Baryon Asymmetry in a Heavy Moduli Scenario

Baryon Asymmetry in a Heavy Moduli Scenario

Masahiro Kawasaki    Kazunori Nakayama Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan
July 3, 2019
Abstract

In some models of supersymmetry breaking, modulus fields are heavy enough to decay before BBN. But the large entropy produced via moduli decay significantly dilutes the preexisting baryon asymmetry of the universe. We study whether Affleck-Dine mechanism can provide enough baryon asymmetry which survives the dilution, and find several situations in which desirable amount of baryon number remains after the dilution. The possibility of non-thermal dark matter is also discussed. This provides the realistic cosmological scenario with heavy moduli.

pacs:
98.80.Cq, 14.80.Ly

I Introduction

Recent cosmological observations have revealed that the ordinary matter contributes to the only small fraction of the energy density of the universe, in terms of the density parameter, and the remainder comes from “dark” components, dark matter and dark energy, whose contributions are represented by and respectively Spergel:2006hy (). The existence of these dark components indicates physics beyond the standard model such as supersymmetry (SUSY) Nilles:1983ge (), but on the other hand, the non-standard physics is also considered to be imprinted in the ordinary matter (baryon) component. From the measurement of cosmic microwave background (CMB) anisotropy and light element abundances predicted by big-bang nucleosynthesis (BBN) Olive:1999ij (), it is known that the baryon density of the universe is almost made only of baryons, and anti-baryons do not exist . But such observed amount of baryon asymmetry can not be generated within the framework of the standard model. Thus, if there is an underlying physics beyond the standard model, the baryon density of the universe should also be explained by violation of the baryon number and CP built in the new physics, as well as the dark matter. In supersymmetric theory, which is one of the best motivated physics beyond the standard model, there is an interesting mechanism to create baryon asymmetry. In SUSY there exist many scalar fields as superpartner of the standard model fermion which carry baryon or lepton number. Along some directions of the configuration of these scalar fields, the scalar potential is flat. Thus scalar fields corresponding to these flat directions can develop to large field value during inflation, and subsequent evolution of the scalar fields naturally leads to baryon asymmetry. This is called Affleck-Dine mechanism Affleck:1984fy (), which we will focus on this paper.

On the other hand, global supersymmetry is naturally extended to the local supersymmetry, which inevitably includes gravity, that is supergravity. In supergravity there appear long-lived massive particles whose lifetime are typically longer than 1 sec and they decay after BBN starts. One is the gravitino, the superpartner of the graviton. Gravitinos are produced in high-temperature plasma via scatterings of the particles in thermal bath and their subsequent decay may greatly affect the standard cosmology Khlopov:1984pf (); Bolz:2000fu (), or may overclose the universe if they are stable Moroi:1993mb (). Another is the Polonyi field, which is a singlet scalar field introduced in order to give the SUSY breaking masses to the superparticles, especially the mass of gauginos, in many models of SUSY breaking. Generally the Polonyi field has the large field value during inflation, and it begins to oscillate coherently with initial amplitude of order reduced Planck scale when the Hubble parameter becomes equal to the gravitino mass . It has extremely large energy density and its decay after BBN has catastrophic effects on the standard cosmology Coughlan:1983ci (). Furthermore, supergravity may be the low energy effective theory of string theory, which is defined in 10 dimensional space-time. In compactification of such extra dimensions, there appear light scalar fields called moduli. Generally moduli have the mass of order through non-perturbative dynamics associated with SUSY breaking. The dynamics of moduli fields and cosmological difficulty they cause are similar to those of the Polonyi field, and we call them moduli problem Banks:1993en ().

There are some suggestions to solve the moduli problem. One possible way is to use late-time inflation and the subsequent entropy production in order to dilute the moduli abundance to cosmologically safe value. Such late-time inflation is realized by thermal inflation models and concrete examples for are found in Refs. Yamamoto:1985rd (); Lyth:1995hj (); Asaka:1999xd (). The other way is to make moduli heavy enough to decay well before BBN. The modulus mass larger than about 100 TeV is safe and such large mass is naturally realized in anomaly-mediated SUSY breaking models, where other SUSY particles obtain the mass of order TeV Randall:1998uk ().

