High Energy Sphalerons for Baryogenesis at Low Temperatures

We discuss baryogenesis in scenarios where the Universe is reheated to temperatures $\lesssim 100\,$GeV by the decay of long-lived massive particles into energetic SM particles. Before its thermalization, the center-of-mass energy in collisions between such a particle and a particle from the ambient plasma can be higher than the typical sphaleron mass, even if the temperature of the plasma itself is much lower. Optimistic estimates for the high energy enhancement of the sphaleron cross section suggest that successful baryogenesis is possible for reheating temperatures as low as $0.1\text{-}1\,$GeV. With a simple extension of the SM, sufficient baryon production can be achieved even if more pessimistic results for the sphaleron rate are correct. In both cases this scenario can be probed in collider and cosmic-ray experiments. We briefly discuss the possible origin of the required CP violation.

We discuss baryogenesis in scenarios where the Universe is reheated to temperatures 100 GeV by the decay of long-lived massive particles into energetic SM particles. Before its thermalization, the center-of-mass energy in collisions between such a particle and a particle from the ambient plasma can be higher than the typical sphaleron mass, even if the temperature of the plasma itself is much lower. Optimistic estimates for the high energy enhancement of the sphaleron cross section suggest that successful baryogenesis is possible for reheating temperatures as low as 0.1-1 GeV. With a simple extension of the SM, sufficient baryon production can be achieved even if more pessimistic results for the sphaleron rate are correct. In both cases this scenario can be probed in collider and cosmic-ray experiments. We briefly discuss the possible origin of the required CP violation.
From the viewpoint of the Sakharov conditions [52] the late-time decays of heavy particles also provide an opportunity: they are inherently an out-of-equilibrium process. 1 Therefore, let us focus on the short period soon after a long-lived particle decays into energetic SM particles. During the first steps of the thermalization of an energetic SM particle, the center-of-mass energy between it and a particle in the (already thermalized) ambient plasma can easily be higher than M sph , even when the reheating temperature 100 GeV. In such energetic collisions, it may be easier to produce suitable sphalerons and therefore fulfill Sakharov's condition of a sufficiently effective baryon number violating process.
the authors considered the possibility that in each decay of the long-lived particles the decay products first heat a small asymmetric region where the plasma temperature is much higher than 100 GeV. Thus less suppressed symmetric-phase sphalerons can be active in this region. Another option is to use symmetric-phase sphalerons at the early stage of reheating [55] (when the plasma temperature is actually higher), the resulting asymmetry is, however, typically diluted by (T R /100 GeV) 5 . In contrast, we will consider the direct sphaleron enhancement by the scattering of an energetic particle from the decay and a plasma particle. At present calculations of highly energetic sphaleron rates are still plagued by uncertainties. Therefore, we first quantify how much enhancement of the sphaleron effect is needed for successful baryogenesis. The relevant cross sections turn out to be of a size that may allow for tests at the LHC, future colliders, and cosmic-ray experiments. Turning to concrete calculations of the sphaleron rate we find that more optimistic estimates give values that allow for successful baryogenesis. If more pessimistic estimates turn out to be correct, we can still obtain successful baryogenesis by considering a simple BSM extension of the gauge group that enhances the gauge interaction and thus the sphaleron rate at high energies.
Finally, turning to the last Sakharov condition -CP violation -we briefly discuss some possibilities how this may be realized in the present context in Appendix A.
Brief review of out-of-equilibrium sphalerons.-The electroweak instanton/sphaleron is the only source of baryon number violation within the SM. As a tunneling process between vacua with different Chern-Simons numbers it is expected that the instanton effect is exponentially suppressed. A sphaleron, being externally supplied with enough energy, M sph ∼ πm W /α w ∼ 7 TeV, does not have this suppression since it is a classical saddlepoint solution connecting the different vacua. Here, m W (α w ) is the weak boson mass (coupling constant). Therefore, cosmological models with a reheating temperature T R M sph are usually argued not to have efficient baryogenesis via sphalerons.
