Gravitino Production Suppressed by Dynamics of Sgoldstino
Raymond T. Co Keisuke Harigaya
hep-ph





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    diag Spin SOØ O SU U Sp SL trM_ PlInt. J. Mod. Phys. Mod. Phys. Lett. Nucl. Phys. Phys. Lett. Phys. Rev. Phys. Rev. Lett. Prog. Theor. Phys. Z. Phys. df BHinf inf evap eq smM_ Pl GeV
    
    Department of Physics, University of California, Berkeley, California 94720, USATheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
    In supersymmetric theories, the gravitino is abundantly produced in the early Universe from thermal scattering, resulting in a strong upper bound on the reheat temperature after inflation.
    We point out that the gravitino problem may be absent or very mild due to the early dynamics of a supersymmetry breaking field, i.e. a sgoldstino. 
    In models of low scale mediation, the field value of the sgoldstino determines the mediation scale
    and is in general different in the early Universe from the present one.
    A large initial field value since the era of the inflationary reheating suppresses the gravitino production significantly.
    We investigate in detail the cosmological evolution of the sgoldstino and show that the reheat temperature may be much higher than the conventional upper bound,
    restoring the compatibility with thermal leptogenesis.
    
    Gravitino Production Suppressed by Dynamics of Sgoldstino
    
        Keisuke Harigaya
        December 30, 2023
    ===========================================================
    
    
    
    
    § INTRODUCTION
    
    
    One of the most challenging puzzles in the standard model is the hierarchy problem, in which the Higgs mass is unstable against quantum corrections at high energy scales. As one of the most motivated solutions, supersymmetry (SUSY) ensures the cancellation of quantum corrections between the SM particles and their superpartners, which considerably relaxes the hierarchy problem <cit.>.
    On the other hand, gauge coupling unification at a high energy scale gives strong hints to the Grand Unified Theories (GUTs). Remarkably, supersymmetric GUTs do not suffer from the proton decay problem faced by the standard model GUTs, and further improve the precision of gauge coupling unification <cit.>. 
    
    Despite all the successes in particle physics, supersymmetry is known to create cosmological difficulties. In the case of low scale mediation of supersymmetry breaking such as gauge mediation, the gravitino is much lighter than the weak scale and is often the lightest supersymmetric particle. The gravitino is abundantly produced from the scattering of the thermalized particles in the early Universe <cit.>.
    In order not to overproduce gravitino dark matter,
    the reheat temperature after inflation T_R must be sufficiently low, T_R 10^6  GeV(m_3/2/ GeV) <cit.>, where m_3/2 is the gravitino mass,
    and this bound strongly restricts the cosmological history including inflation models and baryogenesis. Especially, T_ R < 10^9 GeV is in conflict with thermal leptogenesis <cit.>. This is known as the gravitino problem in low scale mediation of supersymmetry breaking.
     
    
    Several solutions have been considered so far.
    One may assume a non-conventional cosmology model with a large amount of dilution from the decay of a long-lived particle <cit.>. The large entropy production needed to reproduce the observed dark matter abundance also dilutes away the baryon asymmetry created previously,
    which calls for an efficient mechanism of baryogenesis. For example, the observed baryon asymmetry can be explained by thermal leptogenesis only if the reheat temperature is extremely high, T_R 10^16  GeV (m_3/2 /  GeV).
    Refs. <cit.> introduce a low messenger scale and a small coupling between the goldstino component of the gravitino and the messenger.
    The gravitino production is then suppressed at a temperature higher than the messenger scale.
    The suppressed production helps reduce the dilution factor needed  and thus relaxes the stringent lower bound on the reheat temperature from thermal leptogenesis.
    A different solution in Ref. <cit.> involves an additional field whose field value determines the coupling between the messenger and the goldstino.
    By a smaller field value and thus a smaller coupling in the early Universe, the upper bound on the reheat temperature is relaxed.
    
    The interaction rate between the thermal bath and the gravitino is suppressed by the mediation scale, which is given by the field value of the scalar component of the SUSY breaking field (sgoldstino).
    We point out that if the sgoldstino potential is flat enough, the field value may be large in the early Universe, suppressing the gravitino production.
    We study the dynamics of the sgoldstino including thermal effects, and find that the reheat temperature may be much higher than the conventional upper bound.
    The compatibility of our scenario with thermal leptogenesis is also investigated.
    We emphasize that this suppression mechanism is a result of a thorough analysis of the dynamics of the existing fields necessary for low scale mediation, and can be applicable to a broad class of models with a sufficiently flat sgoldstino potential.
    
    
    
    
    
    
    § REVIEW OF THE GRAVITINO PROBLEM IN GAUGE MEDIATION
    
    
    We first review gauge mediation and the production of gravitinos from the thermal bath. The SUSY breaking field S is coupled to the messenger field Q and Q̅ via the superpotential term
    
        W = y S Q Q̅,
    
    which in turn generates the following term in the Lagrangian when Q is integrated out
    
        ℒ = ∑_i ∫ dθ^2 α_i/4πS/v_S W_i^α W_i  α,
    
    where i is summed over (U(1), SU(2), SU(3)) and v_S is the vev of the scalar component of S.
    Here we assume that Q and Q̅ form a complete multiplet of SU(5) GUT.
    We parametrize the F term of S as
    
        F_S = k √(3) m_3/2 M_Pl  ,
    
    where k ≤ 1 parametrizes the fractional contribution to SUSY breaking and M_Pl = 2.4 × 10^18 GeV is the reduced Planck mass. The gaugino mass is then given by
    
        m_i = α_i/4πF_S/v_S = √(3)α_i/4πk m_3/2 M_Pl/v_S .
    
