ECT Trento, September 2009
QCD-TNT
Screened perturbation theory for 3d YM theory
Owe Philipsen
The Linde problem
The QCD pressure with effective field theory methods
Resummation and screened perturbation theory
in collaboration with Daniel Bieletzki, York Schröder
The Linde problem of finite T QCD / 3d YM
contribution to pressure
contribution from
Matsubara 0-mode:
even for weak coupling!
So why bother?
Lattice MC does not work at finite density or real time!
Momentum scales in QGP
High T effective theory: dimensional reduction
(using dimensional regularisation)
= contribution from 3d YM, starting at 4-loop
= 4-loop!
Kajantie et al. 03
idea: pE , pM by (HTL resummed) perturbation theory, pG non-perturbatively, e.g. 3d lattice
N.B.: coefficients N-independent!
Here: resummation schemes
in particular, a mass term for the gluon will regulate the IR divergences
How to do this in a gauge invariant way?
Also resum interactions, such as to maintain ST-identities!
e.g. using Higgs effect, non-linear sigma model:
m iπa (x)T a
φ(x) =
e
gM
Lφ = C Tr[(Di φ) Di φ]
†
local action, but involves auxiliary field
=limiting case of linear model:
Buchmüller, O.P. 95
Meaning of auxiliary field?
Z=
!
DA Dφ ∆F P
gauge transform with
!
integrate
Z=
!
1
exp − (SY M + Sφ − lSφ + Sgf [A])
l
a a
Ai → AU
,
U
=
exp
iπ
T
i
− 1l Sgf
Dφ ∆F P e
(unitary gauge)
=1
1
DA exp −( SY M − m2
l
!
TrA2 + lm2
!
TrA2 )
just YM with gauge-invariantly resummed mass term!
Jackiw, Pi 97
Effective Lagrangian not unique
Lφ =
m2G Tr
1
Fµ 2 Fµ ,
D
1
Fµ = !µαβ Fαβ
2
Chern-Simons eikonal (HTL inspired)
non-local
Jackiw, Pi 97
non-local
Alexanian, Nair 95
....
Different gauge invariant additions/subtractions = different resummations
No small expansion parameter, expansion in dynamically generated number:
2
2
gM
gM
=
2
m
CgM
1
(times factors ~
)
4π
Need to check convergence empirically!
~ Dyson-Schwinger eq., self-consistent
transverse self-energy gauge independent on-shell = pole mass
Alternatively: gap equation from pinch technique, gauge inv. for all p
Results for m, 1-loop SU(2):
m|BP
=
2
0.28gM
m|AN
=
2
0.38gM
m|C
=
2
0.25gM
2-loop: ~15% corrections in NLSM scheme
Cornwall 97
Eberlein 98
Lattice:
gauge fixed propagators
2
m ∼ 0.35 − 0.46gM
gauge invariant correlators (cf. ‘gluelumps’)
Ad
!(Di Fij )a (x)Uab
(x, y)(Dk Flm )b (y)"
∼ exp[−(3m − 2m)|x − y|]
Ad
a
b
!(Fij ) (x)Uab (x, y)(Flm ) (y)"
2
= 0.36(2)gM
Karsch et al.
O.P. 02
Two problems of perturbation theory
Application to electroweak phase transition
g(300MeV) ∼ 1
strong coupling
linear model (SU(2)-Higgs) predicts electroweak crossover,
finite THiggs
infrared
for non-abelian
critical
massproblem
~10% accurate
at 1-loop !theories:
gauge boson self-coupling:
g2
eE/T − 1
E,p!T
∼
Buchmüller, O.P. 97
Linde
g2 T
m
infrared divergent for m = 0 ,
e.g. QCD, symmetric phase electroweak theory
resummation generates magnetic mass scale
m ∼ g2 T
convergence??
finite T QFT has no solid perturbative definition, even for weak coupling!
Application to pressure, general covariant gauge, SU(2)
up to 2 loop, O(l)
Counter terms:
Feynman diagrams through 2 loops
Result for 3d YM contribution
1 loop
1 loop CT
2 loop
Convergence?
The coefficients, numbers for
LO + NLO CT naturally same order of magnitude
NLO contribution ~10% of LO
looks promising(?)
C~N (LO gap equations), coefficients N-independent!
3-loop under way.....
49 diagrams, 13 master integrals...
Conclusions
Gauge invariant resummation methods for screened non-abelian p.t. at high T
More flexibility than lattice, but convergence not guaranteed,
to be observed in higher order calculations
For ‘magnetic mass,’ pressure, NLO-correction ~15% of LO contribution
In case of apparent convergence: apply to dynamical problems!