A mini review on Affleck–Dine baryogenesis

The superpotential for MSSM is given by [5]

$$\begin{equation} W_{\mathrm{MSSM}}= \lambda_{u} {\bf Q} {\bf H_u} {\bf u} + \lambda_{d} {\bf Q} {\bf H_d} {\bf d} + \lambda_e {\bf L} {\bf H_d} {\bf e} + \mu {\bf H_u} {\bf H_d}\,, \end{equation} \tag{ 1 }$$

where H_u,H_d,Q,L,u,d,e are chiral superfields representing the two Higgs fields (and their Higgsino partners), left-handed (LH) (s)quark doublets, right-handed (RH) up- and down-type (s)quarks, LH (s)lepton, doublets and RH (s)leptons respectively. The dimensionless Yukawa couplings λ_u,λ_d,λ_e are 3 × 3 matrices in the flavor space, and we have omitted the gauge and flavor indices. The last term is the μ term, which is a supersymmetric version of the SM Higgs boson mass.

The field space of supersymmetric theories contains many directions along which the D-term contributions to the scalar potential identically vanish. The scalar potential along these 'flat' directions arises from F-terms and SUSY breaking contributions. A very interesting class of flat directions is those made up of SUSY partners of the SM fermions, namely the squarks and sleptons, and the Higgs fields. The D-flat directions in MSSM are parameterized by gauge invariant monomials of the chiral superfields [6, 7]. Many of these directions are also F-flat in the sense that their potential does not receive any contribution from the MSSM superpotential (1). There are nearly 300 such directions [7].

MSSM flat directions are lifted by SUSY breaking terms and higher order terms in the superpotential. The latter arise from the effect of new physics above a high scale M (typically Planck or string scale), which appears as non-renormalizable superpotential terms (n ⩾ 4) of the form

$$\begin{equation} W \sim \lambda_{n}{{\Phi}^n \over n {M^{n-3}}} , \end{equation} \tag{ 2 }$$

where Φ is the superfield comprising the flat direction ϕ and its fermionic partner, and $\lambda _n \sim {\cal O}(1)$. MSSM flat directions typically carry a baryon and /or lepton number. For example, consider the cases when ϕ represents the LLe, udd and H_uL flat directions, respectively. Then the baryon number B and lepton number L carried by ϕ are B = 0 and L = + 1, B = −1 and L = 0, B = 0 and L = + 1, respectively. The superpotential terms that lift ϕ are W ∼ (LLe)², (udd)² and (H_uL)², respectively.

Low-energy SUSY breaking results in soft terms in the scalar potential. In addition, SUSY breaking by the non-zero energy density of the universe contributes to the flat direction potential. The scalar potential along a flat direction can then be written as [6]

$$\begin{eqnarray} &&\fl V(\phi) = \left(m^2_\phi + c_{\mathrm{H}} H^2 \right) {\vert \phi \vert}^2 + \vert \lambda_{n} \vert^2 {{\vert \phi \vert}^{2(n-1)} \over M^{2(n-3)}} + \left[(a_{\mathrm{H}} H + A) {\lambda_n \phi^n \over n M^{n-3}} + \mathrm{h.c.} \right], \end{eqnarray} \tag{ 3 }$$

where n ⩾ 4. The three terms on the right-hand side of this equation represent, respectively, the sum of low-energy and Hubble-induced soft mass terms, the contribution of the non-renormalizable superpotential term and the sum of low-energy and Hubble-induced A-terms. In the above equation, H is the Hubble expansion rate and $m_\phi ,~\vert A \vert \sim {\cal O}(\mathrm {TeV})$.

The role of flat directions in the early universe crucially depends on c_H, see equation (3). If c_H ≳ 1, the flat direction mass exceeds the Hubble expansion rate during inflation H_I. Then, due to fast rolling, the condensate would settle down to the origin of its potential during inflation, and would remain there at all times. Therefore the flat direction will have no interesting consequences in this case. Note, however, that it is possible to find a local minimum for a very large, a_H, such that a²_H > 4(n − 1)λ²_nc²_H. In this case, the condensate can play an interesting role similar to what happens when c_H < 0 [8] (which will be discussed shortly).

