Download pdf - Global versus local cosmic strings from pseudo-anomalous U(1)

26 November 1998

Ž .Physics Letters B 441 1998 52–59

ž /Global versus local cosmic strings from pseudo-anomalous U 1

P. Binetruy a, C. Deffayet a, P. Peter b´a LPTHE, UniÕersite Paris-XI, Batiment 211, F-91405 Orsay Cedex, France´ ˆ

b DARC, ObserÕatoire de Paris-Meudon, UPR 176, CNRS, F-92195 Meudon, France

Received 24 July 1998Editor: R. Gatto

Abstract

Ž .We study the structure of cosmic strings produced at the breaking of an anomalous U 1 gauge symmetry present inmany superstring compactification models. We show that their coupling with the axion necessary in order to cancel theanomalies does not prevent them from being local, even though their energy per unit length is found to divergelogarithmically. We discuss the formation of such strings and the phenomenological constraints that apply to theirparameters. q 1998 Published by Elsevier Science B.V. All rights reserved.

PACS: 98.80.Bp; 98.80.Cq; 98.80.Hw; 11.30.Qc; 02.40.Pc

1. Introduction

Most classes of superstring compactification leadto a spontaneous breaking of a pseudo-anomalousŽ . w xU 1 gauge symmetry 1 whose possible cosmologi-

cal implications in terms of inflation scenarios werew xinvestigated 2,3 . Anomalies are cancelled through a

mechanism which is a four-dimensional version ofthe famous anomaly cancellation mechanism of

w xGreen and Schwarz 4 in the 10-dimensional under-lying theory; the coupling of the dilaton-axion fieldto the gauge fields plays a central role. Cosmic

w xstrings 5 might also be formed in this framework;because of their being coupled to the axion field,such strings were thought to be of the global kindw x2,6–8 . It is our purpose here to show that there

Ž .exists a possibility that at least some of the stringsŽ .formed at the breaking of this anomalous U 1 be

local, in the sense that their energy per unit length

can be localized in a finite region surrounding thestring’s core, even though this energy is formallylogarithmically infinite. It will be shown indeed thatthe axion field configuration can be made to windaround the strings so that any divergence must comefrom the region near the core instead of asymptoti-cally. The cut-off scale that then needs be introducedis thus a purely local quantity, definable in terms ofthe microscopic underlying fields and parameters. Itis even arguable that such a cut-off should be inter-preted as the scale at which the effective model usedthroughout ceases to be valid.

2. The model

The antisymmetric tensor field B which appearsmn

among the massless modes of the closed string playsa fundamental role in the cancellation of the anoma-

0370-2693r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01139-3

( )P. Binetruy et al.rPhysics Letters B 441 1998 52–59´ 53

w xlies. Indeed Green and Schwarz 4 had to introducea counterterm of the form BnFnFnFnF inorder to cancel the gauge anomalies of the 10-dimen-sional theory, F being the gauge field strength.Compactifying six dimensions to get to four space-time dimensions and replacing three of the fieldstrength with compact indices by their backgroundvalues yields a term proportional to BnF. It iswell-known that in four spacetime dimensions anantisymmetric tensor is dual to a pseudoscalara: e E nB lt ;E a. The coupling is then simplymnlt m

A mE a.m

In this formulation, the pseudoscalar belongs toŽ .the same chiral supermultiplet as the dilaton field s

and they form a complex scalar field Sssq ia.This superfield couples in a model independent wayto the gauge fields present in the theory; one has inparticular:

s aa a a a˜LLsy F F q F F , 1Ž .Ý Ýmn mn mn mn4M 4MP Pa a

where M is the reduced Planck scale, the index aP

runs over all gauge groups and

1amn mnrs aF ' ´ F . 2Ž .rs2

Thus a has axion-like couplings and is indeed calledthe string axion. And the vacuum expectation value

² : 2of the dilaton s yields the gauge coupling 1rg .An abelian symmetry with gauge field A maym

Ž .seem to have mixed anomalies: under A ™A qm m

E am

1 a a˜d LLsy d a F F .ÝGS mn mn2a

But this can be cancelled by an appropriate shift ofthe string axion a. Since there is a single model-in-dependent axion, only one abelian symmetry, hence-

