Symmetries and Self-consistency
Vector mesons in the fireball
π, . . .
ρ/ω
γ∗
e−
e+
Hendrik van Hees
Fakultät für Physik
Universität Bielefeld
Symmetries and Self-consistency – p.1
Content
Concepts
Real time formalism
2PI formalism
Symmetries and trouble with 2PI formalism
Symmetries and Self-consistency – p.2
Content
Concepts
Real time formalism
2PI formalism
Symmetries and trouble with 2PI formalism
Application to the πρ-system
Kroll-Lee-Zumino (KLZ) model
Perturbative results
2PI: Trouble in paradise (due to violation of symmetries)
First way out: Projection formalism
First toy calculations:
Outlook
Symmetries and Self-consistency – p.2
Real time formalism
Initial statistical operator ρi at t = ti
Time evolution
"
Z
hO(t)i = Tr ρ(ti ) Ta exp +i
|
ti
}
anti time–orderd
OI (t)
Tc
dt0 HI (t0 )
{z
|
t
exp −i
Z
t
dt0 HI (t0 )
ti
{z
time–ordered
#
.
}
Symmetries and Self-consistency – p.3
Real time formalism
Initial statistical operator ρi at t = ti
Time evolution
"
Z
hO(t)i = Tr ρ(ti ) Ta exp +i
|
exp −i
Z
t
dt0 HI (t0 )
ti
{z
}
#
.
}
time–ordered
Contour ordered Green’s functions
ti
ti
anti time–orderd
OI (t)
Tc
dt0 HI (t0 )
{z
|
t
K−
tf
t
K+
t
C = K− + K+
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 contour
Im t
ti
t−
1
K−
tf
Re t
M
−iβ
K+
C = K− + K+ + M
t+
2
+
Correlation functions with real times: iGC (x−
1 , x2 )
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
Z
Dφ exp iS[φ] + i {J1 φ1 }1 +
i
K12 φ1 φ2
2
12
Generating functional for connected diagrams
Z[J, K] = exp(iW [J, K])
The mean field and the connected Green’s function
δW
δ2 W
δW
1
ϕ1 =
, G12 = −
⇒
= [ϕ1 ϕ2 + iG12 ]
δJ1
δJ1 δJ2
δK12
2
|
{z
}
standard quantum field theory
Legendre transformation for ϕ and G:
IΓ[ϕ, G] = W [J, K] − {ϕ1 J1 }1 −
1
{(ϕ1 ϕ2 + iG12 )K12 }12
2
Symmetries and Self-consistency – p.5
2PI-formalism: The Φ-functional
Exact closed form:
o
i
i n −1
−1
IΓ[ϕ, G] =S0 [ϕ] + Tr ln(−iG ) +
D12 (G12 − D12 )
12
2
2
+ Φ[ϕ, G] ⇐ all closed 2PI interaction diagrams,
D12 = − − m
Equations of motion
δIΓ
!
= −J1 − {K12 ϕ2 }2 = 0,
δϕ1
2 −1
δIΓ
i
!
= − K12 = 0,
δG12
2
Equation of motion for the mean field ϕ and the “full” propagator G
−ϕ − m2 ϕ := j = −
δΦ
,
δϕ
−1
−i(D12
− G12 −1 ) := −iΣ = 2
δΦ
δG21
Integral form of Dyson’s equation:
G12 = D12 + {D110 Σ10 20 G20 2 }10 20
Closed set of equations of for ϕ and G
Symmetries 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] = 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
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 correlation
functions
Dyson’s equation as functional of ϕ:
δIΓ[ϕ, G] ≡0
δG G=Geff [ϕ]
Define new effective action functional
Γeff [ϕ] = IΓ[ϕ, Geff [ϕ]]
Symmetry analysis ⇒ Γeff [ϕ] symmetric functional in ϕ
Stationary point
δΓeff =0
δφ ϕ=ϕ0
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 of
symmetries
External Propagator
(G−1
ext )12
δ 2 Γeff [ϕ] =
δϕ1 δϕ2 ϕ=ϕ0
Gext generally not identical with Dyson resummed propagator
Problem: Calculation of Σext needs resummation of Bethe-Salpeter ladders
−iΣext
=
iΦϕϕ
+
iΓ(3)
iI (3)
Symmetries and Self-consistency – p.9
Gauge theories
Abelian massive gauge boson: Do not need Higgs! (Stueckelberg formalism)
Gauge invariant classical Lagrangian:
1
1
1
L = − Vµν V µν + m2 Vµ V µ + (∂ µ ϕ)(∂µ ϕ) + mϕ∂µ V µ
4
2
2
Gauge invariance:
δVµ (x) = ∂µ χ(x), δϕ = mχ(x)
Quantisation: Gauge fixing and ghosts
LV = −
1
m
1
Vµν V µν + Vµ V µ −
(∂µ V µ )2 +
4
2
2ξ
ξm2 2
1
µ
ϕ +
+ (∂µ ϕ)(∂ ϕ) −
2
2
+ (∂µ η ∗ )(∂µ η) − ξm2 η ∗ η.
