Stability of superconducting strings coupled to cosmic strings

Alexandru Babeanu* and Betti Hartmann†

School of Engineering Science, Jacobs University Bremen, 28759 Bremen Germany(Received 1 November 2011; published 13 January 2012)

We study the stability of superconducting strings in a Uð1Þlocal � Uð1Þglobal model coupled via a gauge

field interaction term to U(1) Abelian-Higgs strings. The effect of the interaction on current stability is

numerically investigated by varying the relevant parameters within the physical limits of our model. We

find that the propagation speed of transverse (resp. longitudinal) perturbations increases (decreases) with

increasing binding between the superconducting and Abelian-Higgs string. Moreover, we observe that for

small enough width of the flux tube of the superconducting string and/or large enough interaction between

the superconducting and the Abelian-Higgs string superconducting strings cannot carry spacelike, i.e.

magnetic currents. Our model can be seen as a field theoretical realization of bound states of p F-strings

and q superconducting D-strings and has important implications to vorton formation during the evolution

of networks of such strings.

DOI: 10.1103/PhysRevD.85.023518 PACS numbers: 98.80.Cq, 11.27.+d

I. INTRODUCTION

Cosmic strings are very massive topological defectswhich could have formed via the Kibble mechanism [1]during one of the phase transitions in the early Universesuch as the Grand Unification phase transition or theelectroweak phase transition. They are analogous to de-fects in condensed matter physics, such as flux tubes insuperconductors or vortex filaments in superfluid helium[2] being thus intimately connected to spontaneous sym-metry breaking. The study of cosmic strings is currentlymotivated by the possibility that their production is relatedto inflation models resulting from string theory [3]. Braneinflation is a popular inflationary model that can be em-bedded into string theory and predicts the formation ofcosmic string networks at the end of inflation [4]. Forexample, in the framework of type IIB string theory theinflaton field corresponds to the distance between twoDirichlet branes with 3 spatial dimensions (D3-branes)and inflation ends when these two branes collide andannihilate. The production of strings (and lower dimen-sional branes) then results from the collision of these twobranes. Each of the original D3-branes has a U(1) gaugesymmetry that gets broken when the branes annihilate. Ifthe gauge combination is Higgsed, magnetic flux tubes ofthis gauge field carrying Ramond-Ramond (R-R) chargeare D-branes with one spatial dimension, so-calledD-strings. When the gauge combination is confined thefield is condensated into electric flux tubes carryingNeveu Schwarz-Neveu Schwarz (NS-NS) charges andthese objects are fundamental strings (F-strings) [5].D-strings and F-strings are so-called cosmic superstrings[3] which seem to be a generic prediction of supersym-

metric hybrid inflation [6] and grand unified based infla-tionary models [7]. D- and F-strings, however, havedifferent properties than the usual (solitonic) cosmicstrings. The probability of intercommutation of solitonicstrings is equal to one but less than one in the case ofcosmic superstrings. Therefore, solitonic strings do notmerge, while cosmic superstrings tend to form boundstates. When p F-strings and q D-strings interact, theycan merge and form bound states, so-called (p, q)-strings[8] whose properties have been investigated [9]. Eventhough the origin of (p, q)-strings is type IIB string theory,their properties can be investigated in the framework offield theoretical models [10–13]. The influence of gravityon field theoretical (p, q)-strings has been studied in [14].The field theoretical model most frequently used to

study cosmic strings is the U(1) Abelian-Higgs model[15], which possesses static, linelike solutions. These so-lutions are straight strings in the sense that their fields donot depend on the z coordinate. These straight strings canbe interpreted as a local description for potentially curvedobjects, when viewed at large scales. In fact, it is believedthat they are part of intricate, dynamical networks ofinteracting cosmic strings [16], which may bend, collideand eventually form a multitude of loops. If strings have nointernal structure, as in the U(1) Abelian-Higgs model,these cosmic string loops would rapidly disappear throughself-gravitational collapse, radiating all their energy away[17]. Nonetheless, string loops may survive if they carry acurrent in their core. Such current carrying string loops arecalled vortons and, in the simplest scenario, they reach astable, ringlike shape [18–20]. Further studies showed thatvortons could have been produced from cosmic stringsappearing during the electroweak phase transition[21,22]. In order to understand the cosmological implica-tions of such objects it is essential to know about theirstability and is has been suggested that vortons can be

*ababeanu@jacobs-alumni.de†b.hartmann@jacobs-university.de

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unstable under certain conditions [23]. A current carryingstring can be most simply described using a Uð1Þ � Uð1Þmodel [24]. In this case the U(1) Abelian-Higgs model iscoupled to a U(1) Abelian-scalar field model through apotential term. The U(1) symmetry of the latter is unbro-ken, which leads to a locally conserved Noether currentand a globally conserved Noether charge. If the U(1)symmetry is gauged as has been the case in the originalproposal [24], the energy density per unit length divergesdue to the logarithmic falloff of the electromagnetic field.It has hence been suggested to consider the unbroken U(1)symmetry to be global or equivalently the correspondinggauge coupling to be small in order to have localizedsuperconducting strings [25]. In this limit it is possible toapply the formalism developed in [26–28]. This distin-guishes between superconducting strings with timelikecurrents (‘‘electric’’ regime) and spacelike currents (‘‘mag-netic’’ regime) and relates the longitudinal and transversalperturbations on the string to the energy and tension perunit length giving explicit criteria for the stability of theseobjects. In [25] it was shown that within the full Uð1Þ �Uð1Þ model longitudinal perturbations always propagatemore slowly than transverse ones. Moreover, a generallogarithmic equation of state was suggested in [25] whichhas been confirmed numerically to hold [29].

