Stabilization of Chromomagnetic Fields at High Temperature?

5 Pages·0.1 MB·English

by  David Persson

STABILIZATION OF CHROMOMAGNETIC FIELDS
AT HIGH TEMPERATURE?
9 DAVIDPERSSON
9
Department of Physics and Astronomy
9
The University of British Columbia
1
Vancouver, B.C. V6T 1Z1, Canada

n E-mail: [email protected]
a
J It is well known that a tachyonic mode appears in the spectrum of Yang—Mills

7 theory witha static uniformmagnetic field, andthat the free energy has an (un-
2 stable)minimumatfinitemagneticfield. Itisarguedthatspontaneousgeneration
ofmagneticfielddoesnottake placeathightemperatureduetononperturbative
1 magnetic screening. Furthermore, the dispersion relation for gauge field fluctu-
v ations in an external magnetic field at high temperature is solved. The lowest
energy mode is stable against spontaneous generation of magnetic fields since it
3
acquires a thermal mass. However, the resummed free energy (by necessity com-
1
puted in the imaginary time formalism) still shows an instability, unaffected by
4
theresummation,sincetheself-energyisvanishingatstaticMatsubarafrequency.
1
0
9
1 The Instability and how it is Screened at High Temperature
9
/
h For non-abeliangaugetheories there areclasses of gaugeequivalentpotentials
p corresponding to the same field strength. We shall in this letter only consider
- the abelian like class, that is the only one reasonable in an early universe
p
scenario,andforsimplicityonlySU(N =2). Letusthereforeconsiderastatic
e
h uniform (chromo-) magnetic field in z direction in space, with the potential
: Aa =δa3(0,0, Bx,0),wherethefieldpointsinthe3-directioningroupspace.
v µ −
i TheSU(2)Lagrangianwiththisbackgroundfieldisthenrewrittenintermsof
X
the charged Vector Field W =(W1+iW2)/√2. The energy spectrum reads
µ µ µ
r
a
E2(k ,l,σ)=k2+(2l+1)gB 2σgB , (1)
z z −
where the term (2l+1)gB, l =0,1,2,..., comes from the orbital angular mo-
mentum and the term 2σgB, σ = 1, is the spin energy. The momentum
±
parallell to B is k . Obviously, we have an instability in the lowest Landau
z
level (LLL), l = 0, σ = 1. At T = 0, this leads to a spontaneous generation
of a magnetic field1 gB = Λ2 = λ2exp 48π2 , where λ is the renor-
0 (cid:16)−11Ng2(λ)(cid:17)
malizationscale,andg2(λ)is runningsothatΛis independentofλ. However,
the free energy acquires an imaginary part, so this minimum is unstable2, A
possible groud-state has to be varying over the non-perturbative scale 1/Λ—
“Copenhagen Vacuum”3.
1
Athightemperature,wehaveforanasymptoticallyfreetheorythefollow-
ing hierarchy of scales (in due order)
T: The temperature is the typicalenergy of particles in the plasma and the
inter-particle distance is 1/T.
∼
gT: The interaction of soft particles (p gT) with hard particles (p T)
∼ ∼
generatesa thermalmassofordergT. Static electric(but notmagnetic)
fields are screened over the length scale 1/gT.
g2T: On this momentum scale Yang-Mills theory becomes non-perturbative.
Theoretical arguments and lattice simulations show that non-abelian
magnetic fields are screened over the length scale 1/g2T.
Λ: The strong coupling scale below which the vacuum theory becomes non-
perturbative.
Duetothenon-perturbativemagneticscreeningtheextensionoftheLLLis
muchlargerthanthelength-scaleoverwhichthemagneticfieldcanbeconstant
1/Λ 1/g2T. The would be unstable mode thus will not see a uniform
≫
field,andtheSaviddymechanismforspontaneousgenerationofmagneticfield
cannot operate.
2 External Chromomagnetic Fields
If we instead assume an external magnetic field, generated by some other
mechanism, we may consider the hierarchy of scales:
N
T2 2 (gT)2 gB (g2T)2 . (2)
≫M ≡ 9 ≫ ≫
Inordertoinvestigateiftheinstabilityisscreenedathightemperature,wenow
needtoconsiderthedispersionrelationobtainedfromtheeffectiveLagrangian
d4xd4x′W†(κ,x)[ δ(x x′) gµνD2 DµDν 2igFµν
Z µ − − − −
(cid:0) (cid:1)
Πµν(x,x′)]W (κ′,x′) , (3)
ν
−
whereΠisthegluonself-energyinamagneticfieldathightemperature. With
wave-functions corresponding to the LLL, unstable at T = 0, the dispersion
relation reads in momentum space
k2+gB k2+Π (k ,k )=0 , (4)
0 − z LLL 0 z
2
where we integrate over perpendicular momenta
Π (k ,k )= ∞ 2p⊥dp⊥e−p2⊥/gBwLLLΠµν (k ,k ,p )wLLL , (5)
LLL 0 z Z gB µ HTL 0 z ⊥ ν
0
and w is the polarization vector in LLL. It turns out that the leading high
µ
temperature correction comes only from the ordinary hard thermal loop ap-
proximation of the gluon self-energy tensor, conveniently separated in its lon-
gitudinal (Π ), and transverse (Π ) parts. To leading order, the magnetic
L T
field only enters through the external states being Landau levels. In order to
investigate the instability, let us consider k = 0. For k √gB, we
z 0
