Leptogenesis in Cosmological Relaxation with Particle Production

Cosmological relaxation of the electroweak scale is improved by using particle production to trap the relaxion. We combine leptogenesis with such a relaxion model that has no extremely small parameters or large e-foldings. Scanning happens after inflation--now allowed to be at a high scale--over a sub-Planckian relaxion field range for an $\mathcal{O}(100)$ TeV cut-off scale of new physics. Particle production by the relaxion also reheats the universe and generates the baryonic matter-antimatter asymmetry. We propose a realisation in which out-of-equilibrium leptons, produced by the relaxion, scatter with the thermal bath through interactions that violate CP and lepton number via higher-dimensional operators. Such a minimal effective field theory setup, with no new physics below the cut-off, naturally decouples new physics while linking leptogenesis to relaxion particle production; the baryon asymmetry of the universe can thus be intrinsically tied to a weak scale hierarchy.

Introduction. -There is no doubt that the Standard Model (SM) is a low-energy effective field theory (EFT) approximating a more fundamental theory at higher energies. That theory presumably gives us a deeper understanding of a plethora of arbitrary features currently fixed by hand in the SM, not least of which are the Higgs sector's Yukawa couplings and electroweak-symmetrybreaking potential. For the latter, in particular, there is good reason to believe that such an explanation should lie close to the weak scale as it is well-known that, in any calculable Higgs potential, a light Higgs mass is sensitive to large ultra-violet quantum corrections. This fact is, however, in tension with null results in the search for new physics at the LHC. They could well be indicating that the underlying theory is decoupled at higher energies. It is then interesting to explore ways of naturally obtaining a little hierarchy between the weak scale and this intermediate energy scale representing the SM EFT cut-off, with no new physics in between. Indeed, a new class of models to achieve this using early universe dynamics has emerged under the name of cosmological relaxation [1].
The general idea is as follows. Consider an axion (the so-called relaxion), φ, whose shift symmetry is softly broken by some dimensionful parameter g; this can arise, for example, in axion monodromy [2] and clockwork constructions [3]. Below the SM EFT cut-off Λ, including all interactions not forbidden by symmetries, the most general Lagrangian contains the terms, where h is the Higgs doublet with mass µ 2 ∼ Λ 2 , and the ellipses denote higher-order terms in the soft-breaking potential V (gφ). This potential causes a slope along which φ rolls during the early universe. As it rolls, it scans an effective Higgs mass µ 2 | eff. ≡ Λ 2 − gφ. At negative values of µ 2 | eff. , the Higgs' vacuum expectation value v is non-zero. All that is then needed to explain why v Λ is for a backreaction to switch on and trap the relaxion when µ 2 | eff. is small and negative.
In the original GKR mechanism [1], the trapping backreaction acted on the relaxion's periodic potential, whose barriers Λ 4 c Λ 3 QCD v will grow with a linear dependence on v until they are sufficiently large to compensate for the slope. Unfortunately this creates several problems: if φ is the QCD axion, it no longer solves the strong CP problem without some additional mechanism, and if the periodic potential is due to the condensate of another gauge group, then it reintroduces new physics near the weak scale. Moreover, for the barrier to trap the relaxion at the weak scale requires g ∼ 10 −31 GeV. Despite being technically natural (the shift symmetry is restored as g goes to zero) such a tiny value leads to conflict with the weak gravity conjecture [4] and exponentially long e-foldings of super-Planckian scanning during inflation.
An improvement comes from trapping using particle production [5,6] 1 . The relaxion's shift symmetry permits an anomalous coupling to gauge bosons and a derivative coupling to fermions, where F µν is a gauge field strength and J 5µ a fermionic current. The exponential production of gauge bosons is an efficient source of friction. In the context of the original GKR model, it was used by Ref. [8] to prevent rolling past the weak scale during reheating and in Ref. [9] to reheat the universe by Schwinger pair production in strong electric fields. In other cosmological relaxation models [5,6], it is instead an intrinsic part of the backreaction mechanism, since the periodic potential barriers no longer depend on v. In this case the relaxion initially has sufficient kinetic energy to roll over them. In Ref. [6], dark gauge boson production was used to slow the relaxion into being trapped: the backreaction of v switching on triggered the end of inflation which caused Hubble to fall below the gauge boson production threshold. In this work we shall follow instead the approach of Hook and Marques-Tavares (HMT) [5], where the vdependence of the backreaction mechanism resides more directly in electroweak gauge boson production. After describing the essential features of the HMT model in the next Section, we then show how it can be combined in a natural way with leptogenesis during reheating.
