Dark Matter in Gauge Mediated Supersymmetry Breaking using Metastable Vacua
aa r X i v : . [ h e p - ph ] O c t DESY 07-156
Dark Matter in Gauge Mediated Supersymmetry Breakingusing Metastable Vacua
Motoi Endo and Fuminobu Takahashi
Deutsches Elektronen Synchrotron DESY,Notkestrasse 85, 22603 Hamburg, Germany
Abstract
We point out that, in a class of gauge mediation models using metastable supersymmetry break-ing vacua, the minimum of the supersymmetry breaking field in the early universe is dynamicallydeviated from the one in the low energy. The deviation induces coherent oscillations of the super-symmetry breaking field, which decays into the gravitinos. For certain parameters, it can producea right amount of the gravitinos to account for the observed dark matter. . INTRODUCTION Gauge mediation (GM) of supersymmetry (SUSY) breaking [1] is a natural solution tothe phenomenological problems such as excessive flavor-changing neutral currents. In spiteof the successes, building a realistic model is a rather non-trivial task. According to theargument of Nelson and Seiberg, an exact U (1) R symmetry is required if the superpotentialis generic [2]. This observation strongly limits possible models, and a lot of efforts have beendevoted to building a realistic one.Recently, Murayama and Nomura has proposed drastically simplified models, focusing onmetastable vacua in the SUSY breaking sector [3]. Even though the entire superpotentialdoes not possess the exact U (1) R symmetry, an accidental one exists near a local SUSYbreaking minimum [4]. Such a scenario is viable as long as the metastable vacua have asufficiently long life time.In a class of the metastable SUSY breaking models, there are the local SUSY breakingvacua near the origin of the SUSY breaking fields, where the accidental U (1) R symmetryexists. On the other hand, the breaking of U (1) R symmetry is necessarily involved in themessenger sector to mediate the SUSY breaking to the gauginos in the supersymmetricstandard model (SSM). Since the SUSY breaking field (denoted by S ) linearly couples tothe messenger fields, the breaking of the U (1) R symmetry induces a linear term of S in theK¨ahler potential.Such a linear term forces the SUSY breaking field S to deviate from its minimum in thelow energy, while the inflaton field dominates the energy of the universe. When the Hubbleparameter becomes comparable to the mass of S , it starts to oscillate coherently aroundthe SUSY breaking vacuum, and then decays into the gravitinos. In the GM models, thegravitino is stable and behaves as dark matter (DM) in the universe as long as the R-parityis preserved. Thus, DM is generally produced from the SUSY breaking field in a class of theGM models using the metastable vacua. In this letter, we study the gravitino production inthis scheme.Before proceeding to the details, let us comment on the previous works. The cosmologicalevolution of the SUSY breaking field has been studied in the models with the metastableSUSY breaking vacua in Refs. [5]. Those literatures assumed so high reheating temperatureas to make the SUSY breaking sector to be in thermal equilibrium. Then the gravitino is2xpected to reach thermal equilibrium, and as a result, its abundance exceeds the observedDM abundance or it erases the density fluctuation too much, unless the gravitino mass issmaller than 16 eV [6]. For a wide range of the gravitino mass, i.e., from 16 eV to O (10) GeV,therefore, the SUSY breaking sector should not be thermalized as long as the standardthermal history of the universe is assumed. In this letter, we do not pursue this possibilityand assume that the SUSY breaking sector never reaches thermal equilibrium. On the otherhand, Ibe and Kitano discussed the gravitino production from the SUSY breaking field in theGM models using the metastable vacua [7]. They assumed a different thermal history takinga different set of the parameters; in particular, the reheating temperature in our scenario isas high as 10 − GeV, while they considered the low reheating temperature. Furthermore,the interaction that shifts the SUSY breaking field from its minimum in the low energy isdifferent from that considered in Ref. [7].
