Theory of Magnetic Seed-Field Theory of Magnetic Seed-Field Generation during the Cosmological First-Order Electroweak Phase Transition

0.94 MB·English

by  Trevor Stevens

Theory of Magnetic Seed-Field Generation during the Cosmological
First-Order Electroweak Phase Transition
0
1 Trevor Stevens
0
Department of Physics, West Virginia Wesleyan College, Buckhannon, West Virginia 26201
2
Mikkel B. Johnson

n Los Alamos National Laboratory, Los Alamos, New Mexico 87545
a
J
We present a theory of the generation of magnetic seed fields in bubble collisions during a first-

0 orderelectroweak phasetransition (EWPT) possible forsomechoicesofparametersintheminimal
2 supersymmetric Standard Model. The theory extends earlier work and is formulated to assess the

importanceofsurfacedynamicsinsuchcollisions. Weareledtolinearizedequationsofmotionwith
]
O O(3)symmetryappropriateforexaminingcollisionsinwhichtheHiggsfieldisrelativelyunperturbed
from its mean value in the collision volume. Coherent evolution of the charged W fields within
C
the bubbles is the main source of the em current for generating the seed fields, with fermions
. also contributing through the conductivity terms. We present numerical simulations within this
h
formulation to quantify the role of the surface of the colliding bubbles, particularly the thickness
p
of the surface, and to show how conclusions drawn from earlier work are modified. The main
-
o sensitivityarisessuchthatthesteeperthebubblesurfacethemoreenhancedtheseedfieldsbecome.
r Consequently,the magnetic seed fields may be several times larger and smoother over the collision
t
s volumethanfoundinearlierstudies. Ourworkthusprovidesadditionalsupporttothesupposition
a that magnetic fields produced during the EWPT in the early universe seed the galactic and extra-
[
galactic magnetic fieldsobserved today.

1
v PACSnumbers: 98.62.En,98.80.Cq,12.60.-i
4
9
6 and for certain minimal extensions of the Standard
3 Keywords: Cosmology; Electroweak Phase Transi- Modeltherecanbeafirst-orderphasetransition[4–
. 6]. Limitsonparameter-spaceoftheminimalexten-
1 tion; Bubble Nucleation
sion of the standard model (MSSM) placed by elec-
0
0 tric dipole moment measurements and dark matter
1 searchesallow a first-orderEWPT which couldlead
: I. INTRODUCTION to successful electroweak baryogenesis [7] and the
v
possibility that we are exploring, namely that seed
i
X Identifying the source of the observed large-scale fields responsible for the large-scale magnetic fields
r galactic and extra-galactic magnetic fields remains seentodayarecreatedduringthe eraofthe EWPT.
a
an unresolved problem of astrophysics [1]. One of Interest in these issues has led to quantitative
the interestingpossiblesourcesiscosmologicalmag- studies of EWPT magnetogenesis based on the so-
netogenesis, where the seed fields would have arisen lution of equations of motion (EOM) derived from
duringoneoftheearly-universephasetransitions. In specific models. In the Abelian Higgs model, the
this work our interest is seed field production dur- first-order phase transition developed as the Uni-
ing the electroweak phase transition (EWPT) dur- verse condensed into bubbles consisting of localized
ingwhichtheHiggsandtheotherparticlesacquired regions of space filled by Higgs field in a broken
their masses. phase. This model was one of the earliest attempts
If magnetogenesis occurred during the EWPT, it to describeseedfield productionduring the EWPT.
most likely required a first-order phase transition. The EOM related the seed fields to gradients in the
A first-order phase transition proceeds by a process phase of the Higgs field that were produced when
in which bubbles of matter in the broken phase nu- bubbles merged following nucleation [8–10]. Simple
cleate within the unbroken phase, similar to the fa- andtransparentsolutions to the EOMevolvedfrom
miliar process of steam condensing to water as the specific field configurations applied at the point of
water-vapormixtureiscooled. Althoughitisgener- collision in a relativistic O(1,2) symmetric model.
allybelievedthattherecanbe nofirst-orderEWPT The production mechanism of seed fields within
in the Standard Model [2], there has been a great an EOM approach has been pursued more recently
deal of activity in supersymmetric extensions [3], within the framework of the MSSM [11, 12] along
2
the lines of the Abelian Higgs model in Ref. [8–10] Out theory is applied in a specific model along
in which O(1,2) symmetric solutions evolved from the lines of Ref. [12, 14] in Sect. IV. Because in our
specific field configurations applied at the point of presentformulationboundaryconditionsareapplied
collision. In this work,new EOMwere derivedfrom beforethecollisionoccurs,weareabletoexaminein
the MSSMLagrangian,andaccordinglythebubbles addition to collisions, the nucleation process, which
developed a coherent mode of charged W± fields. is the evolution of the bubbles before the collision
As the bubbles merged, the chargedgauge fields re- takes place. The numerical solutions of nucleation
placed gradients in the phase of the Higgs field as andcollisionsinthismodelarethenpresentedinthe
the source of the electromagnetic em currents pro- following two sections.
ducing the magnetic seed fields, and the mechanism In Sect. V we examine the W fields, the current
by whichthis occurredwasdevelopedindetail. Nu- produced in collisions, and the magnetic field. For
mericalresults[12]showedthattheMSSMproduced collisions,findthemagneticfieldtobelargerinboth
seed fields of a size similar to the Abelian Higgs scaleandmagnitudecomparedtoourearlierO(1,2)
model even though the source of the current in the results [12]. In Sect. VI we estimate the sensitivity
two approaches was quite different. of the seed fields to the steepness of the surface of
Althoughtheseearlierstudiesgaveinsightintothe thescalarfield,andinSect.VIIthesensitivitytothe
production mechanism of seed fields in a first-order bubble wall speed and conductivity of the medium.
