Shapes of cosmic strings and baryon number violation

23 May 19%

PHYSICS LETTERS B

ELSEWIER Physics Letters B 376 (1996) 41-47

Shapes of cosmic strings and baryon number violation Masatoshi Sato ’

Depurtment of Physics. Kyoto University, Kyoto 606-01, Jupan

Received 29 August 1995; revised manuscript received 12 February 1996 Editor: M. Dine

Abstract

The relation between shapes of cosmic strings and baryon number violation is investigated. If there exist fermionic zero modes on the string, using bosonization technique, it is possible to obtain the effective action which describes the fermion coupled to an arbitrarily shaped cosmic string. The relation between baryon number and the sum of the writhing number and linking number of the cosmic strings is rederived. Baryons are created on the strings as the shapes of the cosmic strings change. Furthermore we discuss implications of this baryon number violating process to baryogenesis.

PACS: I I .27.+d; I I .3O.Fs Krywor-rls: Cosmic string; Baryon number violation; Writhing number; Linking number; Zero mode; Bosonization; Baryogenesis

1. Introduction

Toward a better understanding of the physics of the string-like objects, it is important to search for typi- cal phenomena of their structures of extent. It is es- pecially interesting to investigate the entanglement of the string-like objects, which is inherent in the string- like objects in four dimensional space-time. Recently the present author and Yahikozawa attempted to find such phenomena for the Nielsen-Olesen vortices [ I]. It was found that the expectation value of s dx4 FpyFp,, under the background of the Nielsen-Olesen vortices becomes the difference between the sum of the linking number and the writhing number of them at the initial time and the one at the final time. This means that if chiral fermions are coupled to the vortices, the fermion number is violated as vortices change their

’ E-mail address: [email protected]. Fellow of the Japan Society for the Promotion of Science for Japanese Junior Scientists.

shapes, because of the anomaly. In this letter we in- clude fermions in the system explicitly and closely examine this mechanism of fermion number violation.

Another purpose of this letter is related to baryogenesis. Baryogenesis has become a topic of much recent activity. Not to contradict inflation scenarios, it is preferred that a baryon asymmetry is produced at lower temperature. In many GUT theories, the re- heating temperature after inflation must be lower than the GUT scale [ 2-51. This infers that baryogenesis at GUT scale [6-lo] is incompatible with inflation. Furthermore it is known that the EW sphaleron erases the baryon number asymmetry if the baryon number violating process at GUT scale preserves B-L [ 121. This also infers the baryon asymmetry is realized at lower temperature. To consider alternative scenarios of baryogenesis, we must introduce a baryon number violating process at lower than GUT scale without contra- dicting proton decay experiments. In most alternative baryogenesis scenarios, the EW sphaleron [ II] was

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42 M. Sum/Physics Lerrers B 376 (1996) 41-47

used [ 12- 141. Our mechanism of fermion number violation is another way to introduce a baryon number violating process without contradiction to proton decay experiments. We also investigate implications of our fermion number violating process to baryogenesis.

Table I U( I ) charge assignment

matter

charge rf 112 r- 112 i -- 112 1’+1/2 fl

There are related papers to this work [ 15- 171. They discussed the relation between the linking number of the strings and baryon number. In [ 171, using a restricted configuration of the strings, inclusion of fermions into the system was attempted.

In order to discuss baryon number from the writhing number, it is necessary to treat the fermion coupled to arbitrarily shaped strings. In the following, we will construct the effective action of the fermion coupled to arbitrarily shaped strings and show explicitly that baryons and leptons are created on the strings as they change their shapes. Furthermore we discuss its implications on baryogenesis. Units in which kB = h = c = 1 will be used and mpI is the Planck mass.