Cosmological Relaxation with Particle Production. -The HMT model [5] can work either before, during, or after inflation. Relaxation during inflation requires a very low Hubble scale, H < v; we therefore focus on the variation in which scanning happens after inflation ends, which has the added benefit of allowing high-scale inflation 2 . After inflation ends, the inflaton decays to a hidden sector; this ensures the relaxion will be scanning the zero-temperature Higgs potential. The relaxion is initially displaced with an initial field value φ 0 > Λ 2 /g where the effective Higgs mass is negative and v is large. As it scans down to smaller values, over a typical field range ∆φ ∼ Λ 2 /g, the value of v decreases until the electroweak gauge bosons are light enough to be produced 3 . This happens when v ∼φ v /f V , whereφ v φ 0 , and determines the weak scale hierarchy v Λ in a technically natural way. Through particle production, the kinetic energy of the relaxion is converted into the temperature of the visible sector's thermal bath, T 4 φ2 v , thus reheating the universe in the process.
Note that, unlike in the original GKR model, the condensate Λ c of the periodic potential is due to a hidden sector and does not depend on the Higgs' vacuum 2 A non-minimal Higgs coupling to gravity keeps the Higgs located at its minimum. Note that there are models of spontaneous baryogenesis that use a displaced Higgs relaxing back to its minimum after inflation to generate an effective chemical potential, e.g. [10]. These are called Higgs relaxation models and are not to be confused with the similarly named cosmological relaxation mechanism. 3 A coupling to photons must be sufficiently suppressed from the start, for example in an ultra-violet completion with SU (2) L × SU (2) R left-right symmetry [5]. Alternatively, we note that the relaxion coupling to the Higgs will also exponentially produce scalar boson modes that could trap the relaxion in an even more minimal and generic setup. expectation value v. The relaxion must initially have enough kinetic energy to overcome the barriers,φ 2 0 > Λ 4 c , or, as discussed in Ref. [5], it can also start from rest, φ 0 ∼ 0, if there is a 10 % coincidence in the scales of the slope and periodic potential. The initial condition for a non-zero relaxion velocity can be set in several ways after exiting high scale inflation: the inflaton preheating into hidden sector gauge bosons or fermions to which the relaxion couples could temporarily act as a background source in its equation of motion; alternatively, an inflaton-relaxion coupling κσφ can act as a faster effective slow-roll slope during inflation so that it exits with a velocityφ ∼ κσ/H I . If the inflationary scale is low enough, H I gΛ 2 /Λ 2 c , then the shift-symmetry breaking slope alone gives sufficient slow-roll velocity.
The tachyonic condition for exponential gauge boson production is set by the equation of motion for the gauge boson A µ ≡ {Z µ , W ± µ }. In unitary gauge, ∂ µ A µ = 0, and assuming spatial homogeneity, ∇ · A = 0, they can be written in terms of the transverse circular polarisation modes A ± asÄ where We therefore see that only the solutions A ± (k) ∝ exp(iω ± t) with low momentum modes k < m A experience exponential growth once the gauge boson mass m A ∝ v drops below the dissipation threshold ∼φ v /f V . We neglected Hubble, since the dissipation and trapping will happen on a shorter timescale, and assumed zero temperature. In a plasma at finite temperature T, Eq. 5 can be shown to be approximately given by [5] In this case there can always be exponential production . Since the end of relaxation will occur in a thermal bath, the inverse of the tachyonic plasma frequency gives the relevant timescale τ for particle production to lose enough energy to reach the trapping threshold atφ c ∼ Λ 2 c , where we substituted f V ∼φ v /v. This timescale is taken to be faster than Hubble, τ 1/H (note that this is not Hubble during inflation but when the relaxion is able to scan the entire field range ∆φ ∼ Λ 2 /g at H ∼ g). Moreover, the relaxion must not roll past the Higgs mass scale before being trapped, gφdt < v 2 , which leads to the constraint For g ∼ Λ 2 /M p (the value that saturates the bound of the sub-Planckian field range requirement ∆φ M p ) Eq. 8 places an upper limit on the cut-off Λ, This can be maximised forφ v ∼ Λ 2 c ∼ T 2 . However, a stronger bound comes from requiring the relaxion energy density to be sub-dominant during scanning, To maximise the reheating temperature for leptogenesis we shall take this as our typical upper limit for a sub-Planckian field range.
