Properties of the electroweak vacuum versus the QCD-vacuum in strong magnetic fields

1.8 MB·English

by  Jos Van Doorsselaere

Properties of the electroweak vacuum versus the
QCD-vacuum in strong magnetic fields
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a Jos Van Doorsselaere
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GhentUniversity,Belgium
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E-mail: [email protected]

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In strongly magnetized backgrounds, electrically charged vector bosons can condense. When
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p consideringtheρ–mesonsectorofQCD,thisresultsintheexistenceofasuperconductingstate
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h withAbrikosovvortices.Asanextordereffect,alsotheneutralmesonscondenseandoneexpects
[ theresultinglatticetobeamixofneutralandchargedcondensates, openingapossibilitytothe

1 existenceofasuperfluidproperty. Wewillshowhowthedetailedstructureofthevacuumcanbe
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calculatedandwhichimplicationsithasonthenatureoftheexoticsuperconductingstate.
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InternationalWorkshoponQCDGreen’sFunctions,ConfinementandPhenomenology
5-9September2011
Trento,Italy
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
ElectroweakversusQCD-vacuuminstrongmagneticfields
1. Introduction
Itwasproposedlastyear[1]thatinastrongmagneticfieldthevacuummaybecomeanelectro-
magneticsuperconductor. Forverystrongmagneticfields,thelightestofthechargedvectormesons
are likely to condense in a similar way as the electroweak vector bosons [2]. The ρ–mesons have
apolarizationwiththerightquantumnumberstoseeitsenergyreducedbyanincreasingmagnetic
field, a property that is not shared with the lighter π–mesons. Hence the validity of the restric-
tiontotheρ-mesonsectorandthestabilityofthecondensates,asπ–mesonsarenolongerfavored
energetically.
While it is a hope to observe the effects of such condensates experimentally, the required
static magnetic field is way beyond experimental capabilities. However, a beam collision with
very small impact parameter could create a magnetic field with the required strength [3], about
1016T. Thissettingispromising,butdoessharetheabsenceofelectricalfieldswiththetheoretical
approximation. It is unclear whether the new vacuum could still appear in this setup and if it
could create a detectable signal. If not, the relevance may be in Big Bang scenarios, as the early
universemayhavebeenstronglymagnetized[4]andhavesufferedsomeeffectsoftheexoticstate
this implies, as we will show. From a theoretical point of view there’s also a strong incentive
to study this effect, as the results seem to have a broader application base than the ρ–mesons
only. For example,W-condensation in the electroweak model [2] can be better understood by the
very same –or at least similar– calculations and in particular one finds that the condensates imply
superconductivity [5]. The obtained solution to the equations of motion joins other well known,
non-trivial such solutions like monopoles, cosmic strings and sphalerons, all interesting in their
ownright[6].
Aside from obtaining a better picture of the vacuum configuration and its superconducting
nature,thecalculationspresentedgivesomeinsightinthepotentialsuperfluidstructureofthevac-
uum. Whilenoproofofsuperfluiditywasfound,andinparticularthesuperfluidLondonequations
are not satisfied, we find a substructure in the Abrikosov lattice consisting of ’superfluid’ vortices
[7]. Thesevorticesgiveanon-trivialwindingofthephaseofneutralfields,justlikeinsuperfluids,
butitremainsunclearwhetherourexoticmediumisasuperconducting/superfluidhybridorjusta
very peculiar superconductor. The superconductivity, at least, was confirmed in numerical lattice
calculations[8]andalsohintedfromholographicapproaches[9].
As a last interesting fact, we found the superconductivity to be strongly influenced by the
neutral condensate such that, in the core of the Abrikosov-vortices the London-equation is still
satisfied, unlike the ordinary case where it vanishes at the core. This means in some sense the
superconductivityisstrongerthanintheordinarycase.
In these proceedings we will go into detail on how the Abrikosov lattice can be obtained
starting from nothing but the Lagrangian of the theory. It corresponds roughly to the paper [7],
workdoneincollaborationwithMaximChernodubandHenriVerschelde.