r QB = --•-_E k gr2 1.1 s

d3xZ;&

i QL = -E;‘k

89s .’ d3xZ;Z,,k

When the U( 1) symmetry is spontaneously broken, cosmic strings form. Under the background of the strings, the flux is concentrated on the strings and this can be expressed as

where

2. The construction of effective action of fermions coupled to arbitrarily shaped strings

The model which we will consider is

C=C gnuge - Higgs + &rmion 3

c gauge-Higgs = -$ZPyZPy + (D,qo)*DPq

- A( 1q/* - u2>“,

Here Xr denotes the position of the I-th cosmic string and 7 is the coordinate parameterizing the world sheet swept by the cosmic string. Solving this equation under the gauge conditions 7’ = @ and &Z; = 0, Z, become

+&.i$EL.+&i$ER-yl(&qo*EL+h.c.), (1)

where Z,, = a,Z, - &Z, and D,qo = (3, - iA,)q. The C/( 1) charges are assigned as shown in Table 1. The anomaly cancellation condition requires that r and i must satisfy the relation r* = i*. Because of the anomaly, the “baryon” number current j,” = uyp CJ and the “lepton” number current j,” E Eyfi E do not conserve and satisfy the following relations: Using this and (2)) we obtain

A a,jf = &ZByZpB. (2)

where Zp” = &‘“pnZp,,/2 and e”‘23 = -Co123 = 1. From these equation, we obtain The definition of W, and Lk are

UR UL ER EL P

(6)

cW,(X,) +~Lk(xI,~J) I I+J

~w,(&) + cLk(xI,xJ) (7) I I+J

M. Sato / Physics Letters B 376 (19%) 41-47 43

E:(x,y,z,t) =l(z,t)~~(x,y)

E~,(X,y,z,t) =~(zvf)PRa(X,Y) (10)

axi axj (x(7) - ~(7’))~ x E”kF&- IX(T) - X(T’)/3 ’

Substituting these configurations for Crehon and inte- grating over x and y, we obtain

dr; L&f = iq”(& - irzo)q+ iq*(d3 - irz3)q

ax’&; (X(7) - Y(T’))k x Tga7; IX(T) - Y(7’)13 .

(8) + il*(& - iizo)l- il*(& - iiz3)Z, (11)

where we define Leff as SdxdyGf,tii,,. The gauge fields za are p = 0,3 components of the ZFtex. For the z-directed string they can be zero, but for arbitrarily shaped strings they are not zero.

Lk is the Gauss linking number and it takes an integer value. Wk is called the writhing number and in general not takes integer value [ 181. This is a geo- metrical quantity and measures the twist of the cosmic string. Unless the cosmic string intersects with itself, the value of the writhing number changes continuously aa the shape of the string change. When the string intersects with itself, the value of the writhing number changes by 2. An explicit example of the writhing number was given in [ 191.

From (7)) it is found that the “baryon” number and the “lepton” number change according to the shapes of cosmic strings. To examine this more closely, we will analysis the fermionic part of the system.

On the cosmic string, the vacuum expectation value of the Higgs field 40 becomes zero. Since the fermions get masses by the Yukawa coupling, massless fermionic modes exist on the string. In the following, we will derive the effective action of these massless modes and show that these massless modes are produced on the strings as “baryon” or “lepton”.

First we will derive the effective action of the massless modes on the z-directed cosmic string. The field configuration of 9 and Z, are given as the Nielsen-Olesen solution ~“ortex and Zr@-’ [ 201. On this background, it is known that there exists a solution of the Dirac equation which is indepen- dent on z and t and normalized in the x-y plane [21,22]. We denote this as (U~,UR~,E~,ER~) = (.PP(x,,‘)~P~n(x.~).&(x>~)$~a(x~y)). P and p satisfy U3PR = PR, dL = -pL, u&R = -pR

and asj?~ = BL. We take the normalization as s dxdy)Pl’ = l/2. Using this solution we can express the fermion constrained on the string as

~2(x,y,z,t) =q(z,t)Pf(x,y) URm(x,y,ztt) = dz,t)PRu(x,y) ’

(9)

If we use qr = (q, O)r and Zl E (0, 1)r, this effective lagrangian can be rewritten as the following more familiar form

L,ff = ig,,” (a,, - irza) qr+i~ya(da-ifiztr)Z~, (12)

where a = 0,3 and the 2-dimensional gamma matrix is defined as y” = g’ and y3 = -iu2.

