′ 2 6 where the Lichnerowicz operator on de Sitter is, µνρσh = 1a2 ∂2hµν ηµν∂2h+ηµν∂ρ∂σh +∂µ∂νh 2∂ρ∂(µhν) L ρσ 2 − ρσ − ρ h i +a3H ηµν∂ h ∂ hµν 2ηµν∂ρh +2∂(µhν) +3a4H2ηµνh .(30) 0 − 0 − ρ0 0 00 h i With the stress tensor Tµν set to zero one can study plane wave gravitons, κh = ǫ ei~k~xu(t,k) , (31) µν µν · where transverse-traceless and purely spatial polarization tensor ǫ is the µν same as on flat space background and the tree order mode function is the same as (23).(37) −→ × −→ × This means that taking the conformal factor Ω equal to the inverse scale factor 1/a(t), and working in conformal coordinates, erases any dependence on the scale factor when the conformally transformed fields φ and g are µν used, e = 1∂ φ∂ φgµν g 1(D 2)φ2R g .(73) L −2 µ ν − − − Its leading logarithms can be captured by replacing the Heisenberg equation for the (dimensionally regulated) full field φ(x), with the Langevin equation for the UV finite (hence D = 4) stochastic random variable ϕ(x) [83], ¨ ˙ 2 φ+(D 1)Hφ φ = V (φ) 3H[ϕ˙ ϕ˙ ] = V (ϕ) .First, its expectation value agrees (at leading logarithm order) with exact, dimensionally regulated and fully BPHZ renormalized computations at 1- loop [28] and 2-loop [95] orders, Ω (t,~x) Ω = λH2ln(a) + λ3H5ln2(a) +O(λ5) .(89) µν − 2 µν µν µ ν − 2 µν ρ σ n o 17 Because the propagator of the graviton field h has the same logarithmic µν tail as the nonlinear sigma model scalars of the previous subsection, the differentiated scalar φ in equation (89) needs to be evaluated in the presence of a constant h background.For de Sitter the result would be just −4π − 3 GH4, and explicit computations show that making the 1-loop graviton − sel4 fπ -energy conserved requiresethat precisely this term must be canceled by a combination of the Eddington (R2) counterterm (which makes only a fi- nite contribution to the 1-point function for de Sitter) plus an extra, finite renormalization of the cosmological constant [81].(93) This agrees with (92) because the general Ricci scalar, R = 12g00H2 +6Ha 1gαβΓ0 +a 2R , (94) − − αβ − coincides with 12g00H2 feor constant g .e e e µν − e e 18 3.4 Pure Gravity It is desirable to express the Lagrangian of gravity (10) in terms of the conformally rescaled fields [45], = aD 2 ggαβgρσgµν 1h h 1h h LEinstein − − 2 αρ,µ νσ,β − 2 αβ,ρ σµ,ν p n +1h h ee1he eh +(D 2)aD 1H ggρσgµνh h .The old gauge [45,46], = 1aD 2ηµνF F , F = ηρσ[h 1h +(D 2)aHh δ0 ] , (97) LGF −2 − µ ν µ µρ,σ −2 ρσ,µ − µρ σ involves factors of the Minkowski metric ηµν and ηρσ which must be general- ized to gµν and gρσ.where replaced by H. The various tensor factors are, e [ TA] = 2γ γe 2 γ γ , [ TB] = 4u γ u ,(106) µν ρσ µ(ρ σ)ν − D 3 µν ρσ µν ρσ − (µ ν)(ρ σ) − [ µνTe ρC σ] =(121) µν ≡ µν (µ ν) µ ν µν αβ α The first two terms of (116) represent the action of the gauge-fixed kinetic operator on the graviton field, ∂ a2 ggαβ[1gµρgνσ 1gµνgρσ]κh +2H2a4 gu(µgν)(ρuσ)κh α − 2 −4 ρσ,β − ρσ n p o p e −e → [ 21 egµρ egνσ −e 41gµ eνgρσ]D AA ρσ +eu(µgν)ρDeBB ρe+uµuνD CC.Putteing every ething together in D = 4 gives the final form, ∂ a2 ggαβ[1gµρgνσ 1gµνgρσ]κh +2H2a4 gu(µgν)(ρuσ)κh α − 2 −4 ρσ,β − ρσ n p o p ee −→e −e[gµρgeνσ −e1 2gµνgρσ]Ha3 e −guα∂ αe(A ρσe −a ρσ) p e e e e e 2H2a4 e g[u(µBν)+uµuνC].Adding a point source J(t,~x) = Ka(t)δ3(~x) to the linearized effective field equation for the field B and taking the late time limit results in a solution which involves powers of ln(Hr) [28], B(t,r) KHln(Hr) 1 λ2H2ln(Hr) +O(λ4) .