But the scenario is not complete. In late-time entropy production scenario, the preexisting baryon asymmetry is also diluted. The reheating temperature after thermal inflation is typically less than a few GeV, and hence almost all baryogenesis mechanisms which rely on high energy physics do not work. The variant type of Affleck-Dine mechanism after thermal inflation may work Stewart:1996ai () and to the best of our knowledge it is only possibility to create enough baryon asymmetry in the presence of thermal inflation. In heavy moduli scenario, significant entropy released by the decay of moduli also dilute the preexisting baryon asymmetry. In previous works Moroi:1994rs (); Moroi:1999zb (), it was assumed that Affleck-Dine mechanism can create enough baryon asymmetry which survives the dilution from moduli decay. However, in fact, for large ordinary Affleck-Dine mechanism does not work due to the non-trivial potential minima of the Affleck-Dine field Fujii:2002kr (); Kawasaki:2000ye ().

In this paper we study whether the sufficient baryon asymmetry is created in heavy moduli scenario, such as anomaly-mediated SUSY breaking models or mixed modulus anomaly mediation (or mirage mediation) models Choi:2004sx (). Mirage mediation models are based on the concrete model of KKLT flux compactification Kachru:2003aw () and the lightest modulus mass is predicted as . Thus in the following we consider the typical situations where the modulus mass is of the order and as reference values, although we do not specify the concrete SUSY breaking models. The only assumption on which our analysis is based is the hierarchical relation between the gravitino mass and other SUSY particle masses. In both scenarios (anomaly- or mirage mediation models) other SUSY particles have mass of order and hence the gravitino is considered to be as heavy as 100 TeV. As we will explain later, Affleck-Dine mechanism for such large is highly non-trivial. One possible way to incorporate Affleck-Dine mechanism in anomaly- or mirage-mediation model is to introduce the gauged U(1) symmetry Fujii:2001sn (). But we pointed out that Affleck-Dine mechanism with large works without any additional assumption if the reheating temperature is relatively high Kawasaki:2006yb (). We examine these two scenarios and find possible parameter region in which the desired amount of baryon asymmetry is generated.

This paper is organized as follows. In Sec. II, we briefly review the cosmological dynamics of moduli fields. In Sec. III, Affleck-Dine baryogenesis in high reheating temperature scenario is discussed. In Sec. IV, Affleck-Dine baryogenesis in gauged U(1) model is discussed. Somewhat similar to this case, but the case without superpotential and Q-ball domination is discussed in Sec. V. The decay of moduli may cause another difficulty, especially LSP overproduction from the decay of moduli-induced gravitinos. We give possible solutions to this problem in Sec. VI. We conclude in Sec. VII.

Ii Dynamics of modulus fields

In general, a modulus field has the mass of order in the presence of SUSY breaking. In the early universe, the inflaton dominates the energy density and this vacuum energy also breaks supersymmetry. As a result the modulus field obtains Hubble-induced SUSY breaking mass of order Dine:1995uk (). Thus, during inflation the modulus has very large mass and sits at the origin. However, this high-energy minimum does not need to coincide with the low-energy true minimum of the potential. In general, these two minima are expected to be separated by the Planck scale and hence when becomes smaller than the modulus mass , the modulus field begins to oscillate with the initial amplitude . This coherent oscillation of the modulus field has the energy density initially, and the total energy density of the universe is given by . We can see that the modulus has inevitably large energy density comparable to the total energy density and dominates the universe as soon as inflaton decays (in the case of , where denotes the decay rate of the inflaton) or the modulus field begins to oscillate (in the case of ).

In the case of where inflaton decays after the moduli start to oscillate, the moduli-to-entropy ratio is given by

(1)

where denotes the reheating temperature from inflaton decay. On the other hand, in the case of , it is given by

(2)

In both cases the moduli abundance largely exceeds the critical density of the universe, GeV. If we parametrize the decay width of the modulus as

(3)

the decay temperature of moduli is given by

(4)

Hence, the decay temperature of modulus typically takes a value from a few MeV to a few GeV for TeV TeV. The late decay of moduli with such large abundance has significant effects on BBN Kawasaki:1994af (), CMB Fixsen:1996nj () and diffuse X-ray background Kawasaki:1997ah (), which results in strong disagreement with observations. But for the modulus mass larger than about TeV, the moduli decay before BBN and do not spoil the success of standard BBN. The universe is finally reheated by the moduli with reheating temperature 111If the modulus field does not have the Hubble mass and obtain unsuppressed quantum fluctuation during inflation, it can be the interesting candidate of the curvaton Moroi:2001ct (); Hamaguchi:2003dc (). But we do not go into the details of this issue..