Symmetries and Self-consistency – p.10
Gauge theories
Free vacuum propagators
∆µν
V (p)
(1 − ξ)pµ pν
g µν
+ 2
=− 2
p − m2 + iη
(p − 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
Dµ π = ∂µ π + igVµ π + ieAµ
Quantisation of free photon as usual
Minimal coupling and KLZ-interaction:
LπV = LV + (Dµ π)∗ (Dµ π) − m2π π ∗ π −
λ ∗ 2
e
Aµν V µν
(π π) −
8
2gργ
Eqs. of motion: Vector meson dominance (Kroll, Lee, Zumino) Photons couple to
pions only over “mixing” with (neutral) ρ-mesons!
Adding Leptons like in usual QED:
Leγ = ψ̄(iD
/ − me )ψ with Dµ ψ = ∂µ ψ − ieAµ ψ
Symmetries and Self-consistency – p.12
The Feynman rules
Propagators
p
ig µν
i(1 − ξρ )pµ pν
ν= −
+ 2
2
2
p − mρ + iη
(p − m2ρ + iη)(p2 − ξm2ρ + iη)
ρ-meson: µ
p
ig µν
i(1 − ξγ )pµ pν
ν= −
+
p2 + iη
(p2 + iη)2
Photon: µ
Pion:
Elektron:
p
=
i
p2 − m2π + iη
=
i(/
p + m)
p2 − m2l + iη
p
Symmetries and Self-consistency – p.13
The Feynman rules
The vertices
r
q
µ
r−q
µ= −igρπ (q + r )
µ
µ
µ= −ie(q µ + rµ )
ν
= ie2 g µν
ν
= 2iegρπ g µν
µ
r−q
µ
ν
2
g µν
= igρπ
µ
r
q
ν= i
e
gργ
=−
iλ
8
p2 Θµν (p)
Symmetries and Self-consistency – p.14
Pion form factor
Definition
π−
e−
π+
e+
F(k2 ) =
π−
e−
π+
e+
Measurable quantity: Pion form factor
2
|F (s)| =
m4ρ
|s − m2ρ − Πρ (s)|2
Πρ : ρ-self-energy!
Symmetries and Self-consistency – p.15
Pion form factor
F 2
F 2
40
40
30
30
20
20
10
10
0
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
s GeV
“Strict” vector-dominance, i.e.,
gργ = g
1
0.4
0.5
0.6
0.7
0.8
0.9
1
s GeV
Including corrections from ρ-ω mixing
Fit with gργ 6= g
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
1
1
140
120
100
80
60
40
20
0
300
400
500
600
700
800
900
1000
s MeV
Plot with data from form-factor fit (without ω mixing)
data: C. D. Frogatt, J. L. Petersen. Phase-Shift Analysis of π + π − Scattering
between 1.0 and 1.8 GeV Based on Fixed Transfer Analyticity (II). Nuclear Physics
Symmetries and Self-consistency – p.17
B129 (1977) 89
Big trouble in paradise!
