Primordial galactic magnetic fields from domain walls at the QCD phase transition

5 Pages·0.13 MB·English

by  Michael McNeil Forbes

Primordial Galactic Magnetic Fields from Domain Walls at the QCD Phase
Transition.
Michael McNeil Forbes and Ariel Zhitnitsky
Department of Physics and Astronomy, University of British Columbia
Vancouver, British Columbia, Canada V6T 1Z1.
We propose a mechanism to generate large-scale magnetic fields with correlation lengths of 100
kpc. DomainwallswithQCD-scaleinternalstructureformandcoalesceobtainingHubble-scalecor-
relations while aligning nucleon spins. Because of strong CP violation, thewalls are ferromagnetic,
whichinduceselectromagneticfieldswithHubblesizecorrelations. TheCPviolation alsoinducesa
maximalhelicity(Chern-Simons)whichsupportsaninversecascade,allowingtheinitialcorrelations
to grow to 100 kpc today. We estimate the generated electromagnetic fields in terms of the QCD
parameters and discuss the effects of the resulting fields.
1
0
0
2 Introduction. The source of cosmic magnetic fields Evolution of Magnetic Fields. As suggestedbyCorn-
with large scale correlations has remained somewhat of wall [3], discussed by Son [4] and confirmed by Field
n
a mystery [1]. There are two possible origins for these andCarroll[5],energyinmagneticfieldscanundergoan
a
J fields: primordial sources and galactic sources. Primor- apparent “inverse cascade” and be transfered from high

4 dial fields are produced in the earlier universe and then frequency modes to low frequency modes, thus increas-
1 evolve and are thought to provide seeds which gravita- ing the overall correlation length of the field faster than
tional dynamos later amplify. Galactic sources would thena¨ıvescalingbytheuniverse’sscaleparameterR(T).
4
produce the fields as well as amplify them. Many mech- Therearetwo importantconditions: turbulence mustbe
v
anisms have been proposed [2–5], however, most fail to supported as indicated by a large Reynolds number Re,
1
5 convincingly generate fields with large enough correla- and magnetic helicity (Abelian Chern-Simons number)
0 tionlengthstomatchtheobservedmicrogaussfieldswith H = A~ ·B~d3x is approximately conserved. The im-
4 ∼ 100 kpc correlations. We present here a mechanism portanRceofhelicitywasoriginallydemonstratedby Pou-
0
which,althoughprobablyrequiringadynamotoproduce quet and collaborators [7]. The mechanism is thus: the
0
0 microgauss fields, generates fields with hundred kilopar- smallscalemodesdissipate,buttheconservationofhelic-
/ sec correlations. We presentthis mechanismasanappli- ity requiresthat the helicity be transferedto largerscale
h
cationofourrecentunderstandingofQCDdomainwalls, modes. Some energyistransferedalongwiththe helicity
p
- which will be described in detail elsewhere [6]. and hence energy is transported from the small to large
p
scale modes. This is the inverse cascade. The reader is
e 1. Sometime near the QCD phase transition,TQCD ≈ referred to [3–5] for a more complete discussion.
h 1 GeV, QCD domain walls form.
: Intheearlyuniverse,Reisverylargeandsupportstur-
v 2. These domain walls rapidly coalesce until there re- bulence. This drops to Re ≈ 1 at the e+e− annihilation
i mains,onaverage,onedomainwallperHubblevol-
X epoch,T ≈100eV[4]. Afterthispoint(andthroughout
ume with Hubble-scale correlations. 0
r the matter dominated phase) we assume that the fields
a 3. Baryons interact with the domain walls and align
are “frozen in” and that the correlation length expands
their spins along the domain walls. as R while the field strength decays as R−2. Note that
4. The magnetic and electric dipole moments of the
theinversecascadeisonlysupportedduringtheradiation
baryons induce helical magnetic fields correlated
dominated phase of the universe.
with the domain wall.