The “baryon” number current and the “lepton” number current on the string can be obtained in the similar manner. These are defined as J$ = s dxdyjg= and .I; = s dxdyjr=“ and become

J; = gry”qr,

J;: = f&V/. (13)

To discuss the effective action on the arbitrarily shaped string, bosonization technique is useful, since in the bosonized fields the anomaly can be treated classically [23,24]. Bosonization rules are given by

$r”&q + $%~,a~&, 4y’q + -$fbab& [2X261. USiIIg the relation qy’ysq = -Eab@bq, we obtain

i2 + GZUZ“, (14)

and

db)ab+y - f-pub - $‘)Zb,

--kub + ?fb)6$&$ + &(t.h + gb)& (15)

44 M. Suto/ Physic~s Letters B 376 (1996) 41-47

Here & and 41 are bosonized fields of q and 1 respec- tively. The terms which do not depend on +y or & are determined by gauge invariance. For the bosonization fields, gauge transformation can be defined as

(16)

This gauge transformation is expected from the oper- ator relation of q and @(/, and only when the anomaly cancellation condition is satisfied the bosonized action becomes gauge invariant.

Now we will consider the effective action of the massless modes on an arbitrarily shaped string Xp (7). This action is obtained from the above action by the replacement ~7~” --f ,,@gb and (z, t) -+ (TO, r’), where g,b = c?,,X~&,X, is the induced metric on the string and g = det gab. The effective action is

+ & ( 8’ + J-gg”” 1 zaab&

-2 + ~~~*b&,zb ( 17)

and the “baryon” number current and the “lepton” number current on the arbitrarily shaped string are

+ &(&lb + J-gga”) zb. (18)

Considering the massless modes on the I-th COS-

mic string, Z,,(T) are given by z<,(r) = (dXr/a+‘) x Zvortex X 7)). When the radius of curvature of the fi ((

string is much greater than the string thickness, ZTrtex are given as (6)) therefore

(19)

Note that za ( G-) are not singular at 7 = 7’. As far as the string does not intersect, ~~(7) are well-defined functions. This non-singularity of za is not depend on the gauge condition we take. Even if we use the a,Zfi = 0 gauge, they are not singular. This enables us to tell what occurs on the string definitely and makes the effective action ( 17) useful.

Using the equation of motion, we obtain

a,,J--gJff = -&Pz‘,hr A

a,J-gJ;: = bbZab. 497 (20)

From ( 19), we find that the “baryon” number on the string Q, - $ (‘) = &‘J-gJi and the “lepton” number

on the string Qi2’ 3 $ dr’ fiJ2 satisfy

AQr’ = -rA (

WAX/) + c &(&,XJ) , J( +I) )

AQL2’ = ?A (

W,(X/) + c L~(XI,XJ) . J(+l) )

(21)

These relations are consistent with (7) and show that “baryons” and “leptons” are produced on the strings according to the changes of their shapes.

Although we have treated only a toy model in this section, the conclusions do not change for more com- plicated models. If all the fermions get masses by Yukawa coupling, the effective action of the massless modes can be constructed in the same manner and it

M. Safe /Physics Lerrers B 376 (1996) 41-47 45

is found that baryons and leptons are produced on the string as the cosmic strings change their shapes. In- deed in the cases of the electroweak string [28,29] and cosmic strings in supersymmetric model with an extra U ( 1) [ 3 11, we have constructed the effective actions of the string-fermion system except the neu- trino part [ 271. If massless fermions exist, the above analysis does not apply in general, however, since (7) is irrelevant to the fermion mass, we believe that the conclusions do not change in this case.

3. Implications to baryogenesis

In the previous section, we have shown that as cosmic strings change their shapes, baryons are generated on them. In the following we will discuss baryogenesis due to this baryon number violation mechanism.

