Renormalization of Conserving Selfconsistent Dyson equations

Transcription

1 Renormalization of Conserving Selfconsistent Dyson equations Hendrik van Hees Darmstadt Motivation Thermodynamics of strongly interacting systems Conservation laws, detailed balance, thermodynamical consistency Finite width effects (resonance, damping, ) Concepts Real time quantum field theory The Φ-derivable scheme (example O(N)) Renormalization Restoration of symmetries Gauge Symmetries and Vector Mesons

2 Schwinger-Keldysh Formalism #2 Initial statistical operator ρ i at t = t i Time evolution O(t) = Tr { [ t ]} [ρ(t i ) T a exp +i dt H I (t ) } t {{ i } anti time orderd O I (t) [ t ]} ] T c {exp i dt H I (t ). } t {{ i } time ordered Difference to vacuum: Contour-ordered Green s functions t i = t t f t Im t In equilibrium: ρ = exp( βh)/z with Z = Tr exp( βh) Imaginary part of the time contour t i t 1 = t f + t + 2 iβ Correlation functions with real times: ig (x 1, x+ 2 ) Fields periodic (bosons) or anti-periodic (fermions) Feynman rules time integrals contour integrals Re t

3 The Φ-Functional #3 Introduce local and bilocal auxiliary sources Generating functional Z[J, K] = N Dφ exp [ { } ] i is[φ] + i {J 1 φ 1 } K 12φ 1 φ 2 12 Generating functional for connected diagrams Z[J, K] = exp(iw [J, K]) The mean field and the connected Green s function ϕ 1 = δw, G 12 = δ2 W δj 1 δj 1 δj }{{ 2 } standard quantum field theory Legendre transformation for ϕ and G: δw δk 12 = 1 2 [ϕ 1ϕ 2 + ig 12 ] IΓ[ϕ, G] = W [J, K] {ϕ 1 J 1 } {(ϕ 1ϕ 2 + ig 12 )K 12 } 12 Exact closed form: IΓ[ϕ, G] =S 0 [ϕ] + i 2 Tr ln( ig 1 ) + i { D (G 12 D 12 ) } 12 + Φ[ϕ, G] all closed 2PI interaction diagrams D 12 = ( m 2) 1

4 Equations of Motion #4 Physical solution defined by vanishing auxiliary sources: δiγ δϕ 1 = J 1 {K 12 ϕ 2 } 2! = 0 δiγ δg 12 = i 2 K 12! = 0 Equation of motion for the mean field ϕ ϕ m 2 ϕ := j = δφ δϕ for the full propagator G Dyson s equation: i(d 1 12 G 12 1 ) := iσ = 2 δφ δg 21 Integral form of Dyson s equation: G 12 = D 12 + {D 11 Σ 1 2 G 2 2} 1 2 Closed set of equations of for ϕ and G

5 Diagrammar #5 O(N)-theory = 1 2 ( µ φ)( µ φ) m 2 2 φ 2 λ 4! ( φ 2 ) 2 2PI Generating Functional iφ = mean field part Correlations Mean field equation of motion i( + m 2 )ϕ = + + x + Self-energy iσ 12 = mass terms damping width (momentum dependent)

6 Properties of the Φ-derivable Approximations #6 Why using the Φ-functional? Truncation of the Series of diagrams for Φ Expectation values for currents are conserved Conserving Approximations In equilibrium iiγ[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

7 Problem of Renormalization #7 Why renormalization? Diagrams UV-divergent Control the physical parameters in vacuum Temperature dependence from theory alone How to renormalize self-consistent diagrams? In terms of perturbation theory: Resummation of all selfenergy insertions in propagators Self-consistent diagrams with explicit nested and overlapping sub-divergences Hidden sub-divergences from self-consistency How to manage it numerically? Power counting (Weinberg) valid for self-consistent diagrams At finite temperatures: Self-consistent scheme rendered finite with local counterterms independent of temperature Analytical properties subtracted dispersion relations BPHZ-renormalization Subtracting the integrands Advantage: Clear scheme how to subtract temperature independent sub-divergences Φ-functional consistency of counterterms

