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Baryon number violation, baryogenesis and defects with extra dimensions


SIT-HEP/TM-9

Baryon number violation, baryogenesis and defects with extra dimensions

arXiv:hep-ph/0204307v3 6 May 2002

Tomohiro Matsuda

1

Laboratory of Physics, Saitama Institute of Technology, Fusaiji, Okabe-machi, Saitama 369-0293, Japan

Abstract In generic models for grand uni?ed theories(GUT), various types of baryon number violating processes are expected when quarks and leptons propagate in the background of GUT strings. On the other hand, in models with large extra dimensions, the baryon number violation in the background of a string is not trivial because it must depend on the mechanism of the proton stabilization. In this paper we argue that cosmic strings in models with extra dimensions can enhance the baryon number violation to a phenomenologically interesting level, if the proton decay is suppressed by the mechanism of localized wavefunctions. We also make some comments on baryogenesis mediated by cosmological defects. We show at least two scenarios will be successful in this direction. One is the scenario of leptogenesis where the required lepton number conversion is mediated by cosmic strings, and the other is the baryogenesis from the decaying cosmological domain wall. Both scenarios are new and have not been discussed in the past.

1

matsuda@sit.ac.jp

1

1

Introduction
Although the quantum ?eld theory made a great success, there is no consistent sce-

nario in which the quantum gravity is included. The most promising framework that could help in this direction would be the string theory, whose consistency is maintained by the requirement of additional dimensions. At ?rst the sizes of extra dimensions had
?1 been assumed to be Mp , however it has been observed later that there is no obliga-

tion to consider such a tiny compacti?cation radius[1]. The large compacti?cation scale may solve or weaken the traditional hierarchy problem. If we denote the volume of the n-dimensional compact space by Vn , the observed Planck mass Mp is obtained by the
2 n+2 relation Mp = M? Vn , where M? is the fundamental scale of gravity. If one assumes

more than two extra dimensions, M? can be very close to the electroweak scale without con?icting any observable bounds. Although such a low fundamental scale considerably improves the standard situation of the hierarchy problem, the scenario requires some degrees of ?ne tuning, which is apparently a modi?cation of old hierarchy problems. The most obvious example of such ?ne tuning is the largeness of the quantity Vn . There are other aspects of ?ne tuning, which are common to conventional scenarios of grand uni?ed theories. In particular, the problem of obtaining su?cient baryon number asymmetry of the Universe while keeping protons stable should be one of the most serious problems in models with low fundamental scale. In standard GUT theories, the proton stability is usually achieved by increasing the mass of the GUT particles that mediate the baryon number violating interactions. At the same time, the GUT-scale heavy particles assist the non-equilibrium production of the baryon number asymmetry. However, in theories with low fundamental scale, the suppression can not be achieved by merely increasing the mass scale. In this respect, some non-trivial mechanism is needed to solve this problem. Recently an interesting mechanism was suggested in ref.[2], where a dynamical mechanism of localizing fermions on the thick wall is adopted to solve the problem of fast proton decay. In this scenario, leptons and baryons are localized at displaced positions in the extra space, where the smallness of their interaction is insured by the smallness of the overlap of their wavefunctions along the extra dimension.

2

On the other hand, to explain the observed baryon asymmetry of the Universe, baryon number violating interactions must have been e?ective but non-equilibrium in the early Universe, because the production of net baryon asymmetry requires baryon number violating interactions, C and CP violation and a departure from the thermal equilibrium[3]. If the fundamental mass scale is su?ciently high, the ?rst two of these ingredients are naturally contained in conventional GUTs. The third can be realized in an expanding universe where it is not uncommon that interactions come in and out of equilibrium, producing the stable heavy particles or cosmological defects. In the original and simplest model of baryogenesis[4, 5], a heavy GUT gauge or Higgs boson decays out of equilibrium producing a net baryon asymmetry. In models with large extra dimensions, however, the situations are rather involved because of the low fundamental scale and the requirement for the low reheating temperature. Such a low reheating temperature makes it much more di?cult to produce the baryon asymmetry while achieving the proton stability in the present Universe.2 In this respect, it is very important to propose ideas to enhance the baryon number violating interactions that can take place even in the models with low reheating temperature. In this paper we propose a mechanism where the enhancement of the baryon number violating interaction is realized by the cosmological defect. The plan of our paper is the following. In section 2 we show how to enhance the baryon number violating interaction in the background of a string. Although the mechanism might seem similar to the one in the standard GUT string, there is a crucial di?erence. In the standard scenario of the GUT string, the cross section for the baryon number violating interactions mediated by the string is usually enhanced by the factor
MGU T 2 . Mproton