The instanton/sphaleron-induced cross-section for → m a n y N in → 0 2 → many 2 → m an y Figure. 1. Different estimations of the holy grail function depending on Ecmαw/mW . The optimistic estimation (red) is from [67][68][69]. The pessimistic ones are from [70,71] (black) and [72] (green). We also show the low energy analytic result [73] (gray dotted) and a result with Nin = N sph = 1/αw translated from [70,71] (blue). While the latter is in principle a pessimistic estimate, we note that applying it to αw = O(1) and Nin = N sph = 2 might be optimistic (see the discussion below Eq. (14)).
highly energetic quarks has been studied in the context of collider and cosmic ray phenomenology (cf, e.g., [56][57][58][59][60][61][62][63]), but also for a better conceptual understanding of the sphaleron itself. Unfortunately, the size of the crosssection for this non-perturbative process is still under debate. Let us briefly review the current status. The instanton effect can be estimated from the t 'Hooft operator [64] L eff ∼ e − 2π αw Q 9 L 3 v 14 + h.c., with Q and L being the left-handed quarks and leptons, and v ∼ 170 GeV the Higgs vacuum expectation value (VEV). Importantly, [65,66] pointed out that by emitting multiple bosons in 2 to many processes QQ → 7Q + 3L + n Z Z + n W W + n H H the cross section can be enhanced exponentially in N = n Z + n W + n H , fitting the picture that sphalerons are more efficient than instantons at high energy scales since N can be larger.
The inclusive cross section is often given in terms of the so-called holy grail function, F , specifying the exponential suppression, where E cm is the center-of-mass energy. The holy grail function has been calculated by various authors and using a range of different methods and limits. We summarize the different estimations in Fig. 1 as functions of α w E cm /m W . Broken phase sphaleron from reheating.-For concreteness, we discuss the thermalization process during reheating caused by a modulus decay. Our conclusions do not change if we replace the modulus by the inflaton, a long-lived fermion, e.g. gravitino/Peccei-Quinn fermion, or interaction with macroscopic objects, as, e.g. a bubble wall collision with the ambient plasma [47,74,75] or with another bubble wall [76], as long as they provide energetic daughter SM particles 2 and reheat the Universe. Let φ be the long-lived modulus. A simple case is that φ decays into a pair of energetic SM particles charged under SU(2) L 3 φ → χχ ≡ LL, QQ, WW , ZZ, or/and hh. ( If the non-relativistic modulus dominates the Universe, the reheating temperature is given by GeV is the reduced Planck scale, g (and g s used below) is the number of relativistic degrees of freedom (for entropy) in the SM (we use the numbers from Ref. [77]).
With Γ φ = m 3 φ /M 2 , m φ being the mass of the modulus and M the typical scale of the coupling we ob- Let us focus on what happens within a Hubble time around T = T R , where the energy density in the moduli and in radiation is roughly equal. Baryons produced much earlier will be strongly diluted [55] and much later the sphaleron is inefficient as usual. In general, F depends on E cm and the sphaleron contribution during reheating may depend on the behavior of F . However, for large enough E cm , F is thought to be approximately constant. Thus, we consider a simplified situation where this holds. We will briefly revisit this assumption below.
