Dynamic freeze-in: impact of thermal masses and cosmological phase transitions on dark matter production

Journal of High Energy Physics, Mar 2018

Michael J. Baker, Moritz Breitbach, Joachim Kopp, Lukas Mittnacht

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Dynamic freeze-in: impact of thermal masses and cosmological phase transitions on dark matter production

HJE Dynamic freeze-in: impact of thermal masses and cosmological phase transitions on dark matter production Michael J. Baker 0 1 3 Moritz Breitbach 0 1 2 Joachim Kopp 0 1 2 Lukas Mittnacht 0 1 2 Johannes Gutenberg University 0 1 Field Theory 0 Staudingerweg 7 , 55099 Mainz , Germany 1 8057 Zurich , Switzerland 2 PRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics 3 Physik-Institut, Universitat Zurich The cosmological abundance of dark matter can be signi cantly in uenced by the temperature dependence of particle masses and vacuum expectation values. We illustrate this point in three simple freeze-in models. The rst one, which we call kinematically induced freeze-in, is based on the observation that the e ective mass of a scalar temporarily becomes very small as the scalar potential undergoes a second order phase transition. This opens dark matter production channels that are otherwise forbidden. The second model we consider, dubbed vev-induced freeze-in, is a fermionic Higgs portal scenario. Its scalar sector is augmented compared to the Standard Model by an additional scalar singlet, S, which couples to dark matter and temporarily acquires a vacuum expectation value (a twostep phase transition or \vev ip- op"). While hSi 6= 0, the modi ed coupling structure in the scalar sector implies that dark matter production is signi cantly enhanced compared to the hSi = 0 phases realised at very early times and again today. The third model, which we call mixing-induced freeze-in, is similar in spirit, but here it is the mixing of dark sector fermions, induced by non-zero hSi, that temporarily boosts the dark matter production rate. For all three scenarios, we carefully dissect the evolution of the dark sector in the early Universe. We compute the DM relic abundance as a function of the model parameters, emphasising the importance of thermal corrections and the proper treatment of phase transitions in the calculation. Beyond Standard Model; Cosmology of Theories beyond the SM; Thermal - 1 Introduction thresholds 2.1 2.2 2.3 3.1 3.2 3.3 Toy model The e ective potential Results and discussion 3 Vev-induced production with a vev ip- op Toy Model The e ective potential & the vev ip- op Dark matter freeze-in and relic abundance 4 5 Vev-induced mixing with a vev ip- op Summary and conclusions A Computation of the e ective potential B Boltzmann equations B.1 Freeze-in via decay: B1 ! B2 B.2 Freeze-in via annihilation: B1B2 ! B3 2 Kinematically induced freeze-in: temperature-dependent masses and 1 Introduction \In order for the light to shine so brightly, the darkness must be present." This quote, attributed to Sir Francis Bacon, subsumes much of modern day cosmology. The Universe as we know it, with its abundance of bright galaxies, could not have formed without the presence of large amounts of dark matter (DM). DM drives the formation of structure; the gravitational collapse of primordial density uctuations leads to dense objects like galaxy clusters, galaxies, and stars, with at least one of the latter harbouring life. Even though one of the lifeforms likes to describe itself as intelligent, it is still very much in the dark about dark matter, its origin and its nature. For a long time, the best-motivated scenario to understand DM has been the Weakly Interacting Massive Particle (WIMP) scenario: DM particles are hypothesised to be heavy (mDM & 100 GeV) and to have weak, but non-negligible interactions with Standard Model (SM) particles. In the very early Universe, these interactions keep the DM and SM sectors { 1 { multi-pronged search program [2{8], motivates the study of alternative mechanisms [9{20]. Freeze-in models assume that the initial DM abundance after in ation was zero and that DM particles couple so weakly to the particles in the thermal bath that they never reach thermal equilibrium [17{21]. Consequently, DM annihilation does not determine the relic abundance. Instead, the observed abundance is the result of processes with DM in the nal state, typically at x ' 1{5. The small coupling between DM and SM particles also implies a signi cantly greater challenge for all experimental probes of the nature of DM. In this paper, we consider the impact of nite temperature e ects on the DM abundance in freeze-in models. Such e ects are manifold: rst, particle masses receive corrections from thermal loops, implying that the kinematics of DM production is in general T -dependent. Certain production channels may be open during some epochs of cosmological history, but kinematically closed during others. Closely related to this e ect is the T dependence of the e ective scalar potential V e , which implies that not only scalar masses (the second derivatives of V e ), but also their vacuum expectation values (vevs) change with time in the early Universe. These vevs, in turn, will a ect gauge boson and fermion masses (of course, gauge boson and fermion masses also receive direct corrections from self-energy diagrams evaluated at T 6= 0, although the contribution to fermion masses will be unimportant in the scenarios we discuss). The most interesting case is where the scalar potential develops several disjoint minima and transitions from one to another in a phase transition. The phase transition, for which the scalar vev is an order parameter, can be rst order, second order, or a mere cross-over. The latter (and perhaps least interesting case) is realised in the SM [22{26]. In standard freeze-out scenarios, thermal e ects are usually negligible since they are small at x 1, i.e., at temperatures much lower than the masses of the involved particles. For freeze-in at x ' 1{5, on the other hand, they can be large and have a decisive impact on the DM abundance. Demonstrating this with several examples is the main topic of this paper. To focus on the important e ects and avoid unnecessary complications, we consider a toy model consisting of the standard model, a new gauge singlet scalar, and one or two gauge singlet dark sector fermions. Since we focus on freeze-in, we imagine some couplings to be 1. As discussed below, these small couplings will often be technically natural (i.e., protected from large radiative corrections). In other cases, we assume a small coupling to simplify the analysis, but in these cases we do not expect a larger coupling to spoil the overall picture. The scenarios we consider could be motivated from a wide range of UV theories, including SUSY and extra dimensional scenarios [19]. Scenarios with gauge singlets at the weak scale are notoriously di cult to test at colliders and will not be ruled out at the LHC. The scenarios discussed in sections 3 and 4 will be best probed through their O( 1 ) Higgs portal couplings, but the full parameter space will not be probed until a { 2 { in section 2 is even harder to exclude, due to the small Higgs portal coupling. The impact of thermal e ects in the early Universe on the DM abundance has been previously discussed for instance in [28, 29], where it was argued that DM could be temporarily unstable in the early Universe, so that its abundance would