EWPT, applying boundary conditions at the time ComparedtoresultsgiveninRefs.[14],wefindthat
of the collision as implemented in the O(1,2) for- the magnetic seed fields are not only larger in mag-
mulations did not permit an assessment of the role nitude but extend over substantially larger spatial
of the dynamics of the bubble surface in the seed scales than the results shown there.
field generation. To explore the role of the surface
it is necessary to specify the values of the W fields
II. MAGNETIC FIELD CREATION
and their time derivatives on a surface (t = t ,r,z)
0
DURING A FIRST-ORDER EWPT IN THE
before the collision occurs. Thus, incorporating the
MSSM
initial stages of evolution of individual bubbles on
the collision is an aspect of physics absent from the
treatments found in Refs. [8–12]. The EOMof this workare basedon the same un-
derlyingMSSMLagrangianasthatofRef.[12]. This
Inourmorerecentstudies[13,14]resultsofcalcu-
Lagrangianisassumedtosupportafirst-orderphase
lations were presented in which bubble surface dy-
transition and is of the form
namics were taken into account. We identified a
source of quantitative sensitivity to the bubble sur- MSSM = 1+ 2
face, and the results presented therein showed that L L L
+ leptonic, quark, and supersymmetric
the magnetic fields produced could be as large as,
and possibly even larger than, those calculated in partner interactions
their absence. Encouraged by these results, in the 1 1
1 = Wi Wiµν B Bµν
present work we develop and extend the EOM for- L −4 µν − 4 µν
mulation within the MSSM upon which this earlier g g′
work was based, along lines identified there. 2 = (i∂µ τ Wµ Bµ)Φ2
L | − 2 · − 2 |
We begin, in Sect. II, by reviewing our EOM ap- V(Φ,T) , (1)
proach. We also discuss briefly the nature of a first- −
orderEWPTandsomeofthe issues associatedwith where T is the temperature and
the dynamics of the bubble surface that our theory
is intended to address. In Sect. III we present our Wµiν = ∂µWνi−∂νWµi −gǫijkWµjWνk
extended theory developed in 3 + 1 dimensions in B = ∂ B ∂ B . (2)
µν µ ν ν µ
a regime where the bubble collisionsmay be consid- −
ered”gentle”[11,12]. ForgentlecollisionstheEOM Here Wi, with i = (1,2), are the W+,W− fields,
linearizeanddisplayatransparentconnectiontothe Φ is the Higgs field, and τi is the SU(2) generator.
earlier work of Refs. [8–10, 12]. Various theoretical Fermions are not explicitly considered since earlier
considerationsnecessaryforassessingtheroleofthe work to which we want to compare likewise ignored
bubble surface in magnetic field generation, includ- them. Becauseweareworkingwithintheframework
ingtheimportanceofestablishingappropriateinitial of the MSSM, the bubbles that form as the phase
conditions, are developed. transitionprogressesnaturally consist of a regionof
3
space filed by the Higgs field along with a cloud of ing the phase transition. Although oversimplifiedin
other constituents of the MSSM Lagrangian in the that it lacks medium effects, which can lead to an
broken phase. asymptotic wall speed v < 1 depending on the
wall
As in Ref. [12], we first derive“exact”EOM using pressure difference in the true and false vacuum, it
an effective Lagrangian at the classical level from is seminal in that it was one of the earliest EOM
which the supersymmetric partners have been pro- descriptions of the physics of a first order phase
jected out as explicit degrees of freedom, but whose transition. In his model, prior to nucleation, the
effectisretainedbyarenormalizationoftheeffective dynamics of bubbles is formulated in the Euclidean
potentialtomaintainthepropertiesofthefirst-order metric, which is O(4) symmetric. After nucleation,
phase transition. These EOM are complicated non- thebubbleexpandsinO(1,3)-symmetricMinkowski
linear partial differential equations coupling the W, space-time[16]. Asthephasetransitiondevelopsthe
B,andΦfields. Fromtheirsolutiononemayobtain bubblesstarttomergeor”collide”. Eventuallythey
em
the physical Z and A fields, completely merge, at which point the phase tran-
sition is completed. In subsequent work magnetic
1
Aem = (g′W3+gB ) fieldshavebeenunderstoodtobecreatedinthecol-
µ g2+g′2 µ µ
lisions of bubbles.