Finally, the decay constant f p must also allow for multiple minima, Λ 4 c fp gΛ 2 , each separated by less than the weak scale, gf p < v 2 . A benchmark point for parameter values that satisfy all constraints is given in Table I. Leptogenesis.
-We now turn to the task of implementing leptogenesis from cosmological relaxation with particle production. The relatively low temperatures achievable in the HMT model restrict the possible scenarios if we wish to avoid reintroducing new physics below the SM EFT cut-off. Reheating to the threshold of some new, heavy particle whose out-of-equilibrium decay is responsible for generating the baryon asymmetry requires reheating above the cut-off or adding new physics below it. Other baryogenesis approaches are also severely restricted by the particular requirements of the relaxion mechanism. Here, we instead make use of leptogenesis generated by inelastic scattering between leptons from the relaxion and leptons in the thermal bath [11,12]. All three of Sakharov's conditions are satisfied-the leptons produced by the relaxion are out-of-equilibrium, and scattering proceeds via higher-dimensional operators that violate lepton number, with CP-violating interactions. The resulting lepton asymmetry number density n L will then be converted to a baryon asymmetry n B by the electroweak sphaleron process, The asymmetry is normalised to the entropy density s = (2π 2 /45)g * T 3 , where g * ∼ 10 2 is the number of relativistic degrees of freedom. An order of magnitude estimate of the baryon asymmetry is sufficient for our purpose, as we typically neglected O(1) factors in our relaxion estimates.
Remarkably, all of the ingredients for this mechanism are already present in the relaxion setup. The derivative lepton current coupling in Eq. 3 not only respects the shift symmetry, but was previously necessary to allow the thermal abundance of the relaxion to decay away below the decoupling temperature. Also, operators of higher mass dimension suppressed by the scale of new physics are generically expected to be present in a low energy effective theory. The most general effective Lagrangian for the SM EFT can be written as The unique dimension 5 operator is the Weinberg operator, where we kept SU (2) L and flavour indices implicit, h is the Higgs doublet field and L the left-handed lepton doublet. It generates a Majorana mass for neutrinos when the Higgs field gets a vacuum expectation value. The light neutrino mass bound m ν ∼ 0.1 eV implies that the corresponding operator scale is Λ 5 ∼ 10 14 GeV for an O(1) Wilson coefficient c (5) . The contribution of this operator to the baryon asymmetry is then typically negligible for the low reheating temperatures of the relaxion, so we focus on dimension-7, lepton-number-violating operators generated at a scale Λ 7 . As an illustrative example we shall take the operator In the notation of Ref. [14], the field e is a right-handed lepton and u, d are up-and down-type right-handed quarks, respectively, and the corresponding coefficient c ab contains indices a, b representing the flavour of L and e respectively. Note that there is a lower bound on the scale Λ 7 coming from the contribution of dimension-7 operators to the neutrino mass, as studied for example in Refs. [13,14]. For the operator of Eq. 13 this bound is low enough to be negligible; other operators have stricter bounds.
