Volume 208, number 1 PHYSICS LETTERS B 7 July 1988

BARYON NUMBER GENERATION FROM COSMIC STRING LOOPS

Masahiro KAWASAKI Department of Physics, Faculty of Science, University of Tokyo, Tokyo 113, Japan

and

Kei-ichi MAEDA NASA/Fermilab Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, 1L 60516 and Department of Physics, Faculty of Science, University of Tokyo, Tokyo 113, Japan

Received 30 November 1987

USA

Baryon number generation due to the decay of particle-antiparticle pairs created from cosmic string loops is studied. In the situation when cusp evaporation occurs a significant baryon asymmetry (baryon/entropy ~ 10 4 ~: E is a net baryon asymmetry generated from one pair of particle-antiparticle) can be generated.

1. Introduction

Cosmic strings [ 1 ] ~ are topological defects which are generated at the grand unification phase transi- tion in the early universe. In particular, cosmic string loops, which are produced by an intersection of two infinite strings or a self-intersection of an infinite string, can work as seeds in the process o f galaxy for- mation [ 3 ]. Cosmic string loops oscillate with period ~ L / 2 (L: size of a loop) and their sizes are reduced by emitting gravitational radiation. The radiation rate is estimated as ~ Gp: where it is the line density o f the string and is expressed by using the grand unifi- cation scale a ( ~ 10- 3 Mp~ ) as p ~ a 2.

Recently Srednicki and Theisen [4] and Brandenberger [5] have discussed energy loss by particle-antiparticle emission from string loops. They have considered the coupling between the Higgs field responsible for strings and another "Higgs" field and have found that "Higgs" particle-antiparticle pairs are emitted mostly from the so-called cusps. But they concluded that energy loss due to particle-antiparti-

Permanent address. ~l See ref. [2] for a review.

cle emission is negligible in comparison with that by gravitational radiation.

In this paper we study baryon number generation from cosmic strings assuming that emitted "Higgs" particles decay into quarks and leptons with viola- tion of baryon number conservation. Although par- ticle-antiparticle emission is energetically negligible, it may generate a significant baryon number asym- metry in our universe. The baryon number genera- tion from cosmic strings was studied by Bhattacharjee, Kibble and Turok [6] but from a dif- ferent context in which the baryon number is gener- ated in the collapsing phase of string loops. Their scenario is based on the assumption that the string loops collapse rapidly by the self-intersections but this assumption is in contradiction with the galaxy for- mat ion scenario by cosmic strings.

The existence o f cusps where strings move with the velocity o f light is very important in considering par- ticle-antiparticle emission. In reality cusps cannot exist because strings have a finite width ( ~ a - l ) but a part moving close to the speed of light may exist. Furthermore, in such a region microscopic interac- tions may dominate over the macroscopic dynamics o f the strings and the cusp region may be evaporated by particle-antiparticle emission.

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Volume 208, number 1 PHYSICS LETTERS B 7 July 1988

It seems that strings without cusps do not play any important role in the emission, but it is not the case for a new type of cosmic string loops which were found by Garfinkle and Vachaspati [ 7 ]. Those loops have kinks instead of cusps. We will show that kinky loops emit particle-antiparticle pairs at a rate similar to the loops with cusps.

The particle-antiparticle emission rate is esti- mated in two ways. First we calculate an emission rate by a perturbative method when the string state is changed little by the emission. However, non-pertur- bative processes may be more important. We cannot estimate the emission rate precisely in this case, but we can obtain a rough estimate by considering the evaporation of the cusp region [5 ]. Baryon number generation in both processes is considered in this pa- per and it is concluded that a significant baryon num- ber can be generated when cusp evaporations occur.

2 . P e r t u r b a t i v e p r o c e s s

We first consider the perturbative emission from cosmic loops following Srednicki and Theisen [4] and Brandenberger [5]. A complex scalar field which is responsible for strings may couple to various other fields. Here we assume that 0 couples to an- other scalar field ~u via the interaction lagrangian

Lim = -) . lOl2g t2. (1)

Let us introduce a real field q~ defined by

~ 0 = 1 0 1 - ~ , (2)

where ~0 = 0 far from the string and ~0 = - a at the cen- ter of the string. The amplitude for a string state IS) to emit two "Higgs" particles and to become a state IS' ) is given by

( S t ; ~Jl~2 I S )

d4x exp[i(k~ + k 2 ) x ] ( S ' I (02IS). (3)

