NYUTH/99/3/02
August 4, 2021
Nonconservation of Global Charges in the Brane Universe
and Baryogenesis
Gia Dvali^{1}^{1}1Also, International Center for Theoretical Physics, 34100 Trieste, Italy, Gregory Gabadadze
Department of Physics, New York University, New York, NY 10003, USA
email: dvali,
Abstract
We argue that global charges, such as baryon or lepton number, are not conserved in theories with the Standard Model fields localized on the brane which propagates in higherdimensional spacetime. The globalcharge nonconservation is due to quantum fluctuations of the brane surface. These fluctuations create “baby branes” that can capture some global charges and carry them away into the bulk of higherdimensional space. Such processes are exponentially suppressed at lowenergies, but can be significant at high enough temperatures or energies. These effects can lead to a new, intrinsically highdimensional mechanism of baryogenesis. Baryon asymmetry might be produced due either to “evaporation” into the baby branes, or creation of the baryon number excess in collisions of two Brane Universes. As an example we discuss a possible cosmological scenario within the recently proposed “Brane Inflation” framework. Inflation is driven by displaced branes which slowly fall on top of each other. When the branes collide inflation stops and the Brane Universe reheats. During this nonequilibrium collision baryon number can be transported from one brane to another one. This results in the baryon number excess in our Universe which exactly equals to the hidden “baryon number” deficit in the other Brane Universe.
1. Introduction
The only candidate theory for consistent quantum description of all the interactions in Nature can be formulated in spacetime with dimensionality greater than four. One possible scenario, of how our fourdimensional world emerges in this picture, is based on the assumption that all the observed particles are localized on a 3dimensional brane that propagates in higherdimensional spacetime [1]. The absence of supersymmetry in the observable world can be related to a nonBPS nature of the brane [2]. Within the field theory context the simplest localization mechanism for fermions is due to the index theorem in the solitonic background [3] which was first used in Ref. [1] for a fivedimensional Brane Universe scenario.
As shown in [4], the localization of spin1 gaugefields in the field theory context is less straightforward. The mechanism requires the gauge group to be confining away from the brane [4]. This has an important modelindependent consequence for the gaugecharge conservation on the brane, which is nothing but gauge flux conservation in the confining medium. The similar argument applied to spin2 particles within the field theory context shows why localization of gravitons is hard to achieve. Thus, gravitons, unlike other particles must live in the brane as well as in the bulk of extra compactified dimensions.
One of the motivations for the Brane Universe scenario is that it allows to lower the fundamental scale of quantum gravity all the way down to TeV or so, thus, providing a novel view on the Hierarchy problem [5, 6]. The observed weakness of gravity at large distances is because gravitational fluxes spread into the extra dimensions.
The relation between the observed Planck scale, , and the socalled fundamental Planck scale is then given by [5]:
(1) 
where 7]. , is the volume of extra compactified spatial dimensions. The size of compactified radii in this picture can be in a submillimeter range without conflicting with any present astrophysical or laboratory constraints [
Perhaps, the best motivated framework for the Brane Universe scenario can be found within superstrings and theory. In fact, Dbranes [8] provide natural candidates for objects on which the observed particles could live. Thus, the Standard Model particles can be identified with massless modes of open strings which are “stuck” to a set of overlapping Dbranes. Gravity, on the other hand, is given by the closed string sector of the theory. It was shown in Refs. [6, 9] that the TeV scale quantum gravity models can be embedded in superstring theories (this approach was named “Brane World” in Ref. [10]) ^{2}^{2}2The idea to lower the the Planck scale to the unification scale originates in [11]; An attempt to lower the string scale to the electroweak scale was discussed in Ref. [12].. Various phenomenological aspects of these models are studied in the literature (for an incomplete list of references see, e.g., [13, 14, 15, 16, 17]).
