Lepto-axiogenesis and the scale of supersymmetry

If the Peccei-Quinn field containing the QCD axion undergoes rotations in the early universe, the dimension-five operator responsible for neutrino masses can generate a lepton asymmetry that ultimately gives rise to the observed baryon asymmetry of the Universe. This lepto-axiogenesis scenario requires a flat potential for the radial direction of the Peccei-Quinn field, naturally realized in supersymmetric models. We carefully compute the efficiency of this mechanism for the Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) and Kim-Shifman-Vainshtein-Zakharov (KSVZ) axion models and place lower bounds on the masses of scalar superpartners required to reproduce the observed baryon asymmetry. For the KSVZ model, we find an efficiency for generation of the asymmetry six times larger than the previously extant computation after including scattering channels involving superpartners. In this case, the superpartner scale should be above $\sim$ 30 TeV for a domain wall number of one; the lower bound weakens for larger domain wall numbers. We find that the superpartner mass scale may also be as low as 30 TeV for the DFSZ model. In all cases, the lower bound on the superpartner masses is inversely proportional to the sum of the squares of the neutrino masses and so can strengthen as the upper bound on the neutrino mass improves. We identify the parameter space where the axion rotation can simultaneously produce axion dark matter via kinetic misalignment; in this case it is possible to put an upper bound of order PeV on the masses of scalar superpartners.


INTRODUCTION
The Peccei-Quinn (PQ) symmetry [1,2] provides an attractive solution to the strong CP problem. The pseudo Nambu-Goldstone boson associated with this symmetry, the axion [3,4], can have important implications for cosmology. It is a cold dark matter candidate, and it can also play a central role in the generation of the matter-antimatter asymmetry.
One possibility is that axion dark matter can be generated by the misalignment mechanism [5][6][7], wherein the axion field is displaced from the zero-temperature minimum of its potential in the early universe. In this case, the axion begins its motion from rest when the mass generated by the QCD anomaly becomes comparable to the Hubble expansion rate. However, similar to fields in models of Affleck-Dine baryogenesis, the complex PQ field that contains the axion may receive a kick at early times and rotate in field space. This has ramifications for cosmology. First, axion dark matter may be produced not from the misalignment mechanism, but rather the so-called "kinetic misalignment mechanism" [8,9], wherein the energy contained in the motion in field space is converted to axions. The observed abundance of dark matter points to heavier, less weakly-coupled axions than in the conventional misalignment case. Second, there is a PQ charge associated with the angular momentum in field space. This is analogous to the baryon/lepton number carried by Affleck-Dine fields. In the presence of chirality-and baryon/lepton number-violating interactions, the PQ charge is converted to baryon number, a mechanism known as axiogenesis [10].
In its minimal form, axiogenesis does not simultaneously explain the dark matter and baryon abundances; once the dark matter abundance is fixed, too little baryon asymmetry is produced. A successful simultaneous prediction requires additional physics beyond the Standard Model [10][11][12][13][14][15] to increase the efficiency of the transfer of PQ charge to baryon number. A particularly simple solution takes advantage of lepton-number violation present when neutrino masses are explained by a Majorana mass, a scenario known as lepto-axiogenesis [16,17]. The Majorana mass allows transfer of the PQ charge to baryon minus lepton number B − L, which can eventually be converted to baryon number by weak sphalerons.
In this paper, we revisit lepto-axiogenesis, considering both Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) [18,19] and Kim-Shifman-Vainshtein-Zakharov (KSVZ) [20,21] axion models. We focus on the case where lepto-axiogenesis is embedded in a supersymmetric model. As we will discuss below, supersymmetric scenarios provide the most natural setting for axiogenesis. As in the original lepto-axiogenesis proposal, lepton-number violation is provided by the supersymmetric generalization of the ∆L = 2 Weinberg operator [22] that is responsible for neutrino masses, (LH u )(LH u ).
In the DFSZ case, the PQ field couples directly to the Higgs fields. Then, the nontrivial dynamics of the PQ field can impact the masses of the Higgs fields present in the Weinberg operator and therefore the transfer of the lepton asymmetry. On the other hand, in the KSVZ case the PQ field couples to heavy quarks and not directly to the fields of the Standard Model, so the above effect is absent.
The precise baryon asymmetry depends on the details of the cosmological history, including the reheat temperature T R of the universe following inflation. In our discussion, we pay attention to constraints placed on T R from, for example, avoiding disruption of Big Bang Nucleosynthesis (BBN) by superpartner decays [23,24]. We also carefully account for whether various Yukawa interactions are in equilibrium throughout the thermal history.
This can affect the efficiency of the asymmetry transfer.
In Sec. 2, we review the dynamics of the rotating field and how dark matter is produced in the kinetic misalignment mechanism. We then discuss the computation of the baryon asymmetry in Sec. 3. In comparison to Ref. [16], we take special care to account for the presence of superpartners, which impacts the rate at which the lepton asymmetry is generated. We then present detailed results for the DFSZ model including the thermalization of the PQ field in Sec. 4. The outcome of our analysis is a prediction for the minimum scale of supersymmetry-breaking scalar masses. We also find parameter space where dark matter and the baryon asymmetry may be simultaneously explained. The scalar superpartner masses are bounded from above (≲ 300 TeV), and the axion decay constant is predicted to be approximately 10 9 GeV. We also discuss the possible production of a non-topological soliton, which in principle could disrupt the prediction of the baryon asymmetry. In Sec. 5, we summarize the results. The scale of supersymmetry breaking required by this mechanism is consistent with that indicated by the observed Higgs boson mass. 4

DYNAMICS OF THE ROTATING FIELD
We define our complex PQ field P containing the axion as Here f a is the decay constant, N DW is the domain wall number, S is the radial direction which we call the saxion, and θ = a/f a is the angular direction. We assume that the potential of S is nearly quadratic. This assumption allows large field values for S in the early universe. This is necessary for initiating the rotation in field space, as we will discuss below. A nearly quadratic potential can be naturally realized in supersymmetric theories, where the potential can be flat up to supersymmetry-breaking corrections. This is the case for a two-field model, with superpotential and soft masses given by Here, X is a chiral multiplet whose F -term potential fixes the PQ-charged fields P andP to PP = v 2 PQ . Without loss of generality, we take |P | ≫ v PQ ≫ |P | in the early universe. We may then consider effective single-field dynamics for P with a nearly quadratic potential m 2 P |P | 2 , whileP is fixed to a small field value byP = v 2 PQ /P and is irrelevant. X will be fixed near the origin because of the large mass ≃ λP . A nearly quadratic potential is also achieved by a one-field model with logarithmic corrections [25] V (P ) = 1 2 with m S the mass of the saxion. The logarithmic corrections arise from the quantum corrections due to a Yukawa coupling of P , which can be that with the KSVZ quark in the KSVZ model, while extra fields are required in the DFSZ model.
In the one-field model, the mass of the fermionic superpartner of the axion, the axino, is generated by one-loop corrections and is suppressed relative to the typical scale of scalar soft masses. This tends to make the axino the lightest supersymmetric particle (LSP).
In the two-field case, R-or supersymmetry-breaking effects will induce an axino mass of order the gravitino mass, and an axino LSP is less likely. If stable, an axino LSP has the 5 potential to be problematic because it will typically overclose the universe. 1 The axino may decay if R-parity violation is introduced, or an axino LSP could be avoided if a bino and/or Higgsino were sufficiently light. See Appendix C for details, where we also discuss potential constraints from BBN.
In both the one-and two-field models, assuming the simplest mediation scheme of supersymmetry breaking by Planck-suppressed interactions, the saxion mass is expected to be of the same order as the soft scalar masses of the Minimal Supersymmetric Standard Model (MSSM). We will see that this curvature impacts the rotation of the axion in field space and the generation of the baryon asymmetry, and so the scalar mass may be constrained or predicted. In the one-field model, the curvature of the potential depends logarithmically on the field value of S. When we present results, we neglect this logarithmic dependence.
So, they apply directly to the two-field case, but a small correction should be applied when interpreting results in the context of the one-field model, see Sec. 4.4.

