Notes on superconducting cosmic strings

Volume 194, number 1 PHYSICS LETTERS B 30 July 1987

NOTES ON SUPERCONDUCTING COSMIC STRINGS

Mukunda ARYAL, Alexander VILENKIN Physics Department, Tufts University, Medford, MA 02155, USA

and

Tanmay VACHASPATI Bartol Research Foundation of the Franklin Institute, University of Delaware, Newark, DE 19716, USA

Received 27 March 1987

Several problems in string electrodynamics are discussed. Electric currents induced in strings oscillating in stationary electric and magnetic fields are calculated. In some cases the currents grow linearly with time. The rate of pair production in the electromagnetic field of strings is estimated.

I. Introduction. Superconducting cosmic strings [ 1 ] predicted in some grand unified models can have a variety of astrophysical applications [2-6]. The basic equations describing the currents and the electromagnetic fields produced by the strings have been derived in refs. [7,8], where they have been used to calculate the rate of electromagnetic radiation. In the present paper we further discuss the electromagnetic properties of superconducting strings, concentrating on the physical effects and leaving the astrophysical applications for a future analysis.

The equations of string electrodynamics are sum- marized in the next section and their solutions for the simplest case of a straight string are derived in section 3. In sections 4 and 5 we solve the equation for the electric current induced in a closed loop of string by external electric and magnetic fields. The solutions exhibit charge and current oscillations of amplitude that grows linearly with time. The exis- tence of such solutions was suggested in ref. [ 8 ]. We point out the possibility that the fields produced by the string itself can induce exponentially growing currents (section 6). Finally, we show in section 7, that electromagnetic oscillations in the strings result in copious production of particle-antiparticle pairs and that the electromagnetic properties of strings can

be substantially modified by this pair production.

2. Basic equations. Witten has shown [ 1 ] that the current in a superconducting string can be expressed in terms of a scalar field, ¢((a), which lives on the two-dimensional world sheet of the string, xU((a). The field equations can be obtained from the action [7,8]

S= I d2( (-I~x/------g + ½N/f~--ggab(~,aO,b--eAux'U.ag'abO, b

1 -- | d 4 x Fu~,F ~ (1 )

16rt : - -

Here, gab =XU,aXu,b is the metric on the string world sheet; (o and (~ are a timelike and a spacelike param- eter on the sheet; Greek indices take values from 0 to 3, Latin indices from the beginning of alphabet take values 0 and 1, and Latin indices from the mid- dle of alphabet take values 1, 2, 3.

In the present paper we shall disregard the back reaction of 0 and A, on the motion of the string. For currents much smaller than the symmetry breaking scale, q, this is a good approximation everywhere, except in the vicinity of the cusps. With the gauge conditions

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Jc"x'a=O, /c"J%,+x'"x;,=O, OaAa=O, (2)

the field equations for x u, ¢ and A ~ take the form

[3zx l' = 0 , (3a)

[32¢= - ½ee"hF~,,x",~x%, , (3b)

[--]4 A~' =4n j" . (3c)

Here, e is a model-dependent coupling (0.01 < e 2 < l ) , dots and primes stand for deriva- tives with respect to G ° and (1, the indices are raised using the lorentzian metric rf b, D2--rfbOaOb, 734 =- ~OuO~ and

f ' = --ee ah j d2~ (~(4)(X--X(~))XlZ,a¢, b (4)

is the current density. The charge per unit length of string is - e ¢ ' / I x ' l and the current is i=e(~.

The general solution of (3a) can be written as

t = ( °, x ( ( , t ) = ½ [ a ( ( + ) + b ( ( _ ) ] , (5)

where ( _+ = ( _ t, ( -= ( ~ and the gauge conditions (2) give the following constraints for the otherwise arbitrary functions a and b:

a'2 =b '2 = 1 . (6)

For a closed loop in its center-of-mass frame these functions must be periodic with period equal to the invariant length of the loop, L. (In other frames, a and b have additional non-periodic terms, v(_ and v(+, respectively, where v is the center-of-mass velocity. Below we shall assume, unless indicated otherwise, that the loop is viewed from its center-of- mass frame.) Some general properties of the loop solutions (5) are reviewed in ref. [9].

3. A straight string. We first consider the simplest case of a straight string in the absence of external fields (F~,, = 0). For a string parallel to the x-axis, we can take ( = x , and the solution of eq. (3b) is

¢= ¢1 ( x - t) + ¢2(x+ t ) . (7)

The corresponding current density is

f ' =6(y)6( z)J"( x, t ) , (8)

where J~' =e( - ~ , ¢ ' , 0, 0). Now it is easily verified that

A ~' = 2J~'(x, t) In r , (9)

where r = (y2+ Z 2)1/2 is the distance from the string, solves the Maxwell's equations (3c) with the current (8). The electric and magnetic fields of the string are given by

E = - 2 e ¢ ' r l(O, n v, n~) ,

B=2e~r ~(0, n=, -ny) , (10)

where n = r/r. It is interesting to note that these fields produce no back reaction on the current, since the right-hand side of eq. (3b) is - eEx = 0. This means that current and charge oscillations on a straight string persist indefinitely, without dissipation. We will see later in section 7 that this conclusion is modified when pair production effects are taken into account.

