in strong and eletroweak interations

Tomas Brauner

Nulear Physis Institute

Aademy of Sienes of the Czeh Republi

I amindebted tomy PhD advisor JirHosek for hisguidane through the lastfour years

and for sharing with me his insight into physis during hours and hours of ountless

disussions. I am also grateful to Professor Jir Horejs for his advie during the early

stagesof my PhD study and for hisontinuous support.

IamobligedtoMihaelBuballaforhiskindhospitalityduringmyvisitattheInstitutefor

NulearPhysis,Tehnial University Darmstadt,andforthe amount oftime hedevoted

todisussions with me.

IamthankfultoJirNovotn yforenlighteningdisussions,andtoJirHosekandJirAdam

forthe arefulreadingof themanusriptof thethesis and several improvingremarks and

suggestions. Iwould alsoliketoaknowledgemyfellowgraduatestudentsPetr Benesand

Adam Smetana for the ontinuing ollaboration.

Thepresented workwassupportedby theGACRgrantsNo. 202/02/0847,202/05/H003,

and by the ASCR researh plansNo. K1048102, AV0Z10480505

My last and very personal word of thanks is reserved for

Stepanka: Thank you for all

your love and enouragement, without whih this thesis and my whole life wouldn't be

what they are.

Delaration of originality

This dissertationontains the results of researh ondutedat the Nulear Physis

Insti-tuteof the Aademy ofSienes ofthe Czeh Republiin

Rez, inthe periodbetween fall

2002 and spring 2006. With the exeption of the introdutory Chapter 2, and unless an

expliit referene is given, itis based on the published papers whose opies are attahed

atthe end of the thesis. I delarethat the presented results are original. In some ases,

they have been ahieved inollaborationas indiated by the authorship of the papers.

1 Introdution 4

2 Spontaneous symmetry breaking 6

2.1 General features . . . 6

2.1.1 Realizationof broken symmetry . . . 7

2.1.2 Goldstone theorem . . . 8

2.2 Toy example: Heisenberg ferromagnet . . . 10

2.2.1 Groundstate . . . 10

2.2.2 Goldstone boson . . . 12

2.3 Desription of spontaneous symmetry breaking. . . 14

2.3.1 Linearsigma model . . . 15

2.3.2 Nambu{Jona-Lasiniomodel . . . 16

3 Goldstone boson ounting in nonrelativisti systems 18 3.1 Review of known results . . . 18

3.1.1 Nielsen{Chadha ounting rule . . . 18

3.1.2 Otherpartial results . . . 19

3.2 Linear sigma modelat nite hemialpotential . . . 21

3.2.1 SU(2)U(1)invariant sigma model . . . 21

3.2.2 Linearsigma modelfor SU(3) sextet . . . 23

3.2.3 Generalanalysis. . . 26

4 Dynamial eletroweak symmetry breaking 30 4.1 Toy model: Global Abelian hiral symmetry . . . 31

4.1.1 Ward identities: general . . . 31

4.1.2 Spetrumof salars . . . 33

4.1.3 Ward identities for dynamiallybroken symmetry . . . 34

4.2.2 Phenomenologialonstraints . . . 39

5 Quantum hromodynamis at nonzero density 40 5.1 Single-avorolor superondutor with olor-sextet pairing . . . 41

5.1.1 Kinematisof olor-sextet ondensation. . . 42

5.1.2 Ginzburg{Landaudesription . . . 42

5.1.3 Nambu{Jona-Lasiniomodel . . . 43

5.2 Two-olor QCD:Chiral perturbation theory . . . 45

5.2.1 Symmetry . . . 45

5.2.2 Cosetspae . . . 47

5.2.3 EetiveLagrangian . . . 48

6 Conlusions 50

List of publiations 52

Referenes 53

Introdution

The priniple of spontaneous symmetry breaking underlies muh of our urrent

under-standingof the world aroundus. Although it has been introdued and developed in full

generalityinpartilephysis, itsappliationsalsoovera largepart ofondensed matter

physis, inluding suh fasinating phenomena as superondutivity, superuidity, and

Bose{Einstein ondensation.

Eversinetheverybirthofsiene,philosophers,andlaterphysiists, admiredthebeauty

of the laws of nature, one of their most appealing features always being the symmetry.

Indeed,itwassymmetryonsiderationsthat leadEinsteintothereationofhistheory of

gravity, the general relativity, and it is symmetry that is the basi building blok of the

moderntheoriesofthe otherfundamentalinterations aswellasallattempts toreonile

themwith Einstein's theory.

Symmetry is not only aestheti, it is also pratial. It provides an invaluable guide

to onstruting physial theories and one applied, imposes severe onstraints on their

struture. This philosophy has, in partiular, lead to the development of methods that

allowus to exploit the symmetry ontent of the system even if we atually annot solve

theequationsof motion. The theoryof groupsand theirrepresentationswas rstapplied

in quantum mehanis to the problem of atomi and moleular spetra, and later in

quantumeld theory,startingfromthe quark modelandurrentalgebraand evolvingto

the ontemporarygauge theoriesof strong and eletroweak interations, and the modern

onept of eetive eld theory.

There are many physial systems that, at rst sight, display asymmetri behavior, yet

thereisareasonablehopethattheyaredesribedbysymmetriequationsofmotion. Suh

abeliefmaybebased,forinstane, ontheexisteneofanormal,symmetriphase,likein

the ase of superondutors and superuids. Another nie example was provided by the

historialdevelopment of the standard modelof eletroweak interations. Bythe sixties,

itwasknown thatthe onlyrenormalizablequantumeldtheoriesinludingvetor bosons

were of the Yang{Mills type. It was, however, not lear how to marry the non-Abelian

gaugeinvarianeof the Yang{Millstheory with the requirement enforedby experiment,

that the vetor bosons be massive.

Alltheseissuesareresolvedbytheingeniousoneptofaspontaneouslybrokensymmetry.

Theatualbehaviorofthe physialsystemisdeterminedby thesolutionof theequations

Theinternalbeauty ofthe theoryis thuspreserved and, moreover, oneis abletodesribe

simultaneouslythenormalphaseandthesymmetry-breakingone. Justhoosethesolution

whih isenergetially more favorableunder the speiedexternal onditions.

Thisthesispresentsamodestontributiontothephysisofspontaneoussymmetry

break-ingwithinthestandardframeworkforthestrongandeletroweakinterationsandslightly

beyond. Theoreofthethesis isformedby theresearhpaperswhoseopiesareattahed

atthe end. Throughoutthe text, these artilesare referred toby apital romannumbers

insquarebrakets,whiletheworkofothersisquotedbyarabinumbers. Thealulations

performed in the published papers are not repeated. We merely summarize the results

and provide a guide forreading these artilesand, to some extent,their omplement.

The thesisis a olletionofworks on diverse topis, rangingfromdynamial eletroweak

symmetrybreakingtoolorsuperondutivityofdensequarkmatterandGoldstoneboson

ounting indenserelativistisystems. Ratherthangivinganexhaustivereview ofeahof

them, we try to keep lear the unifying onept of spontaneous symmetry breaking and

emphasizethe similarity of methods used todesribesuh vastly dierent phenomena.

Ofourse,suhatextannot(andisnotaimedto)beself-ontained,andthebibliography

annot over all original literature as well. In most ases, only those soures are quoted

thatwerediretlyusedintheourseofwriting. Forsakeofompletenesswequoteseveral

reviewpaperswheretheoriginalreferenesanalsobefound. Thelessexperienedreader,

e.g. astudentoranon-expertintheeld,isprovidedwithaoupleofreferenestoleture

notes onthe topis overed.

The thesis is organized as follows. The next hapter ontains an introdution to the

physisof spontaneous symmetry breaking. We try to be as generalas possible toover

both relativistiand nonrelativistisystems. Thefollowingthree haptersare devoted to

the three topis investigated during the PhD study. Chapter 3 elaborateson the general

problemofthe ounting ofGoldstone bosons,inpartiularinrelativistisystemsatnite

density. The eletroweak interations are onsidered in Chapter 4 and an alternative

way of dynamial eletroweak symmetry breaking is suggested. Finally,in Chapter 5 we

study densematteronsisting ofquarksof asingleavorand proposea novelmehanism

for quark pairing, leading to an unonventional olor-superonduting phase. After the

summary and onluding remarks, the full list of author's publiations as well as other

referenes are given. The reprints of the researh papers published in peer-reviewed

Spontaneous symmetry breaking

Inthis hapter wereview the basiproperties ofspontaneously broken symmetries. First

we disuss the general features, from both the physial and the mathematial point of

view. To illustrate the rather subtle tehnial issues assoiated with the implementation

ofthe broken symmetryonthe Hilbert spaeof states,asimple exampleisworked out in

some detail{ the Heisenberg ferromagnet.