Ž .forth referred to as U 1 , may be pseudo-anoma-X

lous.The kinetic term for the dilaton-axion fields is

described at the supersymmetric level by the D-termŽ .of the Kahler function Ksyln SqS . This may¨

be modified to include the Green–Schwarz termw x Ž .A E a 1 : Ksyln SqSy4d V where V is am m GS

Ž .vector superfield describing the anomalous U 1 X

vector supermultiplet. This D-term includes otherterms, such as a mass term for the A gauge field.m

The constant d may be computed in the frame-GS

work of the weakly coupled string and is found to bew x1 :

1d s X , 3Ž .ÝGS i2192p i

where X are the charges of the different fields underiŽ .U 1 .X

We are therefore led to consider the general classŽof models described by the Lagrangian restricted

.here to the bosonic fields

† mLLsy D F D FŽ .Ž .m i i

1 amn mn ˜y F F y F Fmn mn2 ž /M4 g P

12 2 m m myd M A A qd M A E ay E aE aGS P m GS P m m4

1 2 2y m a yV F , 4Ž . Ž .i4

where we have set the dilaton field to its vacuum² : 2expectation value s s1rg , we have included a

mass term for the axion, without specifying its ori-gin, and we have introduced scalar fields F carry-i

Ž .ing the integer charge X under the U 1 symme-i X

try; the covariant derivative is defined by

D mF ' E my iX A m F , 5Ž .Ž .i i i

Ž .and the potential V F byi

g 22† 2V F ' F X F qd M . 6Ž . Ž .Ž .i i i i GS P2

The Green–Schwarz coefficient d and the axionGS

field a have been rescaled by a factor g 2.Ž .The lagrangian 4 is invariant under the follow-

ing local gauge transformation with gauge parameterŽ m.a x

F ™F ei X i a , A ™A qE a ,i i m m m

a™aq2 M d a , 7Ž .P GS

2 mn ˜Ž .the transformation of the term ar4 g M F FP mn

cancels the variation of the effective lagrangianŽ 2 .due to the anomaly, namely d LLsy 1r2 g d aGS

mn ŽF F assuming we are also transforming themn

Ž ..fermions of the theory not written explicitly in 4 .Making a rigid gauge transformation with parameteras2p without changing a as a first step but

( )P. Binetruy et al.rPhysics Letters B 441 1998 52–59´54

Ž .transforming the other fields including the fermions ,leads us to interpret a as a periodic field of period4pd M through the redefinition a ™ a yGS P

4pd M which leaves the lagrangian invariant. ItGS P

is also manifest in the following rewriting of thekinetic term and of the axionic u-term in LL where itis clear that a behaves like a phase:

1 amn mn ˜LL sy F F y F Fkin ,u mn mn2 ž /M4 g P

2mE him 2 2 myE f E f yf X yAi m i i i ž /Xi

2mE a2 2 myM d yA , 8Ž .P GS ž /2 M dP GS

ih i Žwhere we have set F 'f e f being the modu-i i i.lus of F .i

Let us now work out the Higgs mechanism in thiscontext. We consider for the sake of simplicity asingle scalar field F of negative charge X and wedrop consequently the i indices, LL can bekin,u

rewritten

2 2 2 2LL sy M d qf Xkin ,u P GS

=

21 m 2 mM d E aqf XE hP GS2mA y 2 2 2 2M d qf XP GS

22 2 2 2 m mf M d X E a E hP GSy y2 2 2 2 2 M d XM d qf X P GSP GS

1mny F Fmn24 g

2 2d f X a hGSq y2 2 2 2 2ž 2 M d X2 g M d qf X P GSP GS

1 2M d aqf XP GS h2 mn˜q F F . 9Ž .mn2 2 2 2 /M d qf Xp GS

The linear combination appearing in this lastequation

a hy 10Ž .

