Out-of-the-box Baryogenesis During Relaxation

We show that spontaneous baryogenesis occurs automatically in relaxion models if the reheating temperature is larger than the weak scale, provided the Standard Model fields are charged under the U(1) of which the relaxion is a pseudo-Nambu-Goldstone boson. During the slow roll, the relaxion breaks CPT, biasing the thermal equilibrium in favor of baryons, with sphalerons providing the necessary baryon number violation. We calculate the resulting baryon asymmetry, explore the possible constraints on this scheme and show that there is a swath of parameter space in which the current observations are matched. Successful baryogenesis can be achieved for a range of relaxion masses between $10^{-10}$ and $10^{-5}$ eV. The mechanism operates precisely in the region of parameter space where recent work has shown relaxion oscillations to be a dark matter candidate.


I. INTRODUCTION
An interesting explicit scenario has been suggested [1] to explain why the Higgs mass could naturally be much smaller than the fundamental scale. Adapting a longstanding idea of Abbott that attempted (unsuccessfully) to explain the smallness of the cosmological constant [2], the relaxion mechanism incorporates an ingenious interplay of two explicit/anomalous breakings of a Goldstone shift symmetry, to relax the Higgs mass dynamically to values close to the weak-scale. Initially the smaller breaking drives the pseudo-Nambu-Goldstone mode (PNGB), the so-called relaxion, which samples over Higgs masses, while the larger breaking comes in the form of a periodic axion-like potential that is proportional to the Higgs mass itself: hence the dynamical evolution of the relaxion stops as soon as the Higgs mass turns on. With a suitable choice of parameters the resultant Higgs mass can be of the correct order.
The central achievement of [1] was to relate the Higgs mass to a small technically natural parameter, overcoming the lack of any direct equivalent to the chiral symmetry breaking that protects quark masses. The proposal can be made natural in the colloquial sense as well, by adapting clockwork-like scenarios [5]. This paper is motivated by the fact that, being a PNGB, the rolling of the relaxion represents a spontaneous CPT violation. Successful baryogenesis scenarios based on such dynamical evolution were proposed in [6][7][8][9][10][11][12] (also see [13] for a recent review), and were dubbed spontaneous baryogenesis (SBG). One is naturally led to ask if spontaneous baryogenesis can be incorporated into the relaxion mechanism.
We will demonstrate that in fact SBG is virtually generic. It requires only the most minimal augmentation of the relaxion mechanism, namely the addition of a single term to the Lagrangian: where φ is the relaxion and J µ is a current of matter fermions that has a component orthogonal to both electromagnetic charge Q, and B − L. Moreover we will see that the term in Eq. (1) can be generated simply by charging the matter fields under the U (1) corresponding to the relaxion shift symmetry.
Although the term in Eq. (1) is invariant, a homogenous time-derivative of φ amounts to the aforementioned spontaneous violation of CPT, and it implies an effective chemical potential for baryon number [13]. As described in [6][7][8][9][10] this biases sphaleron transitions for temperatures around the electroweak scale where they are active and dominant. In the context of this discussion the time-dependence is a function of the relaxion dynamics itself, which also ultimately quenches B +L violation while there is still a nett baryon asymmetry. Thus remarkably all the required ingredients for SBG are already present in the relaxion mechanism, provided the term in Eq. (1) is generated with sufficient strength.
Successful SBG occurs if the cosmological evolution follows the pattern shown in fig. 1. There is a period of inflation (red) and reheat to temperature T r (black). After reheat and for a period until the temperature drops below T sph ∼ T ew ∼ 130 GeV [14], B + L violating transitions (blue) are active in the plasma, dying away exponentially fast below T sph . During this period the relaxion evolves homogeneously (green) towards its final value which it reaches at temperature T c . Thus one has to satisfy the constraint T r > T sph > T c , which as we shall see can be satisfied without violating any bounds. Subsequently as the temperature drops further the relaxion starts to oscillate about the local minimum and becomes a dark matter candidate [15]. It is striking that the baryon asymmetry and the dark-matter matter relic density turn out to be correct in roughly the same region of parameter space.  Figure 1: The pattern of evolution required for successful baryogenesis during cosmological relaxation.

A. The Relaxion
Let us begin by describing in more detail the cosmological relaxation mechanism itself, proposed in Ref. [1].