We stress that one cannot reproduce this enhancement in models with extra dimensions merely extrapolating the analyses on the standard GUT string. We also make a brief
2

Aspects of baryogenesis with large extra dimensions are already discussed by many authors. For

example, in ref.[6, 7], it is argued that the A?eck-Dine mechanism can generate adequate baryogenesis. In ref.[8], the global-charge non-conservation due to quantum ?uctuations of the brane surface is discussed. The baryogenesis by the decay of heavy X particle is discussed in ref.[9, 10]. In ref.[11], it is argued that a dimension-6 proton decay operator, suppressed today by the mechanism of quark-lepton separation in extra dimensions can generate baryon number if one assumes that this operator was unsuppressed in the early Universe due to a time-dependent quark-lepton separation.

3

comment on the scenario for leptogenesis with low reheating temperature. If the maximum temperature is lower than the temperature for the electroweak phase transition, sphalerons cannot convert existing leptons into baryons, which is a serious problem for leptogenesis. Even if sphalerons are not activated, strings with enhanced baryon number violation can mediate the baryon number production. In section 3 we comment on baryogenesis from the decaying cosmological defect. Cosmic strings are not e?ective in generic cases, but domain walls are promising.

2

Defects and enhanced baryon number violation
Our ?rst task is to review the old issues of enhanced baryon number violation due

to conventional GUT strings. Then we extend these analyses to models with extra dimensions, where the fast proton decay is suppressed in the present Universe due to the localized wavefunctions along extra dimension. One can easily ?nd why the naive application cannot reproduce the enhancement of the baryon number violation that appears in standard GUT strings. We solve this problem, and construct strings with enhanced baryon number violation in models with extra dimensions. Our idea is based on the idea of ref.[10]. We stress that our mechanism works without GUT symmetry, but some extensions are required for the standard model.

2.1

Enhanced baryon number violation in standard GUT strings
In the interior of a string formed after GUT symmetry breaking, there are ?elds

that carry both baryon and lepton number. Thus a quark comes into the core of such strings will interact with the background core ?elds, scattered to emerge as a lepton, and vice versa. A GUT string therefore is a candidate source for baryon number violating processes in the early Universe. This reminiscents of the Rubakov-Callan e?ect[12], where the baryon number violating interaction is mediated by monopoles. To be more speci?c, inside the core of the string there are quark-lepton transition mediated by X ? and Y ? gauge bosons, or Yukawa couplings to charged scalar ?elds φX that may condensate within the string core[13]. Denoting the scalar ?eld that forms the string by φstring , a perturbative calculation 4

of the scattering cross-section per unit length (dσ/dl) for a scalar particle φ coupled as
1 ′ λ |φstring |2 φ2 2

reveals that [14] dσ dl ? λ′ λ
2

E ?1 ,

(2.1)

where λ denotes the self-coupling constant of φstring . Fermions have di?erent couplings and di?erent phase spaces, and require more discussions. With a simple coupling, gφ? string ψ ψ , the cross-section per unit length is calculated as [14], dσ dl ? g λ
2 c