Under the assumption that the field is maximally he-
5. The domain walls decay, leaving a magnetic field. lical, these conditions imply the following relationships
6. As the universe expands, an “inverse cascade” between the initial field B (T ) with initial correlation
rms i
mechanism transfers energy from small to large l(T ) and present fields today (T ≈ 2 × 10−4 eV)
i now
scalemodes,effectivelyincreasingtheresultingcor- B (T ) with correlation l(T ) [4,5]:
rms now now
relation length of the observed large scale fields.
−2 −7/3
T T
0 i
Weshallstartbydiscussingthe“inversecascade”mecha- Brms(Tnow)= Brms(Ti) (1)
(cid:18)T (cid:19) (cid:18)T (cid:19)
nismwhichseemsto be the mostefficientmechanismfor now 0
increasingthe correlationlengthofmagnetic turbulence. T0 Ti 5/3
l(T )= l(T ). (2)
Afterpresentingsomeestimatestoshowthatthismecha- now (cid:18)T (cid:19)(cid:18)T (cid:19) i
now 0
nismcanindeedgeneratefieldsoftheobservedscales,we
shall discuss the domain wall mechanism for generating As pointed out in [4], the only way to generate turbu-
the initial fields. lenceiseitherbyaphasetransitionTi orbygravitational
1
2
instabilities. We consider the former source. As we shall it couples to the anomaly and is the dominant player in
show, our mechanism generates Hubble size correlations aligning the magnetic fields. The potential
l at a phase transition T . In the radiation dominated
eipoch, the Hubble size sciales as Ti−2. Combining this V = 1Tr MUeia˜+h.c. −Ecos iln(det(U)) (5)
with (2), we see that lnow ∝ Ti−1/3; thus, the earlier the 2 (cid:0) (cid:1) (cid:18) Nc (cid:19)
phase transition, the smaller the possible correlations.
was first introduced in [9]. It should be realized that
The last phase transition is the QCD transition, Ti = iln(det(U)) ≡ iln(det(U))+2πn is a multivalued func-
T ≈ 0.2 GeV with Hubble size l(T ) ≈ 30 km.
QCD QCD tion and we must choose the minimum valued branch.
We calculate (9) the initial magnetic field strength to be
Detailsaboutthispotentialarediscussedin[6,9]butsev-
B (T ) ≈ eΛ2 /(ξΛ ) ≈ (1017G)/(ξΛ ) where
rms i QCD QCD QCD eral points will be made here. All dimensionful parame-
ξ is a correlation length that depends on the dynamics
ters are expressed in terms of the QCD chiral and gluon
of the system as discussed below and Λ ≈ 0.2 GeV.
QCD vacuum condensates, and are well known numerically:
With these estimates, we see that M = −diag(mi|hq¯iqii|) and E = hbα /(32π)G2i. This
q s
10−9G potential correctly reproduces the Veneziano-Witten ef-
Brms ∼ ξΛ , l ∼100 kpc (3) fective chiral Lagrangianin the largeNc limit [10]; it re-
QCD producestheanomalousconformalandchiralWardiden-
today. One could consider the electroweak transition tities of QCD; and it reproduces the known dependence
which might produce 100 pc correlations today, but in θ for small angles [10]. We should also remark that
this presupposes a mechanism for generating fields with the qualitative results do not depend on the exact form
Hubble-scale correlations. Such a mechanism does not of the potential: domain walls form naturally because of
appear to be possible in the Standard Model. Instead, the discrete nature of the symmetries [8,11,12].
