UT-06-09

The Polonyi Problem and Upper bound on Inflation Scale in Supergravity

M. Ibe, Y. Shinbara and T.T. Yanagida

Department of Physics, University of Tokyo,

Tokyo 113-0033, Japan

Research Center for the Early Universe, University of Tokyo,

Tokyo 113-0033, Japan

## 1 Introduction

In gravity-mediation models for supersymmetry (SUSY) breaking, gaugino masses in the SUSY standard model (SSM) are given by a singlet field -term in a hidden sector. This singlet field called as Polonyi field should be an elementary field, since the gaugino masses are suppressed by higher powers of the Planck scale GeV, otherwise. This singlet field is completely neutral of any symmetry and the origin of the field has no enhanced symmetry. Thus, the minima of its potential during inflation and at the true vacuum are different from each other. The distance between those minima is likely of the order of the Planck scale, . Therefore, after the inflation the Polonyi field starts a coherent oscillation around the true minimum when the Hubble parameter becomes of the order of the mass of and its energy density dominates the early universe if its lifetime is much longer than that of the inflaton. As long as there is no physical scale besides the Planck scale at high energies, the Polonyi field has a mass of the order of the gravitino mass TeV and it decays to the SSM particles at very late times. The energetic photons and hadrons produced by the decay deconstruct the light nuclei created by the big-bang nucleosynthesis (BBN). This is called “the Polonyi problem” [1].

Possible solutions to the above problem may be found if one introduces a new high energy scale besides the . There are two possibilities to use this new scale. One is to increase interactions between hidden and observed sectors to make the decay of faster, for instance, in a Kähler potential. Here, denotes quarks in the SSM. However, such interactions also increase the gaugino masses and one should decrease the gravitino mass by a factor to keep the gaugino masses at the order of TeV. Then, the Polonyi mass is reduced also by the same factor, which results in even worse situation. The second possibility is to increase the interactions among fields in the hidden sector. Namely, one introduces for instance. Then, the mass of the Polonyi field becomes larger than the gravitino mass and the can decay before the BBN.

However, even in the second possibility, there arises another kind of Polonyi problem. The Polonyi field is much heavier than the gravitino and hence it decays mainly into a pair of gravitinos which, in turn, results in “the gravitino overproduction problem” [2]. Thus, there is still a severe upper bound on the relic abundance of the Polonyi field. (Notice that, even if one increases the interactions with the SSM particles as above, as well as the mass of the Polonyi field, the decay into gravitino is still a dominant decay mode of the Polonyi field.)

Despite the above gravitino overproduction problem, the second possibility is very interesting. The presence of the new cut-off scale will stop to run away from the origin and the distance between the potential-minimum points of during the inflation and at the true vacuum becomes of the order of , that is . Hence, the coherent oscillation of the Polonyi field does not necessarily dominate the universe, if is sufficiently smaller than the Planck scale . Thus, as shown in the next section, the gravitino overproduction problem from the Polonyi decay can be evaded for small values of the new scale as GeV. This result strongly suggests that the new scale is nothing but the dynamical scale of the SUSY breaking.

The purpose of this letter is to examine if the Polonyi problem is solved when SUSY is dynamically broken by some strong gauge interactions. In the above argument, we have assumed . However, we find that it is not always the case, since the potential of the is flatter than the mass term above . Thus, a careful analysis on the potential of is required during the inflation. We show as a result that there is a stringent constraint on the Hubble parameter of the inflation or on the reheating temperature for a successful solution, that is, GeV or GeV. This concludes that the gravity-mediation models favor relatively low-energy scale inflations as realized naturally in new inflation models. We consider that the present conclusion is quite generic, although we derive it in a class of SUSY-breaking models.

## 2 Upper bound on the new scale

Before going to discuss the dynamical SUSY-breaking models, we show that there is an upper bound on the new energy scale . As we have discussed in Introduction, we assume, for a moment, that the presence of the new scale may set the distance between minimum points of during the inflation and at the true vacuum to the order of , that is .

After the inflation, the value of the Polonyi field is fixed at the until
the Hubble parameter falls to the mass of the Polonyi field, , and
then, the Polonyi field begins a coherent oscillation around its true minimum.
Here the mass of the Polonyi field is enhanced by a factor of compared to the gravitino
mass as discussed in Introduction,^{1}^{1}1Here and henceforth, we have taken the unit with the
reduced Planck scale, .