    
    The viable parameter space is as follows. To prevent Q from being tachyonic, we require 
    
        y ≥( 4π/α_i)^2 m_i^2/k √(3) m_3/2 M_Pl,
    
    while to ensure that the quantum corrections to the S mass do not exceed its vacuum mass
    
        Δ m_S^2 = y^2/16π^2k^2 m_3/2^2 M_Pl^2/v_S^2 < m_S^2,
    
    we impose the condition that 
    
        y ≤α_i m_S/m_i  .
    
    The consistency between Eqs. (<ref>) and (<ref>) holds only if 
    
        k > 16π^2/α_i^3m_i^3/√(3) m_3/2 M_Pl m_S .
    
    In the class of models where the low energy effective superpotential of S is given by W ≃√(3) k m_3/2 M_PlS,
    [This is not the case, for example, in a model of indirect gauge mediation with a superpotential W = λ S ϕ_1 ϕ_2 and the fields ϕ_1 and ϕ_2 obtain negative soft masses by a coupling with a SUSY breaking sector. The masses of ϕ_1 and ϕ_2 are as large as that of S, and we may not integrate them out.]
    the supergravity effect generates a tadpole term of S, V(S) = - √(3) k m_3/2^2 M_Pl S.
    The tadpole term places a minimum on the vev today, v_S, unless the vev is fine tuned. This translates into a lower bound on the S mass
    
        m_S  10 ( m_3 m_3/2)^1.011/2≃ 300  ( m_3/ TeV)^1.011/2( m_3/2/)^1.011/2.
    
    
    
    
    The functional form of the gravitino abundance produced  at a temperature T is derived as follows;
    
        ρ_3/2/s≃m_3/2 n_i^2 σ_i v/H s≃m_3/2 k^2 M_Pl/v_S^2 T ≃(4π/α_i)^2 m_i^2/3 m_3/2M_Pl T ,
    
    where ρ and s are the energy and entropy density respectively, while σ_i refers to the scattering cross section between the gravitino and the gaugino/gauge boson, which follows the thermal equilibrium number density n_i.
    Here we assume that the temperature is sufficiently small so that the gravitino is not thermalized.
    As can be seen in Eq. (<ref>), the production mode by thermal scattering is dominated at higher temperature, which we call “UV dominated," and peaked at the reheat temperature after inflation T_R.
    The precise result of Eq. (<ref>) is derived e.g. in Refs. <cit.>, which translates into the constraint on T_R
        T_R ≤ 5 × 10^6  ( m_3/2/) (  TeV/m_3)^2 ≡ T_ co.
    
    For m_3/2 1 MeV the upper bound is smaller than the typical gaugino mass, invalidating Eq. (<ref>).
    
    The spin-3/2 component of the gravitino is also produced from the thermal bath via Planck-scale suppressed interactions.
    Using the result in Ref. <cit.>, we obtain an upper bound on T_R,
    
    
        T_R ≤ 2× 10^12  GeV(  GeV/m_3/2).
    
    Although the constraint is much weaker than the one in Eq. (<ref>), it will be important in our mechanism where the production of the spin-1/2 component is suppressed.
    
    
    
    
    § SGOLDSTINO DYNAMICS AS A SOLUTION
    
    
    We propose a new cosmological scenario of gauge mediation where the gravitino problem is much milder. In Eq. (<ref>), it is assumed that v_S has been a constant from the inflationary reheating until today. This is, however, not necessarily the case.
    In this section, we explore the possibility that the field value v_S(T) of the sgoldstino evolves with the temperature according to its potential energy V(S). In particular, we consider the case where the initial field value of the sgoldstino, v_S0, is much larger than today's vev v_S. Based on Eq. (<ref>), a large initial field value results in the suppression of the gravitino interaction with the thermal bath in the early Universe. 
    We refer readers to Ref. <cit.> and the references therein for discussions on the evolution of a scalar field in the early Universe including thermal effects.
    
    We can parametrize the temperature dependence of the sgoldstino oscillation amplitude as v_S(T) ∝ T^n. It is striking that the gravitino production from thermal scattering given in Eq. (<ref>) is dominated at a lower temperature, which we call “IR dominated," for any n>1/2, which is easily satisfied by the typical polynomial and logarithmic potentials. As a result, the gravitino production is insensitive to the reheat temperature.
    In the case with no dilution from entropy production, the conventional constraint on the reheat temperature, T_ co, can be evaded as long as the combination T/v_S^2(T) in Eq. (<ref>) never exceeds T_ co/v_S^2 for any T. In general, the constraint with dilution is
    
        max(T/v_S^2(T)) 1/D≤T_ co/v_S^2 ,
    
    where max(f(T)) refers to the maximum value of f(T) throughout the cosmological evolution.
    This is more likely the case for quadratic and logarithmic potentials because steeper potentials lead to a smaller initial field value of S as well as an earlier onset of the oscillation.
    