thefieldsproducedarecorrelatedatthescaleT−1 which The result is that two different types of axion domain
i
can produce only ∼1 km correlations today. walls form [6]. One is almost identical to the one dis-
′
Thesearecrudeestimates,andgalacticdynamoslikely cussed in [8] with small corrections due to the η . We
amplify these fields. The important point is that we can shallcallthis the axion/pion(aπ) domainwall. The sec-
generate easily the 100 kpc correlations observed today ondtype,whichweshallcalltheaxion/eta’(aη′)domain
provided that the fields were initially of Hubble size cor- wall is a new solution characterized by a transition in
′
relation. Unless another mechanism for amplifying the both the axion and η fields. The boundary conditions
′
correlations of magnetic fields is discovered, we suggest (vacuum states) for this wall are a˜(−∞) = η˜(−∞) = 0
that, in order to obtain microgauss fields with 100 kpc and a˜(∞) = η˜′(∞) = ±π with π0 = 0 at both bound-
correlationlengths, helicalfields must be generatedwith aries. The main difference between the structures of the
Hubble-scale correlations near or slightly after the QCD two walls is that, whereas the aπ domain wall has struc-
tphheasreeletrvaannscietioofnthTeQCQDC.DTshcealesafmoretchoisncplruosbiolenmrewgaasrdailnsog tauηr′ehaosnlayscoanlethoefmhu−η′g1e∼scΛa−QleC1oDf.mT−ah1e,rethaseoηn′istrtahnasti,tiionnthine
reachedin [4,5]. The rest of this work presents a mecha- presenceofthenon-zeroaxion(θ)field,thepionbecomes
′
nismthat canprovidethe desiredHubble size fields,jus- effectively massless due to its Goldstone nature. The η
tifyingtheestimate(3). Weshallexplainthemechanism is not sensitive to θ and so its mass never becomes zero.
and give simple estimates here. See [6] for details. ItiscrucialthatthewallshaveastructureofscaleΛ−QC1D:
Magnetic field generation mechanism. The key play- there is no way for the aπ wall to trap nucleons because
ers in our mechanism are domain walls formed at the of the huge difference in scales but the aη′ wall has ex-
QCD phase transition that possess an internal structure actlythisstructureandcanthereforeefficientlyalignthe
of QCD scale. We shall presenta full exposition ofthese nucleons.
walls in [6] but to be concrete, we shall discuss an axion The model we propose is this: Immediately after the
wall similar to that described by Huang and Sikivie [8]. phase transition, the universe is filled with domain walls
We start with a similar effective Lagrangian to that of scale TQ−C1D. As the temperature drops, these domain
usedbyHuangandSikivieexceptweincludedtheeffects walls coalesce, resulting in an average of one domain
of the η′ singlet field which they neglected: wall per Hubble volume with Hubble-scale correlations
[11,13]. Itistheseaη′ domainwallswhichalignthedipole
L = fa2 ∂ eia˜ 2+ fπ2Tr|∂ U|2−V(U,a˜) (4) moments of the nucleons producing the seed fields.
eff 2 µ 4 µ The following steps are crucial for this phenomenon:
(cid:12) (cid:12)
(cid:12) (cid:12) 1) The coalescing of QCD domain wall gives the fields
where a˜ =fa−1a is the dimensionless axion field and the πf, η′ Hubble-scale correlations. 2) These fields interact
matrix U = exp(iη˜′ +iπ˜fλf) contains the pion and η′
withthe nucleonsproducingHubble-scale correlationsof
fields (to simplify the calculations, we consider only the
nucleonspins residing in the vicinity ofthe domain wall.