Initiation and evolution of rotation
During inflation, the presence of a Hubble-induced mass term can induce a large field value for P [26]. Then, at these early times, operators that explicitly break the PQ symmetry of the form can be enhanced, where q is an integer. Even if these operators are suppressed today so as to not spoil the solution to the strong CP problem, they can have important implications in the early universe.
The potential of P is, for S ≫ f a , where H is the Hubble expansion rate, A is a constant coming from R-symmetry breaking, and c H , the coefficient of the Hubble-induced mass term, is a constant expected to be 1 In general, we expect that the axino will thermalize via the supersymmetric analog of the couplings that thermalize the saxion (see Sec. 4.1), which would overproduce axinos. It is conceivable that saxion thermalization might not occur until temperatures near the EW scale, in which case supersymmetrybreaking masses would be non-negligible, and the axino might not thermalize even if the saxion does. However, we have checked that even in this case the suppressed freeze-in abundance of an axino LSP would be problematically large.
O(1) [26]. Here, m S is a soft supersymmetry-breaking mass, which in the two-field case would be identified with m P . We focus on gravity-mediated scenarios, where A is of the same order as m S . The superpotential in Eq. (2.4) preserves a linear combination of PQsymmetry and R-symmetry, so the explicit breaking of the U (1) symmetry of P requires R-symmetry breaking. Assuming c H > 0 and that the Hubble scale during inflation is larger than m S , P is driven to a large field value where the Hubble-induced mass term and the |P | 2q−2 term balance with each other. After inflation, P follows the minimum where two terms balance with each other [26,27]. When 3H ≃ m S , P begins oscillations. At the same time, the A-term provides a kick for P in the angular direction, and P begins to rotate.
This occurs at a temperature T osc , where T R is the reheat temperature after inflation and g * denotes the number of relativistic degrees of freedom in the bath with a full MSSM value of g MSSM = 228. 75. We assume that inflationary reheating proceeds via perturbative inflaton decay, and thus the scale factor R obeys R 3 ∝ T −8 during reheating [28]. The PQ charge density associated with the rotation is (2.7) We normalize the charge density so that it coincides with −θf 2 a for S = 0. The charge density normalized by the entropy density for T R < T osc can be computed as follows. The inflaton energy density ρ inf scales in the same way as n θ after the initiation of the rotation The A/m S is the ratio of the potential gradient in the angular and radial directions at T osc .
There is also energy density ρ S stored in the oscillations of the radial mode S. Whether ρ S is of importance depends on the cosmological history and at what temperature T th this mode is thermalized. The saxion energy density may come to dominate the energy of the universe 7 if this thermalization is late. We comment on this scenario further at the end of this section.
Following thermalization, the motion of the PQ field becomes circular due to PQ charge conservation: the radial mode dissipates, but much of the axial motion remains-for while part of the charge can be transferred into a charge asymmetry of particles in the thermal bath, it is free-energetically favored to keep almost all of the charge in the rotation [10,29].
The field will rotate around the body of the potential, with the radial direction eventually settling down to its minimum N DW f a . The energy in rotation ρ θ , accounting for both the potential and kinetic energy, is given as −θn θ . Before the radial direction S reaches its minimum, which occurs at a temperature denoted by T S ,θ is a constant, and conservation of the PQ charge implies the energy density of the rotation scales as matter, ρ θ ∝ R −3 . For T < T S , the scaling of the rotational energy density resembles that of kination, ρ θ ∝ R −6 .
This scaling can be derived by noting that conservation of charge n θ R 3 at constant radial field value impliesθ ∝ R −3 . When the saxion settles to its minimum at T = T S , we know both θ ≃ N DW m S and the PQ yield Y θ = −θf 2 a /s, so we can derive If the energy of the rotation dominates the energy of radiation at this time and if the saxion has already undergone thermalization (i.e. T th > T S ), then this T S is also the temperature T MK at which the universe transitions from a matter-dominated to a kination-dominated one. This history is illustrated in Fig. 1. We denote the temperature at which the universe transitions from radiation domination to matter domination as T RM . We emphasize that the matter domination we refer to here is domination by an energy density of rotation that scales as matter, not ordinary matter. This occurs at temperature This expression is general as long as no entropy is produced after T RM , and Y θ refers to the charge yield evaluated at T RM . In particular, this result applies whether or not there was an era where the saxion came to dominate the energy density of the universe prior to T RM . The kination-dominated era ends by the redshift of ρ θ =θ 2 f 2 a /2 = n 2 θ /(2f 2 a ) at temperature (2.11) A matter-dominated era followed by a kination-dominated one would modify the primordial gravitational wave spectrum in a way that potentially provides a unique signal [30][31][32].
It is also possible that the energy density due to rotation remains subdominant to the thermal bath. As we will see, in this case the temperature T S where the saxion reaches its minimum is still of significance for determination of the baryon asymmetry, as it marks the time whereθ changes its scaling. However, this temperature would no longer mark the onset of a kination era because radiation remains dominant.
We now comment on the possibility that saxion thermalization is late so that the saxion comes to dominate the energy density of the universe. We define the ratio r of the axion rotation to the saxion oscillation energy densities, which is in turn determined by the ratio of the potential gradients between the angular and radial modes, This r is inversely related to the ellipticity of the initial motion and r = 1 corresponds to nearly circular rotations. Here it is assumed that the angular direction is not accidentally 9 close to the minimum of the potential in Eq. (2.5); otherwise r becomes smaller than A/m S .
The thermal bath created from the saxion is at a temperature T th upon completion of thermalization. This fact allows us to predict T RM because ρ S × r = ρ θ should hold at T th .

Kinetic misalignment and production of axion dark matter
In the conventional misalignment mechanism, the value of the axion field is initially frozen by Hubble friction. But once 3H < m a (T ), the axion field begins to oscillate around the minimum of its potential. In the axiogenesis framework, on the other hand, the axion is not frozen, rather the PQ field is already rotating with high velocity. This qualitatively changes the dark matter production story. The kinetic misalignment mechanism (KMM) occurs when the kinetic energy of the axion field is greater than the potential energy. The KMM delays and the temperature at the transition from radiation to matter domination using Eq. (2.10) T RM, KMM ≃ 2.9 × 10 6 GeV m S 10 TeV f a 10 9 GeV N DW 3 . (2.17)

COMPUTATION OF THE BARYON ASYMMETRY
In this section, we describe the computation of the baryon asymmetry in lepto-axiogenesis.

Basics of lepto-axiogenesis
The axion rotation couples to the thermal bath via the gluon in the KSVZ theory and via the Higgs fields in the DFSZ theory. The PQ-charge is transferred to a particle-antiparticle asymmetry of particles in the thermal bath, and in equilibrium the charge asymmetry in the bath is of the order ofθT 2 [16]. The total B − L charge vanishes in the absence of B − L violation, and the baryon asymmetry is fixed at the electroweak phase transition [10].
The B − L symmetry is broken if the observed non-zero neutrino mass is explained by a Majorana mass term. We consider the Majorana mass given by the Weinberg operator [22], whose supersymmetrization is given by the superpotential which can be UV-completed by the seesaw mechanism [43][44][45][46]. This operator gives rise to neutrino mass terms, in terms of the vacuum expectation value v Hu of the up-type Higgs related to eigenvalues through the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix The Weinberg operator will transfer the particle-antiparticle asymmetry of L i and H u to B−L through scattering between the lepton and Higgs fields (and their superpartners) in the bath. The scattering is not in equilibrium for temperatures T < ∼ 10 12 GeV × (0.03 eV/m ν ) 2 , so the transfer of the PQ charge to B − L is suppressed by a factor of Γ L /H with Γ L the lepton-number violating rate. That is, B − L asymmetry is produced by a "freeze-in" process [10]. This B−L asymmetry is ultimately further processed by electroweak sphalerons to give a baryon asymmetry n B ≃ (28/79)n B−L [47].
To calculate the rate of B − L asymmetry production, we must account for all scattering processes due to the operator in Eq. (3.1). Each contribution takes the forṁ where a, b, c, and d are field labels, momenta {p a , p b } are incoming and {p c , p d } are outgoing, X with E X the energy of field X. The initial factor of two is because the processes have ∆L = 2. Here, µ X is the chemical potential of field X. See the Appendix of Ref. [14] for discussion in a similar context. For chemical potentials much smaller than temperature, the sum over all scattering processes giveṡ with i and j running over the three generations. We have assumed that processes involving scattering between Higgsinos H u , gauginos λ, and Higgs bosons are in equilibrium (and similarly for sleptons), which is typically the case. We have included scattering processes involving superpartners, which were neglected in Ref. [16]. We also go beyond the onegeneration approximation used there; this has a smaller effect. We have also assumed that the masses of ℓ and H u are smaller than T . As we will discuss in Sec. 3.3, this is not true for the DFSZ model for sufficiently high temperatures, since a large field value of S may impart a mass > T to H u and H d .
The coefficient of the Weinberg operator can be related to the neutrino masses and mixings as in Eqs. (3.2) and (3.3), so the production rate of the B − L asymmetry may be recast asṅ We obtain an analytic result where T i and T f mark the initial and final temperatures of the era when ∆Y B−L is a constant.
Reproducing the observed baryon asymmetry, Y obs B = 8.7×10 −11 [48], requires a saxion mass For T > T R (or for T < T RM ), the universe is not radiation-dominated, and production becomes IR (or UV)-dominated. Again, this is summarized in Table IV and illustrated in the left panel of Fig. 2.
If the reheat temperature is lower than the temperature where the saxion settles to its minimum, i.e., T R < T S , then Eq. (3.9) does not hold becauseθ is never constant during the radiation domination era, insteadθ ∝ T 3 . In this case, B − L production peaks at T S , which is illustrated in the right panel of Fig. 2. Then, the asymmetry may be obtained by first computing the redshift-invariant quantityṅ B−L /(Hρ inf ), with the inflaton energy density denoted by ρ inf . This quantity is readily evaluated at T S , see Eq. (3.6), recalling that θ ≃ N DW m S at this time. Then we can normalize the quantity to n B−L /s at T R : where we have assumed inflationary reheating by perturbative decays of the inflaton so The result depends on the choice of the neutrino spectrum. We will show results for a normal hierarchy (NH), or inverted hierarchy (IH), assuming the lowest mass eigenvalue is negligible, so the overall mass scale is given by the mass differences determined by oscilla-14 tions. Even if we saturate the upper bound m ν < 0.12 eV from the Cosmic Microwave Background along with data from Baryon Acoustic Oscillations [48], the predictions for this case are not so different from those of the inverted hierarchy case. Precisely speaking, the values of C i m 2 ν i with the upper bound saturated are 8% (normal hierarchy) and 16% (inverted hierarchy) larger than that for the inverted hierarchy with a negligible lightest neutrino mass.