2. Effectivepotentials
To find an effective potential that can give a description of the Abrikosov lattice we will ba-
sically need three steps: we describe the model in the transversal (to the external magnetic field)
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ElectroweakversusQCD-vacuuminstrongmagneticfields
plane,thenmakeasmallfieldapproximationandeventuallysolvethesimplifiedequationsofmo-
tions and write the energy density as a function of one scalar field only. We will repeat the steps
for three interesting cases: the ordinary Ginzburg-Landau model for comparison, the electroweak
theorystudiedbyAmbjornandOlesenandtheρ–mesonsectoroftheQCD-vacuum,describedby
theDSGSmodel[10]. Manyofthecalculationscanbefoundalreadyin[1]andmorerecent[7].
2.1 Ginzburg-Landaumodel
TheLagrangianwestartfromisgivenby
1
L=|D φ|2−V(|φ|2)− F2 ; (2.1)
µ 4 µν
1
V(x)= λ(x−φ2)2; D φ =∂ φ−ieA φ. (2.2)
4 0 µ µ µ
Withtheidentification
O =O +iO ; O¯ =O −iO ; (2.3)
1 2 1 2
foranyoperator,wecanrewritethisasanenergydensity,puttingalllongitudinal((0,3)–directional)
dependencetozeroandloweringall(spacial)indices.
1
E =|D¯φ|2+eF |φ|2+V(|φ|2)+ F2. (2.4)
12 2 12
Heresurfacetermsareneglectedandweused[D,D¯]φ =−2eF φ.
12
Now we are ready to make an approximation near the critical point for a sufficiently high
externalfieldeB=e(cid:104)F (cid:105)= λφ2,wherealltermsintheaboveexpressionbecomepositive.
12 2 0
λ
eF > φ2⇒φ =0 (2.5)
12 2 0
λ
eF (cid:46) φ2⇒D¯φ =O(φ2)∼0 (2.6)
12 2 0
Neglectingthecontributionofthecovariantderivativeinawell-chosengaugeaswellasaperiod-
icity requirement for the lattice, gives an Abrikosov ansatz [11] for test-space of solutions for φ,
thatwillbeusedforavariationalmethod:
D¯φn=∂¯φn− eBz¯φn=0 ⇒ φn=e−4eB|z|2e−2πnLz¯B−4eBz¯2−πn2, (2.7)
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(cid:113)
with L = 2π. It is easily seen that with this form translations in real and imaginary directions
B eB
give(n,N ∈N):
φn(z+iLB,z¯−iLB)=eiz+2z¯φn(z,z¯) (2.8)
iπiz−iz¯
φn(z+NLB,z¯+NLB)=e 2LB φ(n+N)(z,z¯) (2.9)
Neglectingthenindependentgaugetransformation,wefindaperiodicsolution
φ(z,z¯)=∑C φ (z,z¯) (2.10)
n n
n
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ElectroweakversusQCD-vacuuminstrongmagneticfields
ifC =C forsomeNandforalln. FixingN,wenowhaveanN–parameteransatztobeplugged
n+N n
intotheenergyfunctional.
Lastlywedeterminetheappropriateenergyfrom(2.4). Thefirsttermcanalreadybedropped,
as our test function makes it negligible. The equation of motion for the gauge field gives further-
more:
∂(F +e|φ|2)=0 (2.11)
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whichimpliestheargumentisconstantandequaltoitsaveragevalue,thus:
F =B+e(cid:104)|φ2|(cid:105)−e|φ|2. (2.12)
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Thelaststepisstraightforward:
λ e2 λ e2
(cid:104)E(cid:105)=(eB− φ2)(cid:104)|φ|2(cid:105)+ (cid:104)|φ|2(cid:105)2+( − )(cid:104)|φ|4(cid:105)+φ4 (2.13)
2 0 2 4 2 0
Pluggingin(2.10)allowsanumericalminimizationintheN–dimensionalparameterspace.