Afterthegeneralintrodutionweturnourattentiontothemethodsofdesriptionof

spon-taneouslybroken symmetries. We startwith ashortdisussion ofthe model-independent

approahof the eetive eld theory, and then reall two partiular models that we take

upinthefollowinghapters{thelinearsigmamodelandtheNambu{Jona-Lasiniomodel.

An extensive review of the physis of spontaneous symmetry breaking is given in Ref.

[1℄. A pedagogial introdutionwith emphasis onthe eetive-eld-theory desription of

Goldstone bosons may be found in the leturenotes [2, 3,4, 5℄.

2.1 General features

We shall be onerned with spontaneously broken ontinuous internal symmetries, that

one meets in physis most often. The reason for suh a restrition is twofold. First,this

is exatlythe sort of symmetries we shall deal with in the partiular appliations to the

strongandeletroweakinterations. Seond,onthegeneralground,spontaneousbreaking

of disrete symmetries does not give rise to the most interesting existene of Goldstone

bosons, while spaetime symmetriesare more subtle, see Ref. [6℄.

As already noted in the Introdution, a symmetry issaid to be spontaneously broken, if

it is respeted by the dynamial equations of motion (or, equivalently, the ation

fun-tional),but isviolatedby theirpartiular solution. 1

In quantumtheory we use, however,

operatorsand their expetationvalues ratherthan solutions tothe lassial equations of

motion. Sine virtually all information about a quantum system may be obtained with

the knowledge of itsground state, it is only neessary to dene spontaneous breaking of

asymmetry in the groundstate or, the vauum [9℄.

1

2.1.1 Realization of broken symmetry

Consider the group of symmetry transformations generated by the harge Q. If the

symmetry were a true, unbroken one, it would be realizedon the Hilbert spae of states

by aset ofunitary operators. In suhaase, their existeneisguaranteed by the Wigner

theorem[10℄ andwespeakof theWigner{Weylrealization ofthesymmetry. Thevauum

isassumed tobe a disrete, nondegenerate eigenstate of the Hamiltonian. Consequently,

it bears a one-dimensional representation of the symmetry group, and therefore also is

an eigenstate of the harge Q. The exited states are organized into multiplets of the

symmetry,whihmaybehigher-dimensionalprovidedthesymmetrygroupisnon-Abelian.

By this heuristi argument we have arrived at the denition of a spontaneously broken

symmetry: A symmetry is said to be spontaneously broken if the ground state is not an

eigenstate of its generator Q. A very lean physial example is provided by the

ferro-magnet. Below the Curie temperature, the eletron spins align to produe spontaneous

magnetization. While the Hamiltonian of the ferromagnet is invariant under the SU(2)

group of spin rotations (not to be mixed up with spatial rotations { see Setion 2.2 for

moredetails),this alignmentlearlybreaks allrotationsexept thoseabout the diretion

of the magnetization.

Notethatasaneessary onditionforsymmetry breakingitisusual todemandjust that

thegenerator Qdoesnot annihilatethe vauum. Suhariterion, however, doesnot rule

out the possibility that the ground state is an eigenstate of Q with nonzero eigenvalue.

On the other hand,the vauumharge an alwaysbeset to zero by aonvenient shift of

the harge operator.

A distinguishingfeature of broken symmetry is that the vauumis innitely degenerate.

Intheaseoftheferromagnet,thedegenerayorrespondstothehoieofthediretionof

themagnetization. Ingeneral,the groundstatesarelabeledbythevaluesofa

symmetry-breaking order parameter. Formally, the various ground states are onneted by the

broken-symmetrytransformations.

Withthis intuitive piture in mind a natural question arises, whether a physialsystem

atually hooses as its ground state one of those with a denite value of the order

pa-rameter, or their superposition. To nd the answer, we go to nite volume and swith

onaweak externalperturbation(suhas amagneti eld). The degeneray isnowlifted

and there is a unique state with the lowest energy. This mehanism is alled vauum

alignment.

After we perform the innite volume limit and let the perturbation go to zero (in this

order),weobtaintheappropriategroundstate. Inorderforthisargumenttobeonsistent,

however, the resultingset ofphysiallyaeptablevaua shouldnot depend onthe hoie

of perturbation. Indeed, it follows from the general priniples of ausality and luster

deompositionthat thereisabasisinthespaeof stateswiththelowestenergy suhthat

allobservables beomediagonaloperatorsin the innitevolume limit[11℄.

We have thus ome to the onlusion that the orret ground state is one in whih the

order parameter has a denite value. The superpositions of suh states do not survive

the innite volume limit and therefore are not physial. Moreover, transitions between

individualvaua are not possible. This means that rather than being a set of ompeting

spaeofitsown,allbearinginequivalentrepresentationsof thebroken symmetry. Thisis

alledthe Nambu{Goldstonerealization of the symmetry.

To summarize, when a symmetry is spontaneously broken, the vauum is innitely

de-generate. The individual ground states are labeled by the values of an order parameter.

Intheinnitevolumelimitthey giverisetophysiallyinequivalentrepresentationsof the

broken symmetry. Transitionsbetween dierent spaes are only possible upon swithing

on an external perturbation. This lifts the degeneray and by varying it smoothly, one

an adiabatiallyhange the order parameter.

This proedure an again be exemplied on the ase of the ferromagnet. To hange

the diretion of the magnetization, one rst imposes an external magneti eld in the

original diretion of the magnetization. The magneti eld is next rotated, driving the

magnetizationtothe desired diretion, and afterwards swithed o.

The issue of inequivalent realizations of the broken symmetry has rather subtle

mathe-matialonsequenes [1℄, whih we nowshortly disuss and later, in Setion2.2,

demon-strateexpliitlyontheaseof theferromagnet. Asalreadymentioned,theHilbertspaes

with dierent values of the order parameter are onneted by broken-symmetry

trans-formations. The reason why they are alled inequivalent is that these broken-symmetry

transformations are not represented by unitary operators. They merely provide formal

mappingsbetween the various Hilbert spaes. Bythe same token, the generator Qisnot

awell dened operator inthe innitevolume limit. What is well dened isjust its

om-mutatorswith other operators, whih generate innitesimal symmetry transformations.

Sinethe broken symmetry isnot realized by unitary operators,it isalsonot manifested

in the multiplet struture of the spetrum. This is determined by the unbroken part of

the symmetry group. Let us, however, stress the fat that the broken symmetry is by no

meanssimilar toan approximate, but spontaneously unbroken one. Even though itdoes

not generate multiplets in the spetrum, it still yields exat onstraints whih must be

satisedby, e.g., the Green's funtionsof the theory.

2.1.2 Goldstone theorem

Oneofthemost strikingonsequenesofspontaneoussymmetrybreakingistheexistene

of soft modes in the spetrum, ensured by the elebrated Goldstone theorem [12, 13℄.

In its most generalsetting appliable to relativisti as well as nonrelativisti theories, it

an be formulated as follows: If a symmetry is spontaneously broken, there must be an

exitation mode in the spetrum of the theory whose energy vanishes in the limit of zero

momentum. In the ontext of relativisti eld theory this, of ourse, means that the

so-alledGoldstone boson is a masslesspartile.

Several remarks tothe Goldstone theorem are in order. First,in the generalase itdoes

not tell us how many Goldstone modes there are. Anyone who learned eld theory in

the framework of partile physis knows that in Lorentz-invariant theories, the number

of Goldstone bosons is equal to the number of broken-symmetry generators [11℄. In

the nonrelativisti ase, however, the situation is more omplex and there is in fat no

ompletely general ounting rule that would tell us the exat number of the Goldstone

E

(

k

)

k

1

2

3 4

Figure2.1: DispersionrelationsoftheGoldstonebosonsinfourphysiallydistintsystems,

onveniently normalized to have the same slope at the origin. 1. The Goldstone boson

in a relativisti eld theory. 2. The aousti phonon in a solid. 3. The phonon-roton

exitationin thesuperuidhelium. 4. Thephononintherelativistilinearsigmamodelat

nitehemialpotential(seeChapter3).

Seond, there are tehnial assumptionswhih,in some physiallyinteresting ases, may

be avoided, thus invalidating the onlusions of the Goldstone theorem. A suÆient

ondition for the theorem to hold is the ausality whih is inherent in relativisti eld

theories. The nonrelativisti ase is, again,more ompliated. In general,the Goldstone

theorem applies if the potential involved in the problem dereases fast enough towards

the spatial innity. An example in whih this ondition is not satised is provided by

the superondutors where the long-range Coulomb interation lifts the energy of the

low-momentumwould-be Goldstone mode, produinga nonzero gap [14℄.