We also have a four-fermion dimension-6 operator, with complex coefficient c (6) abcd and scale Λ 6 , whose contribution to the one-loop diagram is responsible for CP violation in the interference term with the tree-level dia- c c E z J + R S 6 o / k + s l 7 M 5 2 A Q i w V 8 V 0 S V P V / Q Z J U K u 6 7 P H D 5 a a U z X q D Q n C J x c m R S U R C b Q I P V 8 c U C H V c 6 Q b U N Y Q O D F i w 1 4 m 1 d Z F B b l R p 0 u k E q p 4 g w k 9 1 d 7 Z D A k O E t c s 0 5 3 n m j M 1 c C M 0 3 V 9 u t K r X R k 0 B b W B / y v c r s I 3 y u 7 c v / s V k V I p C B q 1 O w 2 4 l I Z P b B o 6 2 b + / t e X 2 F v 0 B 9 7 I q 5 e z b / i N M b C a d b n o / 5 p F H J c Z M I k p E m L q e 7 m c q + Y q d P p S Q I 7 w E i U w 1 S Z D G Y i 5 q u e v c o a a R E 7 M C / 0 y 6 d R 0 O 0 K h T I h 1 F m p l h m Q q d n 0 G / s s 3 L W U 8 n i v C 8 l I 3 E W 8 K x S W t B 0 s P s x O R A r C k a 2 0 g X B C 9 V w e n q E B Y T 2 W 7 i p n q L K 8 b 4 + + 2 Y d + 4 C k a + N / I / e Y P z c d O i A + u 5 9 c I 6 s X z r j X V u f b Q u r Y m F u 2 + 7 Y X f Z p f b E V v Z X + 9 t G 2 u 0 0 M c + s 1 r K / / w a J F z 7 W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " o W F S z v d K k f r 9 f Z / D X z 0 h 4 K L l k e U = " > A A A D 0 3 i c d Z L N j t M w E M f T h o + l f G w X j l w i u p U W F F V J l h U 9 g L Q S H D j s Y Z H o 7 k p t V T n O J L H q 2 F H s V F R W L o g r j 8 e B F + A 5 s N P A N i 1 Y i T L 5 z X 9 m 7 P G E O S V C e t 7 P T t e + c / f e / Y M H v Y e P H j 8 5 7 B 8 9 v S w q U p 1 X v 1 l W D V 9 X J 2 H P P v J e 9 o f M B Y l 3 R g S 8 S C o a o o 2 t I g k E 4 c c E z J + R S 6 o / k + s l 7 M 5 2 A Q i w V 8 V 0 S V P V / Q Z J U K u 6 7 P H D 5 a a U z X q D Q n C J x c m R S U R C b Q I P V 8 c U C H V c 6 Q b U N Y Q O D F i w 1 4 m 1 d Z F B b l R p 0 u k E q p 4 g w k 9 1 d 7 Z D A k O E t c s 0 5 3 n m j M 1 c C M 0 3 V 9 u t K r X R k 0 B b W B / y v c r s I 3 y u 7 c v / s V k V I p C B q 1 O w 2 4 l I Z P b B o 6 2 b + / t e X 2 F v 0 B 9 7 I q 5 e z b / i N M b C a d b n o / 5 p F H J c Z M I k p E m L q e 7 m c q + Y q d P p S Q I 7 w E i U w 1 S Z D G Y i 5 q u e v c o a a R E 7 M C / 0 y 6 d R 0 O 0 K h T I h 1 F m p l h m Q q d n 0 G / s s 3 L W U 8 n i v C 8 l I 3 E W 8 K x S W t B 0 s P s x O R A r C k a 2 0 g X B C 9 V w e n q E B Y T 2 W 7 i p n q L K 8 b 4 + + 2 Y d + 4 C k a + N / I / e Y P z c d O i A + u 5 9 c I 6 s X z r j X V u f b Q u r Y m F u 2 + 7 Y X f Z p f b E V v Z X + 9 t G 2 u 0 0 M c + s 1 r K / / w a J F z 7 W < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " o W F S z v d K k f r 9 f Z / D X z 0 h 4 K L l k e U = " > A A A D 0 3 i c d Z L N j t M w E M f T h o + l f G w X j l w i u p U W F F V J l h U 9 g L Q S H D j s Y Z H o 7 k p t V T n O J L H q 2 F H s V F R W L o g r j 8 e B F + A 5 s N P A N i 1 Y i T L 5 z X 9 m 7 P G E O S V C e t 7 P T t e + c / f e / Y M H v Y e P H j 8 5 7 B 8 9