−48π2 × D 4 48π2 × D 4 8π2 × D 4 − − −(162) 28 The gamma function needed for the Callan-Symanzik equation (159) — with ln(µ) replaced by ln(a) — to explain the scalar mode function (42) and exchange potential (43) is, Needed γ = κ2H2 = δZ = κ2H2 µD−4 .4.3.3 Einstein-Dirac Renormalizing single graviton loop corrections to the fermion self-energy re- quires three counterterms (53), ∆ α ψ(iγµ∂ a 1)3aψ +α D(D 1)H2ψiγµ∂ ψ Dirac 1 µ − 2 µ5 The Gauge Issue The numerous matter loop corrections to gravity reported in expressions (32-35), (44-46), (56-58) and (65-67) involve no graviton propagators and are completely gauge independent.This section is devoted to explaining how gauge dependence can be re- moved from the effective field equations by accounting for the quantum grav- itational correlations of the source which excites the effective field and the observer who detects it [116].1 The procedure is to build the same diagrams that would go into an S-matrix element, and then simplify them with a series of relations derived by Donoghue [8,9,129] to capture the infrared physics.32 5.1 How It Works on Flat Space Consider graviton corrections to a massless, minimally coupled scalar on flat space background (g η +κh ) in the 2-parameter family of covariant µν µν µν ≡ gauge fixing functions, = 1 ηµνF F , F = ηρσ h βh .(182) LGF −2α µ ν µ µρ,σ − 2 ρσ,µ (cid:16) (cid:17) In the Schwinger-Keldysh formalism the renormalized self-mass is [116], M2(x;x) = (α,β) κ2∂8 θ(∆t ∆r) ln[µ2(∆t2 ∆r2)] 1 ,(183) 0 ′ −C0 × 128π3 − − − n (cid:16) (cid:17)o C0(α,β) = +3 4 − 3 4 ×α − 23 × β1 2 + 43 × (( βα − 23 )) 2 , (184) − − where ∆t t t and ∆r ~x ~x .i 1 α β1 2 (( βα − 23 )) 2 Description − − 0 +3 3 3 +3 scalar exchange 4 −4 −2 4 1 0 0 0 +1 vertex-vertex 2 0 0 0 0 vertex-source,observer 3 0 0 +3 2 vertex-scalar − 4 +17 3 0 1 source-observer 4 −4 −4 5 2 +3 3 +1 scalar-source,observer − 2 −2 2 Total +3 0 0 0 Table 1: The gauge dependent factors Ci(α,β) for each contribution to the invariant scalar self-mass-squared.5.3 What Remains to Do There can be little doubt that this technique is correct on flat space back- ground owing to the close connection between it andthe S-matrix, and to the explicit cancellation of dependence on the parameters α and β for a mass- less, minimally coupled scalar [116] and electromagnetism [131].When this technique is implemented on flat space background one can explicitly demonstrate the cancellation of gauge dependence for graviton corrections to a massless, minimally coupled scalar [116] and to electromagnetism [131].".Your output should use the following template:\n\n## {Subtitle01}\n- {Bulletpoint01}\n- {Bulletpoint02}\n## {Subtitle02}\n- {Bulletpoint03}\n- {Bulletpoint04}\n\nSummarize the giving topic to generate a mind map (as many subtitles as possible, with a minimum of three subtitles) structure markdown. 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