If 0 < c_H ≪ 1, quantum fluctuations are accumulated along the flat direction according to a random walk giving rise to a coherent state. The flat direction VEV is in this way displaced from the origin and its variance follows 〈ϕ²〉^1/2 ∼ H²_I/m_ϕ. This, however, assumes that other fields that are coupled to the flat direction are not excited in this fashion. If one considers the growth of quantum fluctuation along all independent fields, the maximum VEV that the flat direction can reach will be much smaller [9].

The case with c_H < 0 is more interesting. A negative c_H can arise at the tree level [6] or as a result of radiative correction [10, 11]. The flat direction VEV is in this case driven away from the origin and quickly settles at a large value $\phi \sim \left (H_{\mathrm {I}} M^{n-3}\right )^{1/n-2}$.

After inflation, the flat direction VEV slides down to an instantaneous value: $\phi (t) \sim (H(t) M^{n-3})^{1/n-2}$. Once H(t) ≃ m_ϕ, the soft SUSY breaking mass term in the potential takes over and the flat direction starts oscillating around its origin with an initial amplitude

$$\begin{eqnarray} &&\phi_0 ~ \sim ~ (m_\phi M^{n-3})^{1/(n-2)} . \end{eqnarray} \noindent \tag{ 4 }$$

If M = M_P, we will have ϕ₀ ∼ 10¹⁰ GeV for n = 4, and ϕ₀ ∼ 10¹⁶ GeV for n = 9. The mismatch between the phases of A-terms from low-energy and Hubble-induced SUSY breaking yields a torque and gives rise to rotational motion of the flat direction VEV [6]. If the flat direction has a non-zero baryon and/or lepton number, the rotating condensate will carry a baryon and/or lepton asymmetry [2, 6].

The dynamics of the AD condensate is non-trivial as shown in figure 1. The baryon/lepton number density is given by $n_{\mathrm {B,L}}=\beta \mathrm {i}(\dot {\phi }^{\ast }\phi - \phi ^{\ast }\dot {\phi })$, where β is the corresponding baryon/lepton charge of the AD field. The time evolution of n_B,L follows

$$\begin{eqnarray} \dot n_{\mathrm{B,L}} + 3 H n_{\mathrm{B,L}} & = & 2\beta \,\mathrm{Im} \left[\frac{\partial V(\phi)} {\partial \phi} \phi \right] = 2 \beta \frac{1}{M_{\mathrm{P}}^{n-3}} \,\mathrm{Im} \left[(a_{\mathrm{H}} H + A) \phi^{n}\right]. \end{eqnarray} \noindent \tag{ 5 }$$

The comoving density of the baryon/lepton number is given by

$$\begin{eqnarray} &&a^3(t)n_{\mathrm{B,L}}(t) = 2\beta |\alpha| ~ \frac{m_{\phi}}{M_{\mathrm{P}}^{n-3}} ~ \int_{t_{\mathrm{osc}}}^{t}a^3(t^{\prime})|\phi(t^{\prime})|^{n} \,\mathrm{sin} (n \theta) ~ \mathrm{dt}^{\prime} , \end{eqnarray} \noindent \tag{ 6 }$$

where a(t) is the scale factor of the universe, and t_osc ∼ m⁻¹_ϕ is the time at the onset of ϕ oscillations. 'α' denotes the CP-violating phase that arises because of the mismatch between the phases of the low-energy and Hubble-induced A-terms, and can be parameterized by sin δ.

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Note that in the above equation the integrand falls off very quickly since the amplitude of oscillations is redshifted according to |ϕ(t)|ⁿ∝a⁻ⁿ(t) (recall that n > 3). As a result, the generated asymmetry will reach its final value after a few oscillations, which is given by [6]

$$\begin{eqnarray} &&n_{\mathrm{B,L}}(t_{\mathrm{osc}}) = \beta {2 (n-2) \over 3 (n-3)} ~ m_\phi \phi^2_0 \,\mathrm{sin}(n \theta) \,\mathrm{sin}(\delta) . \end{eqnarray} \tag{ 7 }$$

After using equation (4), with M = M_P, and for $\mathrm {sin} (n \theta ) \sim \mathrm {sin} (\delta ) \approx {\cal O}(1)$, we find that

$$\begin{eqnarray} n_{\mathrm{B,L}}(t_{\mathrm{osc}}) & \approx & \beta {2 (n-2) \over 3 (n-3)} ~ m_\phi ~ (m_\phi M^{n-3}_{\mathrm{P}})^{2/(n-2)} . \end{eqnarray} \tag{ 8 }$$