Z = p 1 (gW3 g′B ). (3) Inthe LagrangianofEq.(1),theEWPTisdriven
µ g2+g′2 µ − µ by nucleation of the scalar Higgs field just as imag-
p ined in Coleman’s model [16]. In this picture, the
In the picture we are developing, the Higgs field
vacuum state of the Universe corresponds to a local
plays a central dynamical role in EW bubble nucle-
minimum in V(Φ,T). The phase transition occurs
ationandcollisions,andthereforethe effective(now
as the temperature T is lowered through the tran-
appropriatelyrenormalized)HiggspotentialV(Φ,T)
sition temperature T when V(Φ,T) develops a de-
c
is an essential element in the theory. Although this
generatesecondminimumatalargervalueof<Φ>
potential is not known at the present time, depend-
separated from this minimum by a barrier. As the
ing as it does on the unknown parameters of the
universe continues to expand and cool, the depth
MSSMaswellasthepropertiesoftheplasmainthe
of the second minimum increases, meaning that the
early Universe at the time of the EWPT, its spe-
Universe can lower its energy by moving from the
cific form is not relevant for the purposes of this
original, now metastable, false vacuum to the lower
paper. We require only that it should produce a
energy true vacuum. Because the two minima are
first-order phase transition, consistent with certain
separated by a barrier, the transition from the false
MSSM extensions including for example those with
to the true vacuum is delayed as the temperature
a light right-handed Stop [15].
continues to drop, a process referred to as super-
The various parameters are discussed in many
cooling. This delay influences bubble characteris-
publications[4]. Forourcalculationsweusethelab-
tics,andafirst-orderphasetransitionisaccordingly
oratory values,
classifiedasweakorstrongdepending onthe degree
g = e/sinθ =0.646 , of supercooling. A comprehensive phenomenologi-
W
′ cal study of the kinetics of cosmological first-order
g = gtanθ =0.343 ,
W phase transitions, such as the EWPT, in terms of
m = 80.4 GeV , (4)
W such an effective potential is given for example in
where m is the mass of the W± bosons, and we Ref. [17].
W
define
G gg′/ g2+g′2 =0.303 . (5)
≡ B. Bubble dynamics and magnetic field
p
In this section and throughout the paper units are creation with O(1,2) Symmetry
such that ~ = c = 1, with distance and time ex-
pressed in units of m . IntheiranalysisoftheAbelianHiggsmodel,Kib-
W
bleandVilenkin[8]obtainedmagneticfieldsasbub-
bles merge in a regime of gentle collisions. They
A. First-order electroweak phase transition obtained EOM in this case by making an expan-
sion about point ρ(x) = ρ (the “Kibble-Vilenkin
0
Coleman’s model [16] provides a conceptual point”). From these EOM, expressed in terms of
framework based on a Lagrangian for understand- the variables (z,τ = √t2 r2), where t is the time
−
4
andr = x2+y2isthedistanceofapointfromthe C. Effects of the medium: bubble surface
z-axis,thpeyobtainedO(1,2)symmetricsolutionsus- motion and conductivity
ingjumpboundaryconditionsappliedatthetimeof
collision. They demonstrated that when the phase Because effects associated with the surface break
of the Higgs fields is initially different within each O(1,2) symmetry, a theory formulated with this
bubble an axial magnetic field forms as the bubbles symmetry and the associated (z,τ) variables such
merge and that this field has the structure of an as that of Ref. [12] is unsuitable for exploring the
expanding ring encircling the overlap region of the consequences of surface dynamics. To explore such
colliding bubbles. effects, the theory has to be formulated in 3+1 di-
In our earlier work in the MSSM [12], EOM were mensions, and the appropriate symmetry group is
obtainedbymakinganexpansionaboutthe Kibble- O(3).
Vilenkin point. First, ρ(x) was expressed as AdditionaldrawbacksofO(1,2)symmetryinclude
the restriction that v = c and difficulty includ-
wall
ing electrical conductivity in Maxwell’s equations.
ρ(x)=ρ +aδρ(x) , (6)
0 Values of v < 1 have been obtained in diverse
wall
studiesincludingmodelingcollisionsofbubbleswith
with ρ the magnitude of the mean scalar field at constituents of the plasma [18] and solving EOM
0
the center of a single bubble and aδρ fluctuations including nonlinear terms based on an MSSM La-
ofthemagnitudeinthescalarfieldoncethebubbles grangian[19].
merged. MakinganexpansioninaasintheAbelian A value v = 1 and finite conductivity both
wall
6
Higgsmodelweobtainedlinearizedequationswithin directly affect the seed field. This has been dis-
the bubble overlap region. Collisions in which the cussed comprehensively and their effect estimated
Higgs field is relatively unperturbed from its mean in Refs. [8–10] within the context of the Abelian
valuewhenthebubblesmergeweretermed“gentle.” Higgs model. It was found there that finite con-
ductivity would lead to the decay of the currents
Then, assuming as in the Abelian Higgs model
(and therefore the magnetic field) with a character-
that the collision begins at time t = t (called t
0 c
istic time t σ/m [8] with m the gauge boson
in Rf. [12]), when the bubbles first touch at z = d
≈
mass. An additional consequence of the large con-
0, we used jump boundary conditions to determine
the charged W± fields in O(1,2) symmetry and the ductivity arisesas follows. Since the magnetic fields
propagatewiththespeedoflight,forslowlyexpand-
magneticfield. Theseboundaryconditionsrecognize
ing bubbles these fields would very quickly escape
that in some collisions the sign of the z-components
of the W+ field at leading order in a is opposite from the region of the bubble collision in the ab-
senceofconductivity. However,because ofthe large
in the two colliding bubbles while the z-components
for W− has the same sign, and that for others the conductivity the magnetic fields become ”frozen”or
phasesarethesameforW± inthetwobubbles. For confined to the region of the bubbles, hindering the
escape of magnetic flux into the surrounding false
the first case, referred to with a superscript I, the
vacuum. Kibble and Vilenkin showed that the loss
boundary condition was
offluxisnegligibleprovidedthatσR v >>1,where
c
R is the bubble radius at collision time. With val-
wzI(τ =t0,z) = w ǫ(z) uecs of conductivity that are believed to characterize
∂ the plasma, currents and magnetic fields persist on
wzI(τ =t ,z) = 0 , (7)
∂τ 0 time scales that are long compared to those of the
symmetry breaking scale.