KSVZ
The KSVZ model includes a coupling with Ψ a new colored quark charged under the PQ symmetry such that charges P Q Ψ + P QΨ + P Q P = 0. This coupling is the origin of the mixed PQ-QCD anomaly which allows the axion to solve the strong CP problem. The λ Ψ coupling plays an important role in the thermalization of the rotation.
The KSVZ model was carefully examined in Refs. [16,17]. We refer readers to these references for details, including the thermalization of the rotating PQ field. Here we focus on the implications of a factor of six enhancement in the baryon asymmetry production efficiency compared to Ref. [16]. This factor of six is the result of supersymmetrizing the Weinberg operator in Eq. (3.1), allowing lepton asymmetry production from scattering involving superpartners. This factor is independent of the UV completion of the axion and applies to the DFSZ case as well. The existence of superpartners in the bath also changes the efficiency of baryon asymmetry production by affecting the equilibrium Boltzmann equations and conserved quantities given in Appendix A.
As a benchmark, the observed baryon asymmetry is reproduced for a saxion mass where we use C i = 0.0106 based on Table III. In the determination of C i , we have gone beyond the one-generation approximation of Ref. [16]. This value of C i = 0.0106 applies when the anomaly coefficients for the weak and strong interaction are identical c W = c g (= 1) and when all Yukawa couplings are in equilibrium. To get the benchmark value 5.3, we take GeV and T f = T RM = 10 7 GeV.

DFSZ
In the DFSZ case, the effective µ-term depends upon the value of the scalar field. This effective µ-terms arises from the superpotential coupling 14) The idea of relating the µ-term to the scale of Peccei-Quinn symmetry is sometimes known as the Kim-Nilles mechanism, which was originally explored for the n = 2 case in [49].
Because the value of P changes during the universe's history, so too will the masses of the Higgs fields. As discussed below, this can impact the way in which the lepton asymmetry is transferred to the bath via Eq. (3.1).
The superpotential of Eq. (3.14) gives a temperature dependent µ(T ) = λP n /M n−1 . At temperatures before P settles to its minimum, this scales as R −3n/2 , which is proportional to T 3n/2 during radiation domination. We define a temperature T µ at which the temperature and the effective µ(T ) are equal, where µ is the present-day value to which µ(T ) settles for temperatures below T S . For temperatures T > T µ , scattering via the Weinberg operator is ineffective as the leptonnumber violation is limited to even higher dimension operators generated by integrating out the Higgs superfields.
So, the earliest temperature at which the chiral asymmetry may be effectively transferred to B − L is T µ . However, the reheat temperature T R is sometimes limited by BBN constraints [23,24] to values that are smaller than T µ . In this case, the earliest temperature relevant for transfer to B − L is T R . Based on Eq. (3.10) and N DW = 3n, a benchmark prediction of the saxion mass is To get the benchmark value of 5.3 in the parentheses, we have taken T i = T R = 2 × 10 9 GeV and T f = T S = 10 7 GeV. The benchmark value of C i corresponds to the case where all Yukawa interactions and the gaugino mass are in equilibrium; see Table II in the Appendix.
The saxion mass, which we assume to be of the same order as the soft scalar masses of the MSSM, may be O(10) TeV; this is consistent with the observed Higgs boson mass if the ratio of the Higgs field vacuum expectation values tan β ≫ 1. Larger m S is also possible, which could reproduce the Higgs boson mass for more modest values of tan β.

DETAILED ANALYSIS OF THE DFSZ MODEL
We now analyze the DFSZ model in detail. We discuss the thermalization of P via the coupling with the Higgs superfields in Eq. (3.14). We then show the allowed parameter space, determining both the minimum values of m S consistent with the generation of the baryon asymmetry and also the values of m S predicted by the production of both the baryon asymmetry and the dark matter abundance. We analyze the cases where the asymmetry is generated during reheating or the subsequent radiation-dominated era and the case where the saxion eventually dominates the energy. We discuss complications that may arise from the possible fragmentation of the rotation into Q-balls and explain how they can be avoided.

Thermalization
If the saxion does not thermalize sufficiently early, it will come to dominate the energy density of the universe. In this case, when it ultimately decays, it will produce entropy which can dilute the baryon asymmetry.
We assume that the dominant interactions of the saxion are via the coupling in the superpotential that gives the effective µ-term, Eq. (3.14). Then the saxion can be thermalized via its interaction with the Higgsino at a rate given by where the term with T or m S corresponds to the scattering or the decay rate, respectively.
For n = 1, the rate is independent of the evolution of the saxion field value S. The thermalization temperature is found by setting this rate equal to 3H, and is given by which is valid for T th ≫ m S , often the case for parameters of interest. The above expression assumes radiation domination. If the reheat temperature is below this T th , thermalization instead occurs during the period of inflationary reheating, and the actual thermalization temperature becomes lower than that in Eq. (4.2) (but above T R ) due to an enhanced Hubble rate with respect to that for radiation domination. However, thermalization of the saxion during inflationary reheating will not create more entropy than already created by the inflaton. So, the precise value T th will be irrelevant; instead, the value of T R will be important for analysis of the baryon asymmetry.
For n > 1 and S > N DW f a , the thermalization rate depends on S, so we need the scaling of S with temperature. Conservation of S number implies that S scales as R −3/2 . During increasing T faster than a radiation-dominated H ∝ T 2 does for any n > 1. Therefore, for n > 1 the saxion may thermalize at a high temperature but then decouple from the thermal bath when Γ S H H drops below the Hubble expansion rate. However, there is a maximum temperature at which the saxion can thermalize via Higgsino scattering, namely T ∼ T µ . Above this temperature, Higgsinos are out of equilibrium because their mass exceeds the temperature. To test whether thermalization occurs at this point, we equate Γ SHH | T =Tµ = 3H(T µ ). Using Eq. (3.15), for n = 2 we find that T S drops out from this relation, and the following constraint on f a may be derived f a ≲ 2 × 10 9 GeV µ 10 TeV For f a larger than this critical value, the coupling of saxion is too weak to thermalize at T µ .
Instead, thermalization waits until after T S and occurs at the lower T th given in Eq. Another possible thermalization channel is via the saxion scattering with the W gauge boson. This occurs with a rate given by where b ≃ 10 −5 [50][51][52]. Even when the saxion-W scattering does not completely thermalize the saxion, such scattering can play an important role in generating the thermal bath necessary for the Higgsinos to thermalize the saxion. This will be discussed in Sec. 4.3.2.