l a t e x i t s h a 1 _ b a s e 6 4 = " E 2 r t r 2 A V J X t u P M B y 7 O X G Y 9 C 7 S I w = " > A A A D 8 H i c d Z L N b t N A E M e d h I 8 S P p r C k Y t F W q k g K 7 J d K n K p V A k O H H o o E m m L 0 i h a r 8 f x q u t d a 3 c d E V k W V 3 g D b o g r b 8 Q L 8 A A 8 A W P H l D g p K 1 s e / 2 b m P 7 u z E 6 S c a e O 6 P 1 v t z q 3 b d + 5 u 3 e v e f / D w 0 X Z v 5 / G Z l p m i M K K S S 3 U R E A 2 c C R g Z Z j h c p A p I E n A 4 D 6 5 e l / 7 z O S j N p H h v F i l M E j I T L G K U G E T T n d b v y w B m T O R R E k W M Q 5 F H s B A z R d K 4 6 P 5 z V e B F s T 9 0 n U P 3 e X f P f g M R V r T h o w E l C L e x h m E U t B 0 p m d i B N A Y / R u K T d i 9 R g E N k c u Y 5 z C + q f 8 V m s c m l 5 0 j f k Q c F K p 6 Q o D z F z E 5 J K c V B L x N L n O + e T M l u g Q L F K o Q l 9 B s w Q y S b c W G J m l F x i b B u x f K U E y Z K e W d e Z 1 4 j f w 0 5 5 U m O 3 M G h Y 0 C U b U X 7 p V O p H u E u A 9 S d o 4 z f y K l O + 7 8 k u D m p P M U G m m O 3 / q K Q 6 B j 0 k h 0 s W S h N X q W A C F e u 7 f q / u u H u t N d 3 B 2 6 1 7 E 3 D q 4 2 + V a / T a e / X Z S h p l o A w l B O t x 5 6 b m k l e 3 x P K Z x p S Q q / I D M Z o C p K A n u T V c B b 2 H p L Q j q T C V x i 7 o q s Z O U m 0 X i Q B R i b E x H r d V 8 K b f O P M R M N J z k S a Y V P p s l C U 8 W r q c N L t k C m g h i / Q I F Q x 3 K t N Y 6 I I x Z F t V i l H P k m r x n j r b d g 0 z v y B 5 w 6 8 d 2 7 / e F i 3 a M t 6 a j 2 z 9 i 3 P e m U d W 2 + t U 2 t k 0 f a H 9 q f 2 5 / a X j u p 8 7 X z r f F + G t l t 1 z h O r s T o / / g B C 0 0 h 3 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " E 2 r t r 2 A V J X t u P M B y 7 O X G Y 9 C 7 S I w = " > A A A D 8 H i c d Z L N b t N A E M e d h I 8 S P p r C k Y t F W q k g K 7 J d K n K p V A k O H H o o E m m L 0 i h a r 8 f x q u t d a 3 c d E V k W V 3 g D b o g r b 8 Q L 8 A A 8 A W P H l D g p K 1 s e / 2 b m P 7 u z E 6 S c a e O 6 P 1 v t z q 3 b d + 5 u 3 e v e f / D w 0 X Z v 5 / G Z l p m i M K K S S 3 U R E A 2 c C R g Z Z j h c p A p I E n A 4 D 6 5 e l / 7 z O S j N p H h v F i l M E j I T L G K U G E T T n d b v y w B m T O R R E k W M Q 5 F H s B A z R d K 4 6 P 5 z V e B F s T 9 0 n U P 3 e X f P f g M R V r T h o w E l C L e x h m E U t B 0 p m d i B N A Y / R u K T d i 9 R g E N k c u Y 5 z C + q f 8 V m s c m l 5 0 j f k Q c F K p 6 Q o D z F z E 5 J K c V B L x N L n O + e T M l u g Q L F K o Q l 9 B s w Q y S b c W G J m l F x i b B u x f K U E y Z K e W d e Z 1 4 j f w 0 5 5 U m O 3 M G h Y 0 C U b U X 7 p V O p H u E u A 9 S d o 4 z f y K l O + 7 8 k u D m p P M U G m m O 3 / q K Q 6 B j 0 k h 0 s W S h N X q W A C F e u 7 f q / u u H u t N d 3 B 2 6 1 7 E 3 D q 4 2 + V a / T a e / X Z S h p l o A w l B O t x 5 6 b m k l e 3 x P K Z x p S Q q / I D M Z o C p K A n u T V c B b 2 H p L Q j q T C V x i 7 o q s Z O U m 0 X i Q B R i b E x H r d V 8 K b f O P M R M N J z k S a Y V P p s l C U 8 W r q c N L t k C m g h i / Q I F Q x 3 K t N Y 6 I I x Z F t V i l H P k m r x n j r b d g 0 z v y B 5 w 6 8 d 2 7 / e F i 3 a M t 6 a j 2 z 9 i 3 P e m U d W 2 + t U 2 t k 0 f a H 9 q f 2 5 / a X j u p 8 7 X z r f F + G t l t 1 z h O r s T o / / g B C 0 0 h 3 < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " E 2 r t r 2 A V J X t u P M B y 7 O X G Y 9 C 7 S I w = " > A A A D 8 H i c d Z L N b t N A E M e d h I 8 S P p r C k Y t F W q k g K 7 J d K n K p V A k O H H o o E m m L 0 i h a r 8 f x q u t d a 3 c d E V k W V 3 g D b o g r b 8 Q L 8 A A 8 A W P H l D g p K 1 s e / 2 b m P 7 u z E 6 S c a e O 6 P 1 v t z q 3 b d + 5 u 3 e v e f / D w 0 X Z v 5 / G Z l p m i M K K S S 3 U R E A 2 c C R g Z