8 Self-consistent Renormalization #8 Example: Tadpole-Renormalization Φ = iσ = Temperature dependent effective mass: M 2 = m 2 + Σ ren Resummation of the Dyson series Renormalized self-energy iσ ren = = l λ 2 G(l) λ 2 G v(l) λ 2 G2 v (l)σ ren Renormalization: Subtraction of Vacuum (sub)divergences Result: Finite Gap equation M 2 = m 2 + Σ ren =m 2 + λ (M 2 32π 2 ln M 2 ) m 2 Σ ren + + λ d 4 p 2 (2π) 4 2πδ(p2 M 2 )n(p 0 ) }{{} 0 for T 0 n(p 0 ) : Bose Einstein distribution

9 Self-consistent Renormalization #9 First step: Vacuum Power-counting for self-consistent propagators as in perturbation theory: δ = 4 E Usual BPHZ-renormalization for wave function, mass and coupling constant renormalization In practice: Use Lehmann-representation and dimensional regularization Closed self-consistent finite Dyson-equations of motion Numerically treatable Second step: Finite Temperature Split propagator in vacuum and T-dependent part = + ig ig (vac) ig (T) Expand self-energy around vacuum part + Γ (4) + iσ (vac) iσ (0) Need further splitting of propagator iσ (r) ig (T) = Γ (4) + ig (r)

10 Self-consistent Renormalization #10 Third step: 4-point vertex renormalization Σ 0 linear in G (r) Γ (4) = Λ Equation of motion Γ (4) = + Λ Γ (4) s-channel Bethe-Salpeter equation Γ (4) Λ cuts more than three lines! BPHZ Boxes in ladder-diagrams do not cut inside Γ (4). Asymptotics + BPHZ-formalism: Γ (4) (l, p) Γ (4) (l, 0) = O(l α ) with α>0 Renormalized eq. of motion for Λ: Λ(p, q) =Λ(0, 0) + Γ (4) (p, q) Γ (4) (0, 0) d 4 l + i (2π) 4 [Γ(4) (p, l) Γ (4) (0, l)][g vac ] 2 (l)λ(l, q) d 4 l + i (2π) 4 Λ(0, l)[gvac ] 2 (l)[γ (4) (l, q) Γ (4) (l, 0)] Self-energy finite with vacuum counter terms

11 Example: Tadpole+Sunset #11 Approximation of the Φ-functional iφ = + iσ = + iγ (4) = + Renormalization (vacuum) iσ = overall In practice: Use dispersionrelations for propagators Kernels, can be calculated analytically with standard formulae of dimensional regularization Finite Self-consistent integral equations of motion Solved iteratively Calculate also Γ (4) and Λ(0, q)

12 Example: Tadpole+Sunset #12 Renormalization (Finite Temperature) iσ (T) (p) = p p Λ Only finite integrals Numerics for three-dim integrals on a lattice in p 0 and p Equations of motion solved iteratively

13 Results for the Vacuum Sunset Diagram #13 ReΣ for T=0, λ=20 Pert. ImΣ for T=0, λ=20 Re Σ [GeV 2 ] p[gev] p [GeV] Im Σ [GeV 2 ] p[gev] p [GeV] Vacuum: m = 140MeV, λ = 20

14 Perturbative Result for Sunset + Tadpole at T > 0 #14 Pert. ReΣ for T=150MeV, λ=20 Pert. ImΣ for T=150MeV, λ=20 Re Σ [GeV 2 ] p[gev] p [GeV] Im Σ [GeV 2 ] p[gev] p [GeV] T = 150MeV

15 Self-consistent Result for Sunset + Tadpole at T > 0 #15 ReΣ for T=150MeV, λ=20 ImΣ for T=150MeV, λ=20 Re Σ [GeV 2 ] Im Σ [GeV 2 ] p[gev] p [GeV] p [GeV] p[gev] T = 150MeV