Iii Affleck-Dine baryogenesis with early oscillation

iii.1 The model

In minimal supersymmetric standard model (MSSM), there exist some configurations of the scalar fields (flat directions) along which scalar potential vanishes in supersymmetric limit within the renormalizable terms Gherghetta:1995dv (). A flat direction is parameterized by the single complex scalar , which we call Affleck-Dine field. The Affleck-Dine field feels potentials from non-renormalizable superpotentials represented as

(5)

where is the effective cutoff scale and . Including SUSY breaking effects, the potential for the Affleck-Dine field is written as

(6)

where is numerical coefficient. There are other sources for the scalar potentials. As explained in sec.II, the scalar fields obtain Hubble-induced SUSY breaking terms such as

(7)

where and are coefficients 222In some inflation models such as -term inflation Halyo:1996pp (), Hubble-induced term does not arise during inflation Kolda:1998kc (). . Here we assume . Furthermore, in high-temperature environment of the early universe, thermal corrections to the scalar potential also arise. These are Allahverdi:2000zd (); Anisimov:2000wx ()

(8)

where is a constant of order unity, denotes gauge or Yukawa couplings relevant for the Affleck-Dine field, and is a constant of order unity assumed to be positive, which is determined by the two-loop finite-temperature effective potential for the Affleck-Dine field. Then the total scalar potential for the Affleck-Dine field is sum of them,

(9)

Let us summarize the dynamics of Affleck-Dine field. First, it is trapped by the minimum determined by the balance of negative Hubble-induced mass term and non-renormalizable term,

(10)

and tracks this minimum as becomes small. The important fact is that without the finite-temperature effect, the scalar potential has the global minimum at

(11)

if is much greater than , which is the situation we are interested in. Thus the Affleck-Dine field is eventually trapped by this minimum and leads to charge or color breaking vacuum, which is a disaster. But finite-temperature effects can save the situation. Including finite-temperature effects, when becomes equal to determined by

(12)

the Affleck-Dine field begins to oscillate around its minimum of the potential. The important fact is that if the thermal log term dominates the potential and oscillation begins by this term, the Affleck-Dine field will be taken to the origin without trapped by the global minimum Kawasaki:2006yb (). Through the process of field evolution, the Affleck-Dine field receives angular kick from -terms and results in elliptical motion around the origin. Hence the baryon number is generated and conserved in comoving volume. In fact, as we will see, for high reheating temperature from inflaton and high field value, the oscillation starts when thermal logarithmic term dominates the potential and Affleck-Dine mechanism works well. Note that although the resultant vacuum is meta-stable, the lifetime of the false vacuum is much longer than the age of the universe Kawasaki:2000ye ().

iii.2 Baryon asymmetry

Next we estimate the baryon asymmetry in the presence of a heavy modulus field. In the case of early oscillation due to thermal logarithmic potential, is given by

(13)
(14)

Hereafter we consider and case only, because flat directions with are lifted by the superpotential of the form where represents scalar field other than Affleck-Dine field and can not generate baryon asymmetry via Affleck-Dine mechanism Gherghetta:1995dv (). We need following constraints for this scenario to work. One is the condition that early oscillation occurs (), this leads to

(15)

for , and

(16)

for . The other is the condition that thermal correction hide the valley of the potential around the true charge breaking minimum and this thermal logarithmic potential leads Affleck-Dine field to the origin. This condition is written in the form , explicitly,

(17)

for and

(18)

for . We can see that in general the latter condition is severer whenever , which is always satisfied in anomaly-mediated SUSY breaking.