If the string state is not changed by the emission pro- cess, we can replace (S ' I with (S I. Since strings are very thin, ~0 deviates from zero only on the strings. Hence ( S I ~o 21S ) is expressed by

L

(Sl~021S) ~ fd~ls'(~,t)12~3(x-s(~,t)), (4) 0

where s is the vector describing the string location and is a function of two parameters ~ and t. s satisfies the equation o f motion and gauge conditions:

,~'= S", ( ~ $ ' ) 2 = 1. (5)

Eq. (3) is expressed by using the periodicity o f s as

( S ' ; ~', q/2 IS) -~292 ~ an~(E-4zcn/L), (6) n

L 2f a,,= Z d¢ls'(~,t)12exp[-ik.s(~,t)l, (7)

0

where E and k are the total energy and momentum of a particle-antiparticle pair. The total number of par- ticle-antiparticle pairs from one loop with size L is given by

N=~Nn, rl

f d3kl d3k2 Nn=2~r d (2g)3(2kO) (2g)3(2kO) l a . I 2

×d(E-4~rn/L), (8)

where kL and k2 are the momenta of the emitted par- ticle and antiparticle. The dominant contribution to N comes from a cusp for an ordinary loop and N is estimated approximately by

N ~ 1 1 1 F/ma x 1 L ~ n nmin L ' - - - ~ ~ l n

nm,x~aL, nmi,~(mvL) 3/2, (9)

provided that L<~ cr2/m 3, i.e. the min imum energy of emitted particles (m 3 L ~/2 ) [ 3 ] is smaller than ~r.

3. K i n k y l o o p s

There are families of string loops which satisfy eq. (5) and have no cusps. Among them, a family of loops with kinks is very interesting because those may be generated naturally by an intersection of two strings. The kink runs around the loop with the velocity of light in one direction and is present at all times. The simplest kinky solution of eq. ( 5 ) is [ 7 ]

s(~, t) = ½ [ a ( f f - t) +b(~+t) ], (1o)

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Volume 208, number 1 PHYSICS LETTERS B 7 July 1988

a = [ ( L / 2 ~ ) ~ - ~LIA,

= [( - L / 2 r c ) ¢ + ]L]A,

b= [ ( L / 2 r Q q - ~L ]B,

-- [( -L /2~z) t /+ ~L]B,

0~<~<g,

~ ¢ ~ < 2 g ,

0~<t/~<g,

~<q~<2~, ( 1 0 cont 'd)

where A and B are constant unit vectors and ~ and t/ are defined by ¢=(2g /L) (~- t ) and q=(2~/L) × (¢+ t). Substituting this into eq. (7) we get

L ( n2+~q tc2 a,=2--- 5 (nZ_x2)(n2 x~) ( 1 - A . B )

n 2 - - I¢1N2 (1 + A - B ) ) + (n2_K~) (n2_~c~)

× sin( ½7~(n+ x~ ) ) sin( ½n(n-K2) ), ( 11 )

where x~ and x_~ are defined by K~ =L(k.A)/4~r and x2=L(k.B)/4zt. Since n>>xl, K2, [a,[ 2 is approxi- mately given by

l a,, 12= (L/2rr 2)2( 1///4)

X sin2( ½rr(n+x~ ) ) sin2 (½ n(n + x, ) )

= ¼ (L/2n2)2 ( 1/n4). ( 12 )

Hence from eq. (8) we get as an expression of N

1 N= Y, (1/12rc4L//)~ ~,. (13)

n

This is the same order of magnitude as that obtained in the string loops with cusps.

4. Cusp evaporation

So far we have neglected the width of the string. This is because the width of strings is very thin ( ~ a -~) and may not affect their global motion. However, in the region where cusps arise the width of the string cannot be neglected and the cusp region may be smoothed. Near the cusp the string solution is given by [ 8 ]

(Cl +c22 + c 2) (}t3+~zt) - c , c2 (t2~+ -~ 3), sx=t - 1 2

s .... c, ½ (t2+ff z) +c2¢t+c4(}t3+~zt) +c5(t2¢+ _~3),

s: =c3(t+c6( }t3+ (2t) +c7 ( t2(+ ~ 3 ) , (14)

where cb c2 and c3 are constants of the order 1/L and

C4, C5, C 6 and c 7 are constants of the order 1/L 2. As is seen from eq. (14) at the cusp (t, ~ )= (0 , 0) two branches of the string are very close. Therefore if the string width is taken into account, two branches are overlapped in the cusp region. In such a region the particle-antiparticle production may proceed very rapidly through the interaction of eq. ( 1 ). This pro- cess cannot be treated perturbatively. We, however, do not yet know how to treat this non-perturbative process properly.