    
    
    
    
    
     §.§ Evolution of the Sgoldstino Field
    
    
    
    We first consider the case where the sgoldstino field begins to oscillate via thermal effects.
    Through the coupling with S in Eq. (<ref>), Q obtains a large mass from the large field value of the sgoldstino and further generates the thermal logarithmic potential for S
        V_ th(S) = a_0  α_3(T)^2 T^4 ln(y^2 S^2/T^2) ,
    
    where a_0 is a constant of order unity <cit.> and the logarithmic temperature dependence of α_3(T) will be neglected for simplicity. Here it is assumed that the messenger mass is larger than the temperature and we verify that this is true in the entire allowed parameter space.
    
    
    The condition for the onset of the oscillations during inflationary reheating is given by V_ th” (v_S0)  H^2, which leads to
    
        α_3 T^2/v_S0√(π^2 g_*/90)T^4/T_R^2 M_Pl,
    
    where g_* is the effective number of relativistic species.
    [Here it is assumed that the radiation produced by the decay of the inflaton is thermalized and follows thermodynamics. See <cit.> and the references therein for discussion on the thermalization process.]
    The oscillation temperature reads
    
        T_ osc≃ T_R ( 90/π^2 g_*)^1.011/4√(α_3 M_Pl/v_S0) .
    
    We define  
    
        δ≡(π^2 g_*/90)^1.013/8v_S0/α_3 M_Pl
    
    to parametrize the initial field value and this particular definition of δ simplifies the numerical pre-factors in the following derivations.
    Here it is implicitly assumed that
    
    
        v_S0α_3 M_Pl√(90/π^2g_*)        and         T_R > √(m_S v_S0/α_3),
    
    so that the sgoldstino begins its oscillation by the thermal logarithmic potential. If 
    one of these conditions is violated, the sgoldstino begins its oscillation via its temperature independent potential.
    The evolution of the sgoldstino for that case is discussed later.
    
    
    The amplitude of the oscillation, v_S(T), evolves as follows. The mass of S is given by α_3 T^2 / v_S(T). Then the number density of S is proportional to T^2 v_S(T), which decreases with a^-3.
    During the inflaton dominated era and the radiation dominated era a^-3∝ T^8 and T^3, and hence v_S(T) ∝ T^6 and T, respectively.
    The field value of the sgoldstino at the reheat temperature is then given by
    
        v_S(T_R) = v_S0( T_R/T_ osc)^6 ≃δ^4 α_3 M_Pl  .
    
    After reheating, the field value evolves as
    
        v_S(T) = v_S(T_R) T/T_R≃δ^4 α_3 M_PlT/T_R.
    
    
    In the above analysis, we assume that reheating is caused by a perturbative decay of the inflaton. It is also possible that the reheating is caused by other dynamics such as the scattering with the thermal bath. In this case the relation between the initial field value of the sgoldstino and the field value at T_R is different from Eq. (<ref>). It is also possible that a large Hubble induced mass term of the sgoldstino causes non-trivial dynamics of the sgoldstino before the completion of reheating. For those cases, one may still use δ^4 to parametrize v_S(T_R) without changing the discussion below.
    
    
    As the temperature drops, the thermal logarithmic potential in Eq. (<ref>) becomes less effective and eventually becomes subdominant to the vacuum potential.
    To be concrete, we assume that the vacuum potential is given by a simple quadratic one,  
    
        V_ vac(S) = m_S^2 |S- v_S|^2  (≃ m_S^2 |S|^2  for v_S(T) ≫ v_S).
    
    The transition to the quadratic potential occurs at the temperature T_2 defined by V_ th(v_S(T_2)) = V_ vac(v_S(T_2)),
    
        T_2 ≃δ^4 m_S M_Pl/T_R .
    
    Note that T_2 < T_R as long as the conditions in Eq. (<ref>) are satisfied.
    We now quantify v_S(T_2) in relation to v_S. This will tell us whether the gravitino production actually becomes enhanced instead by v_S(T) < v_S because the sgoldstino oscillates around the minimum at the origin set by V_ th(v_S(T)). 
    
        v_S(T_2)/v_S =     v_S(T_R)/v_ST_2/T_R = δ^8 4π/√(3)M_Pl m_S m_3/k T_R^2 m_3/2
        ≃     5 δ^8/k( 10^12 /T_R)^2 (m_S/300  ) ( m_3/ TeV)^1.011/2( /m_3/2).
    
    When this ratio is larger than unity, which is the case for the most of the allowed parameter space, before v_S(T) drops to v_S, S starts to follow V_ vac(S) and oscillates around the minimum today v_S. After T_2, v_S(T) continues to decrease as T^3/2 until the temperature T_S, at which the amplitude is as large as the vev v_S. Using v_S(T_S) = v_S, one obtains
    
        T_S = T_2 ( v_S/v_S(T_2))^1.012/3≃ 2 × 10^8  ( k/δ^2)^1.012/3( T_R/10^12 )^1.011/3(m_S/300  )^1.011/3(  TeV/m_3)^1.012/3( m_3/2/)^1.012/3 .
    
    When the ratio is smaller than unity, v_S (T) drops below v_S. After T_2, S follows V_ vac(S) and oscillates around the minimum today v_S. After the few oscillations by V_ vac(S), v_S(T) increases and quickly becomes as large as v_S.
    
    When the initial field value of the sgoldstino is large or the reheat temperature is small, the sgoldstino begins its oscillation by the quadratic potential V_ vac(S), rather than the thermal potential V_ th(S). This occurs if the condition in Eq. (<ref>) is violated, namely,
    
        v_S0 > α_3 M_Pl√(90/π^2g_*)        or         T_R <  √(m_S v_S0/α_3).
    