Now let us estimate the baryon asymmetry in the presence of modulus fields. It is convenient to express the baryon-to-entropy ratio as

(19)

The ratio is fixed when both the Affleck-Dine field and modulus field begin to oscillate. When , this ratio is fixed at the onset of the oscillation of the moduli, . On the other hand, when , the ratio is fixed at the beginning of Affleck-Dine field oscillation, . From Eqs.(14) and (17) or (18),

(20)

must always hold. Thus if we assume we can safely focus on the case . But in some models based on string theory, might be possible. In mirage mediation model, the modulus mass is predicted as Choi:2004sx (). Although in such a model is still possible, we mainly focus the case and briefly discuss about the modification in the case . The condition is rewritten as

(21)

for and

(22)

for . High reheating temperature from inflaton is not a problem as far as the moduli decay well before BBN and non-thermal LSPs associated with modulus decay do not overclose the universe. In fact it is possible that non-thermal LSPs from the decay of moduli account for the present matter density of the universe (see Sec. VI).

iii.2.1

In this case the baryon-to-moduli ratio is fixed at where the modulus field begins to oscillate with amplitude ,

(23)

where . In order to get correct estimation, we must specify the decay epoch of inflaton, whose decay rate is denoted as . Thus depending on , three scenarios are available: (a) , (b) , (c) . Note that in case (a), at the beginning of oscillation of the Affleck-Dine field the universe already enters radiation dominated era and estimation of baryon number is somewhat different from the other two cases. Before the estimation, we see the conditions when the case (a), (b) and (c) are realized. The condition that can be written as

(24)

On the other hand, the condition can be rewritten as follows,

(25)

which is satisfied for natural range of parameters. In other words, unless reheating temperature is unnaturally high, case (a) is not realized. The conditions (21) (or (22)), (24) and (25) determine which of the following scenario is realized.

In the case (a), early oscillation begins in radiation dominated regime and the baryon-to-moduli ratio (23) is written as

(26)

where denotes the effective CP phase. Note that as far as the initial amplitude of the Affleck-Dine field is smaller than , it never dominates the universe at the instant of oscillation. We can estimate and as

(27)

and

(28)

where . Substituting these values and using Eq. (4), we finally obtain the baryon-to-entropy ratio after decay of the modulus field as

(29)

in the case of flat direction. Clearly this is too small and it is impossible that we obtain a proper amount of baryon asymmetry. For , we obtain

(30)

It also seems too small, but dependence of the cut-off scale is rather large, so that if we assume the desired amount of baryon asymmetry can be obtained. It may seem peculiar that the cut-off scale is bigger than Planck-scale, but our definition of includes some coupling constant, e.g., even if physical cut-off scale is , the effective cut-off scale can be if the relevant coupling constant is .

In the case (b), the modulus oscillation begins in radiation dominated era. The baryon-to-moduli ratio (23) is expressed as

(31)

A straightforward calculation yields

(32)

for case. Obviously, this is too small. On the other hand, for case we obtain

(33)

It seems possible that a proper amount of baryon asymmetry after choosing cut-off scale appropriately. But is constrained from the condition [Eq.(22)]. Substituting Eq. (22) into the above equation, we obtain the upper bound on ,

(34)

Thus we need to obtain enough baryon asymmetry. If this is the case, must also be as high as GeV. This also satisfies the constraint .

In the case (c), the modulus starts to oscillate in inflaton-dominated regime and then inflaton decays resulting in brief radiation dominated era followed by moduli dominated universe. The baryon-to-moduli ratio (23) in this case is expressed as

(35)

For the case, we obtain

(36)

which is extremely small compared with the present baryon density. When we apply to the flat direction, the baryon-to-entropy ratio is estimated as

(37)

which seems successful. However, we need rather high reheating temperature which suppress the baryon-to-entropy ratio, due to the condition [Eq.(22)]. Substituting Eq.(22), the upper limit for baryon asymmetry is obtained,

(38)

If we can obtain desired baryon asymmetry, and this indicates that reheating temperature should be higher than GeV. On the other hand, GeV is necessary in order to satisfy . Thus and GeV GeV are the possible parameter region (see Fig. 1).

iii.2.2

Now let us turn to the case . As we explained, early oscillation to avoid charge or color breaking minima requires and hence this particular possibility arises only when modulus mass is much heavier than 333As we explain in Sec. VI, although moduli decay into gravitinos may cause cosmological difficulty, here gravitinos are also heavy enough to decay well before the BBN. Furthermore LSPs produced by decay of moduli effectively annihilates and do not overclose the universe (or they become dark matter). However, the subsequent decay of non-thermally produced gravitinos may pose a cosmological difficulty. See Sec. VI.. In this case, we can classify the cosmological scenario depending on the inflaton decay rate : (d) , (e) , (f) . Note that baryon-to-moduli ratio is fixed once the Affleck-Dine field starts to oscillate, but resulting formula for is the same as eq.(23). Therefore the results of case (d) and (f) are the same as(a) and (c) respectively. Only the case (e) slightly differs from (b).