Interestingly, the third term in the above expression can be rewritten in the vacuum solution
as:
λ e2
β ( − )(cid:104)|φ|2(cid:105)2 (2.14)
A
4 2
withβ theAbrikosov-parameterdeterminingthelatticepacking,unlike(cid:104)|φ|2(cid:105)thatisindependent
A
of it. When β =0 this ordering becomes degenerate, indicating a phase transition. Indeed, this
A
corresponds to the Bogomolnyi limit at which type II superconductivity goes to type I and vice
versa.
Alastpointwecanlookatisanexplicitproofthatwehavesuperconductivity. Thereforewe
needthecovariantformoftheLondon-relation
∂Ji n e2
= s Ei (2.15)
∂t m
with E the electrical field and σ the conductivity, essentially depending on the density of cooper-
pairs. We can determinge equations of motion assuming a very small electrical field in the longi-
tudinaldirection(paralleltotheexternalmagneticfield, butallothercomponentsstilltransversal.
WegetforthecurrentJ =∂νF :
µ νµ
∂ J3−∂3J =2e2|φ|2E3 (2.16)
0 0
2.2 Electroweakvacuum
Nowthatwehavegonethroughthesimplestcaseinmuchdetail,wecanapplythistoamore
complicatedmodel. TheLagrangianoftheelectroweakmodelintheunitarygaugereads:
1 1 g
L=− (Wa )2− X2 +|D φ|2−V(|φ|2), D φ =∂ φ−i Z φ, (2.17)
4 µν 4 µν µ µ µ 2cosθ µ
with the photon field A = cosθX +sinθW3, Z = −sinθX +cosθW3 and W− = (W1+
√ µ µ µ µ µ µ µ µ
iW2)/ 2.
µ
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ElectroweakversusQCD-vacuuminstrongmagneticfields
TakingtheconventionW =W−+iW−,neglectingthecomponentW¯ andgaugingtotheunitary
1 2
gauge,theexpressionfortheenergydensityfortransversalcomponentsonlyis:
1 1 1 g2φ2 g2
E = F2 + |D¯W|2+ ( −eF −gcosθZ )|W|2+ |W|4
2 12 2 2 2 12 12 8
1 g2φ2
+ Z2 + |Z|2+|∂φ|2+V(|φ|2) (2.18)
2 12 4cos2θ
withD¯W =∂¯W+ieA¯W+igcosθZ¯W. Knowingthatintheordinaryvacuumφ givesamasstothe
W, we observe the same compensating effect of the magnetic field in the mass term as we did in
theordinarysuperconductor. Veryimportantishoweverthatnowthesignofthistermisreversed,
adirectconsequenceofthespin-1natureoftheW-field. ItiseasytocalculatethatforpositiveF
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thiseffectisabsentfortheW¯ componentandthatitwillnotcontributeavacuumexpectationvalue.
Leavingitoutwasthusjustified.
Nowwehave
g2
eF < φ2⇒W =0 (2.19)
12 2 0
g2
eF (cid:38) φ2⇒D¯W =O(W2)∼0 (2.20)
12 2 0
WecaninfactassumealltermstobethesamesmallorderO(W4),astheyareinanapproximation
uptoO(W2)allstillvanishing. WithZ∼φ−φ ∼O(W2),wecansolvetheirrespectiveequations
0
ofmotionandtheoneforthephotonfieldandfind:
g 1 g2φ 1 e e
Z=i cosθ∂ |W|2; φ−φ = 0 |W|2; F =B+ |W|2− (cid:104)|W|2(cid:105);
2 ∂¯∂−M2 0 4 ∂¯∂−M2 12 2 2
Z H
(2.21)
withM2= g2φ02 andM2 =λφ2. ThisisverysimilartotheAbeliancase,butwithadifferentsign
Z 2cos2θ H 0
for|W|2,indicatingthemagneticfieldandthesuperconductingcondensateareinfactproportional
toeach-other,insteadoftheusualinversecorrelation. Thispropertyiscalledanti-ferromagnetism
and it indicates that the superconducting state, rather than being destroyed at a sufficiently high
magneticfield,iscreatedatacertaincriticalfieldstrength.