The baryon and/or lepton asymmetry carried by the AD field will be eventually transferred to fermions when it decays. The rotating condensate induces a large mass ∼y|ϕ| for the fields coupled to it (y being a gauge or Yukawa coupling). The one-particle decay will be kinematically forbidden as long as y|ϕ| ≳ m_ϕ. Therefore the condensate VEV will only be subject to redshift by Hubble expansion initially. The AD field eventually decays, yielding a net baryon/lepton-to-entropy ratio that is given by [3]

$$\begin{eqnarray} &&\frac{n_{\mathrm{B,L}}}{s} = \frac{1}{4} ~ \frac{T_{\mathrm{R}}}{M_{\mathrm{P}}^2 H(t_{\mathrm{osc}})^2} ~ n_{\mathrm{B,L}}(t_{\mathrm{osc}}) \approx \beta \frac{n-2}{6(n-3)} ~ \frac{T_{\mathrm{R}}}{M_{\mathrm{P}}^2 m_{\phi}} \left(m_{\phi} M_{\mathrm{P}}^{n-3}\right)^{2/(n-2)}. \end{eqnarray} \tag{ 9 }$$

Here s = (2π²/45)g_*T³_R is the entropy density of the universe when transition from inflation to a radiation-dominated universe is complete, with g_* being the number of relativistic degrees of freedom and T_R being the reheat temperature of the universe. We have used H(t_osc) ≈ m_ϕ and equation (8) to obtain the final result. For n = 4, we have

$$\begin{eqnarray} \frac{n_{\mathrm{B,L}}}{s} & \approx & 10^{-10} \times \beta \left(\frac{1\,\mathrm{TeV}}{m_{\phi}}\right) \left(\frac{T_{\mathrm{R}}}{10^{9}\,\mathrm{GeV}}\right), \end{eqnarray} \tag{ 10 }$$

and for n = 6

$$\begin{eqnarray} \frac{n_{\mathrm{B,L}}}{s} & \approx & 10^{-10} \times \beta ~ \left(\frac{1\,\mathrm{TeV}}{m_{\phi}}\right)^{1/2} ~ \left(\frac{T_{\mathrm{R}}}{100\,\mathrm{GeV}} \right). \end{eqnarray} \tag{ 11 }$$

This ratio remains frozen during the subsequent evolution of the universe unless there is additional entropy production at a later time.

We note that any lepton asymmetry is partially converted by electroweak sphalerons into a baryon asymmetry according to n_B = (8/23)n_B−L [12]. Therefore, all that is actually needed is that the AD field carries a non-zero B − L so that the generated asymmetry survives washout by the sphalerons. An interesting scenario is AD leptogenesis driven by the H_uL flat direction for which B = 0 and L = + 1 [13]. As mentioned above, the non-renormalizable superpotential term that lifts the flat direction is (H_uL)², which also contributes to the light neutrino masses. It is therefore a rather predictive scenario for obtaining the observed baryon asymmetry, yielding [14]

$$\begin{eqnarray} {n_{\mathrm{B}} \over s} &\approx& 10^{-10} \times \beta ~ \left(\frac{1\,\mathrm{TeV}}{m_{\phi}}\right) ~ \left(\frac{T_{\mathrm{R}}}{10^8\,\mathrm{GeV}}\right) ~ \left({10^{-6} \,\mathrm{eV} \over m_{\nu_{\mathrm{l}}}} \right) , \end{eqnarray} \tag{ 12 }$$

where m_νl denotes the lightest neutrino mass.

We close this subsection by making a comment on the decay of the AD field. It has been shown that a rotating condensate can give rise to non-perturbative particle production [15]. Although efficient, this however does not lead to a fast decay of the AD field [16]. The momenta of particles thus produced are of the order of m_ϕ, which implies that they quickly become non-relativistic due to the Hubble redshift. These particles will decay perturbatively at late times, as mentioned above. Therefore non-perturbative effects do not alter the picture presented here in a significant way. This is very different from non-perturbative particle production after inflation, which typically results in a very rapid decay of the inflaton oscillations [17]. The difference can be understood based on the trajectory of the scalar field motion in the two cases. Oscillations of the inflaton field can result in the production of quanta whose momenta are much higher than the frequency of oscillations. On the other hand, a rotating AD condensate carries a baryon/lepton number whose conservation along with the conservation of energy inhibits populating high-momentum modes and thus a fast decay, via non-perturbative effects [16].