be controlled by its decay rates and by the temperature of the phase transition that stabilises it. Similar thermal e ects have also been considered in, e.g., [30, 31]. The outline of the paper is as follows. In section 2 we discuss a scenario, dubbed \kinematically induced freeze-in", in which the kinematics of DM production is controlled by the T -dependent masses of a new scalar S and a new dark sector fermion . DM is kinematically forbidden. However, as S transitions from a phase with hSi = 0 to a phase with hSi 6= 0, its mass drops close to zero and DM production becomes kinematically allowed for a short period of time. We emphasise that, in a more economical version of this model, S could be replaced by the SM Higgs itself. In section 3 we consider an alternative freeze-in model | a variant of the fermionic Higgs portal scenario | in which the DM production rate in the dominant channels is proportional to hSi. We call this scenario \vev-induced freeze-in". There is a large region of parameter space where its scalar potential undergoes a two-step phase transition (\vev ip- op"), i.e., the Universe starts out in a hSi = 0 phase, followed by an epoch with hSi 6= 0. The electroweak phase transition ends this epoch and reverts the Universe to hSi = 0 (but a non-zero vev for the SM Higgs). It is thus the two phase transitions that control the DM abundance today. In a third model, which will be the topic of section 4, hSi controls mixing between the DM particle and a second new fermion. This mixing, in turn, opens up production channels that are otherwise inaccessible, thus boosting freeze-in production. Hence we call this scenario \mixing induced freeze-in". We summarise and conclude in section 5. 2 Kinematically induced freeze-in: temperature-dependent masses and thresholds In thermal quantum eld theory, particle masses can receive temperature-dependent corrections from self-energy diagrams and thus become functions of T themselves. For instance, the SM Higgs mass is mh(T = 0) = 125 GeV today, but was much larger in the very early Universe and close to zero around the time of the electroweak cross-over at TEW ' 165 GeV. In this section we discuss a scenario where the kinematics of the DM freeze-in rate are controlled by the mass of a new real scalar S. 2.1 Toy model We consider a simple toy model | a variant of the fermionic Higgs portal scenario | whose particle content is given in table 1. Besides the real scalar S and the Dirac fermion , which is the DM candidate, we introduce a second new Dirac fermion . All new particles are SM singlets, and and are charged under a Z2 symmetry. We remark already here that { 3 { one could imagine a variation of the model in which S is replaced by the SM Higgs eld itself. The relevant terms in the Lagrangian at dimension four are L 1 2 m ) + y S + h:c: + y S + y S V (H; S) (2.1) with V (H; S) = 2H HyH + H4 (HyH)2 12 2SS2 + 4! S4 S4 + S3 3! SS3 + p3 S S(HyH) + 2 p4 S2(HyH) : (2.2) The rst line of eq. (2.1) contains the standard kinetic terms and the fermion mass terms. In the second line of eq. (2.1), we identify the Yukawa couplings between S, and . We assume y and y to be tiny to avoid full thermalisation of and thus allow for DM production via freeze-in rather than freeze-out. The smallness of y and y could be motivated, for instance, by extra-dimensional scenarios where could be localised far away from and S along the fth dimension. The coupling y on the other hand, is assumed to be sizeable so that and S remain in thermal equilibrium until T m ; mS. Alternatively, one can also introduce an extra particle | for instance a second new scalar S0 with negligible couplings to the DM particle , but appreciable couplings to and to the SM sector | to achieve the same goal. In fact, in the numerical results shown below we will assume this second possibility because it simpli es the dynamics of the temperaturedependent e ective scalar potential and opens up larger regions of parameter space than the vanilla scenario from eq. (2.1). satis ed for p4 assumption that could be relaxed at the expense of unnecessarily complicating our analysis. The smallness of p3 and p4 could again be motivated in extra-dimensional scenarios by localising S and H far from each other along the fth dimension. We hypothesise, however, that p4 is still large enough to keep S in thermal contact with the SM particles at temperatures T ' mS, when DM freeze-in happens. We nd that these conditions are Where necessary, we will decompose H into its components according to H = G+; (h+ iG0 =p2), where h is the neutral CP even SM-like Higgs boson and G , G0 are the Goldstone modes. Moreover, we will often use the de nitions vS the vacuum expectation values of S and h. hSi and vH hhi for Freeze-in of can proceed through the decays masses), and through the 2 ! 2 reactions S ! S { 4 { ( ! S) = (S ! ) = 16 y 2 8 q q y 2 (m +m ) 2 m2S(T ) m3 m2 (m +mS(T ))2 m2 (m mS(T ))2 ; m2S(T ) (m +m ) 2 m3S(T ) m2S(T ) (m +m )2 m2S(T ) (m m )2 ; (S ! S ) ' 32 s2(s m2 )3 (s y2 y2 " Spin 0 1 2 1 2 +1 1 1 mS(T = 0) ' 5 GeV m m ' 50 GeV ' 50 GeV All new particles are SM gauge singlets. . The Feynman diagrams for these four processes are depicted in gure 1. the very early Universe, however, mS receives large thermal corrections / T , which can lift its value above m + m and thus open up the channel S ! when S develops a non-zero vev, mS(T ) approaches zero and the channel . Later, around the time In the last two expressions, we have taken the limit m ' m and mS ' 0. Moreover, we have set the width of to = 0. In eq. (2.6), we have also set vS = 0 because the full expression is fairly lengthy, and we will see below that the channel SS ! very subdominant when vS 6= 0. In our numerical analysis below, we of course use the is full expressions. { 5 { m2 )(5s3 +55m2 s2 +3m4 s+m6 ) 2s2(s2 18m2 s 15m4 ) log 0 s+ qs(s 4m2 ) 1 s qs(s 4m2 ) 4(s+8m2 )qs(s 4m2 )5 : m2 !# s ; A 3 (2.3) (2.4) (2.5) (2.6) ψ S S ¯ ψ ψ S ψ χ S S ψ χ ¯ ψ S S χ ¯ ψ , we do not show the diagrams with crossed S lines. It is important that, thanks to the non-zero hSi at low temperatures, S can decay through its Higgs portal coupling p4 (or also through p3). If this decay was not present, S would have a relic abundance that would be too large. For mS below the W and Z thresholds, the decay rate is (S ! f f ) = X y f f2 h2S(T ) mS(T ) 8 1 4mf2 m2S(T ) 3=2 mS(T ) 2mf : (2.7) Here, the sum runs over light fermions, f = e; ; ; u; d; s; c; b, and yf = p 2mf =vH are the corresponding Yukawa couplings. The h{S mixing angle is tan hS(T ) = ( p3 S + p4vS(T ))vH (T ) m2H (T ) In these expressions vS(T ) and vH (T ) are the S and Higgs vevs, respectively. S also couples to SM particles via annihilations. After the electroweak phase transition, the main annihilation process is SS ! h ! f f , with cross section vrel(SS ! f f ) = X f Cf yf2 2p4vH2 (T ) 32 (m2H (T ) 4m2S(T ))2 1 m2 f m2S(T ) 3=2 mS(T ) mf (2.9) in the non-relativistic limit. In this expression, Cf is a colour factor. We will choose p4 ' 10 3, which makes S decays, inverse decays, and annihilations fast enough to keep S in thermal equilibrium with the SM at all T . m , where DM freeze-in dominantly occurs. We have veri ed that this is always possible for m > mS & 3 GeV. If S were not in thermal equilibrium during DM freeze-in, the model would not be invalidated, but the dark and visible sector temperatures would di er, complicating the analysis. We will also assume that, while