where ǫ(z)is the signof z,and for the second,iden-
tified with a superscript II,
III. EQUATIONS OF MOTION IN THE
MSSM WITH O(3) SYMMETRY
wzII(τ =t ,z) = w
0
∂
wzII(τ =t ,z) = 0 . (8) In this section we develop a general framework
0
∂τ in O(3) symmetry extending our earlier work in
Refs. [12, 14] based on the MSSM. Our formulation
ComparingtotheAbelianHiggsmodelwefoundthe is intended to be capable of following the evolution
two magnetic fields to be of similar size. of bubbles with a given wall speed v in 3 + 1
wall
5
dimensions, starting at time of nucleation, and de- region the size of a and hence the accuracy of the
termining the magnetic field generated in collisions expansion will depend on how well the mean field
including effects of finite conductivity. ρ¯(x)thatappearsinthe EOMapproximatesthe ex-
We begin the development of our theory in act scalar field for the colliding bubbles. We come
Sect.IIIAbyextendingtheconceptofagentlecolli- back to this issue below.
siontothecasewherethesurfaceisexplicitlyconsid-
ered. Wethenderive,inSect.IIIB,EOMbymaking
an expansionof the scalarfield for a pair of bubbles B. EOM in electroweak theory for gentle
aboutthemeanscalarfield. Theexpansion,justified collisions
for gentle collisions, leads to linearized EOM, thus
simplifyingthetheory. InSectIIICwediscusssome The fact that ψ and Wd (for d = (1,2)) enter
ofthenewissuesthatareencounteredinsolvingthe
quadraticallyinthe ρ-equation(Eq.(8)ofRef. [12])
EOM when the bubbles are initially separated. As
placestwoimportantconstraintsonthesequantities:
showninSect.IIID,thesameexpansionleadstoan (1)ψandWdmusthaveanexpansioninoddpowers
expressionforthe emcurrentintermsoftheW and of a1/2, if we require the square of these quantities
to a corresponding Maxwell equation.
be analytic in a; and, (2) expanding this equation
toleadingorderina1/2,wefindthatthetermsψ(0),
w(0)1, and w(0)2 must vanish. This is most easily
A. Gentle collisions in electroweak theory
seen in the Euclidean metric, from the fact that the
square of each enters with the same sign. However,
When the surface is considered, we generalize the same must be true in the Minkowski metric as
Eq. (6) by writing well by analytic continuation. In view of these con-
siderations, ψ and Wd for d = (1,2) have the fol-
ρ(x)=ρ¯(x)+aδρ(x), (9) ν ν
lowing expansion
withρ¯(x)asimplefunctionapproximatingthemean
scalarfield atany point x in the medium in the col- ψν(x)=a1/2ψν(1)+a3/2ψν(3)+ ... (10)
lision. The quantity aδρ(x) is, as above, the change
of the magnitude in the mean scalar field induced
by the collision. In this paper we will obtain, as in Wνd =a1/2wν(1)d+a3/2wν(3)d+ ... . (11)
Ref. [12], linear approximations to the exact non-
linear EOM by expanding them in terms of the pa- It is natural that an expansion in the same param-
rametera appearingin Eq.(9). The resulting EOM eter a1/2 remains appropriate for d = 3. However,
aresimilarto thoseofRef.[12], butbecausewenow thereisnorequirementthattheleadingtermvanish,
have the surface to consider they differ in a num- so we take
berofessentialwaysandrequirethedevelopmentof
completely new techniques to solve. Wν3 =wν(0)3+a1/2wν(1)3+a3/2wν(3)3+ ... . (12)
The justification ofthe expansioninterms of a in
the presentcase arisesas follows. Clearly,whentwo With a 0 we may take ρ¯(x) ρ(x), and the
≈ ≈
colliding bubbles are completely separated ρ¯(x) for B-, Θ-, and W-equations then give, to first order in
these bubbles is, to a very good approximation, the a1/2,
sum of the scalarfields of independent bubbles, and
ρ(x)2
there are no significant fluctuations that need to be [∂2+ (g2+g′2)]ψ(1) =0 (13)
considered (a 0) to the extent that one has con- 2 α
fidence in the ≈choice made for the scalar field in an ∂αψ(1) =0 (14)
α
individual bubble. Additionally, for two completely
interpenetrating bubbles ρ¯(x) within the central re- where now
gionis approximatelythe same asthe scalarfieldat
the center of a single one of the colliding bubbles, ψ(1)(x)=∂ Θ (g2+g′2)1/2Z(1) . (15)
and the justification of the expansion is the same α α − 2 α
there as it was in Ref. [12].