We will consider cosmic strings on which baryons are generated by our mechanism of baryon number violation. For example, the electroweak string is of this type of the cosmic strings. Another example is cosmic strings in a supersymmeric model with an extra CJ( 1).

As first pointed out by Sakharov, three conditions need to be satisfied in order to generate a net baryon number [ 321. First, baryon number violating process must exist. Second, these process must violate C and CP And third, they must occur out of thermal equilibrium. C and CP are violated in the usual particle physics models and once cosmic strings form, they become rapidly out of equilibrium. Therefore we know that the second and third conditions are satisfied. In the following we will concentrate on the first condition.

It is expected that our mechanism of baryon number violation is the most efficient at string formation, If cosmic strings form by the Kibble mechanism [ 331, the correlation length of the strings 5 at formation is given by t( 7) N (hr])-‘, where r] is the scale of the symmetry breaking by which cosmic string forms and A is a Higgs self-coupling constant or a gauge coupling constant. Since the strings form by randomly assigning values of the phase of the Higgs field, it is expected that the string twists once per ( Av) -3 volumes. Therefore the number of baryons or anti-baryons created on the strings is about ( AT~)~ particles per unit volume. This is comparable to the entropy of the massless particles at T = 7.

Although there exists a baryon number violating

process, it is not obvious to generate a net batyon number because the baryon number is violated due to the anomaly. In this case we find that a linear combination of the baryon number and the Chern-Simons form preserves. This gives a restriction to generate a net baryon number. For example, in our toy model,

QB + $eijk J’ dX3ZiZjk (22)

preserves. Since the system is in thermal equilibrium before cosmic strings form, using the CPT theorem, it is shown that the expectation value of the integra- tion of the Chern-Simons form vanishes before the strings form. Z;j becomes zero after the strings vanishes, therefore this tells us that if there is no net baryon before the cosmic strings form, there also remains no net baryon after the cosmic strings vanish. This is because in the toy model cosmic strings forms by spontaneous breaking of U( 1) gauge symmetry. Since U( 1) gauge symmetry has only one vacuum, no net baryon number is produced by the anomaly after cosmic strings vanish.

One possibility to obtain a net baryon number is to make a combination with an extra baryon number violating process. A candidate for the extra baryon number violating process is the sphaleron (or instanton) transition. The explicit baryon number violating process in GUT is another candidate, but if we consider baryogenesis scenarios after inflation, the sphaleron transition is more attractive since it does not contradict with experiments of the proton decay.

We assume that an extra baryon number violating process exists and the rate of the process is high at string formation. We further assume that after T = v’, this process becomes suppressed because of a phase transition. Until T = r)‘, the baryon number produced by the strings is erased by the extra baryon number violating process. Thus QB in (22) is zero at T = q’. After T = q~‘, the linear combination of the baryon number and the Chern-Simons form (22) preserves, because the extra baryon number violating process becomes irrelevant. Therefore we find that a net baryon number remains after the strings vanish and this is given by the sum of the writhing number and the linking number of the strings at T = Q’.

To estimate the sum of the writhing number at T = q’, considerations about the evolution of the string is needed. After formation, strings experience a sig-

46 M. SUIO /Physics Letters B 376 (1996) 41-47

nificant damping force from the high plasma background density. A heuristic argument [ 331 shows that in the friction dominated epoch the correlation length 5 changes as

l(T) = ($ -$)]“*. (23)

This infers that the structures of the string smaller than ’ (mpr/rl- ) ‘I* are straightened when T becomes slightly

lower than 7. This result is confirmed by solving the equation of motion. We take the metric of the Fried- man universe as ds* = n2( 7) (d? - dx2), where r is conformal time dr = dt/a( t). In the gauge conditions ra = r and 2 . x’ = 0, the string evolves as

I ’