DM freezes in, remains in thermal equilibrium with the SM sector, either through $ SS or through interactions with a second new scalar S0 as explained below eq. (2.2). The cross section for ! SS in the non-relativistic limit is vrel( ! SS) = vr2ely4 qm2 24 m2S(T ) 2m4S(T )m 8m2Sm3 + 9m5 where vrel is the relative velocity of the annihilating particles. Note the vr2el suppression of the annihilation cross section. Eventually, will freeze out, and we ensure that this { 6 { happens late enough for its relic abundance to make up a subdominant contribution to the DM density (less than 1%). Note that is not absolutely stable, but can decay even at T = 0 through (m m )=m 1, and mf = 0) given by ! (S =H ! f f ). The rate of this decay is (for m m > mS, ( ! f f ) ' X f Cf yf2y2 h2S(T ) m5 (m2h + m2S)2 Here, the sum runs over all kinematically accessible SM fermion species. Since we treat fermions as massless, eq. (2.11) will not be accurate near any of the fermion mass thresholds. Even though freezes out with only a subdominant relic abundance, its decays may violate constraints if they happen around the time of recombination [32{34]. We therefore demand . 105 sec [33]. We have veri ed that for the parameter region we will discuss in the following, p4 and thus hS(T ) can indeed be adjusted such that & 1011 sec. Even in this case, care must be taken that the residual abundance of at the time of decay is tiny (. 10 10 ( =1015 sec) times the DM abundance) to avoid anomalous reionization [35]. We have checked that this can be automatically achieved for DM masses . 1 GeV. In this case, is large because the only decay channels available to are suppressed by the small Yukawa couplings of light quarks and leptons. At larger DM mass, the easiest way of ensuring compatibility of the model with reionization constraints is to introduce an auxiliary fermion 0, with m0 m , and with a Yukawa coupling to S and chosen such that the decay 0f f is much slower than freeze-in of , but still ! occurs signi cantly before recombination ( . 105 sec [33]). 2.2 The e ective potential To correctly describe the evolution of the scalar sector of the model from eqs. (2.1) and (2.2) in the hot early Universe, we must go beyond the tree level potential and consider the nite temperature e ective potential V e which includes radiative corrections and thermal e ects. Since we assume the portal couplings p3 and p4 to be small, we can treat the evolution of the dark sector potential as independent from that of the visible sector potential (we will consider the case of large portal couplings in sections 3 and 4). We begin with the approximate tree level potential in the dark sector, V tree(S) 2 2 S S2 + 4! S4 S4 : (2.12) tum corrections, is obtained from needs to be computed perturbatively. The e ective potential V e is de ned in the usual way [36]: one rst rewrites the generating functional, E[J ] = i log Z[J ], as a functional of vS(x) instead of the external source eld J (x). (Here, Z[J ] is the partition function.) This is achieved by relating J (x) with vS(x) via vS(x) = E[J ]= J . Note that, in the presence of an x-dependent external source, vS(x) becomes a function of x as well. The e ective action is in turn the spacetime integral of the e ective potential V e , is given by a Legendre e , which transform: property that e [vS] = R d4x V e [vS] R d4y J (y) vS(y). We see that e has the e [vS]= vS = 0, that is, the vacuum con guration vS, including all quane using a variational principle. Of course, e itself { 7 { 0 GeV, (left). A constant term has been subtracted at each T so that V e (0) = 0. The evolution of the new scalar mass and vev with temperature, (right), with the temperature of the phase transition indicated. Note that we neglect contributions from loops to V e , assuming y . 0:01. As we outline in more detail in section A, the leading corrections that distinguish V e from V tree are the Coleman-Weinberg term V CW that corresponds to resummed 1-loop diagrams at T = 0 [37], the thermal one-loop contribution [38], V T , and the resummed series of higher order \daisy" diagrams, V daisy [39{42]. With our assumption that y , p3, and p4 are tiny, loops involving H and are negligible. Loops involving could be relevant at temperatures not too far below m if y & 0:01. In the following we will assume y . 0:01 to simplify the analysis. As explained below eq. (2.2) this will require a di erent mechanism for keeping in thermal equilibrium throughout DM freeze-in. We have veri ed that our toy model can be phenomenologically successful also for larger y . For y . 0:01, the only relevant diagrams contributing to V e are those involving the quartic coupling S4. In other words, the sums in eqs. (A.2), (A.7) and (A.8) run only over S. The coe cient ni, which can be interpreted as counting degrees of freedom (although see [42]), is nS = 1. As a function of the eld value, the mass of S is given by V daisy also depends on the thermal, or Debye, mass, which is given by the 1-loop self energy at non-zero T . The Debye mass of S is given by [39] m2S ' 2S + 2 S4 S2 : T 2 24 With the e ective potential in hand, we can now consider the evolution of mS and vS hSi as a function of T . This allows us to describe the phase transition, which is analogous to the electroweak phase transition in the SM. We use the program CosmoTransitions [43{ 46] to track the minimum of the e ective potential, to nd hSi as a function of T , and to { 8 { determine the mass of S as the second derivative of V e (S). Although in our particular toy model it would be easy to do the computation without invoking CosmoTransitions, we still use it for consistency with sections 3 and 4. The e ective potential at several temperatures is shown in gure 2 (left), while the behaviour of hSi and mS(T ) is shown in gure 2 (right). In the left panel we see the well known behaviour of a second order phase transition or cross-over: at high temperatures, Tc, the e ective potential has its minimum at S = 0. At the critical temperature, Tc, the minimum begins to move away from S = 0 and a non-zero vev begins to develop. At the present time, near T = 0 GeV, the e ective potential has its minimum at S ' 9 GeV. In gure 2 (right), we similarly see that at high temperatures, S has no vev, and its e ective mass is large thanks to thermal corrections / T . As the temperature drops, the mass of S drops as well and approaches zero at the phase transition. This behaviour can be understood also from the green curve in gure 2 (left): at the transition temperature, its second derivative at the minimum is zero. After the phase transition, the vev and the mass of S grow and quickly approach their present day values, mS(T = 0) = 5 GeV and hSi ' 9 GeV. As well as this temperature dependence of the S mass, the fermions may also have temperature dependent mass contributions. Since we take the tree level fermion masses to be much larger than the S mass, self-energy diagrams evaluated at T 6= 0 will only give a small contribution near the phase transition (fermions have no zero Matsubara mode, so the self-energy contributions for fermions are smaller than those for bosons). However, if y is large, then there can be signi cant corrections to the mass from the Lagrangian term y S when S obtains its vev. The T dependent mass is then m (T ) = m y hSi : (2.15) The crucial point for us is that, thanks to the behaviour of mS(T ) and m (T ), the ! freeze-in channel S is kinematically closed long before and long after the dark sector phase transition, while around the transition temperature, it is open and DM freeze-in can proceed e ciently. 