The new issue is to justify the expansion in the Equationsforw(1)dmaybeobtainedbyexpanding
ν
peripheral region when two bubbles first begin to theB-andW-equationsthroughordera1/2. Ford=
merge. The criticalpoint torecognizeis thatinthis 1 or 2 (corresponding to d′ = 2 or 1, respectively),
6
we obtain the pair of equations beenderivedandconsistsofcoupledpartialdifferen-
tial equations for the relevant fields. Strictly speak-
0 = ∂2wν(1)d−∂ν∂·w(1)d+m(x)2wν(1)d ing, the set is non-linear because the solution of
2[∂µ(w(0)3w(1)d′ w(1)d′w(0)3) Eq.(21) is coupled to the W field throughthe mass
− ν µ − ν µ of the W, evident in Eq. (17).
+ (wµ(1)d′∂µwν(0)3−wµ(0)3∂µwν(1)d′) However, because Eq. (21) does not depend on
− (wµ(1)d′∂νw(0)µ3−wµ(0)3∂νw(1)µd′)] wρ(νax,)tthoetchoeupchlianrggoedftWhefimealdgsniwtu±d1e(xo)fitnheEsqcsa.l(a2r0fi,2e1ld)
− 4[(w(0)3)2wν(1)d−w(0)3·w(1)dwν(0)3] , (16) is particularly simple and allowνs ρ¯(x) for a system
of colliding bubbles to be determined once and for
where m(x) is the mass of the W field,
all. Equation(20)istheneffectivelyuncoupledfrom
Eq.(21)andmaybesolveddirectlytoobtainwa(x)
ρ(x)2g2 ν
m(x)2 = . (17) for all x, effectively linearizingthe EOMand result-
2
ing in an enormous simplification. That the EOM
The corresponding equation determining w(1)d for areeffectivelylinearimpliesthatforgentilecollisions
ν the coupling of the W to the Higgs dominates the
d=3 is
self-coupling of the W fields.
∂2w(1)3 ∂ ∂ w(1)3 =ρ(x)2gψ(1) , (18)
ν − ν · ν
which can be solved once the driving term ψ(1)(x) C. Surface effects and W± fields in bubbles
has been independently determined from the solu-
tion of Eqs. (13,14). The considerations for fixing With the solution of Eq. (21) nearly constant at
theboundaryconditionsforw(1)d andw(1)3 aresim- ρ(x) = ρ in the broken phase comprising the in-
ν ν 0
ilar and discussed below. terior region of single or overlapping bubbles, our
This field w(0)3 appearing in Eq. (16) is found to previous work [12] was simplified. However, now
ν
be the solution of that we are considering as well the surface, where
ρ(x) begins dropping to its value in the symmetric
∂2wν(0)3−∂ν∂·w(0)3 =0 . (19) phase ρ(x) = 0 outside we cannot ignore the spa-
tial dependence of the mass as given in Eq. (17).
Because no mass appears in this equation, W occu-
The spatial dependence in the surface not only in-
pying this mode propagateatthe speed of light and
troduces a few technical challenges but also, as we
experienceno interactionwiththe scalarfieldofthe
will see, new physics with quantitative significance
bubble to lowest order in a, unlike the W described
not present in [12].
by wν(1)d. Because of this, there is no appreciable Oneoftheconsequencesofthespatialdependence
coupling to w(0)3, and Eq. (16) becomes of the W mass can be seen by taking the four-
ν
divergence of Eq. (20). By so doing, we obtain the
0=∂2wνa−∂ν∂·wa+m(x)2wνa . (20) auxiliary condition
We see that for sufficiently gentle collisions, all rele- χa(x)=0 , (22)
vant equations are linear in W.
Simplifying the non-linear ρ equation (Eq. (8) of where
Ref. [12]) using the fact that wa and ψ in leading
ν ν
order go as a1/2, we find that to leading order in a ∂µ(m2wa) wa ∂m2
χa(x) µ =∂ wa+ · . (23)
ρ(x) satisfies the equation ≡ m2 · m2
∂V Equations (22,23) require
0 = ∂2ρ(x)+ρ(x) , (21)
∂ρ2
wa ∂m2
andthe solutionofthis equationis clearlyidentified ∂ wa = · . (24)
· − m2
with ρ¯(x) appearing in Eq.(9). Methods for solving
Eq. (21) are discussed in many places, for example Thus,incontrastto the calculationinRef.[12],itis
Ref. [20]. nolongertrue that∂ wa =0, andas aconsequence
·
In O(3) the complete set of EOM for describing wefindthattheequationsofmotionfortheW fields
bubblecollisionswiththesurfaceconsideredhasnow more complicated.
7
Thephysicsbecomesclearerbyusingtherelation- D. Maxwell’s equations
ship in Eq. (24) to rewrite the EOM in Eq. (20) as
wa ∂m2 WemayfindtheMaxwellequationfortheemfield
0=∂2wνa+∂ν m· 2 +m2wνa . (25) Aeνm(x)bytakingthelinearcombinationoftheW(3)
and B indicated in Eq (3). An expression for the
The solutionto this setof equationsis equivalentto
corresponding em current j (x) consisting of terms
ν
the set in Eq. (20) provided the auxiliary condition quadraticandcubic inthe three fields Wi(x) imme-
Eq. (22) is maintained for all (t,~x).
diately follows [12].