By assuming the very same ansatz (2.10) for W as before, we can find a variational lattice
solutionifwerewritetheenergyfunctionalintermsoftheW-condensateonly. Thisinvolvessome
carefulbookkeepingincoefficients,butisfairlystraightforward:
e2 1 1 g2 (cid:28) (cid:18) 1 1 (cid:19) (cid:29)
(cid:104)E(cid:105)= (cid:104)|W|2(cid:105)2+ (M2 −eB )(cid:104)|W|2(cid:105)+ B2 + M2 |W|2 − |W|2 .
8 2 W ext 2 ext 8 W ∆−M2 ∆−M2
H Z
(2.22)
Clearly,theBogomolnyilimitappearsforM =M ,exactlyaswasalreadyshowninAmbjornand
Z H
Olesen’soriginalwork.
2.3 QCDvacuum
Armedwithinsightfromthepreviousmodels,wecanputthemachinerytoworkincaseofan
effective model for the ρ–meson sector of QCD. As argued in the introduction, this will hold the
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ElectroweakversusQCD-vacuuminstrongmagneticfields
dominatingcontributiontoavacuumcondensateinastrongmagneticfield. TheDSGSLagrangian
isgivenby[10]:
1 1 m2 e
L=− (ρa )2− F2 + ρ(ρa)2+ ρ(0)Fµν. (2.23)
4 µν 4 µν 2 µ 2g µν
s
For consistency, we will assume the same notation for the charged mesons as we did previously
forthevectorbosonswhenrewritingthetheoryinthetransverseplane. Itisgoodtoknowthatthe
conventionintheoriginalpaper[1]isfoundbyρ¯ (cid:55)→2ρ intheformulabelow.
Neglecting the ρ-polarization in favor of the ρ¯ for the same reason as we droppedW¯ previ-
ously, but the other way around due to the conventions in [10, 1]. We obtain for the transversal
degreesoffreedom:
1 1 1(cid:16) (cid:17) g2 1 1 e
E = F2 + |Dρ¯|2+ m2 +g f(0)−eF |ρ¯|2+ s|ρ¯|4+ (f(0))2+ m2|ρ(0)|2− F f(0)
2 12 2 2 ρ s 12 12 8 2 12 2 ρ g 12 12
s
(2.24)
with Dρ = ∂ρ¯ −ieAρ¯ +ig ρ(0)ρ¯ and f(0) = −i(∂¯ρ(0)−∂ρ¯(0)). The similarity with the elec-
s 12 2
troweakmodelisobviousandweproceedinexactlythesameway,makingtheappropriateestimate
forallfieldsassuminge(cid:104)F (cid:105)=eB(cid:38)m2. Theequationsofmotiongiveus
12 ρ
g 1 e e e 1
ρ(0)=−i s∂ |ρ¯|2; F =B+ |ρ¯|2− (cid:104)|ρ¯|2(cid:105)− ∂¯∂ |ρ¯|2; (2.25)
2 ∂¯∂−m2 12 2 2 2 ∂¯∂−m2
0 0
using(cid:104)f(0)(cid:105)=0andm2=m2/(1−e2). Thisleadstediouslyto
12 0 ρ g2
s
(cid:42) (cid:43)
1 e2 1 g2 1
(cid:104)E(cid:105)= B2 + (cid:104)|ρ¯|2(cid:105)2+ (m2 −eB )(cid:104)|ρ¯|2(cid:105)+ sm2 |ρ¯|2 |ρ¯|2 , (2.26)
2 ext 8 2 ρ ext 8 ρ −∂∂¯ +m2
0
wherewereplaceρ¯ bytheAbrikosovansatzillustratedbeforeandproceedtothenumericalmini-
mization.