(1) |

The Polonyi field and the inflaton decay into radiation when the Hubble parameter becomes at the decay rate of the Polonyi field,

(2) |

and at that of the inflaton,

(3) |

respectively.
In Eq. (2), we present a decay rate of the Polonyi field
into a pair of gravitinos, since the Polonyi field mainly decays into a gravitino
pair.^{2}^{2}2Recently, it has been extensively discussed that
the decay into a pair of gravitinos of the moduli fields [3] and the inflaton [4]
is much enhanced by even small mixings of those fields with the Polonyi field.
In Eq. (3), we have parametrized the decay rate of the inflaton by using
a reheating temperature after the inflation, and
denotes the effective massless degrees of freedom of the SSM.
In the following discussion, we assume that the Polonyi field decays fast enough not to
dominate the energy density of the universe after the inflaton decay.

The ratio of the number density of the gravitino to entropy is given by (after the inflaton decay),

(4) |

Here, we have assumed that most of the gravitinos are produced by the Polonyi decay. and denote the number densities of the Polonyi field and the inflaton at , the branching ratio of the Polonyi decay into a pair of gravitinos, and the mass of the inflaton. The factor comes from the dilution of the gravitino by the entropy production of the inflaton decay. (The equality in Eq. (4) holds as long as the Polonyi field decays before its domination of the energy density.) The number density of the Polonyi field and the inflaton at are given by (when the Polonyi field starts the oscillation),

(5) | |||||

(6) |

Thus, the yield of the gravitino can be expressed by,

(7) |

where we have used and Eq. (1).

To keep the success of the BBN, the gravitino abundance must satisfy [2],

(8) |

for TeV. ¿From Eq. (7), we find that an upper bound on the new scale ,

(9) |

Notably, the above upper bound on is close to the scale of the dynamical SUSY breaking,

(10) |

Therefore, the above constraint (9) strongly suggests that the new scale is nothing but the dynamical scale of strong interactions for the SUSY breaking, where the Polonyi field obtains its mass .

## 3 Upper bound on the Inflation scale

As we have seen in the previous section, the solution to the Polonyi problem using the new cut-off scale suggests the dynamical SUSY breaking by strong interactions. In this section, we discuss the Polonyi problem to examine if the dynamical SUSY breaking model can indeed solve the problem. As we have warned in Introduction, we cannot apply the result of the previous section directly, since the potential of is very flat above and our assumption is not guaranteed automatically. Thus, we have to arrange the inflaton potential to keep during the inflation. We will show in this section that there is a stringent upper bound on the inflation scale.

### 3.1 The scalar potential of a flat direction

Before going to discuss the dynamics of the Polonyi field during the inflation, we consider the scalar potential of the Polonyi field in the dynamical SUSY breaking model [5, 6]. To see it explicitly, we adopt a dynamical SUSY breaking model based on the SUSY gauge theory with four fundamental fields and six singlet fields . The tree-level superpotential is given by [6],

(11) |

Here, ’s denote coupling constants and we have omitted the gauge indices and the summations over . The equations of motion for , , set , which contradict with the quantum modified constraint Pf [7]. Here, denotes the dynamical scale of the gauge interactions. Hence, the SUSY is broken dynamically.

In this model, there is a flat direction which is a linear combination of the singlets, , which corresponds to the Polonyi field in the previous section. For near the origin, , the superpotential Eq. (11) can be effectively written as,

(12) |

by means of the quantum constraint Pf. Here, we have used a naive dimensional counting [8], and denotes an appropriate linear combination of . On the contrary, for large values of , , the ’s become massive and can be integrated out. Thus, the theory exhibits a gaugino condensation which produces an effective superpotential,

(13) |

Therefore, the scalar potential for all range of is given by,

(14) |

and is a flat direction.