    The oscillation temperature becomes independent of the initial amplitude and reads
    
        T_ osc = ( 90/π^2 g_*)^1.011/8( m_S M_Pl T_R^2 )^1.011/4,
    
    where T_ osc > T_R is assumed.
    The field value of the sgoldstino at the reheat temperature is given by
    
        v_S(T_R) = δ( π^2 g_*/90)^1.011/8α_3 T_R^2/m_S  .
    
    After reheating, the field value evolves as
    
        v_S(T) = v_S(T_R) (T/T_R)^1.013/2  =  δ( π^2 g_*/90)^1.011/8α_3 T^3/2 T_R^1/2/m_S
    
    and reaches v_S at the temperature
    
        T_S = T_R ( v_S/v_S(T_R))^1.012/3≃ 8 × 10^7  ( k/δ)^1.012/3( 10^10 /T_R)^1.011/3(m_S/300  )^1.012/3(  TeV/m_3)^1.012/3( m_3/2/)^1.012/3 .
    
    
    The sgoldstino eventually delivers all its energy to radiation by scattering with (decaying to) thermal particles at the destruction (decay) temperature T_ des (T_ dec), which is discussed in Sec. <ref> (<ref>). 
    
    
    
    
    Fig. <ref> summarizes the sgoldstino evolution and serves as the schematic picture of the suppression mechanism.
    The vertical axis shows the gravitino abundance produced per Hubble time at a given temperature in the horizontal axis.
    The conventional production is UV dominated and thus we need T_R ≤ T_ co to avoid overproduction. In the case where v_S(T) is decreasing faster than T^1/2, the suppressed production becomes IR dominated, as seen in high temperature behavior.
    The kinks of the colored lines occur at T_2, as the temperature dependence of v_S(T) changes from Eq. (<ref>) to Eq. (<ref>) at the transition from the thermal logarithmic potential to the quadratic one. The abrupt change of the lines at T_ des orginates from the fact that the field value of the sgoldstino is suddenly set to the vev today when the condensate is destroyed by thermal scattering.
    In the case of the orange line, the observed dark matter abundance is reproduced. On the other hand, the blue line represents underproduction.
    
    
    
    
    
     §.§ Destruction of Sgoldstinos by Thermal Scattering
    
    
    The discussion in Sec. <ref> assumes that the sgoldstino condensate is intact throughout its evolution. However, due to its coupling with the messenger Q, the sgoldstino scatters with thermalized particles at the following rate given in Refs. <cit.>
        Γ_ scatt = ( T(Q)/16π^2)^2 (12π)^2/ln(α_3^-1)α_3^2 T^3/v_S^2(T)≡ b α_3^2 T^3/v_S^2(T) ,
    
    where T(Q) is the index of Q's representation of SU(3) and we take T(Q) = 1/2. The condensate is destroyed whenever the scattering rate becomes larger than the Hubble rate. The temperature at which such destruction occurs is called T_ des.
    
    
    
    
      §.§.§ Sgoldstino Oscillations Driven by Thermal Effects
    
    
    
    We first explore the case where the sgoldstino begins to oscillate via the thermal logarithmic potential.
    Overproduction of gravitinos excludes the possibility where the sgoldstino condensate is destroyed before the quadratic potential dominates, i.e. T_ des > T_2. This is because for such a case the field S is trapped at the origin, making Q massless and thermalized and greatly enhancing the gravitino production rate.
    [If the Yukawa coupling y is sufficiently small, the produced gravitino abundance is not necessarily very large and thus the entropy production by the sgoldstino trapped at the origin may have sufficient dilution for the gravitino abundance <cit.>. We however do not consider this scenario in this paper.]
    Requiring Γ_ scatt(T_2) < H(T_2) gives
    
        T_R   10^14  δ^4 (m_S/300  )^1.011/3 .
    
    As the condensate is destroyed, the sgoldstino is driven to the local minimum of the potential. In order for S=v_S to be the local minimum at the temperature T_ des, the thermal mass from the messenger should be small enough, 
        y   T_ des<m_S .
    
    This upper bound on y should be consistent with the lower bound in Eq. (<ref>).
    
    
    Below T_2, on the other hand, Γ_ scatt(T)/H is IR dominated only before T_S. This implies either that T_ des > T_S or that there is no destruction by scattering. In order to distinguish our mechanism from other solutions of the gravitino problem, we first explore the parameter space where the sgoldstino condensate does not produce entropy.
    We take v_S(T) = v_S (T/T_S)^3/2 because T_2 > T_ des > T_S and derive the destruction temperature
    
        T_ des≃  6 × 10^5   δ^-2( T_R/10^12 )^1.011/2(m_S/300  )^1.011/2 .
    
    To be consistent with T_ des > T_S so that the sgoldstino is successfully destroyed, one requires
    
        k  10^-4 δ^-1( T_R/10^12 )^1.011/4(m_S/300  )^1.011/4( m_3/ TeV)  ( /m_3/2) .
    
    In the case where k < 1, there should be another SUSY breaking field. If the scalar component of that SUSY breaking field is excited in the early Universe, its decay may also produce gravitinos. To avoid cosmological complications, we assume that this scalar component has a positive Hubble-induced mass and/or efficiently decays into hidden sector fields other than the gravitino.
    
    According to Eq. (<ref>), it is required that T_ des≤ T_ co to avoid overproduction of gravitinos. This  condition is satisfied when
    
        T_R   10^13  δ^4 (300  /m_S) ( m_3/2/)^2 (  TeV/m_3)^4.
    