E , 2 MGU T

(2.2)

which is “not enhanced”. However, there is a crucial factor in the calculation for gauge strings. If the charge of the string ?eld φstring is e, the resulting cross-section depends strongly on the ratio q/e, where q is the charge of the scattered particle under the generator of the broken U (1). The elastic scattering cross-section depends crucially on the ratio q/e, since it controls the ampli?cation of the scattering wave-function at the core of the string[15]. In general, the cross section also depends on the type of the interaction, whether it is mediated by a scalar or a vector ?eld. A vector ?eld produces its maximum e?ect for integer q/e, then the baryon-number violating scattering cross section per unit length σ
B /

becomes ? E ?1 . The baryon-number violating cross-section from a core scalar

?eld takes it maximum value for half-integer q/e, then it reaches ? E ?1 . We shall consider a simple example. Here we follow the settings and analysis in ref.[16] and show the outline of the argument. We consider a straight string formed by a U (1)S charged scalar ?eld φstring , which condensates outside the string, i.e., < φstring >? η . This U (1)S is orthogonal to electromagnetic U (1)em . We include a second scalar ?eld φX that is not charged under U (1)S , carries baryon number, and condensates within the string core. The Lagrangian includes Yukawa interactions L = ?λ(φX χψ + φ? X ψχ) (2.3)

where ψ and χ are both fermions of U (1)S charge. In ref.[16], the cross section for the production of χ in the scattering of ψ fermions o? a string is calculated. The Dirac equations for ψ and χ have o?-diagonal element λ < φX >, which vanishes outside the

5

string but becomes non-zero in the string core. Schematically, the Direc equations are written by
? ? ?

i ? /?i A / ? mψ λ < φ X >?
? ? ? ? ? ? ?

λ < φX >

?

i ? /?i A / ? mχ

?? ?? ??

ψ ? χ
?

?

=0

(2.4)

where

A=

1 gr 2

, λ < φ X >= ?y ? ? ? x
? ? ?

0 ? ?

?

? ? ?

0, f or [r > R] v, f or [r < R].

(2.5)

The result is interesting, since the cross sections for these processes are generically enhanced by the large factor over the naive geometrical cross section. They have explicitly calculated the cross section for quarks and leptons, which are denoted by ψ and χ in eq.(2.3), and found that the enhance factor becomes up to
MGU T 2 . Mproton

In ref.[16], the

baryon-number violating processes including couplings to core gauge ?elds X ? and Y ? are also calculated and found that these scattering cross sections are enhanced by the same factor. With this in mind, it is natural to ask whether a similar e?ect occurs for cosmic strings in models with extra dimensions. In the models with localized wavefunctions along the extra dimension, the baryon-number breaking interactions, such as λ′ φX ψχ must be suppressed exponentially in the true vacuum so as to ensure the proton stability. In this case, even if the charged scalar φX condenses in the core of the string, λ′ < φX > remains negligibly small. Thus in the naive setup, the baryon number violating interaction mediated by the string with extra dimensions is exponentially suppressed and phenomenologically negligible. In the next subsection we show that we can improve this disappointing result if the idea of ref.[10] is taken into account.

2.2

Defects in models with extra diemnsions
Now we want to construct viable models for cosmic strings that can mediate baryon

number violation in models with extra dimensions. Although the TeV scale uni?cation is already discussed by many authours[17, 18], a naive extrapolation of the standard GUT string cannot madiate baryon-number violating interactions, from the reason that we have 6

discussed above. In the following, we do not assume explicit realization of GUT, since one may include baryon-number breaking couplings without assuming GUT embeddings. These baryon-number violating couplings must be suppressed in the vacuum to prevent fast proton decay. In ref.[10], we have proposed a new scenario of baryon number violation, which can be utilized to solve this dilemma. To show the elements of our idea, here we limit ourselves to constructions with fermions localized within only one extra dimension[2] and show how the tiny couplings can be enhanced in the defect. To localize ?elds in the extra dimension, it is necessary to break higher dimensional translation invariance, which is accomplished by a spatially varying expectation value of the ?ve-dimensional scalar ?eld φA of the thick wall along the extra dimension. If the scalar ?eld φA couples to the ?ve-dimensional fermionic ?eld ψ through the ?vedimensional Yukawa interaction gφA ψψ , whose expectation value < φA > varies along the extra dimension but is constant in the four-dimensional world, it is possible to show that the fermionic ?eld localizes at the place where the total mass in the ?ve-dimensional theory vanishes. For de?niteness, we consider the Lagrangian L = ψi (i ? /5 + gi φA (y ) + m5,i ) ψi 1 + ?ν φA ? ν φA 2 ?V (φA ),