Z j h c p A p I E n A 4 D 6 5 e l / 7 z O S j N p H h v F i l M E j I T L G K U G E T T n d b v y w B m T O R R E k W M Q 5 F H s B A z R d K 4 6 P 5 z V e B F s T 9 0 n U P 3 e X f P f g M R V r T h o w E l C L e x h m E U t B 0 p m d i B N A Y / R u K T d i 9 R g E N k c u Y 5 z C + q f 8 V m s c m l 5 0 j f k Q c F K p 6 Q o D z F z E 5 J K c V B L x N L n O + e T M l u g Q L F K o Q l 9 B s w Q y S b c W G J m l F x i b B u x f K U E y Z K e W d e Z 1 4 j f w 0 5 5 U m O 3 M G h Y 0 C U b U X 7 p V O p H u E u A 9 S d o 4 z f y K l O + 7 8 k u D m p P M U G m m O 3 / q K Q 6 B j 0 k h 0 s W S h N X q W A C F e u 7 f q / u u H u t N d 3 B 2 6 1 7 E 3 D q 4 2 + V a / T a e / X Z S h p l o A w l B O t x 5 6 b m k l e 3 x P K Z x p S Q q / I D M Z o C p K A n u T V c B b 2 H p L Q j q T C V x i 7 o q s Z O U m 0 X i Q B R i b E x H r d V 8 K b f O P M R M N J z k S a Y V P p s l C U 8 W r q c N L t k C m g h i / Q I F Q x 3 K t N Y 6
I I x Z F t V i l H P k m r x n j r b d g 0 z v y B 5 w 6 8 d 2 7 / e F i 3 a M t 6 a j 2 z 9 i 3 P e m U d W 2 + t U 2 t k 0 f a H 9 q f 2 5 / a X j u p 8 7 X z r f F + G t l t 1 z h O r s T o / / g B C 0 0 h 3 < / l a t e x i t > FIG. 1. Tree-level and one-loop Feynman diagrams involving the lepton-number-violating dimension-7 operator (13) and the four-fermion operator (14), whose interference violates CP. gram 5 , shown in Fig. 1. The labels a, b, c, d are flavour indices. The efficiency factor for the asymmetry in the scattering can be parametrised as the difference between the interaction rate of the processesLe →hud and Lē → hūd, Since the phase space factor cancels in the ratio, we only need to evaluate σ ∝ |M| 2 where the amplitude M can be written as We defined c aa ≡ c a and c (6) abab ≡ c (6) ab , where the lepton fields are diagonalised in O (7) with real coefficient, while c (6) is complex in general. I is a loop factor that must also have an imaginary part from on-shell particles running in the loop for Eq. 15 to be non-zero, since it evaluates to where ImI = p 2 /(8π), and p 2 is the square of the fourmomentum sum of the initial state leptons. After perturbative decay of the relaxion with mass m φ , and before reaching a thermal distribution, the out-of-equilibrium leptons will have energies distributed from m φ /2 to T as they are upscattered by the thermal bath, corresponding to 6m φ T p 2 T 2 . Since the dominant contribution to the asymmetry will come from the higher energy non-thermal leptons we mainly use p 2 max T 2 as an upper bound in our estimates, while also showing the p 2 min 6m φ T points as a conservative lower bound case. In a standard approximate picture for the perturbative decays of the relaxion, since m φ > H at the time of trapping, we may treat the oscillations of φ in its local minimum as a gas of non-relativistic particles whose equation of state is that of matter. Its number density is given by and the perturbative decay rate into leptons is Γ D ∼ m 3 φ /f 2 L . This decay rate is sub-dominant to the rate of condensate scattering with the thermal bath, which goes as Γ S ∼ T 2 m φ /f 2 L (we included an additional m φ /T suppression in the naive rate to account for boseenhancement [5]), so the available number density for producing out-of-equilibrium leptons in