Collision of cosmic superstrings

Collision of cosmic superstrings

E. J. Copeland*School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

H. Firouzjahi†

Physics Department, McGill University, 3600 University Street, Montreal, H3A 2T8, Canada

T. W. B. Kibble‡

Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom

D. A. Steerx

APC, University Paris 7, 10, Rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France(Received 7 January 2008; published 19 March 2008)

We study the formation of three-string junctions between �p; q�-cosmic superstrings, and collisionsbetween such strings and show that kinematic constraints analogous to those found previously forcollisions of Nambu-Goto strings apply here too, with suitable modifications to take account of theadditional requirements of flux conservation. We examine in detail several examples involving collisionsbetween strings with low values of p and q, and also examine the rates of growth or shrinkage of strings ata junction. Finally, we briefly discuss the formation of junctions for strings in a warped space, specificallywith a Klebanov-Strassler throat, and show that similar constraints still apply with changes to theparameters taking account of the warping and the background flux.

DOI: 10.1103/PhysRevD.77.063521 PACS numbers: 98.80.Cq

I. INTRODUCTION

One of the most plausible scenarios leading to the for-mation of cosmic strings [1,2] is that of brane inflation [3].In these models, the inflaton field is the distance between aD3-brane and an anti-D3-brane. There is an attractiveCoulombic potential between them, which may lead to aslow-roll inflation. In models of warped brane inflation [4],the fine-tuning problem associated with the flatness of thepotential in the original models of brane inflation may beless severe. Furthermore, due to warping the tension spec-trum of cosmic superstrings, �, is much smaller than thenaive expectation, m2

s , where ms is the mass scale of stringtheory. Consequently,G�, the dimensionless number mea-suring the tension of cosmic superstrings in units of theNewton constant G, can be lowered below the currentobservational bounds [5,6].

A brane collision creates both fundamental (F-) stringsand D-strings (D1-branes), as well as �p; q�-strings, com-posites of p F-strings and q D-strings [7,8]. When cosmicsuperstrings of different types collide, they cannot inter-commute, instead they exchange partners and form a junc-tion. This is a simple consequence of charge conservationat the junction of colliding �p; q�-strings and is in contrastto ordinary Abelian gauge strings, where two collidingstrings usually exchange partners and intercommute withthe intercommutation probability of order unity. Most work

done to date on cosmic strings has involved such Abelianstrings because of the possibility that they could haveformed in the early universe as a result of phase transitionspossibly associated with the grand unified theory (GUT)scale. The interest in strings with junctions has been moreat the level of a curiosity although earlier work has con-sidered them in relation to monopoles connected to strings[9] and in terms of non-Abelian gauge theories [10,11].However, it has been the recent realization that there couldbe superstrings of comic length that has prompted peopleto take seriously the implications of junction formingstrings, in particular, could there be observational smokingguns that would enable us to determine whether they reallyexist? It is primarily this prospect which motivates us toconsider the kinematics associated with �p; q� strings.

Previous studies [12–14] dealt with three-string junc-tions between Nambu-Goto (NG) strings. However�p; q�-strings carry fluxes of a gauge field and are thereforedescribed instead by the Dirac-Born-Infeld (DBI) action.Constraints arise not only from conservation of energy andmomentum but also from charge conservation. The con-ditions for the formation of junction between �p; q�-stringsat the static level was studied in [15,16]. For the system tobe stationary and supersymmetric, the orientations ofstrings are controlled by charge conservations.

Our aim here is to study how this difference affects thebehavior of strings at junctions. A very important feature ofthe DBI action is that, as in the NG case, it is reparamet-rization invariant; because of that, many of the previousresults carry over, but with important changes. Throughout,we work in the probe-brane approximation, neglecting

*[email protected]†[email protected]‡[email protected]@apc.univ-paris7.fr

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backreaction on the space-time metric. The layout of thepaper is as follows: in Sec. II we derive the requiredequations of motion and charge conservation conditionsfor �p; q� strings at a junction; and in Sec. III we show thatit is consistent to write everything in the temporal gauge.This is followed in Sec. IV by a detailed analysis of thecollision of strings with different charges, meeting at anarbitrary angle and relative velocity. In particular we deriveanalytic expressions which show the conditions underwhich junctions can form in terms of the string tensions,as well as their angle and speed of approach, includingsome specific examples. In Sec. V we extend our analysisto include the collision of cosmic superstrings in a warpedthroat, making the calculation applicable to the stringmotivated models of warped brane inflation. Finally weconclude in Sec. VI, where some comments on averageproperties of the network are also given.

