Cosmic strings and baryogenesis

Volume 218, number 3 PHYSICS LETTERS B 23 February 1989

COSMIC STRINGS AND BARYOGENESIS

Robert H. BRANDENBERGER Department of Physics, Brown University, Providence, R1 02912, USA

Anne-Christine DAVIS Department Qf Applied Mathematics and Theoretical Physics and King's College, University q/Cambridge, Cambridge CB3 9EI4~ UK

and

Andrew M. MATHESON Department of Physics, Illinois Institute of Technology, Chicago, 1L 60616, USA

Received 25 August 1988; revised manuscript received 25 October 1988

Grand unified cosmic strings catalyze baryon decay, albeit only with a geometrical cross section. Nevertheless, the density of strings in the early universe could be so large that catalysis processes could wash out a substantial fraction of a primordially generated baryon-to-entropy ratio.

1. Introduction

Grand unified gauge theories provide ways to explain the observed net baryon-to-entropy ratio of the universe. Since baryon number is not conserved in such theories, the universe may evolve from an initial state with no net baryon number to a state with the presently observed baryon-to-entropy ratio

10-1°< nb--n-------E <10 s, (1) s

where nb and n6 are the baryon and antibaryon number densities and s is the entropy density. The first mechanism proposed was based on out-of-equilibrium decay of heavy gauge particles [ 1 ] in a theory with CP violation.

Topological defects which are produced during the phase transition when the grand unified symmetry is broken, however, provide channels via which the primordially generated baryon-to-entropy ratio could be washed out. The effect involves baryon decay via catalysis by the topological defects.

It is well known [2,3 ] that cosmic strings like mon-

opoles can catalyze baryon decay. In a previous pa- per [3] we calculated the cross section for this process. In contrast to monopole catalyzed proton decay, the cross section is not enhanced. Nevertheless, in the very early universe the density of strings is high enough that catalysis could wash out a significant fraction of the net initially produced baryon-to-entropy ratio.

The catalysis processes, an initial fermion b scat- ters offofthe core of a topological defect and emerges as a final fermion ~ with a different baryon number. In the early universe, initial and final states will be in thermal equilibrium. By detailed balance (or by the C P T theorem ), the amplitude for the process b - ~ is the same as the amplitude for ~ b . Hence it is im- possible to generate a net baryon-to-entropy ratio by catalysis processes ~. However, these processes can partially or completely erase an initial asymmetry.

For monopoles, this issue has been studied in ref. [4], where it was shown that the effect is too weak to erase an initial baryon-to-entropy ratio in the range

~ We thank H. Tye and J. Distler for stressing this point to us.

304 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


(1). In this letter, we shall consider this issue for cosmic strings. In section 2 we calculate the maximal baryon-to-entropy ratio which can be erased by catalysis. The result depends on the coupling constants of the grand unified theory. If these constants are large, then the effect can be important. In section 3 we explain why catalysis can be important for cosmic strings when it is unimportant for monopoles and when the cross section is much smaller for strings.

We consider a grand unified theory with scale of symmetry breaking M which admits cosmic strings. We assume that CP is violated. The catalysis cross section per unit length of cosmic string is [ 3 ]

da 1 m( T) dE2 dl - 0 ( 1 ) ~ r ~ ' (2)

where m (T) is the finite temperature mass of the fermions ~2. At high temperatures, m (T) ~ T. Then, as we shall calculate in section 2, the maximal baryon- to-entropy ratio which can be erased by catalysis is

nb--n6 ~_Dg2p X 10_6 , (3) s

where t'2p is the fraction of the total energy density in baryons and antibaryons ~3, and D is a function of the coupling constants.

A word concerning our notation. We use units in which c=kB=h= 1. G is Newton's constant and me1 the Planck m a s s . teq is the time of equal matter and radiation, z( teq ) is the corresponding redshift.

2. Baryogenesis from catalysis processes

As mentioned in the introduction, the cross section for baryon and antibaryon catalysis by cosmic strings is very small. However, the density of strings in the early universe is large, and thus catalysis could be an important effect in determining the final ( n b - rib)/S ratio.

Consider a simple example of a model with cosmic

~2 The last factor on the RHS of (2) comes from the spin sums and represents the correction for fermions to the classical geometrical cross section M - ~.