In the above estimation of baryon asymmetry, we have ignored the finite-temperature corrections to the flat direction potential. As a source of SUSY breaking, they can have important effects on the dynamics of flat directions and hence must be included. Thermal corrections are inevitable if the inflaton decays dominantly into the visible sector fields and produces MSSM degrees of freedom, as required by observations. Since the AD condensate is made up of MSSM scalars, the finite-temperature corrections would only involve those degrees of freedom that are coupled, directly or via loops, to these scalars. The VEV of the AD condensate naturally induces a mass ∼y|ϕ| to the coupled fields, where y is an SM gauge or Yukawa coupling between the fields. Depending on whether y|ϕ| ≲ T or y|ϕ| ≫ T, with T being the temperature of the thermal bath, different situations can arise.

Thermal effects from light fields. If y|ϕ| ≲ T, the fields that are coupled to the AD field have a mass smaller than temperature and hence reach thermal equilibrium with the thermal bath. The backreaction from these fields then results in a thermal correction to the flat direction mass

$$\begin{equation} m^2_{\phi,\mathrm{eff}} \sim m^2_\phi + y^2\,T^2 {\vert \phi \vert}^2 , \end{equation} \noindent \tag{ 13 }$$

where typically yT ≫ m_ϕ. The flat direction ϕ starts oscillating once m_ϕ,]eff ≃ H. Therefore, if y|ϕ| ≲ T and yT > H are both satisfied when $H \gg {\cal O}(\mathrm {TeV})$, thermal effects trigger early oscillations of the flat direction [18]. This requires that y be not too large (so that yϕ ≲ T) nor too small (so that yT > H). It turns out that for y ∼ 10⁻⁴–10⁻², thermal effects can lead to early oscillations of flat directions that are lifted by superpotential terms of order 4 ⩽ n ⩽ 6 [18].

When early oscillations start at $H_{\mathrm {osc}} \gg {\cal O}(\mathrm {TeV})$, the flat direction is in the minimum of its potential along the angular direction set by the Hubble-induced A-term, see equation (3). Rotational motion, which is a prerequisite for the generation of baryon asymmetry, can occur if a new A-term with a different phase from that of the Hubble-induced A-term appears at this time. Thermal effects [18, 19] and supergravity effects [19] can give rise to such new A-terms:

$$\begin{equation} A_{\mathrm{th}} \sim y H \left({\lambda_n \phi^n \over n M^{n-3}_{\mathrm{P}}} + \mathrm{h.c.} \right) , \quad A_{\mathrm{H}} \sim b_{\mathrm{H}} {H^2 \over M_{\mathrm{P}}} \left({\lambda_n \phi^n \over n M^{n-3}_{\mathrm{P}}} + \mathrm{h.c.} \right). \end{equation} \noindent \tag{ 14 }$$

However, A-terms do not receive an enhancement similar to the flat direction mass. As a result, in the cases when thermal masses lead to early oscillations of the flat direction, the generated baryon asymmetry is typically suppressed compared to that at zero temperature [18, 19].

Thermal effects from heavy fields. If y|ϕ| ≫ T, the fields that are coupled to the AD field have a mass larger than temperature and will decouple from the thermal bath. For the same reason, they are also decoupled from the running of gauge coupling constants at finite temperature. The leading effect can then be determined by considering the free energy as a function of the coupling constant g:

$$\begin{equation} \delta \Omega \propto g^2(T) T^4 . \end{equation} \noindent \tag{ 15 }$$

Decoupling of the heavy fields from the running of g(T) amounts to a logarithmic correction to the flat direction potential [19]:

$$\begin{equation} V_{\mathrm{eff}}(\phi) \sim V(\phi) + a \alpha^2(T) ~ T^4 \,\mathrm{ln} ({\vert \phi \vert}^2), \end{equation} \noindent \tag{ 16 }$$

where V (ϕ) is given by the expression in equation (3) and α(T) is the gauge fine structure constant at temperature T. The prefactor a can either be positive or negative depending on whether the gauge/gaugino or the scalar/fermion fields are decoupled from the thermal bath. A positive logarithmic correction triggers early oscillations of the AD field.