    Furthermore, to identify the parameter space with no dilution, we need to ensure that the sgoldstino condensate is destroyed before its energy density dominates over radiation. We can estimate the temperature T_M^ (th) at which the matter energy density dominates over that of radiation
    
        π^2/30 g_* (T_M^ (th))^4 = m_S^2 v_S^2(T_2) ( T_M^ (th)/T_2)^3
    
    and the result reads
    
        T_M^ (th) = 30/π^2α_3^2/g_* T_2 = δ^4  30/π^2α_3^2/g_*m_S M_Pl/T_R≃  10^5   δ^4 ( 10^12 /T_R) (m_S/300  ) ,
    
    where we use α_3^2 T_2^4 = m_S^2 v_S^2(T_2) (the definition of T_2). No entropy is produced when T_ des > T_M^ (th), which is the case for
    
        T_R  10^11  δ^4 (m_S/300  )^1.011/3  .
    
    
    If the scattering is inefficient, the sgoldstino dominates the energy density of the Universe.
    After the sgoldstino dominates, destruction occurs via scattering with the thermal bath created from the condensation of the sgoldstino. The destruction temperature is derived in App. <ref> and reads
    
    
        T_ des≃ 3 × 10^5  ( m_S/300  )^1.012/3.
    
    The dilution factor D via entropy production is given by
    
        D = T_M^ (th)/T_ des≃ 3  δ^4( m_S/300  )^1.011/3( 10^11 /T_R).
    
    The condition from Eq. (<ref>) is T_ des /D ≤ T_ co so the upper bound on T_R is relaxed to
    
        T_R  10^12  δ^4 ( m_3/2/) ( 300  /m_S)^1.011/3(  TeV/m_3)^2.
    
    
    
      §.§.§ Sgoldstino Oscillations Driven by Vacuum Potential
    
    
    We next explore the case where the sgoldstino begins to oscillate via the vacuum mass term.
    In the case where the condensate is a subdominant component, the destruction temperature is given by
    
        T_ des = 3^3/4 10^3/8√(b)/π ^3/4δ g_*^3/8m_S √(M_Pl)/√(T_R)≃ 3 × 10^5  δ^-1( m_S/300  ) ( 10^9  /T_R)^1.011/2 .
    
    One needs to require T_ des > T_S to ensure successful destruction, which limits 
    
        k  10^-4 δ^-1/2( m_S/300  )^1.011/2( m_3/ TeV) ( /m_3/2) ( 10^9  /T_R)^1.011/4.
    
    When the scattering rate is inefficient, the Universe enters the matter-dominated era at the temperature 
    
        T_M^ (vac) = v_S0^2 T_R/3 M_Pl^2 = ( 90/π^2g_*)^1.013/4δ^2 α_3^2 T_R/3 .
    
    With the destruction temperature given in Eq. (<ref>), the dilution factor can be computed
    
        D = T_M^ (vac)/T_ des =  10^1/4α _3^4/3δ^2 / 3^1/6√(π) g_*^1/4  b^1/3T_R/( M_Pl m_S^2 )^1/3≃ 9  δ^2   ( T_R/10^10 ) ( 300  /m_S)^1.012/3 .
    
    Therefore, the sgoldstino does not produce entropy when 
    
        T_R  10^9  δ^-2( m_S/300  )^1.012/3 .
     
    The condition from Eq. (<ref>) is T_ des /D ≤ T_ co and places the following upper bound on T_R
        T_R  2 × 10^10  (0.6/δ)^2 ( 200   MeV/m_3/2) ( m_S/ TeV)^1.014/3( m_3/2   TeV)^2.
    
    
    
    Let us discuss the compatibility with thermal leptogenesis. The maximal baryon asymmetry Y_B,  max that can be obtained from thermal leptogenesis in the units of that observed today Y_B,  obs is given in Refs. <cit.>
        Y_B,  max≃T_R/10^9   Y_B,  obs.
    
    This baryon asymmetry may be subject to dilution from subsequent entropy production, which leads to a more stringent lower bound on T_R,
    
        T_R/D(T_R)  10^9  .
    
    
    
    There is a further constraint from the production of the fermion component of S, ψ_S for the following reason. Since we are currently concerned with the case where the sgoldstino condensate is destroyed by thermal scattering, a small k given in Eq. (<ref>) is assumed.
    For a small k, the production of ψ_S is enhanced by 1/k^2 compared to that of the gravitino.
    To avoid the gravitino overproduction from the decay of ψ_S, we require that the mass of ψ_S is larger than that of the lightest observable supersymmetric partner (LOSP) so that ψ_S can decay into the LOSP.
    [This requires a direct coupling between the SUSY breaking sector containing S and other SUSY breaking sector. Otherwise, ψ_S is a pseudo-goldstino and obtains a mass only as large as the gravitino.]
    We find that the LOSP from the decay of ψ_S immediately annihilate to the SM particles with a negligible amount decaying to the gravitino. 
    The mass of the sgoldstino would not be much smaller than that of ψ_S, and thus we fix m_S = 1 TeV. 
    
