Self-consistent Conserving Approximations and Renormalization in Quantum Field Theory at Finite Temperature Hendrik van Hees in collaboration with Jörn Knoll Contents Schwinger-Keldysh real-time formalism The Φ-derivable scheme Tadpole resummation for φ 4 -theory The self-consistent sunset diagram Thermodynamical quantities Outlook
Schwinger-Keldysh Formalism #2 Initial statistical operator ρ i at t = t i Time evolution of expectation values of observables: O = Tr[ρ(t)O(t)] Feynman rules Difference to vacuum: Contour-ordered Green s functions t i t f t = + + + In equilibrium: ρ = exp( βh)/z with Z = Tr exp( βh) Imaginary part of the time contour Im t t i t f Re t = + + + + iβ Fields periodic (bosons) or anti-periodic (fermions)
The Φ-Functional #3 Introduce local and bilocal auxiliary sources Generating functional Z[J, K] = N Dφ exp [ { } ] i is[φ] + i {J φ } + 2 K 2φ φ 2 2 Generating functional for connected diagrams Z[J, K] = exp(iw [J, K]) The mean field and the connected Green s function ϕ = δw δj, G 2 = δ2 W δj δj 2 δw δk 2 = 2 [ϕ ϕ 2 + ig 2 ] Legendre transformation for ϕ and G: Γ[ϕ, G] = W [J, K] {ϕ J } 2 {(ϕ ϕ 2 + ig 2 )K 2 } 2 Exact saddle point expansion: Γ[ϕ, G] =S[ϕ] + i 2 Tr ln( ig ) + i 2 2 (G 2 2) } 2 + Φ[ϕ, G] all 2PI diagrams with at least 2 loops ( m 2 λ ) 2 ϕ2 δ(x x 2 ) 2 = {
Equations of Motion #4 Physical solution defined by vanishing auxiliary sources: δγ δϕ = J {K 2 ϕ 2 } 2! = δγ δg 2 = i 2 K 2! = Equation of motion for the mean field ϕ ϕ m 2 ϕ λ 3! ϕ3 i 2 ϕ {G(x, x)} x + δφ δϕ = for the full propagator G Dyson s equation: 2 δφ δg 2 = i( 2 G 2 ) := iσ Closed set of equations of for ϕ and G
Diagrammar #5 φ 4 -theory = 2 ( µφ)( µ φ) m2 2 φ2 λ 4! φ4 2PI Generating Functional iφ = + + + 8 2 3! 2 4! Mean field equation of motion i( + m 2 )ϕ = x 3! 2! 3! + x + x + Self-energy iσ 2 = x = x 2 2! + x 2! x 2 + x 3! x 2 +
Properties of the Φ-Functional #6 Why using the Φ-functional? Series of diagrams for Φ truncated at a certain loop order Linearly realized Noether symmetries are respected Conserving Approximations (mod. anomalies) In equilibrium iγ[ϕ, G] = ln Z(β) (thermodynamical potential) consistent treatment of Dynamical quantities (real time formalism) and thermodynamical bulk properties (imaginary time formalism) like energy, pressure, entropy Real- and Imaginary-Time quantities glued together by Analytic properties from (anti-)periodicity conditions of the fields (KMS-condition) Self-consistent set of equations for self-energies and mean fields
Problem of Renormalization #7 Infinities and Renormalization UV-Divergences Need a renormalization technique for numerical solution of the self-consistent equations In terms of perturbation theory: Resummation of all selfenergy insertions in propagators Self-consistent diagrams with explcit nested and overlapping sub-divergences Additional nested and overlapping sub-divergences from selfconsistency Conjecture from Weinberg s theorem, BPHZ-renormalization At finite temperatures: Self-consistent scheme rendered finite with local counterterms independent of temperature Analytic properties subtracted dispersion relations Φ-Functional technique Consistency of counterterms
Self-Consistent Tadpole I #8 The tadpole approximation Φ = iσ = Here: Only time-ordered propagator needed The renormalized tadpole d = 2ω = 2(2 ɛ): iσ = iλ d d p 2 (2π) d µ2ɛ ig(p) + CT Self-energy constant in p temperature dependent effective mass Dyson s equation can be resummed: ig(p) = i p 2 M 2 + iη + 2πn(p )δ(p 2 M 2 ) with M 2 = m 2 + Σ, n(p ) = exp(β p ) Use standard formulae for dimensional regularized Feynman integrals: Σ inf = λ [ ( )] 4πµ 32π 2 M 2 2 ɛ γ + ln M 2 () Does one need temperature dependent counter terms ( λm 2 /ɛ)?