X--E -I 0 ( f + 2!+E5 (1-32)~4 a lu >

(24)

where E is given by

/ _I2 \ ‘12

c= (&2) (25)

and ,!L = v2 is the string tension and p is a numerical constant of order unity [34]. Here dots and primes denote derivatives with respect to 70 and 71. Let us examine small perturbations on the static straight string. If we represent x as x = erl +6x( 70) eik+l, where e is a unit vector, and neglect the Hubble damping term, the equation of motion becomes

Si + pT”,S_t + k2Sx = 0. cc

(26)

In the radiation era, a and T are given as a - mptra and T N 2/7n. In order to examine this equation at 70 N 77-l, it is sufficient to solve the equation where the coefficient of Sk is replaced with a constant @npt. The solution can be easily obtained and we find that if k/u is larger than (v3/mpi)‘/*, the amplitude of Sx is damped within bra - l/v. This means that structures of the strings smaller than ( mpt/q3) 1/2 are straightened within AT - 7. This result agrees with the heuristic argument 2.

We have found that a net baryon obtained is determined by the sum of writhing number and the linking number of the strings at T = 7’. The sum of the

2 Same conclusion was obtained in /35,36J

writhing number per unit volume is expected to be &(v’) -3. If CP violation bias parameter is denoted as E, we obtain a net baryon number per unit volume as ES($) -3. Since the entropy of massless particles is given by s N T3, the baryon to entropy ration will be

n” - - 5($;“s’J* s (27)

This gives a severe constraint to work the scenario. Unless 7 is close to mpt or E is very large, we find that v’ must almost coincide with 7 to obtain a baryon asymmetry of the magnitude required to explain the present baryon to entropy ratio. If we consider cosmic string forming at O(TeV) and the EW sphaleron as the extra baryon number violating process, it is impossible to obtain the baryon asymmetry.

There is another possibility to obtain a net baryon number. Let us consider the string-formingphase transition G -+ H. If ~1 (G/H) is non-abelian, a topological argument tells that the strings corresponding to non-commuting elements of n-1 cannot pass through each other [ 37,381. Therefore the Iinking number of the cosmic strings almost cannot change and a net baryon number remains. Vachaspati and Field esti- mated numerically that the linking number of the cosmic strings per unit volume is lop4 ( Ar]) -3 [ 16 1. This gives the baryon to entropy ratio as 10-4~A-3.

We comment on the evolution of this type of strings. It was pointed out that this type of cosmic strings form a tangled network and if they form by the Ribble mechanism, they dominate the universe very soon after formation [ 391. This is an obstacle to the baryogenesis scenario. However, the conclusion changes if we take into account intercommuting of the string at crossing itself. Since an element of ~1 commutes with itself, the topological argument gives no constraint when the string crosses itself. Then this intercommuting is possible. This enables not to form a tangled string network, since by intercommuting, tangled long strings become loosely linked long strings with many linked tiny strings. Therefore too early string-domination in universe does not occur.

4. Conclusions

In this letter we have presented the effective action of fermions coupled to arbitrarily shaped strings. As

M. Sate/Physics Letters B 376 (1996) 41-47 47

was suggested in [ 11, the fermions are created as the strings change their shapes. Furthermore we have discussed its implication to baryogenesis. We have considered two possibilities to obtain a net baryon number. The first is to combine with an extra baryon number violating process. After a careful consideration of the evolution of cosmic strings in the friction dominated epoch, it has been shown that the string-forming scale must be very close to the scale where the extra baryon number violating process begins to suppress. This is because the structures of the strings smaller than (~npi/$) ‘I2 are straighted within AT N 7. This constraint is not inherent in our baryon number violating process. Defect-mediated electroweak baryogenesis [ 30,311 also suffers from this constraint” . As a second possibility, we have considered the cosmic string with non-abelian ~1 (G/H). It is possible to produce a baryon asymmetry without too early string- domination in universe.

Acknowledgments

The author would like to thank S. Yahikozawa for reading the manuscript and useful comments. He also thanks to M. Nagasawa for a useful discussion.

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s Eq. (3. IO) in I 301 is not correct and their conclusion must be changed.

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Shapes of cosmic strings and baryon number violation