perturbative decays is However, this approximation does not account for the full production of fermionic modes [18], similarly to the case of bosons. Though the occupation number of fermions cannot be exponentially enhanced due to pauli blocking, the generation of fermionic modes from a rolling scalar field has also been shown to give large effects [18]. We therefore expect this to give a large contribution of non-thermal leptons whose typical energy will be of or- Since a full numerical investigation of fermionic preheating-like process including backreaction and thermal effects is beyond the scope of this work, we simply account for this extra contribution by allowing the effective number density of the condensate that is available for out-of-equilibrium leptons, n φ , to vary between the minimum perturbative contribution of Eq. 19 and the total condensate number density Eq. 18., For the case of perturbative decays, they are only active when Γ D > H, which sets an upper bound on the decay constant of the fermion coupling, and decaying to muons sets a minimum relaxion mass m φ > 10 −1 GeV. The fermion current in Eq. 3 is the combination of muon and tau current, J 5µ = J 5µ µ − J 5µ τ , to avoid reintroducing a coupling to the FF and evade astrophysical constraints on couplings to electrons. The constraint on the mass and couplings is relaxed when the dominant contribution to out-of-equilibrium leptons is from rolling, where a lighter relaxion with weaker couplings can still source the production of heavier fermions [18].
The number density of the net lepton asymmetry can now be estimated as the fraction of the number density n φ converted into pairs of out-of-equilibrium leptons with a branching ratio B that undergo lepton-numberviolating inelastic scattering at a rate Γ LN V , relative to the thermal elastic scattering rate Γ th. , with an efficiency : This expression approximates within an order of magnitude the numerical results of solving the Boltzmann equations including washout effects [12]. The lepton-numberviolating and thermal scattering rates are given by where α 2 ( p 2 = 10 5 GeV) ∼ 0.03 is the SU (2) L structure constant. Putting this all together, and setting c (7), (6) = O (1), we obtain the following baryon asymmetry entirely from perturbative decays assuming p 2 max for the benchmark point of Table I, Table I also shows a benchmark point for p 2 min . We set Λ ∼ Λ c,6,7 ∼ T for simplicity, though one should bear in mind that they only appear to be equal within an order of magnitude and can be varied independently. The strong dependence on the dimension-7 operator scale for giving sufficient lepton number violation implies this cannot be too decoupled. However, as discussed above, the parameter space and scales involved can be relaxed by allowing the number density n φ to be larger than the minimum contribution from perturbative decays. For example, the second benchmark point for p 2 max in Table II shows that for the maximum number density n φ = n φ the scale Λ 7 can be pushed up to O(10 7 ) GeV: For p 2 min this can only be achieved by relaxing the constraint from sub-Planckian field excursions on the cut-off, though this case is overly conservative as the dominant contribution will be from higher energy leptons.