2.3 Results and discussion In gure 3 (top-left) we show the instantaneous DM yield from each freeze-in channel in gure 1, including (solid lines) and ignoring (dashed lines) nite temperature contributions to the scalar masses and vevs and to the fermion masses. If we ignore the nite temperature corrections, only two channels (S ! S and SS ! ) contribute to the abundance. These contributions are largest at high temperatures (or small x, where x = m =T ), where the abundance of is not Boltzmann suppressed and where S have enough energy to produce the heavier states and . The instantaneous freeze-in yield reduces smoothly as the temperature reduces, except for small steps where SM particles freeze-out and the e ective number of relativistic degrees of freedom in the Universe, ge , changes. If nite temperature e ects are included, then two new channels contribute at di erent times. At very high temperatures, S has a su ciently large mass that it can decay to HJEP03(218)4 (normalised to the abundance when nite temperature e ects are included) through each channel as a function of m for mS(T = 0) = 5 GeV (bottom-left) and mS(T = 0) = 10 GeV (bottom-right). As and are much heavier than S at T = 0 GeV, this channel is only open at very high temperatures (x . 0:1), and it no longer contributes at lower temperatures. As the Universe approaches the phase transition in the dark sector, the mass of S reduces until it becomes smaller than m (T ) m obtains its vev). At this point, the decay at x ' 1:7 (the mass of is constant before S ! S becomes kinematically possible and is produced. This happens at a rate much larger than that via the S ! S and SS ! channels because the latter channels are suppressed by the o -shellness of the intermediate ! propagator. The S channel reaches its maximum rate around the dark phasetransition, where mS(T ) goes to zero. As the temperature further reduces, the mass of S increases and the mass of reduces until the channel closes at x ' 4:5. Comparing the rates with and without including nite temperature e ects, we see that these e ects are relevant in all channels. The rate of S S shows a peak at the dark phase-transition because the intermediate s-channel propagator in the third diagram of gure 1 can go nearly on-shell when mS(T ) is small around the transition temperature. This is essentially a manifestation of the infrared divergence of the corresponding amplitude. The resulting relic abundance of extrapolated to zero redshift is shown in gure 3 ! (top-right). The extrapolated abundance at a given x m =T is obtained by rescaling the number density at this time by the subsequent expansion of the Universe and normalising to the critical density today. Here we clearly see that the dominant contribution to the relic abundance comes from the S channel. We emphasise that if nite temperature e ects were not included, the calculated relic abundance would be incorrect by a factor O(104). For the benchmark parameters chosen, the resulting observed relic abundance for y = 2:57 In gure 3 (bottom-left) we show the abundance produced through each channel as a function of m , normalised to the abundance when nite temperature e ects are included. We keep the mass di erence between at T = 0 and xed at 4 GeV. The fraction of produced through channels other than S is always small, but it is smallest for ! m ' 50 GeV. At lower values of m , S no longer requires signi cant thermal energy to produce via S ! S and SS ! , so the ! S channel produces a smaller fraction of . As the value of m increases, processes which occur at lower temperatures receive more Boltzmann suppression than those occurring at higher temperatures. This means that the amount of produced through S ! S and SS ! (which is important at high temperatures) is mildly reduced whereas the amount produced via ! S (which is important around the phase transition) has greater Boltzmann suppression, reducing its relative importance. Finally, gure 3 (bottom-right) shows the freeze-in abundance of through the di erent channels for mS = 10 GeV, where now the phase transition occurs at T = 36 GeV. We see that the picture is qualitatively similar to gure 3 (bottom-left). For ms = 10 GeV, there is a milder reduction in the relative importance of the S channel at large m , due to a milder Boltzmann suppression of the abundance at the phase transition. In both gure 3 (bottom-left) and (bottom-right), the yields due to S S are similar ! including or ignoring nite temperature corrections. This channel predominantly produces ! at high temperatures, where the particle momentum dominates over the particle masses. For the small Yukawa coupling y = 0:01 and large quartic coupling S4 = 1, the SS ! production rate is much larger when S has a vev, which explains the di erence seen between the curves that include or ignore the nite temperature corrections in this channel. For de niteness in our numerical calculations, we x a reheating temperature TR = 1 TeV which is su ciently large so that any freeze-in at higher temperatures will produce negligible abundance of . We nish this section by noting a particularly simple model which shares many features with the toy model discussed above. If the SM is extended by two dark sector fermions, a SM gauge singlet, , and an su( 2 )L doublet with hypercharge, , then the SM Higgs can play the role of S above. The Lagrangian term H + h:c: can lead to processes which produce via freeze-in. If m m < mh(T = 0) and m < m , then the channel ! H will be open only when the mass of the SM Higgs is reduced during the second order electroweak phase transition. For lighter than 1.1 TeV, it freezes-out as a subdominant component of the dark matter abundance [47]. The remaining relic abundance can then be provided by the freeze-in of . The calculation of the abundance is somewhat complicated and we defer this to later work. 3 Vev-induced production with a vev In the previous section we have highlighted the potential importance of including nite temperature corrections to particle masses in calculations of DM freeze-in. In this section, we consider a model with a fermionic DM candidate and with a scalar sector identical to the one in eq. (2.1), but focussing on a di erent region of parameter space. Namely, we now take the Higgs portal couplings so large that a two-step phase transition (or \vev ipop" [28]) is realised [27, 48{54] (see also [55]). In other words, the Universe goes through a phase where the new scalar S obtains a non-zero vev, but this vev jumps back to zero in a rst order electroweak phase transition. The value of hSi will control the DM freeze-in rate, so DM can only be e ciently produced during a relatively short time interval. The nal DM abundance is determined not only by the relevant coupling constants, but also by the length of this time interval. We dub this mechanism \vev-induced production." 