Around the relevant time we can estimate the total number density of highly energetic SM particles, χ, produced in the decay of the moduli, as, with H being the Hubble parameter at that time. In the last approximation we have used Γ φ ∼ H and the definition of the end of reheating, n φ m φ ∼ g π 2 T 4 R /30. Now, let us look at the thermalization of the energetic SM particles, χ. To this end, we focus on a single φ particle. At this time, the Universe is comprised of the other φ particles who will decay later, ambient thermalized plasma from the φ decays at earlier times, and the energetic χχ from the φ decay of interest. Initially χ carries an energy, Thermalization proceeds via the gauge interactions. The time-scale on which χ looses an O(1) fraction of its energy, is t th . This is estimated [54,[78][79][80] (see also Refs. [81,82]) from the Landau-Pomeranchuk-Migdal effect [83,84] to be, Here, α χ = α 3 or α e , i.e. the coupling constants for the gluon and the photon, depending whether χ is a lefthanded quark or charged lepton. The scattering rate (denominator in Eq. (5)) also receives contributions of order If this rate is larger than the denominator on the r.h.s of (5), we should use it in the evaluation of t th . A similar t-channel 2 → 2 scattering process obtained by exchanging a soft W-boson is particularly important when χ is a neutrino. In this case, the neutrino is translated into a charged lepton at a time scale (see e.g. [85]) After this the resulting charged lepton quickly thermalizes. In addition, if the sphaleron reaction is too fast, we need to consider its effect on the thermalization. Since Eq. (5) will be dominant in most of the parameter region of interest, we will mainly focus on that time-scale in our discussion. In the result of Fig. 2, we also include the other contributions. The important thing is that during t th we have energetic SM particles, χ, whose collision center-of-mass energy can be much larger than T R .
The most significant process for the baryon number violation is that χ scatters with the ambient plasma. 4 The typical center-of-mass energy for these interactions is E cm ∼ E χ T R . For each χ, the probability to interact via a sphaleron can be estimated as, Here, denotes the thermal average, and n r is the number density of the ambient plasma particles that are charged under the SU(2) L . 5 Then we obtain the 4 The number density of direct sphaleron-interactions between two highly energetic χ is further suppressed by t th H and nχ/nr compared to the χ scattering with the ambient plasma. We expect this to be small unless the higher Ecm increases the sphaleron cross section significantly. 5 Assuming, again, that F depends only weakly on Ecm and the type of incoming particles, we obtain the thermal average of the reaction σ sph nr g log 48(EχT R ) T 3 R with g being the number of fermion scatterars in the plasma, and the IR cutoff in the angular integral. It is roughly sphaleron events per unit volume per unit Hubble time as n sph P sph n χ .
Normalizing this to the entropy density, s = 2gs π 2 T 3 R

45
, we obtain (using Eq. (5)), In the case where the sphaleron interaction dominates the thermalization of χ, it is this rate that determines, which is independent of σ sph as long as the reaction is faster than the other thermalization processes.
The baryon to entropy ratio should satisfy [86] ∆n B s CPV n sph s 8.7 × 10 −11 (10) with CPV being a model-dependent parameter representing the amount of the CP violation, which will be discussed in Appendix A. Effectively it is the number of baryons produced per sphaleron interaction. Usually, we expect CPV < 1. In Fig. 2 we show the required sphaleron cross section (red dashed) and values for the holy grail function (black) that allow for a sufficient baryon asymmetry in the T R − E χ ( m φ /2) plane (for CPV = 10 −2 ). We have taken into account the various thermalization processes assuming that χ is a charged lepton, and using t −1 th = t −1 LPM + t −1 sph . The lower shaded region corresponding to T R E χ TeV may be inconsistent with the collider search [62,63]. In the blue region, the sphaleron interaction dominantly thermalizes χ and not enough baryons are generated. Fig. 3 in Appendix B depicts the case where χ is a neutrino where the soft W boson exchange can be important.
Exploring the parameter space (cf. also Figs. 2, 3) we find that successful baryogenesis is possible even for temperatures around ∼ 100 MeV, if the sphaleron cross sections are large enough. If we adopt the most optimistic estimation of Fig. 1, F ≈ 0 for E cm 9 TeV. In this case, our scenario allows for T R 100 GeV with CPV 10 −6 . As mentioned before, realistically F is not a simple constant. In the optimistic scenario of Fig. 1, F changes rapidly when E cm < 9 TeV. If T R E χ 9 TeV, the reaction during reheating with the dilute plasma temperature T > T R is actually more important despite the later dilution. It is dominant at T ∼ T * satisfying T * · E χ ∼ 9 TeV if the reheating lasts long enough. if T R < T * . 6 From this we may obtain additional allowed parameter space. However, if we adopt the pessimistic estimations in Fig. 1, we get 4π/α w F = O(100), and the scenario is difficult in the minimal setup. Therefore, let us briefly consider BSM effects that may sufficiently enhance the sphaleron such that successful baryogenesis is also possible in the pessimistic case.