(2.6)

where y is the ?fth coordinate of the extra dimension. For the special choice φA (y ) = 2?2 y , which corresponds to approximating the kink with a straight line interpolating two vacua, the wave function in the ?fth coordinate becomes gaussian centered around the zeros of gi φA (y ) + m5,i . It is also shown in ref.[1] that a left handed chiral fermionic ?eld in the four-dimensional representation can result from the localization mechanism. The right handed part remains instead delocalized in the ?fth dimension. When leptons and baryons have the ?ve-dimensional masses m5,l and m5,q , the corresponding localizations
m5,q ,l are at yl = ? 2gl5? 2 and yq = ? 2g ?2 , respectively. The shapes of the fermion wave functions q m

along the ?fth direction are Ψ l (y ) = ?1/2 exp ??2 (y ? yl )2 1 / 4 (π/2) 7

Ψ q (y ) =

?1/2 exp ??2 (y ? yq )2 . (π/2)1/4

(2.7)

Even if the ?ve-dimensional theory violates both baryon and lepton number maximally, the dangerous operator in the e?ective four-dimensional theory is safely suppressed. For example, we can expect the following dangerous operator in the ?ve-dimensional theory, O5 ? d5 x QQQL 3 M? (2.8)

where Q, L are ?ve-dimensional representations of the fermionic ?elds. The corresponding four-dimensional proton decay operator is obtained by simply replacing the ?vedimensional ?elds by the zero-mode ?elds and calculating the wave function overlap along the ?fth dimension y . The result is O4 ? ? × d4 x qqql , 2 M? (2.9)

where q, l denotes the four-dimensional representation of the chiral fermionic ?eld. The overlap of the fermionic wavefunction along the ?fth dimension is included in ?, ?? dy e??
2 (y ?y 2 q)

3 ??2 (y ?y )2 l

e

.

(2.10)

For a separation r = |yq ? yl | of ?r = 10, one can obtain ? ? 10?33 which makes this operator safe even for M? ?TeV. On the other hand, however, this suppression prevents the required baryon-number violating interaction of the form LφX lq = λ′ φX lq, where λ′ is exponentially suppressed. To construct defect con?gurations, we extend the above idea to include another scalar ?eld φB that determines the ?ve-dimensional mass m5 . This additional ?eld φB determines the position of the center of the fermionic wavefunction along the ?fth dimension. We assume that φB does not make a kink con?guration along the ?fth dimension, but does make a defect con?guration in the four-dimensional spacetime. For de?niteness, we consider the Lagrangian L = ψi (i ? /5 + gi φA (y ) + m(φB )5,i ) ψi 1 + ?ν φk ? ν φk 2 ?V (φk ), 8 (2.11)

(2.12)

where indices represent i = q, l and k = A, B . Here φA makes the kink con?guration along the ?fth dimension while φB develops defect con?guration in the four-dimensional spacetime. We note that φB plays the role of φstring in the previous arguments. Now we consider the simplest case where m5,i are given by m(φB )5,i = ki φB , and the potential for φB is given by the double-well potential of the form;
4 VB = ?mB φ2 B + λB φ B .