3.1 The eld content of the dark sector in this toy model is shown in table 2. As in section 2, our dark matter candidate is a Dirac fermion, , which is a SM gauge singlet. We assume that it is stabilised by a Z2 symmetry. The DM mass m and the scalar mass parameter j Sj are taken to be ' 100 GeV. The relevant terms in the Lagrangian are y S V (H; S) ; with V (H; S) again given by eq. (2.2). DM will freeze-in via the Yukawa coupling y ; consequently, this coupling needs to be tiny. As in section 3.1, this could be motivated in extra-dimensional scenarios by localising far away from the other elds along a fth dimension. S3 and p3 are assumed to be small as well to simplify the analysis. Note that an extra global Z2 symmetry is restored if y , S3, and p3 are set to zero. Therefore, small values for these couplings are natural in the 't Hooft sense [56]. Setting p3 1 means we can ignore mixing between the SM Higgs boson and S at T = 0. In this limit, the mass of S is given by m2S(T = 0) = 2S + 2 p4 vH2 (T = 0) at tree level and T = 0. We see that for can be realised where p4vH2 (T = 0)=2 > 2 S S ' vH (T = 0) and . In this case, vS(T ) p4 ' 1, a situation hSi 6= 0 as long as vH (T ) = 0 (barring thermal corrections for the moment), but when vH (T ) becomes signi cantly di erent from zero, the term quadratic in S in the scalar potential experiences a sign ip, making vS(T ) = 0 energetically favourable in the broken phase of electroweak (3.1) (3.2) Field S Spin 0 1 2 +1 1 symmetry. vS(T ) = 0 is also realised at very early time thanks to thermal corrections to V e . These corrections are large especially when S4 & 1. This behaviour is the gist of the vev ip- op, and it de nes the parameter region we will be interested in the following: S ' vH (T = 0), p4 ' S4 ' 1. The e ective potential & the vev ip- op To quantitatively compute the e ective potential for the model de ned in eq. (3.1), the same methods as in section 2.2 can be applied, but since the Higgs portal coupling p4 is no longer negligible, we need to consider the joint evolution of the visible and dark scalar sector. In other words, we need to treat V e as a function of both S and H. As explained in Secion 2.2 and appendix A, the one-loop contribution to the e ective potential, V CW(h; S)+V T (h; S), depends on the eld dependent masses of all particles with couplings to the scalars. In the model from eq. (3.1), the eld dependent masses of W i; B and t are the same as in the SM. In particular the gauge boson mass eigenvalues are m2W m2Z = 14 (g2 + g02)h2, m2 = 0, and mt = yth=p2. Here, g and g0 are the su( 2 )L and u( 1 )Y = 14 g2h2, gauge couplings, respectively, and yt is the top quark Yukawa coupling. The mass matrix of the neutral CP-even Higgs bosons is m(2h;S)(h; S) = 2H + p3 SS + 12 p4 S2 +3 H4 h2 ( p3 S + p4 S)h ( p3 S + p4 S)h 2S + S3 S S + 12 S4 S2 + 12 p4 h2 ; while the mass of the neutral CP-odd and the charged component of H are given by m2G+;G0 (h; S) = 2H + H4 h2 + p3 S S + 1 2 p4 S2 : We can see that in regions where both h and S are non-zero, there will be mixing between the associated particles. Note that with our simplifying assumption p3 0, we can neglect h{S mixing in the hSi = 0, hhi 6= 0 phase at T = 0. The sums in V CW(h; S) and V T (h; S) (see eqs. (A.2) and (A.7)) now run over i 2 fh; S; G0; G+; W i; B; tg. For h and S, this is understood to mean summing over the neutral CP-even mass eigenstates, determined by diagonalising the mass matrix in eq. (3.3). The coe cients ni for the SM elds are nh = ns = nG0 = 1, nG+ = 2, nW i = nB = 3, and nt = 12 [42]. The Debye masses of h and S, relevant in the computation of V daisy (see eq. (A.8)) are h;G0;G+ = (24 H4 + 9g2 + 3g02 + 12yt2 + 2 p4) ; (3.3) (3.4) (3.5) (3.6) -100 -200 200 100 0 production scenario de ned in eq. (3.1). The black cross indicates the phase the Universe is in at the given temperatures. The two-step phase transition (\vev ip- op") is clearly visible: at T = 500 GeV (top left), both hSi and hhi vanish, as V e is dominated by thermal corrections. At a lower temperature, the Universe transitions to the hSi 6= 0, hhi = 0 phase (top right), but eventually the minimum with hSi = 0, hhi 6= 0 becomes the global one, so the Universe transitions into it and remains there (bottom left and bottom right). The Debye masses of W i, B and t are the same as in the SM (see section A) as these particles do not couple to S. The sum in eq. (A.8) runs over i 2 fh; G0; G+; S; W i; Bg. The contribution of to the e ective potential is negligible due to the smallness of y and is therefore dropped in our calculations. The behaviour of the e ective potential is illustrated in gure 4 for a parameter point featuring a two-step phase transition or vev ip- op. At early times (top left panel of gure 4), V e is dominated by the nite temperature and therefore approximately parabolic. Consequently, the SM Higgs and S have zero vacuum expectation values. As the universe expands and cools down (top right panel of gure 4), the nite temperature corrections become similar in magnitude to the tree-level terms and the e ective potential develops minima at hSi 6= 0. There is typically a second order phase transition, so the Universe immediately enters a phase where S has a non-zero vev. After further cooling, new minima at hhi 6= 0 develop. These become the global minima at some critical temperature Tc. However, there will now be a barrier between the hSi 6= 0 and the hhi 6= 0 minima, so the Universe cannot immediately transition into the global minimum, but undergoes a short period of supercooling. The subsequent phase transition is rst order and proceeds via bubble nucleation, when at the nucleation temperature Tn it becomes energetically point in the vev-induced DM freeze-in scenario. mH denotes the mass of the SM Higgs doublet above electroweak symmetry breaking, while mh is the mass of the SM-like physical Higgs boson below. favourable for bubbles of the new phase to expand and ll the entire universe. Typically, one nds Tn ' Tc, but in narrow regions of parameter space, Tn may also be signi cantly below Tc. As in section 2, we have used CosmoTransitions [43{46] to determine the nucleation temperature. Numerically, CosmoTransitions computes Tn by determining the temperature at which SE (T )=T drops below a critical value of 140. Here SE (T ) is the minimum Euclidean action corresponding to a transition between the two potential minima [43]. The e ective potential at T = Tn is shown in the bottom left panel of gure 4. The h 6= 0 minima then deepen as T goes to zero, and the universe remains in a phase where hSi = 0 and hhi 6= 0. In gure 5 we show the masses and vevs of the new scalar S and the SM Higgs doublet as a function of temperature. At high temperatures hSi is zero, but at lower temperatures it obtains a non-zero value. This situation persists until the rst order electroweak phase transition at Tn = 136 GeV. At this point, hSi goes to zero while the SM Higgs vev becomes non-zero. hhi then gradually increases until it attains its T = 0 value of 246 GeV. We can see that, as in gure 4, the scalars receive large nite temperature corrections to their masses at high temperatures. For the parameters chosen here, both mS and mh become smaller than their T = 0 values between the phase transitions, similar to what we found for mS in section 3.2. 