BSM with faster sphalerons.-Let us consider a simple renormalizable model to enhance the sphaleron rate. The key idea is to enhance the gauge coupling for the SM fermion at a high energy scale. One possibility is SU(2) 1 × SU(2) 2 → SU(2) L broken by a bi-fundamental Higgs field. (See also models with a bi-fundamental Higgs for GUT breaking [87,88] and generating axion/ALP potentials [89][90][91][92][93].) The Lagrangian is given as with L H b being the Lagrangian for a bi-fundamental Higgs field, which transforms as is the field strength (gauge transformation) for the gauge field A i of SU(2) i . We omit the Lorentz indices. A 1 (A 2 ) does not (does) appear in L SM [A 2 ] which is the SM Lagrangian with the usual SU (2) L fields and coupling, A L , g L , replaced by A 2 , g 2 .
It is straightforward to find a renormalizable potential such that H b = v b δ ij ,. This spontaneously breaks SU(2) 1 ×SU(2) 2 → SU(2) L , absorbing 2×(2 2 −1)−(2 2 − 1) would be Nambu-Goldstone bosons. For simplicity, we further restrict the field space with the condition, such that the number of real degrees of freedom is four, three of which are eaten by the gauge bosons (see Appendix C for a brief discussion of the situation where we do not impose this constraint).
If we have g 2 g 1 , we get g L g 1 , A L A 1 , as well as A H A 2 and g H g 2 . Thus we can have a strong coupling of the SM fermions/Higgs to A 2 . To evade collider bounds we need M H O(1) TeV.
Another efficient, but not renormalizable, model to enhance the sphaleron rate may be a large volume extradimension where the SU(2) L gauge boson lives in the bulk. In this case, the small instanton of the SU(3) can be significantly enhanced in the context of the heavy QCD axion [94,95]. However, the CP-violating higher dimensional terms should be somehow made small to solve the strong CP problem [96,97]. In the context of our SU (2) sphalerons, the CP-violating nature is good news since it naturally satisfies one of the conditions for baryogenesis.
To conclude.-Depending on whether more optimistic or more pessimistic estimates for the sphaleron rate at high energies are correct, successful baryogenesis via sphalerons induced by scattering energetic decay products on the ambient plasma may be achievable for reheating temperatures as low as ∼ 100 MeV with or without a BSM modification. This scenario can be probed in collider or cosmic-ray experiments by searching for the sphaleron reaction. In the BSM models, we can also search for heavy gauge bosons that couple to SM particles. In addition the φ decays may also produce weakly-coupled out-of-equilibrium BSM particles, e.g. right-handed neutrinos [85], WIMP-like particles [85] or axion-like particles [20-22, 27, 28, 34, 35, 98-102], that reach Earth. Our scenario could then also be probed by measuring those "primordial cosmic-rays".
That said, a further understanding of the sphaleron interaction with multiple incoming particles in the regime of large coupling and including fermions is quite important for clarifying the allowed parameter region of this simple scenario.
Acknowledgments.-WY was supported by JSPS KAKENHI Grant Number 20H05851, 21K20364, 22K14029, and 22H01215. JJ is happy to be a member of the EU supported ITN HIDDEN (No 860881).