To elucidate, let us consider the H_uL flat direction. This flat direction VEV breaks the electroweak symmetry down to a U(1), but does not touch the strong interactions. It also induces a large mass to the top (s)quarks, thus making them heavy, through the large top Yukawa coupling. Gluon/gluino fields and those (s)quarks that are not heavy give rise to a contribution to the free energy as in equation (15), with g_S(T) being the strong coupling constant at temperature T. The decoupling of top (s)quarks from the running of g_S(T) results in a positive logarithmic correction to the flat direction potential:

$$\begin{equation} V_{\mathrm{eff}}(\phi) \sim V(\phi) + \alpha_S^2 (T) T^4 \,\mathrm{ln} ({\vert \phi \vert^2}) , \end{equation} \tag{ 17 }$$

which leads to early oscillations of the flat direction.

In summary, the exact effect of thermal corrections depends on the nature of the AD condensate. In general, however, they must be taken into account for a precise calculation of the baryon asymmetry via the AD mechanism.

Our discussion of thermal effects has been based on the existence of a thermal plasma from inflaton decay at arbitrarily early times. This implicitly assumes that particles produced from decay of the inflaton immediately reach thermal equilibrium. However, it is known that reheating after inflation can be a very complicated process involving various perturbative and non-perturbative phenomena [17]. Particles produced from inflaton decay, typically, have a non-thermal distribution, and the time scale of their equilibration is model dependent. Assignment of a temperature T to the reheat plasma is only justified after full thermal equilibrium is achieved.

When in thermal equilibrium, the number density n and the energy density ρ of a relativistic species are given by n∝T³ and ρ∝T⁴, with T being the temperature of the thermal bath. A thermal distribution is therefore characterized by the relation n ∼ ρ^3/4. Right after inflaton decay, we typically have n ≪ ρ^3/4 or n ≫ ρ^3/4 [17]. If inflaton decay products thermalize immediately, reheat temperature of the universe will be given by T_R ∼ (Γ_dM_P)^1/2, where Γ_d is the inflaton decay rate. However, thermalization may take a much longer time.

Let us consider the situation after perturbative inflaton decay, for which n ≪ ρ^3/4. This implies that the number density of particles is smaller than that of a thermalized distribution, while the average energy of particles is larger than the thermal value. In order to reach full thermal equilibrium, one therefore needs inelastic reactions that increase the number of particles as well as elastic reactions that redistribute the energy among particles. It has been shown that the dominant processes for thermalization are 2 → 2 and 2 → 3 scattering processes with gauge boson exchange in the t-channel [20, 21]. Thermal equilibrium is established when these processes are efficient, i.e. their rates Γ_2→2 and Γ_2→3 are larger than the Hubble rate H.

Since the AD condensate is made up of fields that carry gauge quantum numbers, its VEV spontaneously breaks gauge symmetries. This induces a large mass ∼g|ϕ| to the gauge fields via the Higgs mechanism, thereby suppressing the rate of processes relevant for thermalization. If the entire SM gauge group is broken, thermalization can be delayed substantially [22]. This can happen, for example, when the udd and LLe flat directions independently acquire large VEVs. In fact, flat directions that are lifted by superpotential terms of order n ⩾ 6, see equation (3), can result in reheat temperatures well below the naive estimate T_R ∼ (Γ_dM_P)^1/2, perhaps as low as ${\cal O}(\mathrm {TeV})$ [22]. Moreover, the inflaton decay itself may be delayed as a result of kinematic blocking due to the large masses induced by the flat direction VEV [23].

Therefore flat directions can affect the thermal history of the universe significantly, which has interesting and important cosmological consequences [22].

