Symmetries and Self-consistencyhees/publ/santo-semi.pdf · 2011. 7. 1. · Content Concepts Real...

Symmetries and Self-consistency

Vector mesons in the fireball

ρ/ω γ∗e−

e+

π, . . .

Hendrik van Hees

Fakultat fur Physik

Universitat Bielefeld

Symmetries and Self-consistency – p.1

Content

ConceptsReal time formalism2PI formalismSymmetries and trouble with 2PI formalism

Application to the πρ-systemKroll-Lee-Zumino (KLZ) modelPerturbative results2PI: Trouble in paradise (due to violation of symmetries)First way out: Projection formalismFirst toy calculations:

Outlook


Real time formalism

Initial statistical operator ρi at t = ti

Time evolution

〈O(t)〉 = Tr

[

ρ(ti)Ta

{

exp

[

+i∫ t

ti

dt′HI(t′)

]}

︸︷︷︸

anti time–orderd

OI(t)

Tc

{

exp

[

−i∫ t

ti

dt′HI (t′)

]}

︸︷︷︸

time–ordered

]

.

Contour ordered Green’s functions

K+

C = K−

+ K+

K−

ti

tt

tf


Real-time formalism: Equilibrium

In equilibrium

ρ = exp(−βH)/Z with Z = Tr exp(−βH), β = 1/T

Can be implemented by adding an imaginary part to the contourIm t

Re t

K−

K+

−iβ

C = K−

+ K+ + M

tfti

M

t−1

t+2

Correlation functions with real times: iGC (x−1 , x+2 )

Fields periodic (bosons) or anti-periodic (fermions) in imaginary time

Feynman rules ⇒ time integrals → contour integrals


2PI-Formalism: The Φ-functional

Introduce local and bilocal sources

Z[J,K] = N

∫

Dφ exp

[

iS[φ] + i {J1φ1}1 +

{i2K12φ1φ2

}

12

]

Generating functional for connected diagrams

Z[J,K] = exp(iW [J,K])

The mean field and the connected Green’s function

ϕ1 =δW

δJ1, G12 = −

δ2W

δJ1δJ2︸︷︷︸

standard quantum field theory

⇒δW

δK12=

1

2[ϕ1ϕ2 + iG12]

Legendre transformation for ϕ and G:

IΓ[ϕ,G] = W [J,K] − {ϕ1J1}1 −1

2{(ϕ1ϕ2 + iG12)K12}12


2PI-formalism: The Φ-functional

Exact closed form:IΓ[ϕ,G] =S0[ϕ] +

i2

Tr ln(−iG−1) +i2

{

D−112 (G12 −D12)

}

12

+ Φ[ϕ,G] ⇐ all closed 2PI interaction diagrams, D12 =(−� −m2

)−1

Equations of motion

δIΓ

δϕ1= −J1 − {K12ϕ2}2

!= 0,

δIΓ

δG12= −

i2K12

!= 0,

Equation of motion for the mean field ϕ and the “full” propagator G

−�ϕ−m2ϕ := j = −δΦ

δϕ, −i(D−1

12 −G12−1) := −iΣ = 2

δΦ

δG21

Integral form of Dyson’s equation:

G12 = D12 + {D11′Σ1′2′G2′2}1′2′

Closed set of equations of for ϕ and GSymmetries and Self-consistency – p.6

2PI-formalism: Features

Truncation of the Series of diagrams for Φ

Expectation values for currents are conserved⇒ “Conserving Approximations”

In equilibrium iIΓ[ϕ,G] = lnZ(β)

(thermodynamical potential)

consistent treatment of Dynamical quantities (real time formalism) andthermodynamical 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


Symmetries

Problem with Φ–Functional: Most approximations break symmetry!

Reason: Only conserving for Expectation values for currents, not for correlationfunctions

Dyson’s equation as functional of ϕ:

δIΓ[ϕ,G]

δG

∣∣∣∣G=Geff[ϕ]

≡ 0

Define new effective action functional

Γeff[ϕ] = IΓ[ϕ,Geff[ϕ]]

Symmetry analysis ⇒ Γeff[ϕ] symmetric functional in ϕ

Stationary pointδΓeff

δφ

∣∣∣∣ϕ=ϕ0

= 0

ϕ0 and G = Geff[ϕ0]: self–consistent Φ–Functional solutions!


Symmetries

Γeff generates external vertex functions fulfilling Ward–Takahashi identities ofsymmetries

External Propagator

(G−1ext )12 =

δ2Γeff[ϕ]

δϕ1δϕ2

∣∣∣∣ϕ=ϕ0

Gext generally not identical with Dyson resummed propagator

Problem: Calculation of Σext needs resummation of Bethe-Salpeter ladders

−iΣext = iΦϕϕ + iI(3)iΓ(3)


Gauge theories

Abelian massive gauge boson: Do not need Higgs! (Stueckelberg formalism)

Gauge invariant classical Lagrangian:

L = −1

4VµνV

µν +1

2m2VµV

µ +1

2(∂µϕ)(∂µϕ) +mϕ∂µV

µ

Gauge invariance:δVµ(x) = ∂µχ(x), δϕ = mχ(x)

Quantisation: Gauge fixing and ghosts

LV = −1

4VµνV

µν +m

2VµV

µ −1

2ξ(∂µV

µ)2+

+1

2(∂µϕ)(∂µϕ) −

ξm2

2ϕ2+

+ (∂µη∗)(∂µη) − ξm2η∗η.