Appendix A: CP violation
So far we have simply parametrized the necessary CP violation by employing a free parameter CPV in the sphaleron rate. Let us now list some exemplary scenarios how such a CP violation could arise. In fact, there are various viable models for CP violation e.g. [37, 39-47, 103, 104]. In principle, these should also work in our scenario of sphalerons being triggered in the broken phase. Baryogenesis via CP-violating sphaleron reactions We can directly produce the baryon asymmetry by including the CP violation during the thermalization. For instance, we can slightly extend Eq. (11) without imposing the restriction of the field space. Note that we can have the couplings (λ with H being the Higgs doublet that gives masses to the SM fermions. In this case, the couplings λ , A , λ generically involve CP phases that cannot be removed by the redefinition of the H b phase. (They are absent if we impose the restriction.) Using naïve dimensional analysis we expect very roughly that With the field space restriction a possibility is to introduce another heavy Higgs doublet,H, whose VEV is small. If it transforms via (2, −1/2) under SU(2) 1 and U(1)Y , the Higgs potential can have a contribution V ⊃ AH * · H b · H +λ(H * ) 2 H 2 . By noting that the phase of H b is fixed by the reality condition (12), the phase argλA * cannot be removed by field-redefinitions. Then very roughly we get CPV ∼ | [λA * ]| M sph .
Interestingly, in this setup the flavor structure of the SM may be partially explained by noting that some families may be charged under SU(2) 1 while some are charged under SU(2) 2 . For instance, let us consider that only the 3rd generation fermions are charged under SU(2) 2 and thus obtain mass via the VEV of H. Then the first two generation masses are naturally suppressed by the mixing betweenH and H. The flavor structure is similar to the two Higgs doublet model specifying the top quark in Ref. [105]. The only difference is that the flavor-dependent discrete symmetry in the reference is replaced by the gauge symmetry in our case. Therefore the flavor structure is quite similar: i.e. we have suppressed flavor-changing neutral currents among the first two generation quarks as well as an electron electric dipole moment (EDM) which is generated at the 2-loop level [105]. In particular, with a BSM mass scale around 1 − 10 TeV the EDM may be probed in the not too distant future. If the sphaleron is enhanced by a large α 2 , there is a small instanton contribution in the low energy physics. In this case we have an effective coupling , for the 3rd generation fermions. Via CKM mixing V C , we then obtain an operator that involves only first generation quarks, e.g.
being the order of magnitude estimate of the RG effect and y b the bottom quark Yukawa coupling. The effective suppression scale for this higher dimensional operator is around the GUT scale for α 2 ∼ 0.2 and v b ∼ 10 TeV. Thus proton decay may be a useful tool to probe this scenario. On the other hand, it is already excluded for α 2 ∼ 1. We can also use different charge assignments. For example if the first two generations are charged under SU(2) 2 the proton does not decay. This is because then the operator generated by the instanton is , which preserves a Z 2 baryon number symmetry, which stabilizes the proton. This model can also explain the small masses of the first two generations ifH is the SM-like Higgs field. Leptogenesis One can also first produce a lepton asymmetry from the φ decays. Later this is transferred into the baryon asymmetry by the broken phase sphalerons discussed in this Letter. This is the famous idea of leptogenesis in the symmetric phase [48][49][50][51]. One can easily produce the lepton asymmetry from lepton flavor oscillations [53,103,104,106]. Alternatively, we can assume that φ carries lepton number and employ the Affleck-Dine mechanism [36,107]. The SM leptons are then obtained from a coupling φHLHL. In any case, we need a much larger neutrino/lepton asymmetry ∼ CPV than the baryon asymmetry because only a small fraction can be transferred by the sphalerons induced by highly energetic particles (cf. Eq. (10)). This is in contrast to the conventional symmetric phase leptogenesis which predicts that the two asymmetries are comparable. The predicted large neutrino asymmetry may be probed, e.g., in future CMB experiments [108,109]. Spontaneous baryogenesis (may not work) We may also consider a coupling of αw . This is similar to the spontaneous baryogenesis picture [110] (cf. also [111][112][113][114][115][116][117][118][119][120][121][122][123][124][125][126] for a variety of interesting implementations) in which the baryon asymmetry is proportional to the chemical potential µ =Ȯ Λ d . However, this also requireṡ O ∼ E cm Λ d in order to have efficient CP violation. Assuming Λ E cm , this implies that there is a large kinetic energy density contribution from the operator E 4 cm (at least for d = 1). This is inconsistent with the assumption in our setup that the Universe has an energy density T 4 R E 4 cm . Thus the spontaneous baryogenesis scenario may not work.