No saxion domination
The analysis of baryon asymmetry and dark matter production from axion rotations depends on whether the saxion comes to dominate the energy density and creates entropy upon its thermalization. In this section, we focus on the case where the saxion is thermalized sufficiently early so this does not occur. Then, for much of the parameter space, the baryon asymmetry production dominantly occurs during the radiation domination era following inflationary reheating. We also consider the possibility that the dominant production occurs during inflationary reheating, which can happen for low reheat temperatures. The case with saxion domination is analyzed in Sec. 4.3.

n = 1
In this section we give results for the n = 1 case, where the µ-term arises through a renormalizable coupling defined in Eq. (3.14). First, we discuss whether both the baryon asymmetry and dark matter may be generated by the dynamics of the axion field. In this n = 1 case, consistency with bounds on the axion decay constant from red giant cooling significantly constrains the ability to simultaneously realize the baryon asymmetry and axion Moreover, if additional thermalization channels beyond those described in Sec. 4.1 are present, then it would be possible to increase T th and hence the maximal yield. This could allow the KMM to reproduce the observed DM abundance for larger f a ; see Ref. [8].
Here, we do not include such channels. So, above this green line, an additional source of dark matter would be required. We assume that whatever produces the balance of the dark matter budget does not disturb the prediction of the baryon asymmetry. This would be the case, for example, if the dark matter were produced by thermal freeze-out of an LSP. The optimal cosmological evolution to obtain the smallest m S can be obtained as follows.
It is best to minimize T f = max(T S , T RM ) so the logarithmic enhancement in Eq. (3.9) is maximized, but this should be done while avoiding entropy production that would dilute the asymmetry. Thus, the maximum baryon generation efficiency is achieved if neither the saxion nor the rotation comes to dominate the total energy density. This is accomplished if T RM = min(T th , T S ). This ensures saxion thermalization (which occurs at T th ) happens early enough to avoid entropy production. It also ensures that the rapid redshift of the energy of rotation (which begins at T S ) occurs early enough that the rotation does not come to dominate; this will make the radiation-dominated era-and hence the period of logarithmically enhanced baryon production-as long as possible. This procedure also minimizes T f .
To clarify this cosmological history that produces the maximum asymmetry, we have shown relevant temperatures as functions of f a for m S = 30 TeV in Fig. 4. The left (right)  Fig. 3. The subscript "optimal" indicates the cosmological scenario where Y B is most efficiently produced, leading to the smallest required m S , corresponding to the curve segments above the green dotted line in Fig. 3. The optimal scenario is achieved by choosing the charge yield Y θ such that T RM, optimal = min(T S, optimal , T th ) so as to avoid rotation or saxion domination. For f a > 1.5 (7.5)×10 8 GeV, T RM, optimal follows T S (T th ), while T µ, optimal and T S, optimal change accordingly. Temperatures marked T y d indicate when interactions involving the down-Yukawa coupling come into equilibrium for different choices of tan β; different C i will apply above and below these lines, see (III). Temperatures denoted T µ show when Higgsinos come into thermal equilibrium.
panel is for T R = 2 × 10 9 GeV (10 7 GeV). As discussed above, the optimal scenario for efficient baryon asymmetry production is to choose the PQ charge Y θ (and hence the energy density in the saxion and rotation) so that T RM = min(T S , T th ). For f a > 7.5 × 10 8 GeV, this imposes T RM = T th , and we refer to this T RM as T RM, optimal , according to Eq. (2.13).
Once we have fixed T RM in this way, T S and T µ may be determined. We have shown these as the T S, optimal and T µ, optimal curves, which deviate from the values required by the KMM shown in dashed lines.
There is one final minor complication. The region above the orange line in Fig. 3 would lead to a period of matter domination followed by kination domination had we assumed axion dark matter from the KMM, i.e., T RM, KMM > T S, KMM using Eqs. (2.16) and (2.17). However, the goal for Fig. 3 is to derive the minimum m S rather than requiring the KMM. The optimal choice for T RM to find this minimal m S is not T RM, KMM , but rather T RM, optimal = T S, optimal , and this choice is applied in the region between the green dotted and the orange lines in Fig. 3 with m S ≳ 10 TeV. This is the case between f a = (1.5-7.5) × 10 8 GeV in both panels of Fig. 4, where T RM, optimal = T S, optimal . This optimal cosmology corresponds to Fig. 1 but with the blue curve shifted downwards and to the left so that the radiation energy density is equal to that of rotation at the kink in the rotation energy density.  Table II and Appendix A for details. See possible that for sufficiently low T R , the baryon asymmetry is dominantly produced during the period of inflationary reheating. We will discuss such effects below after commenting on the constraints from BBN.
A reheat temperature T R = 2 × 10 9 GeV with m S ∼ TeV requires either i) R-parity violation, ii) a small gravitino mass m 3/2 ∼ 100 GeV, or iii) a sneutrino as the next-to-LSP that is nearly degenerate with the gravitino LSP. For m 3/2 ≳ 7 TeV the bound on T R rapidly weakens, so we expect that the dashed and dotted curves are valid without any additional assumptions. See Appendix C for more details on BBN constraints.
As T R decreases, the generation of the baryon asymmetry is less efficient, and higher values of m S are needed to reproduce the observed baryon abundance. At minimum, this is because lowering T R reduces T i , the onset of the radiation-dominated era that is responsible for the logarithmic enhancement in the generation of the asymmetry in Eq. (3.9). This explains why the dashed and dotted curves are shifted to higher m S than the solid curves at low values of f a . For higher f a , the curve bends further and becomes a straight line because, although T RM still follows T th , eventually the resultant T S, optimal exceeds T R . When this occurs, Y B is no longer dominantly produced during an radiation-dominated era but rather during inflationary reheating. The logarithmic enhancement disappears, and the baryon asymmetry is diluted by entropy produced from the reheating, as in Eq. (3.11). The result is that the baryon asymmetry is sensitive to T S and therefore f a . This cosmological evolution may be clarified by examining the right panel of Fig. 4. There, T S, optimal can be seen to deviate from T S, KMM at f a ≃ 1.5 × 10 8 GeV (when T RM, optimal starts to track T S, optimal ), change its slope at f a ≃ 7.5 × 10 8 GeV (when T RM, optimal starts to track T th ), and then eventually exceed T R = 10 7 GeV for f a ≳ 2 × 10 9 GeV. hierarchy. The blue and red contours show the reheat temperatures required to reproduce both the baryon asymmetry from lepto-axiogenesis and dark matter from kinetic misalignment. The kinetic misalignment mechanism predicts a period of matter domination followed by kination domination in the entire parameter space shown here. The green region leads to underproduction of axion dark matter from kinetic misalignment because of entropy production from saxion domination, i.e., T RM > T th using Eqs. (2.10) and (4.2). The purple region is excluded by the red giant brightness observations. The brown region is excluded because the required PQ charge leads to an energy density of the PQ field ρ P exceeding that of the inflaton, while the brown dotted contours show lower values of ρ P /ρ inf . and the saxion ρ P ≡ ρ S + ρ θ (≃ 2ρ θ for A ≃ m S according to Eq. (2.12)), exceeds that of the inflaton. The origin of this region may be understood by noting that larger values of m S require less efficient production of Y B , which may be achieved by a smaller logarithmic enhancement during radiation domination by decreasing the ratio between T R and T RM . For fixed T R , this requires a larger T RM in Eq. (2.10). However, eventually T RM becomes as large as T R /2, at which point ρ P ≃ 2ρ θ = ρ inf , and an inconsistency arises because P would instead drive an epoch of inflation. Two brown dotted curves are shown for ρ P /ρ inf = 0.1 and 0.01, which are perhaps more realistic energy densities for P . In summary, the saxion mass is now predicted to have a strict upper limit of 125 (240) TeV for inverted (normal) hierarchy with lower m S preferred for a realistic ρ P . This mass range is intriguingly consistent with the supersymmetry-breaking scale determined from the observed value of the Higgs boson mass.

n = 2
We now move to the case of n = 2 and N DW = 6. Our focus in this case will be on the parameter space where both the baryon asymmetry and dark matter abundance result from the axion rotation. This case is presented in the left panel of Fig. 6 for µ = m S /5. The blue curve shows the minimum values of m S compatible with axion dark matter and the baryon asymmetry both arising from axion rotations. This minimum m S is achieved when T R is sufficiently high, as discussed below. The blue curve assumes the inverted hierarchy neutrino mass spectra as labeled, while the normal hierarchy case overproduces the baryon asymmetry in this parameter space. We will see that the requirement of the successful generation of both the baryon asymmetry and dark matter prefers a relatively small region of m S ranging from around 60 TeV to nearly 100 TeV and f a = (1-2) × 10 9 GeV.
Above the orange line in the left panel of Fig. 6, a period of matter domination followed by kination domination exists because T RM > T KR . In this case, the era of logarithmically enhanced baryon production ends at T RM , when the matter domination begins. The region above the orange line also gives a potential signal in the modification of primordial gravitational waves [30][31][32]. This is discussed around Eq. (3.9) and illustrated by the red shaded region in the right panel of Fig. 6. To accurately obtain the final Y B , rather than using the analytic estimate given in Eq. (3.9), we numerically integrateṅ B−L R 3 from T µ to a temperature much lower than T RM ; this improves the precision of the prediction on m S and changes the prediction by a factor as large as 2.
In the green shaded region, axions from kinetic misalignment cannot account for all of dark matter because the necessary axion yield Y θ = 3rT th /4N DW m S requires a T th value that is higher than can be achieved from saxion-Higgsino scattering given in Eq. negatively-sloped boundary of the green region, we have T th = T µ and T th > T RM so that the saxion does not create entropy upon thermalization. Lastly, as can be seen in the right panel, the era that dominates the production of the baryon asymmetry begins at T µ , and therefore the result is independent of T R as long as T R > T µ . Using T µ from Eq. (3.15) and T S from Eq. (2.16), one finds T µ ≃ 6 × 10 7 GeV (f a /10 9 GeV) 1/2 (m S /5µ) 1/2 . The calculated asymmetry will be valid for all T R larger than this value.
In other words, by lowering T R , one can explain the baryon asymmetry and dark matter in the region to the right of the magenta line. However, there is a limit on how low T R can be: in the low T R and high m S limits, T R and T RM approach each other, and when T R = 2T RM , the energy density of the complex field ρ P ≡ ρ S + ρ θ is equal to that of the inflaton ρ inf if A ≃ m S , i.e., ρ S ≃ ρ θ based on Eq. We do not analyze the baryon asymmetry in the green shaded region, where axion dark matter is underproduced by kinetic misalignment, because we find that in some of the parameter space the onset of the P rotation can be initiated by the saxion thermal mass. This is because the thermal mass is proportional to µ, which is in turn enhanced at high temperatures by S n−1 . Rotations initiated by the thermal mass complicate the determination of the optimal cosmological evolution for the most efficient baryon asymmetry production.
The thermal mass also leads to potential formation of Q-balls whose presence makes the baryon asymmetry calculation uncertain; see Sec. 4.5.