2 M d XP GS

is the only gauge invariant linear combination of h

Ž .and a up to a constant . The other one1 2M d aqf XP GS h2

ll' 11Ž .2 2 2 2M d qf XP GS

has the property of being linearly independent of theprevious one and of transforming under a gauge

Ž .transformation 7 as ll™ llqa . We now assumeexplicitly that F takes its vacuum expectation value² † : 2 Ž .F F 'r in order to minimize the potential 6 :

r 2 syd M 2rX . 12Ž .GS P

We are left, among other fields, with a massivescalar Higgs field corresponding to the modulus of F

of mass m given byX

m2 s2 g 2r 2 X 2 sy2d Xg 2M 2 13Ž .X GS P

and we define

'a h 2 rM d XP GSa' y 14Ž .ˆ

2 2 2 22 M d X (M d qr XP GS P GS

and

1 M 2 g 4P2 2 2 2 2F s M d qr XŽ .a P GS4 2 2128p r X

21 m 1X2 4s M g 1q 15Ž .P4 2 2ž /ž /M128p 2 g XP

so that with r being set:

22 2 2 2 m mw xLL sy M d qr X A yE llkin ,u P GS

a dˆ GS1 m mny E aE aq q ll F Fˆ ˆm mn2 2 232p F 2 qa

1mny F F 16Ž .mn24 g

we can now make a gauge transformation to cancelE m ll by setting asyllqb where b is a constantparameter. This leaves us with

m2A 1m mLL sy A A y E aE aˆ ˆkin ,u m m222 g

a 1ˆmn mn˜q F F y F F , 17Ž .mn mn2 232p F 4 ga


where m given byA

2 2 2 2 2 2m s2 g r X qM dA P GS

2m 1X2sm 1q 18Ž .X 2 2ž /M 2 g XP

is the mass of the gauge field after the symmetrybreaking. The remaining symmetry

32p 2Faa™aq d b 19Ž .ˆ ˆ GS22 g

is the rigid Peccei–Quinn symmetry which compen-sates for the anomalous term arising from a rigidphase transformation of the parameter b.

To summarize we have seen that in the presenceof the axion the gauge boson of the pseudo-anoma-lous symmetry absorbs a linear combination ll ofthe axion and of the phase of the Higgs field. We areleft with a rigid Peccei–Quinn symmetry, the rem-nant axion being the other linear combination a ofˆthe original string axion and of the phase of theHiggs field.

( )3. Pseudo-anomalous U 1 strings

Cosmic strings can be found as solutions of theŽ .field equations derivable from Eq. 4 provided the

Ž .underlying U 1 symmetry is indeed broken, whichimplies that at least one of the eigenvalues X isi

negative. This is the first case we shall consider here,so we shall in this section assume again only onefield F with the charge X, with X-0. Assuming a

w xNielsen–Olesen-like solution along the z-axis 9 , weset, in cylindrical coordinates,

Fsf r eih , hsnu , 20Ž . Ž .for a string with winding number n. This yields thefollowing Euler–Lagrange equations

1m mn 2˜

I as2d M E A y F F qm a ,GS P m mn22 g MP

21Ž .2 2 2 2

Ifsf E hyX A qg Xf Xf qd M ,Ž . Ž .m m GS P

22Ž .

2 m mE f E hyX A s0 , 23Ž . Ž .m

1 amn mn n 2 2 n˜E F yF sd M E ay2d M Am GS P GS P2 ž /Mg P

q2 Xf 2 E nhyX An ,Ž .24Ž .

from which the string properties can be derived.Ž .Eq. 24 can be greatly simplified: first we make

˜mnŽ .use of Eq. 2 , which implies E F s0, and thenm

Ž . nwe derive Eq. 24 with respect to x . This gives,Ž . Ž .upon using Eqs. 21 and 23 ,

˜mn 2 2F F s2m M g a , 25Ž .mn P

Ž .and, with the help of Eq. 24 ,

1 1mn mn n n˜E F s F E aqJJ qJ , 26Ž .m m2 Mg P

where the currents are defined as

J m sy2 Xf 2 E mhyX A m , 27Ž . Ž .

and

JJm syd M E may2d M A m . 28Ž .Ž .GS P GS P

Ž . Ž .Eqs. 21 and 23 then simply express those twocurrents conservation EPJsEPJJs0, when account