(2.13)

In this simplest example, because of the e?ective Z2 symmetry, the resultant defect is the cosmological domain wall. One can easily extend the model to have the string or the monopole con?guration in four-dimensional spacetime, if the appropriate symmetry is imposed on the scalar ?eld φB . For example, one can consider the form m(φB )5,i = ki where φB is charged with U (1)S . In any case, the center of the fermionic wavefunction in the ?fth dimension can be shifted by the defect con?guration in the four-dimensional spacetime. Because of the volume factor suppression, the largest contribution is expected in the quasi-degenerated vacuum surrounded by the cosmological domain-wall that interpolates between φB = ±v . Let us consider the case where the wavefunctions of quarks and leptons are localized at the opposite side of the φA kink. It can be realized if their couplings to φB have opposite signs. In the core of the φB string, the centers of the quarks and leptons move toward the origin. Then the distances between quarks and leptons become r = 0 in the core of the string, which drastically modi?es the magnitude of the baryon number violating interactions mediated by the string. If one assumes GUT uni?cation, the origin of such strings becomes clear. For the simplest example, we shall follow ref.[2, 18] and show how one can realize the localization in GUT embeddings. The embedding of SM gauge groups in some grand uni?ed group is an interesting issue. However, the most serious obstacle for the low energy uni?cation is the problem of proton stability, which prevents such embedding at low GUT scale. If the quarks and leptons are uni?ed into the SU (5) multiplets 5 and 10, then the heavy gauge bosons X and Y must mediate the fast proton decay. In ref.[2, 18], GUT models with 9 | φB | 2 M? (2.14)

stable proton are constructed by using the mechanism of localized wavefunctions. In the bulk, Dirac masses residing in quintuplets and decuplets can be split as a result of GUT symmetry breaking according to the following equations: 5 (< Σ > +M5 + φA ) 5 = 0 10 (< Σ > +M10 + φA ) 10 = 0. (2.15) where < Σ >= v × diag (2, 2, 2, ?3, ?3). In this model any transition between quarks and leptons in four dimensions is naturally suppressed by the exponential factor. In this GUT example, < Σ > plays the role of m5,i in the previous discussions. Although it requires some extensions to larger gauge groups than SU (5), GUT string can be formed as in the standard GUT scenarios, and the GUT symmetry is restored inside the core of the string. Thus in the GUT models for wavefunction localization, it is natural to expect that the positions of the fermionic wavefunctions along the ?fth dimension are modi?ed in the background of the GUT defects in the four-dimensional spacetime. Although it seems rather di?cult to produce these defects merely by the thermal e?ect after in?ation, nonthermal e?ect may create such defects during preheating period of in?ation. Nonthermal creation of matter and defects has raised a remarkable interest in the last years. In particular, e?cient production of such products during the period of coherent oscillations of the in?aton has been studied by many authors[19].3 Thermal e?ect becomes important if one considers the supersymmetric extension with intermediate mass scale, i.e., M? >> T eV , where the position of the localized matter ?eld can be parameterized by the ?at direction. In this case, one can expect thermal symmetry restoration at the temperature much lower than the cut-o? scale, which is accessible in realistic scenarios. During thermal symmetry restoration, if the ?ve-dimensional mass terms m5,i is determined by a ?eld φB that parameterizes the ?at direction, the center of the wavefunction is shifted along extra dimension, which results in the huge enhancement in the exponential factor. Then the baryon number violation can become e?ective till
3

There is an another possibility that the defects are generated after the ?rst brane in?ation, while

the reheat temperature after the second thermal brane in?ation is kept much lower than the electroweak scale[20].

10

T ? O (102 GeV ). In this case, the defect formation is not the sole candidate that enhances the baryon number violation.

3

Comments on baryogenesis
In this section we make some comments on baryogenesis that is mediated by cosmo-

logical defects in models with extra dimensions.