Gauge theories

Free vacuum propagators

∆µνV

(p) = −gµν

p2 −m2 + iη+

(1 − ξ)pµpν

(p2 −m2 + iη)(p2 − ξm2 + iη)

∆ϕ(p) =1

p2 − ξm2 + iη

∆η(p) =1

p2 − ξm2 + iη.

Usual power counting ⇒ renormalisable

Partition sum: Three bosonic degrees of freedom!


The π-ρ-system: KLZ-action

Adding π± and γ

Gauge–covariant derivative

Dµπ = ∂µπ + igVµπ + ieAµ

Quantisation of free photon as usual

Minimal coupling and KLZ-interaction:

LπV = LV + (Dµπ)∗(Dµπ) −m2ππ

∗π −λ

8(π∗π)2 −

e

2gργ

AµνVµν

Eqs. of motion: Vector meson dominance (Kroll, Lee, Zumino) Photons couple topions only over “mixing” with (neutral) ρ-mesons!

Adding Leptons like in usual QED:

Leγ = ψ(i/D −me)ψ with Dµψ = ∂µψ − ieAµψ


The Feynman rules

Propagators

ρ-meson:p

µ ν= −igµν

p2 −m2ρ + iη

+i(1 − ξρ)pµpν

(p2 −m2ρ + iη)(p2 − ξm2

ρ + iη)

Photon:p

µ ν= −igµν

p2 + iη+

i(1 − ξγ)pµpν

(p2 + iη)2

Pion:p

=i

p2 −m2π + iη

Elektron:p

=i(/p+m)

p2 −m2l

+ iη


The Feynman rules

The vertices

q

r − qrµ= −igρπ(qµ + rµ) q

r − qrµ= −ie(qµ + rµ)

µ ν

= ig2ρπgµν

µ ν

= ie2gµν

µ ν

= 2iegρπgµν = −

iλ8

µ ν= ie

gργ

p2Θµν(p)


Pion form factor

Definition

F(k2) =

π−

π+

π−

π+

e+

e−

e−

e+

Measurable quantity: Pion form factor

|F (s)|2 =m4

ρ

|s−m2ρ − Πρ(s)|2

Πρ: ρ-self-energy!


Pion form factor

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

s GeV

0

10

20

30

40

F 2

“Strict” vector-dominance, i.e.,gργ = g

Fit with gργ 6= g

0.4 0.5 0.6 0.7 0.8 0.9 1

s GeV

0

10

20

30

40

F 2

Including corrections from ρ-ω mixing

data: L. M. Barkov, et al., Electromagnetic Pion Form Factor in the Timelike Region,Nucl. Phys. B256 (1985) 365


Pion phase-shift δ11

Phase shift in the s = 1, t = 1-channel of pion scattering

300 400 500 600 700 800 900 1000

s MeV

0

20

40

60

80

100

120

140

11

Plot with data from form-factor fit (without ω mixing)

data: C. D. Frogatt, J. L. Petersen. Phase-Shift Analysis of π+π− Scatteringbetween 1.0 and 1.8 GeV Based on Fixed Transfer Analyticity (II). Nuclear PhysicsB129 (1977) 89 Symmetries and Self-consistency – p.17

Big trouble in paradise!

Kroll–Lee–Zumino interaction: Coupling of massive vector bosons to conservedcurrents ⇒ gauge theory

Symmetry breaking at correlator level

Internal propagators contain spurious degrees of freedom

Negative norm states

Numerically instable due to light cone singularities


Digression: Classical transport picture

Classical picture (Fokker–Planck–equation):

Πµν(τ, ~p = 0) ∝ 〈vµ(τ)vν(0)〉

„One–loop” approximation in the classical limit

Πµν(τ, ~p = 0) ∝ exp(−Γτ)

1/Γ: Relaxation time scale due to scattering

Exact behaviour due to Conservation law:

Π00(τ, ~p = 0) ∝ 〈1 · 1〉 = const, Πjk(τ, ~p = 0) ∝⟨

vjvk⟩

∝ exp(−Γxτ)

For Πjk: If Γ ≈ Γx ⇒ 1–loop approximation justified

Classical limit also shows: Πjk only slightly modified by ladder resummation

For self–consistent approximations: Use only pjpkΠjk and gjkΠjk

Construct ΠT and ΠL


First toy calculations

Lagrangian: Lint = + +

Φ = + +

Self–energies:

Πρ = +

Πa1=

Σπ = + +


Results: Broad pions in the medium

Avoided renormalization problem: Took only imaginary parts of the self-energies!