Ž .is taken of Eq. 25 .The standard paradigm concerning the strings ob-

tained in this simple model states that the presenceof the axion makes the string global in the followingsense: even for a vanishing a, A behaves asymptot-m

ically in such a way as to compensate for the HiggsŽ .field energy density i.e., A ™yE hrX and there-m m

fore yields an energy per unit length which divergesasymptotically. It should be clear however that thebehaviour of a could be different; indeed, it could aswell compensate for this divergence as we will nowshow. In this case then, a divergence is still to befound but this time at a small distance near the stringcore, so that the total energy is localised in a finiteregion of space. This is in striking contrast with thecase of a global string where the divergent behaviourarises because the energy is not localized and a largedistance cut-off must be introduced.


In order to examine the behaviour of the fieldsand the required asymptotics, we need the stressenergy tensor

d LLm mg mT sy2 g qd LL , 29Ž .n ngnd g

which reads explicitely

21mn m n mnT s2 E fE fy g EfŽ .2

11rm n mnq F F y g FPFŽ .r 42g

1 2 2 mny m a g4

21 2 mn 2 2y g g Xf qd MŽ .GS P2

11m n mn 2q JJ JJ y g JJ22 22d MGS P

11m n mn 2q J J y g J 30Ž .22 22 X f

where account has been taken of the field equations.The energy per unit length U and tension T will thenbe defined respectively as

Us du rdrT t t and Tsy du rdrT z z , 31Ž .H HThe question as to whether the corresponding stringsolution is local or global is then equivalent toasking whether these quantities are asymptotically

Ž .convergent i.e. at large distances .Ž .It can be seen on Eq. 30 that only the last two

terms can be potential source of divergences. Thew xNielsen–Olesen 9 solution for the very last term

consists in saying that A is pure gauge, namelym

lim D F s 0, so that, as already argued,r ™` m

lim A syE hrX. With this solution, settingr ™` m m

Ž .as0 implies that the second to last term in Eq. 30should diverge logarithmically for r™`. However,at this point, it should be remembered that a can beinterpreted as a periodic field of period 4pd MGS PŽas long as a cosine-like mass term is not included asis usually the case at very low temperatures if this

.axion is to solve the strong CP problem of QCD and

therefore can be assigned a variation along h. Infact, setting

2d MGS Pas h , 32Ž .

X

a perfectly legitimate choice, regularizes the integralsŽ .in Eqs. 31 , at least in the r™` region.

Ž .The solution 32 turns out, as can be explicitlyŽ . Ž .checked using Eqs. 21 and 25 , to be the only

possible non trivial and asymptotically convergingsolution. In particular, no dependence in the string

Ž .internal coordinates z and t in our special case canbe obtained. This means that the simple model usedhere cannot lead to current-carrying cosmic stringsw x Ž .10,11 . Moreover, the stationary solution 32 shows

˜mnthe axion gradient to be orthogonal to F , i.e.,˜mn Ž .E aF s0. Therefore, Eqs. 21-26 reduces to them

w xusual Nielsen–Olesen set of equations 9 , with theaxion coupling using the string solution as a sourceterm. It is therefore not surprising that the resultingstring turns out to be local.

Ž .The total energy per unit length and tension ishowever not finite in this simple string model for itcontains the term

2dr d M nGS PUs f.p. q2p yd M A ,H GS P už /r X

33Ž .Ž .f.p. denoting the finite part of the integral so that,since A must vanish by symmetry in the stringu

core, one ends up with

2d M n RGS P A

Us f.p. q2p ln , 34Ž .ž / ž /X ra

where R is the radius at which A reaches itsA m

asymptotic behaviour, i.e., roughly its ComptonŽ .wavelength m given in 18 , while r is defined asA a

Ž .the radius at which the effective field theory 4ceases to be valid, presumably of order My1 ; theP

correction factor is thence expected to be of order< Ž 2 . <ln d g , which is of order unity for most theo-GS

ries. Hence, as claimed, the strings in this model canbe made local with a logarithmically divergent en-ergy. The regularization scale r is however a shorta

distance cut-off, solely dependent on the microscopicstructure and does not involve neither the interestingdistance nor its curvature radius. In particular, the


gravitational properties of the corresponding stringsw xare those of a usual Kibble–Vilenkin string 12 ,

given the equation of state is that of the Goto-Nambustring UsTsconst., and the light deflection is

w xindependent of the impact parameter 13 .