The transformed EOM Eq. (25) reveal that the
The result for j (x) may also be simplified by ex-
spatial dependence of the W± mass provides a panding the Aemνand W fields in powers of a1/2.
perhaps unexpected sensitivity to the bubble sur-
Letting a(n)(x) refer to the terms in the expansion
face. The sensitivity occurs through the term wa em ν em
∂m2/m2, which becomes in fact divergent in the· of Aν (x), we find that to leading order Aν (x) =
a(2)(x)andsatisfiesthe followingMaxwellequation,
limit of an infinitely sharp bubble surface. At this ν
point one cannot rule out significant modifications
∂2a(2) ∂ ∂ a(2)
to results obtained in O(1,2), where the surface is ν − ν ·
ignored. = Gǫab3(w(1)b∂ w(1)a
ν ·
ToseehowtheauxiliaryconditionEq.(22)maybe
w(1)a∂ w(1)µb+2w(1)a ∂w(1)b)
maintained for all (t,~x), note that Eq. (25) requires − µ ν · ν
em
χa(x) to satisfy the Klein-Gordon equation ≡ 4πjν (x) , (29)
∂2χa(x)+m(x)2χa(x)=0. (26) wherewehaveintroducedthecouplingparameterG
By choosing the initial configuration of wa(x), at defined in Eq. (5). From this we learn that the first
non-vanishing contribution to the em current is of
time t=t , to satisfy
0 order a3/2 and that it depends on the components
χa(t0,~x)=0 (27) wν(1)iofthechargedW fields(i=1and2),calculated
and atorder a1/2. Expressingthe currentin terms of G,
we find
∂χa(t ,~x)
0
=0 , (28)
∂t 4πjem(x) = Gǫab3(w(1)b∂ w(1)a
ν ν ·
we assure that χa(x) = 0 for all future times since w(1)a∂ w(1)µb+2w(1)a ∂w(1)b)(3.0)
Eqs. (27,28) are boundary conditions for the trivial − µ ν · ν
solutionχa(t,~x)=0 ofEq.(26). Thus, Eqs.(27,28) It is easy to prove that this current is conserved,
provide constraints on the initial conditions for the
W± fields. ∂ jem(x)=0 (31)
·
To establish the initial conditions requires the
choice of a time t at which the initial values of the using the fact that at the classical level
0
W fields in the bubble arespecified. InRef. [12]the [wa(x),wb(x′)] = 0 and the fact that the W
µ ν
counterpart of t was the point of first contact of fields appearing in Eq. (30) satisfy the EOM,
0
the bubbles. In the current approach t may be in Eq. (20).
0
fact much earlier, in particular it could be as early So far the bubble has been considered to consist
as the time of nucleation t . The choice of initial purely of the scalar Higgs field and the associated
n
conditions is further discussed in the context of a cloudofchargedW gaugebosonscoupledtoit. Ob-
our model in Sect. IVE3 below. taining the contribution of the charged gauge W±
These observations make it natural to distinguish fields, Eq. (30), to the em current for collisions of
two categoriesof initial conditions when the surface such bubbles is one of the important results of this
effects are considered. The first, which we will refer derivation.
to as boundary conditions, consists of the initial W However, fermions also contribute to the em cur-
fieldsthatmaybechosenfreely. Thesecondconsists rentand havea significantimpact onmagnetic seed
of the set determined by Eq. (27,28), which we will fieldproduction. Onecontributionwasdiscussedre-
refer to as the constrained initial conditions. centlyinRef.[19]andestimatedthereforthe nucle-
The definition of χa given in Eq. (23) requires ation phase of the collision. Another, discussed in
m(x)2 > 0 everywhere as it would be in the mean- Sect. IIC, occurs through the conductivity of the
field approximation adopted in Sect. IVD. medium σ. This is one of the most important and
8
best-knowncontributionsandmaybetakenintoac- Higgs couples to the other fields as it does in the
count through its associated current~j (x), MSSM through the Lagrangian of Eq. (1), strong
c
andhighlynon-perturbativecouplingsariseforming
~jc(x)=σE~(x) , (32) a tightly coupled many-body system as the phase
transition develops. Specific consequences of this
where the usual assumption that ~j (x) is propor-
c weresuggestedandembodiedinRef.[12],andthese
tional to the electric field E~ has been made. De- apply as well to the present work.
tailedcalculationsofσintheearlyuniverseareavail- Oneoftheidentifiedconsequencesofthiscoupling
able [21]. is that as the bubble growth occurs, the W± (and
To find Maxwell’s equation for the magnetic field other constituents that we are ignoring here) that
B~, enterthe bubble fromthe plasmagaintheirmassat
theexpenseofthermalenergy,acoolingprocessthat
B~ = ~ A~em , (33)
continues as volume available to the W increases
∇×
withthebubbleexpansion. Anotheristhatthecou-
we multiply Eq. (29) by ǫ ∂ , obtaining
ijk j
pled fields tend to follow the evolution of the Higgs
ǫ ∂ ∂2Aem ǫ ∂ ∂ ∂ Aem fieldcoherently. As they losethermalenergy,the W
ijk j k − ijk j k · passing into the bubble enter a single mode, a solu-
= ǫ ∂ jem . (34)
ijk j k tion to the EOM discussed in the previous section.
Expresing Eq. (33) in components, This mode plays a special role, and it is quite
different from the more familiar incoherent thermal
Bi = ǫijk∂jAekm, (35) modes outside the bubbles. One may think of it as
a coherent state, much like a state of electrons in
we immediately find the desired result,
a superconductor (except that the W are bosons).
