Volume 200, number 4 PHYSICS LETTERS B 21 January 1988

S U P E R C O N D U C T I N G C O S M I C S T R I N G S W I T H C O U P L E D Z E R O M O D E S

Mark H I N D M A R S H Theoretical Division, T-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received 26 June 1987

The physics of fermion zero modes on bosonic superconducting strings is investigated. The charged field at the core couples left movers to right movers, giving an effective theory of massive fermions on the string world sheet. Despite their mass, the fermions still contribute to the bosonic current through the two-dimensional anomaly. Even when the Higgs field is neutral and there is no anomaly, the string remains superconducting provided the sum of the charges of the zero modes do not vanish. The mechanism is closely analogous to a theory of superconductivity proposed by Fr6hlich.

The existence of zero modes on vortex lines or strings [ 1,2] are now thought to have some excit ing consequences for cosmology [ 2 -8] . It is possible that large currents can be carr ied by charged part icles t r apped on the string in a superconduct ing state, and superconduct ing strings have been invoked, because o f their abi l i ty to produce enormous electromagnet ic fluxes, to account for as t rophysical phenomena such as "explosive" galaxy format ion [4] and quasars [7].

The charge carriers may ei ther be bosons or fer- mions, and there are many models in which the charged Higgs field that carries the bosonic current is coupled to the fermion zero modes. In this let ter I shall examine this s i tuat ion using a s imple U ( I ) XU(1 ) model , and find a number of surprises. Naively, we might expect that since the fermions are now massive on the string as well as off, applying an electric field parallel to the string cannot create any part icles unless eE is greater than the mass squared o f the fermions on the string, and hence fermionic currents cannot be exci ted by the field strengths we expect to f ind in the recent his tory o f the universe. In general this expecta t ion is not fulfilled. We shall see that when the fermions gain a mass on the string through a charged Higgs field they contr ibute to the current through an anomaly, effectively renormal- ising the response to an appl ied electric field of the pure bosonic string. When the Higgs field is neutral there is no anomaly. However , we f ind that the string is still superconduct ing i f the sum of the charges of

the fermion zero modes does not vanish. The current is now carr ied by a different mechanism, a phenom- enon analogous to one in certain quasi -one-dimen- sional conductors, where it is known as Fr6hl ich conduct ion [ 9 ]. Currents can also be carr ied by mas- sive fermions t rapped on the string at its formation. They turn out to be insignificant for astrophysical purposes.

Let us introduce the model, which is essentially the one considered by Witten. Let there be 2n pairs o f fermions (~u~,~u~), with U ( 1 ) × U ( 1 ) charges (q~,R, C]~,R ). They are coupled to two complex Higgs fields 0 and a, with charges (0, 0) and (0, 0) re- spectively. We shall take the left handed and right handed members o f each pair to have the same U(1 ) charge (which will be ident i f ied with electromag- netic charge), and we shall further d iv ide the fer- mions into n pairs (~u2L,~,~,~m) and (~'~m+l, ~/2Rm + ~ ) which are coupled to each other ( and not to members of different pairs) by a. Lastly, we shall take the ( 2 m ) t h pair to get a mass from 0 and the ( 2 m + l ) t h pair to get a mass from 0". This con- strains the charge assignments:

q~,, • - q 2 , , + l =e=qRm--qRm+ ~

and

O2L,,,-q2R, n = e = - (qzL,,,+,--qRm+ l ) .

Indeed, once we allow two complex Higgs fields, the ground state o f the string has an expectat ion value

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Volume 200, number 4 PHYSICS LETTERS B 21 January 1988

for one of them at the core for a large class of pa- rameters [ 10,11 ], and so we must consider what happens to fermions coupled to these fields. The specified couplings are simple enough not to make the analysis too complicated, and have the possibil- ity of realistic applications. For example, 0 and a could be members of an SU (2) doublet, in which case we might be considering ordinary quarks and lep- tons bound to the string. The four-dimensional La- grangian is therefore

2n-- l ~ = ~ ((u~iy'D~u~ + ~ i y ' D v , R)

1=0

n- - l (/~2 m 0 !//2m ~//Rm * - L R - - 2 - L "{- '~ '2m+lO U r / 2 m + l ~ / 2 ; n + l

m = 0

- L R ~ * - L +g,,,a~'EmV/2,,,+ I +gma ~v2,,,+ 1 ~Rm +h.c.)