The ground state of the AD condensate is determined by the coherent oscillations in a potential V (ϕ) with a frequency that is large compared to the Hubble expansion rate. In our case the motion of the condensate is not simply oscillatory; instead, similar to the case of a rotating trajectory with a phase, one obtains an elliptical orbit. For a sufficiently small amplitude, the trajectory follows a simple potential:

$$\begin{equation} V(\phi)=\textstyle\frac 12 m_{\phi}^2|\phi|^2\left({\phi^2/\mu^2}\right)^x. \end{equation} \tag{ 18 }$$

The equation of state for such an oscillatory trajectory is given by γ ≡ p/ρ = (1 + x)/(1 + x/2), p = x/(2 + x). This gives rise to a negative pressure whenever x < 0. This is a sign of an instability of the scalar field oscillations under arbitrarily small perturbations. When the effective mass of the scalar field is much larger than the Hubble rate, the quantum fluctuations in the condensate grow according to

$$\begin{equation} \ddot{\delta}_{{\bf k}} = -K{\bf k}^2\delta_{{\bf k}}. \end{equation} \tag{ 19 }$$

If K = 2x < 0, quantum fluctuations of the condensate at the scale, λ = 2π/|k|, will grow exponentially in time:

$$\begin{equation} \delta\phi_{{\bf k}}(t)=\delta\phi(0)\mathrm{exp}(-K{\bf k}^2 t). \end{equation} \tag{ 20 }$$

In reality, the onset of nonlinearity sets the scale at which the spatial coherence of the field can no longer be maintained and the condensate fragments. The initial fluctuations in the condensate are set by perturbations of the AD field generated during the inflationary era.

If the condensate carries a global charge, the energy-to-charge ratio changes as the condensate fragments. This is what happens in the case of MSSM flat directions [29–33]. The AD condensate responsible for generating the baryon asymmetry at the first instance could fragment into non-topological solitons with a fixed charge, called the Q-ball [34, 35]. Typically, fragmentation of the AD condensate is a violent process, and can generate observable gravitational waves as shown in [36].

The stability of Q-balls produced from fragmentation of the AD condensate depends on their energy-to-charge ratio. The crucial point is the presence of a global U(1) charge carried by the Q-ball, which is denoted by Q. The configuration that has the lowest energy-to-charge ratio will then be the stable one.

For a sufficiently large Q, the Q-ball energy is given by E = |νQ| [34, 35, 37]. Therefore, if |νQ| < m_ϕ|Q|, the Q-ball will not decay into plane waves corresponding to ϕ quanta. The stable configuration will instead be a soliton whose profile is given by $\phi ({\vec x},t) = \mathrm {e}^{\mathrm {i} \omega t} \phi ({\vec x})$, where $\phi ({\vec x}) \simeq \phi _{0}$ inside and $\phi ({\vec x}) \simeq 0$ outside the soliton. Note that the global U(1) symmetry is broken inside the soliton by the non-zero value of ϕ₀, but remains unbroken outside the soliton.

In models of gauge-mediated SUSY breaking the Q-balls can be absolutely stable, and hence a candidate for cold dark matter (CDM) [38]. In these models the potential for the AD field takes the form $V(\phi ) \approx m^4_{\phi } \ln (1+{|\phi |^2}/ {m_{\phi }^2} )$, where $m_{\phi } \sim {\cal O}(\mathrm {TeV})$ represents the soft mass of sparticles. The profile of a Q-ball follows $\phi (r) \propto \exp (-m_{\phi }r)$, and its energy E in terms of the charge Q is given by [38]

$$\begin{equation} E \approx \frac{4\sqrt{2}}{3}\pi m_{\phi}|Q|^{3/4}. \end{equation} \tag{ 21 }$$

Typically, Q represents the baryon number B carried by the Q-ball. This allows for the existence of some entirely stable B-balls, provided that B is sufficiently large. Indeed, the energy per baryon number, given by ∼(1 TeV) × B^−1/4, is less than 1 GeV for B > 10¹². Such large B-balls cannot decay into protons and neutrons and are entirely stable. If they were produced in the early universe, they would exist at present as a form of dark matter [38]. The astrophysical and terrestrial constraints [39, 40], as well as bounds from direct searches for Q-balls [41], place a lower limit of |B| > 10²² on the size of B-balls.

In models of gravity-mediated SUSY breaking, the B-balls are not stable. In these models the potential for the AD field has the form V (ϕ) = m²_ϕ|ϕ|²(1 + Kln(ϕ²/μ²)) [30]. K can be calculated for a given flat direction, and the potential is flatter than |ϕ|² when K < 0. Due to the smallness of K, the B-balls are not stable in this case regardless of how large |B| is. Instead they decay via surface evaporation. The VEV of ϕ near the surface is low enough, and so are particles that are coupled to the ϕ, to kinematically allow the B-ball decay to the latter.