3.3 Dark matter freeze-in and relic abundance In the model de ned in eq. (3.1), the coupling p4 between the new scalar eld, S, and the SM Higgs doublet needs to be of order one for the vev ip- op to occur (see discussion below eq. (3.2)). Sizeable p4 in turn means that at T ' mS; mH , the scalar S is in thermal equilibrium with the SM sector. , on the other hand, never comes into thermal equilibrium because of our assumption that y is tiny. Instead, a small abundance of is S ¯ χ λp3µ S, λp4vS S ¯ χ λS3µ S, λS4vS S ¯ χ produced via freeze-in, facilitated by the processes S ! The corresponding Feynman diagrams are shown in , HyH ! , and SS ! gure 6, and the decay rates and annihilation cross sections are The last expression should be understood as the cross section for one of the two components of the doublet H to annihilate with its antiparticle into . The rst process, S ! is kinematically forbidden when mS(T ) < 2m , but since thermal corrections may drive mS(T ) to large values at T mS(T = 0), it may be allowed at early times. Whether or not this production channel is important will thus depend on the reheating temperature. We will be particularly interested in TR not too far above the electroweak scale, as in this case the dynamics of the vev ip- op are most important for DM physics. The other two freeze-in processes can be mediated either by the cubic scalar couplings S3, p3, or by the quartic couplings S4, p4 if vS 6= 0. We will focus on the parameter region where the cubic couplings are small because this is the region where freeze-in depends most strongly on vS and thus on the dynamics of the vev ip- op. Moreover, as explained in section 3.1, S3; p3 1 is technically natural in the 't Hooft sense. We note, however, that p3 should not be exactly zero. Otherwise, the small relic abundance of S could not decay away and would violate direct detection limits. The small vev that S has even at T = 0 when p3 6= 0 would not a ect our results. It is also important to note that the restrictions we impose here on the parameters of the scalar potential and on TR are not necessary to make the model phenomenologically viable. There are other large regions of parameter space where the DM relic abundance can be successfully generated, albeit without strong involvement of the vev ip- op. The DM production rate and the resulting abundance (extrapolated to zero redshift) are shown in gure 7 for two illustrative parameter points of the vev-induced freeze-in scenario. The rst parameter point shown in gure 7 (top panels) is characterised by a low reheating temperature TR = 500 GeV. In this case, DM production is entirely dominated by 2 ! 2 processes proportional to vS, so the dynamics of the vev ip- op are crucial in this case. We see that the DM production rate dY =dx rises rapidly after S develops a vev (3.7) (3.8) (3.9) . , HJEP03(218)4 (right) as a function of x m =T for two di erent parameter points in the vev-induced freeze-in scenario de ned by eq. (3.1). Solid curves correspond to the di erent production mechanisms shown in gure 6. The dashed blue line indicates the result one would obtain if thermal corrections and the vev ip- op were neglected. We see that for a reheating temperature TR just above the electroweak scale (top panels), the DM abundance is entirely dominated by the processes HyH ! and SS ! process S ! , whose rates are greatly enhanced when hSi 6= 0. For larger TR (bottom panels), the , which is independent of hSi, becomes allowed thanks to thermal corrections, even though at T = 0, mS < 2m . at T ' 250 GeV. Between the two phase transitions, dY =dx follows the evolution of vS, and the contribution from the vS-dependent channels is 2{3 orders of magnitude larger than the contribution from vev-independent SS ! vev-dependent channels, SS ! dominates over HyH ! annihilation via S3. Among the mainly because S4 > p4. The small drop in dY =dx immediately before the electroweak phase transition is due to the onset of Boltzmann suppression of H and S. After the electroweak phase transition at T ' 136 GeV, the vS-dependent production processes cease. Before and after the two phase transitions, only the vS-independent channel SS ! via S3 is active, but its overall temperature e ects in the vev-induced freeze-in scenario. We show a cut through the parameter space of the model in the plane spanned by the zero temperature mass of S, mS(T = 0), and the quartic Higgs portal coupling p4. The white outline indicates the point considered in gure 7. contribution is tiny. At x . 0:5, this channel is suppressed as the s-channel mediator, S, is very heavy at these high temperatures. Beyond the electroweak phase transition this channel gradually reduces, due to the Boltzmann suppression of S. At the second parameter point shown in gure 7 (bottom panels) the behaviour of the vev-dependent DM production channels and of vev-independent production via SS ! (mediated by 3) is similar to the top panels. However, as the second parameter point features a larger TR = 10 TeV, the decay channel S ! , which does not depend on vS and on the vev ip- op, is kinematically allowed at x . 0:2, when thermal corrections lift mS above 2m . In this case, this production channel dominates the nal abundance. We conclude that, for low TR, it is the dynamics of the vev ip- op that determines the DM abundance today. For high TR, it is the thermal corrections to mS(T ), which in turn depend on the couplings in the scalar sector, especially S4. In either case, the inclusion of thermal e ects in the computation of the DM relic density is essential. To emphasise this point, we show in gure 7 also the production rate and abundance that would be obtained if thermal corrections to V e (and thus the two-step phase transition) were neglected in the calculation (dashed blue lines). In this case, only the processes SS ! and HyH ! with cross sections proportional to the small couplings contribute. For our choice p3 = 0, only SS ! 2 S3 and 2p3, respectively, would is open. The production rate in this channel is non-zero at TR (or even during preheating [57], which we neglect here assuming it is very rapid) and rst rises slightly as there is more time to freeze-in at greater x. At x & 0:5, the rate begins to drop as Boltzmann suppression becomes signi cant. Eventually, S3-mediated production via SS ! freezes out. We further explore the crucial importance of thermal corrections in gure 8, which shows a cut through the model's parameter space in the plane spanned by the zero temField S Spin 0 1 2 1 2 +1 1 1 +1 m m perature mass of S, mS(T = 0), and the quartic Higgs portal coupling p4. The pixelated region shows where the two-step phase transition occurs. The colour coding quanti es the ratio of the DM abundance obtained including thermal corrections to the abundance if these corrections were neglected for a low TR. We see that thermal e ects dominate the abundance by up to four orders of magnitude. For xed m , they are largest at small mS(T = 0), where thermal corrections to mS are most important, and at large p4, where the hSi 6= 0 phase lasts longer. For a high TR inated by the S ! four orders of magnitude over the whole parameter space shown. In the white area at the top of gure 8, the global minimum of V e at T = 0 would be the one with hSi 6= 0, i.e., electroweak symmetry would never be broken. In the white region at the bottom of the plot, S never acquires a non-zero vev. 10 TeV, the freeze-in abundance is domchannel and thermal e ects dominate the abundance by around 4 Vev-induced mixing with a vev Let us now move to a third scenario illustrating the importance of thermal e ects on the DM abundance in the Universe. The scenario discussed in the following, which we dub mixing induced freeze-in, is based on the same particle content as the kinematically induced freeze-in model from section 2 with an extra discrete symmetry. The model's Lagrangian is thus given by eqs. (2.1) and (2.2), with the two forbidden Yukawa terms removed. The most important term for DM