In the present work we argue that global charges are not conserved in the Brane Universe. The nonconservation of global charges is due to quantum fluctuations of the brane on which the Standard Model lives. These fluctuations can produce baby branes which can capture global charges and carry them away from the brane. At high enough temperatures or energies comparable with the brane tension the process of baby brane creation becomes significant. This leads to the globalcharge transport from our brane. The corresponding process will look as nonconservation of global charges for a fourdimensional observer living on the brane^{3}^{3}3These processes somewhat resemble the loss of quantum coherence in quantum gravity [18, 19, 20].. These nonconservation mechanisms are significant in the cosmological context and can lead to new sources of baryogenesis. We discuss a possible cosmological scenario based on recently proposed Brane Inflation mechanism [21]. This scenario results in the baryon number access in our fourdimensional Brane Universe.
The terminology we adopt in this work is as follows.
The threedimensional brane
on which we live will be referred as the mother brane.
The dimensional Universe accommodated on the mother brane
will be called the Brane Universe.
The Brane Universe is able to
communicate gravitationally and/or via some other
bulk fields with other Brane Universes which also
propagate in the original higherdimensional (presumably
ten or elevendimensional) spacetime.
This feature is going to be decisive for what we discuss below.
In section 2 we discuss a number of mechanisms of nonconservation
of global charges in the Brane Universe.
We show (subsection 2.1) that at high enough temperatures, or
in scatterings at energies of order TeV or so,
the Brane Universe could produce baby branes which would carry the
Standard Model global quantum numbers
off the Brane Universe.
For an observer living in the Brane Universe
these processes will look as nonconservation of
global quantum numbers,
such as nonconservation of baryonic or leptonic charges.
In subsection 2.2 a fieldtheoretic scenario
of the global charge nonconservation
in the hot Brane Universe is proposed.
The Brane Universe, at some finite temperature
produces confining bubbles with net global charges
and liberates them into the bulk of higherdimensional space.
This scenario is complimentary to the one discussed in subsection 2.1.
In section 3 we discuss applicability of these mechanisms
to the baryon asymmetry problem.
We argue that within the cosmological scenario
proposed in [21] these nonconservation
processes should lead to the baryon
asymmetry in the Brane Universe.
Discussions and conclusions are
presented in section 4.
2. Nonconservation of Global Charges in the Brane Universe
In this section we propose mechanisms for
nonconservation of global charges in the Brane Universe.
The objective is to demonstrate that there are intrinsically
highdimensional
phenomena which lead to baryon and lepton number nonconservation on
our brane.
2.1. Baby Branes and Nonconservation of Global Charges
Let us start with the case when our dimensional Brane Universe is embedded in higher dimensional spacetime. We will think of the brane as being a dynamical object with the tension . This brane can fluctuate. The fluctuations are stronger at high temperature. In fact, there is a probability for the Brane Universe to wiggle strongly and create a baby brane (see Fig. 1). At high enough temperature the baby brane will be able to pull off the mother brane and propagate into the bulk of higherdimensional space. This probability is nonvanishing if the temperature of the Brane Universe is nonzero (the same process could also be seen at high enough energies). The rate of this process is exponentially suppressed and can be estimated as [22]:
(2) 
where stands for the surface energy of the baby brane, which can be determined via its surface area and tension , . stands for temperature of the Brane Universe. Thus, for high temperatures of order TeV or so, creation of baby branes should be an appreciable effect, while it should drop off rapidly as the brane cools down.
Once the baby brane is formed, it can capture some particles which happen to be nearby and carry them away from the mother brane. What will happen if the captured particles carry a net global charge, let us say baryon or lepton number? To clarify the issue, let us consider the case when the captured state is a gauge singlet combination of quarks (it might be any other combination of the Standard Model states which carries a nonzero baryon or lepton number but has strictly zero gauge charge). For a fourdimensional observer living on our brane this process will look as follows:
(3) 
where ”NOTHING” stands for the baby brane which got separated from the mother brane and carries away the corresponding global charge. Since this object can gravitate, it will look for a fourdimensional observer as a piece of dark or hidden matter. Thus, the baby brane will carry its own baryon number . The value of will exactly equal to the baryonic charge that is lost on the mother brane. If there are no bulk particles which can carry well defined baryon number, an observer on the mother brane will not be able to measure . Thus, the process will look as disappearance of baryonic charge on the mother brane and appearance of the same charge on the baby brane^{4}^{4}4If there are some bulk particles that can carry baryon number, the issue becomes more subtle. In this case can be measured via the exchange of these bulk modes. If these modes are heavy the induced effective interaction between the baryons localized on different branes will be exponentially suppressed by the Yukawa factor . In this section we will assume that there are no such particles in the bulk..