4. Local string genesis

Forming cosmic strings during a phase transitionis a very complicated problem involving thermal and

w xquantum phase fluctuations 14 . As it is far frombeing clear how will a and h fluctuations be corre-

Ž .lated even though they presumably will , one canconsider to begin with the possibility that a networkof two different kinds of strings will be formed rightafter the phase transition, call them a-strings andh-strings, with the meaning that an a-string is gener-

Žated whenever the axion field winds ordinary axion.string while an h-string appears when the Higgs

field F winds. Both kinds of strings are initiallyglobal since for both of them, only part of thecovariant derivatives can be made to vanish. Wehowever expect the string network to consist, aftersome time, in only these local strings together withthe usual global axionic strings.

Let us consider and axionic string with no Higgsm wwinding: as A /0, the vacuum solution Fsr Eq.

Ž .x12 is not a solution, and thus the axionic stringfield configuration is unstable. As a result of Eq.Ž .22 , the Higgs field amplitude tends to vanish in thestring core. At this point, it becomes, near the core,topologically possible for its phase to start windingaround the string, which it will do since this mini-mizes the total energy while satisfying the topologi-cal requirement that A flux be quantized. Such am

winding will propagate away from the string.Conversely, consider the stability of an h-string

with as0. The conservation of JJ implies, as onecan fix E A m s0, that I as0, whose generalm

Ž < < .time-dependent solution is asa r " t . Gives thecylindrical symmetry, this solution can be further

Ž .separated into as f ry t u . This means that havinga winding of a that sets up propagating away fromthe string is among the solutions. As this configura-tion ultimately would minimize the total energy,provided lim fsyd M r2 X, this means thatt ™` GS P

the original string is again unstable and will evolve

into the stationary solution that we derived in theprevious section. Note finally that if, instead ofconsidering the variables a and h, one had decidedto treat the formation problem by means of the newdynamical variables a and ll , then it would haveˆbeen clear from the outset that the resulting stringconfigurations could consist in two different cate-gories, namely global axionic strings with a windingof a, and local strings with ll winding.ˆ

It should be remarked at this point that these timeevolution can in fact only be accelerated when onetakes into account the coupling between a and h: ifany one of them is winding, then the other one willexhibit a tendency to also wind, in order to locallyminimize the energy density. Indeed, it is not evenreally clear whether the string configurations westarted with would even be present at the stringforming phase transition. What is clear, however, isthat after some time, all the string network wouldconsist of local strings having no long distance inter-actions. This means in particular that the relevantscale, if no inflationary period is to occur after thestring formations, should not exceed the GUT scalein order to avoid cosmological contradictions.

5. Conclusions

Ž .Spontaneous breaking of a pseud-anomalous U 1gauge symmetry leads to the formation of cosmicstrings whose energy per unit length is localizedaround their cores, contrary to what the presence ofthe axion field in these theories might have sug-gested. This happens in the simple case we’ve con-sidered here, namely that of a single scalar fieldacquiring a VEV at the symmetry breaking. In orderto be general, this result should be generalized to thecase where more than one field gets a VEV; this wenow prove.

Ž .The potential we consider is that given by Eq. 6which, in fully generality, can be rewritten in theform

g 22† 2V F s F XFqd M , 35Ž . Ž .Ž .i GS P2

where X is an N=N hermitian matrix, and F takesvalues in an N-dimensional vector space VV . We


denote by p the number of negative eigenvalues ofX and f the restriction of F to that subspaceVV gVV spanned by the eigenvectors of X withp

negative eigenvalues. Once diagonalized, X can bewritten as

M 0Xs , 36Ž .ž /0 P

with M and P containing respectively the negativeand positive eigenvalues.