freeze-in is again the Yukawa coupling y S . However, we now assume the DM candidate and the new scalar S to have masses around the electroweak scale, with mS > m . The auxiliary new fermion is assumed to be much heavier (see table 3). The idea is that the reheating temperature is low, TR < m , so that never comes into thermal equilibrium and DM production via ! S (the channel we had focused on in section 2) does not occur. Instead, the main DM production channels will be S ! , facilitated by { mixing through the S coupling, and SS ! mediated by a t-channel (see gure 9). The former process is of particular interest to us because { mixing depends on the vev of S. For the parameters of the scalar potential, we consider values similar to the ones we chose (and motivated) in section 3: negligible S3, p3, but sizeable Higgs portal and dark sector quartic couplings, p4; S4 ' O( 1 ), to induce a vev ip- op. Thus, DM production via S ! will be open for a limited amount of time while vS 6= 0. In the following, we will study the interplay of the two production processes, focusing in particular on the importance of vS and the vev ip- op. ¯ χ ¯ χ denotes the hSi-dependent mixing angle between and . The decay width for S ! and the cross section for SS ! θ is ) = (S ! (SS ! reads y 4 (4.1) (4.2) (4.3) where we have taken the limit m assumption m TR. The mixing angle between and is ps; mS(T ); m , which is justi ed in view of our The dynamics of freeze-in via mixing are illustrated in gure 10. In analogy to gure 7 in section 3, this gure shows the DM production rate and the extrapolated DM abundance as a function of x m =T . We again consider one parameter point with a low reheating temperature, TR = 150 GeV (top panels), and one parameter point with a higher reheating temperature, TR = 500 GeV (bottom panels). For low TR, the DM abundance today is dominated by the decay S ! , which is only possible for vS 6= 0. For the parameters shown in the top panels of gure 10, this phase is already realised at TR. The rate for S ! increases along with vS(T ) and then drops sharply to zero at x ' 0:7 as the electroweak phase transition switches vS o in favour of non-zero vH . Similar behaviour is also seen for higher reheating temperature (bottom panels of gure 10), but in this case, the overall importance of S ! does not dominate that of SS ! . As the latter process is independent of vS, it leads to DM production immediately after reheating (or already during preheating, which we neglect here), while S ! vS becomes non-zero at x ' 0:2. Note from eq. (4.2) that for TR < m is only activated when (the case realised in gure 10), the DM production rate via SS ! is ultraviolet-dominated [58]. The dependence of SS ! scales / T 3=m2 . In other words, freeze-in on thermal corrections (namely the temperature dependence of mS(T )) is relatively weak. It is re ected for instance in a jump in the rate at the electroweak phase transition. For comparison, we show also the production rate and DM yield in the absence of thermal corrections (dashed lines). We see that a calculation neglecting these corrections would fairly accurately predict freeze-in via SS ! , but would completely miss the mixing-induced channel S ! . z = 0 (right) as a function of x m =T for two di erent parameter points in the mixing induced scenario from section 4. Solid curves correspond to the di erent production mechanisms shown in gure 9. The dashed curves indicates the result one would obtain if thermal corrections and the vev ip- op were neglected. We see that for a reheating temperature TR just above the electroweak scale (top panels), the DM abundance is dominated by the mixing-induced decay S ! is facilitated by the vev ip- op. For larger TR (bottom panels), the process SS ! , which , which is independent of hSi, becomes relevant. We further explore the dependence of thermal e ects on the parameters of the mixing induced scenario in gure 11. The two panels in this gure show di erent slices through the parameter space: one in the mS(T = 0){ p4 plane (cf. gure 8) and one in the mS(T = 0){ S4 plane. We observe that thermal e ects | in this case the vS-dependent process S ! | are most relevant both when the Higgs portal coupling p4 is large and when the quartic self-coupling S4 or the T = 0 S mass is small. For larger p4 or smaller mS(T = 0), the vS 6= 0 vacuum becomes deeper and the phase during which S has a vev and S ! is open becomes longer. For small S4, the vS 6= 0 phase begins earlier, again implying that there is more time for DM production via mixing. In any case, we see that nite temperature e ects in the mixing induced scenario. We show a cut through the parameter space in the plane spanned by the zero temperature mass of S, mS(T = 0), and the quartic Higgs portal coupling p4 (left) and a slice of parameter space in the mS(T = 0){ S4 plane (right). We see that the impact of thermal e ects is largest at large portal coupling p4 and small self-coupling lasts longer. The black outline indicates the point considered in gure 10. S4, where the hSi 6= 0 phase thermal e ects are at most O( 1 ) in most of the parameter space, but in some regions can modify the DM abundance today by an order of magnitude. 5 Summary and conclusions In this work we have considered the impact of nite temperature corrections on the freezein of dark matter. We have highlighted several e ects which can have a dramatic impact on dark matter production. We have illustrated the impact of these e ects in three toy models, which demonstrate `kinematically induced', `vev induced' and `mixing induced' freeze-in, respectively. In `kinematically induced freeze-in', the dominant production channel of dark matter may be closed at zero temperature, but may be open in the early universe as temperaturedependent particle masses vary. Although calculationally complex, a simple realisation of this is the SM Higgs coupling to two dark sector fermions. We have highlighted the analogous e ect in a realistic toy model consisting of a new scalar which is weakly coupled to the SM, and two dark sector fermions. We show that a calculation ignoring the temperature-dependent scalar mass produces an estimate of the dark matter abundance which is incorrect by a factor of O(10). If instead, the new scalar couples signi cantly to the SM Higgs, the Higgs portal coupling can induce a two-step phase transition (or \vev ip- op"). In this case the new scalar may obtain a vev for some time, which then disappears when the SM Higgs obtains its vev. This can lead to `vev induced' production, where dominant channels of dark matter production only open when the new scalar has a vev. We illustrate the e ect in a phenomenologically viable toy model. For reheating temperatures around the electroweak scale, dark matter production occurs mainly via `vev induced' production, whereas for higher reheating temperatures (TR & a few TeV) `kinematically induced freeze-in' dominates the production. In both cases, these nite temperature e ects can easily change the relic abundance by several orders of magnitude. Finally we consider a scenario where the temporary vev of a new scalar leads to mixing between fermionic dark matter and another fermion, which we call `mixing induced freezein'. This mixing may then o er a dark matter production mode which is not available at zero-temperature. We again consider a realistic toy model and show that dark matter can be dominantly produced through this