Recent studies have explored a possibility of couplingtwo U(1) Abelian-Higgs models (with both U(1) symme-tries broken) via a gauge field interaction term [30]. Thisinteraction has been motivated by the possible existence ofa separate, dark matter sector in the Universe [31], whichallows for the existence of so-called ‘‘dark strings’’ [32].This dark sector would weakly interact with the standardmodel sector through a small gauge field interaction and ismotivated by recent astrophysical observations [33] thathave shown an excess electronic production in the galaxywith electrons having energies between a few GeV and afew TeV. In [30], the interaction of dark strings with U(1)Abelian-Higgs strings was studied and the possibility offorming bound states was explored. This study was ex-tended to semilocal strings [34] which are solutions of aSUð2Þ � Uð1Þ model with an ungauged SU(2) symmetry.

In this paper we study bound states of a Uð1Þlocal �Uð1Þglobal superconducting string and an Abelian-Higgs

string. We will put the emphasize on the study of thestability of the superconducting strings when these arecoupled via an attractive gauge interaction to Abelian-Higgs strings. Our model can be seen as a toy model for(p,q)-strings in which the attractive interaction mediated bythe scalar fields as suggested in the original model [10] isreplaced by an attractive interaction mediated by the gaugefields. D-strings can carry currents. In [35] the formation ofvortons from superconducting D-strings carrying chiralfermion zero modes has been discussed, while it hasbeen suggested that D-strings could also carry bosoniccurrents in special cases, e.g. if a D-string would be coin-

cident with a D7-brane [3]. In the following we will henceassume that the superconducting string in the bound state isa superconducting D-string.Our paper is organized as follows: we give the model in

Sec. II. We present our results in Sec. III and conclude inSec. IV.

II. THE MODEL

From a qualitative perspective the field theoreticalmodel studied in this paper is a combination of the modelsstudied in [25,29,30]. For the vanishing current limit it canalso be seen as an alternative to the field theoretical toymodel for (p,q)-strings suggested in [10]. In the latter case,the attractive interaction between the p- and the q-string ismediated via a scalar field potential term, while here theattractive interaction is mediated via a gauge field term.The model without currents has been studied in [30]. Here,we extend this investigation assuming that one of thestrings possesses a current. Our model could hence be atoy model for (p,q)-strings where the q D-strings aresuperconducting.The field theoretical model has the following action:

S ¼Z

d4xffiffiffiffiffiffiffi�g

pL; (1)

where g denotes the determinant of the metric with signa-ture (þ���) andL is the Lagrangian density given by

L ¼ � 1

4Fð1Þ��F

��ð1Þ �

1

4Fð2Þ��F

��ð2Þ þ

"22Fð1Þ��F

��ð2Þ

þ ðD��1ÞyðD��1Þ þ ðD��2ÞyðD��2Þþ ð@��3Þyð@��3Þ � �1

4ðj�1j2 � �2

1Þ2

� �2

4ðj�2j2 � �2

2Þ2 ��3

4ðj�3j2 � 2�2

3Þj�3j2

� �4j�2j2j�3j2; (2)

where FðaÞ�� ¼ @�A

ðaÞ� � @�A

ðaÞ� is the field strength tensor

of the U(1) gauge field AðaÞ� , a ¼ 1, 2 and D��a ¼

@��a � ieaAðaÞ� �a is the covariant derivative of the com-

plex scalar field�a, a ¼ 1, 2. The current carrying field�3

is ungauged in order to have localized current carryingstrings. ea, a ¼ 1, 2 and �i,�i, i ¼ 1, 2, 3 denote the gaugecouplings, the self-couplings and the vacuum expectationvalues, respectively. The index (1) refers to the fields andcoupling constants associated to the U(1) Abelian-Higgsmodel describing e.g. the p F-strings in a (p, q)-string. Theindices (2) and (3) are related to the fields and couplingconstants of the Uð1Þlocal � Uð1Þglobal model describing

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superconducting strings, e.g. the q D-strings in a (p, q)-string. In the following (see Ansatz below) the index (2)will denote the fields of the broken U(1) symmetry, whilethe index (3) will denote the unbroken symmetry ofthe superconducting string. "2 is the gauge interactioncoupling between the U(1) Abelian-Higgs string and thebroken U(1) symmetry of the superconducting string. Itcan only take values "2 2 ½0:0; 1:0Þ for mathematical rea-sons as for " � 1 the nature of the differential equation[see (18) and (19)] changes. Moreover, we have "2 2½0:0; 0:001� for experimental reasons described by [30] inthe context of dark strings. The interactions between thetwo scalar fields of the superconducting string, which isresponsible for the existence of the current, is here medi-ated by the constant �4. Note that the constant ð�3�

43Þ=4 in

the quartic potential of j�3j has been eliminated from themodel, as it would simply shift the energies by a constantterm. The Higgs and gauge field masses are mH;i ¼

ffiffiffiffiffi�i

p�i

and mW;i ¼ ei�i, i ¼ 1, 2, respectively. The mass of the

scalar field �3 associated to the unbroken U(1) is given by

mS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�4�

22 � �3�

23

q. The two strings each possess a

scalar core with width �H;i / m�1S;i and a flux/gauge tube

with width �W;i / m�1W;i.

We use the following cylindrically symmetric ansatz forthe scalar fields:

�1ð�;’Þ ¼ �1f1ð�Þein1’;�2ð�;’Þ ¼ �2f2ð�Þein2’;�3ð�;’Þ ¼ �3f3ð�Þeið!tþkzÞ

and the following ansatz for the associated gauge fields

AðaÞ� ð�Þdx� ¼ 1

eaðna � Pað�ÞÞd’ (3)

with a ¼ 1, 2. Here the na 2 Z, a ¼ 1, 2 are integersindexing the vorticity of the fields. Note that if wethink of our model as a toy model for a (p,q)-string withp F-strings and q superconducting D-strings, then n1 � p

and n2 � q. The strings possess magnetic fields ~Ba ¼ Ba ~ezwith Ba ¼ �P0

a=ðea�Þ and magnetic fluxes �a ¼2�na=ea.