Let us try to understand this effect from the point of view of the effective fourdimensional field theory which is seen at distances larger than the size of extra dimensions. Let be a wavefunction describing a state of some group of particles on our brane which carries a net baryonic charge. Likewise, we can define to be a wavefunction of a set of particles on the baby brane. Both and are sharply localized functions in the bulk. The overlap between them is exponentially suppressed as the separation of the branes, , in extra dimensions increases. Baryon number conservation in the theory is a result of the symmetry under the global phase rotations , and , where and denote the charges for the corresponding wavefunctions. When the branes are separated the wavefunctions and get decoupled.
As a result, there are two independent symmetries available: and . However, when branes are close to each other, the functions and overlap. As a result, there is only one unbroken combination of and , call it , which is defining baryon number of the whole system of the overlapping branes. In terms of the effective field theory language, the effective lowenergy Lagrangian contains violating term whose strength depends on the distance between the branes. This term drops exponentially fast as the separation between the branes increases:
(4) 
What is important here is that the interaction should necessarily respect the symmetry. As we mentioned above, for small branes are interacting. Thus, there is only one baryon charge. This charge can be exchanged among the states of and . Suppose denotes the amount of charge which is being transferred from the one set to another one. Once the branes are separated () the overlap term disappears. Thus, there are two separately conserved charges corresponding to and respectively. However, only the charge will be seen in the mother brane and, thus, interpreted as the baryon number of our brane. Summarizing, the charge transport from to will look as disappearance of the amount of the baryon charge on our brane and as appearance of exactly the same amount of the baryon charge on the baby brane.
Evidently, in each individual process can take either sign and, if the system is in equilibrium, the net baryonic charge left on the brane will average to zero. However, it might be possible to generate a net baryon asymmetry on the mother brane if the system was out of equilibrium for some time during its evolution (it also requires C and CP violation [26], see discussions below). In section 3 we address this issue and propose possible scenarios of how the baryon asymmetry could be generated in the Brane Universe.
Before we turn to the next section, let us discuss the fate of local charges in the Brane Universe. Seemingly, the same nonconservation process might be happening with the gauge charges, such as electric charge for instance. However, this cannot be true [4, 5]. Indeed, consider the case when the local charge is attached to the strongly fluctuating region of the mother brane which is about to be pulled off (Fig.2). The local charge, due to the corresponding flux conservation, would necessarily create a flux tube originating at the location of this charge and ending on the mother brane (see Fig. 2). At high enough energies, or temperatures likewise, the flux tube can break apart and the baby brane will eventually be liberated into the bulk of higherdimensional space. However, the liberated baby brane will necessarily be neutral with respect to the local charge under consideration. Indeed, the process of breaking of the flux tube goes through creation of a chargeanticharge pair in the tube. Once this pair is created, the anticharge will get attracted by the original charge siting on the baby brane. Thus, the flux tube will break apart in such a way that the anticharge from the pair will be attached to the baby brane and the charge of the pair will be attached to the mother brane. Hence, the final configuration of the liberated baby brane will be electrically neutral and the local charge will be conserved on the mother brane. Another way of saying this is to recall that all the Standard Model gauge interactions should be in a confining phase in the bulk spacetime [4].
Summarizing the discussions in this section we conclude that
the process of baby brane creation should
lead to nonconservation of
global charges (such as baryon or lepton number) in the Brane Universe.
Moreover, this process
will necessarily respect all the local charge conservation laws.
2.2. Bulk “Glueballs” and “Hadrons”
It is important to emphasize that the existence of generic bulk particles, which may carry baryon number, cannot lead to the baryonic charge nonconservation in the fourdimensional effective field theory. The reason why this was possible for the baby branes is that branes are sharply localized coherent states, wave packets in some sense. Because of this property the baryonic charge localized on such an object is impossible to probe from a distant brane, and, thus, is effectively lost. On the other hand, the wavefunctions of the generic bulk particles, such as KaluzaKlein gravitons for instance, are spread over the whole bulk and their baryonic charge can constantly be measured from our brane at any energies.
