Yang-Mills thermodynamicsRalf Hofmann
Universitat Karlsruhe (TH)
“Symmetry in Nonlinear Mathematical Physics”
Institute of Mathematics, National Academy of Science, Kyiv, Ukraine
June 28, 2007
Yang-Mills thermodynamics – p.1/28
planI brief motivation and preview on phase diagram
[hep-th: 0411214, 0504064, 0609033, 0609172, 0702027]
I deconfining ground-state physics:coarse-grained, interacting calorons
I coarse-grained excitations:Legendre-trafos and loop expansion
I preconfinement:cond. magn. monopoles, dual Meissner effect
I low temperatures:Hagedorn, flip of statistics, Borel summation
I summary, conclusions, mention of applicationsYang-Mills thermodynamics – p.2/28
Why nonpert. YMTD?I infrared instability of PT even for T � Λ
in magnetic sector[Linde 1980]
I highly nonpert. ground-state physics even for T � Λ:
– θµµ ∝ T[Miller 1998]
– spatial string tension: σ ∝ T2
[Philipsen 1998, Korthals-Altes 1998, ...]
I no lattice control at low temperature:
– correlation length larger than linear lattice size
– analytical grasp ⇒ equilibrium violatedYang-Mills thermodynamics – p.3/28
preview: phase diagram SU(2)
� �� �� ����
confining preconfining
ground state:
condensate of magnetic monopoles, collapsing center−vortex loops,negative pressure
excitations: massless and massive gauge modes
ground state: interacting calorons and anticalorons, negative pressure
power−like approach to Stefan Boltzmann limit
deconfining
2nd order likeHagedorn
excitations:
massive dual gauge modes
Cooper−pair condensateof single center−
ground state:
vortex loops, pressureprecisely zero
excitations:
massless (single) and massive (self−intersecting) center−vortex loops (spin−1/2 fermions)
T
P/T4 T 4ρ/ T 4
12 14 16 18 20 22 24
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
12 14 16 18 20 22 24
1.8
2
2.2
2.4
2.6
electr
ic
mag
netic
electr
ic
mag
netic
Stefan−Boltzmann limit Stefan−Boltzmann limit
ρ/Τ =as 2.62P/T4as = 0.86
ρc,M /T
4c,E
ρ /T4c,Ec,E
λE
Tc
γV
+, V
−
(anti) screening
−+
c =2.73 KTdecoupling of , VV
−4, no (anti) screening
Λ∼10 eV
Yang-Mills thermodynamics – p.4/28
deconfining ground stateI coarse-grained (anti)calorons of |Q| = 1
⇒ adjoint scalar field φa, |φ| spatially homogeneous
I strategy:
– thermodynamics ⇒ φa periodic in eucl. timein any admissible gauge ⇒phase φa determined by classical configs.
– stable configs.: |Q| = 1 HS (anti)calorons (BPS)of trivial holonomy (only these enter!)
Yang-Mills thermodynamics – p.5/28
– compute φa ∈(Kernel of D) by respectingisotropy and SHS = 8π2
g2 6= f(T, Λ) ininf.-vol. average over magnetic-magneticcorrelation mediated by single (anti)caloron
– fixes D uniquely ⇒ winding number
– impose BPS ⇒ φa
– average saturates rapidly
⇒ scale Λ and analyticity in φa
⇒ RHS of BPS eq. for φa
⇒ φ’s potential and saturation scale |φ|⇒ inertness of |φ|
Yang-Mills thermodynamics – p.6/28
technically:
(integration over SR=∞3 )
φa(τ) ∈∑
HS (anti)caloron
tr∫
d3x
∫
dρλa
2×
Fµν ((τ, 0)) {(τ, 0), (τ, ~x)} ×
Fµν ((τ, ~x)) {(τ, ~x), (τ, 0)} .