The 4-current can be computed from Noether’s theorem[36], which states that if under an infinitesimal transforma-tion � ! �þ ��, where � is a parameter theLagrangian density is invariant up to surface terms, i.e.

L ! Lþ �@�K�; (4)

then there exists a locally conserved 4-current given by

J� ¼ @L@ð@��Þ�� K�: (5)

The explicit expressions in our model read

Jt ¼ !q23f23; Jz ¼ �kq23f

23; (6)

where the integral of Jt and Jz over x and ’ give theNoether charge and total Noether current per unit length,respectively, which are globally conserved. It can be easilyseen that the value of the condensate f3ð0Þ controls theamount of both charge and current per unit length, whilethe quadratic norm 2 :¼ k2 �!2 determines the natureof the 4-current. According to the criterion in [26–28], wewill call the 4-current timelike (electric) when 2 < 0since k2 <!2, spacelike (magnetic) when 2 > 0 sincek2 >!2 or lightlike when 2 ¼ 0 since k2 ¼ !2. Thisterminology is related to the fact that k2 (!2) vanisheswhen convenient Lorentz boosts are applied in the timelike(spacelike) case. Note also that the lightlike case includes! ¼ k ¼ 0, but is not restricted to it.

The energy-momentum tensor T�� ¼ 2g�� @L

@g�� � ��L

is given by

T�� ¼ �X2

i¼1

F��ðiÞ F

ðiÞ�� þ 2"2F

��ð1Þ F

ð2Þ�� þX2

i¼1

½ðD��iÞyD��i

þ ðD��iÞyD��i� þ ð@��3Þy@��3 � ��L: (7)

It is convenient to use the following rescaled quantities:

x ¼ e1�1�; �i ¼ �i

e21; i ¼ 1; 2; 3; 4; (8)

g2 ¼ e2e1

; q2 ¼ �2

�1

; q3 ¼ �3

�1

(9)

and to rescale 2 ! e21�21

2. Then, rescaling theLagrangian density as L ! L=ðe21�4

1Þ it reads

L ¼ � 1

2x2

�P021 þ P02

2

g22

�þ "2

g2

P01P

02

x2�

�f01 þ

P21f

21

x2

�

� q22

�f02 þ

P22f

22

x2

�� q23½2f23 þ f03�

� �1

4ðf21 � 1Þ2 � q42

�2

4ðf22 � 1Þ2

� q43�3

4ðf23 � 2Þf23 � q22q

23�4f

22f

23; (10)

where the dependence of all functions on the rescaledradial distance x has been omitted for simplicity and theprime now and in the following denotes the derivative withrespect to x. In these rescaled variables the energy E andtension T per unit length of these solutions are given by

E ¼ 2�Z 1

0Tttxdx; T ¼ 2�

Z 1

0Tzzxdx; (11)

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where the energy density Ttt and tension density Tz

z are thett and zz components of the energy-momentum tensor T�

�

and read

Ttt ¼ 1

2x2

�P021 þ P02

2

g22

�� "2

g2

P01P

02

x2þ

�f01 þ

P21f

21

x2

�

þ q22

�f02 þ

P22f

22

x2

�þ q23½ �2f23 þ f03� þ

�1

4ðf21 � 1Þ2

þ q42�2

4ðf22 � 1Þ2 þ q43

�3

4ðf23 � 2Þf23

þ q22q23�4f

22f

23 (12)

and

Tzz ¼ 1

2x2

�P021 þ P02

2

g22

�� "2

g2

P01P

02

x2þ

�f01 þ

P21f

21

x2

�

þ q22

�f02 þ

P22f

22

x2

�� q23½ �2f23 � f03� þ

�1

4ðf21 � 1Þ2

þ q42�2

4ðf22 � 1Þ2 þ q43

�3

4ðf23 � 2Þf23

þ q22q23�4f

22f

23; (13)

respectively, where �2 ¼ 2 for k2 >!2 and �2 ¼ �2

for!2 > k2. According to the formalism developed in [27]the propagation velocities of tangential and longitudinalperturbations on superconducting strings are

c2T ¼ T

E; c2L ¼ � dT

dE(14)

and stable string configurations require c2T > 0 and c2L > 0.In the following we will be interested in these quantitiesand we will explore how c2T and c2L change when couplingthe superconducting string to an Abelian-Higgs string via agauge field interaction. In [25] it was found that in theUð1Þlocal � Uð1Þglobal model c2T and c2L are smaller than 1

and that c2T > c2L such that longitudinal perturbations moveslower than transverse ones.

Note that the energy-momentum tensor possesses an off-diagonal component Tz

t . However, since Tzt is proportional

to the integral of !kf23 we can always choose a reference

frame where Tzt vanishes. This can e.g. not be done when

two coupled currents are present on the string and theenergy-momentum tensor has to be diagonalized accord-ingly [37].