II. MODEL
In this section we provide a model of the gauge mediation using the metastable vacua.To be explicit, we adopt a model by Murayama and Nomura given in Ref. [3]. The K¨ahlerpotential and the superpotential for a gauge singlet chiral field S and the messengers f and¯ f are written as K = | S | − | S | + | f | + | ¯ f | , (1) W = − µ S + κSf ¯ f + M f ¯ f , (2)where the higher order corrections of O ( | S | / Λ ) are omitted for simplicity in the K¨ahlerpotential. In the following we take f and ¯ f to be in + ∗ representation of SU (5) GUT , and µ , κ and M are set to be real and positive without loss of generality. We assign the charges U (1) R symmetry as R [ S ] = 2 and R [ f ] = R [ ¯ f ] = 0. Then one can see that the messengermass term explicitly violates the U (1) R symmetry to Z .For our purpose we do not need to specify the UV physics above a scale Λ that providesthe second term in Eq. (1) as well as the first term in Eq. (2). We simply note here that thereare many explicit models that actually lead to this low energy effective theory. (See Ref. [3]for examples.) In particular, we do not give the SUSY breaking mechanism explicitly here,which is assumed to be such that the first term in Eq. (2) is somehow produced.3rom the K¨ahler potential and the superpotential given above, one can show that thereis a SUSY minimum at S = − Mκ , f = ¯ f = µ √ κ . (3)On the other hand, SUSY is broken at S = f = ¯ f = 0, which is a metastable local minimumas long as M > κµ (4)is satisfied, since otherwise one of the messenger scalars becomes tachyonic. Note that thesecond term in Eq. (1) produces a positive mass of S , m S = µ / Λ, around the origin. TheSUSY breaking scale is dictated by the first term in Eq. (2), and the F -term of S is givenby F S ≃ µ . Requiring a vanishing cosmological constant, we can relate the SUSY breakingscale to the gravitino mass m / as µ = (cid:16) √ m / M P (cid:17) ≃ × GeV (cid:18) m / (cid:19) , (5)where M P = 2 . × GeV is the reduced Planck scale. The SUSY breaking effects aretransmitted to the visible sector by the messenger loops. The integration of the messengersgive rise to the gaugino masses as [1] m i ≃ α i π κµ M for i = 1 , , . (6)Here, m , , and α , , are the gaugino masses and the gauge coupling constants for U (1) Y , SU (2) L and SU (3) C in the SSM. We have used the SU (5) GUT normalization forthe U (1) Y gauge coupling constant. We can express κµ /M in terms of the gluino mass m : κµ M ≃ × TeV (cid:18) α . (cid:19) − (cid:18) m (cid:19) . (7)From (5) and (7), one can express M as M ≃ × GeV κ (cid:18) α . (cid:19) (cid:18) m (cid:19) − (cid:18) m / (cid:19) . (8)We also assume m S < ∼ Λ, or equivalently, µ < ∼ Λ , (9)since we consider the dynamics of S (e.g. coherent oscillations and decay), which should bedescribed within the low energy effective theory. Using (9), we obtain an upper-bound on4 S , m S = µ Λ ≃
40 TeV (cid:18) m / (cid:19) (cid:18) Λ10 GeV (cid:19) − ,< ∼ × GeV (cid:18) m / (cid:19) . (10)Lastly, let us discuss radiative corrections to the K¨ahler potential. Integrating the mes-senger loop, the relevant corrections are given by K (1) = K (1) nh + K (1) h , (11)with K (1) nh = − M π ( (cid:18) κM (cid:19) | S | ( S + S † ) − (cid:18) κM (cid:19) | S | ( S + S † ) + · · · ) , (12) K (1) h = − M π (cid:26) κM ( S + S † ) + · · · (cid:27) , (13)where we have separated the holomorphic terms and the non-holomorphic ones. Note that S is assumed to be much smaller than M/κ so that S sits far away from the SUSY minimum(see (3)). As explained in Introduction, the reason why such corrections appear is that the R -symmetry is explicitly broken by the messenger mass term.One can check that the radiative corrections (12) reproduce the result of the Coleman-Weinberg potential for S given in Ref. [3], up to the order explicitly shown in (12): V (1) nh = 5 µ π ( κ M ( S + S † ) − κ M ( S + S † ) + · · · ) (14)To avoid the mass of S to become tachyonic due to the radiative corrections, we require [3] M > ∼ κ π Λ , (15)throughout this letter.As far as the SUSY breaking sector is concerned, the linear term in the K¨ahler potential(13) does not modify the scalar potential significantly. In the very early universe, however,such a linear term makes the minimum of the scalar potential to deviate from the origin.Therefore, it is crucial for cosmological evolution of S to take into account the linear termin the K¨ahler potential, as we will show in the next section.5 II. COSMOLOGY