Here we focus on models with Hubble friction; for alternative models involving particle production see Ref. [16,17] and Ref. [18] where the latter also implements leptogenesis in this scenario. The defining feature of relaxion models is that the Higgs mass squared term, µ 2 , is dynamical and depends on the classical value of the slowly rolling relaxion, φ. We will consider the following typical potential, (2) where M is a cut-off scale. Suppose that initially the relaxion field has a value φ > ∼ M 2 /g such that µ 2 > 0, and electroweak symmetry is unbroken. During inflation, the relaxion field rolls due to the linear term, and in the process scans the value of µ 2 . As the relaxion field crosses the point φ = M 2 /g , µ 2 becomes negative triggering electroweak symmetry breaking. At this point, the Higgs gets a vacuum expectation value (VEV), turning on a crucial ingredient of relaxion models, the so-called backreaction potential, Here j < 4 is a model-dependent integer. We will describe how such a potential is generated below, but for the moment we note that the important feature is its periodicity, which causes the relaxion comes to a halt at φ = φ 0 satisfying, Due to quantum spreading, when it stops, the relaxion field is spread across multiple vacua with weak scale Higgs vacuum expectation value. While, generally the phase at these minima φ0 fw ∼ 1, it can be arbitrarily close to 0 or π/2 in a small fraction of these vacua (see for eg. Ref. [19]). A large hierarchy between the Higgs VEV and the cutoff scale can therefore be achieved if g is very small. This smallness of g is radiatively stable, because one recovers the discrete symmetry φ → 2πnf w , n ∈ Z, in the limit g → 0. Note that the relaxion travels a distance F ∼ M 2 /g f w in field space which can be problematic as discussed in Ref. [20]. This feature can, however, be incorporated in a consistent way in clockwork models [3][4][5] where the g-dependant parts of the potential arise from periodic terms with the larger periodicity 2πF .
Let us briefly now discuss how the backreaction contribution can arise by describing the non-QCD model of Ref. [1]. In this model, φ is the axion of a new strong sector, and the Higgs couples to two vectorlike leptons charged under the strong group, as follows: Here (L, N ) have the same electroweak quantum numbers as the SM lepton doublet and right-handed neutrino, respectively, and (L c , N c ) are in the conjugate representations. Integrating out L, L c below their mass, we obtain a Higgs contribution to the mass of N , ∆m N = y1y2 H 2 m L . As long as m N 4πf c m L , the fermion N forms an electroweak preserving condensate, and the relaxion, being the axion of the new strong group, gets the following contribution to its potential due to non-perturbative effects: There are also radiatively generated, Higgs independent, contributions to the potential. To ensure that these are subdominant relative to Eq. (5), we must have . Taken together these conditions give the upper bound, The period immediately after inflation is when baryogenesis can take place and appears to be somewhat generic. Our main assumption is that the reheat temperature after inflation, T r , is higher than the electroweak scale, v. Since f c < v one can conclude that T r > T c ∼ √ 4πf c , where T c is the critical temperature corresponding to the chiral phase transition of the new strong group. Note that a separation of scales Λ c f c ∼ T c is technically natural. Therefore under this assumption the backreaction potential vanishes after inflation, and the relaxion enters a second short period of rolling.
As the universe cools and the temperature drops below T c again, the 'wiggles' of the backreaction potential reappear. During the second phase of rolling the relaxion obeys the attractor solution V 5Hφ and gets trapped again by the backreaction potential, provided the temperature induced Hubble friction is always larger than its mass, i.e. m φ < ∼ H ∼ T 2 /m pl [21]. In Refs. [15,22] a more precise condition for the relaxion to get trapped was derived from a full solution to the equations of motion, namely This condition was obtained by demanding that the relaxion does not pick up enough kinetic energy to overshoot the barriers once the backreaction potential reappears.
In the second phase of rolling the relaxion is displaced from its original stopping point by a misalignment angle given by [15], One can check that this changes the Higgs mass by a very small amount. Once the temperature drops to a value such that H(T osc ) < m φ /3, the relaxion starts oscillating about the local minimum. As shown in Ref. [15], these oscillations give rise to a relic abundance given by,

B. Spontaneous Baryogenesis
We now proceed to show that baryogenesis is a generic consequence of the relaxion mechanism discussed in the previous section. We begin with an explanation for the operator in Eq. (1), and then in the following subsection estimate the baryon asymmetry that it generates.