A. Equations and boundary conditions

The equations of motion result from the variation of theaction (1) with respect to the gauge and scalar fields. Theseread

2ðxf01Þ0 ¼ �1xðf21 � 1Þf1 þ 2P21f1x

; (15)

2ðxf02Þ0 ¼ �2q22xðf22 � 1Þf2 þ 2�4q

23xf

23f2 þ 2

P22f2x

;

(16)

2ðxf03Þ0 ¼ �3q23xðf23 � 1Þf3 þ 2�4q

22xf3f

22 þ 22xf3

(17)

for the scalar field functions and

ð1� "22Þx�P01

x

�0 ¼ 2P1f21 þ 2"2g2q

22P2f

22; (18)

ð1� "22Þx�P02

x

�0 ¼ 2g22q22P2f

22 þ 2"2g2P1f

21 (19)

for the gauge field functions, respectively. This system ofcoupled, ordinary differential equations needs to be solvednumerically subject to appropriate boundary conditions.The requirement of the regularity at x ¼ 0 leads to thefollowing conditions:

f1ð0Þ ¼ 0; f2ð0Þ ¼ 0; f03ð0Þ ¼ 0;

P1ð0Þ ¼ n1; P2ð0Þ ¼ n2:(20)

Moreover, we want solutions that have finite energy perunit length. This leads to the following conditions at x ¼ 1

f1ð1Þ ¼ 1; f2ð1Þ ¼ 1; f3ð1Þ ¼ 0;

P1ð1Þ ¼ 0; P2ð1Þ ¼ 0:(21)

Note that the trivial solution f3ðxÞ � 0 is a solution to theequations of motion fulfilling the above boundary condi-tions. In order to force the f3ðxÞ to be nontrivial, we imposea further boundary condition f3ð0Þ ¼ � 0. This in turnoverdetermines the problem or to state it differently thechoice of will fix the value of . Practically, we willmake a function ðxÞ and implement a differentialequation for which is of the form 00 ¼ 0. The numericalprogram will then determine for a given value of .Finally, to ensure that the U(1) symmetry associated to

the fields f2, P2 is broken and the U(1) symmetry associ-ated to the field f3 remains unbroken we have to fulfill anumber of conditions for the coupling constants in themodel. To understand these conditions let us look at thepotential associated to the scalar fields of the supercon-ducting string which in rescaled variables reads

V 23ðf22; f23Þ ¼ q42�2

4ðf22 � 1Þ2 þ q43

�3

4ðf23 � 2Þf23

þ q22q23�4f

22f

23: (22)

The conditions then are [29]

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�3q43 <�2q

42; (23)

�3q23 < 2�4q

22; (24)

ffiffiffiffiffiffiffiffiffiffiffiffi�2�3

p< 2�4; (25)

2C�2�4

�23

<q43q42

: (26)

Condition (23) is given by the inequality between theminima: V 23ð1; 0Þ<V 23ð0; 1Þ when x ! 1, such thatthe system chooses the former. Condition (24) is givenby the requirement that the mass of f3ðxÞ given in rescaledquantities by m2

S ¼ 2�4q22 � �3q

23 is positive. Condition

(25) is a restrictive lower bound which ensures that bothV 23ð1; 0Þ and V 23ð0; 1Þ are local minima. Condition (26)was found numerically [29] with C � 1:4 and makes thecondensate stable. When � 0 more general symmetrybreaking conditions were found by replacing the potentialin Eq. (22) with an effective potential, as described in [38].These conditions read:

�3

�q23 � 2

2

�3

�2<�2q

42; (27)

�3q23 � 22 < 2�4q

22; (28)

2 <�3q

23

2� ffiffiffiffiffiffiffiffiffiffiffiffi

�2�4

pq22: (29)

Thus, condition (27) is the generalization of (24), while(28) is the generalization of (24) ensuring that the currentdoes not ‘‘create’’ �3 particles. Also, (29) is a condensatestability condition, found from a theoretical analysis ofsmall perturbations in the field j�3j within the effectivepotential by considering the analogue of a quantum har-monic oscillator. It is interesting to note here that, whengoing back to 2 ! 0, (29) reduces to (26) for C ¼ 2. Thismeans that restriction (26) had been in the literature, in aslightly modified and generalized version and was onlyrediscovered by [29] through a different approach. Notethat condition (27) together with condition (25) leads to thecondition discussed in [25], the so-called phase frequencythreshold condition beyond which no stationary solutionsexist and energy and tension diverge.

III. NUMERICAL RESULTS

This section presents the numerical results obtainedusing a damped Newton method [39] to solve the systemof coupled ordinary differential equations subject to theappropriate boundary conditions. This method uses a qua-

silinearization such that at each iteration step a linearizedproblem is solved by using a spline collocation at Gaussianpoints.The parameters defining the potential are kept constant

for the entire study:

�1 ¼ 2:0; �2 ¼ 2:0; �3 ¼ 80:0;

�4 ¼ 10:0; q2 ¼ 1:0; q3 ¼ 0:37(30)

such that they simultaneously satisfy, with reasonable mar-gins, (23)–(29) in the 2 ¼ 0 limit. Furthermore, the cur-rent was also restricted by the validity of (27) and (28) forall 2.In order to investigate the stability of our solutions we

will show tension-energy characteristics TðEÞ. The currentis controlled by the value f3ð0Þ � . In Fig. 1(a) we showhow the quadratic norm 2 principally depends on thevalue of the condensate for such strings. In Fig. 1(b)we show how the tension T depends in principle on theenergy E. In a generic way, we shall refer to these diagramsas current nature diagram and tension-energy diagram,respectively. Note that all tension-energy diagrams includethe contribution of the Abelian-Higgs string, even whenthis is completely decoupled from the superconductingstring, i.e. for "2 ¼ 0. In this case, this contribution onlyshifts the values of E and T by a constant value.For a convenient future use, we define five important

points, or ‘‘current markers’’, in these diagrams: point ‘‘e’’denotes the lower limit of the electric region, where 2

attains its lowest, negative value, just before falsifyingcondition (27). This corresponds to the phase frequencythreshold discussed in [25] and marks the value of 2

beyond which no stationary solutions exist with the energyE and tension T diverging there. Note that the solutions arestable in the full electric regime with T and E positive anddTdE < 0 between points ‘‘o’’ and ‘‘e’’. Point ‘‘o’’ marks the