4. A loop in homogeneous electric and magnetic fields. Consider a closed loop oscillating in constant, homogeneous external fields E and B, disregarding the fields produced by the loop itself. Using a three- dimensional notation, we can rewrite eq. (3b) as

[72¢=ex' .(E+ x X B ) . (11)

This equation can be solved exactly,

¢= le[(a-b).Et+ (a×b).B] +¢o, (12)

where a ( ( _ ) and b ( (+ ) are defined in eq. (5) and ¢o is a solution of the homogeneous equation. ¢o can be represented as

¢o=(io/e) t+¢l( (_ )+¢2((+ ) , (13)

where Ct and ¢2 are arbitrary periodic functions with period L. The first term in (13) represents the DC current, i= io, while the last two terms describe current and charge waves travelling along the loop at the speed of light in opposite directions.

From eq. (12) we see that current and charge oscillations induced by the magnetic field have constant amplitude while those induced by the electric field linearly grow with time. A loop exposed to an electric field for a time period At develops a current of the order [ 8 ]

i~ ~o~E At, (14)

where o~=e 2. The fields E and B in eq. (12) should be taken in the center-of-mass frame of the loop. For a loop moving in a magnetic field B with E = 0 we

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have i ~ aBvAt, where v is the velocity of the loop.

5. A loop in an inhomogeneous magnetic field. Suppose now that E= 0 and

B=]~x, (15)

where/~ is a constant matrix. From 17-B=0 it follows that Tr/~=0. It is convenient to represent 0 as a function of (_ and (+; then eq. (3b) takes the form

02~ - t e ( a ' Xb') l~(a+b) . (16) o~ + o(_

Using the identity

aia'j = ½ (aiaj)' + ½eijk(a×a')k ,

we can express the current as

i= eO = e( a¢/a( ÷ - a¢/O~_ )

= ¼cz((a×b')l~(a+2b) - (a' ×b)/~(b + 2a)

- f d ( _ (a×a ' )~b '+ f d ( + ( b × b ' ) l ~ a ' )

+e0o, (17)

where ~o is a solution of the homogeneous equation. The first two terms on the right-hand side ofeq. (17) are strictly periodic, while the last two terms can have linearly growing components. For example, if we take Burden's family of string trajectories [10],

a ( ~ ) = p - l ( e l s inp(+e3 cosp() ,

b(O =q-~[(e~ cos q/+ez sin q/) sin q(

+ e 3 c o s q~] , ( 1 8 )

then aXa ' =P-~e2, bXb ' =q-~ (e2 c o s w--el sin q/) --q-~e~,, and the growing part of the current is

i--, ~o¢( q - ~ e~,fla ' - p - t e21~b')t . (19)

Here, p=2rtm/L, q=2nn/L, m and n are relatively prime integers, and ei is a unit vector along the xi- axis. If the typical inhomogeneity scale of the magnetic field is 2 > L, then the current can be estimated as

i~ ( a/8zO( L/2 )BAt . (20)

As an example, suppose there is a loop of string in our Galaxy. Then B ~ 3 × 1 0 -6, 2~100 pc, At~ to ~ 10 l° yr is the age of the universe and the maxi- mum size of the loop is determined by the galactic scale at the horizon crossing, L ~ 100 pc. With a ~ 10 -2 we obtain i~ 1030 e s u s - l ~ 1014 GeV. More realistically, as the loop moves through the randomly varying magnetic field, the sign of di/dt will change and the growth of the current will be similar to a ran- dom walk. Then i,,, (ol/87t)(L/)OB(2/o)(Oto/~) 1/2, where o is the velocity of the loop. With 0~200 kms -~ this gives i~ 10 ~ GeV, which is still much greater than the estimate of ref. [ 2 ], where the growth of the current was not taken into account.

If superconducting strings exist in Nature, we expect them to be imbedded in cosmic plasmas, and so eqs. (3) and all their consequences have to be modified to include the plasma effects. In section 7 we are going to argue that a substantial modification of the equations may be needed even in vacuum, due to the production of particle-antiparticle pairs.

6. Possibility of a string dynamo. The linearly growing currents induced in the strings by external fields suggest an intriguing possibility that superconducting strings can be unstable with respect to a spontaneous generation of charge and current oscillations with an exponentially growing amplitude. In this case the role of an external field is played by the field produced by the string itself, F ~ i/L. Eqs. (14) and (20) describing the growth of the amplitude of the current can be written symbolically as (for 2 ~ L)

di/dt ~ k a F ~ k ( a / L ) i , (21 )

where k is a numerical coefficient. The solution of (21) is

ioc exp( ka t /L ) . (22)

The sign of k is, of course, of crucial importance. If k<0, then eq. (22) describes the dissipation of the current oscillations due to the electromagnetic back reaction. However, for k > 0 eq. (22) would corre- spond to oscillations being enhanced by their own electromagnetic field. In this case the energy for the exponentially growing currents has to be drawn from the energy of the moving string.