Tuned “π4-interaction such that pions get an arbitrarily chosen width

In “reality”: pion-width due to interactions with baryons (not included in the toymodel yet!)

1e−07

1e−06

1e−05

1e−04

0 100 200 300 400 500 600 700 800 900

Energy (MeV)

pi−Meson Spectral function, T=110 MeV; p=150 MeV/c

vacuumΓπ = 50 MeVΓπ =125 MeV

0

50

100

150

200

250

300

0 100 200 300 400 500 600 700 800 900

MeV

Energy (MeV)

pi−Meson Width, T=110 MeV; p=150 MeV/c

vacuumΓπ = 50 MeVΓπ =125 MeV


Results: ρ-meson spectral functions

3D transverse and longitudinal quantities: ΠT , ΠL etc.

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0 200 400 600 800 1000

Im Π

ρ (G

eV2 )

Energy (MeV)

Rho−meson: Im Π, T=110 MeV, p=150 MeV/c

self consistent (T)self consistent (L)

on−shell loop (T)on−shell loop (L)

0

50

100

150

200

250

300

0 200 400 600 800 1000Γ ρ

(MeV

)inv. Mass (MeV)

Rho−meson: Γρ, T=110 MeV, p=150 MeV/c

self consistent (T)self consistent (L)

on−shell loop (T)on−shell loop (L)


Results: ρ-meson properties

Averaged quantities: (2ΠT + ΠL)/3

rho−meson spectral function, T=150 MeV

0p

p0

400800

12001600

400800

12001600

02468

10

rho−meson width, T=150 MeV

p0

p0

400800

12001600

200400800

1200

00.40.81.21.6

1600


Results: ρ-meson properties

Averaged quantities: (2ΠT + ΠL)/3

1e−07

1e−06

1e−05

1e−04

0 100 200 300 400 500 600 700 800 900

inv. Mass (MeV)

Rho−meson Spectral fct., T=110 MeV, p=150 MeV/c

from vacuum spectral fct.a: on−shell loopb: Γπ = 50 MeVc: Γπ =125 MeV

1e−07

1e−06

1e−05

0 200 400 600 800 1000inv. Mass (MeV)

Rho−Meson Spectral Fnct., T=150 MeV, p=150 MeV/c

self consistent totalpi−pi−annihilation

from pi−pi−scat.a_1 Dalitz−decay


Results: Dilepton rates

1e−10

1e−09

1e−08

1e−07

1e−06

0 100 200 300 400 500 600 700 800 900

inv. Mass (MeV)

Dilepton rate (arb.units), T=110 MeV, p=150 MeV/c

from vacuum spectral fct.a: on−shell loopb: Γπ = 50 MeVc: Γπ =125 MeV

1e−10

1e−09

1e−08

1e−07

1e−06

0 100 200 300 400 500 600 700 800 900

inv. Mass (MeV)

Dilepton rate (arb.units), T=110 MeV, p=150 MeV/c

self consistent: Γπ = 50 MeVπ−π−annihilation

from π−π−scat.a1 Dalitz−decay


Summary and Outlook

Self–consistent Φ–derivable schemes

Renormalization

Symmetry analysis

Scheme for vector particles

Numerical treatment

“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


References[Hee97] H. van Hees, Quantenfeldtheoretische Beschreibung des πρ-Systems, Master’s

thesis, Technische Hochschule Darmstadt (1997), URLhttp://theory.gsi.de/~vanhees/index.html

[Hee00] H. van Hees, Renormierung selbstkonsistenter Näherungen in derQuantenfeldtheorie bei endlichen Temperaturen, Ph.D. thesis, TU Darmstadt(2000), URL http://elib.tu-darmstadt.de/diss/000082/

[HK01] H. van Hees, J. Knoll, Finite Width Effects And Dilepton Spectra, Nucl. Phys.A683 (2001) 369, URL http://arXiv.org/abs/hep-ph/0007070

[HK02a] H. van Hees, J. Knoll, Renormalization of Self-consistent Approximations II:applications: Applications to the sunset diagram, Phys. Rev. D 65 (2002)105005, URL http://arxiv.org/abs/hep-ph/0111193

[HK02b] H. van Hees, J. Knoll, Renormalization of Self-consistent Approximations III:Symmetries, Phys. Rev. D (2002), accepted

[HK02c] H. van Hees, J. Knoll, Renormalization of self-consistent approximations:theoretical concepts, Phys. Rev. D 65 (2002) 025010, URLhttp://arXiv.org/abs/hep-ph/0107200

[KV96] J. Knoll, D. Voskresensky, Classical and Quantum Many-Body Description ofBremsstrahlung in Dense Matter (Landau-Pomeranchuk-Migdal Effect), Ann.Phys. (NY) 249 (1996) 532


http://theory.gsi.de/~vanhees/index.html

http://elib.tu-darmstadt.de/diss/000082/

http://arXiv.org/abs/hep-ph/0007070

http://arxiv.org/abs/hep-ph/0111193

http://arXiv.org/abs/hep-ph/0107200

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