    We summarize the above discussions in Fig. <ref>. The light gray region is excluded by Eq. (<ref>) because the sgoldstino is destroyed by thermal scattering when the potential is still governed by the thermal logarithmic potential, whose minimum is at the origin of S. This results in a vanishing messenger scale and the gravitino is overproduced by the scattering of thermalized messengers. The orange region is excluded by Eqs. (<ref>) and (<ref>) because the sgoldstino destruction occurs too early and the mechanism fails to suppress the messenger scale.
    The dilution factor D is labeled by the black contours in the right panel, whereas the left panel does not have dilution for the chosen value of δ. The maximal value δ_ max≃ 0.67 is inferred from Eqs. (<ref>) and (<ref>). Above (below) the sharp kink of the orange boundary, the onset of the sgoldstino oscillation is driven by the thermal logarithmic (vacuum) potential, discussed in Secs. <ref> and <ref> respectively. The yellow region is excluded by Eq. (<ref>) because, at the time of sgoldstino destruction, the thermal mass dominates and the destruction will set the field value to the origin, resulting in the gravitino overproduction. Here k_c refers to the critical value in Eq. (<ref>) for successful destruction. The purple region is excluded by Eq. (<ref>) for overproduction of spin-3/2 gravitinos through supergravity interactions.
    We find that in the allowed parameter region the thermal leptogenesis can create an enough amount of the lepton asymmetry.
    
    
    
    
    
     §.§ Destruction of Sgoldstinos by Decay
    
    
    It is pointed out in Sec. <ref> that k has to be smaller than the critical value k_c given in Eqs. (<ref>), (<ref>) or (<ref>) in order for the sgoldstino condensate to be destroyed by thermal scattering. In this section, we assume a sufficiently large k, meaning that thermal scattering is never effective enough and instead the sgoldstino condensate eventually decays to particles in the thermal bath. 
    
    The real and imaginary parts of S may have different decay modes <cit.>. 
    In the phase convention where v_S is real,
    both the real and imaginary components of S can decay to a pair of gluons at a rate
    
        Γ_ dec^gg≃( α_3/4π)^2 m_S^3/8π v_S^2.
    
    The real component of S can also decay to Higgs/electroweak (EW) gauge bosons if kinematically allowed at a rate
    
        Γ_ dec^ h,W,Z≃1/8πm_H^4/m_S v_S^2 .
    
    Assuming m_S∼ TeV, this decay mode is more efficient than the one into gluons.
    The relative abundance of the real and imaginary parts depends on the phase of the initial field value v_S0. As the decay to Higgs is more efficient, the real component will decay before the imaginary one. As a result, the final decay temperature is mainly governed by the decay to gluons if the initial relative abundance is comparable or dominated by the imaginary component.
    
    To find the temperature T_ dec when the sgoldstino decays to gluons, one equates the decay rate with the Hubble rate and obtains
    
        T_ dec^gg≃√(Γ_ dec^gg M_Pl)≃ 
        4   MeV  k^-1( m_S/300  )^1.013/2(  100   MeV/m_3/2) ( m_3/ TeV) .
    
    The decay temperature then allows us to compute the dilution factor using Eq. (<ref>)
    
        D = T_M^ (th)/T_ dec^gg≃ 
        8 × 10^6   k  δ^4 ( 10^12 /T_R) ( 300  /m_S)^1.011/2( m_3/2/ 100   MeV) (  TeV/m_3) .
    
    Since the decay occurs well after the field value of S settles to the minimum today v_S, the gravitino production peaks at T_S. With dilution, the constraint from the gravitino abundance using Eq. (<ref>) becomes T_S /D ≤ T_ co and gives
    
        T_R  
        5 × 10^15   k^1/4 δ^4 ( 300  /m_S)^1.015/8( m_3/2/ 100   MeV)  (  TeV/m_3)^1.017/4 .
    
    
    
    With accidental suppression of the imaginary part of the initial field value or the presence of a CP violating mixing between the real and imaginary components of S,
    the decay temperature T_ dec is now determined by the larger of the rate into gluons and that into EW bosons. For the decays to H, W^±, and Z, we obtain
    
        T_ dec^ h,W,Z≃√(Γ_ dec^ h,W,Z M_Pl)≃ 
        5    k^-1( m_H/ TeV)^2 ( 300  /m_S)^1.011/2(  100   MeV/m_3/2) ( m_3/ TeV).
    
     The dilution factor becomes
    
        D = T_M^ (th)/T_ dec^ h,W,Z≃ 
        6 × 10^3   k  δ^4 ( 10^12 /T_R) (  TeV/m_H)^2 ( m_S/300  )^1.013/2( m_3/2/ 100   MeV) (  TeV/m_3) .
    
    Finally, the constraint of the gravitino abundance from Eq. (<ref>) requires T_S /D ≤ T_ co, giving
    
        T_R  
        3 × 10^13   k^1/4 δ^4 (  TeV/m_H)^1.013/2( m_S/300  )^1.017/8( m_3/2/ 100   MeV)  (  TeV/m_3)^1.017/4.
    
    In the case where the condition in Eq. (<ref>) is violated, the sgoldstino starts oscillating via the vacuum potential V_ vac(S) rather than the thermal potential V_ th(S). The decay temperatures T_ dec calculated above do not change but the matter-domination temperature should become T_M^ (vac) in Eq. (<ref>). The dilution factors are modified as 
    
        D =     T_M^ (vac)/T_ dec^ gg≃ 8 × 10^8   k  δ^2 ( T_R/10^10 ) ( 300  /m_S)^1.013/2( m_3/2/ 100   MeV) (  TeV/m_3) , 
        
        D =     T_M^ (vac)/T_ dec^ h,W,Z≃ 6 × 10^5   k  δ^2 ( T_R/10^10 ) ( m_S/300  )^1.011/2( m_3/2/ 100   MeV) (  TeV/m_3) (  TeV/m_H)^2 ,
    
    for the decays to into gluons and into Higgs and EW bosons, respectively.
    