16 Spectral function of the Meson # Pert. spectral function for λ= Self-consist. Spectral function for λ= A[GeV -2 ] A[GeV -2 ] s[gev 2 ] p=0 p=100 MeV p=200 MeV p=300 MeV s[gev 2 ] p=0 p=100 MeV p=200 MeV p=300 MeV T = 150MeV:Perturbative (left) and self-consistent (right) calculation

17 Symmetries at the correlator level #17 Symmetry restoration Problem with Φ Functional: Most approximations break symmetry! Reason: Only conserving for Expectation values for currents, not for correlation functions Dyson s equation as functional of ϕ: δiγ[ϕ, G] δg 0 G=Geff [ϕ] Define new effective action functional Γ eff [ϕ] = IΓ[ϕ, G eff [ϕ]] Symmetry analysis Γ eff [ϕ] symmetric functional in ϕ Stationary point δγ eff δφ = 0 ϕ=ϕ0 ϕ 0 and G = G eff [ϕ 0 ]: self consistent Φ Functional solutions! Γ eff generates external vertex functions fulfilling Ward Takahashi identities of symmetries External Propagator (Gext) 1 12 = δ2 Γ eff [ϕ] δϕ 1 δϕ 2 ϕ=ϕ0 G ext generally not identical with Dyson resummed propagator

18 Example: Hartree approximation #18 Hatree approximation: External self energy iφ = + + External self energy defined on top of Hartree approximation iσ ext = + } {{ } Σ int RPA Resummation restores symmetry + + +

19 Diagrammar for external vertices I #19 1st step: define Φ and internal propagator iφ = i( m 2 )ϕ= + + iσ = Defines mean field and Dyson resummend internal propagator 2nd step: Derivatives δg eff δϕ = iγ(3) = iγ (3) = ii (3) + ik iγ (3) External self energy iσ ext = iφ ϕϕ + ii (3) iγ (3)

20 Diagrammar for external vertices II #20 Definition of Bethe Salpeter equation elements iφ ϕ,ϕ = + + ii (3) = iφ ig,ϕ = + ik = iφ ig,ig = + + Here: Green s function lines and mean fields: fixed from self consistent Φ Functional solution

21 Application to the π ρ System #21 The free vector meson Gauge invariant classical Lagrangian: = 1 4 V µνv µν m2 V µ V µ ( µ ϕ)( µ ϕ) + mϕ µ V µ Gauge invariance: δv µ (x) = µ χ(x), δϕ = mχ(x) Quantisation: Gauge fixing and ghosts V = 1 4 V µνv µν + m 2 V µv µ 1 2ξ ( µv µ ) 2 + Free vacuum propagators ( µϕ)( µ ϕ) ξm2 2 ϕ2 + + ( µ η )( µ η) ξm 2 η η. µν V (p) = g µν p 2 m 2 + iη + 1 ϕ (p) = p 2 ξm 2 + iη 1 η (p) = p 2 ξm 2 + iη. (1 ξ)p µ p ν (p 2 m 2 + iη)(p 2 ξm 2 + iη) Usual power counting renormalisable Partition sum: Three bosonic degrees of freedom!

22 Application to the π ρ System #22 Gauge covariant derivative Adding π ± and γ D µ π = µ π + igv µ π + iea µ Quantisation of free photon as usual Minimal coupling: πv = V +(D µ π) (D µ π) m 2 ππ π λ 8 (π π) 2 e 2g ργ A µν V µν Eqs. of motion: Vector meson dominance (Kroll, Lee, Zumino) Adding Leptons like in QED: eγ = ψ(i /D m e )ψ with D µ ψ = µ ψ + ieψ (1)

23 Application to the π ρ System #23 The Propagators p ig µ µν ν= p 2 m 2 ρ + iη + i(1 ξ ρ )p µ p ν (p 2 m 2 ρ + iη)(p2 ξm 2 ρ + iη) p µ ν= igµν p 2 + iη + i(1 ξ γ)p µ p ν (p 2 + iη) 2 p p = = i p 2 m 2 π + iη i(/p + m) p 2 m 2 l + iη