Appendix B: Sphalerons from neutrino scattering
In the main text we have shown results for the situation where the moduli decay into charged leptons. Another interesting situation arises if the decay is into neutrinos. This is shown in Fig. 3, where CPV = 1 (top panel), 10 −3 (middle panel), 10 −6 (bottom panel). Here, we use t −1 th = (t ν + t LPM ) −1 + t −1 sph because only after a soft W-boson exchange, the photon interaction becomes active. The contour is flat in the low temperature regime because the thermalization time-scale for a neutrino is determined by the W boson emission, t ν . This gives a longer time for the sphaleron reaction to work. Here we discuss the vacuum structure by stabilizing the potential for the bifundamental SU(2) 1 × SU(2) 2 Higgs. Let us start by commenting on what happens if we do not impose the restriction (12). From gauge invariance, the potential is then given by Here λ b , λ tr , λ denote the dimensionless parameters and µ H , A are the dimensionful parameters. The first line is invariant under an SO(8) symmetry. Here, λ and A may be complex. The second line breaks SO(8) explicitly down to U(1) × SU(2) 1 × SU(2) 2 . Finally, the third line breaks the U(1) explicitly. Therefore, the would-be Nambu-Goldstone (NG) bosons from the symmetry breaking of SO(8) → SU(2) 1 × SU(2) 2 obtain masses via the explicit breakings. In addition, the third line also contributes to the CP violation, as discussed above. In general, we also have the po- c. with additional dimensionless couplings. Including them does not significantly change the discussion as long as the additional couplings are subdominant.
To check the spectrum, we expand by using the Pauli matrices. Here T 0,1,2,3 are the complex fields. We then find If we take µ 2 H > 0, λ b > 0 and neglect the other terms the potential becomes Thus H b = T 0 1 = 1 2 µ 2 H λ b 1 without loss of generality. At this point, we have 7 NG modes.
Next let us introduce λ tr = 0, but |λ tr | λ b /2. Then we can see that the potential features an additional term Now from the first term the VEV is shifted to T 0 = µ 2 H /(λ b + λ tr /2)/2. We note that the second term in Eq. (C5) provides a mass term for the CP-even triplet The CP-odd triplets are the would-be NG bosons that are eaten by the heavy gauge bosons in the SU(2) 1 × SU(2) 2 → SU(2) L breaking. To achieve that the remnant SU(2) remains unbroken, i.e. T * i + T i does not have a VEV, we need λ tr > 0. We also note that given λ tr = O(1) the triplet is as heavy as T 0 = v b .
The determinant terms not only give a VEV shift and the mass shift to the CP-even triplet Higgs but it also gives the mass to the neutral NG boson associated with the U(1) symmetry, under which, H b → e iα H b . We get the potential for the axion a from the term as In summary, having spontaneous symmetry breaking by our bifundamental Higgs and a potential that consists only of the first row of Eq. (C1), we obtain 7 NG modes. They are a neutral "axion" and 2 triplet Higgs fields under SU(2) L . The CP-odd triplet is eaten by the gauge boson. The second row of Eq. (C1) provides mass for the CP-even triplet Higgs. The third row finally provides masses to the CP-even triplet Higgs and the "axion." Our discussion in the main part corresponds to the case that the triplet Higgs and the "axion" are heavy. If they are lighter they may lead to an interesting phenomenology in accelerator experiments.