Another examples of such sharply localized objects can be bulk “glueballs” or “hadrons”. These are the states that appear in the bulk due to the particular mechanism of localization of the gauge fields on the brane. As it was shown in [4], the field theory mechanism for localization of the massless gaugefields on the brane implies that corresponding gauge group is in a confining phase in the bulk. Thus, a pair of test charges places in the bulk should be connected by a flux tube with the tension proportional to , where is a scale of the confining theory in the bulk. The inverse confinement scale, , sets the localization width for the observed gauge fields. For phenomenological reasons should be greater than .
Notice, that the gauge group in the bulk can be bigger than the
Standard Model group. A photon, in this case,
if being emitted into the bulk, becomes a gauge boson of the
bigger confining theory. Thus, the photon can only escape the mother
brane in the form of a heavy bound state, a sort of bulk ‘‘glueball’’.
The similar consideration
applies to fermion states. If the gauge group
in the bulk were not confining, these fermions would have escaped
the mother brane at energies bigger than the localization width.
However, since the bulk is confining,
such states can only escape
within the corresponding ‘‘colorless’’ composite objects, bulk ‘‘Hadons’’.
Since the bulk ‘‘Hadrons’’ might carry off some net global charges,
they can also lead to nonconservation of the global charges on the brane
^{5}^{5}5In all the discussions above and in subsection 2.3
we assume for simplicity that the localization width for
fermions approximately equals to ..
2.3. Confining Bubbles and Nonconservation
of Global Charges
In this section we study a fieldtheoretic model of nonconservation of global charges which is based on confining properties of the bulk gauge group. As we mentioned before, the group is restored and confining outside of the brane [4]. For instance, the electroweak gauge group which is broken in our brane, should be restored (or be a part of a bigger unbroken group) and confining outside of the brane (see Fig. 3). The same applies to color and hypercharge symmetries. should either be a subgroup of bigger confining bulk gauge group, or be the same bulk gauge group with the confinement scale greater than TeV. Likewise, should be a part of a bigger group that is confining in the bulk. For simplicity of arguments we will be assuming that the gauge group within the brane is broken and outside of the brane it is confining (the generalization to the other groups and interactions is straightforward). Let us suppose that within the Brane Universe the confining phase which is realized outside of the mother brane is seen as a local false vacuum state. Then, in our fourdimensional world there is a finite probability to create a bubble ( a sort of “hole”) with a confining phase inside. If some “colorless” states with nonzero global charges are captured inside the bubble, they will be able to “leak” into the bulk. These effects are complimentary (but more modeldependent) to those discussed in the previous subsections. Let us study the bubble creation processes more carefully.
The probability to create a bubble per unit volume per unit time in our world with the confinement phase inside of the bubble is given by [23]:
(5) 
Here, denotes the free energy of the system as a function of temperature of the mother brane , denotes the depth of the scalar potential of the broken theory, is the difference between the energy densities of the confining and the Higgs phases, and stands for some positive constant of order 10100. As is close to this probability becomes significant. The theory inside of the bubble is in a confinement phase. Thus, bound states of particles which might form within the bubble are to be singlets^{6}^{6}6 If all the Standard Model interactions are considered, these states are supposed to be “color singlets” with respect to the whole Standard Model gauge group or w.r.t. the corresponding GUT, if the unification is assumed.. These singlet states will be able to propagate out of the Brane Universe. The most dramatic signature of this propagation is that they will be able to carry global quantum numbers off our Brane Universe. For instance, consider a single lefthanded neutrino. This particle transforms in the fundamental dublet of . Thus, it carries a “weak color” charge and cannot escape the brane. However, in accordance with ’t Hooft’s correspondence principle [24], the neutrino of the theory with a broken can be thought of as a “weak colorless” state, or as a bound state of confining . Indeed, in the confinement picture, the lefthanded neutrino can be presented as follows [24]:
(6) 
Here, stands for the Standard Model Higgs dublet, and stands for the lefthanded dublet of a neutrino and electron, . It is straightforward to see that the “weak colorless” bound state reduces to an ordinary lefthanded neutrino once the Higgs field is given a nonzero vacuum expectation value (VEV). Indeed, in the unitary gauge , where denotes the Higgs VEV and stands for Higgs fluctuations about this VEV. Substituting this expression into the right hand side of Eq. (6) one finds, . Thus, the r.h.s. of Eq. (6) can indeed be thought of as a “weak colorless” state of confining ; moreover, this state corresponds to the lefthanded neutrino of the Standard Model.