Getting O1 from Yukawa couplings
The simplest explanation for the operator O 1 in Eq. (1) is that it arises because the SM fermions are charged under the global symmetry for which the relaxion is a PNGB. In this case the Yukawa couplings of the SM carry non-zero charge under the relaxion shift symmetry. To explore this possibility we will generalize the discussion in Ref. [11] in order to deal correctly with anomalies. We begin with the Yukawa-induced masses for the fermions during the relaxion evolution, which in the unitary gauge take the form where obviously v = H , and a summation over generations is implied (and we neglect flavour mixing). The θ i are time-dependent phases driven by the relaxion mechanism, where the q λi are the charges of the Yukawa couplings (i.e. the difference in global charge between the respective left and right handed fermion) under this symmetry. The Higgs VEV, v(t), is of course also time dependent, although that has only a minor effect on the baryon asymmetry. For simplicity we will for this illustrative example take the phases θ i to be generation independent and also neglect flavour changing. As shown in Ref. [23], a more complicated generation dependent structure, also addressing the SM fermion masses and mixings, can be implemented involving a Froggatt-Nielsen mechanism, with φ f playing the role of the familon. To determine the physical consequences of the evolving phases in Eq. (10), one can remove them at the expense of inducing baryon and lepton currents. Broadly speaking this must be equivalent to a term of the form in Eq. (1) with J µ being some combination of only right-handed currents. We know this because one is free to remove the phases with a rotation on the right-handed fermions only, which avoids topological SU (2) terms.
To see this in more detail, first we remove the phases in Eq. (10) by making the rotations, where we are allowed two arbitrary phases, ω and ω (corresponding to B and L rotations), upon which of course the eventual physics should not depend. The corresponding change in the classical action is, where the ellipsis refers to the accompanying shift in the mass terms that removed the phases. At the same time the rotation in Eq. (12) induces contributions from global anomalies and finite temperature triangle diagrams, which combined (and neglecting all masses except m t,b ) are where the finite temperature pieces are given by At high temperatures, T m, these terms can be approximated as Thus we have implicitly already taken the infinite T limit for all but the top and bottom masses, by setting their Λ's to zero. In the small T limit, one instead finds Here the requirements for baryogenesis during the relaxion mechanism start to diverge from those of electroweak baryogenesis in Ref. [11]. In particular the anomalous GG terms for the strong coupling should be zero in the final vacuum such that an O(1) strong CP phase is not generated once the relaxion stops. This motivates (at least for generation independent phases) models with the particular physical choice, Under this assumption we may neglect the effect of strong sphalerons [24,25]. We now follow [11] and remove the SU (2) topogical piece in the action as well, by setting This particular choice of ω puts all the important physics in the δS 0 term: Note that, since we will ultimately set B − L = 0, the arbitrary phase ω can have no effect on the physics. Inserting Eq. (11), the remaining part can as promised be interpreted as a term of the form in Eq. (1), with It is straightforward to generalise the above discussion to generation dependent charges if we keep in mind some crucial distinctions. First of all, for general charges it is not possible to remove the phases from the Yukawa terms by redefinitions of the singlet quarks alone. The redefinition of the doublet quarks will then generate a φF 2F2 term that can be removed by a B +L rotation. Finally the condition in Eq. (18) for not generating a strong CP phase is generalised to the condition that the global symmetry has no triangle anomaly with QCD. Such an anomaly free charge assignment that, at the same time, explains the SM fermion masses and mixings was presented in Ref. [23] 2. Getting Baryon Asymmetry from O1 Let us now see how the operator O 1 with a current generically of the form, where i goes over all the spin 1/2 particles, leads to a baryon asymmetry. We can continue with the approach of [9][10][11], noting that in the presence of the operator O 1 the background value ofφ shifts the energy of particles and antiparticles differently (implying a spontaneous breaking of CPT symmetry). This is equivalent to where µ i andμ i are the chemical potentials for particles of species i and their anti-particles respectively, Q i is the electromagnetic charge and µ Q,B−L are Lagrange multipliers introduced to enforce conservation of Q and B − L.
As a result, as long as there is a source of B +L violation, the following asymmetry in the species i is generated: where we assumed µ T and the factor g i incorporates colour, as well as two spin degrees of freedom for each Weyl fermion, a factor of two for complex scalars, and three for massive gauge bosons. The chemical potentials µ Q and µ B−L are then determined by solving n Q = n B−L = 0.
For the case at hand, treatingθ i = q λiφ as classical homogeneous background fields, with the q i 's taken from Eq. (21), we find where in order to keep contact with [9][10][11] we have allowed for n charged Higgs scalars with charge ±1. Finally substituting into Eq. (24) gives where g SB = 3 ((33 + 4n)q λe + 9q λu ) (111 + 13n) is a constant of order one.