II. EQUATIONS OF MOTION AND CHARGECONSERVATION

In this section we set up the equations of motion forcharged �p; q�-strings meeting at a junction, and discussthe question of gauge fixing. Consider first a single �p; q�string evolving in a flat target space-time metric �� (� �0; . . . ; 3), with position X��a� where the two world sheetcoordinates are �a � ��;��. The induced metric on thestring world sheet, �ab, is given by

�ab � ��X�;aX�;b: (1)

Similarly to the Nambu-Goto action, the DBI action enjoysworld sheet reparametrization invariance. Thus two of the3 degrees of freedom of �ab can be eliminated by imposingthe conformal gauge; that is choosing f�; �g such that �abexhibits explicitly the conformal flatness common to all 2Dspaces:

_X 2 � X02 � 0; _X � X0 � 0: (2)

The prime and over-dot denote derivatives with respect to� and � respectively and the inner products in (2) is definedvia the metric �� with the signature ��;�;�;��. As weshall show later, it is also consistent to impose the temporalgauge condition, choosing the temporal world sheet coor-dinate � to coincide with the global time coordinate X0. Forthe moment we do not impose this condition.

As discussed in [17,18], a �p; q�-string is a bound stateof p fundamental strings, F-strings, and q D1-brane, D-strings, where p and q are integer coprime numbers.Alternatively one can view the �p; q�-string with q � 0as p units of electric flux dissolved on the world volumeof q coincident D-strings. In flat space-time and in theabsence of the background fluxes the Chern-Simons termsvanish and the action is given by the DBI part. Working inthe conformal gauge (2), the action for the �p; q�-string is[18]

S � ��Zd�d�

��j�ab � �Fabj

q

� ��Zd�d�

��X02 _X2 � �2F2

��

q; (3)

where � � jqj=�gs�� is the tension of q coincident D-strings, � � 2�0 and gs is the perturbative string cou-pling. The electric flux p is given by p � @L=@F�� whereS �

Rd�L.

Now consider three cosmic �pi; qi�-strings (labeled byindex i � 1, 2, 3) meeting at a junction. The action for thesystem is

S � �Xi

�i

Zd�d�

��X02i _X2

i � �2Fi2��

q�si��

�Xi

Zd��fi � �Xi�si��; �� X��

� gi�Ai��si��; �� _siA

i��si��; �� A�� (4)

where f�i and gi are the Lagrange multipliers, and

�i �jqijgs�

: (5)

The f�i term imposes that the position X�i �si��; �� of theendpoint of each of the three strings (as imposed by thetheta function in �) coincides with the position �X�� ofthe vertex. The gi term performs a similar task for the U(1)gauge potential Aia, ensuring that its component tangent tothe vertex worldline coincides with the single componentof the gauge potential �A confined to the worldline. Theaction is invariant under the U(1) gauge transformationAia ! Aia � @a�i for a � ��; ��, provided that we alsomake the transformation

�A! �A� �d=d�� ;

with the identification

�� i�si��; ��;

which ensures that the �i from the different legs match upcorrectly.

The equations of motion for the gauge field componentsAi� and Ai� respectively are (no summation over i)

�i@��2Fi��si�� X02i _X2

i � �2Fi2��

q � �gi��si � ��; (6)

�i@��2Fi��si�� X02i _X2

i � �2Fi2��

q � gi _si��si � ��: (7)

The equation of motion for X� is

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�i

�@�X02i _X�i �si�� X02i _X2

i � �2Fi2��

q � @�_X2i X0�i �si��

�X02i _X2i � �

2Fi2��q �

� f�i ��si � ��: (8)

The electric flux pi along each string, determining thenumber of F-strings on the D-string world volume, is givenby pi � �L=�Fi��, which gives

pi ��2�iFi��

�X02i _X2i � �

2Fi2��q : (9)

From Eqs. (6) and (7) we see that @�pi � @�pi � 0. This isa manifestation of the fact that in 2-D the gauge fieldequations are trivial.