3.1

Particle scattering and baryon number violation
The e?ect of the baryon number violation is greatest when the density of the string

is greatest, which in general occurs soon after the phase transition. Unlike conventional GUT strings, the string with extra dimensions is formed at much lower energy scale, and remains friction-dominated. If the string is the e?ective source of the baryon number violation, any existing baryon number asymmetry in the Universe may be washed out. On the other hand, if there is no e?ective source of the baryon number violation other than the Yukawa coupling φX ql that is enhanced by the string, and also if the reheating temperature is too low to activate the sphalerons, the string cannot washout the existing asymmetry, but converts the existing leptons into baryons, and vice versa. In this case the strings can be utilized for leptogenesis with low reheat temperature.
4

Denoting the correlation length of the string network by ξ (t), and the baryon-number violating cross section per unit length by σ B /, one obtains[21] dnB ? ?vσ dt
B / 2 nL , ξ

1

(3.1)

where nB (nL ) is the baryon(lepton) number density and v is the thermally averaged relative velocity of the particles and strings, which is of order 1. We can take σ and ξ (t) ? tp , where p = 5/4 in the friction-dominated phase[21]. In models with large extra dimensions, the lepton number will be produced by the decay of the relics of the in?ation, such as the A?eck-Dine ?eld[23, 7, 6], or the in?aton[9],
4

B /

? T ?1

Leptogenesis in theories with extra dimensions is already discussed in ref.[22]. In this scenario the

reheating temperature should be higher than 10 GeV or so, which gives rise to constraints on the model parameters and the quantum gravity scale. The observed baryon asymmetry can still be generated by out-of-equilibrium, exponentially suppressed B+L-violating sphaleron interactions.

11

which will have large densities. In this case one can naturally assume that the lepton number production and the formation of the string occur at almost the same time, which makes our scenario plausible. If the lepton number production occurs before the string formation, the conversion is promising because the scattering is greatest just after the string formation. On the other hand, if the strings are formed before the lepton number production, the conversion is not trivial. The conversion is greatest just after the time of the lepton production, tL . Then one can obtain the ratio ?nB 1 ? 1 |tL ? nL T (tL ) ξ0 tL
t0

?

p

?2 ?

tL ,

(3.2)

where t0 and ξ0 are the time of the string formation and the initial correlation length of the string, respectively. The scattering is important when
? nB | nL tL

? O (1). Using the above

relations, one can easily ?nd that the scattering process is activated just after the string formation and then lasts for a short period. If the sphalerons and other baryon-number violating interactions are not activated at the time of the string formation, which may occur in the scenario of defect formation during preheating[19], these strings can convert existing leptons into baryons without washout. We believe that our idea can help to solve the di?culty in leptogenesis for models with large extra dimensions related to the low reheat temperature that suppresses the sphaleron-mediated interactions.5

3.2

Baryogenesis from collapsing strings
At ?rst we shall review the basics of the standard scenario of baryon number gener-

ation from the decaying GUT strings. Baryon number asymmetry produced by strings are important when the number density of existing heavy X particle is less than the one produced by the decaying strings. In the standard GUT baryogenesis it is well known that the predicted baryon to entropy ratio is exponentially suppressed if Td , which is the temperature at the time when the baryon number violating processes fall out of thermal equilibrium, is less than the mass of the heavy X particle mX . For example, if the temperature Td at the time td is greater than the mass mX of the superheavy particles, then
5

In ref.[24], sphaleron transitions that may proceed during matter-dominated phase is discussed. In

our model, the reheating process must be di?erent from the one discussed in ref.[24] to avoid washout of the existing asymmetry.

12

it follows that nX ? s. However, if Td < mX , the number density of X is diluted exponentially, which results in the exponential suppression of the produced baryon to entropy ratio: 1 mX nB , ? ? ? λ2 exp ? s g Td (3.3)

where g ? is the number of spin degrees of freedom in thermal equilibrium at the time of the phase transition, λ is the coupling constant of the baryon number violating process, and ? demotes the e?ective CP violation. If the exponential suppression is tiny, the standard GUT baryogenesis mechanism is ine?ective. However, topological defects may solve this problem[21] in standard GUT. The collapse of cosmic string can produce baryon asymmetry if the φX -boson, which couples to both baryons and leptons through Yukawa coupling of the form λφX ql, forms or couples to the cosmic string. The important ingredient in the quantitative calculation is the time dependence of the correlation length ξ (t), which parameterizes the separation between defects. Just after the phase transition, the separation is expected to be ξ (t0 ) ? λ?1 η ?1 . The time period of relevance for baryogenesis, ξ (t) approaches the Hubble radius according to the following equation[25] t ξ (t) ? ξ (t0 ) t0
5 4

.