+ ID,#I2 + ID, al 2 - V(0, a ) . (1)

We shall take the Higgs potential to be such that sta- ble string solutions of the following form exist:

0=00(P) exp(iq0, a=ao(p) ,

~, = au,(#p) - l IF(p) - 1] . (2)

The boundary conditions are: 00 (0) = 0, 0o (or) = q; ao(0) =q ' , a o ( ~ ) =0; F ( 0 ) = 1, F(ov) =0. The fields tend to their vacuum values exponentially fast out- side core regions which are of order the inverse masses in width. Outside the string the unbroken group is U ( I ) , which we take to be electromagne- tism; this U(1 ) is broken at the core of the string by the a field, which in the absense of fermions would produce the bosonic form of superconductivity on the string (the a field acts like the Cooper pairs in an ordinary superconductor). Let us for a moment ignore the couplings between a and the fermions. The current is carried by excitations of the form exp [i0( t, z)] ao (p), which contribute an extra piece to the lagrangian of the static solution

fd4x laol2(O,~O+eA,O 2 ( a = 0 , 1) , (3)

where ce labels world-sheet coordinates. Integrating out the coordinates transverse to the string we obtain xfd 2x (0,~0 + eAo)2, where x is roughly the inverse of the quartic coupling of the a field [ 2,11 ]. Note that the current J . = 2ex(O,~O + eAo) is conserved, and its dual obeys the equation

e'~a O.Jp = 2e2xE , (4)

which is characteristic of a superconductor. How do the fermions carry currents? Solving the

Dirac equation in the string background we find (for i = 2 m )

iDv ~L,. _ 2>. 0V2 R. = -- iy" D . ~Lm,

iDT ~'~,. - 22,,, 0" q/L,. = -- iy '~ D . ~uR, n ( 5 )

(where DT = y J D l + y2D2 ) with similar equations for i= 2m + 1. It is well known that these equations have solutions of the form [ 1,2]

~u~=o~i(t,z) ~ ' ( p ) , ~'R=o~,(t,z)(i~")~(p) (6)

with

(0, +0~)ce2,,, = 0, (0,-0=)a2, ,+l = 0 . (7)

Hence we may think of the solutions as 2n massless chiral fermions of charge ¢ = q~ = qR trapped on the string, moving either to the right ( i = 2 m ) or to the left ( i = 2 m + 1).

In order to find how this system of fermions be- haves in the presence of an applied background elec- tromagnetic field, we can use the bosonisation technique [ 12,2]. First of all we integrate out the transverse dimensions to obtain an effective two-di- mensional fermionic Lagrangian

= jd x [a*mi(D, + Dz)OL2m

+ c~*m+ l i ( D , - D..)a2m +, ] (8)

(the spinors are normalised so that fd2x I T~I 2= 1 ). We can form a two-component spinor ~,,v = (a2m, a2m +I) with slightly unconventional gauge couplings:

5ev= f d2x ~C'my'(iO

+qzmAP+ +q2m+lAP- )~m , (9)

where P±=½(1-+Ys) . The standard way of boson- ising [12] tells us to replace ~ 7 ~ , ~ by (1/x~)E.a0~0m, ~ Y . 7 5 ~ m by (1/X~)0~0m, and ~miy'0~m by ½0.OmO"Om, giving us

~.=fd2x~(½0.O~O-O,~+ q~ (:o:aA,~O,aO,,,

+ ~A"Oo~O,,, , (10)

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Volume 200, number 4 PHYSICS LETTERS B 21 January 1988

where q'_2 = q2,,-I- q2m+ J. We must now consider the gauge transformation properties of the fields era, and check the gauge invariance of (10). The chiral den- sity O+ =~,, ,P+ ~s,,, =a2,,, + * lo~2,n transforms as O+--+ O+ exp(iq~A ): in the bosonised form this op- erator is equivalent to/Xm e x p ( - , ~ # , , , ) , where in the two-dimensional theory/x,,, is a renormalisation- dependent mass. Therefore era-'-> Cm -- ( q'2 / x / ~ )A, and in order to ensure the gauge invariance of (10) we must add a local term ½~[(q~ )2/47c]A 2 (which we are entitled to do because its coefficient is am- biguous in the fermionic theory [13]) and have Zqt7 q~ =0. This latter condition is just the two-di- mensional anomaly cancellation condition that the sum of the squares of the charges of the left movers be equal to that of the right movers [ 14], which in turn comes from the cancellation of the four-dimen- sional anomaly tr(QeQ) [2]. Thus the bosonised Lagrangian is

q'" .2

The electromagnetic current is

• ' o=z : [ aS- , .L , /4 . ,

qg- ~n- -] + (12)

which is conserved by virtue of the equations of motion

A q';.~--E~,nOSSA<~=O (13) 0. 0o,.+ )+

and the anomaly cancellation condition Yq'g q'_" = O. Just as in the bosonic case the dual current e~pJP is not conserved:

~aflOaJ/~=~ (qm)2 +(q ,~)2 E , (14) m A~

revealing that the fermion zero modes are superconducting.

Now let us take into account the coupling between the fermions and the charged bosonic field at the core

of the string• We must solve the Dirac equations again in this new background:

iDsGs},,, _~2,n~/2Rn _g,,allZ2m +R ~ =iT-D,!U2Lm

iDT~S},n+, --22m+, ¢*lU2Rm+ ~ --~ma*qS~m

• O¢ L =17 Dalp'2m+l ,

iDx qS2R,,, * L - L --Z2m¢ q/2,.--gma~//2m+l = iT~D, VSRm ,

iDv GS2Rm+ ~ --22m+ ~ #~/2L,,,+, --gma*!U}m • O¢ R

=17 Daq/2,n+ I • (15)

(Note that the the coupling to the Higgs field fixes q'2 to be equal to e, so that when e # 0 the anomaly cancellation condition implies that Zq~ = 0. ) These coupled equations have two solutions with opposite eigenvalues of a3=7°y 3, which in the limit qmrf, ~,,q' << 2, t / ( i=2m, 2 m + 1 ) tend to solutions (6). For reasons of space we shall consider the theory in this limit, which corresponds to the fermions being much lighter on the string than off, and present a fuller ex- position elsewhere [ 15 ]. The kinetic part of the bo- sonised Lagrangian is just as in (9), and there is now a mass coming from the coupling to the charged Higgs field. These terms in the two-dimensional Lagran- gian are of the form g?,~P+ ~/m exp(i0) +b.c., so after bosonisation we have

~02 M 2 COS(0 _ X / ~ m ) = Z , . . ( 1 6 ) m

Eq. (16) is a sine-Gordon potential, and we shall eventually want to identify the solitons, which have mass ~Mm, with the fermions of the unbosonised theory [ 12 ]. Hence Mm ~ gmt/' • The electromagnetic current is the sum of the bosonic part J ~ = 2e~c(O,~O+eA~) and the fermionic part (12), and is still conserved through the equations of motion

0"(00m+ qm A ] + q~

- x/~M,2, s i n ( 0 - , f i T 0 m ) = 0 ,

2,~O'(O0+eA)+FM£ s i n ( 0 - ~ 0 m ) = 0 . (17) m

We may combine eqs. (17) into equations for a gauge invariant field ~ = . , / ~ 0 m - 0 , so that

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Volume 200, number 4 PHYSICS LETTERS B 21 January 1988

02( , , ,+( l ~M2,+4nM2,)sin~m+q~_E=O.

(18)

To find the response of the system to an applied electric field we examine the equation for the dual current ~ ~PJp:

{ q'" \2 q'-~

¢~nO=J/s = t ~ ) E+2e2xE+ ~ 02(,,, , (19)

where a sum over m is implied, and we have used Yq'~ =0. ( I f e = 0 this need not be true; this case will be discussed below.) To a good approximation the electric fields are applied adiabatically, since for times well after the phase transition at which a gains an expectation value the frequency of oscillation of a loop of string in any external B fields is small com- pared with the Mm, the characteristic frequency ap- pearing in (18). Hence we may drop oscillatory terms such as 02(m, and (19) can be written

Thus the fermions do contribute to the current, and their effect is to renormalise x. This effect stems of course from an anomaly, and we could have arrived at it through one-loop diagrams of the unbosonised theory [ 14-16] although the bosonisation makes calculations much easier. Its presence even for low momenta (i.e. small fields) is a well-known feature of anomalous processes. Note that (20) is not merely the sum of (4) and (14), the currents in the absence of a coupling between the zero modes.