In the extreme limit the cusp region may evapo- rate. Then the emission power of particle-antiparti- cle pairs is estimated as [ 5 ]

P~ lt~c/L~ #( aL ) -~/3, (15)

where ~c = a-W3L 2/3 is the length of the cusp region and is defined by

Is(0, ~c) - s ( 0 , - ( c ) l -~a - ' . (16)

Then the emission rate N is given by using the aver- age energy of the particle-antiparticle pair (E} as

N~lZ/(aL)~/3(E}. (17)

( E } is expected to be between m,~ and a.

5. Baryon number generation

We can now estimate baryon number generation from string loops. Baryon number asymmetry is pro- duced if emitted particle-antiparticle pairs decay in a way which violates baryon number. Here we do not specify any model or any interaction which violates baryon number conservation. Instead we introduce a parameter e which is the average baryon asymmetry produced by the decay of one particle-antiparticle pair. The mechanism of baryon number generation we consider here is almost the same as that due to Higgs decay in the inflationary universe. The differ- ence is whether Higgs particles are produced by infla- ton decay or by cosmic string decay.

The created particle-antiparticle pairs do not con- tribute to the baryon asymmetry when the tempera- ture is higher than the mass of the particles. This is because the inverse decay processes erase the baryon asymmetry at such a high temperature. When the temperature falls below the mass of the particles, the inverse decay processes are suppressed by the

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Volume 208, number 1 PHYSICS LETTERS B 7 July 1988

Boltzmann factor ~ exp ( - my~ T). Therefore we as- sume that the inverse decay is neglected after the temperature drops below mv/lO. Then the baryon number produced by cosmic string loops is given by

f {,8) nB ~- dt dLeN(L)n~t(t,L) kR(ts) j , ts

where n~(t, L) is the distribution function of loops and R is a cosmic scale factor and t~ is the time at which the temperature becomes my~ 10 and is given by

l~64x53Ng45 [ "~ 2 my &=100 1 - - 1 tp,, (19)

k, Mpl]

where Ng is the effective number of helicity states ( ~ 100).

5.1. Perturbative process

In the case of perturbative emission considered in sections 2 and 3 the particle-antiparticle emission can be neglected energetically in comparison with the gravitational radiation. Then the distribution of string loops is determined by gravitational energy loss [ 3 ] : (1) t , < t < (~G#) - t t ,

nst=vt_3/2L 5/2, t ,<L<t,

=0, L<t, ,

(2) t> (TG#)--~t,

nst = vt-3/2L -5/2, (TG#)t<L< t,

(20)

=v(TG#)-5/2t -4, L<(TG#) , (21)

where t, is the time when effects of friction between strings and surrounding matter become negligible. Before t, the motion of string loops inside the hori- zon is highly damped by the friction and string loops may shrink rapidly. Using eqs. ( 18 ) - (21 ) we get the final ratio r/B of baryon to entropy:

t/B = riB/S__ 87~{V ( 4 ~ ) ' / 4 ( G # ) 3 ( p G # ) -1

( 45 "~ I/4 for my > ml - 10 k ~ j

(22) X (Gp)(7G#) I/2Mpl,

and

~B '~ 8 X 1 0-- 37L'{ P \ ~ , ] m p l

X (G#)3(7G#)-'

f o r m v < m j. (23)

Inserting G # - - l 0 6, 7G#= l0 -4 and Ng= 100 into eqs. (22) and (23), ~/B is estimated as

3 my

t/B ~ 10-'4{V min (1, ( 1 0 T r ~ e V ) ) << 10- 1°. (24)

Therefore this contribution to the present baryon asymmetry ( ~ 10-~o) is negligible.