′
SU(2) flavor group). Although the η field is not light,
(The spins align perpendicular to the wall surface.) 3)
3
Finally, the nucleons, which carry electric and magnetic know that the net magnetic field is proportional to ξ−1
moments (due to strong CP violation), induce Hubble- since the dipole fields tend to cancel, thus for a flat sec-
scale correlated magnetic and electric fields. 4) These tion of our domain wall, the field would be suppressed
magnetic and electric fields eventually induce a nonzero by a factor of (ξΛ )−1. For a perfectly flat, infinite
QCD
helicity which has the same correlation. This helicity domain wall (ξ → ∞), there would be no net field as
enables the inverse cascade. pointed out in [17]. However, our domain walls are far
Quantitative Estimates. As outlined below, we have from flat. Indeed, they have many wiggles and high fre-
estimated the strengths of the induced fields in terms quencymodes,thus,thesizeoftheflatregionswherethe
of the QCD parameters [6]. We consider two types of fieldsaresuppressedisgovernedbyacorrelationξ which
interactions. First, the nucleons align with the domain describes the curvature of the wall. Thus, the average
wall. Hereweassumethatthefluctuationsinthenucleon electricandmagneticfieldsproducedbythedomainwall
fieldΨarerapidandthattheseeffectscancelleavingthe are of the order
classical domain wall background unaltered. Thus, we
1
are able to estimate many mean values correlated on a hFµνi≃ dΨhΨ¯σµνγ5Ψi+µΨhΨ¯iσµνΨi (8)
ξΛ
large scale on the domain walls such as hΨ¯γ σ Ψi and QCD (cid:2) (cid:3)
5 xy
hΨ¯γ γ ΨithroughtheinteractionΨ¯(i6∂−m eiη˜′(z)γ5)Ψ. where ξ is an effective correlation length related to the
z 5 N
To estimate the magnetization of the domain wall, we size of the dominant high frequency modes.
maketheapproximationthatthewallisflatcomparedto To estimate what effective scale ξ has, however, re-
the lengths scalesofthe nucleoninteractions. Byassum- quires an understanding of the dynamics of the domain
ing that momentum is conserved in the wall, we reduce walls. Initially, the domain walls are correlated with
our problem to an effective 1+1 dimensional theory (in a scale of Λ−1 . As the temperature cools, the walls
QCD
z and t) which allowsus to compute easily various mean smoothoutandthelowerboundξ−(t)forthescaleofthe
value using a bosonization trick [14,15]. The result for wallscorrelationsincreasesfromξ−(0)≃Λ−1 . Thisin-
QCD
the mean value hΨ¯γ σ Ψi for example is [6]: crease is a dynamical feature, however,and is thus slow.
5 xy
In addition, the walls coalesce and become correlated
µ
hΨ¯σ γ Ψi≃ Λ2 , (6) on the Hubble-scale generating large scale correlations.
xy 5 π QCD
Thusthewallhascorrelationsfromξ−(t)uptotheupper
where µ ≃ mN is a dimensional parameter originating limit set by the Hubble-scale. We expect that ξ ≪ Hub-
fromthebosonizationprocedureofthecorresponding2D ble size at the time that the fields are aligned and that
systemandtheparameterΛ2QCD ∼ dkxdky comesfrom the suppression is not nearly as great as implied in [17].
counting the nucleon degeneracy inR the x–y plane of a Notethat,eventhoughtheeffectsareconfinedtothere-
FermigasattemperatureTc ≃ΛQCD. Thesemeanvalues gion close to the wall, the domain walls are moving and
are only nonzero within a distance Λ−1 of the domain twisted so that the effects occur throughout the entire
QCD
wall and are correlated on the same Hubble-scale as the Hubble volume.
domain wall. The picture is thus that fields of strength
From now on we treat the expectation value (6) as
a background classical field correlated on the Hubble- hE i≃hB i∼ 1 e mNΛ2QCD ∼ eΛQCD (9)
z z
scale. Once these sourcesareknown,one couldcalculate ξΛ m π ξπ
QCD N
the generatedelectromagnetic field by solving Maxwell’s
aregeneratedwithshortcorrelationsξ,butthendomains
equations with the interaction
arecorrelatedonalargescalebytheHubble-scalemodes
1 of the coalescing domain walls. Thus, strong turbulence
L = (d Ψ¯σ γ Ψ+µ Ψ¯iσ Ψ)F +Ψ¯(iD)2Ψ (7)
int 2 Ψ µν 5 Ψ µν µν is generatedwith correlationsthatrun fromΛ up to
QCD
the Hubble-scale.
where d (µ ) is effective electric (magnetic)dipole mo-
Ψ Ψ Finally, we note that this turbulence should be highly
ments of the field Ψ. Due to the CP violation (nonzero
helical. This helicity arises from the fact that both elec-
θ) along the axion domain wall, the anomalous nucleon
tricandmagneticfieldsarecorrelatedtogetheralongthe
dipolemomentin(7)d ∼µ ∼ e isalsononzero[16].