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We were not able to prove or to disprove the exis- tence of exponentially growing solutions of eqs. (3). If such solutions do exist, then the current can be amplified by an enormous factor during the lifetime of the string. The decay time of a string due to grav- itational radiation is r ~ L/100G#, where/t is the mass per unit length of string. The corresponding ampli- fication factor is exp(ka/lOOGlt) . For G/t~ 10 -6, a ~ 10 2 and k ~ 1 this gives ~ e m° and can be much greater for smaller values of G/t. The growth of the current must terminate when the electromagnetic energy becomes comparable to the total energy of the string. (This happensat i ~ 0.1/~- ~/2.) It may be possible to construct scenarios in which the galactic and intergalactic magnetic fields are generated by string dynamos. The possibility of a string dynamo was first suggested by Witten [ 1 ].

7. Pair production. Pair production in the magnetic field of superconducting strings has been recently discussed by Amsterdamski [ 11 ] who claims that pairs are copiously produced even in the case of a DC current in a straight string. On the other hand, Witten [ 12] suggested a simple argument indicating that there is no pair production in arbitrary static magnetic field. While this issue remains unresolved, we shall assume that static magnetic fields do not produce pairs.

In the general case, current and charge oscillations in superconducting strings generate electric as well as magnetic fields. Pairs are copiously produced in regions where

E 2 2 2 - B > E c . (23)

Here E~ = m 2/eo is the critical field, eo is the particle's charge and m is its mass. For E2-B2<~O there is no pair production and for 0 < E 2 - B 2 <<E~ it is exponentially suppressed. For E 2 - B 2 >~E~, the rate of pair production per unit volume per unit time is dn/dt > m 4.

Taking a straight string as an example, we see from eq. (10) that E > B near the string wherever Iq~'/~l > 1. I f the field ~ of eq. (7) includes only waves propagating in one direction, then I~'/~t = 1, E = B and there is no pair production. But if both components are present with comparable ampli- tudes, then the charge per unit length of string is comparable (by amplitude) to the current, and for

some portions of the string the field is overcritical up to a distance rc ~eo i /m 2. Pairs are produced when rc > m-~, that is i>~ m/eo. (The onset of pair production is signalled by the appearance of bound states with energies ~<- m.) e+e--pai r production begins at i~ 1014 esu s-1. Heavier particles are produced as i increases, and we note that the most efficient pair production is that of the heaviest charged particles with rn < eoi.

The pair production ceases when the number of pairs is sufficient to screen the electric charges on the string. It is easily verified that for any appreciable current (i>> 1/eZoL) this happens on a timescale much shorter than the period of loop oscillations, L/2. (We assume that the characteristic period of current oscillations is comparable to the period of the loop.) The number of pairs produced in one oscillation can be estimated as N ~ iL. On average, the number of pairs produced per unit time is N ~ i. If i is in GeV, then N ~ 1024iGeV S-! and the energy of pairs produced per unit time is ~Nm~< 1020 i~ev ergs -~.

Since the pairs screen the fields produced by cer- tain portions of the string, the classical equations of electromagnetism need to be modified. We do not attempt to derive the modified equations in this paper. To illustrate possible effects of the pairs, consider again the example of a straight string with current and charge oscillations of typical wavelength/l. Pairs are produced in the vicinity of the string (r < re) and are then dispersed to distances r>2. The electric field due to the pairs is E ~ i/2. Unlike the field (10), it has a non-zero component along the string, and eq. (3b) gives d i / d t ~ - ai/2. (Here the sign cannot be positive, since there is no energy source for a growing current.) Hence, the oscillations are dissipated on a timescale ~2/a . This rate of dissipation does not exceed, by order of magnitude, the rate of current growth conjectured in section 5, and so a string dynamo appears possible even with pair production taken into account. Besides, as we already men- tioned, the pair production does not occur if O(x, t) has the form of a wave travelling in one direction.

This work was supported in part by the National Science Foundation, the General Electric Company and by the Department of Energy.

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References

[ 1 ] E. Witten, Nucl. Phys. B 246 (1985) 557. [2] E.M. Chudnovsky, G. Field, D.N. Spergel and A. Vilenkin,

Phys. Rev. D 34 (1986) 944. [3] J.P. Ostriker, C. Thompson and E. Witten, Phys. Lett. B

181 (1986) 243. [4] C.T. Hill, D.N. Schramm and T.P. Walker, Fermilab

preprint. [ 5 ] A. Vilenkin and G. Field, Nature, to be published. [6] A. Babul, B. Paczynski and D.N. Spergel, Princeton

Observatory preprint.

[7] A. Vilenkin and T. Vachaspati, Phys. Rev. Lett., to be published.

[8] D.N. Spergel, T. Piran and J. Goodman, Institute for Advanced Study preprint.

[ 9 ] A. Vilenkin, in: 300 years of gravitation, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge, 1987).

[10] C.J. Burden, Phys. Lett. B 164 (1985) 277. [11 ] P. Amsterdamski, University of Texas at Austin Report

(1986). [ 12 ] E. Witten, private communication.

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Notes on superconducting cosmic strings