Saxion domination
In this subsection, we discuss a different cosmology where both dark matter and the baryon asymmetry may still be produced by axion rotations. We do not optimize the pro-duction of the baryon asymmetry nor utilize T R to explore the parameter space as in Sec. 4.2, but rather study the case where the saxion dominates the energy density before it is thermalized. In this case, the saxion creates entropy that dilutes the baryon asymmetry produced immediately after inflationary reheating. Consequently, the final asymmetry is dominantly produced after saxion thermalization and is therefore independent of the inflationary reheat temperature T R . The predicted m S is generically larger than the optimal cases studied in Sec. 4.2 giving the most efficient baryon asymmetry production. We now comment on the effect of changing µ. If the value of µ is increased, this makes thermalization more efficient and would increase T th . This breaks the relation T RM = rT th .
However, this relation can be restored by going to higher f a since T RM ∝ f a and T th decreases with increasing f a . The predictions for µ = 3m S will be shown and discussed.

n = 1
The thermalization temperature for n = 1 is given in Eq. (4.2). In the saxion domination case, T RM = rT th with T RM determined by requiring dark matter from kinetic misalignment, and this gives a relation between m S and f a for a given r. Furthermore, to accurately derive line is the prediction of kinetic misalignment alone for n = 2 assuming r = 1 (predictions for n = 2, r < 1 are not included due to complications involving thermalization, see text). In the regions above the orange curves, axion dark matter from kinetic misalignment predicts eras with matter and kination domination, which may leave imprints in primordial gravitational waves [30][31][32].

n = 2
We continue to analyze the case where the saxion comes to dominate the energy density of the universe before being thermalized, i.e., T RM > T th , but now for n = 2. The results are given in Fig. 7. The diamonds show the points predicted by requiring both the baryon asymmetry and dark matter abundance, which are in tension with the red-giant observations. We nevertheless analyze this n = 2 scenario to obtain the prediction from the dark matter abundance (black dashed curves) but with an underproduced baryon asymmetry. In what follows, we assume the high T R and/or large initial S limit so the inflationary reheating contribution to the bath is negligible around thermalization. (For instance, for the initial S close to the Planck scale, T R > O(10 7 ) GeV is sufficient for this assumption to hold.) The origin of the bath must be from the saxion-W scattering. The temperature of the bath that originates in this way increases [57] during saxion reheating because the temperature dependence of the rate given in Eq. (4.4).
As the temperature increases, it may eventually become equal to the effective µ(T ) at a temperature we call T th,i , at which point thermalization via Higgsinos is initiated. Thermalization may then suddenly complete via saxion-Higgsino scattering, and the temperature increases abruptly to T th as the saxion energy is suddenly converted to the bath. This occurs as long as Γ S H H > H. We assume the saxion field value does not change significantly after thermalization, S th ≃ S th,i , which is the case if the initial rotation is nearly circular (r ≃ 1). The temperature right after thermalization T th can be computed by requiring 1) conservation of energy ρ S = m 2 S S 2 th,i = π 2 30 g * T 4 th , 2) initial radiation created by W scattering ρ S th,i with the subscript "th, i" indicating evaluation right be-fore the abrupt thermalization, and 3) the condition for Higgsinos to just come into thermal equilibrium T th,i = µ(T th,i ) = µ × (S th,i /N DW f a ) 2 . We obtain (4.5) Using this expression, we find that along the black dashed line of Fig. 7, the PQ charge yield can eventually overtake H. However, in this regime, we find that axion dark matter is underproduced because the low T th gives an insufficient PQ charge Y θ = 3rT th /4N DW m S .
In deriving this black dashed line, we have assumed r = 1. One may be tempted to naively extend the calculation to derive the prediction for lower values of r because r seemingly appears to affect only S th /S th,i . For r < 1, Γ S H H ∝ S 2 may first be larger than H when T reaches T th,i but become smaller than H before reaching complete thermalization at T th .
The condition Γ S H H > H should instead be evaluated at T th with S th rather than at T th,i .
However, we note that thermalization via Higgsino scattering may be further complicated by the fact that the value of S can get close to the origin in some portion of the cycle when the rotations are very elliptical, r ≪ 1. During this portion, the saxion-Higgsino scattering may proceed because T < µ(S) near the origin, while T > µ(S) when P is far away from the origin. We do not pursue this possibility further.  but such low f a is in tension with the red giant constraints. In fact, even the degenerate limit of the neutrino spectrum leads to underproduction of the baryon asymmetry. Lowering µ may increase the predicted values of f a , moving towards compatibility with the red-giant bound, but the black dashed line is truncated at lower f a . As a result, we do not find viable parameter space for a sufficient baryon asymmetry after marginalizing over µ.

Interpretation of results for one-field model
In this subsection, we re-interpret the results presented in Figs In what follows, we define m UV S as the curvature at S = M Pl . We expect this value to be comparable to other scalar masses in the UV. We will discuss how our earlier results are modified under the understanding that the x-axes of the figures will now refer to m UV S . We denote m S (z) as the curvature at lower energy scales; z may refer to the S field value or a temperature to indicate the corresponding field value S(T ) at T , i.e., m S (T ) = m S (S(T )).
The field dependence of m S (S) is shown in Fig. 8. Because the change of m S (z) is only logarithmic, the overall effect on the results is modest. In what follows, we will discuss this effect in detail. Our strategy will be to fix f a and find the value of m UV S that would reproduce the physics of a field-independent m S in each case.
We begin by discussing the effects of the evolution of m S (z) on saxion thermalization.
The green dotted line in Fig. 3, the green boundary of Fig for T i > T > T f . For Figs. 3 and 5 (n = 1), T i is T R and T f is often T S (see Fig. 4). For Fig. 6 (n = 2), T i is T µ , which is not much above T S , and T f is often T S . Since m S is lower at these lower temperatures compared to m UV S , the effect is to reduce the efficiency of Y B−L production. To compensate for this, m UV S needs to increase by a factor of a few.
This results in a shift of the prediction curves to the right. As a result of this shift and the left shift of the green dotted line in Fig. 3, the hierarchical cases shown by blue and red curves are more easily compatible with the red giant bound and the green constraint for dark matter. In other words, a viable parameter space for dark matter would open up with a milder hierarchy between µ and m S than µ = 3m S assumed in Fig. 5. The shift would be more prominent in Fig. 6 than Fig. 3 because the former case involves m S (T ) only at temperatures close to T S .
The brown regions/curves in Figs. 5 and 6 are also affected by a changing m S . As explained in Secs. 4.2.1 and 4.2.2, these constraints occur because a successful production of Y B would require T RM > T R /2, which would result in a period of inflation driven by the saxion. First, as explained above, an evolving m S (T ) decreases the efficiency of production relative to the constant case, so to reproduce the baryon asymmetry, a larger m UV S is required. Second, because m S (T RM ) is smaller than m UV S , the saxion will take longer to come to dominate, and so T RM is smaller in the case where the saxion mass evolves. This means the constraint is relaxed, which also shifts the brown regions/curves to higher m UV S . Lastly, in Fig. 7, the dominant (logarithmically enhanced) era of asymmetry production is present between T th and T RM = rT th . For r = O(1), T RM ≃ T th ≫ T S , so m S (T ) during this era is O(0.5)m UV S , and the predicted points will shift to the right by a factor of 2 or so. The predicted points for r ≪ 1 are excluded by red giants whether or not we account for the effect of m S (z).
On balance, for the one-field model, larger values of m S are preferred than in the two-field case, often by a factor of few.
We discuss the potential domain wall problem in the one-field model. After the initiation but before the thermalization, the rotation is generically not circular. For non-circular motion, fluctuations of the PQ breaking field can be produced by parametric resonance [16,[33][34][35][36][37]58]. The PQ symmetry may be non-thermally restored by the fluctuations and broken again once the fluctuations are reduced by the expansion of the universe. If this actually occurs, a domain wall-string network is produced, which is stable if N DM > 1 and causes a domain wall problem. Unlike the case without angular momentum [59,60], it is not clear if the restoration actually occurs, since the non-zero angular momentum provides an effective potential that strongly disfavors the origin of the field space. We leave the investigation of the dynamics via numerical lattice computation to future work, and only note that the one-field model may require N DW = 1 or explicit PQ breaking that can destroy the domain walls N DW > 1 [61].