Once the bubble is formed, the “weak colorless” state can appear in the confining phase inside of the bubble. This state, as we established above, carries leptonic charge. There is nothing that keeps this “weak colorless state” within the hot Brane Universe. Thus, it will be able to escape out into the higherdimensional space. This process would seem as a leptonic charge nonconserving phenomenon to a fourdimensional observer living in the Brane Universe (see Fig. 3). The same applies to all the other standard model particles. Each of them can be thought of as “weak colorless” bound states [24]. Some of them are listed below:
Here, denotes the lefthanded up and down quark dublet. Some combinations of these states, such as (here we suppress all the Lorentz indexes and gamma matrices)
(8) 
will be created as “Standard Model colorless”
excitations inside of those bubbles
and, as a result, they will escape
our brane at high enough temperatures or energies.
Evidently, they will be able to carry
the corresponding global charges, such as lepton or baryon number,
away from the mother brane.
This will make a fourdimensional observer think that
the global quantum numbers are not
conserved at high temperatures or energies in the Brane Universe.
In the next section we
discuss how these processes might lead to the baryon asymmetry in
the Brane Universe.
3. Baryon Asymmetry in the Brane Universe
In this section we argue that the baryon number nonconservation
mechanisms discussed above might lead to a new
approach to baryogenesis in the Brane Universe.
We discuss two possible mechanisms.
The first one is based on the fact that
C and CP asymmetric branes can treat
baryons and antibaryons differently.
As a result, the rate to capture a baryon on a baby brane differs from
that for an antibaryon. Thus, net baryon charge accumulation is possible
if the system is out of equilibrium.
This scenario is discussed in subsection 3.1.
The second scenario is based on production of the baryon number excess
in a collision of two different Brane Universes after inflation.
This scenario emerges naturally within the
recentlyproposed “Brane Inflation”
framework [21]. The corresponding
discussions are given in subsection 3.2^{7}^{7}7Some more conventional
scenarios of baryogenesis in theories with low were discussed in
[25]. These, however, are not related to the present work..
3.1. Baryon Asymmetry on the Asymmetric Brane
As we discussed above, the baby branes and/or confining bubbles will carry some baryonic charge off our brane. The very same processes will be happening with antibaryons which will be taken away from the brane by the same mechanism. If the theory at hand does not distinguish between baryons and antibaryons, then the net charge carried away from our brane will average to zero. However, there is a possibility that the brane actually do distinguish between baryons and antibaryons if C and CP are broken. In particular, if the rate to capture a baryon on a baby brane differs from the corresponding rate for antibaryons, then the accumulation of the net baryonic charge on our brane will be possible in nonequilibrium processes [26].