The degeneracy of the above flat direction is lifted by quantum-effects in the Kähler potential [5]. For small value of () the effective Kähler potential is expected to take a form,

(15) |

where is a real constant which we expect to be of order one, and hereafter, denotes the coupling constant at the dynamical scale . It leads to a mass term of as,

(16) |

We find the mass of the Polonyi field as

(17) |

where we have used a definition,

(18) |

Note that for , the mass squared of becomes positive and the vacuum expectation value (VEV) is .
On the contrary, for the mass squared of is negative and we expect
, since the effective potential is lifted in the large region
(see Eq. (19)) [9].
In the following, we only consider the case of and for simplicity,
since the following discussion will not be changed significantly for .^{3}^{3}3For small values of , , the mass squared of the Polonyi field
is dominated by calculable one-loop corrections and is shown to be
positive [10].

For large values of (), quantum corrections to the scalar potential come from the perturbative wave function renormalization factor of and the potential is given by [9] ,

(19) |

where denotes the scale of the renormalization group^{4}^{4}4Here, we are assuming that .
Thus, the scalar potential in Eq. (19) is much flatter than
Eq. (16).
The flatness of the potential for the large field value is a generic feature of
any effective O’Raifeartaigh models where flat directions are lifted by
the quantum corrections.

Before closing this subsection, we stress an important feature of the Polonyi field in gravity-mediation models. In gravity-mediation models, the gauginos in the SSM obtain the SUSY breaking masses via direct couplings to the Polonyi field,

(20) |

where ’s denote gauge field strength chiral superfields. Hence, we expect that the Polonyi field must be neutral under any symmetries. This means that, in general, we cannot forbid a linear term of the Polonyi field in the Kähler potential,

(21) |

where is a dimensionful parameter and is expected to be of the order of the Planck scale . Furthermore, even if we set at the tree-level, the interaction terms such as Eq. (20) generate the linear term of order at one loop level, where is the number of the gauge multiplets circulating in the loop diagrams. Thus, we naturally expect that the linear term is at least of order , i.e. . As we see in the following discussion the linear term in the Kähler potential has a serious effect on the dynamics of the Polonyi field during the inflation.

### 3.2 Effects of the Hubble parameter during inflation

Now, let us consider the dynamics of the Polonyi field during the inflation. First, we assume that the Polonyi field is set to the origin at the beginning of the inflation. We do not discuss, here, what physics provides such a desired situation, since it is beyond the scope of this letter. (If the ’s are in the thermal bath before the inflation, the Polonyi field acquires the thermal mass which drives the to the origin . This may be a possible candidate for the physics.)

Once the inflation starts, the effective potential of the Polonyi field and the inflaton are given by,^{5}^{5}5We can easily extend the following discussion to inflation models
with many fields.

(22) |

where we have assumed the Kähler potential as,

(23) |

for simplicity. and are defined by

(24) | |||||

(25) |

In addition, we also assume that the hidden sector and the inflaton sector is separated in the superpotential as,

(26) |

By using a potential of given in Eqs. (16) and (19), a potential of the inflaton which is nearly constant during the inflation, and a Kähler potential for the Polonyi field , the above scalar potential can be rewritten as,

(27) |

During the inflation, the potential of the Polonyi field is changed from the one at the true vacuum due to the first term, . Here, we have neglected , since it is irrelevant to our discussion.

As we have mentioned at the end of the previous subsection, we cannot forbid the linear term in the Kähler potential. During the inflation, such a linear term leads to a slope of the Polonyi potential around the true vacuum ,

(28) |

On the other hand, the Polonyi potential in Eq. (19) is nearly flat for with a height of

(29) |

where, is a numerical constant at most of order one. Thus, if the linear term in Eq. (28) is large, the potential minimum is shifted from the origin .

To investigate the behavior of during the inflation more closely, we approximate the above Polonyi potential Eq. (16) and (29) by,

(30) |

Here, is the Polonyi mass given in Eq. (17) and is a field value where higher order terms of the effective Kähler potential in Eq. (15) become important. In the following analysis, we take

(31) |

with .

Under the above approximation, we find that the true minimum () is substantially shifted unless a condition,

(32) |

is satisfied. By using Eqs. (18) and (31), this condition can be expressed as an upper bound on the Hubble parameter during the inflation,

(33) |

where we also used the relation .^{6}^{6}6We assume inflation models with -term potentials in this letter.
However, if we consider -term inflation models, it depends on details of
the models if we may evade the constraint Eq. (33).
Figure. 1 is a schematic figure of the Polonyi potential around
for various values of .
The figure shows that the minimum at is substantially shifted from the origin when
the above condition is violated.