Q-balls
If the potential of the S field is nearly quadratic, a small correction may make the potential flatter than a quadratic one, for which a non-topological soliton called a Q-ball may be formed [62][63][64][65][66]. Q-ball formation can complicate the thermal history. If formed, Qballs will localize the PQ charge inside them. It is unclear as to what the spatial distribution of theθ will be as the universe evolves in the presence of these Q-balls. This uncertainty would confuse the evaluation of the baryon asymmetry.
Most discussions of Q-balls have taken place in the context of potentials with minima near the origin in field space. It is possible that the symmetry-breaking potential of P allows the Q-balls to decay or even prevents its initial formation. While understanding the dynamics of the Q-balls associated with a symmetry-breaking potential such as the one needed for the axion is of interest, we leave it for future work. For now, we assume that the properties of the Q-balls in the present setup are identical to the more familiar ones associated with potentials that have minima at the origin. We then comment on which cases Q-ball formation might confuse the calculation of the baryon asymmetry, while keeping in mind that future investigations might mitigate these concerns. Histories that include Q-ball formation may actually ultimately prove viable.
For n = 1, the thermal potential, given in the second terms of Eqs. (4.6) and (4.8) below, is flatter than a quadratic one for both µ(S) > T and µ(S) < T , so once Q-balls are formed, they would remain stable until T ≪ m S when the quantum correction to the soft mass of S from interactions with the Higgs fields dominates over the thermal potential. In this case, the estimation of the baryon asymmetry would potentially be rendered invalid, because the Q-balls would be present during the epoch that is important for the generation of the asymmetry. We may avoid the era of a flat potential by coupling P to additional fields, W = y ψ P ψψ. Because we will require a large y ψ , the ψ fields receive a large mass from the large P field value and are not present in the thermal bath. Assuming that it is gauge-singlet, ψ also does not introduce a coupling of P to gauge bosons. So, the effect of ψ is to introduce a modification of the zero-temperature potential. Assuming that the soft mass squared of ψψ is positive, quantum corrections to the soft mass of P induced by this coupling steepens the zero-temperature potential and can destabilize Q-balls. So, with an O(1) coupling y ψ , for µ(S) > T the non-quadratic part of the potential of S is where κ ∼ 1/(16π 2 ). Q-ball solutions exist if V /S 2 is minimized at non-zero S. The above potential has a minimum at S 2 ∼ α 2 T 4 /(κm 2 S ). Requiring self-consistency with the condition µ(S) > T , we obtain The Q-ball solution may also exist in the regime µ(S) < T , for which the non-quadratic part of the potential of S is where y = µ/(f a N DW ) is the coupling between P and H u H d , c T ∼ 1/(16π 2 ). Note that the thermal trilinear term −y 3 S 3 T is absent since the Higgs field obtains a large thermal mass ∼ gT and the IR singularity is removed. The minimum of V /S 2 is at S 2 ∼ κm 2 S /(c T y 4 ). For consistency, this should satisfy yS < T , so we obtain Comparing Eq. (4.7) with Eq. (4.9), the latter gives a slightly stronger condition, so Q-balls disappear when Eq. (4.9) is violated. Unless T R ≫ f a , the production of B − L asymmetry dominantly occurs after Q-balls disappear, so the estimation of B − L asymmetry is not affected by the production of Q-balls. Given current bounds on f a and constraints on T R from BBN, we do not expect T R ≫ f a .
For n = 2, the potential of S is flatter than a quadratic one only for µ(S) > T . Therefore, even if Q-balls are formed, once the field value of S inside the Q-balls is such that µ(S) < T , Q-balls should disappear. However, this can occur only at a temperature below T µ , since the field value of S inside the Q-balls is larger than the average field value. With Q-balls at temperatures below T µ , the estimation of B − L asymmetry may be affected. We may avoid this by a coupling W = P ψψ as in n = 1. So, for n = 2, when the condition in Eq. (4.7) is violated, Q-balls disappear.
So, for both n = 1 and 2, even if Q-balls form at the early stage of the evolution of the axion rotation, they can disappear by the era when B − L asymmetry is produced by lepto-axiogenesis if there exists a coupling to extra fields ψψ. We stress again that this extra couplings may not be necessary because the symmetry breaking potential of S may lead to additional effects that destabilize the Q-balls.
We note that the Q-ball formation may lead to production of domain walls. Indeed, Q-ball formation is a result of the growth of fluctuations. As in the parametric resonance during the oscillation of the PQ symmetry breaking field [59,60], the growth of fluctuations may non-thermally restore the PQ symmetry and produce domain walls. Since N DW > 1 for the DFSZ model, domain walls are stable and will come to dominate the universe. However, we expect that the symmetry restoration would not occur in the two-field model since the PQ symmetry-breaking fields are fixed on the moduli space where the PQ symmetry is broken.
In the one-field model, on the other hand, the symmetry restoration might occur. Whether or not the domain wall production actually occurs should be investigated by numerical computation; it is possible that the non-zero angular momentum in field space tends to expel the field from the center and prevent the symmetry restoration.
In summary, it remains to be seen whether or not Q-balls, if formed, are ultimately problematic, and whether they disturb the calculation of the baryon asymmetry. However, coupling the PQ-field to other fields induces quantum corrections to the saxion potential that steepen it and can avert Q-ball production.

DISCUSSION
In this work we have explored the possibility that the observed baryon asymmetry arises from the interplay of early-universe dynamics of the axion and the origin of neutrino masses.
Under this assumption, we could obtain information on the mass of the saxion, the radial mode of the complex field that contains the axion. In models of gravity mediation, the mass of the saxion would be comparable to the masses of the MSSM particles. So, one can interpret the results as predictions for the masses of the superpartners. We have investigated the DFSZ model in detail including the successful thermalization of the saxion.
For a hierarchical neutrino mass spectrum, the scalar mass may be as low as O(10) TeV.
The observed Higgs boson mass in this case may be explained by moderately large tan β. For the scalar mass of O(10) TeV, the gaugino masses given by the anomaly mediation [67,68] is below O(100) GeV, so singlet SUSY-breaking fields must be present to give phenomenologically viable gaugino masses. This generically leads to the Polonyi problem [69], which can be avoided by a large coupling between the SUSY-breaking fields and the inflaton [70][71][72] or a coupling between the SUSY-breaking fields and a pseudo-flat direction [73].
Successful thermalization of the rotation typically requires µ different from m S by an O(1) factor. For µ > m S , electroweak symmetry breaking requires the soft masses of the Higgs fields to be also larger than m S by an O(1) factor.
If reheat temperatures are somewhat lower than the maximum value considered here, or if the saxion comes to dominate the energy density of the universe at some point in its history, then the scalar mass is required to be larger. Interestingly, after requiring the kinetic misalignment mechanism to explain the observed dark matter abundance, we find the scalar mass is at most 300 TeV. (One can check that the predicted scalar mass is still small enough that the tachyonic instability to create a helical magnetic field is ineffective, so the associated overproduction of the baryon asymmetry recently noted in Ref. [74] is avoided.) The scalar mass of 300 TeV is compatible with the scenario without singlet supersymmetry-breaking fields [68], also known as mini-split SUSY, pure gravity mediation, spread SUSY, etc. In this scenario, the infamous Polonyi problem and the BBN gravitino problem are absent, the SUSY flavor/CP problem mitigated, and the observed Higgs boson mass in this case can be explained with tan β of order unity [75][76][77][78][79][80][81][82][83][84]. The dominant contribution to the gaugino mass is given by anomaly mediation [67,68], and the gauginos may be searched for at the LHC.
As for the axions, we find a preferred region that simultaneously predicts the dark matter and the baryon asymmetry with f a ∼ 10 9 GeV, just above the current bound from observations of red giants. This presents a target for experimental searches including the Broadband Reflector Experiment for Axion Detection (BREAD) [85], the Axion Resonant InterAction Detection Experiment (ARIADNE) [86,87], or other future detectors [88].
Questions regarding the dynamics of the rotating axion field remain. As discussed in Sec. 4.5, Q-balls can form when the saxion potential is flatter than a quadratic one. The spatial distribution of the angular velocity of the axion field after Q-balls form, but prior to their decay, is of importance to accurately estimate the efficiency of axiogenesis scenarios. In Sec. 4.5, we introduced new couplings of the PQ-field to hasten the disappearance of the Qballs, rendering them irrelevant. However, even in the absence of these additional couplings, we expect Q-balls to eventually decay since the zero-temperature potential does not admit isolated Q-ball solutions. When the decay actually occurs requires additional investigation, perhaps with the help of a lattice computation. Because the requirement of a large initial field value constrains the potential of the saxion to be nearly quadratic in axiogenesis, the condition for an epoch of Q-ball formation should be satisfied rather generically. This makes answering the fate of axiogenesis in the presence of Q-balls a particularly interesting question.
number asymmetry in the Boltzmann equations to vanish. The solution to this system of equations depends on the magnitudes of coupling constants. However, because the up-Yukawa coupling is small, it can be set to zero to a good approximation [14]. And while the goal is to find the (quasi)-equilibrium values for the case where the chiral symmetry is completely broken, this procedure, wherein we take the parameter which breaks the chiral asymmetry the least (the up-Yukawa) to vanish, will reproduce the leading contribution to the asymmetry. Then, with this prescription for the up-Yukawa coupling in place, taking the time-derivatives to be zero is equivalent to applying the principle of detailed balance to each scattering process.
The equilibrium conditions for the remaining Yukawa interactions are We have chosen to express equilibrium conditions in terms of the fermionic part of each chiral supermultiplet, and µ λ is the chemical potential of gauginos. Since the doublet quarks and squarks couple to all gauginos, as long as the gauge interaction is in thermal equilibrium, all gauginos have the same chemical potential. The scalar and fermionic chemical potentialsowing to in equilibrium interactions with gauginos-are related by where ϕ and ψ represent the scalar and fermion part of a chiral supermultiplet, respectively. While the charged lepton and up-quark Yukawa interactions may be taken to be flavor diagonal, in general, there will be off-diagonal components for the down-quarks, see where c W and c g are the weak and strong anomaly coefficients of the PQ symmetry. These anomaly coefficients are set to zero in the DFSZ case, but not the KSVZ case. Because ρ θ is given as −θn θ , µ θ must be −θ. Other interactions to consider are chiral-symmetry violation by the gaugino mass and either the standard MSSM µ-term (for KSVZ) or the interaction in Eq. (3.14) (for DFSZ), which give Setting n = 0 in Eq. (A.9) corresponds to taking the standard MSSM µ-term for KSVZ, while taking n ̸ = 0 corresponds to the DFSZ case.
In addition to the above detailed balance relations, we must also impose the conservation laws to determine the asymmetry. With all interactions in thermal equilibrium, the only conservation laws are those of weak hypercharge, Y = 0, and B/3 − L i for each generation i. B/3 − L i is violated by the superpotential in Eq. (3.1), but this interaction is never close to equilibrium, and it is therefore a small perturbation that may be neglected for the computation of chemical potentials. For µ i , m i ≪ T , the net fermion and boson densities are given by n ψ − n ψ † = g 6 T 2 µ ψ and n ϕ − nφ = g 3 T 2 µ ϕ , so the hypercharge and B/3 − L i conservation conditions can be expressed in terms of chemical potentials as