∂2B~ = ∂~ ~jem . (36) Themode ofcourseevolvesintime accordingto the
× EOM,andasthebubbleexpandsanddisplacesmore
of the volume of the plasma the occupation of this
IV. MODELING BUBBLE COLLISIONS IN modealsogrows. Thedynamicsdrivingitisclearlya
O(3) SYMMETRY non-equilibrium component of the phase transition,
and it is a basic assumption of our EOM approach
that these coherent fields give rise to the seed fields
We will assess the importance of surface effects
as bubbles merge before thermal equilibrium is re-
by makingnumericalsimulationsthat canbe mean-
established.
ingfully compared to earlier work in Refs. [12, 14].
Thereareofcoursemanysuchfieldconfigurations
Thecommondynamicalframeworkissummarizedin
that satisfy the EOM, and the one that is realized
Sect. IVA, with common geometrical aspects spec-
inagivenbubbleinthephasetransitiondependson
ified in Sect. IVB. The representation of the mean
the overall history of the process just discussed. To
scalar field for two bubbles ρ¯(x) = ρ (x) in this ge-
2
calculate the net magnetic field produced in bubble
ometry, along with the arguments for its choice, is
collisions properly, one would have to evaluate the
given in Sect. IVC. The presence of other bubbles
fieldcorrespondingtoeachpossibleinitialconfigura-
aretakenintoaccountinamean-fieldapproximation
tionandaverageovertheensembleofconfigurations.
inIVD. InitialconditionsontheW fields,including
The neteffectofthis averagingprocedurewasex-
boundaryconditionsandtheconstraintsimposedby
aminedinRef.[12]andfoundto befactorlessthan,
surface geometry, are discussed in Sect. IVE. We
but the order of, unity. Thus, to get a fair estimate
specialize the theory to cylindrical coordinates in
ofthenetmagneticfielditisnotnecessarytoexplore
Sect. IVF. The model is applicable to both weak
the full range of possible initial conditions; rather,
and strong first-order phase transitions and incor-
it is sufficient to examine one initial condition char-
poratessomeimportantmediumeffects absentfrom
acteristic of the entire ensemble of possibilities.
our former work.
A. The dynamical framework B. Bubble collision geometry
The familiar conceptual features of our MSSM For the application of our theory to the collision
theory have been presented in Sect. IIA. When the of two bubbles, we will be assuming, as in Ref. [14],
9
that the bubbles are nucleated simultaneously at where
timet=t atpointsz = z locatedsymmetrically
n 0
about the origin on the z±-axis with nucleation radii R(r,z) RR(r,z)θ(z)+RL(r,z)θ( z) , (39)
≡ −
r . Additionally, here we assume that the bubbles
ns
with
are well separated and non-overlapping before they
collide.
R (r,z)= r2+(z z )2 (40)
The scalar field of a single bubble may be repre- R − 0
p
sented in general as
being the distance fromthe center ofthe right-hand
ρ(t,~x)=ρcfs(t,~x) , (37) bubble at z0 to the point (r,z) on the surface, and
where fs(t,~x) is the shape of the field and ρ is its
c R (r,z)= r2+(z+z )2 (41)
magnitude at the center. The fs(t,~x) that we will L 0
p
use in this paper, essentially equivalent to that in beingthedistancetothesamepointfromthecenter
Ref.[14],isdefinedinAppendixAintermsthebub- of the left-hand bubble at z .
0
blenucleationtimetn,nucleationradiusrns,surface After the bubbles merge−, the right-hand and left-
speed vwall, and surface diffuseness as. The surface handsurfacesintersectonthe x y plane ina circle
diffuseness is approximately half the distance over −
of radius b(t) = R (t)2 z2 that expands at a
which the scalar field falls from its 10% to 90% at q 1/2 − 0
rate
t for large bubbles, i.e. bubbles with r &2a .
n ns s
The bubble nucleated at z =z0 >0 is referred to db(t) R (t)dR (t)
1/2 1/2
as the right-hand (R) bubble and the one nucleated = . (42)
dt b(t) dt
at z = z < 0 as the left-hand (L) bubble. We
0
−
assume axial symmetry so that the relevant spatial Ourmodelofscalarfieldinthe collisionoftwobub-
coordinates(r,z)for apointareits axialcoordinate bles, including the region of coalescence is given in
r = x2+y2, its distance from the z-axis, and its Sect. IVC.
longiptudinal coordinate z, its distance from y z Sincethebubblecollisiongeometryisaxiallysym-
−
plane that passes through the origin at z =0. metric with the relevant spatial coordinates being
Itishelpfultothinkofthecollisionintermsofthe (r,z), where z is the distance of a point from the
evolution of spatial surfaces that separate regions y z plane on the z-axis, and r = x2+y2 being
of the collision occupying true vacuum from those its−distance from the z-axis, cylindrpical coordinates
of false vacuum. The connectedness of the surfaces is the natural coordinate system for expressing re-
change as the collision process proceeds. sults. In this coordinate system, the W fields may
Initially, for times tn < t < tc, where tc is the be taken to have the following form,
collisiontime or time ofmerging of the bubbles, the
boundary consists of two disconnected surfaces, one wa(t,~r,z) = wa(x), ν =0
ν 0
for the left-hand bubble i = L and the other for wa(t,~r,z) = x wa(x), ν =(1,2)
the right-hand bubble i = R. For times t t the ν ν
≥ c wa(t,~r,z) = wa(x), ν =3 , (43)
bubbles coalescetoformaregioni=c witha single ν z
boundary surface, S .