transition between the electric and magnetic regions,where 2 ¼ 0. ‘‘c’’ is the critical point in the magneticregion where TðEÞ attains its minimum, with dT

dE ¼ 0, at the

transition between the stable (o-c, dTdE < 0) and unstable

(c-m, dTdE > 0) magnetic regions. ‘‘m’’ gives the upper limit

of the magnetic region. For 2 larger than the value at ‘‘m’’the current carrier field decouples from the Higgs field ofthe cosmic string and we are left with a simple Abelian-Higgs string coupled to another Abelian-Higgs string.Finally, ‘‘s’’ is the point where condition (29) is falsified– region s-m is thus also unstable with respect to (29).These points divide the diagrams into multiple ‘‘currentregions’’. In this line of thought, we define the followingenergy differences: Ee-o :¼ Ee � Eo—energy width of theelectric region; Em-o :¼ Em � Eo—energy width ofthe magnetic region; Ec-o ¼ Ec � Eo—energy width ofthe stable, magnetic region (contained within Em-o);Es-c ¼ Es � Ec—energy width between ‘‘s’’ and ‘‘c’’ de-fined above. The definitions of these four current regions

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are essential for evaluating how the main features of thecurrent evolve when changing various parameters, and willbe extensively used below. We also use the derivative ðdTdEÞsas a measure of the instability of point ‘‘s’’ with respectto (14).

We start by mentioning some limiting cases of ourmodel, which are equivalent to those used in formerstudies—see Sec. III A. We then proceed by investigatingthe effects of the gauge interactions on the current andpresent a detailed analysis of the "2-mediated interactionbetween the broken symmetry of the Abelian-Higgs stringand the broken symmetry of the superconducting string—see Sec. III B.

For convenient future use, we choose to denote the threesectors of our field theoretical model in the following way:‘‘sector 1’’—the broken symmetry of the Abelian-Higgs(dark) string; ‘‘sector 2’’—the broken symmetry of thesuperconducting string; ‘‘sector 3’’—the unbroken sym-metry of the superconducting string. The labeling of ourfunctions: f1, f2, f3, P1, P2 is consistent with this notation.Furthermore, note that the values of E, T, Ee-o, Em-o, Ec-oand Es-c are given in units of 2� in all plots. We use EA�B

as a generic notation for the above mentioned energywidths of the four current regions.

A. Limiting cases

We have checked the consistency of our numericaltechniques by going to certain limiting cases, which havebeen studied before. We have checked the known resultsand found that they agree (for more details see [40]). In[30] the model introduced above was studied for �4 ¼ 0

and for vanishing current f3ðxÞ � 0 describing the (attrac-tive) interaction between a dark string and a cosmic string.In this limiting case, this model can also be seen as analternative field theoretical model to describe (p, q)-strings.In the original proposal [10] the two Abelian-Higgs stringshad been coupled via a potential interaction term, whilehere the interaction is mediated via the gauge fields.The "2 ¼ 0 limit corresponds to an Abelian-Higgs stringand a superconducting string that do not interact.Superconducting strings in a Uð1Þlocal �Uð1Þglobal model

have been studied e.g. in [24,25,29].

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

func

tion

log(x+1)

f1f2f3

P1P2

FIG. 2 (color online). We show the profiles of the functionsdescribing a superconducting string interacting with a cosmicstring with g2 ¼ 1:0, "2 ¼ 0:01, n1 ¼ 1, n2 ¼ 1, ¼ 0:7ð2 ��0:50Þ.

FIG. 1 (color online). We show the principal dependence of 2 on the value of the condensate f3ð0Þ ¼ [Fig. 1(a)] and the principaldependence of the tension T on the energy E [Fig. 1(b)]. In 1(b) the (red) solid curve from o to m corresponds to the magnetic regime,while the curve from o to e given by the þ is the curve for the electric regime.

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B. Superconducting strings interactingwith cosmic strings "2 � 0

This section deals with the gauge interaction betweensector 1 and 2, which is mediated by the parameter "2coupling the two broken symmetries of our model. In orderto understand the influence of "2 on the current the varia-tion of the tension-energy diagram is studied using thecurrent markers and regions as defined above. The parame-ters of the potentials are kept constant to (30). In Fig. 2 weshow the profiles of a typical solution in this case.

In Sec. III B 1 we will give our results for the behavior ofthe current when changing the gauge coupling constant g2,

while keeping "2 ¼ 0. Then, for several values of thisparameter around g2 ¼ 1 we will demonstrate inSec. III B 2 how the current changes when changing thegauge interaction parameter "2. For both cases, thewindings are kept to their lowest possible values n1 ¼ 1,n2 ¼ 1. Finally, the gauge constant is set to g2 ¼ 1 inSec. III B 3, and the influence of the windings n1 and n2is studied.

1. Effect of gauge coupling constant g2

First of all, we illustrate the influence of g2 on the valueof 2 and on the tension-energy diagram TðEÞ in Fig. 3.

0.94

0.95

0.96

0.97

0.98

0.99

1

-1 -0.5 0 0.5 1 1.5

c T2

χ2

g2=0.1g2=0.3g2=0.5g2=0.7g2=0.9g2=1.0g2=1.1g2=1.3g2=1.5g2=1.7g2=1.9

-1

-0.5

0

0.5

1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

c L2

χ2

g2=0.1g2=0.3g2=0.5g2=0.7g2=0.9g2=1.0g2=1.1g2=1.3g2=1.5g2=1.7g2=1.9

FIG. 4 (color online). We show the influence of g2 on the tangential [Fig. 4(a)] and longitudinal [Fig. 4(b)] propagation velocities asfunctions of 2 for "2 ¼ 0 and n1 ¼ n2 ¼ 1.