Third,theGoldstone theoremgivesusinformationabout thelow-momentumbehaviorof

thedispersionrelationoftheGoldstoneboson. Intheabseneofothergaplessexitations,

the long-distane physis is governed by the Goldstone bosons and an be onveniently

desribed by aneetiveeld theory. This does not tellus, however, anything about the

high-energy properties of the Goldstone bosons. At high energy, the dispersion relation

of the Goldstone mode is strongly aeted by the details of the short-distane physis.

It is thus not as simple and universal as the low-energy limit, but at the same time not

uninteresting, as doumented by Fig. 2.1.

Let us now briey reall the proof of the Goldstone theorem. The starting assumption

isthe existene of a onserved urrent, j

(x). From its temporalomponent, the harge

operatorgenerating the symmetry is formed,

Q(t)= Z

d 3

xj 0

(x;t):

The domainof integration is not indiated inthis expression. The harge operator itself

iswelldened only innite volume, but aslong asitsommutatorswithother operators

omposite) operator exists suh that

h0j[Q;℄j0i6=0: (2.1)

Note that this immediatelyyields our previous intuitive denition of broken symmetry:

The vauum annot be an eigenstate of the harge Q. This vauumexpetationvalue is

preiselywhat we alled anorder parameter before.

InsertingaompletesetofintermediatestatesintoEq. (2.1)andassumingthetranslation

invarianeof the vauum,one arrivesat the representation

h0j[Q;℄j0i= X

n (2)

3

Æ(k

n )

e

iE(kn)t

h0jj 0

(0)jnihnjj0i e iE(kn)t

h0jjnihnjj 0

(0)j0i

:

(2.2)

Usingthe urrentonservationonean showthatthe GoldstoneommutatorinEq. (2.1)

istime-independent provided the surfae term whihomes from the integral,

Z

d 3

x[rj;℄;

vanishes. This is the entral tehnial assumption whih underlies the requirements of

ausality orfast derease of the potential mentioned above.

One this ondition is satised, the time independene of the Goldstone ommutator

fores the right-handside of Eq. (2.2) to be time-independent as well. This is, however,

not possible unless there is a mode in the spetrumsuh that lim

k!0

E(k) =0, whih is

the desiredGoldstone boson.

2.2 Toy example: Heisenberg ferromagnet

Thegeneralstatementsaboutspontaneoussymmetry breakingwillnowbedemonstrated

on the Heisenberg ferromagnet. Consider a ubi lattie with a spin-1

2

partile at eah

site. The dynamisof the spins isgoverned by the Hamiltonian

H = J X

pairs s

i s

j

; (2.3)

whih is invariant under simultaneous rotations of all the spins, that form the group

SU(2).

For simpliity we hoose the nearest-neighbor interation so that the sum in Eq. (2.3)

runsonlyoverthe pairsofneighboringsites. The ouplingonstantJ isassumed positive

so that the interation favors parallel alignment of the spins. In nite volume we shall

take up the periodi boundary ondition in order to preserve the (disrete) translation

invarianeof the Hamiltonian(2.3).

2.2.1 Ground state

The salarprodut of twoneighboring spin operators may be simpliedto

s

i s

j =

1

(s

i +s

j )

2

(s 2

i +s

2

j )

= 1

(s

i +s

j )

2

3

Itisnowlearthat the statewith thelowest energy willbeoneinwhihallpairsof spins

willbe arranged to have total spin one. The salar produt s

i s

j

then redues to 1

4 . In

athree-dimensionalubilattiewithN sitesintotal,there arealtogether 3N suh pairs

sothat the ground-state energy of the ferromagnet is

E 0 = 3 4 NJ:

As we learned in the ourse of our general disussion of broken symmetries, the ground

state is innitely degenerate. The individual states may be labeled by the diretion of

themagnetization,a unitvetor n. Allspins are alignedtopoint inthis diretion, whih

means that the groundstate vetor j(n)i is a diret produt of one-partile states, the

eigenvetors of the operators ns

i

with eigenvalue one half,

j(n)i= N

Y

i=1

ji;ni; where (ns

i

)ji;ni= 1

2 ji;ni:

The one-partile states may be expressed expliitly in terms of the two spherial angles

;'in the basis of eigenstates of the thirdomponent of the spin operator,

jni= os 2 e i' sin 2 : (2.4)

The two vetors ji;ni and ji; ni form an orthonormal basis of the one-partile Hilbert

spae H

i

. The produts of these vetors then onstitute a basis of the full Hilbert spae

of the ferromagnet, H= N N i=1 H i .

In nite volume N, states with all possible diretions n an be aommodated within

a single Hilbert spae. Two one-partile bases fjn

1

i;j n

1

ig and fjn

2

i;j n

2

ig are, as

usual,onnetedbythe unitary transformationorrespondingtothe rotationthat brings

the vetor n

1

tothe vetor n

2

. Likewise, the twoorresponding produt bases of the full

Hilbert spae H are onneted by the indued unitary rotationon this produt spae.

Let us now alulate the salar produt of the ground states assigned to two diretions

n

1

and n

2

. By exploiting the rotational invariane of the system, we may rotate one of

the vetors, say n

1

, tothe z-axis. The expliit expression for the eigenvetors (2.4) then

yieldshn

1 jn

2

i=os n 1 ;n 2 2 ,where n 1 ;n 2

is the angle between the two unit vetors.

The salarprodut of the two ground-statevetors is thengiven by

h(n 1 )j(n 2 )i= os n 1 ;n 2 2 N

and itapparently goes tozero as N !1unless n

1

and n

2

are (anti)parallel.

Usingaslightlydierentformalismwe shallnowonstrut the wholeHilbertspae H (n)

above the ground state j(n)i and show that, in fat, any two vetors, one from H (n

1 )

and the other fromH (n

2

), are orthogonal in the limitN !1.

Reall that the two-dimensional spae of spin 1

2

may be viewed asthe Fok spae of the

fermioni osillator. One denes an annihilation operator a(n) and a reation operator

a y

(n ) sothat

Theseare atually nothingelse thanthe loweringand raisingoperatorsfamiliarfrom the

theory of angularmomentum. In additionto the identities above,they satisfy

[a(n);a y

(n )℄=2ns; sothat ns= a y

(n )a(n)+ 1

2 :

Whenn=(0;0;1),these operatorsare justa=s

x +is

y , a

y

=s

x is

y

,andinthe general

ase they an befound expliitlyby the appropriate unitary rotation.

The Hilbert spae H (n) is set up as a Fok spae above the vauum j(n )i. In the

ground state all spins point in the diretionn, while the exited states are obtained by

the ation of the reation operators a y

i

(n ) that ip the spin at the i-th lattie site to

the opposite diretion. 2

The basis of the spae H (n) ontains all vetors of the form

a y

i

1 (n)a

y

i

2

(n)j(n )i wherea nite numberof spins are ipped.

Itisnow obviousthatinthe innite-volumelimit,allbasis vetorsfromthe spae H (n

1 )

are orthogonal to all basis vetors from the spae H (n

2

) that is, these two spaes are

ompletelyorthogonal.

Toputitinyetanotherway,atniteN anyvetorfromthespaeH (n

1

)maybeexpressed

asalinearombinationofthe basisvetorsofthe spaeH (n

2

),andthusthese twospaes

maybeidentied. This is,however, nolongertrue asN !1,forthe linearombination

inquestion thenontainsaninnitenumberof terms,and isdivergent. Thereisnoother

way out than treating the spaes H (n

1

) and H (n

2

) asdistint, orthogonalones.

Tosummarize,inthelimitN !1onehasaontinuumofmutuallyorthogonalseparable

Hilbert spaes H (n) labeled by the diretion of the magnetization n. In the absene of

expliitsymmetry breakingnotransitionbetween dierentspaesis possibleand onehas

tohoosethevetornoneforallandworkwithinthespaeH (n). Operatorsrepresenting

the observables are then onstruted from the annihilation and reationoperators a

i (n)

and a y

i (n).

The symmetry transformations are formallygenerated by the operator of the total spin,

S= P

i s

i

. It isnow evident that those transformations that hange the diretionof the

magnetizationn ,i.e. thespontaneouslybroken ones,arenotrealizedbyunitaryoperators

sinethey donot operate onthe Hilbert spae H (n). The only operatorthat doesis the

projetionofthetotalspinonthediretionofthemagnetization,nS. Thisgenerates the

unbrokensubgroup. (Yet, thisoperatorisunbound forN !1,butitan benormalized

by dividing by N toyield the spin density, whihis already nite.)

Itisworth emphasizing,however, thatphysiallyalldiretions nare equivalent.

Measur-able eets an only arise fromthe hangeof the diretionof n.