Note the physical interpretation which is clear from Eq. (24): there is a bias for J µ production in the plasma, but J µ does not preserve B − L or Q. Therefore any production of J µ must be accompanied by a compensating production of J µ B−L and J µ Q . As a consequence even J µ = e r γ µ e r , with no baryon content at all, leads to a baryon asymmetry! (One should be mindful of the background assumption that left-handed and right-handed fields remain in chemical equilibrium throughout, via the Yukawa couplings.) As expected, there is no dependence of the answer on ω, so the physics does not care about how the phase absorption is arbitrarily divided between left and right handed fields. As a second check of these expressions, taking θ u = −θ d = −θ e = −2ω gives a current in Eq. (20) proportional to hypercharge, as it should because in this case the Yukawas have relaxion charges proportional to the hypercharges of the Higgses appearing in the couplings: then Eq. (27) yields the answer of [9,10], relevant for Higgs-driven electroweak baryogenesis. (The relaxion charges here are of course completely general.) To conclude this section therefore, a term equivalent to Eq. (1) is generated if the relaxion is entangled with the fermion mass structure, and it leads to a net baryon number density of order where the particular linear combination of currents in J µ determines the value of g SB .

C. Baryogenesis from the relaxion
Having collected the components, we now proceed to see how they fit together in a generic relaxion scenario [1], and derive constraints. Typically the relaxion stops its slow-roll much before inflation ends (we would need extreme fine-tuning in order to guarantee that the relaxion travels far in field space but stops at a time close to the end of inflation). Therefore the first slow roll of the relaxion is not suitable for baryogenesis as any baryon asymmetry generated would be diluted away. As discussed above however, there is a generic scenario in which the reheat temperature T r at the end of the inflation is high enough to restore the chiral symmetry breaking that is the source of the oscillatory potential for the relaxion. Consequently the relaxion undergoes a second stage of rolling, this time in a radiation dominated phase. It is in this secondary period of rolling that the time derivative of the relaxion can drive baryogenesis.
In order for the mechanism to operate, we first need to ensure that there is an operative source of baryon number violation, which here is provided by electroweak sphalerons. Sphaleron transitions must be faster than the Hubble rate for some region above the temperature T sph at which the sphalerons decouple.
Second, in order to break CPT, the relaxion has to couple to a suitable current J µ through an operator such as that in Eq. (1). The presence of this operator together with a slowly rolling φ leads to a baryon asymmetry given by Eq. (28) with T = T sph .
Using Eq. (28) andφ ∼ V (φ)/(5H) (see Sec. II A), the generated baryon number asymmetry is where the experimentally measured value of η = 8.7 × 10 −11 [26] and the relaxion mass is From the first of these expressions one can conclude that (31) Therefore successful baryogenesis requires a large scale separation between f and f w . This is acceptable because the relaxion dynamics is already driven by the much larger scale separation, F/f w ∼ (M/Λ c ) 4 where F ∼ M 2 /g is the total field excursion. The hierarchy f f w F can easily be obtained from the clockwork mechanism [3][4][5] with the operator in Eq. (1) arising from the first site of the clockwork, the backreaction potential from an intermediate site and the rolling potential from the last site. In such a setup f would be the only fundamental physical scale, while f w , F would be fictitious scales.
The prospects for baryogenesis are determined by five unknown parameters (f w , f , f c , Λ c , m φ ) which, due to the two relations in Eqs. (29) and (30), represent three independent variables. Since the sphalerons must decouple before the back-reaction potential re-emerges at temperature T c , we need T sph > ∼ T c , and so we choose f c = T sph / √ 4π to saturate this bound. As we will see, the bounds for smaller f c can easily be determined by rescaling. Hence we can display the results in the plane of m φ and f . The other two variables are then related by: Now, for a region in the m φ , f -plane to successfully produce the baryon asymmetry, it must satisfy the following constraints: a. Relaxion Constraints: In order to realise the relaxion mechanism successfully, the loop consistency condition in Eq. (6) requires Λ 4 c < 16π 2 v 4 , which translates TeV. Finally, the fifth force constraints form mixing with the Higgs particle rule out region in red. We have added contours of the value of tan(φ0/f ) required to obtain the correct relic density as per Ref. [15]. Finally, the grey line indicates the region where fw = m pl . into π 2 810 η 2 g 3 * In addition the slow roll requirement in Eq. (7) provides an upper a bound on m φ : One can verify that this is the only bound that depends on f c = T c / √ 4π. Therefore, changing f c effects only the upper bound on m φ . Finally for a successful clockwork implementation of the relaxion mechanism, it is essential that the explicit/anomalous breaking at the first and last site are much smaller than the decay constant which implies for the cutoff, M < ∼ f . All these constraints are shown in our master plot in Fig. 1.