Kroll–Lee–Zumino interaction: Coupling of massive vector bosons to conserved
currents ⇒ 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) ∝ hv µ (τ )v ν (0)i
„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:
Π (τ, p
~ = 0) ∝ h1 · 1i = const,
00
jk
D
j k
Π (τ, p
~ = 0) ∝ v v
E
∝ exp(−Γx τ )
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 pj pk Πjk and gjk Πjk
Construct ΠT and ΠL
Symmetries and Self-consistency – p.19
First toy calculations
Lagrangian: Lint =
+
+
Φ=
+
+
Self–energies:
Πρ =
+
Π a1 =
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 “π 4 -interaction such that pions get an arbitrarily chosen width
In “reality”: pion-width due to interactions with baryons (not included in the toy
model yet!)
pi−Meson Spectral function, T=110 MeV; p=150 MeV/c
pi−Meson Width, T=110 MeV; p=150 MeV/c
300
vacuum
Γ π = 50 MeV
Γ π =125 MeV
1e−04
vacuum
Γ π = 50 MeV
Γ π =125 MeV
250
200
MeV
1e−05
1e−06
150
100
50
1e−07
0
0
100 200 300 400 500 600 700 800 900
Energy (MeV)
0
100 200 300 400 500 600 700 800 900
Energy (MeV)
Symmetries and Self-consistency – p.21
Results: ρ-meson spectral functions
3D transverse and longitudinal quantities: ΠT , ΠL etc.
self consistent (T)
self consistent (L)
on−shell loop (T)
on−shell loop (L)
250
0
200
−0.05
−0.1
150
100
−0.15
−0.2
Rho−meson: Γρ, T=110 MeV, p=150 MeV/c
300
Γρ (MeV)
2
Im Π ρ (GeV )
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)
50
0
200
400
600
Energy (MeV)
800
1000
0
0
200
400
600
inv. Mass (MeV)
800
1000
Symmetries and Self-consistency – p.22
Results: ρ-meson properties
Averaged quantities: (2ΠT + ΠL )/3
10
8
6
4
2
0
0
rho−meson spectral function, T=150 MeV
400
800
p
0
1200
800
400
p
1600
rho−meson width, T=150 MeV
1.6
1.2
0.8
0.4
0
1200
1600
0
400
800
p
0
1200
1600
400
200
800
p
1200
1600
Symmetries and Self-consistency – p.23
Results: ρ-meson properties
Averaged quantities: (2ΠT + ΠL )/3
Rho−meson Spectral fct., 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−04
1e−05
Rho−Meson Spectral Fnct., T=150 MeV, p=150 MeV/c
self consistent total
pi−pi−annihilation
from pi−pi−scat.
a_1 Dalitz−decay
1e−05
1e−06
1e−06
1e−07
1e−07
0
100 200 300 400 500 600 700 800 900
inv. Mass (MeV)
0
200
400
600
inv. Mass (MeV)
800
1000
Symmetries and Self-consistency – p.24
Results: Dilepton rates
Dilepton rate (arb.units), T=110 MeV, p=150 MeV/c
1e−06
from vacuum spectral fct.
a: on−shell loop
b: Γ π = 50 MeV
c: Γ π =125 MeV
1e−07
Dilepton rate (arb.units), T=110 MeV, p=150 MeV/c
1e−06
1e−07
1e−08
1e−08
1e−09
1e−09
1e−10
1e−10
0
100 200 300 400 500 600 700 800 900
inv. Mass (MeV)
self consistent: Γ π = 50 MeV
π−π−annihilation
from π−π−scat.
a 1 Dalitz−decay
0
100 200 300 400 500 600 700 800 900
inv. Mass (MeV)
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), URL
http://theory.gsi.de/~vanhees/index.html
[Hee00] H. van Hees, Renormierung selbstkonsistenter Näherungen in der
Quantenfeldtheorie 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, URL
http://arXiv.org/abs/hep-ph/0107200
[KV96]
J. Knoll, D. Voskresensky, Classical and Quantum Many-Body Description of
Bremsstrahlung in Dense Matter (Landau-Pomeranchuk-Migdal Effect), Ann.
Phys. (NY) 249 (1996) 532
Symmetries and Self-consistency – p.27