    In addition, the sgoldstino can decay to a pair of gravitinos at a rate given by Ref. <cit.>
        Γ_ dec^3/2 = k^2 m_S^5/96 π^2 m_3/2^2 M_Pl^2,
    
    where k accounts for the mixing of the gravitino with ψ_S. Based on Eqs. (<ref>) and (<ref>), the sgoldstino dominates the energy density in the parameter space of interest. This implies that the sgoldstino decay can give a sizable contribution to the gravitino despite the small branching ratio B_3/2≡Γ_ dec^3/2  / Γ_ tot. For the decays to gluinos and to H, W^±, and Z respectively, the non-thermal gravitino abundance is estimated as
    
        ρ_3/2/s =      2 m_3/2ρ_S B_3/2/m_S s = k^2 m_S^4/48 π m_3/2 M_Pl T_ dec
        ≃   
        0.6   eV( k/0.1)^3 ( m_S/300  )^1.015/2(  TeV/m_3) 
        
        0.04   eV  k^3 (  TeV/m_H)^2 ( m_S/300  )^1.019/2(  TeV/m_3) .
    
    
    
    The various constraints discussed in this section are shown in Fig. <ref>. In the left panels, we take m_S = m_S  min, which refers to the theoretical minimum given in Eq. (<ref>). k_ min is the theoretical lower bound in Eq. (<ref>), which excludes the cyan region. In the right panels, we take k = k_c, where k_c stands for the critical value of k in Eqs. (<ref>), (<ref>), and (<ref>) applicable for different ranges of T_R. In the light (dark) gray region, the sgolstino is necessarily destroyed by scattering at high temperatures because Γ_ scatt(T_2) > H(T_2) (k_c > 1), whose result is previously shown in Fig. <ref>. We use the maximal value δ_ max≃ 0.67 inferred from Eqs. (<ref>) and (<ref>). The dilution factors in Eqs. (<ref>) and (<ref>) are marked with the black contours. The green contours separate different cosmological evolutions, where v_S(T) in Eq. (<ref>) does (not) drop below v_S above (below) the contours. The orange region is excluded by Eqs. (<ref>) and (<ref>) because the sgoldstino field values drops to the today's value v_S too quickly such that the mechanism fails to suppress the gravitino production until the conventional constraint temperature T_ co. The brown region is excluded by Eq. (<ref>) because the gravitino produced from the sgoldstino decay overcloses the Universe. The purple region is excluded by Eq. (<ref>) due to overproduction of spin-3/2 gravitinos via supergravity interactions. The red regions are excluded as the decay of sgoldstinos occurs after and thus spoils Big Bang Nucleosynthesis (BBN) <cit.>. Regions below the blue contours are incompatible with thermal leptogenesis because the baryon asymmetry in Eq. (<ref>) is depleted by too large of a dilution factor. The blue contours do not extend into the light gray regions, where the dilution factor is unity. For a smaller δ, the orange region as well as the blue line shift downward. The lower bound on T_R from thermal leptogenesis is then relaxed, until the orange region catches up with the blue line.
    
    
    
    
    
    § CONCLUSIONS
    
    We have investigated the possibility that the sgoldstino has a large field value in the early Universe. This suppresses the early production of the gravitino and is expected to relax the upper bound on the reheat temperature after inflation. As a proof of principle, we analyze a specific case where the supersymmetry breaking field S and the messenger fields couple minimally via Eq. (<ref>) and the mass term governs the zero-temperature potential of the sgoldstino.
    The constraints on the gravitino mass and the reheat temperature are summarized in Figs. <ref>-<ref>.
    When the field S provides sufficiently subdominant supersymmetry breaking, the sgoldstino condensate is destroyed by thermal scattering without producing (much) entropy. The reheat temperature may be as large as 10^12  GeV, and thermal leptogenesis is viable as long as the reheat temperature is larger than 10^9 GeV.
    On the contrary, if thermal scattering is inefficient, the sgoldstino condensate decays late with entropy production. The gravitino problem is then solved both by the suppression of the gravitino production and by dilution from entropy production. For a given reheat temperature, the dilution factor required to obtain a small enough gravitino abundance is smaller in our mechanism than the conventional scenario with dilution but not suppression. As a result, the reheat temperature can be as high as 3× 10^13 GeV. When the sgoldstino field breaks supersymmetry subdominantly and later decays, thermal leptogenesis is possible with a reheat temperature T_R  10^12`-13 GeV. Hence, there exist regions in the parameter space where thermal leptogenesis is viable and the gravitino problem is absent or much milder than previously claimed.
    
    
    
    § ACKNOWLEDGEMENT
    
    The authors thank Yiannis Dalianis, Lawrence Hall, and Aaron Pierce for fruitful discussions.
    This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contract DE-AC02-05CH11231 and by the National Science Foundation under grants PHY-1316783 and PHY-1521446. The work of R.C. was in part supported by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists, Office of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education for the DOE under contract number DE-SC0014664. The work of K.H. was in part performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.
    