Let us consider a toy model which demonstrates how this asymmetry can arise. Consider a scalar field which forms a fourdimensional “brane” embedded in fivedimensional spacetime. Let us say the profile of this soliton is given by the familiar “kink” solution:
(9) 
where defines the thickness of the brane and stands for the VEV of the corresponding quantum field. Consider two fivedimensional fermions coupled to :
(10) 
where stands for some mass parameter and denotes the charge conjugation matrix in fivedimensional spacetime, . This theory has the symmetry: , . We identify this symmetry group with , thus and carry opposite baryonic charges. It is well known that each of these fermions give rise (in the massless limit) to a single chiral zeromodes localized on the brane:
(11) 
From the point of view of the brane worldvolume field theory these chiral fermions can be identified with the worldvolume baryon and antibaryon (in Weyl notations)^{8}^{8}8 Switching on small mass does not change the qualitative picture.. In the lowenergy theory the “charge conjugation” symmetry is broken since . This results in difference between the localization widths for and which are given by and respectively. For instance, the width for (left handed) baryon can be made smaller than that for antibaryon (). Then, at energies the antibaryon can be “stripped off” the brane, while the baryon would still be localized. This toy example explicitly shows how the brane can be “Casymmetric”. For generating net baryon charge, however, CP breaking is also required. Assuming that this is the case, (i.e. there are some explicitly CPnoninvariant terms in (10)), we expect that the probability for baryons to be captured by a baby brane is different than that for antibaryons (though, this process is more difficult to quantify). As a result, the baby branes will be able to remove from our world more antibaryons than baryons. Thus, the worldvolume observer will eventually see the net baryon asymmetry provided that “evaporation” into the baby branes is an outofequilibrium process. Such a outofequilibrium condition may emerge for instance from the reheating due to collisions of two Brane Universes.
Note that in this toy model there are bulk states which carry baryon number. They are KaluzaKlein states of the original fermions and . These states can mediate baryon number exchange between different branes. However, they are heavy, and the corresponding interactions are exponentially suppressed by the brane separation (see related discussions in [27, 28])^{9}^{9}9In particular, based on the similar effect, N. ArkaniHamed and M. Schmaltz [28] proposed the mechanism for the proton decay suppression.. Moreover, in realistic models due to the bulk confinement (which we have ignored in this toy example) these heavy states can only propagate within the bulk “colorless Hadrons”.
Finally, we would like to discuss the issue of the overclosure of the Universe by baby branes in such a scenario^{10}^{10}10We are grateful to V. Rubakov for rising this issue.. In order to generate the net baryon asymmetry on our brane, not all the baby branes should return to it. If they stay in the bulk, they will look as a sort of dark matter with TeV mass. If we assume roughly one unit of baryon number captured per baby brane, their number density would be so large that they would overclose the Universe. However, there are several ways to avoid this problem. The most straightforward is to notice that the baby branes need not stay in the bulk, but rather can be “discharged” on some other distant brane (like ours, or even larger dimensionality). In such a case the energy density of baby branes will be converted into the distant brane tension and will be absorbed into the effective overall cosmological term
(12) 
where is the bulk cosmological constant and the
summation is over all branes. The probability that the baby brane
encounters a bigger brane and gets discharged there needs further
quantification within more realistic models.
3.2. Brane Inflation and Baryon Asymmetry
In this subsection we discuss the mechanism of baryogenesis which naturally arises within the Brane Inflation framework [21]. According to the general Brane Universe scenario, we live on a brane or a set of overlapping branes. The later possibility is supported by Dbrane constructions in which the existence of a nonAbelian gauge group requires a number of parallel Dbranes sitting on top of each other ^{11}^{11}11Cosmological scenarios within HoravaWitten type theories [29] were recently discussed in Ref. [30]..
Before supersymmetry is broken branes are BPS states with zero net force between them. This is certainly true for two (or more) parallel Dbranes, where the gravitational and dilaton attraction is exactly canceling with the repulsion mediated by RamondRamond fields [31]. Similar examples can be constructed for fieldtheoretic branes, topological solitons [32]^{12}^{12}12For a reviews on supergravity solitons see, e.g., [33], [34].. However, in the real world supersymmetry must be broken and dilaton should be stabilized. Thus, we expect a nonzero net force between branes. The general expression for a potential between two such parallel branes embedded in transverse dimensions at large distances () takes the following form:
(13) 
The constant term comes from the shortrange branebrane interaction. In fact, it accounts for interactions between particles localized on different branes, whose wavefunctions only can overlap if branes intersect. The potential is normalized as , being the brane tension. Yukawa potentials in (13) come from the exchange of heavy bulk modes with masses , and the power law interaction comes from the bulk gravitational attraction. If the Dbrane picture is adopted, then ’s should be understood as masses of dilaton and RamondRamond fields. Regardless of what is the actual realization of branes, be it the Dbrane picture or field theory soliton context, the potential in (13) describes adequately interactions between those objects. The modeldependent quantities are parameterized by coefficients and . These parameters determine the minimal separation at which the branes are stabilized in the lowestenergy state. If , the separation between the branes is smaller than the typical size at which the branes could fluctuate. Thus, the branes effectively sit on top of each other. As a result, the particles localized on these two branes are effectively shared by both of them. Below we will concentrate on the following alternative possibility. Let us assume that . In this case particles localized on two different branes have no overlap. Thus, they belong to either of branes, but are not shared among them. These two worlds can communicate to each other by exchanging bulk fields. If these interaction preserve global charges, and charges are conserved separately on each branes.