Now define

�� i ��2�2

i � p2i

q�

��p2i �

q2i

g2s

s(10)

which, as we show below, is nothing other than the tensionof the ith string (up to a factor of �). Then from Eq. (9) oneobtains that

�Fi�� pi��iX02i ; (11)

where the minus sign comes from the fact that in Eq. (9)both Fi�� and pi have the same sign while from Eq. (2) wehave X02i � � _X2

i < 0. Furthermore

��X02i _X2

i � �2Fi2��

q� �

��i

��iX02i : (12)

Using this in the equation of motion for X� we find

�X �i � X

�00i � 0; (13)

with the following boundary condition at � � si��:

�� i�X�0i � _si _X�i � � ��f

�i : (14)

Similarly, the boundary conditions from Eqs. (6) and (7)imply

gi � pi: (15)

On the other hand, varying the action with respect to �X,�A�, and �A�, respectively, yields

Xi

f�i �Xi

gi � 0: (16)

Using these in Eqs. (14) and (15), one obtains the followingconservation laws:

Xi

��i�X�0i � _si _X�i � � 0; (17)

and

Xi

pi � 0: (18)

Physically (18) indicates the electric flux conservation atthe junction. With this convention, all fluxes are flowinginto to the junction.

Varying the action with respect to the Lagrange multi-pliers f�i and gi respectively imposes the desired con-straints for the fields at the junction � � si��;

X�i �si��; �� X�;

Ai��si��; �� _siAi��si��; �� A��;(19)

where the second equation is subject to the U(1) gaugeinvariance as described previously. Finally, one can checkthat the equation coming from the variation of the actionwith respect to si is automatically satisfied

�S�si�Zd��i

��X02i _X2

i � �2Fi2��

q� fi�X

0�i � giA

0i�

� gi@�A�� 0; (20)

where all quantities labeled by indices i are evaluated at� � si��. To obtain the final result Eqs. (11), (12), (14),and (15) were used.

Motivated by electric flux conservation (18), one expectsthe same conservation law for the D-string charges:Piqi � 0. To see this, recall that D-strings are charged

under the Ramond-Ramond two-form C�2� via the Chern-Simons part of the action [18]

SiCS �qi2�

Zd�d��ab@aX

�i @bX

�i C�2��: (21)

The action also contains the kinetic energy for C�2�. Thisaction is invariant under the gauge transformation

C�2�� ! C�2�� @�� @��: (22)

Under this gauge transformation, the sum of the Chern-Simons terms for the three strings at the junction trans-forms according to

�SCS � �Xi

qi�

Zd��

dX�i �si; ��d�

� �Xi

qi�

Zd��

d �X�

d�:

In order to keep the action invariant under the gauge trans-formation (22), we require

Xi

qi � 0; (23)

as expected.In our analysis so far we assumed that qi � 0 so a

�p; q�-string is obtained by turning p units of electric fluxon the world volume of q D-strings. If q � 0 for somestrings, then in Eq. (3) one replaces � by the tension of pF-strings, p=�, with no gauge field on the world volume.All our results, namely, Eqs. (18) and (23) go through,while for �p; 0� strings, �� is still given by (10) with q � 0.

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III. TEMPORAL GAUGE

So far we have imposed the conformal gauge (2): onemay ask whether this fixes the gauge completely, orwhether (as for NG strings in Minkowski space) it is alsopossible to impose the temporal gauge condition X0 � t ��. In this section we show that the temporal gauge isconsistent with the above equations of motion.