2.2.2 Goldstone boson

One may now ask where is the Goldstone boson assoiated with the spontaneous

break-down of the SU(2) symmetry of the Hamiltonian(2.3). In the general disussion of the

Goldstone theorem we assumed full translation invariane, while this lattie system has

onlyadisreteone. Fortunately,this isnot aproblemin the innite-volumelimit,where

2

Notethat,inthissetting,annihilationandreationoperatorsatdierentlattiesitesommuterather

there is still a ontinuous momentum variable k to label one-partile states. The only

diereneis that only a nite domainof momentum, the Brillouinzone, should be used.

We shall therefore assumethat =`k

x ;k

y ;k

z

+=`, where` is the lattie spaing.

As we emphasized above, all diretions of n are physially equivalent, so we shall from

now on set n = (0;0;1). The salar produt of two neighboring spins may be rewritten

interms of the annihilation and reationoperators,

s i s j = 1 4 (a i +a y i )(a j +a y j ) 1 4 (a i a y i )(a j a y j

)+( a y i a i + 1 2 )( a y j a j + 1 2 )= = 1 2 (a y i a y j )(a i a j )+a

y i a i a y j a j + 1 4 : (2.5)

Notethat the Hamiltonianpreserves the `partile number' that is, the number of ipped

spins generated by the operator P i a y i a i

. This is of ourse, up to irrelevant onstants,

nothing but the third omponent of the total spin, whih is not spontaneously broken

and thus an be used to label physial states. We shall restrit our attention to the

`one-partile'spae, spannedonthebasis jii=a y

i

j(n)i. Thephysialreasonbehindthis

restritionisthat thesoughtGoldstone bosonturns out tobethe spinwave {atraveling

perturbationindued by ipping asingle spin.

On the one-partilespae, the seond term onthe righthand side of Eq. (2.5) gives zero

whilethe onstant 1

4

may bedropped. The one-partileHamiltonianthus reads

H 1P = J 2 X pairs (a y i a y j )(a i a j );

and ats onthe basis states as 3 H 1P jii= J 2

ji+1i 2jii+ji 1i

: (2.6)

The disrete translation invariane is apparently not broken in the ground state. That

meansthat the stationarystates are simultaneously the eigenstates of the shiftoperator,

T : jii ! ji+1i. The eigenvalues of the shift operator are of the form e ik`

. Eq. (2.6)

impliesthat the one-partileHamiltonianisdiagonalized inthe basis of eigenstatesof T.

The orrespondingenergies are

E(k)= J 2 2 e ik` e ik`

=2Jsin 2

k`

2

; (2.7)

andinthreedimensionswewouldanalogouslyndE(k) =2J sin 2 k x ` 2 +sin 2 ky` 2 +sin 2 k z ` 2 .

We have thus found our Goldstone boson, in the ase of the ferromagnet it is alled the

magnon. We stress the fat that we used no approximation, so Eq. (2.7) is the exat

dispersion relation of the magnon, and the eigenstate P

j e

ijk`

jji is the exat eigenstate

of the fullHamiltonian(2.3).

Notealso that there is just one Goldstone mode even though two symmetry generators,

S

x

and S

y

, are spontaneously broken. This may be intuitively understood by ating

3

As thelow-energydynamisoftheGoldstoneboson isisotropi,weworkwithoutlakofgenerality

with either broken generator on the vauum j(n)i. We nd S

x

j(n )i = 1

2 P

j

jji and

S

y

j(n)i= i

2 P

j

jji that is, both operators reate the same state, whih formally

orre-sponds tothe zero-momentum magnon. This fat appears tobe tightlyonneted to the

dispersionrelationofthemagnon,whihisquadratiatlowmomentum. Thephenomenon

isquite generaland its detaileddisussion is deferred toChapter 3.

Havingfoundthe exat dispersionrelation,itissuitabletoommentontheissue ofnite

vs. innitevolume. Stritlyspeaking,thereisnospontaneoussymmetrybreakinginnite

volume. Alltheeetssuhastheunitarilyinequivalentimplementationsofthesymmetry

andthe existene ofa gaplessexitationappear onlyinthe limitof innitevolume. Real

physial systems are, on the other hand, always nite-sized. They are, however, large

enough ompared to the intrinsi mirosopi sale (here the lattie spaing `) of the

theory sothat the innite-volume limitisboth meaningfuland pratial.

In partiular, when the ferromagnet lattie is of nite size N, the periodi boundary

ondition requires the momentum k to be quantized, the minimum nonzero value being

k

min

`=2=N. The energy gap inthe magnonspetrum is then E

min

2

2

J=N 2

, whih

issmallenough forany marosopisystem tobe toset tozero.

2.3 Desription of spontaneous symmetry breaking

Sofar wehavebeen disussing theverygeneralfeatures ofspontaneously broken

symme-tries. Toinvestigatea physialsystem inmoredetail, onenext has toxthe Lagrangian.

Beforegoing intopartiular models we shall make anaside and mention the very

impor-tant onept ofeetive eld theory.

Themethodofeetiveeldtheory reliesonthe fatthat,intheabsene ofothergapless

exitations,the long-distanephysisof aspontaneouslybroken symmetryisgovernedby

theGoldstonebosons. 4

Onethenonstruts themostgeneraleetiveLagrangianforthe

Goldstone degrees of freedom, ompatible with the underlyingsymmetry [11℄.

Thehiefadvantage of this approahis thatit providesamodel-independentdesription

of the broken symmetry. The point is that by exploiting the underlying symmetry, it

essentiallyyields the most generalparametrizationof the observables in termsof aset of

low-energyoupling onstants.

From the physial point of view, adisadvantage of eetiveeld theory is that ittellsus

nothingabout the origin of symmetry breaking { one simply has to assume a partiular

formof the symmetry-breaking pattern.

Toshowthatthesymmetryisbrokenatallandtospeifythesymmetry-breakingpattern,

onehastondanappropriateorderparameter. Itisthereforenotsurprisingthattheissue

ofndingasuitableorder parameterisof keyimportane, andonsiderable diÆulty,for

the desription of spontaneous symmetry breaking.

Inthefollowing,werealltwopartiularmodelsof spontaneoussymmetrybreaking. The

operatorwhose vauumexpetation value providesthe order parameteris anelementary

4

Quitegenerally,theeetiveeld theoryapproahmaybeappliedwheneverthere aretwoormore

energysalesin thesystemwhih anbetreated separately. It isthus notspeial onlyto spontaneous

Figure2.2: TheMexian-hatpotentialin theeverydaylife {theItalianpastasombreroni.

eld in the rst ase, and a omposite objet in the seond one. In both ases, an

approximation is made suh that the quantum utuations of the order parameter are

negleted.

2.3.1 Linear sigma model

Perhaps the most popular and universal approah tospontaneous symmetry breaking is

toonstrutthe Lagrangian sothatitalready ontainsthe order parameter. This isvery

muh analogous to the Ginzburg{Landau theory of seond-order phase transitions. One

introdues a salar eld 5

and adjusts the potential sothat it has a nontrivialminimum.

The result isthe paradigmati Mexian hat, see Fig. 2.2.

The great virtue of this method is that the order parameter is provided by the vauum

expetation value of an elementary salar eld, whih may be hosen onveniently to

ahieve the desired symmetry-breaking pattern. As a partiular example we shall now

review the simplestmodel withAbelian symmetry.

Startingwith a pure salar theory, we denethe Lagrangian for aomplex salar eld

as

L

=

y

+M 2

y

(

y

) 2

: (2.8)

This Lagrangian is invariant under the phase transformations ! e i

that form the

Abelian group U(1). At tree level, the ground state is determined by the minimum of

the stati part of the Lagrangian, whih is found at y

= v 2

=2 = M 2

=2 so that the

symmetry is spontaneously broken. As explained in Setion 2.1.1, there is a ontinuum

of solutions to this ondition (distinguished by their omplex phases) and the physial

vauummaybehosenasanyoneofthem,but nottheirsuperposition. Thisisthereason

why the followinglassial analysisatually works.

It is ustomary to hoose the order parameter real and positivei.e., we set hi =v= p

2.

Thesalareldisnext shiftedtotheminimumandparametrizedas=(v+H+i)= p

2.

Uponthis substitutionthe Lagrangian beomes

L

=

1

2 (

H)

2

+ 1

2 (

)

2

+ 1

4 M

2

v 2

M 2

H 2

vH 3 1

4 H

4 1

2 H

2 1

4 H

2

2 1

4

4

:

TherstthreetermsrepresentthekinetitermsforHandandminusthevauumenergy

density, respetively. There is also the mass term for H, while the eld is massless {

this isthe Goldstone boson.