5.2. Cusp evaporation

Energy loss by cusp evaporation is not negligible in the early stage of string evolution ( ~ t , ) . Let us con- sider the lifetime of loops for this energy loss. Since the energy loss per unit time is #(aL)-1/3, the life- time is then ltL/#(aL)-I/3=L4/3a 1/3. Hence the minimum size of loops which survive more than one cosmic expansion at time t is ~ t3/4a-~/4 while that for energy loss by gravitational radiation is ~ yG#t. As a result the minimum size of loops is determined by cusp evaporation as - t3/4a-1/4 until the cosmic time equals to t3= (7G#)-4(G#)-l/2tpi. The distri- bution function of loops is then modified by: (1) t , < t < (G#)-5/ztm=-&

nst=Vt-3/eL -5/2, t ,<L<l,

=0, L<&,

(2) t2<t<t3

nst=Vt-3/ZL-5/2, t3/46-I/4<L<l,

=0, L<t3/46 1/4,

(3) t>t3

nst=vt 3/2L-5/2, (},Glt)t<L<t,

=v(yG#)-5/21-4, t3/4a-'/4<L<yG#,

=0, L<t3/4~7 1/4 (25)

where t2 is the time at which loops with size t, decay out. Then the final ratio of baryon to entropy is given by

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Volume 208, number 1 PHYSICS LETTERS B 7 Jul2¢ 1988

,,o \ M n I (g 'Lt) 2

( 45 ~ w4 for m~, << m2 - 10 \ 167r3Ng j (Gg)5/4M~,~,

(26) / \ 3 / 4

~/B--~4EU ((E)~-' (~-~e~)\MpI/] (G,/~) 17/.6

1

( (E>'~ (TG,u)3/2(G,u)5/4 + 10eu \ M p I j

( 45 "~ 1/4 for lO \ ~ j (Gp)I/4(yGll)2Mp,

= m 3 << my << m2, (27) / / - , 5 / 3 1

103~Z~V j 1/B---

for m,~ << m3. (28)

If we take Ggt= 10 -6 and 7G#= 10-4 then we get the final ratio of baryon to entropy as is shown in fig. 1. qB increases as the average energy of emitted parti- cle-antiparticle pairs decreases. It turns out that qB~O( 10-4)Ev if the mass m,, is about 101° GeV and ( E ) ~m,,. It is possible to account for the present baryon asymmetry in this case. While t/B reduces to (10-8-10-~°)eu for ( E ) = a and m,~> 1010 GeV. Furthermore, the estimation is based on the uncer- tain assumption that cusp evaporation really occurs.

-2

-4

-6

-8

Log ( ~ ) - 1 0

-12

-14

-16 -11

<E> =,5

<E> = M#

' i i , i i , i , i

10 -9 -8 -7 -6 -5 -4

Log ( m l M i )

Fig. 1. Final baryon to entropy ratio r/B as a function of"Higgs" mass m~ for ( E ) =m,¢, aand Mr, i.

We should regard this estimation as an upper limit for non-perturbative emission from string loops.

6. Conclusion

As was seen in the previous section the perturba- tive emission has a negligible effect on the baryon asymmetry in our universe. Only the non-perturba- tive effect may contribute to the present baryon asymmetry. It is, however, difficult to perform a pre- cise estimation for the non-perturbative effect.

In this paper, assuming that cusp evaporation oc- curs, we have estimated baryon number generation, and have found that a singificant baryon number can be generated. Unfortunately, so far, we have no po- sitive evidence of cusp evaporation. If this really oc- curs those cosmic string loops may account not only for the galaxy formation but also for the baryon asymmetry in our universe.

Acknowledgement

The authors would like to thank M. Izawa for use- ful discussion and Professor K. Sato for continuous encouragement. One of the authors (K.M.) acknowl- edges R. Brandenberger, E. Copeland and N. Turok for valuable discussion and Rocky Kolb and the The- oretical Astrophysics Group at Fermilab for their hospitality. He is also grateful to the Yamada Science Foundation for financial support for his visit to Fer- milab. This work was supported in part by the DOE and by the NASA at Fermilab.

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2175; H. Sato, Prog. Theor. Phys. 75 (1986) 1342; A. Stebbins, Astrophys. J. 303 (1986) L21.

[4] M. Srednicki and S. Theisen, Phys. Lett. B 189 (1987) 397. [5] R.H. Brandenberger, Nucl. Phys. B 293 (1987) 812. [6] P. Bhattacharjee, T.W. Kibble and N. Turok, Phys. LeU. B

119 (1982)95. [7] D. Garfinkle and T. Vachaspati, Phys. Rev. D 36 (1987)

2229. [8] D.N. Spergel, T. Piran and J. Goodman, Nucl. Phys. B 291

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