In the temporal gauge, conditions (2) reduce to

_x 2i � x02i � 1; _xi � x0i � 0; (24)

where X�i � �t;xi�. Consider first the� � 0 component of(17) which, in the temporal gauge (with � � d=dt) givesthe constraint

Xi

��i _si � 0: (25)

This is nothing other than energy conservation at thevertex: the energy required to create new strings balancesthe energy recovered from the disappearance of the old. Tosee this explicitly, let us calculate the Hamiltonian of thesystem. Starting first for a single �p; q� string, from theLagrangian S �

Rd�L defined from Eq. (3), and using

(11) and (13), one obtains

H � pF�� _x@L@ _x�L �

��: (26)

Thus the tension of the �p; q� string is given by T � ��=�,while the total energy of the 3-string system with junctionis

E �1

�

Xi

Z si�t�d� ��i: (27)

Equation (25) is simply dE=dt � 0.Now we show that Eq. (25) can be satisfied without

imposing extra constraints on the system, and hence thatthe temporal gauge can always be imposed. Denote thespatial component of �X�t� by �x. On multiplying the spatialcomponents of (17) by _�x one obtains

Xi

��i�x0i � _si _xi� � _�x � 0: (28)

Now, the derivative of xi�si�t�; t� � �x is

_x i � _six0i � _�x (29)

which, combined with Eq. (24), gives

x 0i � _�x � _six02i ; _xi � _�x � _x2i : (30)

Using these identities in Eq. (28), one obtains

Xi

��i _si�x02i � _x2i � �

Xi

��i _si � 0: (31)

This indicates that Eq. (25) can be satisfied automaticallyand one can choose the temporal gauge without imposingextra constraints on the system.

Finally we note that the spatial components of (17) yieldthe following equations for ��i�1� _s2

i �:

Xi

��i�1� _s2i �x0i � 0: (32)

One immediate corollary of this equation is

x 01 � �x02 x03� � 0; (33)

which indicates that the x0i are coplanar at the point of thejunction.

IV. CONSTRAINTS ON �p; q�-STRING JUNCTIONFORMATION

In this section we apply the above formalism to study thecollision of strings with charges �p1; q1� and �p2; q2�meet-ing at an angle and traveling with equal and oppositevelocities v. When the strings collide, they may becomelinked by a string with charges �p3; q3� � ��p1 �p2; q1 � q2�. Here we show that strong kinematic con-straints apply to this process, coming from the requirementthat the length of the joining string must increase intime. We then give specific examples in the followingsubsections.

In [12,13] the same question was addressed for thecollision of NG strings with tensions �1 and �2 (the NGlimit for �p; q� strings in the following analysis can beobtained by setting p � 0 and �� ). There the proce-dure was to write the solution of the wave equation (13) interms of ingoing and outgoing waves;

x i�t; �� 12�ai�� t� � bi�� t��; (34)

where, from the temporal gauge conditions (24)

a 02i � b02i � 1: (35)

The amplitudes of the ingoing waves b0i are fixed by theinitial conditions and it was shown that one can solve for _siin terms of these. Causality (or more specifically the re-quirement that j _sij 1) imposes that the �i satisfy thetriangle inequalities, namely, that for all i

�i � 0 (36)

where, for example,

�1 � �2 ��3 ��1 (37)

(and cyclic permutations).Using the equations derived in previous sections, it is

straightforward to see that the procedure detailed in [13]goes through in exactly the same way provided (a) onemakes the replacement

�i ! ��i; ��i ! ��i� (38)

in the equations of [13], where ��i is given in terms ofpi; qi, and gs in Eq. (10); and (b) one imposes the twocharge conservation conditions (18) and (23). An immedi-


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ate corollary of this is that �p; q� strings meeting at ajunction automatically satisfy the triangle inequalities be-tween the ��i, namely ��i � 0.

Also we note that if gs � 1 then

�� i ’jqijgs

�1�

g2sp2

i

2q2i

�; qi � 0: (39)

As expected, in this case the D-strings are much heavierthan F-strings, by a factor of g�1

s . Furthermore, the bindingenergy of a �p; q�-string is almost equal to the total tensionof p F-strings:

�� 0;q� � ��p;0� � ��p;q� ’ ��p;0� �O�gs�:

Now consider two strings of tension ��1 and ��2 parallelto the xy-plane but at angles to the x-axis, and ap-proaching each other with velocities v in the z-direction.Before the collision (for t < 0) we take

x 1;2��; t� � ��1� cos;��1� sin; vt�; (40)

where ��1 ��1� v2p

. As in [12,13], we may distinguishtwo cases: the formation of an x-link in which the twosegments of the original strings in the positive-x region arejoined to one end of the third, linking string, and a y-link,where it is the two segments in the positive-y region thatare joined. We consider here the formation of an x-link andthe signs are chosen such that � increases towards thejunction. Thus

a01;2 � ��1 cos;��1 sin; v�;

b01;2 � ��1 cos;��1 sin;�v�: (41)