It isinstrutive to evaluatethe U(1)Noetherurrent in terms of the new elds,

j

=i( y

y

) = v

+(

H H

): (2.9)

We an see that the Goldstone boson is annihilated by the broken-symmetry urrent,

as predited by the Goldstone theorem. The orresponding matrix element is given by

h0jj

(0)j(k)i / vk

, the onstant of proportionality depending on the normalization of

the one-partilestates.

In the standard model of eletroweak interations, the salar eld is in fat added just

for the purpose of breaking the gauge and global symmetries of the fermion setor. The

same may be done in our toy model. We start with a free massless Dira eld whose

Lagrangian, L =

i=

, is invariant under the U(1)

V

U(1)

A

hiral group. The mass

term of the fermion violates the axialpart of the symmetry and thus an be introdued

onlyafter this isbroken.

To that end, we add the salar eld Lagrangian L

and an interation term L =

y(

L R +

R L

y

). The fullLagrangian,L=L +L

+L , remainshirallyinvariant

provided the salar is assigned a proper axial harge. The nontrivial minimum of

the potential in Eq. (2.8) now breaks the axial symmetry spontaneously and, upon the

reparametrizationof the salar eld, the fermion aquires the mass m =vy= p

2.

2.3.2 Nambu{Jona-Lasinio model

In ontrast to the phenomenologial linear sigma model stands the idea of dynamial

spontaneous symmetry breaking. Here, one does not introdue any artiial degrees of

freedom in order to break the symmetry by hand but rather tries to nd a

symmetry-breakingsolution tothe quantum equations of motion.

Physially, this is the most aeptable and ambitiousapproah. Unfortunately, it isalso

muhmore diÆultthan the previous one. The reasonis that one oftenhas todeal with

stronglyoupled theoriesand,moreover, the alulationsalways have tobe

nonperturba-tive. Asarule,itisusuallysimplyassumedthatasymmetry-breakingsolutionexists and

after it is found, it is heked to be energetially more favorable than the perturbative

vauum.

Bythissortofavariationalargument,oneisabletoprovethatthesymmetriperturbative

vauumisnot thetruegroundstate. Onthe otherhand,itdoesnotfollowthatthefound

solution is, whih might be a problem in omplex systems where several qualitatively

dierent andidates for the groundstate exist [16℄.

As an example, we shall briey sketh the model for dynamial breaking of hiral

sym-metryinvented by Nambu andJona-Lasinio[17, 18,19℄. As the samemodelwillbe used

inChapter 5todesribeaolor superondutor[20℄,weshall takeup this opportunityto

introdue the mean-eld approximation that we later employ.

The Lagrangian of the originalAbelian NJL modelreads

L=

i =

+G

(

) 2

(

) 2

Its invariane under the Abelian hiral group U(1)

V

U(1)

A

is most easily seen when

the interation is rewritten in terms of the hiral omponents of the Dira eld, L =

i= +4Gj R L j 2 .

FollowingtheoriginalmethodduetoNambuandJona-Lasinio,weantiipatespontaneous

generationofthefermionmassbytheinterationandsplittheLagrangianintothemassive

freepart and an interation, L=L

free +L int , where L free = (i =

m) ; L

int =m +G ( ) 2 ( 5 ) 2 :

At this stage already, we are making the hoie of the ground state by introduing the

massterm and requiringthatm be realand positive. The generalparametrization of the

massterm would be (m 1 +im 2 5

) with realm

1 ;m

2

. Thephysial massof the fermion

would then be p m 2 1 +m 2 2 .

Theatualvalueof the mass mis determinedby the ondition ofself-onsisteny, thatit

reeives noone-loopradiativeorretions. This givesrise tothe gap equation

1=8iG Z d 4 k (2) 4 1 k 2 m 2 : (2.11)

ThesameresultmaybeobtainedwithamethodduetoHubbardandStratonovih,whih

keeps the symmetryofthe Lagrangian manifestatallstages ofthe alulation. One adds

totheLagrangianaterm j 4G

R L j

2

=4G. Inthepathintegrallanguage,thisamounts

toanadditionalGaussianintegrationover thatmerely ontributesanoverallnumerial

fator. Eq. (2.10)then beomes

L= i = 1 4G ( 2 1 + 2 2 )+ ( 1 +i 2 5

) ; (2.12)

the

1 ;

2

being the real and imaginary parts of , respetively.

The Lagrangian is now bilinear in the Dira eld so that this may be integrated out,

yieldingan eetive ationfor the salar order parameter ,

S e = 1 4G Z d 4 x( 2 1 + 2 2

) ilogdet i = +( 1 +i 2 5 ) (2.13)

Withthis eetive ationone an evaluate the partitionfuntion,or the thermodynami

potential, in the saddle-point approximation. This means that we have to replae the

dynamialeld witha onstantdeterminedasasolutiontothe stationary-point

ondi-tion, ÆS e Æ 1 = ÆS e Æ 2 =0:

Looking bak at Eq. (2.12) we see that the onstant mean eld yields preisely the

eetivemass ofthe fermion,and the stationary-pointondition,

1=8iG Z d 4 k (2) 4 1 k 2 y ;

isidential tothe gap equation(2.11).

In the Nambu{Jona-Lasiniomodel, the Goldstone boson required by the Goldstone

the-orem is a bound state of the elementary fermions. In the simple ase of the Lagrangian

(2.10)itisapseudosalarandmayberevealedasapoleinthetwo-pointGreen'sfuntion

of the omposite operator

Goldstone boson ounting in nonrelativisti

systems

This hapter is devoted to adetailed disussion of the issue raised inSetion 2.1.2: How

many Goldstone bosonsare there,given the patternof spontaneous symmetry breaking?

As already mentioned, in Lorentz-invariant theories the situation is very simple: The

number of Goldstone bosons is equalto the numberof the broken-symmetry generators.

In nonrelativistisystems, however, these two numbers maydier.

We have already met an example where this happens { the ferromagnet. Historially,

this was perhaps the rst ase inwhih the `abnormal' number of Goldstone bosons was

reported, anditstillremainsthe onlytextbook one. Nevertheless, the samephenomenon

has reently been studied in some relativistisystems at nite density [21, 22,23, 24℄ as

wellasinthe Bose{Einstein ondensed atomigases [25, 26℄, and itistherefore desirable

toanalyze the problemof the Goldstone boson ounting ona generalground.

Westart withareview ofthe generalounting ruleby Nielsenand Chadha [27℄and some

otherpartialresults. The mainbodyof thishapter thenonsistsof thedisussion of the

Goldstoneboson ountinginthe frameworkofthe relativistilinearsigmamodelatnite

hemialpotential. The presented results are based on the paper [III℄, where the details

of the alulationsmay be found.

3.1 Review of known results

3.1.1 Nielsen{Chadha ounting rule

FollowingloselythetreatmentofNielsenandChadha[27℄,weonsideraontinuous

sym-metry, some of whose generators, Q

a

, are spontaneously broken. The broken-symmetry

assumption(2.1) now generalizes to

det h0j[Q

a ;

i

℄j0i6=0; a;i=1;:::;# of broken generators :

In addition,it is assumed that the translation invariane is not entirely broken and that

forany two loaloperators A(x)and B(x)a onstant >0 exists suh that

jh0j[A(x;t);B(0)℄j0ij!e jxj

It is then asserted that there are two types of Goldstone bosons { type-I, for whih the

energyisproportionaltoanoddpowerofmomentum,andtype-II,forwhihtheenergyis

proportionaltoaneven powerofmomentuminthelong-wavelengthlimit. Thenumberof

Goldstonebosons of therst type plus twiethe number of Goldstonebosons of the seond

type is always greater or equal to the number of broken generators.

The dierenebetween the two types of Goldstone bosons isniely demonstrated on the

ontrast between the ferromagnet and the antiferromagnet. In the ferromagnet, there is

asingle Goldstoneboson (themagnon). TheNielsen{Chadha ountingrule thenenfores

thatitmustbeoftypeIIandindeed,itsdispersionrelationisquadratiatlowmomentum,

see Setion 2.2.2. In the antiferromagnet, on the other hand, there are two distint

magnonswith dierent polarizations. Their dispersionrelation islinear.

Note that the result of Nielsen and Chadha does not restrit in any way the power of

momentum towhih the energy is proportional. As far asthe ounting of the Goldstone

bosons isonerned, it onlymatterswhether this power is anodd or aneven number. It

seems, however, that there are infat nosystems of physialinterestwhere the power is

greaterthan two.

It is also worthwhile to mention that the Nielsen{Chadha ounting rule is formulated

asan inequality, in most ases of physial interest this inequality is, however, saturated.