Notice that we have derived an upper bound on the Hubble parameter Eq. (33) based on a specific model of the dynamical SUSY-breaking. However, by using a naive dimensional counting [8], we can approximate Polonyi potentials by Eq. (30) with and for any effective O’Raifeartaigh models where flat directions are lifted by the quantum corrections. Thus, we consider that the above upper bound on the Hubble parameter is a quite generic result for any effective O’Raifeartaigh models.

Once the minimum at disappears during the inflation, rolls down to immediately.
In such a case, we suffer from the recurrence of the Polonyi problem,
since is fixed at until the Hubble parameter becomes very small.
As we see in the next section, the spilled Polonyi field causes a gravitino
overproduction problem for ,
if GeV (see Eq. (44)).
Thus, for , the above upper bound on the Hubble parameter,
GeV,
is a necessary condition for the inflation in the gravity-mediation models,
unless the reheating temperature is extremely low, GeV.^{7}^{7}7For the leptogenesis [11] to work, we need
a reheating temperature higher than the critical temperature for the
electroweak phase transition [11, 12].
Besides, it seems rather difficult for the inflation to achieve the low reheating temperature such as GeV.

This result shows that the SUSY chaotic inflation [13] (where typical Hubble parameter is GeV) and the SUSY topological inflation [14] ( GeV) are disfavored. The SUSY hybrid inflations are also disfavored since the typical Hubble parameters are GeV [15].

On the contrary, among inflation models constructed in SUGRA, a new inflation model in [16], is one of the most attractive candidates. The model has a flat inflaton potential,

(34) |

where is the energy scale of the inflation, the coupling constant in the superpotential, and is the quartic coupling constant in the Kähler potential. ¿From the COBE normalization, the inflation scales are determined for [17],

(35) | |||||

(36) | |||||

(37) | |||||

(38) |

and increases for larger .
Thus, we find that the new inflation model with is disfavored.
Interestingly, the favored new inflation model of predicts the spectral index
[18]
which is well consistent with the recent WMAP result,
(68%C.L.) [19].^{8}^{8}8The detailed analysis on the new inflation model of will be
discussed elsewhere [17].

## 4 Fate of spilled Polonyi field

As we have seen in the previous section, if the Hubble parameter is too large and it does not satisfy the condition Eq. (33), the minimum of the Polonyi potential is shifted away from the origin. Then, the Polonyi field falls to the minimum of the potential, , during the inflation. In this section, we consider the Polonyi problem for such a case.

In general, there are many local minimal points of the Polonyi potential around , and has a mass of the order of the gravitino mass at each minimal points. On the contrary, the curvature of the Polonyi potential Eq. (19) at is given by,

(39) |

with , which is at most the gravitino mass when is close to . Hence, for , we do not expect that the Polonyi field returns to after the inflation, since the attractive force towards the origin is weaker than those towards the minimal points at . In this case, the Polonyi field is attracted to one of the local minimal points at and starts to oscillate around the local minimal point when the Hubble parameter becomes at after the inflation. Since the typical distance between the point and the local minimal points at is of the order of the Planck scale, such a late time coherent oscillation causes nothing but the original Polonyi problem [1]. Therefore, once the Polonyi field is spilled out of the minimum, we again suffer from the original Polonyi problem for .

On the other hand, if the linear term is somewhat smaller than the Planck scale,
the Polonyi field is attracted to after the inflation, since the curvature of
Eq. (39) exceeds the gravitino mass.
Then, the Polonyi field starts to oscillate around its true minimum from
when the Hubble parameter falls to ,^{9}^{9}9We confine ourselves to the inflation model with the reheating
temperature GeV, since otherwise we have the gravitino
overproduction problem from the scattering process of the thermal background after the
inflation [2].
For such reheating temperature,
we can safely neglect thermal effects to the Polonyi potential from plasma.
and it eventually decays dominantly into a pair of gravitinos
(see section 2.).