When a B-ball decays, for each unit of baryon number (corresponding to the decay of three squarks to quarks), at least three lightest supersymmetric particle (LSP) are produced. Depending on the nature of the cascade produced by the squark decay, and the LSP mass, more LSPs could be produced. The baryon-to-dark matter ratio right after the decay is given by

$$\begin{equation} r_{\mathrm{B}} \equiv {\Omega_{\mathrm{B}} \over \Omega_{\mathrm{DM}}} = \frac{m_{n}}{N_{_{\mathrm{LSP}}}f_{\mathrm{B}}m_{_{\mathrm{LSP}}}} , \end{equation} \tag{ 22 }$$

where m_n and m_LSP are the nucleon and LSP mass, respectively, N_LSP ⩾ 3 is the number of LSPs produced per baryon number, and f_B is the fraction of the total baryon asymmetry contained in the B-ball. It is therefore rather natural to obtain r_B < 1 from the decay of B-balls.

The subsequent evolution of the LSPs thus produced depends on their nature. If the LSP is a neutralino, the produced quanta will annihilate with a thermally averaged rate 〈σv〉 = a + b(T/m_LSP), where a and b depend on the identity of the neutralino and annihilation final states [42]. Moreover, the neutralinos collide with the SM particles in the background, which keeps them in kinetic equilibrium with the thermal bath down to a temperature $T \sim {\cal O}(\mathrm {MeV})$ [43]. As a result, the LSPs produced in B-ball decays will spread out by a random walk with a rate ν determined by the collision frequency divided by a thermal velocity v_th ≈ (T/m_LSP)^1/2. Since the decay is spherically symmetric, it is very likely that the LSPs have a Gaussian distribution. Assuming an efficient LSP production, so that f_B ≈ 1, the dark matter abundance in terms of the density parameter can be written as [44]

$$\begin{eqnarray} &&\Omega_{\mathrm{LSP}} \simeq 0.5 ~ \left({m_{\mathrm{LSP}} \over 100 \,\mathrm{GeV}}\right) \left({10^{-7} \,\mathrm{GeV}^2 \over \langle {\sigma v} \rangle}\right) \left({100 \,\mathrm{MeV} \over T_{\mathrm{d}}}\right) \left(\frac{10} {g_*(T_{\mathrm{d}})} \right)^{1/2} , \end{eqnarray} \tag{ 23 }$$

where T_d is the decay temperature of the B-ball given by

$$\begin{equation} T_{\mathrm{d}} \ll \left({m_\chi \over 100\,\mathrm{GeV}}\right)^{3/16}\left( {10^{20}\over N_{\mathrm{LSP}}^{\mathrm{tot}}}\right)^{1/8} \left(\frac{100}{g_*(T_{\mathrm{d}})}\right)^{3/16} (20 \,\mathrm{GeV}), \end{equation} \tag{ 24 }$$

with g_*(T_d) being the effective number of relativistic degrees of freedom at T_d. Sufficiently large B-balls decay below the freeze-out temperature such that T_d ≪ T_f ∼ m_LSP/20.

A similar analysis can be performed when gravitino is the LSP. The annihilation of gravitinos with each other and their collision with the SM particles are negligible as they proceed through interactions of gravitational strength. In this case the dark matter abundance from the decay of B-balls is given by [45]

$$\begin{equation} \Omega_{\mathrm{LSP}} \simeq 0.2 ~ \left(\frac{m_{\mathrm{LSP}}}{1\,\mathrm{GeV}}\right), \end{equation} \tag{ 25 }$$

which is valid for reheating temperatures below 10⁷ GeV. A similar expression can be derived for the abundance of axino dark matter from decaying B-balls [46].

We note that dark matter connection to stable or unstable B-balls occurs in a non-thermal fashion, i.e. dark matter relic abundance is not set by thermal freeze-out of its annihilation. Therefore a possibility arises for explaining the dark matter–baryon coincidence puzzle [47] in the context of the AD mechanism. Hidden sector scenarios of addressing this coincidence problem via the AD mechanism have recently been proposed [48]. These scenarios, however, involve more uncertainties as they rely on assumptions about the hidden sector.