This happens not only for the ferromagnet and the antiferromagnet. To the best of the

author's knowledge, all exeptions where a sharp inequality ours, happen at a phase

boundaryof the theory [22, 28℄. Later inthis hapterwe shall see a generilass of suh

exeptions: The phase transitionto the Bose{Einstein ondensed phase of the theory, at

whihthe phaseveloity ofthe superuidphononvanishesand the phononthusbeomes

atype-II Goldstone boson.

Itisnaturaltoaskwhatisthedierenebetweentheferromagnetandtheantiferromagnet

thatausessuhadramatidisrepanyintheirbehavior. Theanswerliesinthenonzero

netmagnetizationofthe ferromagnet. Ingeneral,itisnonzerovauumexpetationvalues

of some of the harge operators that distinguish the type-II Goldstone bosons from the

type-I ones. At a very elementary level, one an say that nonzero harge densitiesbreak

time reversal invariane and thus allow for the presene of odd powers of energy in the

eetiveLagrangian forthe Goldstone bosons [2℄. Theissue of hargedensities, however,

deserves more attention beause they are usually easier todetermine than the Goldstone

boson dispersionrelations.

3.1.2 Other partial results

As we have just shown, the issue of Goldstone boson ounting is tightly onneted to

densitiesofonserved harges. Wethusdealwiththree distintfeatures ofspontaneously

broken symmetries that are related to eah other: The Goldstone boson ounting, the

hargedensitiesinthegroundstate, andthedispersionrelationsoftheGoldstonebosons.

The onnetion between the Goldstone boson ounting and the dispersion relations is

enlightened by the Nielsen{Chadha ounting rule. In general, littleis known about the

diretrelationoftheGoldstonebosonountingandthehargedensities. Thereisapartial

ommutators of all pairs of broken generators have zero density in the ground state.

Aneessary onditionforanabnormalnumberofGoldstonebosonsisthusanonvanishing

vauum expetation value of a ommutator of two broken generators. The value of this

resultisthatitshowsthatthepatternofsymmetrybreakingmustinvolvethenon-Abelian

struture ofthe symmetrygroup. Forinstane, the Goldstone boson ounting is usualin

allolor-superondutingphases of QCDinwhihonlythe net baryonnumberdensity is

nonzero. The reasonisthatthebaryonnumberorresponds toaU(1)fatorof theglobal

symmetry group and therefore does not give rise to anorder parameter for spontaneous

symmetry breaking.

Intuitively,the neessity tomodify the ounting of the Goldstone bosons inthe presene

ofhargedensitiesanbeunderstoodasfollows[III℄.Assumethatthe ommutatorof the

hargesQ

a

and Q

b

develops nonzeroground-stateexpetationvalue. Wemay theninEq.

(2.2) set Q =Q

a

and take the harge density j 0

b

(x) inplae of the interpolating eld for

the Goldstone boson, . We nd

if

ab h0jj

0

(0)j0i =h0j[Q

a ;j

0

b

(x)℄j0i =2iIm X

n (2)

3

Æ(k

n )h0jj

0

a

(0)jnihnjj 0

b

(0)j0i; (3.2)

where f

ab

are the set of struture onstants of the symmetry group. Two points here

deserveaomment. First,itisagain learthatanon-Abeliansymmetrygroupisneeded.

Onlythen may the vauum harge density be treated as an order parameter for

sponta-neous symmetry breaking. Seond, it follows from the right hand side of Eq. (3.2) that

asingleGoldstonebosonouplestotwobrokenurrents,j

a andj

b

. Wehavealreadyseen

in Setion 2.2 that this happens in the ase of the ferromagnet. This suggests the way

howthe ounting rule forthe Goldstone bosonsshould be modiedone nonzero density

of a non-Abelian harge is involved. Nevertheless, it still remains to turn this heuristi

argument into amore rigorous derivation of the properounting rule.

Finally,the onnetion between the harge densitiesand the Goldstone boson dispersion

relations was provided by the work of Leutwyler [29℄. Leutwyler analyzed spontaneous

symmetry breaking in nonrelativisti translationally and rotationally invariant systems.

Hedeterminedtheleading-orderlow-energyeetiveLagrangianfortheGoldstonebosons

as the most general solution to the Ward identities of the symmetry. His results show

that when a non-Abelian generator develops nonzero ground-state density, a term with

asingletime derivativeappears inthe eetiveLagrangian. The time reversal invariane

isthenbrokenandtheleading-orderLagrangianisoftheShrodingertype,resultinginthe

quadratidispersion relation of the Goldstone boson. It should perhaps be stressed that

when this happens, the eetive Lagrangian is invariant with respet to the presribed

symmetry onlyup to a total derivative.

Weshallnowgiveasimpleargument,alsoduetoLeutwyler, explaininghowsuha

single-time-derivativetermintheLagrangianaetstheGoldstonebosonounting. Theeetive

Lagrangianis onstruted onthe oset spaeof the broken symmetry. Consequently, the

number of independent real elds appearing in the Lagrangian is always equal to the

number ofbroken generators.

Now if the single-time-derivative term is absent inthe Lagrangian, the Goldstone boson

dispersion relation islinear and omes, at tree level, inthe formE 2

/k 2

. This equation

real salar eld (similar to the Klein{Gordon eld). There is therefore a one-to-one

orrespondene between the Goldstone bosons and the elds inthe Lagrangian.

On theother hand, ifthere isa termwith asingle time derivativeinthe Lagrangian, the

Goldstone boson dispersion relation is quadrati and appears as E /k 2

. This equation

has, of ourse, only positive energy solutions, very muh like the Shrodinger equation.

As a result, the type-II Goldstone boson is to be desribed with a omplex eld or,

equivalently,withapair ofreal elds. This shows why thetype-IIGoldstone bosonshave

tobe ounted twie, when omparing their numberto the numberof broken generators.

Now and again, this intuitive piture easily aommodates only the Goldstone bosons

with linear or quadrati dispersion. The question of the existene of Goldstone bosons

with energy proportional to higher powers of momentum remains open as well as the

possibility of their desription in terms of a low-energy eetive Lagrangian. Note that

toahieve the appropriate powerof momentum in the dispersion law, one would have to

get rid of the standard bilinear kineti term in the Lagrangian, whih would invalidate

the onventional perturbation expansion aswell asthe power-ounting sheme.

3.2 Linear sigma model at nite hemial potential

TherestofthishapterisdevotedtothestudyofapartiularlassofLorentz-noninvariant

systems{relativistitheoriesatnitedensity. The mirosopidynamisofsuhsystems

is Lorentz-invariant, Lorentz symmetry being violated only at the marosopi level, by

mediumeets. This suggests that muhmore ould besaid about the patterns of

sym-metrybreakingandpropertiesoftheGoldstonebosonsthantheNielsen{Chadhatheorem

does, by exploiting the underlyingLorentz invariane.

Inthe following,we shallstay inthe frameworkof the relativistilinear sigmamodeland

derivean exat orrespondene between the Goldstone boson ounting, harge densities,

andthe Goldstone boson dispersionlaws. Thedisussion of the possibleextension of the

ahieved results is postponed tothe Conlusions.

3.2.1 SU(2)U(1) invariant sigma model

Westartwithasimpleexample: ThelinearsigmamodelwithanSU(2)U(1)symmetry,

whih has been used as a toy model for kaon ondensation in the Color-Flavor-Loked

phaseof QCD[21, 22℄. All essential stepsleading tothe nal ounting rule for the

Gold-stonebosonswillberstdemonstratedwithinthismodel,thenwithinamoreompliated

onewith anSU(3)U(1)symmetry,and afterwards generalizedtothe sigmamodelwith

arbitrarysymmetry.

The modelisdened by the Lagrangian,

L=D

y

D

M

2

y

(

y

) 2

; (3.3)

where the salar is a omplex doublet. Nonzero density of the U(1) harge is

imple-mented in terms of the hemial potential , whih enters the Lagrangian through the

ovariantderivative, D

0

=(

0

In the absene of the hemial potential, the Lagrangian (3.3) is invariant under the

extendedgroupSU(2)SU(2) 'SO(4). Thehemialpotentialbreaksitexpliitlydown

toSU(2)U(1). In the ontext of the CFL phase with the kaon ondensate, the SU(2)

grouporresponds tothe isospin and the U(1) tothe strangeness. The eld is just the

(hargedor neutral)kaon doublet.

The hemial potential ontributes a term 2

y

to the stati part of the Lagrangian.

When > M, the perturbative vauum = 0 beomes unstable and a new, nontrivial

minimumappears{theSU(2)U(1)symmetryisspontaneouslybroken downtoitsU(1)

subgroup. This isthe relativistiBose{Einstein ondensation.