Fµν (x) Fµν (x)
Fµν (0)µν (0)Fz0 z
or
Fµν(z) 0
a parallel transport into isotropic case or gauge trafo thereof
isotropic case other path prohibited by isotropy since not
Yang-Mills thermodynamics – p.7/28
saturation:
0 0.5 1 1.5 2
-300
-200
-100
0
100
200
300
0 0.5 1 1.5 2
-2000
-1000
0
1000
2000
0 0.5 1 1.5 2
-200000
-100000
0
100000
200000
(2π/β) τ
A ζ=10ζ=1 ζ=2
Yang-Mills thermodynamics – p.8/28
=⇒
– D = ∂2τ +
(
2πβ
)2
– ∂τφ = ±i Λ3 λ3 φ−1 , (fixed global gauge)where φ−1 ≡ φ
|φ|2
⇒ V (φ) = tr Λ6φ−2 by squaring RHS
⇒ |φ| =√
Λ3
2π T
⇒ unique, coarse-grained action for|Q| = 1 HS (anticalorons)
⇒ φ’s inertness
Yang-Mills thermodynamics – p.9/28
What about Q = 0?I perturbative renormalizability:
[’t Hooft, Veltman 1971-73]
⇒ coarse-graining yieldssame form as fundamental action
I gauge invariance glues Q = 0 to |Q| = 1 ⇒
S = tr∫ β
0dτ
∫
d3x(
1
2GµνGµν + DµφDµφ + Λ6φ−2
)
I subject to offshellness constraints inunitary-Coulomb gauge(coarse-graining down to resolution |φ|)
Yang-Mills thermodynamics – p.10/28
full ground stateI from DµGµν = 2ie[φ,Dνφ]:
– pure gauge abgµ = π
eTδµ4 λ3
⇒ ground-state energy-density and pressure
ρg.s = 4π Λ3 T = −P g.s 6= 0
I rotation to unitary gauge abgµ = 0:
– gauge transformation singular but admissible
(does not affect periodicity of fluct. δaµ)
– but: Pol[abg] = −1GT→ Pol[abg] = +1
⇒ Zel2 degeneracy
⇒ deconfinement
Yang-Mills thermodynamics – p.11/28
excitations and loop expansionI adjoint Higgs mechanism:
2 out of 3 directions massive with m = e
√
Λ3
E
2πT
I T evolution of eff. coupl. e:requiring that P , ρ, ... from partition function
λc12 14 16 18 20
5
10
15
20
25
λE
g e
elec
tric
mag
netic
= 4 π /e
plateau: e=8 1/2 π
=2 π T/ Λ
− )λ λce~−log(
Yang-Mills thermodynamics – p.12/28
I counting of d.o.f.:
fundamentally:3 species (gluons)×2 pols.+1 species (monop)×2 charges =8
after coarse-graining:2 species (gluons)×3 pols.+1 species (gluon)×2 pols.=8
⇒ 8 (fund)=8 (coarse-grained).
same way for SU(3):⇒ 22 (fund)=22 (coarse-grained)
Yang-Mills thermodynamics – p.13/28
I loop expansion:2-loop:[Rohrer,Herbst,RH 2004; Schwarz,RH,Giacosa 2006]
∆P =1
4+
1
8
+
0 20 40 60 80-0.0035
-0.003
-0.0025
-0.002
-0.0015
-0.001
-0.0005
0
HHM + ∆ Pttv ∆ P HHMttc( )/P1−loop
λYang-Mills thermodynamics – p.14/28
irreducible 3-loop:[Kaviani,RH 2007]
+ +481
1 2
3 4
5
34213 1 2 4
p4= p p p1+ _
2 3 p4= p p p1+ _
2 3p4= p p p1+ _
2 3p = p5 1
p_3
A B C
20 40 60 80 100 120 140
5·10-8
1·10-7
1.5·10-7
2·10-7
λ
P|∆ 1−loop| /P> B
Yang-Mills thermodynamics – p.15/28
arguments on loop expansion in general:[RH 2006]
– resummation of 1PI diagrams⇒ no pinch singularities
– irreducible diagrams terminateat finite loop order
(Euler characteristics for spherical polygon,constraints on loop momenta in effective theory⇒number of constraints exceedsnumber of independent radial loop variables
at sufficiently large number of loops)Yang-Mills thermodynamics – p.16/28
preconfining phaseI condensation of monopoles:
– phase of complex scalar =magnetic flux through SR=∞
2
of M-A pair at rest (e → ∞)– modulus as in dec. phase– no change of form of action for
free dual gauge modes by coarse-graining⇒ unique effective action– Polyakov loop always unique– pressure exact at one loop– evol. of magnetic coupling g by requiring der. of
pressure from fund. partition function
Yang-Mills thermodynamics – p.17/28
I counting of d.o.f.:
fundamentally:
1 species (’photon’)×2 pols.+1 species (center-vortex loop) =3
after coarse-graining:
1 species (massive ’photon’)×3 pols.=3
⇒ 3 (fund)=3 (coarse-grained).