Now we consider the cosmological evolution of the SUSY breaking field, S . First letus give a sketch how S is deviated from the origin due to the linear term in the K¨ahlerpotential, and estimate the cosmic abundance. While the F -term of the inflaton dominatesthe universe, the scalar potential of S is approximately given by a V ( S ) ≃ e K (3 H M P ) , ≃ H (cid:18) | S | − κ π M ( S + S † ) + · · · (cid:19) , (16)where we have assumed that S does not couple to the inflaton in the K¨ahler potential forsimplicity b . The scalar potential has a minimum given by S c = 5 κ π M. (17)If this minimum S c exceeds Λ, there is no stable minimum in the low-energy theory in theearly universe. Depending on the UV theory, the system may settle down at the SUSYminimum in this case. To avoid such a situation, we impose S c < Λ in the following.When the Hubble parameter becomes comparable to the mass of S , it starts to oscillatearound the minimum, S ≃
0, with an initial amplitude S c . The abundance of S is estimatedas n S s ≃ T R (cid:18) κ π (cid:19) m S M m S M P , ≃ × − κ (cid:18) α . (cid:19) (cid:18) m (cid:19) − (cid:18) m / (cid:19) (cid:18) T R GeV (cid:19) (cid:18)
Λ10 GeV (cid:19) , (18)where n S is the number density of S , s is the entropy density, T R denotes the reheatingtemperature, and we have used (8) to eliminate M . We have assumed here that the reheatinghas not completed when H = m S . This assumption is indeed reasonable, since otherwisetoo many gravitinos are produced by thermal scatterings in plasma.Several comments are in order. First, we have assumed that the Hubble parameterduring inflation, H I , is larger than the mass of S , i.e., H I > m S . If m S is larger than a The linear term of S generically appear in the scalar potential due to the supergrav-ity effects [7]. One can neglect its effect on the dynamics of S , as long as κ > ∼ .
04 ( α / . / (1TeV /m ) / ( m / / / (Λ / GeV). We assume that this inequality is satisfiedin the following analysis. b Even in the presence of the interactions, the following argument does not change qualitatively. I , the deviation of S is suppressed by ( H I /m S ) . Considering the upper-bound on m S given by (10), however, the assumption H I > m S is valid except for low-scale inflationmodels. Second, although our model is given only below the scale Λ, this does not limit theapplication of the above arguments only to the inflation models with H I < Λ. For H I < Λ,the above scalar potential (16) is obviously valid both during and after inflation. On theother hand, for H I > Λ, one cannot use (16) during inflation. After inflation, however, theHubble parameter decreases and becomes smaller than Λ at certain point. If the systemcan be then described by the low energy effective theory, S will quickly settle down at thepotential minimum (17) c .Next let us consider the decay processes of S . The SUSY breaking field S will decay intoa pair of the gravitinos, and the decay rate is [8, 9]Γ / ≃ π m S m / M P ≃ π m S Λ , (19)where we have used (5). If m S is smaller than 2 M , S decays into the gauge sector throughthe messenger loop d . The relevant interactions are L ≃ − α i π κM S (cid:20) − F ( i ) µν F ( i ) µν + i ǫ µνρσ F ( i ) µν F ( i ) ρσ − κF S M ¯ λ ( i ) P L λ ( i ) (cid:21) + h . c ., (20)where we neglected terms with higher orders of κ h S i /M . In particular, S decays into thegluons and gluinos, and the decay rates areΓ g ≃ α κ π m S M (21)and [7] Γ ˜ g ≃ κ π m m S M , (22)respectively. Note that the decay rate into the gluinos is smaller than that into the gluons,if m S is much larger than the gluino mass [10], m S ≫ πα m , (23) c Depending on the details of the UV theory, it is possible that the position of S is larger than Λ and thesystem cannot be described by the model given by (1) and (2) even for H <
Λ. Then, the abundance of S will generically become larger than (18), and so, our estimate is conservative. d The decay into the scalars does not change our argument significantly. B / ≃
11 + r (24)with r ≡ m Λ m / M P ! + 329 (cid:18) πα (cid:19) m Λ m / M P ! . (25)Therefore, the SUSY breaking field will dominantly decay into the gravitinos if r < ∼