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

χ2

γ

g2=0.8g2=0.9g2=1.0g2=1.1g2=1.2

1.85

1.9

1.95

2

2.05

2.1

1.9 1.95 2 2.05 2.1 2.15 2.2

T

E

g2=0.8g2=0.9g2=1.0g2=1.1g2=1.2

FIG. 3 (color online). We show the influence of g2 on 2 [Fig. 3(a)] and on TðEÞ [Fig. 3(b)] for "2 ¼ 0 and n1 ¼ n2 ¼ 1.

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Note that with our choice q2 ¼ 1 the value of g2 is equal tothe ratio between the two gauge boson masses mW;1 and

mW;2, g2 ¼ mW;2=mW;1 and as such also equal to the ratio

between the flux tube core widths: g2 ¼ �W;1=�W;2. From

Fig. 3(a) it is obvious that an increase in g2 leads to thedecrease of the maximal possible current on the string (at ¼ 0) before it decouples and leaves behind a standardcosmic string without current. Moreover, the value of beyond which no stationary solutions exist—the so-calledphase frequency threshold which is at2

e � �0:848 for ourchoice of parameters (30)—also decreases. The energy andtension decrease when g2 increases, i.e. when decreasingthe radius of the flux tube in sector 2 with respect to the

radius of the flux tube in sector 1. One can also see that thelarger the coupling g2 the more the electric region 2 � 0is favored over the magnetic one, as 2ð Þ is shifted down-wards which can also be seen in Fig. 3(b), where the size ofthe magnetic branch seems to decrease faster than the sizeof the electric branch. This nontrivial effect indicates thatthe width of the flux tube in sector 2 has a significantinfluence on the nature of the current in sector 3.We also illustrate the influence of g2 on the tangential c

2T

and longitudinal c2L propagation velocities in Figs. 4(a) and4(b), respectively. It is clear that c2T increases with g2within both the electric and magnetic regions, exceptwhen 2 ¼ 0 (point ‘‘o’’), where this influence

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

χ2

γ

ε2=0.000ε2=0.001ε2=0.010ε2=0.050ε2=0.100ε2=0.300ε2=0.600ε2=0.800ε2=0.900

1.91

1.92

1.93

1.94

1.95

1.96

1.97

1.98

1.99

2

2.01

1.94 1.96 1.98 2 2.02 2.04 2.06 2.08

T

E

ε2=0.000ε2=0.001ε2=0.010ε2=0.050ε2=0.100

FIG. 6 (color online). We show the influence of "2 on 2 [Fig. 6(a)] and on TðEÞ [Fig. 6(b)] for g2 ¼ 1 and n1 ¼ n2 ¼ 1.

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

χ2

g2

χ2e

χ2o

χ2c

χ2s

χ2m

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

EA

-B

g2

Ee-oEc-oEs-c

Em-o

FIG. 5 (color online). We show the current markers as functions of g2 in 2 space [Fig. 5(a)] and in E space [Fig. 5(b)] for "2 ¼ 0and n1 ¼ n2 ¼ 1.

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vanishes—a simple consequence of the fact that E ¼ T.This means that the decrease of the width of the flux tube ofthe superconducting string with respect to that of thestandard cosmic string leads to an increase of the propa-gation velocity of the transverse perturbation. On the otherhand, c2L decreases with g2 in the magnetic region andpartly in the electric region—a close look at Fig. 4(b)shows that, as 2 < 0 decreases and the curves approacheach other, some of them intersect. This means that in themagnetic region and partly in the electric region the dif-ference between the transverse and longitudinal propaga-tion speeds becomes larger when decreasing the width ofthe flux tube of the superconducting string.

Figure 4(b) also shows that the transition to the unstable,magnetic region (point ‘‘c’’) is attained for lower 2 as g2increases.

In order to better understand these effects, the value atall current markers and the energy intervals of all currentregions are plotted as functions of g2 in Figs. 5(a) and 5(b),respectively. Figure 5(a) shows that the complete magneticand stable magnetic regions both decrease with g2 in 2

space, while the electric region remains the same (becauseboth 2

e and 2o are constant with 2

e � �0:848 for ourchoice of parameters (30) and 2

o ¼ 0 by definition). Thevalue of 2

s is constant [see (29)], but joins the curve for 2m

at g2 ¼ 1:0. Figure 5(b) shows that the energy widths of allcurrent regions decrease with g2. It can be clearly seen thatthe magnetic region would vanish long before the electricone.

2. Effect of "2

We illustrate the influence of "2 on the current nature2ð Þ and on the tension-energy diagram TðEÞ in Fig. 6 forg2 ¼ 1:0. One can see that the influence is qualitativelysimilar to the influence of g2. Figure 6(a) shows that the

increase of "2 leads to the decrease of the maximal possiblecurrent on the string before it decouples ( ¼ 0 and hencef3ðxÞ � 0). The critical value of beyond which no sta-tionary solutions exist (the so-called phase frequencythreshold at 2

e � �0:848) also decreases. For largeenough "2 we find that 2 < 0 always and the magneticregime completely disappears. We find that this happensfor "2 � 0:84 if g2 ¼ 1.In Fig. 6(b) we show the dependence of T on E for

reasonably small values of "2. Increasing "2 decreases thetension T and energy E such that the curves get shifteddownwards along a line T ¼ E for increasing "2.We also illustrate the influence of "2 on the tangential c

2T

and longitudinal c2L propagation velocities in Figs. 7(a) and7(b), respectively. c2T increases with "2 while c

2L decreases,

0.94

0.95

0.96

0.97

0.98

0.99

1

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

c T2

χ2

ε2=0.000ε2=0.050ε2=0.100ε2=0.300ε2=0.500ε2=0.700ε2=0.800ε2=0.900

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

c L2

χ2

ε2=0.000ε2=0.050ε2=0.100ε2=0.300ε2=0.500ε2=0.700ε2=0.800ε2=0.900

FIG. 7 (color online). We show the influence of "2 on the tangential [Fig. 7(a)] and longitudinal [Fig. 7(b)] propagation velocities asfunctions of 2 for g2 ¼ 0 and n1 ¼ n2 ¼ 1.