Self-Consistent Tadpole II #9 Counterterms Solution: We do not only need an overall but also a vertex counter-term for sub-divergence: iσ = iσ reg + + i δλ λ Σ reg iδm 2 How to determine the vertex counterterm? From equations of motion consistency condition of Bethe-Salpeter-type : it (c) 2 = = + = + + + Renormalize the Dinosaur diagram vertex counterterm iγ (4) =
The Finite Equation # The counterterms Renormalization conditions (physical scheme) Γ (4) vac() =, Σ vac (m 2 ) = Counterterms: δm 2 = λ 32π 2 m2 δλ = λ2 32π 2 [ ] 4πΛ2 γ + + ln ɛ m 2 [ ] 4πΛ2 γ + + ln ɛ m 2 Counterterms are independent of temperature and adjusted in vacuum Self-consistent equation (gap equation) M 2 = m 2 + λ 32π 2 M 2 ln M 2 m 2 + λ 2 d 4 p (2π) 4 2πδ(p2 M 2 )n(p )
Numerical Results #.7.6.5 M[GeV].4.3.2...2.3.4.5 T[GeV] Numerical solution of the self-consistent tadpole equation compared to the perturbative result for m = 4MeV and λ = 5
Spontaneously Broken Symmetry #2 = 2 ( µφ)( µ φ)+ µ2 2 φ2 λ 4! φ4 Stable (tree-level) vacuum at φ = ϕ = 6µ2 m 2 = +2µ 2 > λ. Particles have Have to take the Mean Field ϕ into account and solve the coupled equations for both ϕ = const and Σ = const. The renormalization procedure is the same as in the unbroken case. Parameters from linear the σ-model: µ = 4MeV, λ = ϕ [GeV] V eff [GeV 4 ] 23..8.6.4.2..2.3 T[GeV] 25 2 5 5-5 -.2.4.6.8. ϕ[gev] M[GeV].6.5.4.3.2. 23..2.3 T[GeV] The effective potential for T = 8MeV to T = 28MeV
The Sunset Diagram #3 The vacuum part iφ = iσ = 2 4! 3! Overall and sub-divergences to all orders perturbation theory Subtracted dispersion relations for vacuum divergences +2 + +2 Counterterms of tadpole-type const. and the overall subtraction O(p 2 ) (mass- and wave function renormalization) Simple subtractions in dispersion relation
Algorithm #4 Strategy for the vacuum Starting with perturbative propagators calculate imaginary part of the retarded self-energy (finite!): Im Σ R = Σ + Σ + 2i Real part from twice subtracted dispersion relation: Σ R (s) =(s m 2 ren )2 dz π Im Σ R (z) (z s + iɛσ(p ))(z m 2 ren) 2 + + Σ R (m2 ren )(s m2 ren ) + Σ R(m 2 ren ) Dyson s equation for the retarded propagator G R = p 2 m 2 Σ R + iɛσ(p ) and plug it into the next calculation for Im Σ R. Iterate this procedure until the Σ R does not change anymore
The Sunset Diagram for T > #5 Renormalization of vacuum sub-divergence iγ (4) vac = Algorithm Calculate renormalized Γ (4) vac with already given self-consistent vacuum propagator (blue) Renormalization condition: Γ (4) vac(s = ) =. Calculate Im Σ R with the full thermal propagator Subtract the vacuum part and 3 the bad diagram Dispersion relation without subtraction for the rest Only pure vacuum subtractions in this part Full retarded bad diagram with vacuum subtracted Γ (4) - insertion finite Only vacuum sub-divergence subtracted Result: explicit overlapping divergences of the sunset diagram are completely subtracted with pure vacuum counter terms