1, orequivalently, Λ < ∼ × GeV (cid:18) α . (cid:19) (cid:18) m (cid:19) − (cid:18) m / (cid:19) . (26)We will assume this inequality is met in the following analysis. The decay temperature of S is given by T d ≡ π g ∗ ! − q Γ / M P , ≃ (cid:18) g ∗ (cid:19) − (cid:18) m / (cid:19) (cid:18) Λ10 GeV (cid:19) − , (27)where g ∗ counts the relativistic degrees of freedom at the decay.Now we can estimate the gravitino abundance. Since S dominantly decays into a pair ofthe gravitinos, the gravitino abundance is given by Y / ≃ × − κ (cid:18) α . (cid:19) (cid:18) m (cid:19) − (cid:18) m / (cid:19) (cid:18) T R GeV (cid:19) (cid:18)
Λ10 GeV (cid:19) . (28)The density parameter of the gravitino isΩ / h ≃ . κ (cid:18) α . (cid:19) (cid:18) m (cid:19) − (cid:18) m / (cid:19) (cid:18) T R GeV (cid:19) (cid:18)
Λ10 GeV (cid:19) . (29)where h is the present Hubble parameter in units of 100 km/s/Mpc. Therefore, a rightamount of the gravitinos can be produced by the decay of the SUSY breaking field, S . Notethat, for the non-thermally produced gravitinos to be a dominant component of DM, κ should not be suppressed. The reason is as follows. For a small κ , the messenger mass isalso small to keep the size of the soft masses in the SSM sector (see (8)). Since the shift ofthe S field is proportional to the breaking of U (1) R symmetry, i.e., the messenger mass, thegravitino abundance is suppressed for a small value of κ . Furthermore, depending on thereheating temperature and the mass spectrum of the SSM particles, the thermal productionof the gravitinos and the NLSP decay may also give sizable contributions [11, 12, 13, 14, 15].8ote also that the relatively high reheating temperature is favored, which may accommodatethe thermal leptogenesis scenario [17].Finally, let us estimate the free streaming length of the gravitinos produced by the decayof S . The comoving free streaming length λ F S at matter-radiation equality is defined by λ F S ≡ Z t eq t D v / ( t ) a ( t ) dt, (30)where a ( t ) is the scale factor, and t D and t eq ( ∼ × sec) denote the time at the S decayand at matter-radiation equality, respectively. v / is the velocity of the gravitino, given by v / ( t ) = | p / | E / ≃ m S (cid:16) a D a ( t ) (cid:17)r m / + m S (cid:16) a D a ( t ) (cid:17) , (31)where we have approximated m S ≫ m / , and a D is the scale factor at the decay of S .Integrating (30) yields λ F S ≃ H √ z eq X − sinh − X, ∼ (cid:18) g ∗ (cid:19) (cid:18) m / (cid:19) − (cid:18) Λ10 GeV (cid:19) (32)with X ≡ m / m S a eq a D , ∼ (cid:18) g ∗ (cid:19) − (cid:18) m / (cid:19) (cid:18) Λ10 GeV (cid:19) − , (33)where H is the Hubble parameter at present, and z eq and a eq are the red-shift and thescale factor at the matter-radiation equality. In the second equation of (32), we have used H − ∼ × Mpc and z eq ∼ λ F S is expressed interms of the gravitino mass and the scale Λ, and it can be as small as 1 kpc. Interestingly, therecent observations on the dSph galaxies seem to exhibit a sharp cut-off around 100 pc [16]in the smallest size of the galaxies, which may be explained by DM with free streaminglength of O (100 pc) e . Our scenario may be supported by further observations in the nearfuture. e It is also possible to produce a right amount of the gravitino DM from the inflaton decay [18], and thegravitino can have right free streaming length to explain the cut-off in the smallest size of the galaxies. V. CONCLUSIONS AND DISCUSSION
The metastable SUSY breaking vacua provide a drastically simplified scheme of the gaugemediation. The models possess an accidental U (1) R symmetry, which is broken in themessenger sector. The breaking induces a linear term of the SUSY breaking field in theK¨ahler potential. We have pointed out that the SUSY breaking field is forced away fromits minimum due to this linear term while the inflaton field dominates the energy of theuniverse. Then the gravitinos are produced when it decays, and we have shown that a rightabundance of the gravitino DM can be realized for certain parameters. Further, the freestreaming length of the gravitino may explain the recent observations on the smallest sizeof the dSph galaxies. Acknowledgment
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