~3 Strictly speaking, 12p is the fraction of energy density in baryons and antibaryons evaluated at the present t ime assuming no catalysis. Hence g2p will be larger than the present density in baryons g2B.

strings #4: a gauge theory with a complex Higgs field with potential

V(~) = ~2(~ 2 - M 2)2. (4)

There is an additional term CT202 in the finite temperature effective potential, where C is proportional to g2, where g is the constant which couples ~ to the gauge fields. In this model, cosmic strings form during a phase transition at a temperature

1/2

at the time

_ ( _ _ 9 0 "]'/22C MP, M_~ , (6) t o - \ 32n3Nj 2 M

where N is the number of effective spin degrees of freedom in the radiation bath. The mean separation of strings at tc is [6]

~(tc)_~2-~Tg - ' - a M -~ , (7)

and we shall take the mean string length per domain [volume ~(tc) 3 ] to be C/~(tc), where C/is some constant of the order 1. For t> tc the correlation length increases faster than the scale factor of the universe a( t ) [recall that in the early universe a(t) ~ t ~/2 ]. We shall take

with 1 > p > ½. The maximal baryon-to-entropy ratio which can be

erased in the time interval dt is given by

( ~ ) da d = -dl lns@s-I dt . (9)

Here ns is the number density of strings, l is the mean length and q~ is the flux of baryons. We have

lns( t)=Cl~-2( t) (10)

and (taking the relative velocity of the baryons to be l )

qb(t)=nb(t)= l O p p ( t ) . (11) m

#4 Fora review see e.g. ref. [5].

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rtb is the number density o f baryons, m is the mass of the baryons and p (t) is the energy density.

From (9), it is obvious that the total baryon-to- entropy ratio - obtained by integrating from tc to the present time - is dominated by the contribution from earliest times. Hence,

nb--n~ _ ct~_z(tc ) 1 T,. nb(tc) t~. (12) S M M S(tc)

The n b / S ratio is constant and can hence be evaluated most easily at teq. Since

s (teq) = 4Tgqlp(teq) ( 13 )

we can combine (8-13 ) to obtain

nb - n6 _ Ci (2p. 3 oe_ 2 T ( tc ) TeqMte (14) s M m

Using (5) and (6), and replacing a by 2 and C with the help of (7), we obtain

n b -- n b

( ~ ) 1 / 2 ( 90 x~l/2N_l/2mpl Teq (15) 22C/~P \32/t3J M m

The result depends on the scale o f grand unification M, on the mass m of the light fermions, and on the coupling constants. To evaluate the above formula, we shall use C/= 10, ~p = 1, C_~2, and we shall ex- press the result in terms of ml, Ml6 and Nloo, where ml is the light fermion mass in units of 1 GeV, M16 is the scale of grand unification in units o f 1016 GeV, and Nmo is N i n units of 100. With

Teq =Zeq (2.7 K)-~ 5.8h2ff2X 10 -9 G e V (16)

(h is the expansion rate of the universe in units o f 100 km s- 1 M p c - ~ and ~ is the ratio o f energy density to critical density) we obtain

nb --n~ _ 10_6)]2.5h2M76 j m ? 1 N Folo/2 . (17) S

Eq. ( 17 ) gives the maximal fraction of the primordial baryon-to-entropy ratio which can be erased by catalysis. The result is extremely model dependent. However, provided the self-coupling 2 is not too small, our effect could be very important and erase the net baryon-to-entropy ratio produced by out-of-equilib- r ium decay of heavy gauge and scalar particles.

An alternate way to understand our effect is to de- termine if catalysis processes are in thermal equilib- r ium after the phase transition. To do this, we com- pare the catalysis rate

d6 F = ~7 lns (18)

with the expansion rate H of the universe, both calculated at t~.. Using (2) and ( 5 ) - ( 7 ) we find

~l~j.2M~-oJNloJo/2 )< . (19 ) 102

Hence, for large 2 catalysis processes are in thermal equilibrium at tc and can erase a substantial fraction of the net baryon-to-entropy ratio.

3. Comparison with catalysis by monopoles

Our conclusion that baryon decay catalysis by strings can erase a substantial fraction o f the primordial baryon-to-entropy ratio may seem surprising on the following grounds. It has been shown [4] that baryon number violating processes catalyzed by monopoles are always out of thermal equilibrium and hence cannot erase a primordial baryon-to-entropy ratio. It is also known - the Cal lan-Rubakov effect [ 7 ] - that at zero temperature the catalysis cross section for monopoles is enhanced by a factor ( M / m ) 4

compared to the geometrical cross section,

( Lo m ,20, Therefore one might conjecture that if monopoles cannot erase a significant baryon-to-entropy ratio, strings would be even less efficient.