Torevealthephysialontentofthemodelinthespontaneouslybrokenphase,weproeed

inthe standard manneri.e., alulatetheminimumof thepotential,shiftthe salareld,

andexpandtheLagrangianaboutthenewgroundstate. Thesalareldisreparametrized

as = 1 p 2 e i k k =v 0

v+H

; where v 2 = 2 M 2 ; k

beingthePaulimatries. Thethree`pion'elds

k

would,intheabseneofthehemial

potential, orrespond tothe three Goldstone bosons of the oset [SU(2)U(1)℄=U(1).

The exitationspetrum isdetermined by the bilinearpart of the Lagrangian,

L bilin = 1 2 ( k ) 2 + 1 2 ( H) 2 v 2 H 2 +( 1 0 2 2 0 1

)+(H

0 3 3 0 H): (3.4)

Thepresene ofthe hemialpotentialapparentlyleads tonontrivialmixingof the elds

whihannotbe removed by aglobalunitarytransformation. Tondthe dispersionlaws

ofthefourdegrees offreedom,itisthereforemoreappropriatetolookforthe polesof the

propagators. Itturnsout[21,22℄thatthemixingof

1 and

2

givesrise toone Goldstone

boson with the low-momentum dispersion law E(k) = k 2

=2, while the other mode is

gapped, E(k) =2+O(k 2

). On the other hand, the setor(

3

;H) produes one gapless

exitation with E(k) = q 2 M 2 3 2 M 2

jkj+O(jkj 3

), and a massive radial mode with a gap

p 3 2 M 2 .

In onlusion, there are two Goldstone bosons, one with a linear dispersion law (the

phonon) and one with a quadrati dispersion law. This is in aord with the Nielsen{

Chadhaounting rulesine thevauumexpetationvalue hiarries nonzeroisospin. To

seeinmoredetailhowthisfataetsthe strutureofthe bilinearLagrangian(3.4), note

that ( 1 0 2 2 1 )= v 2 k 0 l Im [ k ; l ℄ :

In this form it is obvious how the nonzero density of the ommutator of two broken

harges(3.2) enters the Lagrangianand thusgivesrisetothe existeneof asingletype-II

Goldstone boson instead of two type-I ones.

To understand more deeply the nature of the type-II Goldstone boson, we shall now

investigate the orresponding plane-wave solution of the lassial equation of motion.

Note rst that the unbroken U(1) group is generated by the matrix 1

2 (1+

3

). In order

to keep this U(1) symmetry manifest, we ombine

1

and

2

into one omplex eld,

= 1 p 2 ( 2 +i 1

As far as the quadratiGoldstone boson is onerned, we may drop the elds

3

and H

and rewritethe Lagrangian (3.4) in terms of ,

L =2i y 0 + y :

The eld annihilates the type-II Goldstone and the orresponding lassialplane-wave

solutionis given by =

0 e

ikx

, with the exat (tree-level)dispersion relation

E(k)= q k 2 + 2 :

The SU(2)U(1) symmetry gives rise to four onserved urrents whih, interms of the

doublet, read

j

k

= 2Im y k +2Æ 0 y k ; j

= 2Im y +2Æ 0 y :

Forthe quadrati Goldstone plane wave we nd

j 1 =+(k +2Æ 0 )v p

2Re ; j 2 = (k +2Æ 0 )v p

2Im :

We an immediately see that the isospin density rotates in the isospin plane (1;2) i.e.,

the planewave is irularlypolarized. In this way, a singleGoldstone boson exploitstwo

broken-symmetrygenerators, assuggested by the general form of the ommutator (3.2).

It is notable that the plane wave with the opposite irular polarization orresponds to

the gapped exitationin the setor (

1 ;

2 ).

The remaining two urrents are onveniently expressed in the rotated basis, expliitly

separating the unbroken and broken generator,

1

2 (1+

3 ): j

=2(k +Æ 0 )j j 2 ; 1 2 (1 3 ): j

=Æ 0 v 2 :

It is seen that the isospin wave is assoiated with a uniform urrent of the unbroken

symmetry that is, the Goldstone boson arries the unbroken harge. This seems to be

ageneri feature of type-II Goldstone bosons.

Finally, the broken generator 1

2

(1

3

) gives rise just to nonzero harge density and,

moreover, is independent of the amplitude and momentum of the isospin wave. It is

thereforeto be interpreted asjust a bakground on whih the isospin waves propagate.

3.2.2 Linear sigma model for SU(3) sextet

Asanontrivialexampleofaspontaneouslybrokensymmetrywithnonzerohargedensities

the linear sigma modelfor anSU(3) sextet salar eld willnow be investigated.

The Lagrangian reads

L=Tr (D

y D ) M 2 Tr y

aTr( y ) 2 b(Tr y ) 2 ; (3.5)

andis invariantunderthe global SU(3)U(1)symmetrythat transformsthe salar eld

as!UU T

. AU(1)hemialpotentialisintroduedsothattheovariantderivative

a b SO(3) SU(2)U(1)

unstable potential

Figure 3.1: Phase diagram of the model dened by the Lagrangian (3.5). The ordered

phases are labeled by the symmetry of the ground state. The `unstable potential'region

marksadomainofparameterswherethetree-levelpotentialisnotboundedfrombelow.

This model provides a phenomenologial desription of the olor-superonduting phase

of QCD with a olor-sextet pairing of quarks of a single avor, whih was proposed in

Ref. [I℄. The global SU(3) symmetry is what remains of the olor gauge invarianeafter

the gluonshavebeen `integrated out',while the U(1)orresponds to the baryonnumber.

The salareld is aneetive ompositeeld forthe quarkCooper pairs.

It turns out that this theory has two dierent ordered phases, with dierent

symmetry-breaking patterns and exitation spetra, see Fig. 3.1. The Bose{Einstein ondensation

setsat=M=2. All phasetransitions, between thenormaland anordered phaseaswell

asbetween the ordered phases,are ofseond order.

In general, the exitations are grouped into multiplets of the unbroken symmetry. This

means that the more of the original SU(3)U(1) symmetry is spontaneously broken,

the more ompliated the struture of the spetrum is. Both phases willnow be treated

separately.

Thea>0 phase

Thestati part of the Lagrangian (3.5) isminimizedby a salar eldproportionalto the

unit matrix i.e., =11. The SU(3)U(1) symmetry is thus spontaneously broken to

itsSO(3) subgroup.

Withthis symmetry-breaking patternin mind, the salar eld is parametrizedas

(x) =e 2i(x)

V(x)[11+'(x)℄V T

(x):

Here is the Goldstone boson of the spontaneously broken U(1) and V = e i

k

k

, k =

1;3;4;6;8, ontains the 5-plet of Goldstone bosons of the oset SU(3)=SO(3). The real

symmetrimatrix 'represents six heavy `radial' modes.

Using the notation =

k

k

0 1 2 3 4

0 1 2

Mass = M 2=M 6 6 1 5 6

Figure 3.2: Massesof theexitationsasafuntion ofthehemialpotentialin the

SO(3)-symmetri phase. Degeneraies of the exitation branhesare indiated by the numbers.

Thenumerialdatawereobtainedwitha=b=1.

Lagrangian, L bilin =12 2 ( ) 2 +4 2 Tr( ) 2

+Tr(

') 2 4 2

aTr' 2

+b(Tr') 2

16[

0

Tr'+Tr ('

0 )℄:

We nd that there are six Goldstone bosons, all with linear dispersion relation. Sine

there are six broken generators aswell, this result is in aord with the Nielsen{Chadha

ounting rule. All exitations fall into irreduible representations of the unbroken SO(3)

group. In partiular, there is a Goldstone singlet and a gapped singlet in the setor

(;Tr'). In addition,there are two 5-plets, a gaplessand agapped one, stemmingfrom

mixingof with the traeless part of ', see Fig. 3.2.

Thea<0 phase

In this ase the minimum of the stati potential an be reast to the diagonal form

with a single nonzero entry, = diag(0;0;). The symmetry-breaking pattern is now

SU(3)U(1)!SU(2)U(1). The salar sextet is onveniently parametrizedas

(x)=e i(x) (x) +H(x) e i T (x) :

The matrix eld is again given by the linear ombination of the broken generators,

=

k

k

,k =4;5;6;7;8, isa omplexsymmetri 22 matrix, and H is areal salar.

The bilinearpart of the Lagrangian is

L bilin =Tr( y )+( H) 2 +2 2 ( ) 33 +2 2 ( 33 ) 2 4 2

(a+b)H 2

+2 2

aTr y 16H 0 33 4 2 Im[; 0 ℄ 33

4ImTr y

0 :

The SU(2) singlets H and

8

mix, giving a Goldstone boson with linear dispersion law

0 1 2 3 4

0 1 2

Mass

=

M

2=M 6

6

1

2 3

3

3

Figure3.3: Massesoftheexitations asafuntionofthehemialpotentialin theSU(2)

U(1)symmetriphase. Degeneraiesoftheexitationbranhesareindiatedbythenumbers.