(3.4)

After some algebra, one can obtain the baryon to entropy ratio: nB Td nB ? λ2 |0 s η s where
nB | s 0

(3.5)

is the unsuppressed value of GUT baryogenesis, and η denotes the square root

of the energy density per length. Thus in the standard GUT models, one can see that the superheavy particles produced by the succeeding decay of the cosmic strings can overcome the exponential suppression. The domain wall, which seems much more e?ective for producing particles, is ruled out by the cosmological requirement.6 Then what is the di?erence between the defect mediated baryogenesis in the standard GUT and the one with extra dimensions? The most obvious di?erence is the reheating
6

However, we should stress that in models with extra dimensions the mass scale of the domain wall is

much lowered, thus they can survive till the time period of relevance for baryogenesis. We shall discuss on this issue in the next subsection.

13

temperature, which must be much lower than the scale of the string formation. In scenarios for string mediated baryogenesis, the greatest contribution appears just after they have formed. Regarding to the baryon number production by the decaying strings, later contribution cannot exceed the initial one, because of the rapid growth of their correlation length and the dilution by the redshift. Then the low reheating temperature is problematic, since there should be a large amount of dillution by the decaying ?eld in the time period between string formation and reheating.7

3.3

Baryogenesis from unstable Domain walls

Unlike the strings that we have discussed above, unstable domain walls are the e?cient source of baryon number asymmetry, if they decay after reheating. In most cases, the domain walls are dangerous for the standard evolution of the Universe[27]. However, if some criteria are satis?ed, unstable domain walls that disappear at t < tc ? (Gσ )?1 can exist[28]. When the discrete symmetry is not the exact symmetry, it may be broken by the interactions suppresses by the cut-o? scale. Then the degeneracy is broken and the energy di?erence ? = 0 appears[29]. Regions of the higher density
7

We shall consider two separated situations. In one case[24], preheating is not e?ective and the

maximum temperature during reheating can be much greater than the reheating temperature. This may happen when the reheating is far from being an instantaneous process, and the decay products of the relevant scalar ?eld thermalize rapidly before the Universe becomes radiation dominated. They have calculated the dillution factor of the baryon to entropy ratio, which is produced before the Universe becomes radiation dominated; nB ? B0 s TR TB
5

,

(3.6)

where TB denotes the temperature when the baryon number asymmetry is produced. In the most naive case, when TB ? 10T eV , and TR ? 1GeV , the dillution factor is crucially small. If the potential for φstring is su?ciently ?at, which may occur in supersymmetric extensions of the model, the time of the string formation is delayed and one can improve the result. However, the supersymmetric extension seems far from being attractive for scenarios with TeV scale uni?cation. Otherwise, one should ?ne tune the self-coupling constant, which makes this scenario unsatisfactory. One can consider another scenario for string formation. Preheating after in?ation[26] may lead to nonthermal phase transitions with defect formation. The time of the string formation may be delayed so that the string formation occurs just before the reheating, but it requires the ?ne tuning of the coupling constants, which makes this scenario unattractive.