At the collision t � 0, we suppose that the strings bindwith the formation of an x-link. We suppose the new stringis oriented at an angle to the x-axis and moves along thez-direction with the velocity u; therefore

x 3��; t� � ��1u � cos; ��1

u � sin; ut�; (42)

and

b 03 � ��1u cos; ��1

u sin;�u�: (43)

As shown in [13], one finds that

uv�

tantan

(44)

whereas u is determined by the following equation:

� ��2�sin2�u4 � � ��2

3�1� v2� � ��2

��v2cos2� sin2��u2

� ��2�v

2cos2 � 0 (45)

where �� 1 ��2. It can be checked that this alge-braic equation always has one positive root for u2; further-more u2 < v2, a result which will be used later.

The condition for junction formation is _s3 > 0; that isthe length of the joining string should increase in time. Thiswill only be satisfied in certain domains of the �; v� plane

which we now determine. One easy way to calculate _s3 isas follows. Eliminating a0i in terms of b0i and _�x in Eq. (29),from the spatial component of Eq. (17) one obtains

Xi

��i�1� _si�b0i � � �� _�x; (46)

where �� 1 � ��2 � ��3. Since �x�t� � x3�s3�t�; t� then

_�x � � _s3��1u cos; _s3�

�1u sin; u�: (47)

Using this in the y and z-components of (46) gives

_s 3 �G �� 3

�� G ��3; (48)

where

G ��1 cos

��1u cos

�

��1� v2��v2cos2� u2sin2�

v2�1� u2�

s: (49)

From (45) one calculates u2, which then can be used in (48)and (49) to find _s3. The result is a function of �� , ��3,and .

As mentioned before, u2 < v2, which implies that G<1. From the triangle inequalities the denominator of (47) isalways positive. Hence the condition for junction forma-tion to be kinematically possible, _s3 > 0, is equivalent to

G>��3

��(50)

which can be rewritten as

f��1� � A1��4 � A2�

�2 � A3 < 0; (51)

where

A1 � ��2�cos2� ��2

3 � ��2�sin2� ��2

�cos2�;

A2 � 2 ��2� ��2

�cos2� ��43 � �2cos2� 1� ��2

� ��23;

A3 � �43 � ��2

� ��2�:

(52)

Condition (50) can then be solved straightforwardly toobtain a condition on the values of v for which junctionformation is possible:

0 v2 < v2c��; (53)

where the critical velocity vc also depends on ��1;2 andhence on the charges of the two colliding strings. Let usdenote by vmax

c the maximum value taken by vc as variesfrom 0 to �=2. As was shown in [13], vmax

c 1 only if

�� 23 � ��j ��j � j ��2

1 � ��22j: (54)

If this condition is satisfied by the tensions of the joiningstrings, then there is a velocity vmax

c above which the twocolliding strings will pass through each other rather thanforming a junction.

We now study some physically interesting examples,namely, collisions between �p; q�-strings in the low-lyingenergy states, such as (0, 1), (1, 0), � 1; 1�, . . ..


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A. Example 1: Collision of two strings of equal tension

Initially we consider the simplest case from the point ofview of the formalism, namely, the collision of strings ofthe same tension ��1 � ��2 so that �� 0. Examplesinclude colliding (1,1) and ��1; 1� strings, for example.Note that condition (53) is always satisfied, so that a regionof the �; v� plane is always forbidden. Since �� 0 it isstraightforward to see from (44) and (45) that u � � 0.ThusG � ��1 cos and, provided the triangle inequalitiesare satisfied, the kinematic constraint (50) is that

��1 cos>��3

2 ��1(55)

[which one can verify is identical to condition (51)].Now take the first string, string 1, to have given charges

�p1; q1� which we take to be positive. String 2 has the sametension and if gs � 1, then from (10), its charges in generalare given by �p2; q2� � � p1; q1�. These four cases aresummarized in Table I and Fig. 1.