1.275

1.2755

1.276

1.2765

1.277

1.2775

1.278

1.2785

1.279

1.278 1.28 1.282 1.284 1.286 1.288 1.29 1.292 1.294 1.296

T

E

ε2=0.900

FIG. 8 (color online). We show the TðEÞ diagram when themagnetic region is no longer present, for g2 ¼ 1:0, "2 ¼ 0:9,n1 ¼ 1, n2 ¼ 1.

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-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Em

-o

ε2

g2=0.1g2=0.4g2=0.7g2=1.0

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Em

-o

ε2

g2=1.3g2=1.6g2=1.9

FIG. 10 (color online). We show the energy range of the magnetic region Em-o in dependence on "2 for several values of g2 andn1 ¼ n2 ¼ 1.

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ec-

o

ε2

g2=0.1g2=0.4g2=0.7g2=1.0

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ec-

o

ε2

g2=1.3g2=1.6g2=1.9

FIG. 11 (color online). We show the energy range of the stable magnetic region Ec�o in dependence on "2 for several values of g2and n1 ¼ n2 ¼ 1.

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ee-

o

ε2

g2=0.1g2=0.4g2=0.7g2=1.0

0

0.01

0.02

0.03

0.04

0.05

0.06

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ee-

o

ε2

g2=1.3g2=1.6g2=1.9

FIG. 9 (color online). We show the energy range of the electric region Ee-o in dependence on "2 for several values of g2.

ALEXANDRU BABEANU AND BETTI HARTMANN PHYSICAL REVIEW D 85, 023518 (2012)

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0

2

4

6

8

10

12

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

dE/d

Ts

ε2

g2=0.1g2=0.4

-100

0

100

200

300

400

500

600

700

800

900

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

dE/d

Ts

ε2

g2=0.7g2=1.0

FIG. 13 (color online). The value of the derivative at current marker ‘‘s’’ ðdTdEÞs in dependence of "2 for several values of g2 andn1 ¼ n2 ¼ 1. The bars indicate the errors in the numerical evaluation of the derivative.

-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.2 0.4 0.6 0.8 1 1.2

χ2

γ

ε2=0.000ε2=0.001ε2=0.010ε2=0.050ε2=0.100ε2=0.300ε2=0.600ε2=0.900

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5

4.4 4.5 4.6 4.7 4.8 4.9 5 5.1 5.2 5.3

T

E

ε2=0.000ε2=0.001ε2=0.010ε2=0.050ε2=0.100

FIG. 14 (color online). We show the influence of "2 on 2 [Fig. 14(a)] and on TðEÞ [Fig. 14(b)] for n1 ¼ 2 and n2 ¼ 3 and g2 ¼ 1.

-0.005

0

0.005

0.01

0.015

0.02

0.025

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Es-

c

ε2

g2=0.1g2=0.4g2=0.7g2=1.0

-0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Es-

c

ε2

g2=1.3g2=1.6g2=1.9

FIG. 12 (color online). We show the energy differences between current markers ‘‘s’’ and ‘‘c’’ given by Es-c in dependence on "2 forseveral values of g2 and n1 ¼ n2 ¼ 1.

STABILITY OF SUPERCONDUCTING STRINGS COUPLED . . . PHYSICAL REVIEW D 85, 023518 (2012)

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within both the electric and magnetic regions. It is remark-able that, when "2 ¼ 0:9 and the magnetic region is nolonger present, the velocities c2T ! 1 and c2L ! 0 as thecondensate vanishes ! 0 at 2 � �0:29. In this case,the TðEÞ curve only keeps part of the electric branch, as canbe seen in Fig. 8.

We are mainly interested in how the energy widths of themain current regions change with "2. In Figs. 9–12, weshow the values of Ee-o, Em-o, Ec-o and Es-c, respectively,in dependence on "2 for several values of g2. We observethat all values decrease with increasing "2. The quantitativeform of these curves depends on g2. A larger g2 shifts thecurves downwards correlating with a decrease of the widthof the flux tube in sector 2.

It is also interesting to note that for certain combinationsof ðg2; "2Þ the energy widths Em-o, Ec-o, Es-c vanish, whichimplies that the magnetic region completely vanishes forthese parameter choices. The larger g2, i.e. the smaller thewidth of the flux tube of the superconducting string with

respect to that of the Abelian-Higgs string, the smaller thevalue of "2 ¼ ð"2Þcr at which the magnetic regime disap-pears and no spacelike currents are possible. We find e.g.ð"2Þcr � 0:73 for g2 ¼ 1:3, ð"2Þcr � 0:62 for g2 ¼ 1:6 andð"2Þcr � 0:47 for g2 ¼ 1:9.Even though Es�c decreases with increasing "2 this is

related to the overall shrinking of the entire magneticbranch. In order to understand how the position of point‘‘s’’ changes relative to the other current markers withinthe magnetic region, the tension-energy derivative (nega-tive propagation velocity for longitudinal waves) at thispoint is plotted as a function of "2 in Fig. 13.One can see that the derivative is always positive and

increases with "2 until reaching the upper, magnetic limitwhere ‘‘s’’ coincides with ‘‘m’’. For g2 > 1:0, this alreadyhappens at "2 ¼ 0. Our results hence imply that – at leastfor the potential parameters used here (30)—the condition(29) is falsified (if at all) only in the region where the stringis already unstable according to criterion (14).