In the case (e) the baryon-to-moduli ratio is expressed as

(39)

which is slightly different from the case (b). Note that we have used the approximation that the moduli dominate the universe soon after the oscillation. The following calculations are similar, and the result is

(40)

for case, where we have used the constraint in the second line. Using the same constraint, for case we obtain

(41)

Similar to the case for , for appropriate choice of the cut-off scale and the reheating temperature , it seems that we can obtain a proper amount of baryon asymmetry. However, we should recall that the constraint narrows the allowed parameter range. In fact, the case (e) is not realized in the parameter region we are interested in.

In Figs.1 and 2 we show the resulting baryon-to-entropy ratio in plane in the case of TeV and TeV for . The latter case is naturally realized in mirage-mediation models. We can see that GeV and GeV are required in the former case. In the latter case where the modulus field is much heavier than the gravitino, the constraint is weaker. Note that in such a heavy moduli scenario gravitinos can be efficiently produced by the decay of moduli, and these non-thermal gravitinos also decay before BBN for TeV. LSPs produced by the decay of those gravitinos may be harmful. We will discuss it in Sec. VI.

iii.3 Q-ball formation

Finally we must consider the effects of Q-ball formation. The fluctuations of the Affleck-Dine field with charge grow and result in lumped condensate, called Q-balls Coleman:1985ki (); Kusenko:1997zq (). The Q-ball formation leads to many non-trivial cosmological consequences, and they highly depend on SUSY breaking models Kusenko:1997si (); Enqvist:1997si () (see also Kasuya:2001hg (); Fujii:2002kr ()). As we have seen in the previous subsection, quite large cut-off scale is required. One may wonder this leads to large Q-balls and invalidates the applicability of our scenario. However, as we will see, largeness of Q-balls is suppressed because of early oscillation. The radius of Q-balls is comparable to the hubble horizon scale at the epoch of Q-ball formation. Thus although larger cut-off scale tends to create larger Q-balls, but higher reheating temperature , which causes earlier oscillation, tends to make Q-balls smaller. Now let us estimate .

It is found that that for the Q-balls which have developed via logarithmic potential, the total charge of Q-ball is fitted by the formula Kasuya:2001hg (),

(42)

where . Applying to the early oscillation case for flat direction, it is estimated as

(43)

for , and

(44)

for . It is known that evaporation of Q-balls in high-temperature plasma can efficiently transfer the charge of Q-balls up to almost model independently Laine:1998rg () (see also Kawasaki:2006yb ()). Therefore in the most interesting parameter region, Q-balls formed through Affleck-Dine mechanism can completely evaporate and have no further effects on cosmological evolution of baryon asymmetry.

In Figs.1 and 2, we show the contour of with black dotted line. It can be seen that in the interesting parameter region where is obtained, only small Q-balls are produced and they evaporate in the high-temperature plasma.

Figure 1: Contour plot of as a function of and . We take TeV. In the shaded region Affleck-Dine field is trapped into charge-breaking minima and baryogenesis does not work. Also we show by the dotted line . The left side of the dotted line predicts and Q-balls completely evaporate in high-temperature plasma.
Figure 2: Same as Fig.1, except for TeV.

Iv Affleck-Dine baryogenesis with gauged

Next we turn to another possibility that Affleck-Dine baryogenesis with large gravitino mass works with an extension of MSSM to include some additional fields and gauged symmetry. Because the global symmetry within MSSM is anomaly-free, it can naturally be extended to local symmetry. But from the viewpoint of baryogenesis, it must be spontaneously broken at some high energy scale in order to create baryon asymmetry and not to contradict with terrestrial experiments such as proton decay.

iv.1 The model

We briefly explain the model discussed in Ref. Fujii:2001sn (). First, we introduce MSSM singlet fields which have the superpotential as

(45)

where and have the charge and respectively, and denotes the breaking scale. They induce the scalar potential given by

(46)

where denotes the gauge coupling constant and denotes the charge of the Affleck-Dine field. The second term comes from the -term contribution. In the following, we consider flat directions which are lifted by non-renormalizable superpotential in MSSM, such as or direction. In this model, gauge-invariant superpotential which lifts those flat directions are given by