As noted in Table I, the collision of a �p1; q1� string withan identical �p1; q1� string (case A1) cannot lead to theformation of a joining string with charges �2�p1; q1�: this

is forbidden since from (54) it can only take place for v �0 � . Physically this is expected, since identical colliding�p; q� strings are expected to intercommute with each otherin the usual sense rather than forming a junction. A relatedphenomenon occurs when the collision is with an antistring��p1; q1� (case A2): then ��3 � 0 so that the linking stringhas zero tension. Again we expect the strings to undergo astandard intercommutation, in that when the two identicalstrings meet they intercommute in the usual fashion, sim-ply exchanging partners. Indeed notice that both A1 andA2 are related to each other in that these two initialconfigurations are the same, just rotated by 90�, the dif-ference being that a prospective x-link for A2 is a y-link forA1 and vice versa. The junction forms only in cases A3 andA4 and for small gs � 1, the largest region of the �; v�plane is open to the collision of �p1; q1� with �p1;�q1�(case A3): that is, it is much more likely to form a lightjoining p-string than a heavy joining q-string. The sym-metry between the cases A3 and A4 under p! q and gs !1=gs is evident from the table. Physically this is expected,since F and D-strings are symmetric under string theory S-duality where weak coupling is replaced by the strongcoupling, i.e. gs ! 1=gs.

B. Example 2: Collision of an F-string with a D-string

The collision of F, (1,0), and D (0,1)-strings is probablythe most interesting example in this analysis, as it is alsothe basic building block for general �p; q� string collisions.The third string formed at the junction is a (1,1) string andfrom Eq. (52) one finds

A1 � 2g�3s cos2�2cos2� 1��1� gs�2;

A2 � 2g�3s ��1� g2

s� � 2cos2�1� gs�2�;

A3 � 4g�2s :

(56)

We assume that the junction is an x-link and 2cos2>1. We then see that (51) has two positive roots, one biggerthan unity and one smaller than unity; we denote the latterby ��2

c . The condition (51) for junction formation is thusequivalent to ��2

c < ��2 < 1. In terms of velocity this istranslated into 0< v2 < v2

c where

TABLE I. Equal tension ��1 � ��2 scattering and the formation of an x-link. The charges of the 1st string are fixed and positive:�p1; q1�. The different possible charges �p2; q2� are given in the table, as are the charges of the joining �p3; q3� string. Note that theformation of a y-link in case A1 is equivalent (by rotation by �=2) to case A2; and similarly cases A3 and A4 are related by a rotationby �=2 (see also Fig. 1).

�p2; q2� �p3; q3� ��3 Kinematic constraint Case

�p2; q2� � ��p1;�q1� ��2p1;�2q2� ��3 � 2��p2

1 � �q1

gs�2

qintercommutation allowed with no link A1

�p2; q2� � ��p1;�q1� (0, 0) ��3 � 0 intercommutation allowed with no link A2�p2; q2� � ��p1;�q1� ��2p1; 0� ��3 � 2p1 ��1 cos � 1��

1�q21=�g

2sp2

1�p A3

�p2; q2� � ��p1;�q1� �0;�2q1� ��3 � 2 q1

gs��1 cos � 1��

1��p21g

2s �=~q2

1

p A4

FIG. 1 (color online). Schematic representation of the differentcases considered in the text, all with ��1 � ��2 and for �p1; ~q1�fixed (and positive). Note that the formation of a y-link in ‘‘caseA3’’ is equivalent (by rotation by �=2) to a considering theformation of an x-link in ‘‘case A4.’’


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v2c ��1� g2

s� � 4cos2sin2�1� gs�2 ��1� g2

s�2 � 4cos2sin2�1� g2

s�2

p2cos2�1� gs�

2�2cos2� 1�: (57)

Note also that vc � 0 when � �=4 independent of gs.For > �=4 no x-link can be formed in this case, what-ever the values of gs or v.

Various limits are instructive here. First consider gs !0, in which case vc � 1. In other words, half the �; v�plane is allowed. Physically, this means that a very heavyD-string can always exchange partners with a light F-string. Furthermore, one also finds that u! v and !. The third string moves with a velocity approximatelyequal and in a direction almost parallel to the incomingheavy D-string. A second interesting limit is when gs ! 1in which case the F-string is almost as heavy as the D-string and �� ! 0. This limit was studied above since��1 � ��2 and one finds ��1 cos> 1=

��2p

, and u� � 0.