After the whole reheating process, the yield of the gravitino is given by,

(40) |

where and denote the number densities of the Polonyi field and the inflaton at . To estimate the yield of the gravitino conservatively, we assume that the Polonyi field decays into a pair of gravitinos with its mass in Eq. (17) immediately after it starts to oscillate around . Then, the number density of the Polonyi field and the inflaton at are given by,

(41) | |||||

(42) |

By using these number densities, we obtain the yield,

(43) |

Hence, the BBN constraint [2] in Eq. (8) requires,

(44) |

Thus, for ,
we cannot avoid the gravitino overproduction problem even if the spilled Polonyi field
returns to after the inflation, unless the reheating temperature is extremely low.^{10}^{10}10Again, we consider that the above upper bound on
is a quite generic result for effective O’Raifeartaigh models.

Notice that the above upper bound on is conservative since we have used in Eq. (17) to estimate the decay rate of the Polonyi field (Eq. (2)) and the number density of the Polonyi field (Eq. (41)). Since the amplitude of the Polonyi field is much larger than at the beginning of the oscillation, the effective mass of the Polonyi field is smaller than . Thus, one may consider that the decay rate of the Polonyi field is smaller and is larger, which leads to a larger number density of the gravitinos. However, even if the effective mass of the Polonyi field is much smaller than , it behaves as a particle with mass during a time period of in each oscillation. Thus, the effective decay rate of the Polonyi field (with mass ) at the beginning of the coherent oscillation is given by,

(45) |

As we see from Eqs. (2) and (17), is close to
for , and hence, the effective decay rate is comparable to the Hubble
parameter at the beginning of oscillation, .
Thus, the Polonyi field effectively decays with a mass , immediately after it
starts to oscillate, and the number density of the Polonyi field which
results in the gravitino number density is roughly given by
dividing the energy density in Eq. (30) by (see Eq. (41)).
Therefore, we may use safely the above conservative analysis.^{11}^{11}11
In the model we have considered in section 3, there is a non-anomalous approximate
symmetry.
Then, the -ball and anti--ball [20]
can be formed after starts to oscillate around .
In this case, as long as the annihilation of -balls can be neglected,
the -ball has a long lifetime, which increases the resultant gravitino
abundance.
Thus, the upper bound on can be much severer than Eq. (44)
for the dynamical SUSY breaking model in section 3.
We thank F. Takahashi for pointing out this.

Finally, we comment on possible effects of direct couplings between the hidden sector and the inflaton sector in the superpotential. Although such interactions are highly model dependent (charges of the fields, etc.), we at least expect the terms,

(46) |

since the Polonyi field is completely neutral under any symmetries. Such terms, in general, increase the number of the minimal points around , and hence, the above problems are not improved.

## 5 Conclusions

In this letter, we have considered a solution to the Polonyi problem by assuming dynamical
SUSY breaking.
We have found that even for dynamical SUSY breaking models,
the linear term of the Polonyi field in the Kähler potential may bring us back to the
Polonyi problem or the gravitino overproduction problem.
To avoid the problems for the most natural case,
,
in the gravity-mediation models the inflation in the early universe should have
a very small Hubble parameter, GeV
(Eq. (33)) such as new inflation models, or
a very low reheating temperature, GeV
(Eq. (44)).^{12}^{12}12If there is a late-time entropy production, the constraint on the reheating
temperature can be weakened.
This result is very interesting since the favored new inflation model in SUGRA naturally
predicts the spectral index as , which is very consistent with
the recent WMAP observation.

We comment that there is also no theoretical reason to suppress the Kähler interactions such as . If it exists, the inflaton decay into a pair of gravitinos is enhanced and a stringent constraint on the inflation model is obtained. The recent analysis has shown that the hybrid inflation model is very disfavored [4]. Furthermore, the interactions between the hidden sector and the inflaton sector in the superpotential can enhance the inflaton decay rate into a pair of gravitinos, which may give more stringent constraints. The detailed analysis in the new inflation model [16] including such superpotential interactions will be discussed elsewhere [17].

We should note finally that the Polonyi problem discussed in this letter may not exist in gauge- or anomaly-mediation models for SUSY breaking. This is because the Polonyi field may have charges of some symmetries suppressing the linear term in the Kähler potential or because there is not necessarily present the elementary Polonyi field in those models.

## Acknowledgments

M. I. thanks the Japan Society for the Promotion of Science for financial support. This work is partially supported by Grand-in-Aid Scientific Research (s) 14102004. The work of T.T.Y. has been supported in part by a Humboldt Research Award.

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