Out of equilibrium Yukawa interactions
The conservation law that is broken at the lowest temperature isē 1 number conservation.
This is finally broken when the scattering rate involving the electron Yukawa coupling with rate α 2 y 2 e T overtakes the Hubble expansion rate. Whenē 1 number is conserved, Eq. (A.1) for i = 1 should be replaced by (A.14) Other Yukawa interactions could be out of equilibrium, but this would occur at high enough temperatures that the B − L production by lepto-axiogenesis is subdominant.
The temperatures at which these different conservation laws are broken are shown in RGEs [90] of the MSSM. In Fig. 9, it is assumed that the Hubble parameter is that of a radiation-dominated universe with g * = g MSSM = 228.75. A universe with fewer relativistic degrees of freedom would break the symmetries at a higher temperature, while a universe not dominated by radiation would break them at a lower temperature.
Whether or not the high temperatures where these new conservation laws apply are compatible with constraints from supersymmetric relics, see Appendix C, depends on the details of the spectrum.

Out of equilibrium gaugino masses and µ-term
At sufficiently high temperatures, scattering due to the gaugino mass or the µ-term may be ineffective.
First, we discuss the gaugino mass. The rate of chiral-symmetry violation by the gaugino mass is Γ ∼ m 2 λ /T . Equating this rate with the Hubble expansion rate during a radiationdominated era, we find this interaction goes out of equilibrium for temperatures above  Above these temperatures, the chemical potential associated with the gauginos µ λ will no longer vanish, and it will enter into the equations that result from the Yukawa interactions   To find the new conservation law, we must identify the relevant R-symmetry. It should be non-anomalous with respect to SU (3) and SU (2). One set of R-charge assignments for this symmetry for the MSSM superfields is given in Table I. Because this symmetry is an R-symmetry, the charge of the fermions is Q ψ = Q R − 1. Gauginos have Q λ = 1.
Contributions to would-be anomalies are 2N c from gauginos, and Q R × N from the chiral superfields, where N counts the multiplicity. This allows us to verify that the would-be R-SU (2)-SU (2) anomaly cancels between winos and the leptons (4 − (4/3)N g =0). The would-be R-SU (3)-SU (3) anomaly cancels between gluinos and the right-handed quarks (6+ 2N g (−1) = 0), where we have combined the contributions from the up-type and down-type quarks. In terms of chemical potentials, the conservation condition for this R-symmetry is If the axino-gluino-gluon or axino-Higgs-Higgsino coupling were in thermal equilibrium, the axino would contribute µ λ or µ Hu + µ H d + µ λ to this expression for the conserved charge.
This affects the values of C i by at most a few percent, so we ignore these possibilities.
In the case of the KSVZ axion, the µ-term is ineffective at temperature higher than Eq. (A.15) with m λ replaced with µ. We do not expect that there is such a temperature regime in the DFSZ case because the µ-term itself will increase with temperature, keeping it in equilibrium. If such a temperature regime does exist, this results in yet another conserved symmetry, which can be taken to be a linear combination of the Weinberg-Wilczek Peccei-Quinn (WW) symmetry wherein Q Hu = Q H d = 1, and Q Q = Q L = −1 and two additional symmetries: B + L, and the symmetry under which only the right handed up quark is charged Q u1 , (Q(U c ) = 1). The resulting charges for a non-anomalous symmetry are given where we have added a multiple of the non-anomalous B − L symmetry to give the more convenient charge assignments shown in Table I. In terms of chemical potentials, the W W ′ conservation condition is . Results for C i Solving the relevant system of equations for the µ i allows determination of C i , see Eqs. (3.5) and (3.6).

DFSZ:
In Table II, we show C i for the various cases in the DFSZ model. When all Yukawa interactions and the gaugino mass term are in equilibrium, C 1 = C 2 = C 3 = 0.0459 n N DW , independent of the PMNS mixing angles. When the electron Yukawa is out of equilibrium, the C i coefficients are slightly different from each other and depend on the PMNS mixing angles. But using PMNS mixing angles θ 12 = 34 • , θ 23 = 48 • , θ 23 = 8.5 • , the C i coefficients are still all 0.046 n N DW to two significant digits. Whether the down-Yukawa interaction is out of equilibrium has a more significant impact; in this case the coefficients become C 1 = 0.0229 n N DW , C 2 = 0.0203 n N DW , and C 3 = 0.0182 n N DW , and the resulting asymmetry can be affected by more than a factor of 2. Whether or not the offdiagonal Yukawa interactions with the down quark are in equilibrium has no effect on C i .
Whether the gaugino mass term is in equilibrium has a small effect. For the cases when the down Yukawa is in equilibrium, the difference is roughly 3%, pushing C i to 0.0446 n N DW . The effect on the cases where the down is out of equilibrium is similarly small. Table III  KSVZ m λ and µ Efficient All Yukawas Efficient y e Inefficient y e and y d Inefficient C 1 0.0037c g + 0.0069c W 0.0016c g + 0.0082c W −0.0063c g + 0.0083c W C 2 0.0037c g + 0.0069c W 0.0033c g + 0.0072c W −0.0055c g + 0.0074c W C 3 0.0037c g + 0.0069c W 0.0047c g + 0.0064c W −0.0050c g + 0.0066c W m λ Inefficient All Yukawas Efficient y e Inefficient y e and y d Inefficient C 1 0.0037c g + 0.0089c W 0.0016c g + 0.0098c W −0.0063c g + 0.0083c W C 2 0.0037c g + 0.0089c W 0.0033c g + 0.0091c W −0.0055c g + 0.0074c W C 3 0.0037c g + 0.0089c W 0.0047c g + 0.0085c W −0.0050c g + 0.0067c W m λ and µ Inefficient All Yukawas Efficient y e Inefficient y e and y d Inefficient C 1 −0.0107c g + 0.0063c W −0.0126c g + 0.0072c W −0.0127c g + 0.0071c W C 2 −0.0107c g + 0.0063c W −0.0111c g + 0.0064c W −0.0112c g + 0.0064c W C 3 −0.0107c g + 0.0063c W −0.0098c g + 0.0059c W −0.0101c g + 0.0058c W   TABLE III. C i coefficients in the KSVZ model when different reactions are in equilibrium. The first group of rows corresponds to the case when scattering via the gaugino mass and µ-term are in equilibrium. The second group corresponds to the case where the µ-term is in equilibrium but the gaugino mass is not. The third group corresponds to the case where both the gaugino mass and µ-term are out of equilibrium. The first column corresponds to the low-temperature case when all Yukawa interactions are in equilibrium. The second gives results when only the interactions via the electron-Yukawa is out of equilibrium, and the third also has down-Yukawa interactions out of equilibrium. In the standard normalization of the axion-gluon coupling, c g = 1.