Ψ Ψ mN entire domain wall, hE~i ∼ hA~i/τ where hA~i is the vec-
This is an important point: if no anomalous moments
torpotentialandτ isarelevanttimescaleforthe electric
wereinduced, thenonly chargedparticlescouldgenerate field to be screened (we expect τ ∼ Λ−1 as we discuss
the magnetic field: the walls would be diamagnetic not QCD
below). The magnetic helicity density is thus
ferromagnetic as arguedin [17] and Landau levels would
exactly cancel the field generated by the dipoles. e2 Λ2
Solving the complete set of Maxwell’s equations, how- h∼A~ ·B~ ∼τhEzihBzi∼τπ2 QξC2D. (10)
ever, is extremely difficult. Instead, we use simple di-
mensional arguments. For a small planar region of area Notecarefullywhathappenshere: Thetotalhelicitywas
ξ2 filled with aligned dipoles with constant density, we zerointhequark-gluon-plasmaphaseandremainszeroin
4
thewholeuniverse,butthehelicityisseparatedsothatin anomalous dipole moment induced by the CP violating
one Hubble volume, the helicity has the same sign. The domain walls. The nucleons thus make the wall ferro-
reason for this is that, as the domain walls coalesce, ini- magnetic, not diamagnetic as discussed in [17]. Second,
tialperturbations cause either a solitonoranantisoliton the interaction between the domain walls and nucleons
′
todominateandfilloneHubblevolume. Intheneighbor- are substantial because of the QCD scale of the η tran-
ing volume, there will be other solitons and antisolitons sition. There is no way that axion domain walls with
so that there is an equal number of both, but they are scales ∼ m−1 can efficiently align nucleons at a temper-
a
spatially separated which prevents them from annihilat- ature T .
QCD
ing. Thisissimilartohowaparticleandantiparticlemay We should also note that the magnitudes of the fields
be createdandthen separatedso they do not annihilate. generated by this mechanism are small enough to sat-
In any case, the helicity is a pseudoscalar and thus has isfy the constraints placed by nucleosynthesis and CMB
the same sign along the domain wall: The entire Hubble distortions. Thus, domain walls at the QCD phase tran-
volumehashelicityofthesamesign. Thisistheoriginof sition, in particular those described in [6], provide a nice
the Hubble-scale correlations in the helicity and in B2. methodofgeneratingmagneticfields on100kpc correla-
Thecorrelationparameterξ whichaffectsthemagnitude tions today (3).
of the fields plays no role in disturbing this correlation. This work was supported by the NSERC of Canada.
Eventually,theelectricfieldwillbescreened. Thetime WewouldliketothankR.Brandenbergerformanyuseful
scale for this is set by the plasma frequency for the elec- discussions. AZ wishes to thank: M. Shaposhnikov and
trons(protonswillscreenmuchmoreslowly)ω ∼Λ . I. Tkachev for valuable discussions which motivated this
p QCD
The nucleons, however, also align on a similar timescale study; Larry McLerran and D. Son for discussions on
Λ−1 , and the helicity is generated on this scale too, Silk damping; and M. Voloshin and A. Vainshtein for
QCD
so the electric screening will not qualitatively affect the discussions on the magnetic properties of domain walls.
mechanism. Finally,wenotethattheturbulencerequires
a seed which remains in a local region for a timescale
set by the conductivity σ ∼ cT/e2 ∼ Λ where for
QCD
T = 100 MeV, c ≈ 0.07 [18] and is smaller for higher
T. Thus, even if the domain walls move at the speed of
light (due to vibrations), there is still time to generate
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