Let us see how this picture is affected by the dynamics of the brane inflation [21]. Once the branes are separated by a distance , the nonzero potential energy between the branes gives rise to the fourdimensional effective cosmological constant that drives inflation [21]. This constant can be defined as follows:
(14) 
where is the bulk cosmological constant and is the volume of the extra compactified space. Nearly zero value of the cosmological constant that is observed today implies that
(15) 
Thus, according to Eqs. (13,14), the fourdimensional vacuum cosmological constant will be nonzero for any . This potential energy will drive inflation, the exponential growth of the three noncompact dimensions^{2}^{2}2 The size of the extra dimensions will not be affected by this growth provided that the mass of the radius modulus is at least [21]. Some possibilities of primordial KK inflation with change of were studied in [35].. The next crucial thing is to note that for the potential (13) is a very flat function of . As a result, the branes fall very slowly on each other. Thus, during this process the Universe is dominated by the potential energy which in fact triggers inflation in noncompact dimensions. We should also emphasis that the compact dimensions will not inflate since the effective Hubble size is never smaller than the size of the compact dimensions [21]. From the point of view of an effective fourdimensional theory this process is equivalent to slow rolling of a scalar field, an inflaton
(16) 
This field, according to (13) has a very flat potential. The quantity is just the vacuum expectation value of the inflaton today.
The end of inflation is determined by the value of which breaks either of the standard slowroll conditions (see [21] for details). The epoch in which we are interested in starts right at this point of the evolution. We will argue below that after the branes collide and reheat each other, the net baryonic charge can be induced on our brane.
One possible scenario emerges when the branes get stabilized after the collision at some large distance . This is going to be the case if the branes repeal at short distances. For instance, this condition can be realized within the Dbrane construction if dilaton becomes heavier than the corresponding RamondRamond field . As a consequence, when the RamondRamond repulsion takes over and branes get stabilized at ^{1}^{1}1The value of can control the reheating temperature after “Brane Inflation” [36], this clarifies the issue of reheating temperature discussed by T. Banks, M. Dine and A. Nelson (see the reference in [15])..
Let us follow this scenario more closer. The potential energy of the Universe during inflation can be estimated as follows:
(17) 
This amount of energy will transform into the energy of colliding branes after inflation. Let the wavefunction of a set of particles localized on our brane be , likewise, the wave function of a set of some different particles living on the other brane be . There is a baryon number symmetry on our brane
(18) 
Likewise, there is a similar symmetry on the other brane
(19) 
When branes are separated, these are two different symmetries. In the effective fourdimensional theory, this simply means that the interactions that break are suppressed as follows:
(20) 
However, once the branes come on top of each other, the suppression goes away. As a result, we are left with the only one conserved charge .
During inflation particles are inflated away on both branes and the expectation values of the operators and vanish. When the branes collide part of their energy is spent on creation of particles, baby branes and/or bubbles (see Fig. 4). Since the total charge is conserved, the net charge produced on the both branes should be zero. However, during the nonequilibrium collision process the branes overlap. Thus, and will not be separately conserved, and it might happen that in some reactions . Thus, the net global charges will be left on each branes. In addition to this effect, some charge will be carried away by the baby branes and/or the confining bubbles as discussed in the previous sections. We can briefly summarize the process described above as follows: The branes, while colliding, spend a very little time on top of each other. After that, they just “bounce back” and start to oscillate about the equilibrium point . If C and CP symmetries are broken during the brane collisions, the couplings (20) allow “charges” to be “exchanged” among and during the short time moment of the collision. Thus, it might happen that one charge is produced in inflaton decays in excess and the other one in deficit (see the example below). Once the collision happened, these couplings switchoff almost instantly, and as a result, the values of nonzero charge asymmetries freezeout. This, in particular, happens since the couplings (20) vanish almost instantly and the charges become separately conserved on two different branes.