b. Cosmological constraints: The Universe must reheat to sufficient temperature in order to activate sphalerons, T r > T sph . Furthermore, the sphalerons must decouple before the relaxion stops again, T sph > T c , otherwise CPT is restored while sphalerons are active and they erase any net baryon number. This is satisfied by our aforementioned choice of f c . The combination of these two conditions gives the required second period of relaxion roll. Finally, Ref [13] shows that the backreaction of J µ onφ is sufficiently small if f > T sph ∼ 130 GeV.
c. Experimental constraints: The coupling of φ to SM fields must not be detectable. There are two types of constraint: the first type arises from the mixing of φ with the Higgs boson which results in an emergent fifth force for the relaxion mass range relevant here. This effect does not depend on the choice of J µ , so it can be treated independently. The mixing angle between the relaxion and the Higgs comes from the back-reaction potential in Eq. (5) and is of order The constraints on this angle as function of m φ have been presented in Ref. [27] and are reinterpreted in our master plot as the red exclusion area. The second type of constraint is from the coupling of the relaxion to SM particles via the operator in Eq. (1). In our region of interest, the most important bounds on the coupling of the relaxion to electrons, photons and nucleons arise from the fact that such a coupling allows for a more efficient cooling of stars and supernovae (see for eg. Ref. [28]). While most of these bounds can be evaded if J µ does not contain first generation fermionic currents [29] (recall that a coupling to GG must be absent in any case as discussed in Sec. II B), a coupling to photons would still be induced, where E is the electromagnetic anomaly coefficient for the relaxion shift symmetry and we have ignored terms with a further m 2 φ /m 2 f suppression (m f being the mass of the fermions in the loop). The best experimental constraint on the operator comes from bounds on the rate of cooling of globular cluster stars, f /E > 2 × 10 7 GeV [28]. In our master plot we show this bound for E = 1 as a purple-dashed line, which will to currents such as Clever choices for the global charges can, however, suppress or even lead to a vanishing E. Consider for example where L 2 is the second generation lepton doublet. For both J µ 1 and J µ 2 , E = 0 and the bound can be entirely evaded. For J µ 1 , it might seem that any baryon number generation would cancel between the two terms but, as shown in Sec. II B, the mass difference between the top and charm can become important once one takes into account finite temperature effects. Indeed an analysis following that in Sec. II B readily leads to factor proportional to Λ t − Λ c in the baryon asymmetry corresponding to this current, which is not particularly suppressed.
Note that the above charge assignments have no colour anomaly and are thus consistent with the requirement of no strong CP phase being generated when the relaxion stops (see Sec. II B). Finally we should remark that for a full flavour model, the relaxion can induce flavour violation once the operator in Eq. (1) is rotated to the mass basis [23].
d. Dark Matter: Ref [15] points out that the relaxion oscillations can explain the observed dark matter abundance. At any point in the plane of Fig. 2, the expected dark matter abundance can be computed up to the square of the factor tan(φ 0 /f ) (see Eq. (8) and Eq. (9)). We show in Fig. 2 lines indicating the value of tan(φ 0 /f ) required to match the observed dark matter abundance. Values of tan(φ 0 /f ) much larger or smaller than unity are an indication of the level of tuning required to obtain the correct relic density (see the discussion below Eq. (4)). This is especially true if tan(φ 0 /f ) 1 as the universe will overclose without this tuning whereas in the regions with tan(φ 0 /f ) 1, relaxion oscialltions would only account or a small fraction of the observed dark matter density without tuning. Remarkably, we find that in the small allowed region obtained after imposing all the theoretical and experimental constraints discussed here, the required tuning to obtain observed relic density ranges from none at all to a maximum of ∼ 100.
Finally, the relaxion DM should be stable and not decay to photons. We have checked that, even in the example with the least suppressed couplings discussed above, the relaxion decay time to two photons is much longer than the age of the Universe: τ H −1 0 .

III. DISCUSSION AND CONCLUSION
We have shown that Spontaneous Baryogenesis is an integral feature of relaxion models with a high reheat temperature, which induces a second stage of rolling, and that it can generate the SM baryon asymmetry we observe today with almost no adjustment. Indeed the only additional ingredient is a single operator coupling the relaxion to matter currents, ∂ µ φJ µ , which arises automatically if some SM fields are charged under the U(1) of which the relaxion is a pseudo-nambu goldstone boson. Remarkably one encounters (to adapt a phrase from [15]) a double "relaxion-miracle", because the baryogenesis mechanism operates most effectively, and with least tuning, exactly where the relaxion oscillations are a viable dark matter candidate.