∼ M ≫
find to leading order
2gB
k2+gB 2 1+ =0 , (6)
0 −M (cid:18) 5 2(cid:19)
M
i.e. k (1 3gB/10 2),astablesolution. However,ifweinsteadexpand
0
≃M − M
for small k , we get to leading order
0
3π3/2k 2
k2+gB+i 0M =0 , (7)
0 8 √gB
withtheleadingsolutionk i16/(3π3/2)(gB)3/2/ 2. Thismaythusbe the
0
≃ M
signal of an instability. In Figure 1, we show the two branches obtained by
solving the real part of the dispersion relation for real k , and then determine
0
the imaginary part on this solution. Furthermore, the spectral weight of the
stable mode is shown. For weak magnetic fields it is close to unity, leaving
no phase space for the unstable mode. However, work in progress shows that
when considering the full dispersion relation, there is a branch with purely
imaginary k , indicating the survival of the instability at high temperatures.
0
3 The Resummed Free Energy
In order to avoid the tree level instability, it is necessary to consider the re-
summed free energy, including the leading self-energy term, as well as the
corresponding counter term. However, this corresponds to a sum over dia-
grams with vanishing momenta on the external lines (here rather no external
lines),sothattherealtimeformalismisnotstraightforwardlyapplicable4. Let
us neglectthe spatialmomenta,andfirst considerthe self-energylike function
in imaginary time
1
Θ = T ∆(iω ,k)=T ; ω =2πnT , (8)
IT − n ω2 +k2+Π(iω ,k) n
Xn Xn n n
3
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8
Figure1: Thesolidlinesshowthetwobranchesofthesolutiontothedispersionrelationfor
kz = 0 (fat—stable, thin—unstable). The dashed lines show the corresponding imaginary
parts,andthedotted lineisthespectral weightofthestablemode.
where the self-energy Π, has a branch cut along the imaginary axis. This
equals the real time expression
dk 1 1
0
Θ =Θ = ̺(k ) +f (k ) ; f (k )= , (9)
IT RT Z 2π 0 (cid:26)2 B 0 (cid:27) B 0 eβ|k0|−1
if we only have poles in ∆ along the real axis. The spectral density is
iε(k )̺(k )=Disc∆(k )=∆(k +iǫ) ∆(k iǫ) ,ǫ 0+ , (10)
0 0 0 0 0
− − − →
andεisthesignfunction. Fromthiswemayderivethatthepartitionfunction
like quantity
T
Ψ = ln β2 ω2 +k2+Π(iω ,k) , (11)
IT −2 n n
Xn (cid:8) (cid:2) (cid:3)(cid:9)
rewritten as an integral over real energies, takes the form
T dk β k
Ψ = 0 | 0| +ln 1 e−β|k0| 2k [(1 Reν)̺
IT 0
−2 Z 2π (cid:26) 2 (cid:16) − (cid:17)(cid:27) | | −
iRe∆Discν] T ln 1 e−βk . (12)
− − −
(cid:0) (cid:1)
Obviously this differs from the correspondning real-time expression, that thus
has to be wrong. It is thus necessary to compute the resummed free energy in
the imaginary time formalism. The contribution from LLL reads
1 gB
lnZ = dk ln β2 ω2 +k2 gB Π (k ,k ) .
βV LLL −(2π)2 Z z n z − − LLL 0 z
Xn (cid:8) (cid:2) (cid:3)(cid:9)
(13)
4
The most infra-redsensitivepartis forstatic Matsubarafrequency iω =0,
n=0
in which case the self-energy reads
Π (k =0,k )=0 . (14)
LLL 0 z
When computingthe free energy,the instabilitythus appearsto be unaffected
by the resummation. The instability may be removed by introducing a non-
vanishing Polyakov loop5. This essentially amounts to replacing ω ω
n n
→ −
φ/β, and φ is determined by minimizing the free energy. Confer the recent
studies of an external magnetic field on the lattice6, that did not show any
signsoftheunstablephase,andthefinalremarksinSection2. Itisthepresent
author’s definite opinion that this subject requires further investigation.
Acknowledgments
The firsttwo thirds ofthis workwasdone incooperationwith PerElmfors,as
reported in Ref.7, to which we refer for further details and references. I would
like to thank the organisers of SEWM ‘98. Financial support was obtained
from the Nordic project“Hot Non-PerturbativeParticlePhysics”;andNorFA
under grant no. 96.15.053-O. The author’s research was funded by STINT
under contract 97/121.
References
1. G. K. Savvidy, Phys. Lett. B71 (133) 1977.
2. N. K. Nielsen and P. Olesen, Nucl. Phys. B144 (376) 1978.
3. J. Ambjørn, N. K. Nielsen and P. Olesen, Nucl. Phys. B152 (75) 1979;
H. B. Nielsen and M. Ninomiya, Nucl. Phys. B156 (1) 1979;
H. B. Nielsen and P. Olesen, Nucl. Phys. B160 (380) 1979;
J. Ambjørn and P. Olesen, Nucl. Phys. B170 [FS1] (1980) 60.
4. T. S. Evans and A. C. Pearson,Phys. Rev. D52 (4652) 1995;
A. C. Pearson, in Khanna, Kobes, Kunstatter and Umezawa (eds),
“Banff/CAPWorkshoponThermalFieldTheory”(WorldScientific,Sin-
gapore, 1994)
5. P. N. Meisinger and M. C. Ogilvie, Phys. Lett. B407 (1997) 297;
M. C. Ogilvie, Nucl.Phys.Proc.Suppl.63 430(1998)..
6. K.Kajantie,M.Laine,J.Peisa,K.RummukainenandM.Shaposhnikov,
preprint hep-lat/9809004,unpublished;
M. Laine, these procedings.
7. P. Elmfors and D. Persson, Nucl. Phys. B538 (1999) 309.
5