same way for SU(3):⇒ 6 (fund)=6 (coarse-grained)
Yang-Mills thermodynamics – p.18/28
low temperatureI condensation of center-vortex loops (CVL’s)
– discrete values of phase of complex scalar field= center flux through SR=∞
1 of spin-0 vortex pairat rest (g → ∞)
– spectrum: single and selfintersecting CVL’s
n=2:
n=3:= 4
)
)
2
3
= )1n= 0:
0
n=1: = )11
(
(N
(N2=
N
(N4λφ
multiplicity of mass−n soliton: number of connected bubble diagrams at order n in
large−order counting:
[Bender, Wu 1969−76]anharmonic oscillator
over−exponentially in energy growing density of states
Hagedorn transition topreconfining phase
Yang-Mills thermodynamics – p.19/28
I counting of d.o.f.:
fundamentally:
1 species (’very massive photon’)×3 pols.=3
after coarse-graining:
1 species (massless CVL)+1 species (massive CVL)× 2 charges=3
⇒ 3 (fund)=3 (coarse-grained).
same way for SU(3):
⇒ 6 (fund)=6 (coarse-grained)
Yang-Mills thermodynamics – p.20/28
– potential unique up to inessential,U(1) invariant rescaling
-1
0
1-1
0
1
0
2
4
-1
0
1
-1
0
1-1
0
1
02468
-1
0
1
SU(2) SU(3)
pole
zero
negative tangential curvature
negative tangential curvature
negative tangential curvature
negative tangential curvaturenega
tive t
ange
ntia
l cur
vatu
re
Yang-Mills thermodynamics – p.21/28
– asymptotic-series representation of pressure:
Pas = Λ4
2π2 β−4×
(
7π4
180+√
2π β3
2
∑Ll=0
al
∑
n≥1(32λ)n n! n
3
2+l
)
,
where β ≡ Λ/T and λ ≡ exp[−β].
– Borel transformation and analytic continuation⇒ analytic dependence on Borel parameter t(polylogs) for λ < 0
– inverse Borel trafo:⇒ analytic dependence for λ < 0 and
meromorphic in entire λ-plane except for λ ≥ 0
Yang-Mills thermodynamics – p.22/28
analyticity structure of physical pressure P :
Im λ
Re λP analytic branch cut
isolated poles
λ = e−T/Λ
Yang-Mills thermodynamics – p.23/28
– Re P continuous across cut:– sign-ambiguous Im P grows slower than Re P
– turbulences become relevant for sufficiently high T only
0 0.05 0.1 0.15 0.2 0.25 0.3
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0 0.002 0.004 0.006 0.008 0.01-0.01
-0.008
-0.006
-0.004
-0.002
0
0 0.05 0.1 0.15 0.20
0.0025
0.005
0.0075
0.01
0.0125
0.015
0.0175
0 0.2 0.4 0.6 0.8 1
-0.05
-0.04
-0.03
-0.02
-0.01
0
0 0.00020.00040.00060.0008 0.0010
0.001
0.002
0.003
0.004
(a) (b)
(c) (d)
(e) Yang-Mills thermodynamics – p.24/28
summary and conclusionsI deconf. phase:
– magnetic-magnetic correlations in(anti)calorons generate adj. Higgs field
– rapid saturation of average
– negative ground-state pressure by microscopicholonomy shifts (annihilating M-A pairs)
– thermal quasiparticles on tree level(adj. Higgs mech.)
– very small radiative corrections, terminationof expansion in terms of irreducible loops
Yang-Mills thermodynamics – p.25/28
I preconf. phase:
– averaged magnetic flux of M-A pairthrough S∞
2 generates phaseof complex scalar
– dual gauge field Meissner massive
– loop expansion trivial
Yang-Mills thermodynamics – p.26/28
I conf. phase:– averaged center flux of CVL pair
through min. surface spanned by S∞1
generates discrete values of phaseof complex scalar
– excitations are single or selfintersecting CVL’sof factorially growing multiplicity
– asymptotic-series representation of pressure– Borel summability for complex values of T– analytic continuation: rapidly (slowly) rising
modulus of real (imaginary) part– interpretation: growing relevance of turbulences
with increasing T
Yang-Mills thermodynamics – p.27/28
I physics applications:
– CMB
– late-time cosmology (axion + SU(2))
– electroweak symmetry breaking
Thank you.
Yang-Mills thermodynamics – p.28/28