c with wa = wza and wa = w0a. Correspondingly,
TheradiusofthebubblesurfaceR (t)isdefined z − 0
1/2 wewritetheemcurrentincylindricalcoordinatesas
as the distance from the center of the bubble (its
nucleation point) to the point at which the scalar
j (t,~r,z)=(j (t,~r,z),~rj(t,~r,z),j (t,~r,z)) . (44)
ν 0 z
field has fallen to half its central value. It of course
depends on our choice of scalar field, whose details
are given in Appendix A. The collision time may C. Mean scalar field in two colliding bubbles
then be taken to be the time at which the radius of
either bubble becomes equal to the distance of its
An expression for the mean scalar field ρ¯(x) =
nucleation point from the origin of the coordinate
ρ (x)forapairofcollidingbubblesmaybeexpressed
2
system, i.e. when R (t ) = z . The solution of
1/2 c 0 intermsofthenon-overlappingportionofthescalar
thisequationforourscalarfieldisgiveninEq.(A9).
field, ∆ρ(x),
The spatial points forming the boundary surfaces
are then determined by the equation ∆ρ(x) = ρ (x)(1 ρ (x)/ρ )
L R c
−
R(r,z)=R (t) , (38) +ρ (x)(1 ρ (x)/ρ ) , (45)
1/2 R L c
−
10
where the scalar fields in the left-hand and right-
hand bubbles, while still separated, are taken as
ρ (t,r,z) = ρ fs(t,r,z+z )
L c 0
ρ (t,r,z) = ρ fs(t,r,z z ) (46)
R c 0
−
with fs(t,r,z) and ρ given in Eq. (37). 1
Now, if the scalarficelds were to simply addin the €€·€€€€€ 0.75 40
·0 0.5
overlap region (which we know is not the case [12])
0.25 30
the scalar field would be 0
--110000 20
ρL(x)+ρR(x) , (47) --5500 t mW
00 10
which is as large as twice the size of that when they zz mmWW 5500
are well separated. What we have learned from the 110000 0
calculation in Ref [12] is that the scalar field, when
thebubbleshavecompletelymerged,isactuallyhalf
this value, so if we define FIG. 1: Evolution of scalar field in the collision of two
bubbles. Nucleation is at t = 0. Field is plotted as a
1 n
ρ (x)=∆ρ(x)+ (ρ (x)+ρ (x) ∆ρ(x)) ,(48) functionofz forr=0overthetimeinterval0≤t≤40.
2 L R
2 −
then we obtain our ansatz, which is in agreement
with the results of Ref. [14]. D. Medium containing many bubbles
Because there is not much a priori guidance on
how to define ρ¯(x) in the region where the bubbles In a medium undergoing a first-order phase tran-
begin to overlap, the accuracy of the expansion in sition there are many distinct regions S of scalar
i
terms of a appearing in Eq. (9) is accordingly dif- potential, each of which is either a single bubble or
ficult to establish for any given ρ¯(x). However, be- a collection of mutually overlapping bubbles. The
causetheperipheralregionextendsoverasmallvol- net scalar field ρ(x) for this system may thus be ex-
ume relative to that of the two bubbles, it is likely pressed as a sum over the scalar fields ρ(i)(x) for
thatthecorrectionsforanyreasonablechoicewillbe each of the N regions,
relatively insignificant.
Theevolutionofthescalarfieldinacollisionwith N
ρ (x)definedinEq.(48)isshowninFig.1. Thisfig- ρ(x)= ρ(i)(x) . (49)
2
X
ure shows the scalar field on which the calculations i
of Sect. V are based. The individual bubbles nucle- For such a system, the mass of a W± boson at any
ateattimet =0onthez-axiscenteredatz = z ,
n ± 0 pointxcontinuestobegivenbyEq.(17),whereρ(x)
with z =35 and nucleation radius r =20. Their
0 ns is the net scalar potential at the location of the W.
scalarfields takento have wallthickness a =4 and
s In this paper we are interested in the evolutionof
v = 1. Aside from our choice of surface diffuse-
wall at most two bubbles in this sum. But in tracking
ness and some compensating adjustment in r and
ns their evolution, we should not ignore the presence
z (the parameters z and r are a bit larger than
0 0 ns of the other bubbles. We will account for them in
the ones used there so that with the thicker surface
a mean-field approximation by averaging over their
the bubbles do not overlap at t = t ) the parame-
0 locations and representing their collective effect by
ters and collision geometry closely matches that of
an average scalar field ρ ,
Ref. [14]. Because of the larger surface diffuseness, av
this corresponds to a somewhat weaker phase tran- N′
sitionthanthatofRef.[14]andthusprovidesanin- ρ = ρ(i)(x)
av
terestingcontrasttothecalculationpresentedthere. hX i
i
The collision time is found from Eq. (A9) to be N′
t = tc ≈ 11. The bubble radii at this time are 1 d3x ρ(i)(t,~x) , (50)
R (t )=z =35. For times t<t the scalar field ≡ V Z
1/2 c 0 c Xi
is approximately confined to two isolated regions,
the two individual bubbles, and that after this time where V is the total volume over which the integral
to just one region, the collision region. runs and the prime on the sum means we exclude