(47)

where and are coupling constants, and the resulting zero-temperature scalar potential is written as

(48)

Although the whole dynamics is somewhat complicated and we do not give the details here (see Fujii:2001sn () for a detail), the point is that by using the additional -term potential which does not exist in MSSM, the Affleck-Dine field can be stopped at the breaking scale during inflation. If is smaller than the hill of the potential of the Affleck-Dine field

(49)

Affleck-Dine mechanism works without trapping into the charge or color breaking global minimum. If we assume and we focus on case, this condition is equivalent to

(50)

If the value exceeds this bound, Affleck-Dine baryogenesis can not work due to trapping of the Affleck-Dine field in global charge breaking minima, if thermal effects are neglected.

iv.2 Baryon asymmetry

We saw that in this type of model, the Affleck-Dine field stops at breaking scale until Hubble parameter becomes of the order and oscillation begins. If is smaller than the hill of the potential of the Affleck-Dine field, Affleck-Dine mechanism works. In the case of early oscillation, the result is the same as usual early oscillation scenario considered in the previous section. Thus we consider only the case of no early oscillation in this subsection. The condition to avoid early oscillation is

(51)

Thus we can safely set . The baryon number at the instant of oscillation of the Affleck-Dine field is given by

(52)

with . The baryon-to-moduli ratio is once fixed at the epoch of oscillation of the Affleck-Dine field, , and the final reheating comes from the decay of moduli. The result is

(53)

which depends on seventh powers of . Substituting the upper bound on [Eq.(50)], we obtain an upper bound on the baryon-to-entropy ratio,

(54)

which seems successful. However, it is non-trivial whether Q-ball is small enough to evaporate completely. Charge of Q-ball is given by Kasuya:2000wx ()

(55)

where is order factor which represents the delay of Q-ball formation from the oscillation of Affleck-Dine field and is called the ellipticity parameter given by

(56)

Therefore, using the upper bound of [Eq.(50)], we obtain

(57)

for , which is a little larger than total evaporated charge . For pure leptonic flat direction such as , Q-balls must completely evaporate above the temperature where electroweak phase transition occurs in order to convert lepton number into baryon number by sphaleron effects, and hence is not acceptable. On the other hand for flat directions carrying baryon number such as , is allowed. In such a case, where Q-balls decay below the freeze-out temperature of LSP, we must care about overproduction of LSPs from Q-ball decay. But in our scenario entropy production from moduli decay dilutes them. Therefore for direction 444Actually and direction can have large field value simultaneously. gauged U(1) scenario in the presence of heavy moduli is marginally possible.

In Fig. 3, we show the resultant baryon asymmetry in () plane with constraints. We can see that for TeV, Q-balls become too large. But for larger the correct baryon asymmetry can be obtained without forming too large Q-balls.

Figure 3: The two solid lines show in gauged scenario, the upper (blue) line corresponds to TeV and the lower (red) line corresponds to TeV. We take TeV. In the dark shaded region Affleck-Dine field is trapped into charge-breaking minima and baryogenesis does not work. We also show by the dotted line.

V Affleck-Dine baryogenesis without superpotential

v.1 The model

Next we consider the models of Affleck-Dine baryogenesis with gauged including no non-renormalizable superpotentials due to some symmetry such as -symmetry. In such a case, baryon number violating operators are supplied by higher order effects from Kahler potentials (see e.g., Ref. Fujii:2002aj ()) and the initial amplitude of the Affleck-Dine field can become as large as Planck scale. The dynamics of the Affleck-Dine field is similar to the previous section. As a result, large Q-balls are formed associated with Affleck-Dine baryogenesis and they decay at late time after the freeze-out of LSPs but before BBN. Interestingly, in this type of model late-decaying Q-balls may once dominate the universe Fujii:2002aj (). If this is the case, a nice feature arises when considering the moduli-induced gravitino problem, as explained in Sec. VI.

Now let us investigate the above model. The zero-temperature scalar potential for the flat direction is given by

(58)

where the ellipsis denote the higher order terms, which stabilize the Affleck-Dine field at some value of order the Planck-scale. Note that the potential (58) also has charge and/or color breaking global minimum near the field value at . Similar to the previous section, in order to avoid falling into this minimum, the -term stopping at must satisfy the following condition,

(59)

Here we consider only the case without early oscillation due to thermal effects. This requires