Results for the Vacuum Sunset Diagram #6 ReΣ[GeV 2 ] ImΣ[GeV 2 ] 5-5 - -5 - -5-2 -25.2.4 p[gev].6.8.2.4.8.6 p[gev].2.4 p[gev].6.8.2.4.8.6 p[gev] Vacuum: m = 4MeV, λ = 5
Perturbative Result for the Sunset Diagram at T > #7 ReΣ[GeV 2 ] ImΣ[GeV 2 ] 5-5 - -5 - -5-2 -25.2.4 p[gev].6.8.2.4.8.6 p[gev].2.4 p[gev].6.8.2.4.8.6 p[gev] T = MeV
Self-consistent Result for the Sunset Diagram at T > #8 ReΣ[GeV 2 ] ImΣ[GeV 2 ] 5-5 - -5 - -5-2 -25.2.4 p[gev].6.8.2.4.8.6 p[gev].2.4 p[gev].6.8.2.4.8.6 p[gev] T = MeV
The Analytic Green s Function #9 The imaginary part of the contour So far real-time formalism Entropy, pressure, mean energy, Analytic propagator Branch of analytic continuation of G Matsubara dz ρ(z, p) G C (p, p) = π z with p z R : R ρ(z, p) = ρ( z, p) = Im G R (z, p) Causality structure of G R and G A G C (p ± i) = G R/A (p) for p R Matsubara-propagator G M (iω n, p) = G C (iω n, p) with ω n = 2πi β n = 2πinT, n Z
Matsubara Sums #2 Summing over Matsubara frequencies F (z): analytic in an open strip around imaginary axis β n Z F (iω n ) = 2πi i +ɛ i +ɛ dx[f (x)+f ( x)] F(z): also analytic away from the real axis Im x [ ] 2 + exp(βx) Re x β n Z F (iω n ) = 2πi + 2πi dx [F (x iɛσ(x)) F (x + iɛσ(x))]+ 2 dxn(x)[f (x iɛσ(x)) F (x + iɛσ(x))]
The Entropy #2 Thermodynamical properties Expectation values from thermal quantum field theory: Z(β, V ) = Tr exp( βh) ε = V H = V β ln Z, V d(β ln Z) = εdβ Define thermodynamical quantities: P = ln Z, s = β(p + ε) dp = sdt βv Solution of the real time Φ-derived self-consistent equations ln Z = iγ P = iγ s = i T Γ Stationarity with respect to G R : Need to derive only with respect to explicit temperature dependency s = 2 + i p > d 4 p (2π) 4 T n(p ) { Im ln[ G R (p)] + Im(Σ RG R ) } + { } δφ[ϕ, G] δn G R,ϕ fixed T n Especially for 2-point Φ-functionals d 4 p s = 2 (2π) 4 T n(p ) { Im ln[ G R (p)] + (Im Σ R)(Re G R ) } p >
Result for Tadpole Resummation #22.95 s/s free.9.85.8.75..2.3.4.5 T[GeV] The entropy density for the free gas and for the selfconsistent tadpole resummation. The parameters were m = 38MeV and λ = 5.
Conclusion and outlook #23 Conclusions Self-consistent Φ-derivable models can be renormalized and solved numerically Applications for the consistent treatment of particles and resonances with finite mass widths possible Applicable as well for dynamics as for thermodynamical quantities Consistent schemes for transport equations for such particles and resonances Outlook Problem with Goldstone s theorem and phase transitions (Linear Σ-model) Development of numerics for more complicated cases beyond two-point level (ladder summations) Big fundamental problem: Most important physical theories involve gauge fields Standardmodel (Phasetransitions in QCD and electro-weak Theory) Effective meson theories (vector dominance and dileptons) Does there exist a gauge-invariant scheme?