C. Example 3: Collision of an F-stringwith a ��1; 1� string

A further interesting example is the collision of (1, 0)and ��1; 1� strings, with the third string being a D-string.In this case

�� 1 � 1; ��2 �

��1�

1

g2s

s;

��3 �1

gs) ��2

3 � j ��21 � ��2

2j

(58)

so that the condition in (54) is saturated. As discussed in[13] there is no bound on v and one can show that

sin 2<1

2

�1�

gs��1� g2

s

p �: (59)

The mirror image of the above example is the collisionof (0,1) and �1;�1� strings, where the third string is a lightF-string. Again this case satisfies the bound (54) so thatthere is no constraint on v. This time, however,

sin 2<1

2

�1�

1��1� g2

s

p �: (60)

The symmetry between (59) and (60) under gs ! 1=gs isevident. As explained before, this is a consequence ofsymmetry between F and D-strings under S-duality wheregs ! 1=gs. Now a much larger region of the �; v� plane istherefore open when the joining string is lighter.

V. COLLISIONS IN A WARPED BACKGROUND

Our analysis for the collision of cosmic strings in a flatspace-time can also be generalized to the collision ofcosmic superstrings in a warped throat. This is of greatinterest because in models of warped brane inflation [4] theinflation takes place inside a warped throat and cosmic

superstrings produced at the end of inflation are located atthe bottom of the throat.

To be specific, we study the collision of �p; q�-strings inthe Klebanov-Strassler (KS) throat [19] which is a warpeddeformed conifold. At the tip of the throat, where weassume the strings are located, the internal geometryends on a round three-sphere and the metric is given by

ds2 � h2��dx�dx� � gsM0�d 2 � sin2 d�22�; (61)

where h is the warp factor at the bottom of the throat. Here is the usual polar coordinate on a S3, ranging from 0 to�,and M is the number of Ramond-Ramond F�3� fluxesturned on inside this S3. At the tip of the throat the two-form C�2� corresponding to the three-form F�3� is given by

C�2� � M0� �

sin�2 �2

�sindd : (62)

As studied in [20], (see also [21,22] for its dual pre-scription), a �p; q�-string in the KS throat is constructedfrom a wrapped D3-brane with q units of magnetic fluxesand p units of electric fluxes turned on in its world volume.The D3-brane is wrapped around a S2 inside the S3, whereeach S2 is determined by a slice of constant , given by

��pM: (63)

After integrating the contribution from the wrapped dimen-sions of the brane, and imposing the conformal gauge, theaction for a �p; q�-string extended along the z-direction is

S �Zdtdz��

��h4x02�1� _x2� � �2F2

0z

q��F01�; (64)

where, as opposed to the flat metric, there is now a nonzerocontribution from the Chern-Simons term;

� � ��3

ZS2C�2� �

M�

��pM�

1

2sin�2�pM

��(65)

and

� � �3g�1s

ZS2

��gg � �

2F2

q

� ��1

��M2

�2 sin4

��pM

��q2

g2s

s: (66)

Here g and g are the angular parts of the metric of S2

in (61) and �3 � 1=�2��302 is the D3-brane charge. Notethat if we turn off the Chern-Simons term, � reduces to thetension of q coincident D-strings given by jqj=�gs.

Comparing the action in Eq. (64) with the action of astring in a flat background, � plays the role of the baretension, i.e. �i as in Eq. (4). Furthermore, the term con-


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taining � in Eq. (64) corresponds to a modification to theconjugate electric charge, p � �L=�Ftz, due to the back-ground flux. Constructing the Hamiltonian, one can checkthat the tension of the �p; q�-strings is equal to ��=� where

�� h2

��q2

g2s�M2

�2 sin2

��pM

�s: (67)

In the limit where M ! 1, the formula above reduces tothe tension of �p; q� strings in a flat background given byEq. (10). This is expected, since in this limit the size of S3

at the bottom of the throat is very large and one effectivelydeals with a flat background. Furthermore, the tension ofstring is warped by two powers of warp factor, as expected.