Thenumerialdatawereobtainedwitha= 0:5andb=1.

SU(2). They yieldadoubletof gapped modesand a doubletoftype-II Goldstone bosons

with a quadrati dispersion relation. Finally, the omplex matrix ontains two real

tripletsof massive partiles. Forsummary see Fig. 3.3.

Note that there are now only three Goldstone bosons even though ve generators are

spontaneously broken. This is, however, again in agreement with the Nielsen{Chadha

rulesinetwoofthe Goldstonesare oftheseondtype. Their existeneisonneted with

the fat that in this ase, the generator

8

develops nonzero ground-state density. The

modiedGoldstone boson ounting suggested by Eq. (3.2) thusapplies.

Phase boundary

At the boundary between the two ordered phases the model displays quite remarkable

properties. The Lagrangian (3.5) is then invariant under an extended SU(6) U(1)

symmetryunderwhihtransformsasafundamentalsextet. Theminimaofthepotential

orrespondingtothetwophasesarenowdegenerateandbothleaveunbrokenthe SU(5)

U(1)subgroup meaningthat thereare altogethereleven broken generators.

This enhaned symmetry must, of ourse, be reeted in the number and type of the

Goldstone bosons [28℄. Indeed, by properly performing the limit a !0 it an be shown

on both sides of the phase transition that there are six Goldstone bosons. One is an

SU(5) singlet and has a linear dispersion law { this is the superuid phonon. The other

ve transform as the fundamental SU(5) 5-plet and allhave a quadrati dispersion that

is,are type-II. The Nielsen{Chadha ounting is thussaturated as expeted.

3.2.3 General analysis

Theresultsahievedsofarbythestudyoflinearsigmamodelswithpartiularsymmetries

will now be extended to the general ase. We start with the formulation and a short

generators impliestheexisteneofonetype-IIGoldstonebosonwithaquadrati dispersion

law.

TheexisteneofasingleGoldstone boson orrespondingtotwobroken generators,whose

ommutator has nonzero density, has been expeted on the basis of Eq. (3.2). Here we

expliitlyprove the missingpiee that is, the Goldstone boson is type-II asit must bein

order to satisfy the Nielsen{Chadha ounting rule. We shall alsosee that the statement

formulatedaboveholdsstritlyspeakingonlywhenaonvenientbasisofbrokengenerators

ishosen.

In a sense, this result is onverse to the theorem by Shaefer et al. [22℄. While they

prove that zero density of ommutators of broken harges implies usual ounting of the

Goldstone bosons, here we show that nonzero densities, on the ontrary, lead to the

existeneof type-II Goldstones and thus modiedounting.

Let us onsider the linear sigma modelwith hemial potential assigned to one or more

generators of the internal symmetry group. In general, the hemial potential for a

on-served harge Q is introdued by replaing the Hamiltonian H with H Q. The key

observation is that, as far as exat symmetry is onerned, the hemial potential is

al-ways assigned to a U(1) fator of the symmetry group that is, the harge Q ommutes

with all generators of the exat symmetry group. The reason is that even if the harge

Q is originally a part of some larger non-Abelian symmetry group, by adding it to the

Hamiltonianweexpliitlybreak allgenerators that donot ommute with it.

The Lagrangian forthe generallinear sigma modelis dened as

L=D

y

D

V(): (3.6)

ThesalareldtransformsunderagivenrepresentationoftheglobalsymmetrygroupG

and V() is the most generalG-invariant renormalizablepotential. Finally the hemial

potentialenters the Lagrangian through theovariant derivativeD

=(

iA

) [30℄,

A

beingthe onstantexternal gaugeeldwhihiseventuallyset toA

=(Q;0;0;0)or

the sum of similar terms,when more hemial potentialsare present.

Thepresene of the hemialpotentialdestabilizes the perturbativegroundstate, =0,

and eventually leads to spontaneous symmetry breaking by the Bose{Einstein

onden-sation. We assume that the new minimum

0

breaks the global symmetry group of the

Lagrangian, G, to its subgroup H. All generators, both broken and unbroken, are then

lassiedby irreduible representations of H.

In the spontaneously broken phase the salar eld is parametrizedas

(x)=e i(x)

[

0

+H(x)℄: (3.7)

Thematrix isalinearombinationofthe broken generatorswhile H ontains the

mas-sive(Higgs)elds. UponexpandingtheLagrangian(3.6)intermsoftheeldomponents,

itsbilinearpart beomes

L

bilin =

H

y

H V

bilin

(H) 2ImH y

A

H+

+ y

0

0

4ReH y

A

0

Im y

0 A

[;

℄

0

: (3.8)

Here V

bilin

Eq. (3.8)isthe mainresultwhihontainsessentiallyallinformationabout thespetrum

of the sigma model. To understand better its onsequenes, we resort for a moment to

asimple bilinearLagrangian with just two salar elds,

L

bilin =

1

2 (

)

2

+ 1

2 (

h)

2 1

2 f

2

()h 2

g()h

0

: (3.9)

Oneof the elds, h,possibly hasa mass termand there isalsoa single-derivativemixing

term,both dependingexpliitlyonthe hemialpotential. Thisisthe generiformof the

bilinearLagrangian we met inthe two partiular examplesin the preeding setions.

Asimplealulationreveals that theLagrangian (3.9)desribesa(massive)partilewith

dispersion relationE 2

(k)=f 2

()+g 2

()+O(k 2

), and a gaplessmode with dispersion

E 2

(k)= f

2

()

f 2

()+g 2

() k

2

+

g 4

()

[f 2

()+g 2

()℄ 3

k 4

+O(k 6

): (3.10)

If f() = 0 that is, if both and h are Goldstone elds mixed by the single-derivative

term, we arrive at one type-II Goldstone boson. The expansion of its energy in powers

of momentum starts at the order k 2

. On the other hand, when jf()j > 0, the eld h

represents a massive mode. The mixing of h and then results in a type-I Goldstone

boson with linear dispersion relation.

We an now understand the ontent of Eq. (3.8). There are kineti terms for both the

radialelds H and the Goldstones , and the mass term for H,essentially given by the

urvatureofthestatipotentialattheminimum

0

. Finally,therearethreemixingterms

with asingle derivative,proportionalto the external eld A

.

The analysis of the model Lagrangian (3.9) tells us that mixing of a radial eld with

aGoldstone eld givesrise to one type-IGoldstone boson. The mixingoftwo Goldstone

elds, on the other hand, produes one type-II Goldstone boson. A short glane at the

lastterm ontherighthand sideof Eq. (3.8) shows thatthe Goldstone{Goldstonemixing

termis,asexpeted, proportionaltotheground-stateexpetationvalueof aommutator

oftwobrokengenerators. Wehavethusestablishedthedesiredresultthatnonzerodensity

of aommutatorof twobroken generators gives rise toa single type-II Goldstone boson.

Inorderforthe onlusionsjustreahedtobereliable,wehavetoshowthat theresultsof

the analysis of the simpleLagrangian (3.9) are appliable tothe muhmore ompliated

ase of Eq. (3.8). A detailed proof may be found in Ref. [III℄ and will not be repeated

here. Instead, we limitour disussion toa simplied version where, nevertheless, all the

essentialsteps are provided.

The ruial observation regarding the harge densities is that one may always hoose

a basis of broken generators so that all generators with nonzero vauum expetation

valuemutually ommute. Wegiveasimpleproofof thisstatementforthe aseof unitary

symmetries[31,32℄. Theset ofvauumexpetationvalues h0jQ

a

j0iofthe generatorsmay

byregardedasavetorv

a

inthespaeoftheadjointrepresentationoftheLiealgebragof

the group G. In the fundamental representation of the unitary group, the generators Q

a

are realized by Hermitianmatries, say T

a

. Now v

a T

a

is also a Hermitianmatrix and as

suh an be diagonalized by a properunitary transformation. After this transformation

v

a T

a

We an now take up the generators that have nonzero density in the ground state and

omplement them to the Cartan subalgebra of g. The rest of the generators is grouped

aordingtothestandardrootdeompositionofLiealgebras[33℄. Thepointisthatwithin

this basis, for any generator there is a unique generator suh that their ommutatorlies

inthe Cartan subalgebra. It is nowproved that the broken generators partiipate in the

lastterm of Eq. (3.8) inpairs and the simple two-eld analysis of Eq. (3.9) is therefore

appliable.

It should, of ourse, also be proved that the same onlusion is true for the mixing

of the Goldstone elds with the radial ones, and of the rad