KSVZ: In
are in equilibrium has no effect when scattering via the µ-term is efficient, and an effect only on the level of several percent when the µ-term is inefficient.
Scaling of quantities relevant for the estimation of the B − L asymmetry. Positive (negative) exponents for R in the final two columns indicate IR (UV)-dominated production. The case that scales as R 0 has equal contributions per Hubble time and so receives a logarithmic enhancement; see text for details. We note that ρ matter represents either ρ inf or ρ S depending on which one dominates and creates entropy.
thermalization before dominating the total energy density. If instead the saxion comes to dominate and subsequently creates a large amount of entropy from its thermalization, any previously produced baryon asymmetry can be sufficiently diluted so that the production after saxion thermalization dominates. As discussed in Sec. 4.3, during the non-adiabatic era before the end of thermalization, the relevant thermalization processes are saxion-Higgsino and saxion-W scatterings for n = 1 and n = 2, respectively. Production of n B−L per Hubble time is listed in Table IV for these two cases with the label MD osc NA , and one can see that production is IR-dominated for both cases. (We do not show the scaling for T < T S here; it is never realized in our parameter space.) This verifies that the contribution produced subsequent to thermalization of the saxion dominates over that produced during thermalization.
The production after thermalization is again logarithmically enhanced during a radiationdominated era but now between T th and max(T RM , T S ) with T RM given by Eq. (2.13). The results for the saxion domination scenario are presented in Sec. 4.3.

Appendix C: Constraints from supersymmetric relics
In this supersymmetric framework, there are a number of potentially long-lived relics.
These relics may provide constraints on the theory. The constraints depend upon the identity of both the LSP, and if long-lived, the next-to-lightest supersymmetric particle (NLSP). The predictions of Big Bang Nucleosynthesis (BBN) must not be disturbed, and, if stable, the LSP density may not exceed the dark matter density.
Non-gravitino/axino LSP: We first consider the case of the LSP being a superpartner of a Standard Model particle. The constraint on the mass spectrum and/or the reheat temperature from BBN is discussed in [23]. If the gravitino mass m 3/2 ∼ TeV, late gravitino decays will disturb BBN unless the reheat temperature T R < ∼ 10 6 GeV. The bound can be relaxed if the LSP is a slepton, but a charged LSP is strongly constrained by searches for heavy hydrogen [91] and a sneutrino LSP is excluded by direct detection experiments.
Because this value for T R is close to the typical T RM (or even smaller) this means that any logarithmic enhancement, see Eq. (3.9), is necessarily absent, and the prediction for m S is somewhat modified (increased). If m 3/2 ∼ 10 TeV, the upper bound is T R < ∼ 10 8 GeV.
It is conceivable that m 3/2 is quite large with mass > ∼ 100 TeV, in which case the gravitino decays might be early enough to avoid conflicts with BBN and larger reheat temperatures might be allowed. However, in this case, the scalar mass must be also O(100) TeV; otherwise for m 3/2 ≫ m S , the A-term in Eq. (2.5) becomes much larger than m S , and P is trapped at a minimum with large S. Additionally, in order for the thermal freeze-out abundance of the LSP (say wino or Higgsino) not to be too large, a hierarchy of the type m LSP ≪ m S ∼ m 3/2 is required. Moreover, even if gravitino decays during BBN are avoided and the LSP thermal abundance is not too large, there is still a danger of non-thermal overproduction of the LSP from gravitino decays. For a gravitino mass of 100 TeV and a LSP mass of a TeV, this constrains the reheat temperature T R < 2 × 10 9 GeV [23].
The above upper bounds on the reheat temperature could be relaxed if R-parity is violated. In this case, we may assume a slepton LSP, thereby weakening the BBN constraints from gravitino decays, but without conflicting with heavy isotope searches nor direct detection. If there are no sparticles between the gravitino and the slepton(s), the upper bound becomes T R < 10 9 (10 11 ) GeV for m 3/2 ∼ 1 (10) TeV. For m 3/2 > 100 TeV, the LSP overproduction bound disappears and T R may be much above 10 9 GeV.
Gravitino LSP: To avoid overproduction of a gravitino LSP from thermal processes requires a reheat temperature T R < 2 × 10 9 GeV × (TeV/m 3/2 ). In this case a logarithmic enhancement as in Eq. (3.9) can remain. However, avoiding disruption of BBN via decay of the (visible sector) NLSP puts strong constraints on the parameter space if T R is above the sparticle masses. The strongest constraints arise [23,24] when where the NLSP has a large branching ratio to hadrons; constraints are minimized for a sneutrino NLSP. This can be realized by taking the soft mass of5 to be smaller than that of 10. Then, the dominant constraint comes from a three-body decay involving a weak boson, whose branching ratio is O(10 −2 ). For the gravitino mass of TeV, the sneutrino NLSP lifetime is around 10 5 s, and from the constraint on the decay into weak gauge bosons derived in [24], we obtain mνYν × Br(three-body) < 10 −14 ⇒ mνYν < 10 −12 . (C.1) The freeze-out abundance of a TeV-scale sneutrino would violate this bound. To evade the bound requires mν > 10 TeV, so that the lifetime of the sneutrino is shorter than 100 s. We may also avoid the bound by taking mν − m 3/2 < m Z . In this case, the decay mode relevant for the BBN constraints becomes a four-body decay with a branching ratio ∼ 10 −4 , and the constraint is marginally satisfied.
If the cutoff scale is below the Planck scale, m 3/2 ≪ m NLSP and the NLSP lifetime may be shorter. For example, with the cutoff scale around the string scale ∼ 10 17 GeV, m 3/2 ∼ 100 GeV with m NLSP ∼ few TeV is possible. The lifetime is then shorter than 100 s, and m NLSP Y NLSP < 10 −10 and 10 −7 is required for the NLSP with the leading hadronic decay mode and the sneutrino NLSP, respectively. This is satisfied for m NLSP = O(1) TeV.
The bound on the mass spectrum may be avoided if R-parity is violated, since the NLSP can decay much before BBN. The gravitino may still be long-lived enough to be dark matter.
In this case, R-parity violating couplings could provide an additional source of asymmetry, see [14], but this is small if the couplings are not large.
Axino LSP: The axino should not be the LSP unless R-parity violation is introduced.
To see why this is so, recall the saxion is thermalized. Unless this thermalization occurs below the masses of the sparticles, the axino is also thermalized. Unless the axino mass mã is below O(100) eV, axino dark matter is overproduced. However, even a subdominant component of hot dark matter is constrained, so a stronger bound mã < ∼ O(10) eV applies [92].
While in some of the parameter space saxion thermalization does occur at T th < ∼ m S , and it may be possible to avoid thermalization of the axino, it is nevertheless potentially produced in dangerous amounts via freeze-in at higher temperatures. Indeed, unless the mã ≪ TeV-difficult in gravity mediation-the axino is still overproduced. Indeed, the two-field model gives mã ≃ m 3/2 because a non-zero vacuum expectation value for X is induced by a supergravity tadpole. In the one-field model, although mã vanishes at tree-level, it is still generated by one-loop quantum corrections. The dominant contribution comes from the Yukawa coupling yP ψψ and the associated A-term, where this interaction is also responsible for the generation of the logarithmic potential.
The axino LSP can be viable if R-parity violation allows the axino to decay. For example, for mã above the electroweak scale, the axino can decay before BBN via the LH u operator without giving a too-large neutrino mass. The contribution to the baryon asymmetry from axiogenesis via R-parity violation [14] is subdominant compared to the lepto-axiogenesis contribution. Such a large mã is readily obtained in the two-field model. In the one-field model, generating a loop-induced mã exceeding the electroweak scale places bounds on the supersymmetry-breaking scale. In gravity mediation with a singlet supersymmetry-breaking field, A ∼ m S , so mã above the electroweak scale requires m S > 10 TeV. In gravity mediation without singlets, A ∼ 0.01m S , so m S > 10 6 GeV would be required.
The upper bound on T R from BBN is not relaxed in comparison with other cases. Although the gravitino can have dominant decayG →ãa if it is the NLSP, the axino anyway decays into SM particles, so the BBN constraint still applies.