Let us now examine the e = 0 case, where it is no longer necessarily true that Q = Zq'g =0. The right- hand side of eq. (19) now reads simply q~02~,~/ x ~ , which is not zero. In fact, since 0 and ~m are separately gauge invariant we have an extra equation of motion

( 1 ) QE 4~" (21) 0 2 2xO+ -~2q) , , =

This combination of fields clearly increases quad- ratically with time for a constant uniform E field, which means that the current increases linearly. Making the same assumptions about E that enabled us to ignore 02(,. we obtain

¢~nO.Ja - Q2E (22) 4zc(n+ 8rtx) "

Hence, although the Higgs field is neutral, the string carries current in a closely similar way to certain quasi-one-dimensional metals, where a periodic dis- tortion in one direction in the lattice opens up a gap just above the Fermi surface. The distortion induces a so-called charge density wave in the electron den- sity, and the sliding of these waves accompanied by the distortion constitutes a current, with no electrons excited across the gap [9]. The phases of the lattice distortion and the electron charge density wave are just the 0 and ~ fields respectively. This mechanism was originally invoked (unsuccessfully) by Frrhlich to explain superconductivity in ordinary metals. Hence we might say that this string is a Frrhlich superconductor.

Finally, we note some physical consequences. For a charged Higgs field coupled to fermions that are supermassive off the string, large currents can be car- ried whose effects have been detailed in refs. [2-8, 19]. If the core field is neutral and Zq~ = 0 there is no superconductivity, a result found by Hill and Widrow for a slightly different system [ 17 ] and also found by Davis [ 16 ]. However, when e = 0 there is no anomaly and the sum of the q~ need no longer be zero. In this case, as I have shown, the string is superconducting (by the Frrhlich mechanism [ 9]) and it is possible to excite large persistent currents. I f the string is not superconducting at all, current can still be carried on the string by massive fermions which have charges !2.+,m. The strings will be formed with one fermion of each type laid down at random in initial each correlation length ~o~ (G/t)-~/2r/-I [ 18 ]. Hence on a loop of string of length L the total number of fermions of each species will be N ~ x / ~ o. For a loop chopped of fa GUT scale net- work at, say, teq (roughly 10 ~ s) this number is ap- proximately 1024, and thus at low temperatures the Fermi seas of each particle will be filled up to a mo- mentum of N/L ~ 10 ~ l GeV. These fermions carry a current which is able to relax to zero, because they travel in both directions on the string and left mov- ers can be scattered into right movers, for example by electromagnetic interactions. However, even if there were no scattering there are only enough fer-

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Volume 200, number 4 PHYSICS LETTERS B 21 January 1988

mions to carry minuscu le currents. Suppose they were

all t r ave l l ing wi th the i r m a x i m u m poss ible mo-

men ta , equa l to the i r masses o f f the string, t hen they

w o u l d car ry a cur ren t J~Y~(rnoff/rno,,)(L~o) -I/2~ (moll~mort) X 10- ~ ~ G e V for the a b o v e va lues o f L

and Go. Hence , unless the d i f fe rence in mass scales

is un rea sonab ly large, such cur ren t s w o u l d no t have

any o f the spectacular effects p red ic ted in refs. [ 2 - 7 ] .

I a m i n d e b t e d to Joe Lykken for m a n y helpful dis-

cuss ions and for r ead ing an ear l ie r ve r s i on o f the

manusc r ip t , and to M i k e S tone for p o i n t i n g ou t and

exp la in ing charge dens i ty waves in sol id state phys-

ics. I w o u l d also l ike to t hank the o the r m e m b e r s o f

the Theo re t i c a l Par t ic le Physics G r o u p at Los Ala-

mos , and also T.W.B. Kibble , u n d e r whose super-

v i s ion this w o r k was started.

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[2] E. Winen, Nucl. Phys. B 249 (1985) 557. [3] E.M. Chudnovsky, G. Field, D. Spergel and A. Vilenkin,

Phys. Rev. D 34 (1986) 944.

[4] J. Ostriker, C. Thompson and E. Witten, Phys. Lett. B 180 (1986) 231.

[ 5 ] C. Hill, D. Schramm and T. Walker, Fermilab-pub-86/146- T (1986).

[6] A. Vilenkin and T. Vachaspati, Phys. Rev. Lett. 58 (1987) 1041.

[7] G. Field and A. Vilenkin, Nature 326 (1987) 772. [ 8 ] E. Copeland, M. Hindmarsh and N. Turok, Phys. Rev. Lett.

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