Ž . Ž .Eq. 35 admits an accidental U N symmetry, ofŽ .which the anomalous U 1 is part; this is not a

Ž . Ž .simple U N = U 1 symmetry as each field compo-Ž .nent transforms differently under U 1 as indicated

Ž .on Eq. 7 . The vacuum configuration is now givenby

² † : 2f Mf syd M , 37Ž .GS P

so it would seem that the remnant symmetry wouldŽ . Ž . Ž .be U Nyp , i.e. a scheme U N ™U Nyp , and

w xa topologically trivial vacuum manifold 5 . Hence,one would naively not expect cosmic string forma-tion in such a model. This conclusion is in fact not

Ž .correct, as only part of the original U N is gauged,Ž .namely the anomalous U 1 subgroup, leading to a

Nœther current

m † m m † 2 m †J A ig f ME fy E f Mf q2 g A f Mf ,Ž .38Ž .

which can be made nonzero by imposing a phaseŽ .variation for f as ;exp inu . Once set to a nonzero

value, this current will remain so for topologicalw xreasons 15 , being called a semi-local or embedded

defect. All the previous discussions concerning thesimple ps1 model hold also for these vortices,including their coupling to the axion field.

The cosmological evolution of the network ofstrings formed in these theories also leads to seriousconstraints on the Green–Schwarz coefficient pro-vided no domain wall form connecting the strings;

Žotherwise, the network is known to rapidly i.e. in.less than a Hubble time decay into massive radia-

tion and the usual constraint relative to the axionw xmass would hold 5,16 . If however the string net-

work is considered essentially stable, then its impacton the microwave background limits the symmetrybreaking scale d M through the observational re-GS P

quirement that the temperature fluctuations be notw xtoo large 17 , i.e.

GUQ10y6 ,y2 Ž 2 .with G the Newton constant GsM r 8p .P

Therefore, the cosmological constraint reads

d Q10y2 , 39Ž .GS

a very restrictive constraint indeed. It should beremarked at this point that the usual domain wallformation leading to a rapid evaporation of the net-work does not seem valid for the string solution weconsidered. In fact, the axion a, defined through Eq.ˆŽ . Ž .14 , as the solution 32 set up, vanishes every-where. Therefore, when the Peccei–Quinn symmetryis broken, the axion itself does not have to wind

Žaround these strings it’ll do so however around the.ordinary axionic strings present in the model . Hence,

it has no particular reason for taking values in all itsallowed vacuum manifold so that no domain wallwill form.

The strings that we have discussed here mightappear in connection with a scenario of inflation.

Ž .Indeed, the potential 6 is used for inflation in thew xscenario known as D-term inflation 18 : inflation

Ž .takes place in a direction neutral under U 1 and thecorresponding vacuum energy is simply given by:

1 2 2 4V s g d M . 40Ž .0 GS P2

Ž .The U 1 -breaking minimum is reached after infla-tion, which leads to cosmic strings formation. Suchan inflation era cannot therefore dilute the density ofcosmic strings and one must study a mixed scenariow x7 . It is interesting to note that, under the assump-tion that microwave background anisotropies are pre-dominantly produced by inflation, the experimental

w xdata puts a constraint 3,8,19 on the scale j'1r2 Ž .d M which is stronger than 39 . Several waysGS P

w xhave been proposed 3,19 in order to lower thisscale. They would at the same time ease the con-

Ž .straint 39 .A final remark concerning currents is in order at

this point. The strings whose structure we haveinvestigated here are expected to be coupled to manyfields, fermionic in particular. It is well known thatsuch couplings yield the possibility for the fermionspresent in the theory to condense in the string core in

w xthe form of so-called zero modes 20 which, uponquantization, give rise to superconducting currents


w x Ž10 . Note indeed the presence of anomalies alongthe worldsheet which was suggested to imply current

w x .inflows 6 . Besides, currents tend to raise thestress-energy tensor degeneracy in such a way thatthe energy per unit length and tension become dy-namical variables. For loop solutions, this means awhole new class of equilibrium solutions, namedvortons, whose stability would imply a cosmological

w xcatastrophe 21 . If these objects were to form, Eq.Ž .39 would change into a drastically stronger con-straint. Issues such as whether supersymmetry break-

w xing might destabilize the currents 22 , thereby effec-tively curing the model from the vorton problem,still deserve investigation.

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