The qualitative discussions given above can be made more precise by considering a simplified toy model. Consider two types of fermions, let us call them and . ’s are localized on our brane and carry baryon number (). ’s, on the other hand, are localized on a distant brane and carry the corresponding global charge (). Given the exponential suppression of the overlap of their wavefunctions, a part of the effective fourdimensional Lagrangian for these fermions can be written as follows:
(21) 
Here, denotes a braneseparation modulus field, the inflaton, and ’s and ’s are some polynomial functions of in which C and CP violations are encoded. and stand for charge conjugated fields. Note that interactions of with the fields on the same brane need not be exponentially suppressed. In some cases, these interactions arise after integrating out the bulk modes (e.g. open string modes stretched between two branes) which acquire masses due to the VEV of and have direct couplings to the light modes on each brane. On the other hand, all the overlapping terms which break symmetry explicitly must be exponentially suppressed (since, by the assumption, there are no light bulk modes with these charges).
Thus, when branes are well separated , the overlap terms are suppressed and the Lagrangian has two independent symmetries. One of them acts on ’s and can be regarded as baryon number symmetry in our brane. When branes come closer, however, the overlap terms do not vanish. As a result, we are left with one common fermionnumber Abelian symmetry group . This last conserves the “total charge” of the branes . Let us now turn to the particles which are being created in the inflaton decay. This decay, as we just mentioned, conserves the total charge . However, the individual charges, and are not conserved. Therefore, the rate for baryon number creation (e.g. in twobody decays)
(22) 
and, likewise, the rate for antibaryon number creation, are different. Thus, although is conserved in the inflaton decays, individually and will not be conserved if both C and CP are broken.
Note that there might exist an additional
source of physical CP violation due to “time interface” which
can arise
as a result of the timedependent VEV of
and different dimensional operators present in the and
functions.
These contributions are clearly very modeldependent and we will
not attempt to quantify them here. An important outcome, however, is
that in general, the rate to produce baryons in the inflaton decays
differs from the same rate for antibaryons if C and CP are broken.
Therefore, the nonzero value of
will be produced.
When the branes bounce back after the collision, the inflaton
VEV sharply increases. Thus, the violating terms in (21)
switch off and the baryon generation process stops before the
system equilibrates. As a result, the accumulated net baryonic
charge freezesout.
Thus, our brane will be carrying
the net baryon number after the system comes to equilibrium.
4. Discussions and Conclusions
Summarizing, we have shown that the processes with nonconservation of global charges, such as baryon and lepton number, will be taking place in the Brane Universe. A number of different scenarios of nonconservation of the global charges were proposed. The basic idea is that the global charges can be carried away from the Brane Universe by baby branes and/or confining bubbles. These mechanisms do not depend on a particular model of interactions realized on the brane. Thus, they are modelindependent and inherent for the Brane Universe scenario.
We have also presented two independent scenarios
of baryogenesis in the Brane Universe.
These approaches are based upon
properties of higherdimensional spacetime,
as well as on nonequilibrium nature of evolutionary processes
taking place in the Brane Universe. Highly
nonequilibrium and violent nature of these processes, be it scatterings
of various branes of different dimensionality or
strong quantum fluctuations of hot branes, makes it
complicated to perform precise analytical studies. Hopefully, these
mechanisms can be tested quantitatively in
numerical simulations of various toy and/or realistic models.
Acknowledgments
The authors are grateful to Nima ArkaniHamed, Savas Dimopoulos,
Glennys Farrar and especially Valery Rubakov for valuable discussions
and comments.
The work of GG was supported in part by the grant NSF PHY9423002.
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