14

vacuum tend to collapse when the pressure induced by the energy di?erence becomes dominant. The corresponding force per unit area of the wall is ? ?. The energy di?erence ? becomes dynamically important when this force becomes comparable to the force of the tension f ? σ/Rw , where σ is the surface energy density of the wall and Rw denotes the typical scale for the wall distance. For walls to disappear, this has to happen before they become harmful. On the other hand, the domain wall network is not a static system. In general, the initial shape of the walls right after the phase transition is determined by the random variation of the scalar VEV. One expects the walls to be very irregular, random surfaces with a typical curvature radius, which is determined by the correlation length of the scalar ?eld. To characterize the system of domain walls, one can use a simulation[30]. The system will be dominated by one large (in?nite size) wall network and some ?nite closed walls (cells) when they form. The isolated closed walls smaller than the horizon will shrink and disappear soon after the phase transition. Since the walls smaller than the horizon size will e?ciently disappear so that only walls at the horizon size will remain,
1 2 their typical curvature scale will be the horizon size, R ? t ? Mp /g? T 2 . Since the energy

density of the wall ρw is about ρw ? and the radiation energy density ρr is ρr ? g? T 4 , one sees that the wall dominates the evolution below a temperature Tw Tw ?
? ?

σ , R

(3.7)

(3.8)

σ g? Mp
1/2

?1
2

To prevent the wall domination, one requires the pressure to have become dominant before this epoch, ?> σ2 σ ? 2, Rwd Mp (3.10)

?

.

(3.9)

which is consistent with the criterion in ref.[27, 28]. Here Rwd denotes the horizon size at the wall domination. A pressure of this magnitude would be produced by higher dimensional operators, which explicitly break the e?ective discrete symmetry[29]. The criterion (3.10) seems appropriate, if the scale of the wall is higher than (105 GeV )3 . For the walls below this scale (σ ≤ (105 GeV )3 ), there should be further constraints 15

coming from primordial nucleosynthesis. Since the time associated with the collapsing
1

2 2 temperature Tw is tw ? Mp /g? σ ? 108

(102 GeV )3 σ

sec, the walls σ ≤ (105 GeV )3 will decay

after nucleosynthesis[31]. In this case, one must consider stronger constraint. The cosmological domain wall that we consider in this paper will have the surface
3 energy density σ ? M? ? (102 T eV )3 , which can decay after or just before the reheating.

When they decay, their energy density ρw can be comparable to the energy density of the radiation. Moreover, their baryon number violating decay is enhanced if the center of the wavefunction of baryons and leptons along extra dimension coincides in the core of the defect. There may be other source of the enhancement in the false vacuum domain where the center of the wavefunctions may be shifted by O(1). We can calculate the baryon to entropy ratio as nB Td ρw /mX ?≤ ? ? 3 s Td mX (3.11)

where Td denotes the temperature when walls decay. The e?ective CP violation and the di?erence between the rates of the decays is included in ?. The mass of X boson that mediates the baryon number violating decay is assumed to be mX ? M? . If the baryon number violating interaction is not enhanced in the background of the wall con?guration, the exponential suppression in ? reduces the baryon number asymmetry, which is very similar to the situation discussed in ref.[9]. However, there should be a huge enhancement of the baryon number violating interactions in our model. In the most optimistic case, where ? ? 10?4 , one can obtain the desired baryon to entropy ratio for Td ? 10?1 GeV and mX ? 105 GeV .

4

Conclusions and Discussions
In the standard grand uni?ed theories, various types of baryon number violating

processes are expected in the early Universe. On the other hand, in models with large extra dimensions, it is di?cult to realize baryon number violation even in the background of a string. In this paper we argue that in the background of a cosmic string with extra dimensions the baryon number violating interactions are enhanced if the proton decay is suppressed by the machanism of localized wavefunctions. We also make some 16

comments on baryogenesis mediated by cosmological defects. At least two scenarios will be successful. One is the scenario of leptogenesis where the required lepton number conversion is mediated by the string, and the other is the baryogenesis from the decaying cosmological domain wall. Both scenarios are new and are not discussed in the past. These mechanisms predicts su?cient baryon number production even if the reheating temperature is much lower than the temperature of the electroweak phase transition.

5

Acknowledgment
We wish to thank K.Shima for encouragement, and our colleagues in Tokyo University

for their kind hospitality.

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