It must be noted that the coefficient of the last term is much larger than in the electroweak
case,asthere’snocompensatingsecondterminit. Thismakesthealgoritmmuchmorestable,and
isexactlythereasonweobtainedsofaronlyresultsforthiscase,theelectroweakstructureshould
soonfollow[5].
3. Results
The results show an ordering of vortices in a hexagonal structure, which can be obtained in
a 2–parameter Abrikosov ansatz, reducing the energy of the more obvious 1–parameter square
structure only slightly. More parameters do not improve the result. The ρ¯-condensate lattice is
superimposed with a more involved lattice of much smaller neutral vortex-like structures, that we
callsuperfluid-vortices.
Thesuperconductivityismediatedbythechargedcondensate. Whilewenowhaveproofitis
organizedhexagonally,thestructureofthevorticesisnotessentiallydifferentfrompreviousresults
[1,2]. WhatisinterestingistheLondonrelation,whichbecomesmoreinvolvedbyinteractionwith
theneutralcondensate. Onegets:
(cid:16) 1 (cid:17)
∂ J3−∂3J =e2m2 |ρ¯|2 E3 (3.1)
0 0 0 −∂¯∂+m2
0
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ElectroweakversusQCD-vacuuminstrongmagneticfields
Figure1: Structureofthecharged(left)andneutral(right)condensates.
compared to the result for the ordinary superconductor (2.16) we have that the right hand side
coefficient is much more ’smoothed out’, and in particular does not make the superconductivity
vanishatthevortexcore.
Asecondinterestingdiscoveryistheexistenceofthesuperfluidvortices: configurationswith
a non-trivial winding of the complex phase. The essential difference with the superconducting
vorticesisthatthelattertellsomethingaboutthephaseofthechargedfields,whichisinthiscasea
gaugeparameter. Thisallowsforthewindingpropertytocreateatopologicallystablelattice. Ifthe
gauge parameter were not local, we would get a Goldstone boson and see a massless fluctuation,
rather that seeing it eaten by a photon turned massive. This phenomenon is superfluidity, and can
be seen in cold liquid Helium by the lack of both translational and rotational resistivity for low
velocities.
a superfluid vortex on top superfluid vortex
of a superconductor vortexsuperfluid antivortex
Figure 2: Phase of the neutral condensate: left the argument explicitly plotted, right the vorticity of the
branchpoints. Redcircledarebranchpointstobothρ(0)andρ¯.
Thelowerordervorticesinoursuperconductingbackgroundhaveindeedanon-trivialwinding
of a non-gauged phase of the neutral fields, hence we call them superfluid. But while the phase
of ρ(0) is no symmetry of the DSGS Lagrangian, no massless states are present and there is no
topologicalstability. Ratherthesuperfluidvorticesarestabilizedbythesuperconductingvortices,
and we could for example not have an isolated vortex solution. We can see from the plots clearly
thatthesuperfluidvorticescomeinpairs,anddonothaveanetvorticity.
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ElectroweakversusQCD-vacuuminstrongmagneticfields
4. Outlook
Whilewehavenowcomputedthestructureofthemagnetizedvacuaclosetothecriticalmag-
netization, an approximation for larger magnetic fields would be interesting. The equations are
unfortunately much more involved once the condensates get larger. An interesting limit that may
beinreachcouldbethecaseofmagneticfieldgreatlyexceedingallmassparametersinthemodel.
For the electroweak model this means symmetry breaking is no longer caused by the Higgs field
and a new phase appears. For the ρ–mesons no such effect exists, but as the masses become neg-
ligible, we could approach a phase with more symmetry. The hopes would be then to infer the
intermediateregimefrombothnear-criticalsolutions.
Within the near-critical magnetization considered here, the existence of superfluidity is still
an unsolved question. No straightforward evidence seems to follow from the theory, but further
numerical research could shed light on the effect of translations and rotations of the lattice and
possiblytheexistenceofLondon-likeequations.
I thank Daniele Binosi for the kind welcome at the TNT-II workshop and Maxim Chernodub
andHenriVerscheldeforcollaborationonthissubject.
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