temperature-dependent channel. The error introduced by ignoring this e ect can be as large as a factor of 10, but depends crucially on a reheating temperature around the electroweak scale and a long period of vev induced mixing. When the reheating temperature is much higher or the vev induced mixing is weak, the standard calculation provides a reliable estimate. Acknowledgments It is a pleasure to thank Christophe Grojean, Matthias Konig, and Andrea Thamm for useful discussions. MJB would like to thank CERN for warm hospitality during part of this work. This work has been funded by the German Research Foundation (DFG) under Grant Nos. EXC-1098, KO 4820/1{1, FOR 2239, GRK 1581, and by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 637506, \ Directions"). MJB was also supported by the Swiss National Science Foundation (SNF) under contract 200021-175940. A Computation of the e ective potential In this appendix, we review the computation of the e ective potential V e (h; S; T ) at nonzero temperature. The leading terms in V e (h; S; T ) are the temperature-independent tree level potential V tree(h; S), the Coleman Weinberg correction V CW(h; S) [37], and a counterterm V CT(h; S), as well as the temperature dependent 1-loop thermal corrections V T (h; S; T ) [38] and a contribution from resummed higher order \daisy" dia V e (h; S; T ) ' V tree(h; S)+V CW(h; S)+V CT(h; S)+V T (h; S; T )+V daisy(h; S; T ) : (A.1) The tree level potential V tree is simply read o from the Lagrangian. For the models considered in this paper, it is given by eq. (2.2). The T -independent Coleman-Weinberg contribution is [37, 40] V CW(h; S) = X i 64ni 2 mi4(h; S) log mi2(h; S) 2 Ci + V CT(h; S) ; (A.2) where the sum is over the eigenvalues of the mass matrices of all elds which couple to the scalars, and jnij accounts for their respective numbers of degrees of freedom. ni is positive for bosons and negative for fermions. We take the renormalisation scale to be the SM Higgs vev vH = 246 GeV. In the dimensional regularisation scheme Ci = 5=6 for that vH = gauge bosons and Ci = 3=2 for scalars and fermions. We also add a counterterm to ensure = p , m2h = 2 2 and that mS is given by its tree level value at T = 0. The counterterm is where the factors i are HJEP03(218)4 V CT(h; S) = h2 + 4 1 2 1 4 = = S = 2v 1 2v3 S S2; ; The one-loop nite temperature correction is [38] X niT 4 Z 1 2 2 0 V T (h; S) = dx x2 log 1 exp qx2 + mi2(h; S)=T 2 ; where we sum over the same eigenvalues as for the Coleman-Weinberg contribution. The negative sign in the integrand is for bosons while the positive sign is for fermions. The bosons also contribute to higher order \daisy" diagrams which can be resummed to give [39{42] V daisy = T 12 X ni m2(h; S) + (T ) 2 3 i m2(h; S) i2 : 3 Here, the rst term should be interpreted as the i-th eigenvalue of the matrix-valued quantity [m2(h; S) + (T )]3=2, where m2(h; S) is the block-diagonal matrix composed of the individual mass matrices [59]. The sum runs over the bosonic degrees of freedom. The thermal (Debye) masses in the SM [39] are h;G0;G+ = 116 T 2 3g2 + g02 + 8 H + 4yt2 ; LW 1;2;3 = 161 g2T 2 ; TW 1;2;3 = 0 ; LB = 161 g02T 2 ; TB = 0 : transverse components ( TW;B(T ) = 0). Here, h;G0;G+ denotes the Debye masses of the components of H, while W;B are the Debye masses of the electroweak gauge boson. Note that the latter are non-zero only for the longitudinal components of the gauge bosons ( LW;B(T ) 6= 0), but vanish for the (A.3) (A.4) (A.5) (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) (A.12) (A.13) Boltzmann equations In the following, we discuss the Boltzmann equations governing DM freeze-in. B.1 Freeze-in via decay: B1 ! B2 For a 1 ! 2 decay of the form B1 ! its interaction partners), the general Boltzmann equation is n_ + 3Hn = d B1 d B2 ( 2 ) pB1 pB2 ) Z d jMB1!B2 j2fB1 (1 fB2 )(1 f ) jMB2 !B1 j2fB2 f (1 fB1 ) : (B.1) HJEP03(218)4 B2 (where is the DM candidate and B1, B2 are Here, d i d3pi=( 2 )3 denotes the three-dimensional phase space integral for particle i = ; B1; B2; pi are the particles' momenta and fi their momentum distribution functions in the primordial plasma. MB1!B2 and MB2 !B1 are the transition amplitudes of the freeze-in reaction and its inverse. In freeze-in scenarios, the abundance of remains well below its equilibrium abundance, so f brackets in eq. (B.1) can be dropped. ' 0. Consequently, the second term in square We also neglect Pauli blocking and stimulated emission, so that (1 fBi ) = 1. We can then write Z n_ + 3Hn = d B1 gB1 2mB1 (B1 ! B2 ) fB1 : We assume that B1 is in thermal equilibrium and that EB1 expect this assumption to introduce an error of around 15% in our nal abundances [20]. T so that fB1 ' e EB1 =T . We We obtain n_ + 3Hn = 2 2 gB1 m2B1 (B1 ! B2) T K1 mB1 x m ; where K1 is a modi ed Bessel function of the second kind and gB1 is the number of degrees of freedom of B1. It is convenient to normalise the number densities ni to the entropy density s to factorise the trivial dilution of ni due to the expansion of the Universe. This leads to the yield Yi ni=s. Introducing the dimensionless evolution variable x m =T , the Boltzmann equation takes its nal form dY gB1 m2B1 2 2 m H(x)s(x)x2 (B1 ! B2) K1 mB1 x m ; (B.2) (B.3) (B.4) where H(x) is the Hubble rate. B.2 Freeze-in via annihilation: B1B2 ! B3 Following similar steps as in section B.1, and using the de nition of the M ller velocity vM l [60], the Boltzmann equation for a 2 ! 2 process of the form B1B2 ! B3 reads n_ + 3Hn = gB1 gB2 ( 2 )6 Z dp3B1 dp3B2 (B1B2 ! B3) vM le (EB1 +EB2 )=T : (B.5) We simplify this expression following ref. [61]. To this end, we de ne E+ E s EB1 + EB2 ; EB1 EB2 ; m2B1 + m2B2 + 2EB1 EB2 2jpB1 j jpB2 j cos ; where Ei and pi are the particles' energies and three-momenta, respectively, and angle between pB1 and pB2 . It is straightforward to show that Moreover, we have [61] EB1 EB2 (B1B2 ! B3) vM l = (B1B2 ! B3) pB1B2 ps ; pij p[s (mi + mj )2][s (mi mj )2] p 2 s is the modulus of the momentum of B1 and B2 in the centre of mass frame. The integrand on the right hand side of the Boltzmann equation (B.5) is independent of E and depends on E+ only through the exponential e E+=T . Evaluating the integrals over E+ and E d3pB1 d3pB2 = 2 2EB1 EB2 dE+dE ds : m p x s m (B.6) (B.7) (B.8) is the (B.9) (B.10) (B.11) : (B.13) : (B.14) where yields [61] forms into dY dY dx Z 1 ds n_ + 3Hn = gB1 gB2 (mB1 +mB2 )2 32 4 4p2B1B2 (B1B2 ! B3)T K1 : (B.12) Note that, in counting the degrees of freedom gB1 , gB2 , care must be taken that each degree of freedom counted towards gB1 should be able to annihilate with each degree of freedom counted towards gB2 . This is usually true for spin and colour degrees of freedom (which the cross sections are typically averaged over). Particles and antiparticles, on the other hand, should be treated as di erent initial states, not as di erent degrees of freedom of the same initial state. The same is true for di erent components of an su( 2 )L multiplet. Expressed in terms of particle yields rather than number densities, eq. (B.12) transFor the special case B2 = B1, this simpli es to gB1 gB2 m 32 4 H(x)s(x)x2 Z 1 (mB1 +mB2 )2 ds 4p2B1B2 (B1B2 ! 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Michael J. Baker, Moritz Breitbach, Joachim Kopp, Lukas Mittnacht. Dynamic freeze-in: impact of thermal masses and cosmological phase transitions on dark matter production, Journal of High Energy Physics, 2018, 114, DOI: 10.1007/JHEP03(2018)114