We may also consider other lepton-number-violating dimension-7 operators; qualitatively, the mechanism is not much affected by the details of the specific operator, though the parameter space will be quantitatively different depending on the phase space factor. Phenomenological constraints also vary for each operator, as studied e.g. Λ, Λc, Λ6,7, T fp m φ fL fV g p 2 max 10 5 10 8 100 10 7 5 × 10 7 10 −8 p 2 min 10 5 5 × 10 6 2 × 10 3 10 9 5 × 10 7 10 − 8   TABLE I. A benchmark point in GeV for our relaxion leptogenesis mechanism in which the baryon asymmetry is generated entirely from the relaxion decaying perturbatively into leptons with n φ = n min.
in Refs. [13,14]. The particular choice may also be theoretically motivated; for example, Ref. [13] showed that for lepton-number-violating operators with two leptons and no quarks there is a unique operator corresponding to the specific chirality of the lepton pair which are of dimension 5, 7 and 9 for LL, Le and ee, respectively. The dimension-7 operator in this case is The lifetime of the inverse neutrinoless double-beta (0νββ) decay is T 1/2 > 1.9 × 10 25 years which sets a lower bound on the scale Λ 7 10 5 GeV for c (7) ∼ O(1) [13]. But a stronger bound comes from the operator's one-loop contribution to the neutrino mass, m ν ∼ v 8 √ 2π 2 Λ m e λ 7, ee , which requires Λ 7 10 7 GeV. Here, the phase space factor in Eq. 23 also gives a larger suppression as it involves an additional particle in the final state. To generate sufficient asymmetry with this operator a generic parameter space point could also require a mildly trans-Planckian field range to raise the cut-off above 10 6 GeV. Conclusion.
-We proposed a model of cosmological relaxation of the weak scale with particle production that generates the baryonic matter-antimatter asymmetry while reheating the universe after inflation. For an SM EFT cut-off up to O(100) TeV, the model has several desirable features: it allows for high-scale inflation, scanning with a sub-Planckian field range, has no extremely small parameters, introduces no new physics below the cut-off, and achieves leptogenesis at low temperatures. Moreover, from a conceptual point of view, leptogenesis in relaxation combines two approaches to understanding the smallness of the weak scale: a "dynamical selection" mechanism and a "censorship" approach [19]. The relaxion mechanism ensures its evolution naturally selects a minimum with v Λ, whereas tying its particle production backreaction to reheating and leptogenesis gives a cosmological censorship criteria for us living in the corner of the universe where the relaxion happened to have the right initial conditions for sufficient scanning-if it did not, the universe would be empty.
The model makes use of the thermal bath from bosons produced by the relaxion, and the lepton coupling that was already included to dilute the relaxion's thermal abundance. This necessarily leads to the production of leptons by relaxion rolling as well as through perturbative decays in the misalignment mechanism. These leptons are out-of-equilibrium and scatter with the thermal bath. The scattering will generally involve higher-dimensional operators that violate lepton number, whose effect can be large enough to generate the observed baryon asymmetry of the universe if the scale of these operators are sufficiently close to the cut-off. We have shown that for dimension-7 operators a sufficient matter-antimatter asymmetry can be generated by the relaxion entirely through perturbative decays, though additional production of fermionic modes through rolling is also expected to contribute a sizeable depletion of the relaxion's condensate into the out-of-equilibrium leptons. The result also depends on details of the thermalisation process that warrant further investigation. We leave the study of such numerical computations to future work.
The lack of new physics at the weak scale that was expected to solve the hierarchy problem may mean such a solution is simply postponed to higher energies. The issue has certainly not gone away-on the contrary, it is exacerbated by the experimental null results. It is therefore worthwhile to explore alternative ways of naturally obtaining a hierarchy, with much still to be learned from dynamics in the early universe where scalar fields and Higgs-dependent phenomena can play a major rôle.