The above reasoning is incorrect for the following two reasons. First, the number density of monopoles is small. It is constrained [8] by requiring monopoles not to dominate the energy density of the universe. String loops in contrast decay by gravitational radiation and their density is hence not subject to a similar constraint %

,s If the initial separation of monopoles were given by (( tc ), then the conclusions in ref. [4] would change. Baryon decay catalysis processes would initially be in thermal equilibrium and catalysis could erase a baryon-to-entropy ratio comparable to what we obtain for strings. However, we would be left with the monopole problem [ 8 ].

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Second and more impor tant ly , the difference in cross sections between strings and monopoles nar- rows at high temperatures . As shown in ref. [ 3 ], the monopole cross section is buil t up as follows:

d a 1 ( ~ ) 2 ( M )4

0£2 ~ ~ m ~ " (21)

The first factor is the classical geometr ical cross section. The second gives the correct ion factor for fermions and comes from the spinor sums in the cross section. The th i rd factor represents the ampl i f ica t ion of the fermion wavefunct ionals at the monopole core radius. It comes from solving the Dirac equat ion for s wave fermions in the presence and absence of the monopole . The mass which appears in the Dirac equat ion is the finite t empera ture effective mass as can be seen on physical grounds. Thus, the cross sec- t ion changes f rom being of the order m - 2 at T = 0 to being of the order M -2 at T ~ M . The enhancement disappears .

4. Conclusions

Cosmic strings can catalyze baryon decay and can hence erase a fraction of the p r imord ia l baryon- to- entropy ratio. The maximal rat io which can be de- stroyed is given by (17) . For a scale of symmetry breaking of 1016 GeV and a light fermion mass of 1 GeV, the ratio is

nb --//6 _ D X 10 -6 , (22) S

where D is a funct ion of the coupling constants of the model. Typically, D is of the order 22, and thus 2 must be smaller than 10 .2 i f we want a p r imord ia l ratio in the range (1) (p roduced e.g. by out-of-equi l ibr ium decay of heavy gauge and scalar part icles before cosmic string fo rmat ion) to survive.

Note that our effect arises from a tree order amplitude (it is dr iven by the difference in number densi- t ies) . The effects of catalysis are most impor tan t at the critical temperature . Any mechanism which gen- erates a net baryon- to-ent ropy rat io at low temperatures is unaffected.

The impor tan t condi t ions for our scenario are the existence of baryon number violat ing processes and the existence of cosmic strings. In addi t ion, the gauge

and scalar fields excited inside the string must be among those which media te baryon number violation. Two examples of grand unif ied theories which satisfy these cri teria are SO (10) and E6 [ 9 ]. Using a Higgs field in the 126 representat ion, SO (10) breaks to SU (5) X Z2 giving rise to Z2 strings. Similarly, Z2 strings arise when E 6 is broken to SO(10 ) using a Higgs field in the 351 representat ion. In both cases the full G U T symmetry is present in the core of the strings. As poin ted out recently [ 10], there is an addi t ional requirement which must be satisfied in order for the mechanism discussed above to work: if the fermions have non-integer charges with respect to the string, then their wavefunct ions will be sup- pressed in the core of the string and there will be no catalysis. Thus, we must have integer charges.

Acknowledgement

We wish to thank J.R. Bond for quest ions which s t imula ted this work, and H. Tye, J. Dist ler and T.W.B. Kibble for useful discussions. One o f us (R.B.) would like to thank the Depa r tmen t of Ap- pl ied Mathemat ics and Theoret ical Physics of Cam- bridge Univers i ty and in par t icular Professor J.C. Taylor for hospi tal i ty while this work was started.

References

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Volume 218. number 3 PHYSICS LETTERS B 23 February 1989

[7] C. Callan, Phys. Rev. D 25 (1982) 2141; D 26 (1982) 2058; V. Rubakov, Pis'ma Zh. Eksp. Teor. Fiz 33 (198l) 658~ Nuch Phys. B 203 (1982) 311; F. Wilczek, Phys. Rev. Lett. 48 (1982) 1146.

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Cosmic strings and baryogenesis