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
Symmetries and Self-consistency – p.2
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
Symmetries and Self-consistency – p.2
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
Symmetries and Self-consistency – p.3
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
Symmetries and Self-consistency – p.3
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
Symmetries and Self-consistency – p.4
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
Symmetries and Self-consistency – p.5
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 and Self-consistency – p.7
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 and Self-consistency – p.8
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)
Symmetries and Self-consistency – p.9
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.
Symmetries and Self-consistency – p.10
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!
Symmetries and Self-consistency – p.11
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
Symmetries and Self-consistency – p.12
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
Symmetries and Self-consistency – p.13
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)
Symmetries and Self-consistency – p.14
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!
Symmetries and Self-consistency – p.15
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
Symmetries and Self-consistency – p.16
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
Symmetries and Self-consistency – p.18
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
Symmetries and Self-consistency – p.19
First toy calculations
Lagrangian: Lint = + +
Φ = + +
Self–energies:
Πρ = +
Πa1=
Σπ = + +
Symmetries and Self-consistency – p.20
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
Symmetries and Self-consistency – p.21
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)
Symmetries and Self-consistency – p.22
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
Symmetries and Self-consistency – p.23
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
Symmetries and Self-consistency – p.24
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
Symmetries and Self-consistency – p.25
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
Symmetries and Self-consistency – p.26
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
Symmetries and Self-consistency – p.27
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