The formation of a junction in a KS throat at the staticlevel was studied in [23], where it is shown that the methodof a wrapped D3-brane can be used for the collision of�0; q1� and �p2; q2� strings. Here we would like to study thecollision of an F-string and a D-string in the KS throat,which would be the generalization of F and D-stringscollision studied in previous section.

Following the same strategy as in the flat background,the action of (0,1), (1,0), and ��1;�1� strings forming ajunction is

S � ��i

Zd�d��

��h4x02�1� _x2� � �2Fi2��

q��iFi��si��

�Xi

Zd��fi:�Xi�si��; �� X��

� gi�Ai��si��; �� _siAi��si��; �� A��: (68)

The results in Secs. II and III formally go through with ��inow given in Eq. (67) and

�� 1 � h2g�1s ; ��2 � h2 M

�sin�M;

��3 � h2

��g�2s �

M2

�2 sin2 �M

s:

(69)

Constructing the coefficients A1, A2, and A3 fromEq. (52) one can check that Ai have the same value as inEq. (56) if gs ! �gs, where

�g s �M�

sin��M

�gs: (70)

This indicates that the bound on the incoming velocity vc isgiven by the same expression as in Eq. (57) where now gsis replaced by �gs.

VI. CONCLUSIONS

The very attractive brane inflation scenario leads natu-rally to the formation of cosmic �p; q�-superstrings. In thispaper we have studied the behavior of these strings at athree-string junction, and the constraints that determine

whether colliding strings can exchange partners.Constraints very similar to those found earlier for NGstrings apply here too, but in addition to the energy-momentum conservation constraint (17) we also havetwo other conservation laws, (18) and (23). Moreover theparameter ��i appearing in (17) is the tension of the�pi; qi�-string rather than �i, the tension of qi coincidentD-branes which multiplies the action in (4). We also brieflyexamined the effect of nontrivial background geometry, inparticular the effect of warping in a KS throat. Here weshowed that the kinematic constraints are again of the sameform but with transformed effective-tension parameters,given by (69).

Our results may be important in studies of the evolutionof a network of �p; q�-strings [24]. Indeed, as discussed in[13] in the case of NG strings, the formalism we have set upin Secs. II and III can be used to determine some averageproperties of a network of cosmic superstrings. More spe-cifically, if one neglects Hubble expansion and energy lossmechanisms, in a network containing strings of differenttensions (with corresponding junctions) some strings willshorten and others grow. In [13] it was shown that if thenetwork contains NG strings of tensions �i (i � 1, 2, 3),then one can calculate explicitly h _sii (the network averageof the rate of increase in length of string i at a vertex), aswell as h _x2

i i (the average root mean square transverse stringvelocity at a vertex). It is easy to see that the explicitexpressions given in that paper, for example h _sii, gothrough to �p; q� junctions provided one replaces �i !��i. Therefore we again expect vertices to move along the

strings in such a way as to increase the length of the lighteststrings: This is shown in Fig. 2 where we plot h _sii for

0.48

0.7

0.24

0.5−0.08 g

1.00.90.8

0.4

0.16

−0.16

0.6

0.32

0.08

0.4

0.0

−0.24

0.30.20.10.0

FIG. 2 (color online). h _sii as a function of gs. Red, uppercurve: i � 1. Blue, lower curve: i � 3, green, middle curve:i � 2.


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�p1; q1� � ��1; 0�;

�p2; q2� � �0; 1�;

�p3; q3� � ��1;�1�

(71)

as a function of gs. (Note that these are chosen so that��1 ��2 < ��3 with the equality holding when gs � 1).

There are a number of directions in which this workcould be taken. They include using our new results to lookat the effect of lensing on string junctions, and to look for aclass of exact loop configurations that would allow us to

determine the distinct gravitational wave emission fromsuch objects. Perhaps the most significant use thoughwould be as an input into detailed simulations of a networkof �p; q� strings, because as we have seen they throw upnovel features that are not present in the case of ordinaryAbelian cosmic strings.

ACKNOWLEDGMENTS

We would like to thank K. Dasgupta for useful discus-sions. The work of H. F. is supported by NSERC.

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Collision of cosmic superstrings