24 Application to the π ρ System #24 The Vertices r q r q µ= ig ρπ (q µ + r µ r ) q r q µ= ie(q µ + r µ ) µ ν = ig 2 ρπ gµν µ ν = ie 2 g µν µ ν = 2ieg ρπ g µν = iλ 8 e µ ν= i p 2 Θ µν (p) g ργ

25 Application to Vector bosons #25 Kroll Lee Zumino interaction: Coupling of massive vector bosons to conserved currents gauge theory Symmetry breaking at correlator level Problems: Internal propagators contain spurious degrees of freedom Negative norm states Numerically instable due to light cone singularities Classical picture (Fokker Planck equation): Π µν (τ, p = 0) v µ (τ)v ν (0),,One loop approximation in the classical limit Π µν (τ, p = 0) exp( Γτ) 1/Γ: Relaxation time scale due to scattering Exact behaviour: Π 00 (τ, p = 0) 1 1 = const Π jk (τ, p = 0) v j v k exp( Γ x τ) For Π jk : If Γ Γ x 1 loop approximation justified Classical limit also shows: Π jk only slightly modified by ladder resummation In self consistent approximations: Use only p j p k Π jk and g jk Π jk Construct Π T and Π L

26 The interacting π-ρ-a 1 system #26 The self consistent approximation Lagrangian: int = ρ π π + a 1 + ρ π ππ ππ Φ Funktional: Φ = π ρ π + π ρ a 1 + π π π π Self energies: π π Π ρ = + π a 1 π Π a1 = π ρ ρ π Σ π = ρ + a 1 + π

27 Results for the πρa 1 System #27 Broad pions in the medium pi Meson Spectral function, T=110 MeV; p=150 MeV/c 1e 04 vacuum Γ π = 50 MeV Γ π =125 MeV 1e 05 1e 06 1e Energy (MeV) pi Meson Width, T=110 MeV; p=150 MeV/c vacuum Γ π = 50 MeV Γ π =125 MeV 200 MeV Energy (MeV)

28 Results for the πρa 1 System #28 Im Π ρ (GeV 2 ) ρ meson Polarisation tensor Rho meson: Im Π, T=110 MeV, p=150 MeV/c 0.1 self consistent (T) self consistent (L) 0.05 on shell loop (T) on shell loop (L) Energy (MeV) Γ ρ (MeV) Rho meson: Γ ρ, T=110 MeV, p=150 MeV/c self consistent (T) self consistent (L) on shell loop (T) on shell loop (L) inv. Mass (MeV)

29 Results for the πρa 1 System #29 ρ meson properties I rho meson spectral function, T=150 MeV p p rho meson width, T=150 MeV p p

30 Results for the πρa 1 System #30 ρ meson properties II Rho meson Spectral fct., T=110 MeV, p=150 MeV/c 1e 04 1e 05 from vacuum spectral fct. a: on shell loop b: Γ π = 50 MeV c: Γ π =125 MeV 1e 06 1e inv. Mass (MeV) Rho Meson Spectral Fnct., T=150 MeV, p=150 MeV/c 1e 05 self consistent total pi pi annihilation from pi pi scat. a_1 Dalitz decay 1e 06 1e inv. Mass (MeV)

31 Results for the πρa 1 System #31 1e 06 1e 07 Dilepton rate Dilepton rate (arb.units), T=110 MeV, p=150 MeV/c from vacuum spectral fct. a: on shell loop b: Γ π = 50 MeV c: Γ π =125 MeV 1e 08 1e 09 1e inv. Mass (MeV) 1e 06 1e 07 Dilepton rate (arb.units), T=110 MeV, p=150 MeV/c self consistent: Γ π = 50 MeV π π annihilation from π π scat. a 1 Dalitz decay 1e 08 1e 09 1e inv. Mass (MeV)

32 Conclusions #32 Summary Self consistent Φ derivable schemes Renormalization Symmetry analysis Scheme for vector particles Numerical treatment Outlook Toolbox for application to realistic models Perspectives for self consistent treatment of gauge theories QCD e.g. beyond HTL? Transport equations for particles with finite width