    
    
    
    § DESTRUCTION TEMPERATURE AFTER MATTER-DOMINATED ERA
    
    
    In this section, we derive the destruction temperature in the case where the sgoldstino condensate dominates the energy of the Universe.
    First of all, we need to require again that v_S(T) is still decreasing before the condensate is destroyed, i.e. T_S < T_ des. Otherwise, the scattering rate is insufficient to destroy the sgoldstino condensate. Under this assumption, we first study the temperature dependence of the Hubble rate during the non-adiabatic era when the dominant source of the thermal bath is the scattering products of the sgoldstino as opposed to existing radiation.
    We define the temperature at the beginning of the non-adiabatic era as T_ NA. By conservation of energy transferred from the sgoldstino condensate to radiation, we write
    
        ρ_S Γ_ scatt/H =     3 M_Pl^2 H Γ_ scatt = π^2/30 g_* T^4 , 
        Γ_ scatt =     b α_3^2  T^3/v_S^2(T) = b α_3^2 m_S^2 T^3/3 H^2 M_Pl^2 ,
    
    where we repeatedly use the fact that the total energy density is dominated by the sgoldstino vacuum potential given in Eq. (<ref>), H = √(ρ_S/3) / M_Pl = m_S v_S(T) / √(3)M_Pl. The Hubble rate during this non-adiabatic phase is then given by
    
        H(T) = 30   b  α_3^2 /π^2 g_* m_S^2/T.
    
    This demonstrates that the Hubble rate is inversely proportional to the temperature and that the temperature during the non-adiabatic phase is increasing over time. As  ρ_S ∝ H^2(T) ∝ T^-2, one can compare this new scaling with the usual temperature dependence ρ_S ∝ T^3 during a radiation-dominated epoch and argue that the dilution factor is D = T_M / T_ des = ( T_ des / T_ NA)^5. We consider T_ des as the temperature at which H(T_ des) is determined by the radiation energy density ρ_R(T_ des) ∝ T_ des^4, which leads to
    
        T_ des =  3^2/3√(10)α _3^2/3 b^1/3/π√(g_*)( M_Pl m_S^2 )^1.011/3≃ 3 × 10^5  ( m_S/300  )^1.012/3.
    
    In fact, at the destruction temperature, the energy densities of the sgoldstino and radiation are comparable within a factor of a few, which allows us to compute the field value at T_ des
        v_S(T_ des) = √(π^2 g_*/30)T_ des^2/m_S = 3^5/6√(10)α _3^4/3 b^2/3/π√(g_*)( m_S M_Pl^2 )^1.011/3.
    
    The earlier assumption, T_S < T_ des, is equivalent to v_S(T_ des) > v_S, which places an upper bound on k
        k ≤4 × 3^2/3√(10)α _3^1/3 b^2/3/√(g_*)m_3/m_3/2( m_S/M_Pl)^1.011/3≃ 10^-4( m_S/300  )^1.011/3( m_3/ TeV) ( /m_3/2).
    
    
    
    § NON-PERTURBATIVE EFFECTS
    
    
    As the sgoldstino field oscillates with a large amplitude, v_S(T)>v_S, the messenger field may be produced in a non-perturbative way because of the rapid change of its mass. In the main sections, we assume the non-perturbative effect is negligible, which we will now justify.
    
    The mass of the messenger is given by
    
    
        m_Q^2 ≃ y^2 S^2 + g^2 T^2,
    
    where g is the gauge coupling constant. The adiabaticity of the mass of the messenger is characterized by the following quantity,
    
    
        q≡|ṁ_Q|/m_Q^2≃y^2 |S Ṡ|/(y^2 S^2 + g^2 T^2)^3/2 .
    
    
    When the sgoldstino oscillates with the thermal logarithmic potential, |Ṡ| ≃α T^2, and q is maximized around yS ∼ gT,
    
        q  y / (4π) .
    
    As long as y< 𝒪(1), the non-perturbative effect is negligible.
    
    When the sgoldstino oscillates with the vacuum mass term, |Ṡ|≃ m_S v_S(T).
    We first consider the case where the sgoldstino is destroyed by scattering.
    As v_S(T) > v_S, S may vary and q is maximized around y S = gT,
    
        q≲y m_S v_S(T)/g^2 T^2.
    
    As long as the sgoldstino is the subdominant component of the energy density of the Universe, q<y<1.
    After the sgoldstino dominates, q grows until the thermal bath is dominated by the radiation produced from the sgoldstino at the temperature of T_ NA.
    Using the formulae in App. <ref>, we obtain the maximal q,
    
    
        q =  √(π^2 g_*/30)y/g^2 D^3/5≲ D^3/5m_S/m_3  and 10^-2 D^3/5( m_S/1   TeV)^1.011/3 ,
    
    where in the inequality we use the upper bound on y in Eq. (<ref>) and  y T_ des < m_S. We find that q is smaller than unity for the parameter space considered in Fig. <ref>.
    
    We next consider the case where the sgoldstino decays. For v_S(T) >v_S, Eq. (<ref>) is applicable, and q<1 as long as the sgoldstino is subdominant.
    We find that T_S>T_M in the parameter space where the dilution factor is small enough that thermal leptogenesis is viable.
    For the parameter region, q<1 for v_S(T) > v_S.
    Once v_S(T) < v_S, q is given by
    
        q = y^2 m_S v_S v_S(T)/ max(y^3 v_S^3, g^3 T^3) < m_S/ y v_Sv_S(T)/v_S,
    
    which is smaller than unity as long as m_S < y v_S.
    
    
    
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