0.01

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0.03

0.04

0.05

0.06

0.07

0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ee-

o

ε2

n1=1n1=2n1=3n1=4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ee-

o

ε2

n1=1n1=2n1=3n1=4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ee-

o

ε2

n1=1n1=2n1=3n1=4

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ee-

o

ε2

n1=1n1=2n1=3n1=4

FIG. 15 (color online). We show the energy range of the electric region Ee�-o in dependence on "2 for n1, n2 ¼ 1, 2, 3, 4 and g2 ¼ 1.

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3. Effect of winding numbers n1 and n2

Here we fix g2 ¼ 1. In this case the energy E and tensionT are simply given by E ¼ T ¼ 2�ðn1 þ n2Þ when thecurrent is absent or neutral, i.e. at points ‘‘m’’ and ‘‘o’’,respectively, if "2 ¼ 0. This is related to the fact that thechoice �1 ¼ 2:0 and �2 ¼ 2:0 corresponds to theBogomolnyi-Prasad-Sommerfield limit if the two stringsdo not carry currents and are not coupled.

First of all, consider the current nature and tension-energy diagrams for n1 ¼ 2, n2 ¼ 3 as shown in Fig. 14.When comparing this with Fig. 6 it becomes clear thatincreasing the windings produces qualitative and quantita-tive changes. We observe that an increase in the windings,which corresponds to an increase in the magnetic fluxes�1, �2, leads to an increase in the maximal possiblecurrent on the superconducting string before the currentdecouples ( ¼ 0 and hence f3ðxÞ � 0). Moreover, the

value of at the phase frequency threshold 2e � �0:848

increases. In order to understand the behavior of the currentregions for different combinations of the windings,we have plotted Ee-oð"2Þ, Em-oð"2Þ and Ec-oð"2Þ inFigs. 15–17, respectively, for all combinations ofðn1; n2Þ 2 f1; 2; 3; 4g � f1; 2; 3; 4g and g2 ¼ 1.We observe that the overall energy widths of all current

regions dramatically increase with the increase of n2.Moreover, an increase in n1 produces a shift of all curves,which also increases with "2. However, the ordering of thecurves for a certain n2 is not solely dependent on n1, butalso on n2, even though we could not deduce the exactordering rules (if any) from our numerical data.Nonetheless, these effects should be connected with theobservation made in [30] stating that strings of equal wind-ings bind stronger. From this one is led to conclude that fora given n2 the energy widths are smallest for n1 ¼ n2.

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0

0.002

0.004

0.006

0.008

0.01

0.012

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Em

-o

ε2

n1=1n1=2n1=3n1=4

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Em

-o

ε2

n1=1n1=2n1=3n1=4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Em

-o

ε2

n1=1n1=2n1=3n1=4

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Em

-o

ε2

n1=1n1=2n1=3n1=4

FIG. 16 (color online). We show the energy range of the magnetic region Em-o in dependence on "2 for n1, n2 ¼ 1, 2, 3, 4 andg2 ¼ 1.

STABILITY OF SUPERCONDUCTING STRINGS COUPLED . . . PHYSICAL REVIEW D 85, 023518 (2012)

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However, we only observe this to be strictly true forn2 ¼ 4.

IV. CONCLUSIONS AND OUTLOOK

In this paper we have studied the interaction of super-conducting strings with Abelian-Higgs strings via an at-tractive gauge field interaction and have investigated thestability of the superconducting strings. We have chosenthe unbroken U(1) symmetry of the superconducting stringto be ungauged in order to maintain the localized characterof the solution in accordance with [25]. We find that thechange of the interaction parameter has similar effects thanthe change of the ratio between the radius of the flux tubeof the superconducting and that of the Abelian-Higgsstring. Increasing either the interaction parameter andhence increasing the attractive interaction between thesuperconducting and Abelian-Higgs string or the ratio ofthe widths leads to a decrease in the maximal possiblecurrent on the superconducting string and a decrease of

the condensate on the string axis given by f3ð0Þ � at thephase frequency threshold beyond which no stationarysolutions exist and at which the tension and energy diverge.While the strings are stable with respect to longitudinal andtransverse perturbation if they carry timelike currents,there is only a limited domain in the case of spacelikecurrents where they are stable with respect to longitudinalperturbations. We find that spacelike currents are not pos-sible for sufficiently strong interaction between the super-conducting and Abelian-Higgs string and/or small enoughratio between the widths of the flux tubes. Increasing thewindings and hence the magnetic fluxes on the stringsleads to an increase in the maximal possible current onthe string and an increase of the condensate on the stringaxis f3ð0Þ ¼ at the phase frequency threshold.Our results have important implications for vorton for-

mation in models where cosmic strings are coupled tosuperconducting strings. In the case of (p,q)-strings a fieldtheoretical model has been developed that describes thebound state of p F-strings and q D-strings by two coupled

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ec-

o

ε2

n1=1n1=2n1=3n1=4

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ec-

o

ε2

n1=1n1=2n1=3n1=4

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ec-

o

ε2

n1=1n1=2n1=3n1=4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Ec-

o

ε2

n1=1n1=2n1=3n1=4

FIG. 17 (color online). We show the energy range of the stable magnetic region Ec-o in dependence on "2 for n1, n2 ¼ 1, 2, 3, 4 andg2 ¼ 1.

ALEXANDRU BABEANU AND BETTI HARTMANN PHYSICAL REVIEW D 85, 023518 (2012)

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Abelian-Higgs strings identifying p and q with the wind-ings of the respective strings [10]. The model studied inthis paper is an extension of this model by allowing theD-string to carry a bosonic current which was shown to bepossible in special cases [3].

ACKNOWLEDGMENTS

We are grateful to P. Peter for discussions and commentson this manuscript.

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